guidelines the calibration evaluation of systems for ... for the calibration & evaluation of...

Guidelines for the Calibration & Evaluation ofOptical Systems for Strain Measurement

Guidelines for the Calibration & Evaluation of Optical Systems for Strain Measurement

First edition published on-line 2007, revised version produced May 2010 Guidelines for the Calibration & Evaluation of Optical Systems for Strain Measurement Copyright © 2007, 2008 & 2010 University of Sheffield All rights reserved. This book, or parts thereof, may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without written permission providing the source is acknowledged by citing as: ‘Guidelines for the Calibration and Evaluation of Optical Systems for Strain Measurement, SPOTS, www.opticalstrain.org, 2010’. Electronic copies including the Appendices can be downloaded at www.opticalstrain.org ISBN: 978-0-9842142-2-8 Publisher: Eann Patterson ([email protected]), Composite Vehicle Research Center Department of Mechanical Engineering, Michigan State University 2555 Engineering Building, East Lansing, MI 48824-1226, USA

GUIDELINES for the CALIBRATION & EVALUATION of OPTICAL SYSTEMS for STRAIN MEASUREMENT Specifications and protocols for the use of a Reference Material and Standardised Test Material for respectively calibrating and evaluating optical systems for strain measurement. Prepared as part of the SPOTS project; a shared cost RTD project with the European Commission’s Competitive and Sustainable Growth Programme (Contract No. G6RD‐CT‐2002‐00856 ‘SPOTS’)

Contributors: Richard Burguete, Erwin Hack, Eann Patterson, Thorsten Siebert & Maurice Whelan

Part I: Calibration

These guidelines were developed during the Standardisation Project for Optical Techniques of Strain measurement (SPOTS) that was supported by EU grant: G6RD-CT-2002-00856 ‘SPOTS’ and the Swiss Federal Office for Education and Science. The partners in the SPOTS project were: Airbus, CRF Societa Consortile per Azionil, Dantec Dynamics, EC-JRC-IHCP, Eidgenössische Materialprüfungs-und Forschungsanstalt (EMPA), Honlet Optical Systems GmbH, NPL Management Limited, Optical Metrology Innovations, Politechnika Warszawska, SNECMA and the University of Sheffield.

The following members of the project contributed to the many discussions which led to the production of this volume:

Eann Patterson University of Sheffield, UK (Project Co-ordinator) Daniel Albrecht EC-JRC-IHCP, EU Philippe Brailly SNECMA, France Richard Burguete Airbus Maria Margherita Dugand CRF Societa Consortile per Azionil, Italy Erwin Hack Eidgenössische Materialprüfungs-und Forschungsanstalt

(EMPA), Switzerland Michel Honlet Honlet Optical Systems GmbH, Germany Liam Kehoe Optical Metrology Innovations, Eire Malgorzata Kujawinska Politechnika Warszawska, Poland David Mendels NPL Management Ltd, UK Lescek Salbut Politechnika Warszawska, Poland Qasim Saleem University of Sheffield, UK Hans Reinhard Schubach Dantec Dynamics GmbH, Germany Thorsten Siebert Dantec Dynamics GmbH, Germany Graham Sims NPL Management Ltd, UK Carole Stochmil SNECMA, France Michel Taroni SNECMA, France Rachel Tomlinson University of Sheffield, UK Maurice Whelan EC-JRC-IHCP, EU The document has been reviewed by a number of international experts in the field and their contribution is also acknowledged. Opinions expressed are those of the authors and not necessarily those of the funding agencies. The authors, contributors, publisher, their sponsors and their employers accept no responsibility for the use of the designs and protocols described in this document or for the correctness of the results obtained from them and no legal liability in contract tort or otherwise shall attend to the authors, contributors, publisher, their sponsors or their employers arising out of the use of the information contained in this document. .

Calibration & Evaluation of Optical Systems for Strain Measurement

5

Contents

Part I – CALIBRATION Executive Summary 7 1. Introduction 9 2. Routes for traceability

(a) Introduction 9 (b) Possible routes for traceability of strain measurements 10 (c) Traceability to unit of length 10

3. Description of Reference Material (a) Approach to development of a Reference Material 12 (b) Design notes 14 (c) Strain field equations 15 (d) Methodology for use 15

Appendices A – Design notes for Reference Material 22 B – Derivation of strain field equations for Reference Material 28

Part II – Evaluation 35 Executive Summary 37 1. Introduction 38 2. Description of Standardised Test Material

(a) Design notes 41 (b) Strain field equations 42 (c) Methodology for use 43

3. Functional paths to standardised data sets 46

(a) ESPI 47 (b) Digital Image correlation 47 (c) Grating interferometry 48 (d) Moiré 48 (e) Photoelasticity 49 (f) Thermoelasticity 49

Appendices

C – Design notes for Standardised Test Material 50 D – Derivation of strain field equations for Standardised Test Material 58 E – Explanation of functional paths for standardised data sets 58

Nomenclature 65

Evaluation & Calibration of Optical Systems for Strain Measurement

6

Figure 1 – Three-dimensional view of the Reference Material

(EU Community Design Registration 000213467)


7

Executive Summary

In the last decade there has been a high level innovation in the field of experimental mechanics with advances in both the capabilities of existing techniques such as in the area of digital photoelasticity and the development of powerful new techniques such as digital image correlation. The wider structural analysis community has not embraced these developments at the same rate and continue to almost exclusively utilise computational methods of stress and strain analysis. However, the need for data from experiments to validate numerical models is becoming more important, particularly as new materials and complex structures provide severe challenges to reliable simulations. Calibration procedures are required to provide user and regulator confidence in optical techniques of strain measurement employed in validation. Certification of calibration requires traceability which is derived from an unbroken sequence of comparisons to form a chain to the primary or national standard and in this case the international standard for the metre has been adopted. This booklet describes a Reference Material (Part I) and a Standardised Test Material (Part II) together with methodologies for their use in the calibration and evaluation, respectively, of optical systems for full-field measurement of strain. A beam subject to four-point bending and encased in a monolithic loading frame forms the Reference Material. An analytical description of the strain field in the central or gauge section of the beam as a function of its displacement forms the basis for comparison. The Standardised Test Material consists of a disc contacting a half-plane and provides a substantial challenge against which the capabilities of the most sophisticated system for optical strain measurement can be evaluated. Exemplar studies illustrating their use in practice have been published elsewhere1,2.

1 Whelan, M.P., Albrecht, D., Hack, E., Patterson, E.A., ‘Calibration of a speckle interferometry full-field strain measurement system’, Strain, 44(2):180-190, 2008. 2 Patterson, E.A., Brailly, P., Burguete, R.L., Hack, E., Siebert, T., Whelan, M.P.,‘A challenge for high performance full-field strain measurement systems’, Strain, 43(3):167-180, 2007.


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1. Introduction There have been many novel developments in experimental mechanics in the last decade. Many of these developments involve optical techniques for the evaluation of strain and consist of either substantial advances in the capabilities of existing techniques such as in the area of digital photoelasticity, or innovative techniques such as digital image correlation. The enthusiasm of the experimental mechanics community for these exciting developments has not been uniformly reflected in the wider stress analysis or structural integrity communities. These communities continue to place a heavy reliance on, and preference for, computational methods of stress and strain analysis. However, the need for data from experiments to validate and verify numerical models is being recognized increasingly3. Despite this potential driver, there has been only a slow adoption by industry of modern optical methods for evaluating full-field strain data. One potential cause of this tardiness, and a motivation for the preparation of this document, is the almost complete lack of standards. There are no internationally recognized procedures for calibrating an optical instrument used for full-field strain evaluation or for evaluating such an instrument against its design specification or comparable apparatus. These guidelines represent a potential precursor to an international standard.

The objective of this document is to describe devices and methodologies designed to permit both the calibration and evaluation of optical systems for strain measurement. Exemplar studies of their use have been published elsewhere1,2.

2. Routes for traceability4

(a) Introduction The traceability of a measurement value is established by a hierarchy of calibrations to a primary or national standard. But, no such thing as a “national microstrain" exists. The concept of a primary standard is equally valid for base quantities and derived quantities, e.g. length and strain, respectively. This implies that a primary standard for strain does not necessarily have to be linked to an embodiment of an SI-unit.

Strain is not a material property in a strict sense, but rather a property of the material state. Since most optical methods do not measure residual strain but differences in strain level, a measurement standard for strain must incorporate a device that reproduces in an unchanging manner during its use, one or more known values of strain, e.g. from an unstrained to a strained state of the object. And furthermore, it must represent a strain field that is appropriate to the imaging property of the techniques.

In addition, strain values from very different optical techniques should be comparable against an identical reference material by using a unified methodology, irrespective of the fact that the measurement chain from the test object to the strain measurement value involves different effects and model assumptions for each method and instrument. This implies that the strain value is regarded as the measurement value to be traced back, and

3 ASME V&V 10-2006, Guide for verification and validation in computational solid mechanics, American Society of Mechanical Engineers, New York, 2006. 4 Hack, E., Burguete, R.L., Patterson, E.A., ‘Traceability of optical techniques for strain measurement’, Appl. Mechanics and Materials, 3-4, 391-396, 2005.


9

primary measurands like intensity of an interference pattern need not be traced back to a primary standard.

As soon as traceability is established the Reference Material described here can be certified. An uncertainty budget relaxes the need to have the highest standards of accuracy such as in a primary standard.

(b) Possible routes for traceability of strain measurement Strain is clearly related to deformation. Therefore, to realize a strain reference material, a device must be devised to deform a piece of material in a reproducible manner between two object states. Any effect that causes a material deformation can be envisaged to provide a viable route for traceability.

Mechanical deformation combined with tracing of strain measurement values to a length standard by using a traceable gauge length and displacement values seems a natural choice. It might be mentioned, however, that more exotic chains can be devised. Electrostrictive materials could be used to trace strain values to the volt using the voltage-strain relation. Further, one might think of thermal expansion of a calibration body, and tracing strain values to the temperature scale, or a shape-memory alloy that is cycled between its two states. Table 1 includes some possible routes for traceability of strain values.

(c) Traceability to the unit of length An important issue is acceptance of the traceability procedure by users. Traceability cannot just be claimed by a party; it must be accepted, either by virtue of the fact that an accepted standard agency sustains the calibration chain, or that the technique is accepted as traceable by virtue of its principle of operation. Feedback during the development of reference materials suggested that traceability to the unit of length is advisable and most likely to be accepted by the community. In a poll of the community (see section 3(a)), the reference material attribute "Traceability to international standards via length" had a high score and, therefore, was the primary route to be assessed. This is both because strain is by definition intimately related to length (deformation) and because users are well-experienced in the measurement of length and deformation. In addition, traceability to length is a natural choice for techniques that measure displacement from which strain is deduced.

In the design of the Reference Material mechanical deformation is favoured over thermal or electrical deformation as it allows the direct relationship of deformation to the length scale. The introduction of deformation is displacement controlled in order not to be influenced by force measurement. The Reference Material thus fulfils the requirement to "reproduce, in a permanent manner during its use, one or more values of strain".

Table 1: Possible routes for traceability of strain values

Primary standard Possible realizations, generation of strain field voltage electric field controlled deformation of an electrostrictive or piezoelectric material current current controlled magnetic field induced deformation of a magnetostrictive material temperature in‐plane displacement field by temperature controlled expansion of a body with linear CTE. angle strain field by angle controlled deformation of an elastic material (torsion rod). length strain field by displacement controlled deformation of an elastic material (bending beam,

tensile test specimen).


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3. Description of Reference Material

(a) Approach to development of a Reference Material The provision of a Reference Material is part of a process to enhance the quality and acceptability of full-field optical techniques of strain measurement. The establishment of this process led to the identification of two distinct needs: the requirement for calibration or reference materials and methods linked to international standards; and the desirability of having standardised test materials that allow the performance of systems and sub-systems to be evaluated and compared. Calibration includes making comparisons with a known, recognised criterion or reference material which in turn has been compared via a continuous chain of comparison to an international standard. Formally, calibration is defined as an ‘operation that, under specified conditions, in a first step establishes a relation between the quantity values with measurement uncertainties provided by measurement standards and corresponding indications with associated measurement uncertainties and, in a second step uses this information to establish a relation for obtaining a measurement result from an indication’5. The determination of the distribution of strain, and subsequently stress in a component, is an essential element in engineering design. Thus, the ability to calibrate and provide traceability for instruments employed to determine strain is an important topic to engineers concerned with the safety and reliability of engineering systems. Since strain is closely related to deformation, or change in length it was decided that the standard metre is the most appropriate primary standard for strain. The concepts of standardisation and traceability in this context are discussed in more detail by Hack et al4. The purpose of Part I of this document is to describe a Reference Material that can be used for the calibration of optical systems for full-field strain measurement and which provides traceability to an international standard thereby facilitating the use of such instruments within a regulatory environment. Whereas, part II of this document details a Standardised Test Material, for evaluating and comparing optical systems: a disc in contact with an elastic half-space. In both parts, the work has been restricted to in-plane strain fields occurring due to static or quasi-static loads. This restriction made the first step in the development process viable and it is anticipated that it will be removed in the future by the ADVISE project6.

The strain field generated by the Reference Material needs to be simple to reduce the uncertainty when making comparisons and yet allow the calibration of full-field optical systems for evaluating strain fields. The rational decision making process7 was utilised to design the Reference Material. In this process, the essential and desirable attributes of the design were identified and a number of options or alternative designs were generated. The extent to which each alternative possessed the attributes was assessed in order to highlight the designs which best fitted the requirements. This approach was intended to allow a large search space so that many widely differing solutions could be considered and the inappropriate dominance of previously utilised solutions could be avoided. In 5 ISO/IEC Guide 99: 2007, International vocabulary of metrology – Basic and general concepts and associated terms (VIM). 6 ADVISE: Advanced Dynamic Validation by Integrated Simulation and Experimentation, www.dynamicvalidation.org 7 Olden, E.J., Patterson, E.A., ‘A rational decision making model for experimental mechanics’, Experimental Techniques., 24 (4), 26-32, 2000.


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this case, SPOTS8 consortium members were asked to identify attributes and the wider experimental mechanics community to assign levels of importance to each attribute. Five essential attributes were identified: easy optical access, lack of hysteresis, an in-plane strain field, traceability to international standards and utilisation of the length standard for traceability9. Twenty-four desirable attributes were also defined and used to guide the choice of preliminary design. A number of brainstorming sessions lead to a short-list of candidate designs including a disc subject to compression along one diameter, a Hertzian contact pair and a beam subject to four point bending. The latter choice was preferred for a variety of reasons including its use in previous standards associated with optical measurements10, the simplicity of the geometry and the perceived suitability of the strain field for calibration.

A preliminary round robin test within the SPOTS consortium11 on a tensile specimen with some geometric features had revealed the difficulties associated with generating reproducible results from different optical systems in laboratories located in various countries even when the test specimens were manufactured and supplied from one source and many laboratories used the same model of loading frame. The crucial importance of a reproducible loading system became apparent and lead to the concept of a test specimen and loading frame contained within a monolithic design. The design consists of a central horizontal beam loaded in symmetric four-point bending via an outer loading frame as shown in figures 1 and 2. The outer loading frame is designed to eliminate mis-alignment and positioning errors whilst allowing either compressive or tensile loading. The means of application of load is unrestricted and could be dead-weight loading or via a loading machine. The loading frame is massive relative to the test specimen and contains an interlock in the top left and right corners to protect the test beam from over-loading and plastic deformation. In compressive loading, the bottom surface of the frame can be located on a platen and the load applied through the half-cylinder on the top surface to ensure alignment. In tension, alignment is achieved by loading on pins through the two holes located on the centre-line of the frame. It is intended that displacement loading should be applied to permit traceability to the international standard for the metre and so a series of teeth in the top corners allow the use of a variety of types of displacement transducer with either the internal or external surfaces. The choice of displacement transducer will be a decision for the user but is likely to be influenced by the availability of calibrated transducers and the scale of the test specimen. The monolithic frame dictated that knife-edges could not be used at the loading points on the beam and an alternative mechanism was required that would transmit a load but not a moment. A set of whiffle-trees was selected which maintain the continuity between the

8 SPOTS: Standardisation Project for Optical Techniques of Strain measurement, www.opticalstrain.org 9 Burguete, R.L., Hack. E., Siebert, T., Patterson, E.A., Whelan, M., ‘Candidate reference materials for optical strain measurement’, Proc. 12th Int. Conf. Exptl. Mechanics, Advances in Experimental Mechanics, edited by C. Pappalettere, McGraw-Hill, Milano, pp.699-700, 2004. 10 ASTM C1377-97, Standard test method for calibration of surface stress measuring devices, ASTM International, West Conshohocken, PA, USA 11 Mendels, D. –A., Hack, E., Siegmann, P., Patterson, E.A, Salbut, L., Kujawinska, M., Schubach, H.R., Dugand, M., Kehoe, L., Stochmil, C., Brailly, P., Whelan. M, ‘Round robin exercise for optical strain measurement’, Proc. 12th Int. Conf. Exptl. Mechanics, Advances in Experimental Mechanics, edited by C. Pappalettere, McGraw-Hill, Milano, 695-6, 2004.


12

test specimen and the loading frame but minimise the transmission of lateral and rotational forces.

The design is fully parametric based on the beam depth, W, in the direction of the load and this allows the Reference Material to be manufactured from a range of materials and scales from micro to macro. However, it is important to appreciate that a calibration is only valid for the strain range employed in the Reference Material and at the scale of the gauge section. The design is two-dimensional because the SPOTS consortium decided to focus on the simpler two-dimensional problem as an initial step and this was believed to be reasonable, since in practice, most optical measurements of strain are conducted in two-dimensions at the moment. In addition, the plane geometry also provides scope for manufacturing the Reference Material via a variety of methods.

Figure 2 – Schematic of the Reference Material with normalised dimensions based on the depth of the

beam, W. Superscripts refer to design notes in Appendix A (available in the on-line version at www.opticalstrain.org). Detail A and B are shown in figure 3. Tolerances should be 0.05mm unless

otherwise specified including the parallelism of the front and back faces. (EU Community Design Registration 000213467)

12W

12W

[ix]

W [i]

W/5 [viii]2W [ii]3W [iii][v]

W/5

W/2

W [x

iv]

WW

/2W

W/4

W/4

W/4 [xi]

W[v

ii]3W

5W [vi]

χ

χ

χ [iv]

W [x]

W/2W/2

≥W/4 [viii]

≥W/4

2W[x

ii]2W

2W

0.4W [xii]

7W/10 [xiv]7W/10

7W/10

2W/3

[xiv

]

Detail B

Detail A

W/2

rχ

rχ

rχ

rχ

r

χ

χ [iv]

W/3

W/3W/3

W/3

BA0.005W

B0.005W

A

B0.005W

B0.005W

B0.005W

12W

12W

[ix]

W [i]

W/5 [viii]2W [ii]3W [iii][v]

W/5

W/2

W [x

iv]

WW

/2W

W/4

W/4

W/4 [xi]

W[v

ii]3W

5W [vi]

χ

χ

χ [iv]

W [x]

W/2W/2

≥W/4 [viii]

≥W/4

2W[x

ii]2W

2W

0.4W [xii]

7W/10 [xiv]7W/10

7W/10

2W/3

[xiv

]

Detail B

Detail A

W/2

rχrχ

rχ

rχ

rχ

r

χ

χ [iv]

W/3

W/3W/3

W/3

BBA0.005W

B0.005W

A

B0.005W B0.005W

B0.005W B0.005W

B0.005W B0.005W


13

(b) Design notes The Reference Material consists of a two-dimensional specimen which can be manufactured in any homogeneous and isotropic material at any scale, although it should be noted that the material must be homogeneous at the scale of the measurement. Care should be taken to avoid potential effects arising from a heterogeneous microstructure, i.e. at small scales. The design is monolithic in the sense that it is a single piece of material and can be loaded using any suitable arrangement for applying either a compressive or tensile uni-axial load. The relative displacement of the two portions (top and bottom faces) should be measured using an appropriate and convenient device for the scale of the specimen. Further guidance on the use of the specimen is provided in the section on methodology. A slit of width, χ in an interlock is provided between the upper and lower portions of the monolithic frame on each side in order to protect the beam from plastic deformation through over-loading. At very small scales, the use of the slit for this purpose may not be practical and it can be left unscaled at a conveniently large dimension. Detailed design notes are provided in Appendix A (available in the on-line version at www.opticalstrain.org).

Figure 3 – Details from Reference Material shown in figure 2

In the above figures 0.2W≤B≤W [i] (superscripts refer to sections in Appendix A which is available in the on-line version at www.opticalstrain.org) where B is the out-of-plane thickness. The slit defined by χ and rχ is intended to be a single cut and for this purpose it is recommended that rχ = W/9[xii] and all other radii to be ( )5W unless otherwise specified [xiii]. As noted previously, at very small values of W it may be inappropriate to scale the slit dimension χ in which case it should be noted that the beam will not be protected from over-loading and plastic deformation.


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(b) Strain field equations The strains in the gauge section of the Reference Material are derived analytically in Appendix B (available in the on-line version at www.opticalstrain.org). For an ideal beam subject to four-point bending, the strain field is described by:

26Wvy

xx =ε , 26Wvy

yyυε −= and 0=xyε (1)

where W is the depth of the beam, y is the distance from the neutral axis, v is the applied displacement in the Reference Material and υ is Poisson’s ratio. The origin of the reference frame is the center of the gauge section (figure 5). These equations can be transformed into any other direction, θ using a Mohr’s circle of strain. It is useful to note that the measured strain can be plotted as a function of depth in the beam, i.e. in the y-direction. The gradient of strain across the beam will be:

26Wv

dyd xx =

ε and 26Wv

dyd yy υε

−= (2)

If the gradient of these quantities are plotted as a function of applied displacement, v then the gradient of this new plot will be a property of the Reference Material.

261

Wdyd

vdd xx =⎟⎟

⎠

⎞⎜⎜⎝

⎛ ε and 26Wdyd

vdd yy υε

−=⎟⎟⎠

⎞⎜⎜⎝

⎛ (3)

The constraint imposed by the attachment to the monolithic frame implies that these expressions should be modified to represent more accurately the strain in the gauge section, i.e.

( )ηκε += yWv

xx 26, ( )ηκυε +−= y

Wv

yy 26 and 0=xyε (4)

where κ and η are constants that can be evaluated using either a pair of strain gauges on the top and bottom surfaces of the beam or from finite element analysis as suggested in Appendix B (available in the on-line version at www.opticalstrain.org).

(d) Methodology for use A flowchart that outlines the steps described below for performing a calibration is shown in figure 4 and should be read in conjunction with the following sections.

(i) Manufacture:

An isotropic homogeneous material should be selected from which to manufacture the Reference Material. Care should be taken to ensure that the material is free from defects and residual stresses and that, at small scales, grain sizes will not influence the strain distribution. The Reference Material should be manufactured using the tolerances specified in figure 2 and following manufacture it should be measured to establish the actual dimensions of the beam to be subjected to four-point bending. The thickness B, depth W, distance a between the loading points, and the distance c should all be measured (figure 5) and the uncertainty in these measurements u(f), estimated12. The uncertainty of

12 ISO/IEC Guide 98:1995, Guide to the expression of uncertainty in measurements (GUM) Joint Committee for Guides in Metrology 100:2008 (BIPM, Paris 2008).


15

each of these measurements is equal to the positive square root of the estimated variance13 which may be evaluated using any valid statistical treatment of the data, e.g. by calculating the mean and standard deviation of a series of independent observations. The uncertainty in κ and η (equation (4)) must also be evaluated as suggested in Appendix B(iv) (available in the on-line version at www.opticalstrain.org).

(ii) Experimental set-up:

The Reference Material can be loaded statically by any means that is practical and appropriate for the scale being employed. Tension loads can be applied through the two holes on the axis of symmetry whilst compressive loads should be applied by placing the base on a platen and applying load through the half cylinder on the top to ensure alignment of the load. The loading should be displacement controlled and the displacement should be measured at the flats provided in the top corners. The displacement v is taken as the average of the measurement values from each corner. Any calibrated, traceable displacement transducer which is appropriate to the scale of the Reference Material may be employed.

Figure 4 – Flow chart for performing a calibration with operations shown as rectangular boxes and

quantities as ovals. Quantities that must be reported as part of the calibration are highlighted separately as outputs on the right.

13 Taylor, B.N., Kuyatt, C.E., Guidelines for evaluating and expressing the uncertainty of NIST measurements results, NIST Technical Note 1297, NIST Physics Lab, Gaithersburg, MD. 1993.

Manufacture RM

Measure dimensions of RM& estimate their uncertainty (see figure 5)

Apply a displacement load & measure strain in RM

with instrument to be calibrated

Select material for RM & estimate uncertainty in υ

Assess differences (eq. 5) between measured & predicted strain values

Perform linear least-squares fit to field of deviations & evaluate α & βand their uncertainties (eq. 7 & 9) αk & βk and

u(αk) & u(βk)

Calculate RM uncertainty, uRM (eq. 10)

Select calibrated

instrument to measure

displacement

Assess acceptability of calibration (figure 6)

ucal(ε)

field of deviations, dk(i,j)

u(υ)

u(vk)

Values to be reported

Assess acceptability of calibration (figure 6) & calculate calibration uncertainty, ucal (eq. 12)

u(W) u(a) u(c)

Adjust instrument & repeat for acceptable

calibration

Repeat for 3 increments of load

Manufacture RM

Measure dimensions of RM& estimate their uncertainty (see figure 5)

Apply a displacement load & measure strain in RM

with instrument to be calibrated

Select material for RM & estimate uncertainty in υ

Assess differences (eq. 5) between measured & predicted strain values

Perform linear least-squares fit to field of deviations & evaluate α & βand their uncertainties (eq. 7 & 9) αk & βk and

u(αk) & u(βk)

Calculate RM uncertainty, uRM (eq. 10)

Select calibrated

instrument to measure

displacement

Assess acceptability of calibration (figure 6)

ucal(ε)

field of deviations, dk(i,j)

u(υ)

u(vk)

Values to be reported

Assess acceptability of calibration (figure 6) & calculate calibration uncertainty, ucal (eq. 12)

u(W) u(a) u(c)

Adjust instrument & repeat for acceptable

calibration

Repeat for 3 increments of load


16

(iii) Measurement procedure:

The instrument to be calibrated should be set up so that the gauge section of the Reference Material, i.e. the central section of the beam of dimensions W×W occupies at least 90% of the short dimension of the detector. Within the gauge area there must be at least 10 × 10 pixels or data points so that the number of data points considered, 100≥N . All data points (i,j) in the gauge area must be included in the subsequent analysis. A map of strain in the gauge area must be calculated for at least three loads at nominally equal increments of applied displacement, kv , distributed between 10% and 90% of the maximum allowable load.

(iv) Comparison of measured and predicted values:

Employing the measurement procedure described above, values of strain are obtained at an array of N points within the gauge area. The location of a point (i, j) within the array can be defined by the position co-ordinates (xi, yi) based on an origin at the geometric centre of the gauge section. The corresponding values of the strain field in the Reference Material should be predicted using expressions (4). Maps of the difference between the predicted values and the measured values of strain in the Reference Material, dk(i, j) over the gauge area, Ag should be reported using the following expression for each load step, k:

( ) ( )measuredjikpredictedjikk yxyxjid ),(),(),( εε −= (5)

Figure 5 – Schematics showing measurements (in addition to thickness, B) whose uncertainty must be

estimated (left) & the orientation of the axis system at the beam’s geometric centre (right).

(v) Assessment of the differences or deviations:

In a perfect world there would be no differences between predicted and measured strain so that dk(i, j)=0. In practice, the measurements will contain random uncertainty components and probably systematic uncertainty components. In a high quality measurement system there will be no systematic component that varies with x, the distance along the beam within the gauge section. Thus in the interests of simplicity, for the y-direction only, a linear least-squares fit to the field of deviations, dk(i,j) yields fit-parameters αk and βk for each load step by minimizing the residual below:

[ ]2

,),( jkkk

jiyjid βα −−∑ (6)

where the fit-parameters, αk and βk which must be reported are calculated from:

Wca Wca W/2W/2

x

y

W/2W/2

x

y


17

∑∑

∑ ==

jij

jikj

kji

kk y

jidyjid

N,

2,

,

),(),(1 βα (7)

Also, for each increment of load k, the mean square residual deviation should be evaluated after the linear fit by using:

( ) [ ] ∑∑ −−=ji,

jkkkji,

k yN

ji,dN

du 22222 1)(1 βα (8)

and hence the uncertainties of the fit-parameters can be calculated and also should be reported:

( ) ( ) ( ) ( )∑

==

jij

kk

kk y

duu

Ndu

u

,

2

22

βα (9)

Ideally both kα and kβ are zero and no corrections are required to the calibration. In this case there are no systematic uncertainty components, and if present, other components are random with respect to both x and y. When ( )kk u αα 2> , namely αk is greater than its ‘expanded uncertainty’ then there is a statistically significant offset in the calibration, i.e. there is a systematic uncertainty component present in the measurements. And if βk is zero when ( )kk u αα 2> then the systematic component is constant with respect to the y direction.

When ( )kk u ββ 2> , i.e. βk is greater than its ‘expanded uncertainty’ then there is a statistically significant deviation in the calibration. This implies that the difference between the measured and predicted values of strain is a function of y, the distance from the axis through the geometric centre of the gauge section (figure 5) or a function of strain.

Note that an instrument which produces a measured strain distribution that varies linearly with x and is symmetric about the origin will have values of αk and βk indistinguishable from one where the strain distribution is constant with respect to x. In general, u(αk) will be large when the measured strain distribution varies with x; and u(βk) will be large if the measured strain distribution varies non-linearly with y, because the residual, in expression (8) will be larger.

(vi) Calculation of calibration uncertainty:

The expressions for the combined standard uncertainty of the Reference Material strain values, uRM are deduced in Appendix B (available in the on-line version at www.opticalstrain.org) based on the law of propagation of uncertainty using the root-sum-of-squares (RSS) method13. They are given by Eq. (B41) for the load increment denoted by subscript k as a function of strain, and here as a function of y for convenience of discussion. Note that the uncertainty in the identification of the position on the beam u(y), is omitted, because this uncertainty is reflected in the measured difference dk(i,j). The law of propagation of uncertainty yields different results for different components of strain and expressions associated with the sum and difference of the principal strains as


18

well as other Cartesian components are provided in Appendix B (available in the on-line version at www.opticalstrain.org) as expressions (B48) & (B52), while those associated with the x and y strain components are given below by way of example:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ηυ

υυ

κκεε

ηκ

κεε

22

22

2

2

222

2

2

2

222

22

22

222

2

2

2

222

6416

6416

uWvu

WcuauWuu

vvu

u

uWv

WcuauWuu

vvu

u

k

k

kyykyyRM

k

k

kxxkxxRM

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

++++=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ++++=

(10)

The values of the uncertainty in the dimensions W, a and c (see figure 5) can be calculated as described in section (i) above. The uncertainty in the displacement measurement ( )kvu will be available from the calibration of the device used to measure the applied displacement load. Finally, Poisson's ratio should be known to a specific uncertainty, ( )υu based on the material properties supplied for Reference Material. When the uncertainty is stated explicitly, if confidence levels are provided with material data then these can be converted to standard uncertainty, e.g. ( )υu by assuming a normal distribution had been used to calculate them and using the appropriate factor for such distributions (i.e. 1.960 and 2.576 for 95% and 99% confidence levels respectively)13. If insufficient material data is supplied then measurements must be undertaken and the uncertainty of these measurements, u(υ) is equal to the positive square root of the estimated variance which may be evaluated using any valid statistical treatment of the data, e.g. by calculating the mean and standard deviation of a series of independent observations.

For each strain component of interest, these values should be used to obtain the expanded uncertainty in the Reference Material, URM =2uRM. Then for each load step, k plot ±URM as a function of y and the area between these lines defines the uncertainty arising from the Reference Material. On the same graph plot the mean residual deviations, ( ) ( )kkk duy 2±+ βα from equations (7) & (8) where )(2 kdu± indicates the width of the scatter band around the line ykk βα + . The interpretation of this plot (figure 6) is illustrated in the following two cases:

Case 1: When the scatter band )(2 kdu± on ykk βα + does not overlap the area bounded by (±URM)k for every value of y then there is a significant deviation and re-calibration should be considered by taking into account αk and βk using the guidance provided in section (v) above. If the uncertainties in αk and βk are large then they cannot be used as correction factors and efforts must be made to improve the quality of the measurement made by the instrument being calibrated.

Case 2: When the scatter band )(2 kdu± on ykk βα + overlaps the area bounded by (±URM)k for every value of y then no correction to the calibration of the measurement system is needed. In this case the measurement values coincide well with the reference values and dk(i,j) contains random errors (noise) only. However, this noise could depend on the strain level, i.e. on the position on the beam as well as on the load level and so the


19

calibration is only valid for the range of load at which measurements were performed and the above conclusion is valid.

The procedure described above should be repeated for each increment of applied displacement, kv , until the situation in case 2 is achieved for every increment. Then the calibration uncertainty can be approximated using:

( ) ( ) ( )kxxRMkkxxcal uduu εε 22 += (11)

for a single increment of applied displacement and this is applicable for strains induced at this loading. A ‘worst case’ generalized expression for the full range of applied displacement can be written as, using equations (1) and (10):

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ηκ

κεε uWv

WcuauWuu

vvu

duu k

k

kxxxxcal ⎟

⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +++++= 22

222

2

2

2

2

max 6416

21

(12) where u(d)max is the highest value of u(dk) found for the increments of applied displacement and the first term inside the square bracket is the largest uncertainty for the Reference Material which occurs at the smallest increment of loading. The x component of strain has been used as the exemplar in expressions (11) and (12) and it should be noted that when the difference or sum of the principal strains is the measurand then from expression (B48) in Appendix B (available in the on-line version at www.opticalstrain.org):

Figure 6 – Graph showing expanded uncertainty of Reference Material, ±URM as a function of y, the distance from the geometric centre of the beam in the direction of the applied load together with the mean residual deviations (αk+βky)±2u(dk) for two cases. In case 1, ±URM and (αk+βky)±2u(dk) do not overlap for all values of y and so re-calibration following adjustment of the instrument is appropriate. In case 2, ±URM

overlaps with (αk+βky)±2u(dk) for all y and so no adjustment or re-calibration is necessary.

-0.5 0.25 0.5

300

-300

200

100

-200

0

Dev

iatio

ns (m

icro

stra

in)

-URM

+URM

2u(d

k)

(αk +β

k y)case 1

2u(d

k)

Normalised position on beam, y/W

(αk +βky)case 2

-100

-0.5 0.25 0.5

300

-300

200

100

-200

0

Dev

iatio

ns (m

icro

stra

in)

-URM

+URM

2u(d

k)2u

(dk)

(αk +β

k y)case 1

2u(d

k)

Normalised position on beam, y/W

(αk +βky)case 2

-100


20

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

2/1

2

2

2

222

2

2

2

2

21max21 1416

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

++++±+=±

υυ

κκεεεε

m

uW

cuauWuuv

vuduu

k

kcal

( ) ( )ηυ uWvk

⎟⎟⎠

⎞⎜⎜⎝

⎛+ 26

1m (13)

If there is a significant, non-zero bias or offset, i.e. ( )kk u αα 2> or a significant non-

zero deviation, i.e. ( )kk u ββ 2> for more than one load step, this has to be noted for future reference. Normally, the user will want to adjust the measurement system according to the outcome of the calibration. Since strain measurement with optical methods involves many parameters in the transformation from the primary measurand to the strain value, there is no general way to make adjustments based on fit-parameters α and β. It is the user's responsibility to check and correct the relevant parameters which include both extrinsic parameters such as relative distances, angles of cameras and lights as well as intrinsic parameters such as focal length, gain and linearity. After appropriate correction, it is suggested that the calibration is repeated with the adjusted system in order to check whether the measurement value from the adjusted instrument is now close enough to the reference value for the intended use.

(vii) Future experiments:

The calibrated instrument can be used to measure strains over the range employed in the calibration process; use beyond this range will require a fresh calibration. Extension to large scale strains that induce geometric non-linearities is not viable with the method described in this document.

The measurement uncertainty in any future experiment performed with the unchanged measurement system (i.e. no change in the extrinsic and intrinsic parameters of the optical system) is at least as high as the calibration uncertainty found using the procedure described above. Equation (11) corresponds to the calibration uncertainty ucal and the expanded uncertainty12 Ucal is given by:

calcal uU 2= (14)

Stating the measurement result and documenting its uncertainty is an essential part of a traceability procedure and hence forms an indispensable part of the process. The dependence of uRM and thus ucal on the value of εxx or the position, y in the gauge section implies that the uncertainty will be different for individual data points in a field of strain values. A practical engineering approach to obtaining an indicator of the quality of the instrument is to calculate a relative uncertainty. This can be performed using equation (11) taking εxx as the highest value of strain observed at each load step and then the relative uncertainty is given by:.

(Relative uncertainty)k = ( )

( )kxx

xxcalu⎟⎟⎠

⎞⎜⎜⎝

⎛

max

max

εε

(15)

Again, the x-component of strain has been used as the exemplar in equation (15) and there will be corresponding versions for the other components of strain.


21

Appendix A: Design Notes for Reference Material An initial analysis and design was developed using theory of elasticity and subsequently was refined using results from finite element analyses (FEA); subsequent amendments based on the FEA are shown in italics.

Figure B1 - Initial Generic Design of Monolithic Four-Point Bend Specimen. Note: superscripts on dimensions refer to paragraphs in design notes.

i. Thickness, B of the beam, perpendicular to the load is defined by ISO 12108:200214 to be

0.2W≤B≤W (A1) and in analysis below:

mBW = (A2)

ii. Gauge length, 2c is defined by ISO12108:200213 as WWc 01.022 ±= (A3)

but in analyses below: pWc = (A4)

and the recommended value of p is 1 for the reasons demonstrated in (v) below.

iii. Moment arm, a is defined by ISO12108:200213 as WWa 01.0±= (A5)

and in analyses below: nWa = (A6)

The recommended value of n is 3 for the reasons demonstrated in (v) below.

iv. The gap or slit in the interlocks, χ is designed to limit the maximum displacement in four-point bending of the beam. Considering the displacement, vy(x) in the direction of the applied load, relative to the outer two loading points, then it can be shown that (e.g. Gere15):

14 ISO 12108:2002, Metallic materials – fatigue testing – fatigue crack growth method (first edition). 15 Gere, J.M., Mechanics of Materials, 5th SI edition, Nelson Thornes Ltd., Cheltenham, UK, p.891, 2001.

W [i] bc[viii]

c [ii]a [iii][v] bc

W/2W

WW/2

YT /2dh

dh

Rn[xi]

YL/2 [vii]

YL/2

YT /2 [vi]χ

χ

χ χ

χ [iv]

K [x]

H [ix]W/2

W/4

g1 [xii]

g2

g3

G [xii]

W [i] bc[viii]

c [ii]a [iii][v] bc

W/2W

WW/2

YT /2dh

dh

Rn[xi]

YL/2 [vii]

YL/2

YT /2 [vi]χ

χ

χ χ

χ [iv]

K [x]

H [ix]W/2

W/4

g1 [xii]

g2

g3

G [xii]


22

( ) ( )22336

axxLEI

Paxv By −−= (A7)

where x is the distance along the beam from one of the outer loading points, ( )caLB += 2 is the length of the beam between outer loading points,

P is the load applied to the beam at each loading point, E is the modulus of elasticity of the material, and I is the second moment of area of the beam, 123BWI =

At the position of the inner loading points, ax = and so

( ) ( )32 436

aaLEIPav By −= (A8)

substituting for L and I, and also equations (A2), (A4) and (A6) gives:

( ) ( )npEWPmnavy += 34 2

(A9)

It can be shown from simple bending theory (e.g. Gere16) that:

( )I

Myy =σ (A10)

where σ is the bending stress at a distance y from the neutral axis and M is the applied bending moment. Hence in this case the maximum bending stress will be given by:

2max

6BW

Pa=σ (A11)

So the maximum strain due to bending will be given by:

22max

66EWPmn

EBWPa

==ε (A12)

Namely the load applied at each point on the four-point bending beam will be:

mnEW

P6

2maxε

= (A13)

Substituting equation (A13) in (A9) defines the dimension of the gap to be:

( ) ( )3

32 maxWnpnavy

εχ

+== (A14)

v. The minimum depth, W of the beam is limited by the smallest cut that can be machined for the gap, χ. For instance, the minimum slit that can be spark eroded is typically, mm25.0=χ and re-arranging equation (A14):

( ) max323

εχ

npnW

+= (A15)

Hence for example when Ey 2max σε = in a typical aluminium (i.e. 3max 1065.1 −= xε ), then W =

56.8mm when n = p = 1 as recommended by ISO12108:200214. This is clearly impractical for a minimum value of beam depth. Alternatively, when p = 1 and n = 3, then W = 12.6mm which is more viable. This value will decrease with increasing maximum bending strain.

In practice, to achieve a displacement load of this magnitude on a beam of this stiffness induces a stress concentration at the loading points for the beam that is greater than the ultimate tensile strength of the material. Hence, either the displacement load or the stiffness of the beam has to be reduced to avoid failure. Assuming elastic behaviour, the stress at the loading point σw can be linearly related to the stiffness, St of the beam subject to four-point bending such that:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

tpw S

Pκσ (A15a)

16 Gere, J.M., Mechanics of Materials, 5th SI edition, Nelson Thornes Ltd., Cheltenham, UK, p.323 2001.


23

where κp is a constant of proportionality, and defining stiffness as load per unit displacement, then ( )avPS yt = and from equation (A9):

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+

=npmn

EWpw 34 2κσ (A15b)

Finite element analyses indicate that for an aluminium beam (E=6.9 × 1010N/m2) with W=12.6mm, the stress in the concentration at the loading point is 240MN/m2 when the gap, χ takes the smallest practical value (0.25mm) and hence combining equations (A15a) and (A15b) and substituting these values including p=1 and n=3 gives: EWw 276.0=σ (A15c) Hence to avoid failure at the loading points:

( )E

W allowablew

276.0max

max

σ≈ (A15d)

The range of validity of this equation is unknown, particularly with respect to the gap size, χ. The recommended minimum value for the beam depth is W = 14.5mm based on finite element analyses. The corresponding value for maximum strain from equation (A15) would be 3

max 1044.1 −= xε .

vi. The frame of the monolithic structure must be very stiff relative to the beam subject to four-point bending. In compression and tension modes, the upper bar of the monolithic structure is subject to three-point bending until it makes contact with the sidewalls in the interlock. The maximum displacement of a beam when subject to three-point bending is given by (e.g. Gere16):

EIFl

48

3

max =δ (A16)

where F is the force applied to the beam and l is its length, in this case PF 2= , clB 2= and 123

TBYI = hence substituting these values and equations (A2), (A4) and (A13):

3

4max

3

max 32

TnYWp ε

δ = (A17)

Hence if the deflection of this upper bar is to be 1% of the deflection of the beam subjected to four-point bending then: ( ) 100/max avy=δ (A18) and substituting equations (A14) and (A17) gives:

( )( ) 31

364.4

2 pnnpWYT

+= (A19)

so WYT 9.2= when n = p = 1 as recommended by ISO12108:200214 and WYT 23.1= when n = 3 and p = 1. Based on finite element analysis, the cantilevered ends of the upper bar have WYT 3= and the central portion WYT 2= .

vii. Similarly to the upper bar, the lower bar of the monolithic structure also behaves in three-point bending during tensile loading, but in compressive loading will be rigidly supported on a platen. So applying equation (A16) to the lower bar during tensile loading given, in this case PF 2= ,

( )acLl BB +== 2 and 123LBYI = :

( )( )( ) 3

1

364.42 pnn

WnpYL+

+= (A20)

so WYL 85.5= when n = p = 1 as recommended by ISO12108:200213 and WYL 91.4= when n = 3 and p = 1. FEA demonstrated that WYL 5= was sufficient in both tension and compression.

viii. The loading points for the four-point bending would usually be via pins of diameter, Dp where ISO12108:200214 recommends 2WDp ≥ . It can be shown that the contact half-length, bc for a cylinder on a flat plane is given by (e.g.Young17):

17 Young, W.C., Roark’s Formulas for Stress & Strain, 6th Edition, McGraw-Hill Book Co, New York, Table 33, p.650, 1989.


24

EBPD

b pc 075.1= (A21)

where B is the length of the cylinder, P the applied load and E the modulus of elasticity of both the flat plane and cylinder. Now, substituting equations (A2) and (A13) leads to:

n

Wbcmax

21.3ε

= (A22)

However, in tensile loading the stress in the ligament at the point of loading the beam must not exceed the maximum stress due to bending in the beam, i.e. using equation (A11):

2max6BW

PaBb

Pw

==σ (A23)

and thus

nWbw 6

= (A24)

In most cases the result from equation (A24) will be greater than for equation (A22) and so the former defines the minimum width of the ligament, bw. The stress concentration for a flat bar with two semi-circular notches loaded in tension is given by Peterson18 as 3.1 for a ratio of root radius to ligament width of 0.15. So for bw = W/6, the radius of the notches should be greater than W/40. Finite element analyses showed that these ligament dimensions lead to a high stress concentration which was reduced significantly by using 5Wbw = with a notch root radius of W/4.

ix. Buckling of the monolithic structure in the compressive mode must be avoided, and is likely to be initiated at the ligaments where the four-point bending load is applied to the beam. In-plane buckling will be constrained by the whiffle trees (see xiv). The critical load, Pcr for out-of-plane buckling can be calculated using Euler buckling theory for a pin-pin strut (e.g. Gere19):

2

2

scr l

EIP π= (A25)

where ls is the length of the strut and I is the second moment of area about the axis of buckling ( I = bwB3/12 = W4 / 72nm3 assuming bw = W/6n according to Eq.A24). Consider a strut with a cross-section of the ligament and length equivalent to the distance between the bars in the monolithic frame (i.e. the height of the beam and the upper and lower whiffle trees which is 4W), then the limiting condition is Pcr ≥ P and substituting in (B25) and using equation (B13) gives:

mnEW

nmWEW

6)4(72

2max

32

42 επ≥ (A26)

and

2

2

max 192mπε ≥ (A27)

The maximum strain would be limited to 0.0021 for m = 5 which is the minimum recommended in ISO12108:200214compared to 1.65×10-3 assumed in paragraph (v).

x. The sidewalls of the monolithic structure must be sufficiently rigid to produce an order of magnitude change in stiffness of the whole specimen when the upper bar makes contact with them, i.e.

( ) 10/avyK =δ (A28)

when δK is the deflection of the sidewalls and is given by:

KBEPH

K =δ (A29)

and substituting equations (A28), (A13) and (A14) gives:

( )103

326

maxmax

×+

=Wpnn

nKWH εε (A30)

18 Peterson, R.E., Stress concentration factors, John Wiley & Sons, New York, 1974, figure 16 19 Gere, J.M., Mechanics of Materials, 5th SI edition, Nelson Thornes Ltd., Cheltenham, UK, 2001, p.763


25

hence

( )pnnHK

3410

+= (A31)

so if H = 12W and n = p = 1, then K = 7.5W, and when n = 3, K = 0.55W.

xi. The nipple ensures alignment of the compressive load and based on the diameter of the loading pins for a standard four-point bend specimen (ISO 12108:200214) Rn = W/4.

xii. The interlock between the sidewalls and the upper bar of the monolithic structure is designed to control the displacement load applied to the beam, hence the maximum deflection of the cantilevers, vmax in the interlock must be very small (i.e. 1% of the beam deflection):

( ) 100max avv y= (A32) and the maximum deflection of a cantilever subject to a uniformly distributed load is (e.g. Gere20)

EI

PGv8

3

max = (A33)

where G is the length of the cantilever. In this case if the average bearing pressure on the surface of a cantilever is not to exceed the maximum bending stress in the beam then from equation (A11):

2

6BW

PaBGP

= (A34)

and so, nWG 6= for the worst case of n = 1 then G = W/6. Now, substitute in equation (A33) and (A13) to obtain:

3

4max

max 864ngW

vε

= (A35)

where g (assuming g=g1=g2) is the depth of the cantilever perpendicular to its neutral axis. Finally, substituting into (A32) with (A14) gives:

( )3

2 33558.0

pnnWg

+= (A36)

for n = 1, g ≥ 0.35W. In practice, it was found that to create a significant knee in the load deflection curve, g=2W.

The length of the contact zone was not found to be important and the radius of curvature of the slit which defines the interlocking cantilevers was set at rχ = W/9. It is intended that the slit could be manufactured as a single cut of width χ (see (iv)).

xiii. The stress concentration for a stepped bar in bending is approximately 1.5 for a ratio of 2 in the bar widths either side of the step and a radius to smaller bar width of greater than 0.2, hence for a step down from 2W to W the radius should be greater than 0.2W21.

xiv. The whiffle-trees were originally of length W/2 and were extended to be of length W in order to reduce the bending moment transmitted through the loading point and to allow the beam to bend more freely. The width of the slots and the ligaments between them were set at W/10 so that the total whiffle-tree width was 0.7W. The slot length was selected as 2W/3 in order to avoid interactions between the stresses generated at the ends of the slots with those induced by the change in cross-section of the whiffle-trees.

In-plane buckling is constrained by the whiffle trees. The maximum strain could be limited by a buckling of a single branch of the whiffle tree for which I=BW 3/12000 and l=2W/3 hence substituting into (A25) and (A13) as previously gives εmax=0.011n where n≥1 so this component is not critical when εmax is compared to the value of 1.65×10-3 assumed in paragraph (v).

20 Gere, J.M., Mechanics of Materials, 5th SI edition, Nelson Thornes Ltd., Cheltenham, UK, p.887, 2001. 21 Peterson, R.E., Stress concentration factors, John Wiley & Sons, New York, figure 73, 1974.


26

Appendix B: Derivation of Strain Field Equations for Reference Material

(i) Introduction

The derivation of the equations for the gauge section of the Reference Material is provided below from the principles of equilibrium of stress and compatibility of strains assuming point loading. In part (iii) a modification is introduced to account for the constraint produced by the attachment to the monolithic frame. Although the Reference Material relates to strain, the derivation is based on stress because this is the convention adopted in most texts. Finally, in section (iv) and (v), the uncertainty associated with the strain evaluation is considered.

(ii) Derivation

The central portion of a beam of rectangular cross-section subject to symmetrical four-point bending forms the basis of the Reference Material and corresponds to the gauge section in the Reference Material (RM). It is assumed that the bending is essentially two-dimensional, i.e. the strains will vary through the depth of the beam and along its length but not through its thickness. The standard approach to a plane stress problem may be adopted and the Airy’s stress function, φ used to describe the components of stress22, such that:

2

2

yxx ∂∂

=φσ ,

2

2

xyy ∂∂

=φσ and

yxxy ∂∂∂

=φσ

2 (B1)

where the x-direction is longitudinal or horizontal in the Reference Material, i.e. perpendicular to the applied load and the y-direction is transverse or vertical in the Reference Material, i.e. parallel to the applied load. The harmonic equation, which incorporates both the conditions of compatibility and equilibrium, must also be satisfied (see for instance, Heyman23) thus:

( ) 02 =+∇ xxyy σσ (B2) or in terms of the Airy’s stress function:

022 =∇∇ φ (B3) i.e.

02 4

4

22

4

4

4

=∂∂

+∂∂

∂+

∂∂

xyxyφφφ (B4)

Any polynomial of third order or less will satisfy equation (B4). If we use the general polynomial: 3223 yTxySyxRxQ AAAA +++=φ (B5)

then, from equation (B1) the Cartesian components of stress will be: yTxS AAxx 62 +=σ yRxQ AAyy 26 +=σ (B6)

ySxR AAxy 22 −−=σ The boundary conditions on the top and bottom surfaces of the beam can be employed to solve for the coefficients QA, RA, SA and TA. On these surfaces the normal stress σ yy and shear stress σ xy are zero which implies QA, RA and SA must be zero. So:

yTAxx 6=σ (B7) Now assuming that the stress distribution is linear and zero on the centre line ( 0=y ) then the moment across the beam, M is:

∫−

=2/

2/

W

Wxx BdyyM σ (B8)

using the same notation as in the description of the Reference Material where the depth of the beam is W and the thickness is B. Then substituting expression (B7) gives

AITM 6= (B9) where the second moment of area is defined as: 22 Dugdale, D.S., ‘Elements of elasticity’, Pergamon Press, Oxford, England, 1968. 23 Heyman, J., ‘Elements of stress analysis’, Cambridge University Press, Cambridge, England, 1982.


27

12

3BWI = (B10)

So, I

MTA 6= (B11)

and from equation (B5): 3

6y

IM

=φ (B12)

giving by substitution in equation (B1):

IMy

xx =σ (B13)

In the Reference Material, the total load applied to the monolithic specimen is 2P, so that the moment in the gauge section is:

aPM = (B14) where a is the distance between the inner and outer loading points on the beam. Thus equations (B12) and (B13) become:

33

2 yBW

aP=φ (B15)

and

3

12BW

aPyxx =σ (B16)

In general, strains corresponding to the stresses in equation (B1) can be expressed using Hooke’s law22 as:

( )yyxxxx Eυσσε −=

1 , ( )xxyyyy Eυσσε −=

1 , and xyxy E

συε +=

1 (B17)

where E and υ are the Young’s modulus and Poisson’s ratio respectively. So, in the case under consideration:

312EBW

aPyxx =ε ,

312EBW

aPyyy

υε −= , and 0=xyε (B18)

However, this description of the strains includes the applied load P, which may only be known as a displacement load when using the Reference Material, so consideration must be given to the displacements in the beam. The definition of strains gives rise to the displacements such that

xvx

xx ∂∂

=ε , yvy

yy ∂

∂=ε , and

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

=xv

yv yx

xy 21ε (B19)

So, in the case under consideration for the region between the inner loading points:

yEBW

aPxvx

3

12=

∂∂ (B20)

yEBW

aPyvy

3

12υ−=

∂

∂ (B21)

0=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

xv

yv yx (B22)

where vx and vy are displacements in the x and y directions respectively. It can be shown (e.g. Gere24) that the displacement, vy(x) in the direction of the applied load and for y = 0, relative to the outer two loading points is:

( ) ( )22336

axxLEI

Pax,0v By −−= (B23)

where x is the distance along the beam from one of the outer loading points, ( )caLB += 2 is the length of the beam between outer loading points,

P is the load applied to the beam at each loading point, E is the modulus of elasticity of the material, and

24 Gere, J.M., Mechanics of Materials, 5th SI edition, Nelson Thornes Ltd., Cheltenham, UK, 2001, p.891


28

I is the second moment of area of the beam, 123BWI = At the position of the inner loading points, x = a and so:

( ) ( )acEBW

Paavy += 340, 3

2 (B24)

and this deflection is likely to be equal to the amount by which the monolithic frame closes, v , which is the measured quantity in the Reference Material. Thus, it is appropriate to express the load, P in terms of this displacement, i.e.

)3(4 2

3

caaEBWvP

+= (B25)

and substituting into equation (B18) gives the strain in the gauge section of the Reference Material:

)3(3

caavy

xx +=ε ,

)3(3

caavy

yy +−=

υε and, 0=xyε (B26)

Note that this expression is independent of the actual value of beam depth W. Using the parametric design and noting that in the Reference Material a = 3W and c = W from equations (A4) and (A6), we have

26Wvy

xx =ε , 26W

vyyy

υε −= and, 0=xyε (B27)

The strains in any other direction, θ can be obtained by using a Mohr’s circle of strain or the transformation equations (see for example Benham et al25):

θεθεεεε

ε 2sin2cos22 xy

xxyyyyxxxx +

−−

+=′′

θεθεεεε

ε 2sin2cos22 xy

xxyyyyxxyy −

−+

+=′′

(B28)

θεε

θεε 2sin2

2cos xxyyxyyx

−+=′′

And the maximum and minimum principal strains, ε1,2 are given by

( ) 221 2

12 xyxxyy

yyxx εεεεε

ε +−++

=

( ) 222 2

12 xyxxyy

yyxx εεεεε

ε +−−+

= (B29)

If the measured strain is plotted as a function of depth in the beam, i.e. the y-direction then the gradient of the plot can be considered as a function of applied displacement. Differentiating equations (B27), the gradient of strain across the beam will be:

26Wv

dyd xx =

ε and 26W

vdy

d yy υε−= (B30)

Then the gradient of these quantities as a function of applied displacement will be:

261

Wdyd

vdd xx =⎟⎟

⎠

⎞⎜⎜⎝

⎛ ε and 26Wdy

dvdd yy υε

−=⎟⎟⎠

⎞⎜⎜⎝

⎛ (B31)

Two possible sources of error can be readily identified: first, the measured deflection has been assumed to be equal to the deflection of the centre line of the beam at the inner loading points and second, any longitudinal tension or moment at the loading points in the beam has been neglected. These latter issues are addressed in the next section. (iii) Modification for influence of constraint on bending

The monolithic frame exerts a small constraint on the beam subjected to four-point bending that induces an additional bending moment in the beam. The classical analytical description of a beam subject to bending (in the previous section) assumes that the loads are applied through knife edges, i.e. that no moment can be

25 Benham, P.P., Crawford, R.J., and Armstrong, C.G., ‘Mechanics of Materials’ 2nd edition, Addison Wesley Longman Limited, Essex, England, 1987.


29

sustained at the loading point. This boundary condition is not achieved in the Reference Material. Figure B1 illustrates a bending analysis in which a moment at the loading points is included. In elastic conditions it is reasonable to assume that (Mi+Mo) will be linearly related to the load applied to the Reference Material. Thus, a simple correction factor, κ would be:

)(PaM κ= (B32) and following this through the analysis of the previous section, equation (B27) would become:

26Wvy

xxκε = ,

26Wvy

yyκυε −= and, 0=xyε (B33)

Similarly the gradients in equation (B30) would become:

26Wv

dyd xx κε

= and 26W

vdy

d yy κυε−= (B34)

and

26Wdyd

vdd xx κε

=⎟⎟⎠

⎞⎜⎜⎝

⎛ and 26Wdy

dvdd yy κυε

−=⎟⎟⎠

⎞⎜⎜⎝

⎛ (B35)

Two ways of evaluating the correction factor are available: either using finite element analysis or a strain gauge bonded to either or both of the top and bottom faces of the beam in the gauge section and then:

vW xxε

κˆ

12= (B36)

where xxε̂ is the strain value measured using the strain gauge (i.e. maximum value as a function of y) and v is the applied displacement load. It is recommended that κ is found by plotting xxε̂ as a function of v .

Figure B1 – Bending analysis assuming constraint at the loading points. A second source of constraint is that the beam is not free to extend in the x-direction as it bends under the action of the moments. This constraint is alleviated by bending in the whiffle-trees that will be proportional to the load applied to the beam. However any remaining constraint will induce a tensile strain along the length of the beam. In the gauge section, i.e. away from the constraint points, this tensile strain is likely to be uniformly distributed across the depth of the beam, y, and hence will increase strain by an amount proportional to the applied load, P or v , thus causing the neutral axis to shift towards the centre of curvature due to bending. Thus the expression (B33) for strain in the beam becomes

( )ηκε += yWv

xx 26, ( )ηκυε +−= y

Wv

yy 26 and, 0=xyε (B37)

where κ is a constant to be evaluated and 0=η when there is no constraint. Expressions (B34) and (B35) are unchanged. We can estimate κ and η from either a pair of strain gauges on the top and bottom surfaces of the beam or from the results of finite element analysis. Results from finite element analysis give a value of κ = 0.94 ±0.006 for 15mm ≤ W ≤ 30mm while η increases linearly from 0.10mm to 0.18mm over the same range of W. Hence, a realistic estimate is η = -0.009W ±0.0004W for 15 ≤ W ≤ 30mm.

P

PMo

Mi

P

P

Mo

Mi

P

P

Mo Mi

b a x

Shear force, SF=0

Bending moment, BM

( ) ( ) oi MMbaxPbxPMBM −−−−−−==

Considering a section of beam of length, x

Taking moments about the right-hand end:

( )oi MMPaM +−=

P

PMo

Mi

P

P

Mo

Mi

P

P

Mo Mi

b a x

Shear force, SF=0

Bending moment, BM

( ) ( ) oi MMbaxPbxPMBM −−−−−−==

Considering a section of beam of length, x

Taking moments about the right-hand end:

( )oi MMPaM +−=


30

(iv) Uncertainty in the strain values from the Reference Material

The Reference Material (RM) is designed to repeatably reproduce a strain field by displacement controlled beam deflection. To know the strain value in a given location, one has to measure the position (x,y), the load displacement v , as well as the actual geometry of the beam expressed through a, c, and W. In addition, the corrections κ and η must be known or evaluated. All these quantities have an associated uncertainty which must be taken into account when stating a strain value for a specific location. The derivation of the expressions for these uncertainties is described in the following paragraphs.

Note: the uncertainty in position (x,y), due to the fact that the beam edge and the position of the neutral axis cannot be determined exactly, is not included here because it is embedded in the difference between the measured and predicted values of strain and its uncertainty, as described in methodology section, 3 (d) (iv).

From the analytical description of the strain components given by equation (B37) and by partial differentiation it can be deduced that:

( ) ( ) ( )[ ] ( ) ( ) ( )

( ) ( ) ( ) ( )2

22

xx222

2

22

2

22222

2

2

)()()(

)(4)(6

υυεευε

εεηκε

uyx,yx,uyx,u

WWuyx,

vvuyx,uu

Wvy)(x,u

yyyyanalyt

xxxxxx2analyt

+=

+++⎟⎟⎠

⎞⎜⎜⎝

⎛= y

(B38)

where ( )κu is the uncertainty of the correction factor, ( )ηu is the uncertainty of the position of the neutral axis, ( )vu is the uncertainty in the measurement of the load displacement, and ( )Wu takes into account the variations in beam depth. Note that because the beam depth cancels out from the expressions for the strain values in equation (B26), only deviations from a constant beam depth matter in the uncertainty analysis. Finally, Poisson's ratio will be known to a specific uncertainty, ( )υu based on the material properties data supplied with the raw material.

Table B1 - Example for the quantitative determination of measurement uncertainty

measurand f

nominal value uncertainty u(f)

comment

W 10 mm 0.005 W/3 = 0.017 mm assuming tolerance in parallelism is 3σ-value

a 30 mm u(a) = 0.017 mm assuming tolerance is 3σ-value c 10 mm u(c) = 0.017 mm assuming tolerance is 3σ-value y -5…+5 mm u(y) = 0.02 mm assuming 500 pixels on W v for

%1max =xxε , 1.2 mm

%1.0max =xxε , 0.12mm

( )vu = 0.005mm + 0.005 v ( )vu = 0.011mm ( )vu = 0.006mm

from the calibration sheet of the transducers and the difference between top left and top right measurement

υ 0.33 ( )υu = 0.005 from least significant decimal place

κ 0.94 u(κ) = 0.006 assuming a rectangular distribution within the last digit, i.e. value can be anywhere between 0.935 & 0.945, this gives a standard deviation of (0.01/sqrt(3)=0.006)

η 0.06 mm u(η) = 0.006 mm assuming last digit uncertainty as above In order to estimate the influence of load asymmetry in the specimen, i.e. uncertainty in values a and c,

( )au and ( )cu , return to equation (B26) and by partial differentiation obtain:

( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+⎟⎟⎠

⎞⎜⎜⎝

⎛++

= cucaa

aaucaacau xxxxasymmetry

22

22

22

)3(3

)3(32εε (B39)

Again introduce the design values a = 3W and c = W to yield:


31

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

2

2222

2

2222

22

222

4

421

21

Wcuauu

Wcuaucu

Wau

Wu

yyyyasymmetry

xxxxxxasymmetry

εε

εεε (B40)

Combining equations (B38) and (B40) the expressions for the combined standard uncertainty of the Reference Material strain values can be estimated as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ηυ

υυ

κκεε

ηκ

κεε

22

22

2

2

222

2

2

2

222

22

22

222

2

2

2

222

6416

6416

uW

vuW

cuauWuuv

vuu

uWv

WcuauWuu

vvuu

yyyyRM

xxxxRM

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎥

⎦

⎤⎢⎣

⎡+

++++=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎥

⎦

⎤⎢⎣

⎡ ++++=

(B41)

By way of example using the values from Table B1, for a maximum strain level of %1max =xxε , according to equation (B41) the uncertainty of the Reference Material strain values are:

( ) ( ) ( )

( ) ( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )26262626

262222

222222

1012103610641092

10120036.00064.00092.0

1006006.02.1

1004017.018

94.0006.0

2.1011.0

−−−−

−

×+×+×+×≤

×+++=

⎟⎠⎞

⎜⎝⎛

××

+⎥⎥⎦

⎤

⎢⎢⎣

⎡

××

+⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

xx

xxxxRMu

ε

εε

(B42)

and

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( ) ( )2626262626

2622222

2222222

1041050101210211030

1040152.00036.00064.00092.0

1006006.02.133.0

33.0005.0

1004017.018

94.0006.0

2.1011.0

−−−−−

−

×+×+×+×+×≤

×++++=

⎟⎠⎞

⎜⎝⎛

×××

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

××

+⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

yy

yyyyRMu

ε

εε

(B43)

Whereas, for an applied displacement with a maximum of %1.0max =xxε the Reference Material uncertainties are:

( ) ( ) ( )

( ) ( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )26262626

262222

222222

1011036106410500

1010036.00064.00500.0

1006006.012.0

1004017.018

94.0006.0

12.0006.0

−−−−

−

×+×+×+×≤

×+++=

⎟⎠⎞

⎜⎝⎛

××

+⎥⎥⎦

⎤

⎢⎢⎣

⎡

××

+⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

xx

xxxxRMu

ε

εε

(B44)

and

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( ) ( )2626262626

2622222

2222222

104.0100.5102.1101.21016

104.00152.00036.00064.00500.0

1006006.012.033.0

33.0005.0

1004017.018

94.0006.0

12.0006.0

−−−−−

−

×+×+×+×+×≤

×++++=

⎟⎠⎞

⎜⎝⎛

×××

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

××

+⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

yy

yyyyRMu

ε

εε

(B45)

In order to simplify the expressions, a conservative estimate can be made by noting that ( )222 baba +≤+ from which the measurement uncertainty can conveniently be written as

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ηυ

υυ

κκεε

ηκ

κεε

uW

vuW

cuauWuuv

vuu

uWv

WcuauWuu

vvuu

yyyyRM

xxxxRM

⎟⎟⎠

⎞⎜⎜⎝

⎛++

++++=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++++=

22

2

2

222

2

2

2

2

22

222

2

2

2

2

6416

6416

(B46)

The results of equations (B42) through (B46) are summarized in Table B2.


32

Table B2 – Summary of results

%1max =xxε ( ) xxxxRMu εε 012.01012 6 +×= − ( ) yyyyRMu εε 019.0104 6 +×= −

%1.0max =xxε ( ) xxxxRMu εε 051.0101 6 +×= − ( ) yyyyRMu εε 052.0104.0 6 +×= −

(v) Uncertainty in combined strain values

The Reference Material is providing a strain field from the bending of a beam. The principal axes are given by the direction of the beam (x-axis) and the load introduction (y-axis). Hence, so far the traceability of the strain values ),( yxxxε and ),( yxyyε has been emphasised. Note, however that the Reference Material can also be used for calibrating techniques that provide the sum or difference of principal strains, since in the Reference Material we have

),(),(),(),( 21 yxyxyxyx yyxx εεεε ±=± (B47) The corresponding calibration uncertainty is calculated from the uncertainties of ),( yxxxε and ),( yxyyε by properly taking into account their correlation. If the sum or difference of principal strains is of interest, the uncertainty is given from equations (C37) and (C41) by noting: ( )υεεε m121 xx=± and then:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( )ηυυυ

κκεε

υεευεε

22

22

2

2

2

222

2

2

2

22

21

222221

2

61

1416

1

uWvu

WcuauWuu

vvu

uuu xxxxRM

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎥

⎦

⎤⎢⎣

⎡+

++++±=

+=±

mm

m (B48)

When a rotation angle,ϕ between the measurement system and the Reference Material is introduced, a strain field transformed according to Mohr's circle is obtained (equation (B28)) and the strain components are:

( ) ( )( ) ( )

( )yyxxyx

yyxxyyxxyy

yyxxyyxxxx

εεϕε

εεϕεεε

εεϕεεε

−−=

−−+=

−++=

2sin

2cos

2cos

21

''

21

21

''

21

21

''

(B49)

The corresponding calibration uncertainty is calculated from the uncertainties of ),( yxxxε and ),( yxyyε by properly taking into account their correlation, and must include the contribution due to the measurement uncertainty of the rotation angle. After introducing the independent variables

( ) ( )[ ]( ) ( )[ ]

( )υϕεε

υϕυεευϕυεε

+=

+−−=

++−=

12sin

12cos112cos1

21

''

21

''

21

''

xxyx

xxyy

xxxx (B50)

the uncertainties are given by ( ) ( ) ( )[ ] ( ) [ ] ( )[ ]( ) ( ) ( )[ ] ( ) [ ] ( )[ ]( ) ( )[ ] ( ) [ ] ( )[ ] )(12cos)(2sin12sin

)(12sin)(2cos112cos1

)(12sin)(2cos112cos1

2222224122

41

''2

2222224122

41

''2

2222224122

41

''2

ϕυϕευϕεευϕε

ϕυϕευϕεευϕυε

ϕυϕευϕεευϕυε

uuuu

uuuu

uuuu

xxxxxxyx

xxxxxxyy

xxxxxxxx

++++=

+++++−−=

++−+++−=(B51)

where ( ) ( )),(22 yxuu xxRMxx εε = for short. Replacing xxε by the strain components measured in the rotated system, we have, using (B50) and (B41)

( ) ( ) ( )[ ] ( )

( )

( ) ( ) ( ) ( ) ( )

[ ] ( )[ ]( ) ( )[ ] ⎥

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

++−++−

+

⎥⎦

⎤⎢⎣

⎡ ++++

+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++−=

2

2222

2

222

2

2

2

2

2''

22

22

41

''2

12cos1)(12sin2)(2cos1

416

612cos1

υϕυϕυϕυϕ

κκ

ε

ηυϕυε

uu

WcuauWuu

vvu

uWvu

xx

xx


33

( ) ( ) ( )[ ] ( )

( )( ) ( ) ( ) ( ) ( )

[ ] ( )[ ]( ) ( )[ ] ⎥

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+−−+++

+

⎥⎦

⎤⎢⎣

⎡ ++++

+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−=

2

2222

2

222

2

2

2

2

2''

22

22

41

''2

12cos1)(12sin2)(2cos1

416

612cos1

υϕυϕυϕυϕ

κκ

ε

ηυϕυε

uu

WcuauWuu

vvu

uWvu

yy

yy

(B52)

( ) ( )[ ] ( )

( ) ( ) ( ) ( ) ( ) ( )( )

[ ][ ] ⎥

⎦

⎤⎢⎣

⎡+

++⎥

⎦

⎤⎢⎣

⎡ +++++

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

2

22

2

2

2

222

2

2

2

22

''

22

22

41

''2

2sin)(2cos2

1)(

416

612sin

ϕϕϕ

υυ

κκε

ηυϕε

uuW

cuauWuuv

vu

uWvu

yx

yx


34

GUIDELINES for the CALIBRATION & EVALUATION of OPTICAL SYSTEMS for STRAIN MEASUREMENT Specifications and protocols for the use of a Reference Material and Standardised Test Material for respectively calibrating and evaluating optical systems for strain measurement. Prepared as part of the SPOTS project; a shared cost RTD project with the European Commission’s Competitive and Sustainable Growth Programme (Contract No. G6RD‐CT‐2002‐00856 ‘SPOTS’)

Contributors: Richard Burguete, Erwin Hack, Eann Patterson, Thorsten Siebert & Maurice Whelan

Part II: Evaluation


36

Figure 1 – Three-dimensional representation of the Standard Test Material. (EU Community Design Registration 000213467)


37

Executive Summary

Today the experimental stress analyst has a dazzling array of technology available for measuring strains compared with a decade or two ago. Advances have occurred in both well-established techniques and the development of new techniques such as digital image correlation. New instruments and systems have been pioneered in laboratories and have appeared in the marketplace. A procedure for the certified calibration of optical systems for strain measurement is described in part I of these guidelines. The need to provide traceability to a national standard by a series of comparisons implied the use of a simple strain distribution in the Reference Material. Such a distribution should be very straightforward to measure with most systems and present little challenge to sophisticated systems. Thus, to evaluate a sophisticated system against either its design specification or its competitors there is a requirement for a challenging stress distribution defined within a standardised test material. In this part of the guidelines, a Standardised Test Material (STM) is described that consists of a disc in contact with an elastic half-space whilst subject to a diametral compression. A methodology for the use of the Standardised Test Material is described together with a set of functions which allow standardised data sets to be generated and used to verify the output from each stage of the measurement process. ESPI, image correlation, grating interferometry, moiré, digital photoelasticity and thermoelasticity have been used as exemplars but the methodology could easily be extended to other techniques. Exemplar studies illustrating the use of both the Reference Material and Standardised Test Material have been published elsewhere26,27.

26 Whelan, M.P., Albrecht, D., Hack, E., Patterson, E.A., ‘Calibration of a speckle interferometry full-field strain measurement system’, Strain, 44(2):180-190, 2008. 27 Patterson, E.A., Brailly, P., Burguete, R.L., Hack, E., Siebert, T., Whelan, M.P.,‘A challenge for high performance full-field strain measurement systems’, Strain, 43(3):167-180, 2007.


38

1. Introduction The provision of a standardised test material is part of a process to enhance the quality and acceptability of full-field optical techniques of strain measurement. During the establishment of this process two distinct needs were identified: the requirement for calibration materials and methods connected to international standards; and the desirability of having standardised test materials that allow the performance of systems and sub-systems to be evaluated and compared. Calibration involves making comparisons with a known, recognised reference material which in turn has been compared via a continuous chain of comparison to an international standard. The standard metre appears to be the most appropriate primary standard for strain. The concepts of standardisation and traceability are discussed in more detail by Hack et al28. Part I of this document details a Reference Material (RM) and an analytical description of the strain field in the gauge section of the RM which can be used for calibration.

Figure 2 – A schematic illustrating the generic measurement process (right) for optical strain measurement

systems. The ovals represent data sets and the rectangles represent components of the data evaluation system. The process is exemplified using measurements in a cracked compact tension specimen obtained

using ESPI29 and processed to produce a map of the y-direction (vertical) strain.

A Standardised Test Material (STM) has been designed to allow the performance of a measurement system to be assessed. This is relevant to system developers when designing an instrument and its algorithms, to manufacturers concerned with quality assurance, to instrument purchasers wishing to compare the capabilities of systems, and to end-users for setting up and maintaining the system and training staff in its use. The Standardised Test Material is intended to provide a strain field which is challenging to 28 Hack, E., Burguete, R.L., Patterson, E.A., ‘Traceability of optical techniques for strain measurement’, Appl. Mechanics and Materials, 3-4, 391-396, 2005. 29 Shterenlikht, F.A. Diaz Garrido, P. Lopez-Crespo, P. Withers, P.J., Patterson, E.A., ’SIFs from strain fields using ESPI and image correlation‘, Proc. SEM Int. Congress Exptl. Mech., Costa Mesa, CA., 2004.

μm

%

transformation

phase (modulo 2π) maps

test object subject to strain

fringe maps

scaling

displacement

strain tensor map, εx,εy,γxy

differentiation

device

intensity maps

unwrapping

continuous (non-periodic) maps

convolution / correlation

μm

%

transformation



fringe maps

scaling

displacement

scaling

displacement

strain tensor map, εx,εy,γxy

differentiation

devicedevice

intensity mapsintensity maps

unwrapping


convolution / correlation


39

analyse and to allow each stage in the determination of strain to be evaluated. The strain field was chosen after consultation30 with the experimental mechanics community which suggested that the Standardised Test Material should include a strain concentration, a variation in strain direction, a reversal of the sign of strain, a discontinuity in the strain distribution, and a physical boundary.

In order to provide a uniform approach that is applicable to all techniques of full-field strain measurement, the stages in the measurement process have been classified31 and are described in the generic process chart shown in figure 2. Some ESPI data obtained4 from a compact tension specimen with a crack is shown in figure 2 in order to illustrate the stages. It is clear that not all techniques will involve every step and the schematic in figure 3 maps a number of techniques onto the generic process.

Figure 3 – Process flow chart for image correlation, ESPI, shearography, moiré, photoelasticity and thermoelasticity showing the appropriate route through the map in Figure 2. Operations are shown as black

boxes, data as white boxes and the corresponding standardized data sets as coloured boxes (left).

Figure 4 shows the intended use of the standardised data sets as inputs to an operation in the measurement process and as a comparison for evaluating the outputs. The geometry selected for the Standardised Test Material is a disc in contact with an elastic half-space and subject to diametral compression. This geometry contains a strain concentration, a variation of strain direction, a reversal of the sign of the strain and physical boundaries. Only the discontinuity in the strain distribution suggested by the community is not included and the SPOTS consortium32 proposed a second geometry consisting of a pair of 30 Burguete, R.L., Hack, E., Siebert, T., Patterson, E.A., Whelan,M., ’Candidate reference materials for optical strain measurement‘, Proc. 12th Int. Conf. Exptl. Mech., Bari, Italy, Paper no. 145, 2004. 31 Burguete, R.L., Hack, E., Kujawinska, M., Patterson, E.A., ‘Classification of operations and processes in optical strain measurement’, Proc. 12th Int. Conf. Exptl. Mech., Bari, Italy, Paper no. 130, 2004. 32 SPOTS: Standardisation Project for Optical Techniques of Strain measurement, www.opticalstrain.org

Continuous map (non-periodic)(displacement) (displacement)(displacement difference) (shear strain)(displacement) (surface temperature)

strain map

scaling

unwrapping

convolution/correlation

intensity maps

device


Image Correlation ESPI Moiré*Shearography Photoelasticity* Thermoelasticity

Fringe maps(displacement)(displacement difference) (shear strain)(displacement)

εx, εy, γxy… εx, εy, γxy … (ε1 - ε2 )εx, εy, γxy… (εx + εy )εx, εy, γxy….

Standardised Test Specimen

*intrinsic techniques in which the fringe pattern is recorded directly (naked eye test)

transformation

Phase maps (modulo 2π)

displacement

differentiation differentiation

displacement.

StandardisedData Set 1

Standardised Data Set 2





Continuous map (non-periodic)(displacement) (displacement)(displacement difference) (shear strain)(displacement) (surface temperature)

Continuous map (non-periodic)(displacement) (displacement)(displacement difference) (shear strain)(displacement) (surface temperature)(displacement) (displacement)(displacement difference) (shear strain)(displacement) (surface temperature)

strain map

scaling

unwrapping

convolution/correlationconvolution/correlation

intensity maps

device


intensity maps

device


Image Correlation ESPI Moiré*Shearography Photoelasticity* Thermoelasticity

Fringe maps(displacement)(displacement difference) (shear strain)(displacement)

Fringe maps(displacement)(displacement difference) (shear strain)(displacement)(displacement)(displacement difference) (shear strain)(displacement)

εx, εy, γxy… εx, εy, γxy … (ε1 - ε2 )εx, εy, γxy… (εx + εy )εx, εy, γxy….εx, εy, γxy… εx, εy, γxy … (ε1 - ε2 )εx, εy, γxy… (εx + εy )εx, εy, γxy….

Standardised Test SpecimenStandardised

Test Specimen

*intrinsic techniques in which the fringe pattern is recorded directly (naked eye test)

transformation


transformation


displacement

differentiation differentiationdifferentiation differentiation

displacement.














40

interference-fit cylinders to include this feature but did not develop a physical manifestation of this design. The Standardised Test Material is planar or two-dimensional which was intentional since most measurements follow this form and it was decided to address the challenge of developing a unified approach for all planar techniques before addressing out-of-plane or three-dimensional strain systems.

Serious consideration was given to the inclusion of noise in the standardised tests. Noise tends to be a handicap to obtaining high quality reliable data. Primary data should reflect real scenarios in which temporal variations in the data are usually labelled noise whilst spatial variations are normally labelled experimental artefacts and arise from surface texture, contrast inhomogenieties, surface and, or coating defects (missing data) and offsets (fluctuations in ambient light). There are too many possible weaknesses in each measurement system that could lead to the generation of noise for all of them to be simulated in a standardized data set. A unified approach to simulating noise could not be envisaged and modelling noise sources for each measurement system was believed to be beyond the scope of this first step towards a Standardised Test Material for optical methods of strain measurement. Therefore, it is left for the user to introduce noise in the standardised data sets that is representative of their measurement system.

The following sections describe the essential elements of the Standardised Test Material with design notes and derivations being included in the Appendices (available in the on-line version at www.opticalstrain.org).

Figure 4 – Schematic illustrating the generic measurement process (right) as in figure 2 for optical strain measurement systems and the relationship of the standardized test and data sets (left) to the process. The

ovals represent data sets and the rectangles represent components of the data evaluation system.

theory

theory

theory

theory

theorytransformation


Test object subject to strainStandardised Test Material

device

intensity maps

Convolution / correlation

fringe maps

unwrapping


Standardised Data Set - 2


calibration

Displacement

Strain tensor map, εxx,εyy,γxy

differentiation



theory



input

comparison

comparison

input

input

comparison

input

input

input

comparison

comparison

comparison

theory

theory

theory

theory

theorytransformation


Test object subject to strainStandardised Test Material

device

intensity mapsintensity maps


fringe maps


fringe maps

unwrapping


Standardised Data Set - 2Standardised Data Set - 2


calibration

Displacement

Strain tensor map, εxx,εyy,γxy

differentiation



theory



inputinput

comparison

comparison

inputinput

inputinput

comparison

inputinput

inputinput

input

comparison

comparison

comparison


41

2. Description of Standardised Test Material

(a) Design notes The Standardised Test Material consists of a two-dimensional specimen that can be manufactured in any homogeneous and isotropic material at any scale. At small scales care should be taken to ensure that material defects and microstructure will not influence the strain distribution. The design is monolithic in the sense that it is a single piece of material and can be loaded using any suitable arrangement for applying a compressive load. The relative displacement of the two portions (top and bottom faces) should be measured using an appropriate and convenient device for the scale of the specimen. Further guidance on the use of the specimen is provided in the section on methodology. The specimen includes two pairs of leaf springs which maintain the alignment of the disc and elastic half-space as well as being responsible for the monolithic status. At very small scales these springs should not be scaled. A slit of width, χ is provided between the upper and lower beams of the monolithic frame in order to protect the disc from excessive plastic deformation through over-loading. At small scales this slit could be left unscaled which would leave the specimen unprotected from plastic deformation. Detailed design notes are provided in Appendix C (available in the on-line version at www.opticalstrain.org).

Figure 5 – Diagram of Standardised Test Material showing dimensions normalised on the diameter, D of the disc. All radii should be D/20 unless otherwise specified. Tolerances should be 0.05mm unless

otherwise specified including the parallelism of the front and back faces. Note that it may be inappropriate to scale the slit in the interlock and the leaf springs at very small scales.

(EU Community Design Registration 000213467)

A0.01D0.01D

D

D/8

D/8D/8

D/8

t l

tl(iii)=D/40

D/1

0

j (vi) =D/4

χ=D/20

g(ii) =

D/4

0

D/4

l(iii)=41D/60

c(iv)=D/5

D/8

k(viii) =D/2

5D/8

D/8

D/2

D/4

4D

H=

4D

4D/25

D/2

3D/4

D

7D/8

h(v) =D/4

D/8

0.01D

B0.01D

B

A

B0.01D

D/8

2D

2D

A0.01D0.01D

A0.01D0.01D

D

D/8

D/8D/8

D/8

t l

tl(iii)=D/40

D/1

0

j (vi) =D/4

χ=D/20

g(ii) =

D/4

0

D/4

l(iii)=41D/60

c(iv)=D/5

D/8

k(viii) =D/2

5D/8

D/8

D/2

D/4

4D

H=

4D

4D/25

D/2

3D/4

D

7D/8

h(v) =D/4

D/8

0.01D

B0.01D B0.01D B0.01D

B

A

B0.01D

D/8

2D

2D


42

(b) Strain field equations In order to assess an optical system for measuring strain, the results obtained from the Standardised Test Material can be compared to those predicted using the equations describing the analytical strain field for the same geometry. The analytical solution to the disc being compressed and in contact with an elastic half-space is described in two separate parts. First, the strain field in the central portion of the disc, i.e. at a distance from the points of application of load can be expressed by (see Appendix D, equations (D6) to (D8) which is available in the on-line version at www.opticalstrain.org):

( ) ( ) ( ) ( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡ −−

+−++

−−−−=

RryRyRx

ryRyRx

BEπPyx,xx 2

124

2

32

41

32 υυυε

( ) ( ) ( ) ( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡ −−

+−++

−−−−=

RryRxyR

ryRxyR

BEPyxyy 2

12, 42

23

41

23 υυυπ

ε (1)

( ) ( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡ +−

−+= 4

2

2

41

212,r

xyRr

xyRBE

Pyxxy πυε

where the origin is taken at the centre of the disc and P is the load applied to the disc, E is the Young’s modulus, R=D/2 is the radius of the disc, B is the thickness of the disc, υ is Poisson's ratio, the y axis coincides with the load direction, and

( )2221 yRxr −+= (2)

( )2222 yRxr ++=

For contact of the disc with the elastic half-space the strain field is described by (see Appendix E which is available in the on-line version at www.opticalstrain.org) ):

( )( )

( )( ) ( )⎥⎥

⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

++−+−

+−

2423

2

32212

212 1

11

2n

nnxx

y

xyyx,ζζ

ζζζ

ζζξε

( )( ) ( ) ( )

( )( )⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

++

+−−

+++μ 24

212

212242

12

2 11

121

1n

nn

n

nn

yyx

yyx

ζζζυξ

ζζ

ζζζυξ

( )( ) ( )

( )( )

( )( )

( ) ( )

( ) ( ) ( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−−

++−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

++−

+−

+−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

+⎥⎥⎦

⎤

⎢⎢⎣

⎡

++=

212242

12

2

242

32212

212

24

212

3

24212

2

112

1

1

1

12

1

1

23

ζ

ζ

ζζ

ζυξμ

ζζ

ζζζ

ζ

ζυξ

ζζζξ

ζζ

ζξμε

n

n

nn

n

nn

nn

n

nnyy

xy

yx

y

xy

yy

y

yxyx,

(3)


43

Figure 6 – A schematic diagram illustrating the comparisons used to evaluate the performance of stages in

the measurement chain.

( )( ) ( )

( )( )

( ) ( ) ⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++−+−

+−+

++

−

+=

24212

322

12

212

24212

2

1

1

12

11)(

n

nn

n

nn

xy

y

xy

y

yx

yx,

ζζ

ζζζ

ζ

ζμ

ζζ

ζ

υξε

where

( )22

12 υπξ

−=

ERBP (4)

( ) ( )[ ]⎭⎬⎫

⎩⎨⎧

+−−+−−−=21

2222222 41121

nnnnn yyxyxζ (5)

with c

n bxx = ,

cn b

yy = and µ is the coefficient of friction.

(c) Methodology for use The schematic in figure 6 summarizes the expected approach for employing the Standardised Test Material. Data could be collected from the Standardised Test Material and a map of intensity generated by the system under evaluation. This intensity map, the experimental data set (EDS1) could be compared with the corresponding standardised data set (SDS1). The areas for data analysis in the standardized test material are defined taking account of the desired features in the strain field of the STM and the availability of analytical solutions. The definition of the data zones are shown in figure 7. For the data areas in figure 7, it is recommended that the comparison is quantified, by calculating the

device

intensity

PSTM

SDS1

SDS2comparison

device

Intensity, EDS1

convolution

STM

SDS1comparison

data

collection

SDS2EDS2

input

device

intensity

PSTM

SDS1

SDS2comparison

device

Intensity, EDS1

convolutionconvolution

STM

SDS1comparison

data

collection

SDS2EDS2EDS2

input


44

difference between the two data sets using expression (5) in part 1 of this document and the associated uncertainties can be estimated using a similar methodology. Comparisons in areas of high strain gradients associated with the contact zones are better performed using data along the two broken lines indicated in figure 7. The strain field will be a function of the applied displacement. Subsequently, SDS1 could be supplied to the next algorithm in the process, producing EDS2, and the result compared with SDS2. This sequence of processing and comparison could be commenced at any stage in the measurement sequence. Whilst it is not prohibited to use a processed result from one algorithm as the input to the next algorithm in the system (i.e. following a broken line in figure 6) it is not recommended because it makes it difficult to evaluate the performance of an individual algorithm.

Standardised Test Material can be used either way up, though it should be noted that rigid body displacements make it more challenging to some techniques when employed as shown in figure 1. The choice of force or displacement loading is not specified and for convenience the relationship between force and displacement has been developed below. Once contact has been achieved by the disc with the elastic half-space, then the relative displacement of the upper and lower halves of the frame of the Standardised Test Material, which could also be termed its vertical deflection, can be described by:

ligamentbeamhalfspacediscPSTM δ+δ+δ+δ=δ (6)

where δligament refers to the small ligament connecting the bottom of the disc to the supporting beam whose length is 8D/25 and cross-section area at its narrowest point is DB/5. Then using expressions (D22), (D23) and (C22) respectively from Appendices D and C (available in the on-line version at www.opticalstrain.org):

( ) ( )5BE

PmEh

PDbd

BEP

bR

bR

BEP

sc

c

ccPSTM

121

81627

12ln2114ln4ln12

3

222

++⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+

−=

υυ

πυ

πυδ (7)

Figure 7 – Diagram showing zones (shading) for data analysis. Comparisons should be made along x = 0

from (0, 0) to (0, D) and along ( )cc bDy 8.02 += (broken red lines) as well in the data zones.

8D/7

D/1

4

Region of invalidity for disc solution

8D/7

D/2

D

D/7

D/7

P

(0,D)

(0,0)

(-D/2,D/2+ 0.8bc) (D/2,D/2+ 0.8bc)

8D/7

D/1

4

Region of invalidity for disc solution

8D/7

D/2

D

D/7

D/7

P

(0,D)

(0,0)

(-D/2,D/2+ 0.8bc) (D/2,D/2+ 0.8bc)


45

0

20

40

60

80

100

0 500 1000 1500 2000 2500

Load (N)

Per

cent

age

cont

ribut

ion

dischalf spacebeamligament

0

20

40

60

80

100

0 50 100 150 200 250

Load (N)

Perc

enta

ge c

ontri

butio

n

0

20

40

60

80

100

0 500 1000 1500 2000 2500

Load (N)

Per

cent

age

cont

ribut

ion

dischalf spacebeamligament

0

20

40

60

80

100

0 50 100 150 200 250

Load (N)

Perc

enta

ge c

ontri

butio

n

now 2DR = and DmB s= . Assuming that the contact half-lengths at both ends of the diameter of the disc are equal, i.e. ccc bbb ==

21 and Ddc = then:

( )⎭⎬⎫

⎩⎨⎧

+⎥⎦

⎤⎢⎣

⎡−

−−+−

≈ 1101

22ln61 2

υυ

πυδ

cPSTM b

DBEP (8)

Note that ( ) 21

218⎥⎦⎤

⎢⎣⎡ υ−π

=BEPRbc and with υ =0.3 this gives

BEDPbc 08.1= .

These relationships are shown in figures 8 and 9.

Figure 8 – Load-displacement relationship for the Standardised Test Material following contact of the disc with the half-space, showing the contributions of the various aspects of the geometry based on expressions

(6), (7) & (8). Data for an aluminium specimen of thickness, B=5mm and D=10mm.

Figure 9 – Percentage contribution to total deflection of the Standardised Test Material following contact of

the disc with the half-space, based on the data in figure 8.

0

0.005

0.01

0.015

0.02

0.025

0 500 1000 1500 2000 2500

Load (N)

Def

lect

ion

(mm

)

Total deflectionbeam deflectionhalfspace deflectiondisc deflectionligament deflection

0

0.0005

0.001

0.0015

0.002

0 50 100 150 200 250

Load (N)

Def

lect

ion

(mm

)

0

0.005

0.01

0.015

0.02

0.025

0 500 1000 1500 2000 2500

Load (N)

Def

lect

ion

(mm

)

Total deflectionbeam deflectionhalfspace deflectiondisc deflectionligament deflection

0

0.0005

0.001

0.0015

0.002

0 50 100 150 200 250

Load (N)

Def

lect

ion

(mm

)


46

3. Functional paths to standardised data sets

The strain field described for the standardised test can be used to compute the standardised data sets (SDS) shown in figures 4 and 6 which correspond to the output from each stage of the measurement process, the experimental data sets (EDS). Since the strain fields can be expressed as analytical functions of load and the spatial co-ordinates then it follows that the SDS can be obtained in a similar form via a set of analytical functions based on the underlying principles of each technique. It should be noted that SDS and EDS are intended to be arrays of data that can be viewed as an image.

The remainder of this section contains process charts relating to each of the techniques considered. These show the functions required to achieve the necessary transformations to create the SDS beginning from an analytical function describing the strain field (SDS6) and proceeding from SDS(n) to SDS(n-1). These functions are generic in the sense that they apply any test geometry.

The intermediate standardised data sets are intended for testing the corresponding step in the measurement process (see figure 4). The construction of SDS5, backwards from SDS6, cannot be performed in a unique way for some techniques because there is an unknown integration constant when going from strain to displacement unless the boundary conditions are known. However, the test would consist in feeding SDS5 into the experimental routine and then comparing EDS6 to SDS6 so that the lack of uniqueness is not significant.

Note that the broken lines indicate processes and datasets which do not usually exist for that particular technique but are shown in the interests of clarity and completeness.


47

(b) D

igita

l Im

age

Cor

rela

tion

(a) E

lect

roni

c Sp

eckl

e Pa

tter

n In

terf

erom

etry

devi

ce

inte

nsity

m

aps

conv

olut

ion/

corre

latio

n

SDS 1

≡

fring

e m

aps

trans

form

atio

ns

SDS 2

≡

phas

e m

aps

unw

rapp

ing

SDS 3

≡

cont

inuo

us m

aps

scal

ing

SDS 4

≡

stra

in te

nsor

map

s≡

disp

lace

men

t

diffe

rent

iatio

n

test

obj

ect

≡A

naly

tical

Mod

el

SDS 5

≡

()∗

⋅=

),

,(

),

,(

),

,(

φφ

φd

yx

ESd

yx

ESd

yx

ISD k

nD k

nD k

n

SDS

6

devi

ce

inte

nsity

m

aps

conv

olut

ion/

corre

latio

n

SDS 1

≡SD

S 1≡

fring

e m

aps

trans

form

atio

ns

SDS 2

≡SD

S 2≡

phas

e m

aps

unw

rapp

ing

SDS 3

≡

cont

inuo

us m

aps

scal

ing

SDS 4

≡SD

S 4≡

stra

in te

nsor

map

s≡

disp

lace

men

t

diffe

rent

iatio

n

test

obj

ect

≡A

naly

tical

Mod

el

SDS 5

≡SD

S 5≡

()∗

⋅=

),

,(

),

,(

),

,(

φφ

φd

yx

ESd

yx

ESd

yx

ISD k

nD k

nD k

n

SDS

6

test

obj

ect

conv

olut

ion

/ cor

rela

tion

inte

nsity

map

s

Ana

lytic

al M

odel

fring

e m

aps

devi

ce

trans

form

atio

n

phas

e m

aps

unw

rapp

ing

cont

inuo

us m

aps

scal

ing

disp

lace

men

t

diffe

rent

iatio

n

stra

in te

nsor

map

s

≡

≡ ≡≡≡≡ ≡

SD

S

3SD

S

2

SD

S

4

SD

S

5

SD

S1

()

()

∫=

dxy

x,ε

yx,

uxx

()

()

∫=

dyy

x,ε

yx,

vyy SD

S6

()

()

() y

x,ε

&y

x,ε,

yx,

εxy

yyxx

SD

S

2

Und

isto

rted

patte

rn

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erte

d to

dis

torte

d pa

ttern

bas

ed o

n an

alyt

ical

m

odel

Vec

tor f

ield

of

defo

rmat

ions

is a

pplie

d to

tran

sfor

m u

ndis

torte

d da

ta s

et

test

obj

ect

conv

olut

ion

/ cor

rela

tion

inte

nsity

map

s

Ana

lytic

al M

odel

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e m

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ce

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aps

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ing

disp

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men

t

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rent

iatio

n

stra

in te

nsor

map

s

≡

≡ ≡≡≡≡ ≡

SD

S

3SD

S

2

SD

S

4

SD

S

5

SD

S1

()

()

∫=

dxy

x,ε

yx,

uxx

()

()

∫=

dxy

x,ε

yx,

uxx

()

()

∫=

dyy

x,ε

yx,

vyy

()

()

∫=

dyy

x,ε

yx,

vyy SD

S6

()

()

() y

x,ε

&y

x,ε,

yx,

εxy

yyxx

()

()

() y

x,ε

&y

x,ε,

yx,

εxy

yyxx

SD

S

2

Und

isto

rted

patte

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erte

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ical

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odel

Vec

tor f

ield

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defo

rmat

ions

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pplie

d to

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m u

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ta s

et


48

(c) G

ratin

g In

terf

erom

etry

(d

) Moi

ré

test

obj

ect

conv

olut

ion

/ cor

rela

tion

inte

nsity

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s

Ana

lytic

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odel

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ce

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cont

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aps

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ing

disp

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men

t

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rent

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n

stra

in te

nsor

map

s

≡

≡ ≡≡≡≡ ≡

SD

S

3

SD

S

2

SD

S

4

SD

S

5

SD

S1

()

()

()

yx,

2πm

ody

x,u

2π uϕ

ϕ=

()

()

()

yx,

2πm

ody

x,v

2π vϕ

ϕ=

()

() y

x,u

d4πy

x,u

=ϕ

()

() y

x,v

d4πy

x,v

=ϕ (

)(

)∫

=dx

yx,

εy

x,u

xx

()

()

∫=

dyy

x,ε

yx,

vyy SD

S6

()

()

() y

x,ε

&y

x,ε,

yx,

εxy

yyxx

test

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ect

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nsity

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Ana

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n

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≡

≡ ≡≡≡≡ ≡

SD

S

3

SD

S

2

SD

S

4

SD

S

5

SD

S1

()

()

()

yx,

2πm

ody

x,u

2π uϕ

ϕ=

()

()

()

yx,

2πm

ody

x,u

2π uϕ

ϕ=

()

()

()

yx,

2πm

ody

x,u

2π uϕ

ϕ=

()

()

()

yx,

2πm

ody

x,v

2π vϕ

ϕ=

()

()

()

yx,

2πm

ody

x,v

2π vϕ

ϕ=

()

() y

x,u

d4πy

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=ϕ

()

() y

x,u

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x,u

=ϕ

()

() y

x,v

d4πy

x,v

=ϕ

()

() y

x,v

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=ϕ (

)(

)∫

=dx

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x,u

xx(

)(

)∫

=dx

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xx

()

()

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vyy

()

()

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dyy

x,ε

yx,

vyy SD

S6

()

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() y

x,ε

&y

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yx,

εxy

yyxx

()

()

() y

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&y

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yx,

εxy

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test

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ect

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ion

/ filtr

atio

n

inte

nsity

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s

Ana

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s

≡

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SD

S 3

SD

S 2

SD

S 4

SD

S 5

SD

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()

()

()

yx,

2 πm

ody

x,u

2π uϕ

ϕ=

()

()

()

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ody

x,v

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()

() y

x,u

p2 πy

x,u

=ϕ

()

() y

x,v

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=ϕ

()

()

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dxy

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uxx

()

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yx,

vyy

SD

S6

()

()

() y

x,ε

&y

x,ε,

yx,

εxy

yyxx

()

()

()

[] φ

ϕd

1n

yx,

kco

sb

1y

x,I

2π u

K

1k

km

u−

++

=∑ =

()

()

()

[] φ

ϕd

1n

yx,

kco

sb

1y

x,I

2π v

K

1k

km

v−

++

=∑ =

test

obj

ect

conv

olut

ion

/ filtr

atio

n

inte

nsity

map

s

Ana

lytic

al M

odel

fring

e m

aps

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ce

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S 3

SD

S 2

SD

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SD

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SD

S1

()

()

()

yx,

2 πm

ody

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ϕ=

()

()

()

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2 πm

ody

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2π uϕ

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()

()

()

yx,

2 πm

ody

x,v

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ϕ=

()

()

()

yx,

2 πm

ody

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()

() y

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=ϕ

()

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=ϕ

()

() y

x,v

p2 πy

x,v

=ϕ

()

() y

x,v

p2 πy

x,v

=ϕ

()

()

∫=

dxy

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uxx

()

()

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dxy

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uxx

()

()

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dyy

x,ε

yx,

vyy

()

()

∫=

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x,ε

yx,

vyy

SD

S6

()

()

() y

x,ε

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yx,

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yyxx

()

()

() y

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&y

x,ε,

yx,

εxy

yyxx

()

()

()

[] φ

ϕd

1n

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sb

1y

x,I

2π u

K

1k

km

u−

++

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()

()

()

[] φ

ϕd

1n

yx,

kco

sb

1y

x,I

2π u

K

1k

km

u−

++

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()

()

()

[] φ

ϕd

1n

yx,

kco

sb

1y

x,I

2π v

K

1k

km

v−

++

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()

()

()

[] φ

ϕd

1n

yx,

kco

sb

1y

x,I

2π v

K

1k

km

v−

++

=∑ =


49

(e)

Phot

oela

stic

ity

(f)

The

rmoe

last

icity

devi

ce

inte

nsity

map

s

conv

olut

ion/

corre

latio

n

SD

S 1

≡

fring

e m

aps

trans

form

atio

ns

SD

S 2

≡

phas

e m

aps

unw

rapp

ing

SD

S 3

≡

cont

inuo

us m

aps

calib

ratio

n

SD

S 4

≡

stra

in m

aps

≡

disp

lace

men

t

diffe

rent

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n

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S 5

≡

test

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ect

≡A

naly

tical

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el:

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ce

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nsity

map

s

conv

olut

ion/

corre

latio

n

SD

S 1

≡S

DS

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t

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n

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Exp

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ions

for s

train

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ε xx, ε y

y, ε x

y

test

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naly

tical

Mod

el

Am

plitu

de o

f firs

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in

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riant

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load

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≡

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ce

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nsity

map

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ion/

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SD

S 6

≡S

DS

6≡


50

Appendix C: Design Notes for Standardised Test Material The intention of this design is to provide strain fields with the following features: boundaries, rotation of principal strain directions and a strain concentration. This is achieved through the loading of a disc under diametral compression in such a way that contact with an elastic half-space is available for analysis. The specimen is monolithic in the sense that it is formed from a single planar piece of material and requires simple loading in compression between platens. Two pairs of ‘leaf springs’ provide a physical linkage between the two moving parts of the specimen and offer a negligible resistance to deflection under an applied load. The use of two pairs provides additional resistance to out-of-plane rotation when handling the specimen. The disc is protected from overloading and buckling and or excessive plastic deformation by contact of the upper and lower parts of the monolithic frame at the slit χ.

A preliminary round robin test within the SPOTS consortium33 on a tensile specimen with some geometric features had revealed the difficulties associated with generating reproducible results from different optical systems in laboratories located in different countries even when the test specimens were manufactured and supplied from one source and many laboratories used the same model of loading frame. The crucial importance of a reproducible loading system became apparent and lead to the concept of a test specimen and loading frame contained with a single monolithic design for the Standardised Test Material.

An initial analysis and design was developed using the theory of elasticity and subsequently was refined using results from finite element analyses. Amendments to the notes arising from the finite element analyses are shown in italics.

i Load on the disc For a cylinder of diameter, D in contact with a flat half-plane of similar material, i.e. E1=E2 and

υ1=υ2=0.3, then (Roark34, p.651) gives the maximum compressive stress, cσ̂ , as

DEpl

c 591.0ˆ =σ (C1)

where pl is the load per unit length of the cylinder. The maximum shear stress occurs at (0.8bc) below the surface where bc is the contact half-length, and is given by (Roark6, p.651) as:

( ) 21

218⎥⎦⎤

⎢⎣⎡ υ−π

=BEPRbc

which for υ =0.3 is EDp

b lc 15.22 = (C2)

and the maximum shear stress at this location is

3ˆ

maxcστ ≈ (C3)

So if the maximum shear stress is limited to 80% of yield in order to avoid permanent deformation of the specimen:

3ˆ

28.0max

cY σστ =⎟

⎠

⎞⎜⎝

⎛= (C4)

and substituting equation (C1) with the specimen thickness B, for applied load P, gives:

( )Yc BDPE σσ 8.0

23591.0ˆ == (C5)

let DmB s= to obtain:

222

2 12.48.0591.01

23

Yys DmPE σσ =⎟

⎠⎞

⎜⎝⎛ ××= (C6)

thus E

DmP sY22

12.4 σ= (C7)

33 Mendels, D. –A., Hack, E., Siegmann, P., Patterson, E.A, Salbut, L., Kujawinska, M., Schubach, H.R., Dugand, M., Kehoe, L., Stochmil, C., Brailly, P., Whelan. M, ‘Round robin exercise for optical strain measurement’, Proc. 12th Int. Conf. Exptl. Mechanics, Advances in Exptl Mechs, edited by C. Pappalettere, McGraw-Hill, Milano, 695-6, 2004. 34 Young, W.C., ‘Roark’s Formulas for stress & strain’, 6th edition, McGraw-Hill Book Co., New York, 1989.


51

Figure C1 - Schematic diagram of Standardised Test Material. All radii are D/20 unless separately specified. Roman numeral superscripts refer to design notes below (EU Community Design Registration 000213467).

ii. Gap to contact of disc and half-space Considering the gap between the disc and the rigid beam, g, let the ‘leaf springs’ be analysed as cantilevered beams with a point load, WL on the end, then the maximum deflection for a beam of length l, will be (Roark34, p.100):

EIlWy L

3

3

= (C8)

In practice, for the leaf springs 2Dl > so they will be more flexible than described by this expression when l = D/2. Assume the load is shared between the two beams on each side, where each beam is of section (B × t1) and B = msD so that:

12

3ls DtmI = and

( )3

2

3

3

224

ls

L

ls

L

tEmDW

tEDm

DWy == (C9)

Note that it has been assumed that the beams act independently even though they are joined at both ends. In practice this is unlikely to be the case and the load needed to achieve the required deflection will be higher. This has been confirmed by finite element analysis. If the gap to be closed before the disc makes contact is the smallest slit that can be machined, i.e. g = 0.25mm then for a minimum disc diameter D = 10mm, this implies 40Dg = :

3

2

240 ls

L

tEmDWD

= or D

EmtW s

l

L

203 = (C10)

Now, the maximum stress in each leaf spring due to bending is given by simple bending theory:

12

2ˆˆ 3

max

ls

lL

Dtm

tlW

IyM ⎟

⎠⎞⎜

⎝⎛

==σ l

ls

L tDtm

lW3

6= (C11)

A0.01D0.01D

D

D/8

D/8D/8

D/8

t l

tl(iii)=D/40

D/1

0

j (vi) =D/4

χ=D/20

g(ii) =

D/4

0

D/4

l(iii)=41D/60

c(iv)=D/5

D/8

k(viii) =D/2

5D/8

D/8

D/2

D/4

4D

H=

4D

4D/25

D/2

3D/4

D

7D/8

h(v) =D/4

D/8

0.01D

B0.01D

B

A

B0.01D

D/8

2D

2D

A0.01D0.01D

A0.01D0.01D

D

D/8

D/8D/8

D/8

t l

tl(iii)=D/40

D/1

0

j (vi) =D/4

χ=D/20

g(ii) =

D/4

0

D/4

l(iii)=41D/60

c(iv)=D/5

D/8

k(viii) =D/2

5D/8

D/8

D/2

D/4

4D

H=

4D

4D/25

D/2

3D/4

D

7D/8

h(v) =D/4

D/8

0.01D

B0.01D B0.01D B0.01D

B

A

B0.01D

D/8

2D

2D


52

where WL is the load on the cantilever, M̂ is the maximum value of the bending moment and l is

the length of the cantilever. Allow 10

ˆ yσσ = and set 2Dl = then:

2

310 ls

LY

tmW

=σ or

302Ys

l

L mtW σ

= (C12)

Hence by combining equation (C10) and (C12) to give:

ED

m

EmD

t Y

Ys

sl 3

230

20σ

σ== (C13)

so for Aluminium (7075) alloy 610480×=Yσ Pa and 91072×=E Pa giving t1 = 0.0044D or t1 = 0.044mm for D =10mm which is impractical and so instead let t1 = 0.25mm when D =10mm, i.e. t1 = D/40 (=0.025D). In practice, it was found that large stress concentrations were present with l = D /2 and so their length was increased to 53Dl ≈ .

iii. Load for contact of disc and half-space The load to contact is given by equation (C10), assuming equal loads are taken by each of the four cantilevers:

3

23

)40(52044 DEm

DtEmW sls

L == (C14)

which as a proportion of the applied load, P is given by equation (C7) as:

26

2

225

2

10318.112.4102.34

YsY

s

L E

EDm

DEm

PW

σσ ×=×= (C15)

Again for Aluminium (7075) alloy, 58100874.04 ≈=PWL which means that the load required to achieve contact of the disc with the beam will be about 1.7% of the maximum load applied to the disc. Finite element analyses with 53Dl ≈ gave a ratio of 2.6% for contact to maximum load.

iv. Contact half-length/Disc-beam ligament dimension The contact area for the disc with the upper beam is given by equation (C2) which given DmB s= leads to:

EmPbs

c 15.22 = (C16)

and substituting for the applied load, P from equation (C7) gives:

ED

EDb YY

cσσ 18.212.4075.1 2

22

== (C17)

and for aluminium Dbc 015.0= . This dimension would probably cause failure of the ligament at the base of the disc so instead let 5Dc = . This dimension was changed to allow scaleable dimensions elsewhere in the design, and is a function of the radii in the ligament and the distance between the disc and the beam subject to three-point bending.

v. Dimensions of beam supporting the disc The disc is mounted on a flexible double-cantilevered beam of length D3 and cross-section Bh × . This beam is subject to constrained three-point bending and ISO178:200335 recommends for three-point bending that the ratio of length to thickness should be 120 ±=BL which implies

21/3DB = or 71=sm . Now for the beam, if the maximum stress due to bending is limited to 2Yσ then:

35 Plastics – Determination of flexural properties (Fourth edition), ISO 178:2003


53

IyMY max

ˆ

2ˆ ==

σσ (C18)

where 8

ˆ PLM = from Roark33, p.101 leading to, for a maximum displacement ymax= h/2:

23 49

12

28

32 hm

PBh

hPD

s

Y ==σ (C19)

hence, substituting equation (A7) for the maximum applied load:

ED

ED

h YY σσ3.4

212.4

32

== (C20)

and for aluminium alloy, 3Dh ≅ . Note that ISO178:200335 recommends that 8=hL and the above calculation leads to ( ) 933 == DDhL . The length of this beam was increased to 36D/10 as a consequence of the increase in length of the leaf springs which followed from the finite element analyses.

vi. Deflection of beam supporting the disc In these circumstances the deflection of the beam will be (Roark34, p.101):

EIWL

192

3

=δ (C21)

substituting as appropriate leads to:

3

2

3

3

1627

12192

27hEm

PDBhE

PDjs

===δ (C22)

substituting equation (A7) for the applied load and equation (C20) for h leads to:

ED

ED

EDmE

DDmEmhPDj YY

Ys

sY σσ

σ

σ 088.0791612.427

3.416

12.4271627 5.12

3

2

222

3

2

=⎟⎠⎞

⎜⎝⎛

××

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

×==

− (C23)

and for aluminium, 140D≈δ which is very small. Note that ISO178:200335 recommends a maximum deflection of B5.1 or ( ) 575.15.1 DDDms ≈=× .

vii. Maximum displacement of beam supporting disc Alternatively, let the beam bend until it touches the lower frame and then the deflection will be the smallest gap, j that can be machined, 40D=δ for 10≥D mm. Hence from equation (C22):

3

2

1627

40 hEmPDDj

s

== (C24)

so 3/2

32

32

3 527.62782

135⎟⎠⎞

⎜⎝⎛===

ED

ED

EmPDh YY

s

σσ (C25)

and for aluminium, Dh 23.0= .

viii. Stiffness of monolithic frame The sidewalls of the frame need to be at least an order of magnitude stiffer than the test specimen,

i.e. Frame sidewall stiffness = ⎥⎦

⎤⎢⎣

⎡+

×deflectiongapP10 =

( ) ( ) DP

DDP 200

404010

=+

(C26)

and the sidewall deflection (compression) is given by:

AEFL

=δ or AEPH

=δ (C27)

and j≤δ10 (C28) Hence, with thickness DmB s= and wall width, k substituting in (C28):


54

4010 D

DEkmPH

s

≤ (C29)

from which substituting equation (C26) gives:

EDmPHks

2400≥ (C30)

And substituting the applied load from equation (C7) yields: 222

2 16481648⎟⎟⎠

⎞⎜⎜⎝

⎛=≥

EH

EDm

EDmHk YsY

s

σσ (C31)

and with DH 4= for aluminium, Dk 3.0≥ . In practice, this was found to be too small and lead to unacceptable deflections in the region of the slit and cantilever which were designed to protect the disc from overloading. Thus, based on finite element analyses and experience from practical tests, the wall width was selected to be 2/Dk = .


55

Appendix D: Derivation of Strain Field Equations for Standardised Test Material The compression of the disc and the contact between the elastic half-space can be considered separately. The problem of a circular disc with a compressive load applied across a diameter was solved by Hertz36 in 1883 and by Michell37 in 1900 and has been described in detail by Frocht38. Only the final result is reproduced here:

( ) ( )⎥⎦

⎤⎢⎣

⎡−

++

−−=

RrxyR

rxyR

BP

xx 212

42

2

41

2

πσ (D1)

( ) ( )⎥⎦

⎤⎢⎣

⎡−

++

−−=

RryR

ryR

BP

yy 212

42

3

41

3

πσ (D2)

( ) ( )⎥⎦

⎤⎢⎣

⎡ +−

−= 4

2

2

41

22r

xyRr

xyRBP

xy πσ (D3)

where ( )222

1 yRxr −+= and ( )2222 yRxr ++= (D4)

Now, the components of strain can be readily obtained since from Hooke’s law39:

( )yxxx Eυσσε −=

1 , ( )xyyy Eυσσε −=

1 , and xyxy E

συε +=

1 (D5)

where E and υ are the Young’s modulus and Poisson’s ratio respectively. So in the case under consideration:

( ) ( ) ( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡ −−

+−++

−−−−=

RryRyRx

ryRyRx

BEP

xx 212

42

32

41

32 υυυπ

ε (D6)

( ) ( ) ( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡ −−

+−++

−−−−=

RryRxyR

ryRxyR

BEP

yy 212

42

23

41

23 υυυπ

ε (D7)

( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡ +−

−+= 4

2

2

41

212r

xyRr

xyRBE

Pxy π

υε (D8)

these expressions will describe the strain field in the body of the disc except for small circular regions in the immediate vicinity of the points of application of the load.

The subsurface stresses in an elastic half-space compressed by a cylinder can be solved analytically40. In the case of normal and tangential loading, the subsurface stress field is given for a static or sliding line contact by:

( )( )

( ) ( )

( ) ( ) ( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−−

+++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++−

+−

+−=

21

21

23

21

21

2242

2

242

322

2

112

1

1

1

12

ζ

ζ

ζζ

ζμ

ζζ

ζζζ

ζ

ζσ

n

n

nn

n

nn

o

xx

xy

yx

y

xyp (D9)

( )( ) ( ) ( )⎥

⎥⎦

⎤

⎢⎢⎣

⎡

++−

++

−=242

2

24

23

21

21

1

1

n

nn

n

n

o

yy

y

yxy

yp ζζ

ζμζζ

ζσ (D10)

( ) ( ) ( )( )

( ) ( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++−

+−

+−+

++

−=

242

322

2242

2

23

21

21

21

1

1

12

1 n

nn

n

nn

o

xy

y

xy

y

yxp ζζ

ζζζ

ζ

ζμζζ

ζτ (D11)

36 Hertz, H., Z. Math. Physik, vol. 28, 1883. 37 Michell, J.H., Proc. London Math Soc., vol. 32, p.44, 1900 and vol.34, p.134, 1901. 38 Frocht, M.M., Photoelasticity, vol.II, John Wiley & Sons, New York, 1948, p.121-6. 39 Dugdale, D.S., ‘Elements of elasticity’, Pergamon Press, Oxford, England, 1968. 40 Hills, D. A., Nowell, D. Sackfield, A., Mechanics of Elastic Contacts. Butterworth Heinemann, Oxford, 1993.


56

where

( ) ( )[ ]⎭⎬⎫

⎩⎨⎧

+−−+−−−=21

2222222 41121

nnnnn yyxyxζ (D12)

and the co-ordinate system (x,y) is defined from the centre of contact with the y-axis being positive into the elastic half-space and (xn, yn) are values normalised by the contact half-length bc. p0 is the maximum value of the Hertzian distribution of the pressure over the contact half-length. The Hertzian distribution of pressure is given by41:

21

2

0 1⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

cbxpp (D13)

It can be shown that39:

2

1

02

⎟⎠⎞

⎜⎝⎛=

π=

*πBR*PE

BbPp

c

(D14)

where

2

22

1

21 111

EE*Eυ−

+υ−

= (D15)

but in this case both the cylinder and elastic half-space are manufactured from the same material so:

( )212 υ−=

EE* (D16)

and (1/R*) is the relative curvature given by:

21

11*

1RRR

+= (D17)

but in this case the curvature of the elastic half-space is zero, so R*=R =D/2 for the disc thus substituting in equation (D14) gives:

( )2

1

20 12 ⎥⎦

⎤⎢⎣

⎡−

=υπBR

PEp (D18)

The strain field equations can be obtained by substituting expression (D9) to (D12) into (D5):

( ) ( ) ( )( )

( ) ( )

( )( )

( ) ( ) ( ) ( )( )( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

++

−+−−

++

+−

μ

++−+−

+−

−

24

2122

12

212242

12

221

2423

2

32212

212

21

2

1121

121

112

1

11

212

22

2

2

n

nn

n

nn

n

nnxx

yERBPx

yyx

ERBP

y

xERB

Pyyx,

yζζ

ζυυπζ

ζζζ

ζυυπ

ζζ

ζζζ

ζζ

υπε

(D19)

( ) ( ) ( ) ( ) ( )( )( )

( ) ( )( )

( ) ( )

( ) ( ) ( ) ( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−−

++⎥⎦

⎤⎢⎣

⎡−

−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

++−

+−

+−⎥

⎦

⎤⎢⎣

⎡

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

⎥⎦

⎤⎢⎣

⎡

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++⎥⎦

⎤⎢⎣

⎡−

=

212242

12

221

2

22

242

32212

212

21

2

22

24

2122

1

2

6

24212

221

2

2

112

112

1

1

12

12

112112

23

ζ

ζ

ζζ

ζυπ

μυ

ζζ

ζζζ

ζ

ζυπ

υ

ζζζ

υπζζ

ζυπ

με

n

n

nn

n

nn

n

n

n

nnyy

xy

yxERB

P

y

xERB

Py

yERBPy

y

yxERB

Pyx,

(D20)

41 Johnson, K.L., Contact Mechanics, Cambridge University Press, Cambridge, p.101,1985.


57

( )( )

( ) ( )

( )( )

( ) ( ) ⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

+ζζ+

ζ−ζ

ζ+−ζ+

ζ−μ+

+ζζ+

ζ−

−π+=

24212

322

12

212

24212

2

2

2

11

12

1

121)(

21

n

nn

n

nn

xy

yxy

yyx

ERBPyx,

υυε

(D21)

In both strain field descriptions above the load, P is expressed as a force and in practice will be the same in both cases. The compression of the disc or cylinder along its diameter when subject to compression, P along the diameter has been found to be42:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 14ln4ln12

21

2

ccldisc b

RbR

EP

πυδ (D22)

where bc1 and bc2 are the contact half-lengths at each end of the diameter. Also Johnson41 states that “the compression of a half-space relative to a point at a depth, dc below the centre of the Hertzian contact pressure distribution” is given by:

( )⎥⎦

⎤⎢⎣

⎡

−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=− υ

υπ

υδ1

2ln21 2

c

clspacehalf b

dE

P (D23)

and from equation (D14):

( ) 21

218⎥⎦⎤

⎢⎣⎡ υ−π

=BEPRbc

(D24)

42 Johnson, K.L., Contact mechanics, Cambridge University Press, Cambridge, 1985, p.131.


58

Appendix E: Explanation of Functional Paths for Standardised Data Sets (a) ESPI In this section the steps required to generate the SDS are described. Depending on the test object and the analytical model the calculation of the displacement, SDS5 may vary. For the generation of continuous phase data (SDS4) the type of sensor and configuration is needed. The equations here are based on the use of a 3D-ESPI sensor using two in-plane and one out-of-plane measurement. For the use of other sensor configurations the equations have to be adapted. The fringe maps (SDS2) are based on the method of phase calculation. In general one would use the more stable method not based on the fringe maps but building the phase difference of between the reference and the deformed state for each speckle. For this method the fringe maps are not needed and the intensity maps (SDS1) can be calculated directly.

SDS6 to SDS5 - Strain to Displacement Input: The origin of the co-ordinate system is in the centre of the CCD, the z-axis is in the direction of the optical axis and the viewing direction is the negative z-direction. From the analytical model of the STM, the strain field is given:

),( yxxxε ; ),( yxyyε ; ),( yxxyε (E.a.1)

Procedure: The displacement will be achieved by integration:

dyyxu xx∫= ε),(

dyyxv yy∫= ε),( (E.a.2)

Taking the boundary conditions and ),( yxxyε into account. Note that direct integration may not be easily achieved and numerical methods may need to be employed.

Output: The output is the displacement: ),( yxu and ),( yxv

SDS5 to SDS4 - Displacement to Continuous Phase: Input:

Measured displacement components: a) In – plane x b) In – plane y c) Out-of–plane z

Positions of illumination: ),,( 1111 LQLQLQ zyxLQ ; ),,( 2222 LQLQLQ zyxLQ ;

),,( 3333 LQLQLQ zyxLQ ; ),,( 4444 LQLQLQ zyxLQ (E.a.3)

Wavelength of illumination: λ

Procedure: ),,( zyxP : Vector from origin to the object point

),,( zyxI : Vector from origin to the image point

PH : Vector from origin to the principle point of the imaging (pin hole model)

),,( zyxLQi : Vector from origin to the illumination point i

Determine 4,3,2,1),,(),,(

),,(),,(),( =

−

−= i

zyxLQzyxP

zyxLQzyxPyxL

i

ii

(E.a.4)


59

and ),,(),,(

),,(),,(),(

zyxPzyxI

zyxPzyxIyxO

−

−= or

),,(

),,(),(

zyxPPH

zyxPPHyxO

−

−= (E.a.5)

with ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+=

)1(00

mfPH where f is the focal length and m the magnification of the imaging. The

sensitivity vectors for the different measurement configurations are: ),(),(),( 21 yxLyxLyxSa −=

),(),(),( 43 yxLyxLyxSb −= (E.a.6)

)()()( yx,Oyx,Lyx,S ic −= Assuming a pure in-plane deformation, the deformation ),,( zyxV

r for each point on the object is

given by:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

0),(),(

),,( yxvyxu

zyxV (E.a.7)

The phase value ),( yxkϕ for each measurement configuration can be calculated as:

( ) cbakyxSyxVyx kk ,,),(),(2),( =⋅=λπϕ (E.a.8)

SDS4 to SDS3 - Continuous Phase to Wrapped Phase The wrapped phase value ),(2 yxk

πϕ will be calculated using the expression:

)),((2mod),(2 yxyx kk ϕπϕ π = (E.a.9)

SDS3 to SDS2 - Wrapped Phase to Fringe Maps Input:

4 times π/2 phase shift 4 + 4 method (difference of speckle phase) )( yx,I sp : Intensity of speckle

From FT method: ),(),( yxieayxE φα ⋅= ; aα: constant amplitude; φ(x,y) : randomized phase distribution

The intensity of the speckle field ),( yxISP can be calculated using a virtual aperture ( )ψξ ,pA in the Fourier plane with:

eimpermeablpermeable

Ap 01

),( =ψξ (E.a.10)

The complex electric field can be expressed as ))),(),(((),( 1 ψξApyxEyxEA ⋅ℑℑ= − (E.a.11)

And then the intensity is: ),(),(),( yxEyxEyxI AAsp

∗⋅= (E.a.12) Using the intensity of the speckle field, the intensity field ),( yxIFk for each measurement direction k can be calculated as:

)),(cos(),(),( 2 yxyxIyxIF kspkπϕ⋅= (E.a.13)

with cbak ,,= For the calculation of the phase data the fringe maps can be used. Depending on the phase shift algorithm used, n fringe maps with a phase shift of φd can be calculated as:

...,3,2,1))1()((cos)()( 2 =⋅−+ϕ⋅= π ndnyx,yx,Idy,x,IF kspn

k φφ (E.a.14)


60

SDS2 to SDS1 - Fringe Maps to Intensity Maps Depending on the method for the calculation of the phase data a certain number of intensity maps for the reference and the deformed state are needed. Assuming that the phase data are calculated as the difference of the phase of the speckle in the reference and the deformed state, n intensity maps for each measurement direction for the reference and the deformed state are necessary. Starting with the complex electric field ),( yxEA of the speckle field, the electric fields

),( yxES Rk in the reference and ),( yxES D

k in the deformed state can be derived. Using a phase shift algorithm with n steps of phase shift φd :

φφ dniA

Rk

n eyxEdyxES )1(),(),,( −⋅= ),()1( 2

),(),,( yxidniA

Dk

n keeyxEdyxESπϕφφ ⋅⋅= − (E.a.15)

The intensity maps ),,( φdyxIS Rk

n and ),,( φdyxIS Dk

n are given by:

( )∗⋅= ),,(),,(),,( φφφ dyxESdyxESdyxIS Rk

Rk

Rk

n

( )∗⋅= ),,(),,(),,( φφφ dyxESdyxESdyxIS Dk

nDk

nDk

n (E.a.16) (b) Image Correlation In the image correlation technique a 2-dimensional random pattern (of dots) is recorded in its undistorted and distorted form. The difference between these patterns is directly proportional to the displacement field. SDS6 to SDS5 – Strain to Displacement

Input From the analytical model the strain field is given:

εxx (x,y); εyy (x,y); εxy (x,y) (E.b.1)

Procedure The displacement will be achieved by integration:

∫∫

=

=

dyyxv

dxyxu

yy

xx

ε

ε

),(

),( (E.b.2)

Taking the boundary conditions and εxy(x,y) into account. Note that direct integration may not be easily achieved and numerical methods may need to be employed.

Output The output is the displacement: u(x,y) and v(x,y)

SDS1 – Intensity Maps♣

Intensity maps (consisting of a random dot pattern or speckle pattern) can be generated synthetically using a computer program or manually using paint or ink and photographed using a digital camera, this is the undeformed or reference speckle pattern. The distorted patterns can be generated from SDS5 by applying the distortion field to the reference speckle pattern, or directly by using the analytical model. The reference and deformed patterns can then be combined using the correlation algorithm to determine the displacement.

Of the two different methods for transformation of the undistorted data set, the first is to do a direct transformation using the analytical model on the reference speckle pattern. This can be done using a computer programme to transform the reference speckle pattern.

♣ SDS1 consists of two images a) the undistorted speckle pattern (reference image) and b) the distorted speckle pattern (image of the loaded specimen)


61

The second approach is by using the deformation field to produce a vector map (resultants of the x and y displacements) for each pixel. This can then be used to translate the intensities of the pixels to generate a distorted speckle map. In both cases the un-deformed data set must be sufficiently large to contain the deformed reference data set (i.e. none of the image created by deforming the data set must lie outside the reference data field). The speckle pattern must be subdivided into sub-pixels to perform the distortion, so as to achieve an accurate representation of the sub pixel motion. Sub pixels should be between 1/100 and 1/10 of the size of a pixel.

(c) Grating Interferometry In grating interferometry a signal proportional to light intensity obtained due to interference of two light beams (fringe maps SDS2) is produced from a CCD matrix sensor and this is calibrated to generate displacements (SDS5) and strain (SDS6) maps

SDS6 to SDS5 – Strain to Displacement

Input From the theory the strain field is given:

εxx (x,y); εyy (x,y); εxy(x,y) (E.c.1)

Procedure The displacement will be achieved by integration:

∫∫

=

=

dyyxv

dxyxu

yy

xx

ε

ε

),(

),( (E.c.2)

Taking the boundary conditions and εxy(x,y) into account. Note that direct integration may not be easily achieved and numerical methods may need to be employed.

Output The output is the displacement: u(x,y) and v(x,y)

SDS5 to SDS4 – Displacement to Continuous Phase In grating interferometry the continuous phase can be obtained by a scaling procedure:

)(4)(

)(4)(

yx,vp

yx,

yx,up

yx,

v

u

π=ϕ

π=ϕ

(E.c.3)

where, p is the period of the specimen grating.

SDS4 to SDS3 – Continuous Phase to Wrapped Phase The wrapped phase φ2π(x,y) value will be calculated using the expressions:

)),((2mod),(

)),((2mod),(2

2

yxyx

yxyx

vv

uu

ϕπϕ

ϕπϕπ

π

=

= (E.c.4)

SDS3 to SDS2 – Wrapped Phase to Fringe Maps Depending on the phase shift algorithm used, n fringe maps with a phase shift of dφ can be calculated as:

( )( )( )( ))1)((cos(12)(

)1)((cos(12)((2π2

2π2

φ−+ϕ+=

φ−+ϕ+=

dnyx,Ayx,I

dnyx,Ayx,I

vvn

uun (E.c.5)

where n ≥ 3 and A is the amplitude of the interfering beams (we assume that the contrast of the fringes equals 1). We suggest using a four step phase shift algorithm (n = 4, dφ = π/2).


62

(d) Moiré In digital geometric moiré a signal proportional to light intensity, obtained due to the beat between one periodic and one quasi periodic motif, (fringe map EDS2), is produced from a CCD sensor and this signal is calibrated to generate displacements (EDS5) and strain (ESDS6) maps. The methodology to generate the corresponding standard data sets (SDS) are described below.

SDS6 to SDS5 – Strain to Displacement Input From the theory the strain field is given: εxx(x,y); εyy(x,y); εxy(x,y) (E.d.1) Procedure The displacement will be achieved by integration:

∫∫

=

=

dyyxv

dxyxu

yy

xx

ε

ε

),(

),( (E.d.2)

Taking the boundary conditions and εxy(x,y) into account. Note that direct integration may not be easily achieved and numerical methods may need to be employed. Output The output is the displacement: u(x,y) and v(x,y)

SDS5 to SDS4 – Displacement to Continuous Phase In moirè fringe technique the continuous phase can be obtained by scaling procedure:

),(2),(

),(2),(

yxvp

yx

yxup

yx

v

u

πϕ

πϕ

=

= (E.d.3)

where, p is the period of the specimen grid.

SDS4 to SDS3 – Continuous Phase to Wrapped Phase The wrapped phase φ2π(x,y) value will be calculated using the expressions:

)),((2mod),(

)),((2mod),(2

2

yxyx

yxyx

vv

uu

ϕπϕ

ϕπϕπ

π

=

= (E.d.4)

SDS3 to SDS2 – Wrapped Phase to Fringe Maps Depending on the profile of the specimen and reference grid and the optical filtering process the moiré fringes will have a different number k of high frequency terms:

( ) ( )[ ]∑=

+=K

k 1

2πukmu yx,k cosb1yx,I ϕ and 1 ( ) ( )[ ]∑

=

+=K

1k

2πvkmv yx,k cosb1yx,I ϕ (E.d.5)

where k = 1, …, K and the amplitude of the fringes has been normalized to 1. However, the system should be designed in such a way to minimize the influence of high order terms and to obtain at the output fringes with a cosinusoidal profile:

( ) ( )yx,cos1yx,I 2πumu ϕ+= and ( ) ( )yx,cos1yx,I 2π

vmv ϕ+= (E.d.6) Generally it is possible to analyse non-cosinusoidal fringes but a properly chosen phase shifting algorithm should be applied.

Depending on the phase shift algorithm, n fringe maps with a phase shift dφ can be calculated as:

( ) ( ) ( )∑=

−++=K

1k

2πvkmu d1nyx,coskb1yx,I φϕ

( ) ( ) ( )∑=

−++=K

1k

2πvkmv d1nyx,coskb1yx,I φϕ where n≥3. (E.d.7)

In the case of moiré fringe technique it is suggested that a (K+2) step phase shifting algorithm is used with a phase shift:


63

22+

=K

d πφ (E.d.8)

where K is the highest nonlinear term which can be predicted from the profile of the specimen gratings and the filtering process performed by the imaging system. For K=1 (Eq. C.d.6) the three-step phase shifting algorithm with shift dφ = 120 deg (or dφ = 90 deg) can be used. (e) Photoelasticity The analytical expression for the strain components can be utilised with the strain transformation equations which can be derived via a Mohr’s circle of strain to obtain the principal strain difference (SDS6).

SDS6 to SDS 4 – Strain maps to fringe order map In reflection photoelasticity43:

kopc

fNKt

N ⋅=⋅=−221

λεε (E.e.1)

where N is the fringe order or number of wavelengths of retardation, λ is the wavelength of the tint of passage in white light which is usually taken as 575x10-9m, tc is the coating thickness in metres, Kop is the strain-optic coefficient of the photoelastic plastic and fk is the fringe value of the plastic coating in m/m per fringe. So to calculate the continuous fringe map, SDS 4 the following expression will suffice:

kfN 21 εε −

= (E.e.2)

SDS4 to SDS3 – Fringe order map to phase maps Now, the retardation is λδ Nr = where δr is expressed in metres. Thus, the relative retardation, αr is given by:

( )21222 εεππ

λπδα −===

k

rr f

N (E.e.3)

And note that the isoclinic angle is equal to the polar co-ordinate in the case of the interference fit cylinders, i.e.

xy

i1tan −=θ (E.e.4)

otherwise from the strain transformation equations:

21

1

21

1 sin21cos

21

ε−ε

ε=

ε−ε

ε−ε= −− xyyx

iθ (E.e.5)

SDS3 to SDS1 – Phase maps to intensity maps Now in phase-stepping photoelasticity the light intensity emitted at any point (x,y) in the field of view by a circular polariscope, i0 is given by44:

( ) ( ) ( ) ( )[ ]rpwpirpwAyxi αφβφθαφβ sin2cos2sincos2sin12

,2

0 −−−−+= (E.e.6)

where A is the amplitude of the light emitted from the polariser and βw and φp are the angular positions of the second quarter wave-plate and polariser respectively. In the Wang and Patterson44 algorithm the following pairs of values of β and φp result in the expressions given below:

0,4 == pw φπβ ( ) ( )rAyxi αcos12

,2

1 −=

0,4 =−= pw φπβ ( ) ( )rAyxi αcos12

,2

2 +=

43 Zandman, F., Redner, A.S., Dally, J.W., Photoelastic coatings, SESA monographs no.3, Iowa State Univ., Ames, USA, 1977. 44 Patterson, E.A., Ji, W., & Wang, Z.F., ‘On image analysis for birefringence measurements in photoelasticity’, Optics & Lasers in Engng., 28(1997)17-36.


64

0,0 == pw φβ ( ) ( )θα 2sinsin12

,2

3 rAyxi −= (E.e.7)

4,4 πφπβ == pw ( ) ( )θα 2cossin12

,2

4 rAyxi +=

2,2 πφπβ == pw ( ) ( )θα 2sinsin12

,2

5 rAyxi +=

43,43 πφπβ == pw ( ) ( )θα 2cossin12

,2

6 rAyxi −=

Alternative phase-stepping positions could be used with equation E.e.6 but the most commonly employed at the moment is given here. (f) Thermoelasticity In thermoelasticity a signal proportional to temperature (EDS 1) is produced from a photon detector or sensor and this is calibrated to generate a strain map (EDS 6).

SDS 6 to SDS 1 – Strain to Sensor Signal SDS 6 is obtained by summing any two orthogonal strain components found using the analytical model. SDS1 is simply obtained by using the following equation operated on every point in the field of view:

( )21

303

εερ

γ

ε

+Δ−=cFRD

T*eBS

ST

(E.f.1)

where B* is the integration constant for the photon flux over the wavelength range of the detector, e is the emissivity of the material, DT is temperature responsivity of the infrared detector, RS is a surface correction factor, F is the overall sensitivity of infrared detector, ρ is the material density, cε is the specific heat at constant volume, γ is one of Lamé’s elastic constants and T0 is the ambient temperature. It is common in thermoelastic stress analysis to use a calibration factor, Aε obtained by experiment in which case (E.f.1) can be written as: ( )

ε

εεA

S 21 +Δ= (E.f.2)

Note that thermoelasticity is usually performed using stress rather than strain for which the corresponding expression would be: ( )

σ

σσA

S 21 +Δ= (E.f.3)

The derivation of (E.f.1) follows from Biot’s45 expression for relationship between the first strain invariant, ( )21 εε + in a material subject to load and the temperature change in the material:

( ) εργ

εε cTT

021

Δ−=+Δ (E.f.4)

and the definition of the signal, S generated by the differential infrared thermal cameras used to measure ΔT 46:

FRD

TT*eBSST

Δ= 03 (E.f.5)

45 Biot, M.A., Thermoelasticity and irreversible thermodynamics. J. Appl. Physics., 27(3):240-253, 1956. 46 Dulieu-Barton, J.M., & Stanley, P., Reproducibility and reliability of the response from four SPATE systems. Exptl. Mech., 37(4):440-444, 1997.


65

NOMENCLATURE

a Distance between inner and outer loading points on beam "moment arm" m

aα Amplitude of illumination J1/2 m-1 A Amplitude of light emitted from polariser J1/2 m-1 Ag Gauge area m2 Aσ Thermoelastic calibration factor (stress-based) Nm-2V-1 Aε Thermoelastic calibration factor (strain-based) V-1 AL Constant in Lame’s equations Nm-2

Ap(ξ,ψ) Virtual aperture in the Fourier plane m b Interface radius for interference fit cylinders m bc Contact half-length in STM m bw Minimum width of ligament m B Thickness m

B* Integration constant for the photon flux - BL Constant in Lame’s equations N c Distance from centre to inner loading point in beam m cε Specific heat at constant volume Jkg-1K-1

cp Specific heat at constant pressure Jkg-1K-1 C Width of ligament joining disc to beam (=D/5) m d Difference between analytical and measured strain values m dc Depth below centre of Hertzian contact relative to which

compression of the elastic half-space is assessed m

dh Hole diameter m dφ Phase shift rad D Cylinder or disc diameter m Dp Pin diameter m DT Temperature responsivity of the infrared detector V-1K e Emissivity Jm-2K-4

E Young’s modulus Nm-2 E* Defined by equation (D15) in Part II Nm-2 EA Complex electric field J1/2m-1

),( yxES Rk

Speckle field J1/2 m-1 f General measured quantity fk Fringe value of coating m/m per fringe fl Focal length of lens m F Infrared detector sensitivity setting for strain - g Gap between disc and elastic half-space m

g, g1, g2, g3 Cantilever depth in monolithic frame m G Cantilever length in monolithic frame m h Depth of beam to which disc is attached in STM m H Height of RM or STM m

i0, i2 … i6 Intensity of images collected in phase-stepping Jm-2 I Second moment of area m4

),,( zyxI Vector from origin to the image point m ),( yxISP Intensity of speckle Jm-2

),( yxIFk Intensity field Jm-2 ),,( φdyxIS R

kn ,

),,( φdyxIS Dk

n Intensity maps Jm-2

j Gap between beam and frame in STM m k Side-wall thickness in STM (=D/2) m K Side-wall thickness in RM m


66

Kop Strain-optic coefficient - l Length of leaf springs in STM m lB Length of beam in RM monolithic frame m ls Strut length in RM m L Length of beam in STM m LB Length of beam in RM between outer loading points m

)( zy,x,Li Defined by equation (E.a.4) in Part II m/m ),,( zyxLQi Vectors from origin to illumination point m

m = W/B (recommended value, m =1) in RM m/m ms = B/D in STM - M Bending moment Nm n = a/W (recommended value, n=3) - N Photoelastic fringe order fringe

),,( zyxOi Defined by equation (E.a.5) in Part II m/m

p = c/W (recommended value, p=1) in RM m/m pl Load per unit length of cylinder N/m po Maximum value of Hertzian distribution over contact length N/m2 P Applied load N

Pcr Critical buckling load N )( zy,x,P Vector from origin to the object point m

PH Vector from origin to the principle point of the imaging m QA Constant in Airy’s stress function r Polar coordinate

r1, r2 Defined by equations (2) in Part II m rχ Radius in slit m R = D/2, radius of disc in STM m RA Constant in Airy’s stress function R* Relative curvature defined by equation (D17) m Rn Radius of half cylinder m RS Surface correction factor - s Deformation in radial direction m S Thermoelastic signal V SA Constant in Airy’s stress function St Stiffness N/m

),,( zyxS Sensitivity vector m/m t Time s tl Thickness of leaf springs m T Temperature K TA Constant in Airy’s stress function T0 Ambient temperature K

u, v Displacements in x and y directions respectively m u2(f) Uncertainty in measurand f units of f U2(f) Expanded uncertainty in measurand f units of f vx, vy Displacements in x and y directions respectively m

vy(a,0) Displacement in y direction at (a,0) m v Relative displacement of Reference Material sections or of applied

displacement load m

),,( zyxV Deformation vector m W Depth of beam subject to 4-point bending in RM m WL Load on leaf springs in STM N

x,y,z (xn, yn) Cartesian coordinates (normalised values) m YL Depth of lower horizontal bar of monolithic frame m YT Depth of upper horizontal bar of monolithic frame m α Fit parameter m/m


67

αr Relative retardation - β Fit parameter m/m2 βw Angular position of 2nd quarter waveplate rad γ Lame’s elastic constant Nm-3 δ Beam deflection m δk Deflection of sidewalls in RM m

δmax Deflection of upper bar in RM m δr Retardation m

ε1, ε2, Principal strains - εr, εθ Polar components of direct strain -

εxx, εyy, Cartesian components of direct strain - εx’x’, εyy, Direct strain in any arbitrary directions x’, y’ -

εxy Shear strain - ξ Defined by equation (4) for STM in Part II ζ Defined by equation (5) for STM in Part II - η Correction factor for tensile constraint in RM θ Polar co-ordinate rad θi Direction of principal strain (isoclinic angle) rad κ Correction factor for bending constraint in RM m κp Constant of proportionality λ Wavelength of light m μ Coefficient of friction - υ Poisson’s ratio - ρ Density kg/m3

σ1,2 Principal stresses N/m2 σc Compressive stress N/m2

σr, σθ Polar components of stress N/m2 σw Stress at loading point N/m2

σxx, σyy Cartesian components of stress N/m2 σxy Shear stress N/m2 σY Yield stress N/m2 τ Shear stress N/m2 φ Airy’s stress function N φp Angular position of polariser rad

),( yxkϕ Phase value for kth measurement rad ),(2 yxk

πϕ Wrapped phase value for kth measurement rad χ Slit gap in interlock in RM and STM m ω Ratio of gauge breadth to beam depth (>0.9) in RM


68

NOTES

ISBN: : 978‐0‐9842142‐2‐8

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