Professor Fabrice PIERRON

LMPF Research Group, ENSAM Châlons en Champagne, France

THE VIRTUAL FIELDS METHOD

Introduction and Overview

Paris

Châlons en Champagne

A bit of history

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A bit of history1989 – First paper in “Comptes rendus de l’académie des sciences” (principle)1990 – PhD thesis of Michel GRÉDIAC (thin anisotropic plates, including experiments)1994 – First collaboration between FP & MG (anisotropic in-plane properties, shear)1996 – 98: First application in dynamics (vibration of thin plates, exp. & num.)1998 – 2000: Series of work on in-plane elastic stiffnesses of composites (exp. & num.)2001: first attempt at a non-linear law (anisotropic)2002 – 04: Significant progress on virtual fields selection in elasticity (special virtual fields, minimization of noise effects)

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A bit of history2003: First application in vibration with damping (thin plates)

2005 – 06: Convincing experimental results (in-plane anisotropic composite stiffnesses)

2006 – First application to elasto-plasticity

2006: Theoretical framework in elasticity (relation between FEMU and VFM)

2006: Optimisation of test configuration (with Airbus UK)

2006: First application on heterogeneous materials – stiffness contrast in impacted composite plates (with Bristol Univ.)

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A bit of history2007: Application to elastography (MRI)

2007: First application to viscoplasticity (coll. M.A. Sutton)

2007: First application to heterogeneous plasticity (FSW joints)

OngoingApplication to 3D bulk measurements (composites, biomechanics), project with Loughborough university (Prof. J.M. Huntley, Dr P.D. Ruiz)

Optimisation of virtual fields in plasticity

Friction Stir Welds (collaboration with ONERA, France)

Development of a user-oriented software: CAMFIT

The principle of virtual work

V

*ii

V

*ii

V

*ijij 0dVufdSuTdV

or

Equilibrium equations (static)

0fij,ij + boundary conditions strong (local)

weak (global)

Valid for any KA virtual fields

Illustration of the PVW

01n

Section SF

e1

e2

l

L0

221

112

2221

1211211 dx.edx.e

0

1dx.e.TF

Over element 1

1

F1

211edx

221edx

1

2

3

0

1n

Local equilibrium: 0xx 2

12

1

11

21

Forces exerted by 2 over 1

)xL.(FM

F

0F

10e

12

12

3

F

e1

e2 Section S

L0-x1

Resultant of internal forces

2/l

2/l 2211e

12

2/l

2/l 221

2/l

2/l 211

12

dxxeM

dxe

dxeF

3

1

F1

211edx

221edx

21 F

e1

e2 Section S

L0-x1

Equilibrium

)xL(Fdxxe

Fdxe

0dx

10

2/l

2/l 2211

2/l

2/l 221

2/l

2/l 211

)xL.(FM

F

0F

10e

12

12

3

2/l

2/l 2211e

12

2/l

2/l 221

2/l

2/l 211

12

dxxeM

dxe

dxeF

3

Valid over any section S of the beam: integration over x1

)xL(Fdxxe

Fdxe

0dx

10

2/l

2/l 2211

2/l

2/l 221

2/l

2/l 211

2

FLdxdxxe

FLdxdxe

0dxdx

20

L

0

2/l

2/l 21211

0

L

0

2/l

2/l 2121

L

0

2/l

2/l 2111

0

0

0

Eq. 1

Eq. 2

Eq. 3

Principle of virtual work (static, no volume forces)

0dSu.TdVfV

*ii

V

*ijij

Let us write a virtual field:

0u

xu*2

1*1

e1

Fe2

L0

l

0

0

1

*12

*22

*11

0dSu.TdVfV

*ii

V

*ijij

0L

0

2/l

2/l 2111

V

*1111 dxdxedV 0

0dxdx0L

0

2/l

2/l 2111 Eq. 1

e1

Fe2

L0

l

Let us write another virtual field:

1*2

*1

xu

0u

2/1

0

0

*12

*22

*11

F

e1

e2

L0

l

0dSu.TdVfV

*ii

V

*ijij

0L

0

2/l

2/l 2112

V

*1212 dxdxedV2 0L.F

0

L

0

2/l

2/l 2112 FLdxdxe0 Eq. 2

F

e1

e2

L0

l

F

e1

e2

L0

l

Let us write a 3rd field: virtual bending

2

xu

xxu21*

2

21*1

0

0

x

*12

*22

2*11

0dSu.TdVfV

*ii

V

*ijij

0L

0

2/l

2/l 21211

V

*1111 dxdxxedV 2

L.F 20

2

FLdxdxxe

20

L

0

2/l

2/l 21211

0 Eq. 3

F

e1

e2

L0

l

The Virtual Fields Method

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Basic equations

V

*ii

V

*ii

V

*ijij 0dVufdSuTdV

or

I Equilibrium equations (static)

0fij,ij + boundary conditions strong (local)

weak (global)

II Constitutive equations (elasticity)

klijklij C

III Kinematic equations (small strains/displacements)

)uu(21

i,jj,iij

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The Virtual Fields Method (VFM) Basic idea

0dSuTdVV

*ii

V

*ijij

Eq. I (weak form, static)

Substitute stress from Eq. II

klijklij C

0dSuTdVCV

*ii

V

*ijklijkl

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The Virtual Fields Method (VFM)

0dSuTdVCV

*ii

V

*ijklijkl

valid for any kinematically admissible virtual fields

For each choice of virtual field: 1 equation

Choice of as many VF as unknowns: linear system

Inversion: unknown stiffnesses

Elasticity: direct solution to inverse problem !

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Simple example

Fuuny shaped disc in diametric compressionIsotropic material

-F/2

F

y

x

-F/2

Eps y Eps x Eps s

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0dSuTdVV

*ii

V

*klij

1st virtual field: virtual compression field

yu ; 0u *y

*x

0 ; 1 ; 0 *s

*y

*x

V

y

V

*yy

V

*klij dVdVdV

-F/2

F

y

x

-F/2

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xxyyxxy QQ

V

xxyyxx

V

y dV)QQ(dV

s

y

x

xyxx

xxxy

xyxx

s

y

x

2/)QQ(00

0QQ

0QQ

-F/2

F

y

x

-F/2

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V

xxyyxx dV)QQ(

Homogeneous material

V V

xxyyxx dVQdVQ

Assumption: strain field uniform through the thickness

S S

xxyyxx dxdyeQdxdyeQ

Measurement: uniform strain over a « pixel » (N « pixels »)

S

N

1i

iiyy sdxdy

S

N

1i

iixx sdxdy

-F/2

F

y

x

-F/2

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yd

N

1i

iy

dN

1i

iiy S

N

Ss

« Pixels » are of same area:

N

Sss di

N

1i

iyy N

1Average strain

Finally:

xxyyxxd

V

*klij QQeSdV

-F/2

F

y

x

-F/2

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Virtual work of external forces

)C(u)C(T

)B(u)B(T)A(u)A(TdSuT*

**

V

*ii

Contribution of point A

Coordinates of A:

00A

yu ; 0u *y

*x

00Au*

F

0AT 0)A(u)A(T *

-F/2

F

y

x

-F/2

A

B C

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Contribution of point B

Coordinates of B:

LhB

yu ; 0u *y

*x

L0Bu*

2/F

0BT2

FL)B(u)B(T *

Finally FLdSuTV

*ii

2

FL)C(u)C(T *

-F/2

F

y

x

-F/2

A

B C

Lh

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1st virtual field: uniform diametric compression

dxxyyxx eS

FLQQ

2nd virtual field: transverse swelling

0u ; xu *y

*x

0 ; 0 ; 1 *s

*y

*x

0QQ yxyxxx

-F/2

F

y

x

-F/2

A

B C

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Finally

dxy

xx

xy

yx

eS

FL0

QQ

)(eS

FLQ

)(eS

FLQ

2x

2yd

xxy

2x

2yd

yxx

-F/2

F

y

x

-F/2

Direct solution To inverse problem !!!

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Principal advantagesIndependent from stress distribution

Independent from geometry

Direct identification (no updating)

LimitationsKinematic assumption through the thickness (plane stress, plane strain, bending...)

y

F

-F

x

A

B

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Anisotropic elasticityExample 2

Orthotropic material

s

y

x

2sss

yyxy

xyxx

s

y

x

KQ00

0QQ

0QQ

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Choice of the virtual fields

1. Measurement on S2 only (optical system)

0 ; 0 ; 0 *s

*y

*x Over S1 and S3: (rigid body)

2. A priori choice:

over S1: 0u ; 0u *y

*x

0dSuT1S

*ii

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Unknown force distributionover S1 and S3. Resultant P measured

3. Over S3 (rigid body) : 2 possibilities

0u ; 0u *y

*x 0dSuT

3S

*ii 3.1

3.2

red

*yy

blue

*xx

S

*ii dxutdxutedxdzuT

3

tyi

txi

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tyi

txi

No information on tx

0u*x

rouge

*yy

rouge

*xx

S

*ii dxutdxutedxdzuT

3

Distribution ty unknown ku*y

kPdxtkkdxtdxutred

y

red

y

red

*yy

Filtering capacityof the VF

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4. Continuity of the virtual fieldsConditions over S2

Virtual strain field discontinuous

0uu *y

*x ku ; 0u *

y*x

Choice of 4 virtual fields at least: example

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xu ; 0u *y

*x

Over S2

1 ; 0 ; 0 *s

*y

*x

Over S3 k = -L

Uniform virtual shear

y

x

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V

sdV

Plane stress

S

sdSe

0dSTudVV

*

V

*ijij

s

y

x

ss

yyxy

xyxx

s

y

x

Q00

0QQ

0QQ

Plane orthotropic elasticity

S

sss dSQe

PL

Homogeneousmaterial

S

sss dSeQ

ePLdxdyQ

2S

sss

0dSTudVV

*

V

*ijij

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ePLdxdy)x2L(y6Qdxdy)x2L(y6Q

3

S

yxy

S

xxx

22

y

x

Field n°2: Bernoulli bending

)L3x2(xu

)xL(xy6u2*

y

*x

Sur S2

0 ; 0

)x2L(y6*s

*y

*x

Sur S3 k = -L3

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0dxdy)Lx2(yQ

dxdy)Lx(xQdxdy)Lx(xQ

2

22

S

sss

S

xxy

S

yyy

Field n°3: Global compression

)Lx(xyu ; 0u *y

*x

Over S2

)Lx2(y

)Lx(x

0

*s

*y

*x

Sur S3 k = 0

y

x

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Field n°4: Local compression)3/Lx(xyu ; 0u *

y*x

Over A1

)3/Lx2(y

)3/Lx(x

0

*s

*y

*x

Over S3 k = 0

y

x

)3/L2x)(Lx(yu ; 0u *y

*x Over A2

)3/L5x2(y

)3/Lx)(Lx(

0

*s

*y

*x

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0dxdy)3/L5x2(ydxdy)3/Lx2(yQ

dxdy)3/Lx)(Lx(dxdy)Lx(xQ

dxdy)3/Lx)(Lx(dxdy)Lx(xQ

21

21

21

A

s

A

sss

A

x

A

xxy

A

y

A

yyy

Field n°4: Local compression

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Final system

0

0

ePL

ePL

Q

Q

Q

Q

dS)3L5x2(y

dS)3Lx2(y

dS)3L2x)(Lx(

dS)3Lx(x

dS)3L2x)(Lx(

dS)3Lx(x

0

dS)Lx2(ydS)Lx(xdS)Lx(x0

0dS)x2L(y60dS)x2L(y6

dS000

3

ss

xy

yy

xx

A

s

A

s

A

x

A

x

A

y

A

y

S

s

S

x

S

y

S

y

S

x

s

2

1

2

1

2

1

222

22

AQ = B Q = A-1B If VF independent !!

Pierron F. et Grédiac M., Identification of the through-thickness moduli of thick composites from whole-field measurements using the Iosipescu fixture : theory and simulations,Composites Part A, vol. 31, pp. 309-318, 2000.

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Experimental examples in linear elasticity

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Unnotched Iosipescu test

Material: 0° glass-epoxy (2.1 mm thick)

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Polynomial fitting Noise filtering, extrapolation of missing data

Displacements in the undeformed configuration

Raw data Polynomial fitting Residual

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Strain fields Smooth fields

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

x 10-3

-7

-6

-5

-4

-3

-2

-1

0x 10

-3

-25

-20

-15

-10

-5

0

5

x 10-4

loca

l dif

fere

ntia

tion

FE

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Identification: stiffness

6 specimens

P = 600 N

xxQ yyQ xyQ 0ssQ

Reference (GPa) 44.9 12.2 3.683.86

Coeff. var (%) 0.7 2.8 7.32.4

Identified (GPa) 39.7

6.6

10.4

23

3.65

2.4

3.03

13Coeff. var (%)

Predicted by VFM routine

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Through thickness stiffnesses of thick UD glass/epoxy composite tubes

Optimized positionof measurement area

R. MoulartMaster thesisRef. 10

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Deformation maps

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Strain mapsPolynomial fit, degree 3, transform to cylindrical and analytical differentiation

r

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Strain maps

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Strain maps

s

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rrQ Q rQ ssQ

Reference* (GPa) 10 40 43

Identification results

Identified (GPa) 11.4 44.4 6.83.87Coeff. var (%) – 5 tests 87 66 6959

Problem: not an in-plane test !!!

* Typical values

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Problem with thick ring compression test

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Problem with thick ring compression test

Solution: back to back cameras-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

Load (kN)

Str

ain

(%

)

Front 1

Back 1

Front 2

Back 2

average 1

average 2

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Set-up with two cameras

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Results

rrQ Q rQ ssQ

Reference* (GPa) 10 40 43

Identified (GPa) 11.4 45.4 6.782.62Coeff. var (%) – 9 tests 29 10 429

Moulart R., Avril S., Pierron F., Identification of the through-thickness rigidities of a thick laminated composite tube, Composites Part A: Applied Science and Manufacturing, vol. 37, n° 2, pp. 326-336, 2006.

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ACKNOWLEDGEMENTSProfessor Michel GREDIACBlaise Pascal University, France

Colleagues and students from my research group:Dr Stéphane Avril, Dr Alain Giraudeau, Dr René RotinatDr Hocine Chalal, Mr Baoqiao Guo, Dr Yannick Pannier, Mr Raphaël Moulart

French CNRS network (GDR): « full-field measurements and identification in solid mechanics »

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ACKNOWLEDGEMENTSFunding

French Ministry for Research

French National Research Agency (ANR)

Champagne Ardenne Regional Council

Engineering and Physical Sciences Research Council (UK)

Airbus UK