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Page 1: Prediction of Plastic

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A drop-test is often performed for products used in the auto-motive industry, especially for those made of polymericmaterials, which are vulnerable to drop during the assemblyprocess. They are used in cockpit modules, engine parts,bumpers, electrical housings etc. Using numerical simula-tions for analysing the effects of a drop-test can reduce thenumber of prototypes and development time. Thus, timeand development cost can be reduced and products can bemore competitive on the market.

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Generally engineering plastic materials can be extensivelydeformed and yielding and plastic flow occurs beforefailure. At small strains observed behaviour is predominant-ly linear elastic. Where more precise results are demanded,viscoelastic behaviour, which implies curvature in stress/strain relationship, have to be included. Plastic materialsexhibit stress/strain rate dependent behaviour also. Thus,stress/strain curves are different for different deformationrates. Depending on simulation specifics, appropriate mate-rial propertiessets have to be created. Those that are possi-ble to use in a drop test simulation engineering plasticsproperties are described below.

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Elastic behaviour assumes linear stress/strain dep�����+(#�'+��/�����')+�-+�-�'��'��������'�������'��/�'�)���'

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1)��-������',��-��'�+�)���)�>���'�)υe. It is generallymore accurate to measure properties using tensile tests(Dean and Crocker 2005).

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The effect of strain hardening, which causes subsequentincreases in stress with strain, is described by plastic beha-viour. It occurs when the stress level called the first yieldstress is exceeded. Plastic behaviour is not described bya set of parameters but most commonly by a rel�'�)� /�2'4���'����-�(��-��'����σT���0-��'�+�'����εT

p (Ward andSweeney 2004). The plot showing the dependence is calledtensile hardening curve.

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The viscoelsastic theory was developed to describe polymerbehaviour, when subjected to a history of stress and strain,at modest strains (less than 0.5?). It can be used for exam-ple to model energy dissipation, whichoccurs duringcyclicloading of vibration dumper made of polymer material. Theshear relaxation modulus G(t, T1) measured at some refe-rence temperature T1 is most commonly used to characteriseviscoelastic response. Spring-damper models for exampleMaxwell (Fig. 1) or Kelvin-Voigt (Fig. 2)are often used toapproximate the viscoelastic behaviour. Generally the ideais simple: force applied to the spring – dashpot system re-presents shear stress and the extension represents shearstrain (Bower 2010).

Page 2: Prediction of Plastic

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Elastic-plastic models combine elastic and plastic beha-viour of material. In small strains the material behaviour isdescribed by liner stress/strain relation and after exceedingthe first yield stress the plastic behaviour is taken intoaccount. Thus, in non-linear region total st����ε+��/�0��2���'�������1)*a 0-��'�++)10)���'εp����-��'�++)12

0)���'εe.Because plastic materials exhibit strain-rate dependent

behaviour hardening curve for a set of different strain-rateshave to be measured and elastic-plastic stress-strain curvefor each strain-rate have to be created. Strain-rate depen-dent behaviour has to be calculated in simulations where thegoal is to simulate dynamic situations such as a drop test(Dean and Crocker 2005).

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The most popular and simple yield criterion is that createdby Huber and von Misses. It assumes that effective shearstress σe����-�'��')(��-��'������'����)�σT by equation(1). Measurement data obtained during compression andshear of plastics specimens reveal that yielding is sensitiveto the hydrostatic component of stress in addition to theshear component and thus equation (1) is not true for plas-tics (Dean and Crocker 2005).

σe = σT (1)

where:σe – effective shear stress,σT – yield stress in tension.

Because equation (1) cannot predict behaviour of plas-tics it has to be extended to include hydrostatic stress sensi-tivity. It was done as follows:

33

3e m T Sμ +σ + μσ = σ = σ (2)

where:σe – effective shear stress,

σT – yield stress in tension,

σm – hydrostatic component of stress,

σS – effective shear yield stress,

μ A ,(��)�'�'�+�'���������'�&�'(0���1�'��#

Equation (2) is the basic equation of linear “Drucker-Prager model”. The hydrostatic component of stress σm canbe derived from equation 4.����1�'��μ+��/���'��1����*�)1'��'�����'4)��**����'�'�����'�'��#��',��0�0��(��-�

�'����*�)1'����-�����,���'��'4�� used. Equation (3) wasused to calculate a value )*μ parameter.

33 1s

T

⎛ ⎞σμ = −⎜ ⎟⎜ ⎟σ⎝ ⎠

(3)

where:σT – yield stress in tension,σS – yield stress in shear.

( )1 2 31

3mσ = σ + σ + σ (4)

where: σ1, σ2, σ3 – components of principal stress.

To define material model in Abaqus FEA system, theflow parameter μG���-�)������#�&�-��*)�',��0���1�'��+��/�����&������.�B��'�)�(5).

( )( )

3 1 2

2 1

p

p

− υ′μ =

+ υ(5)

where:μG – flow parameter,

υp – plastic component of Poisson’s ratio.

The plastic component of Poisson’s ratio can be deter-mined using equation (6).

pp t

pT

ευ = −

ε(6)

where:εt

p – true plastic transverse strain,

εTp – true plastic tensile strain.

Material model described above, can be implemented di-rectly into Abaqus FEA system using special parametersshown in the case study (chapter 5). To capture rate depen-dent behaviour during simulation, it is necessary to definematerial data for different strain rates. Abaqus system inter-polates the model for higher speeds, matching interpolationfunction to the data received.

Page 3: Prediction of Plastic

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The choice of the material model determines the experimen-tal tests that have to be performed. For defining rate depen-dent elastic-plastic model with linear Drucker-Prager yieldcriterion tensile and shear test with different strain-rates areneeded. Three specimen shapes were used for these tests.First shape, according to the EN-ISO 527 standard (EN-ISO527), presented in the Figure 3 4�� ���� *)� ��'��1����.��'� ��+�����( *)� +�-+�-�'��. '����-� 1)��-�� - ��� ',��-��'�+�)���)�>� ��'�)υe. Figure 4 presents specimen usedfor tensile test with different strain rates (Dean and Crocker2005). Geometry of specimen presented in the Figure 5, ac-cording to the ASTM D 732-02 standard (ASTM D 732-02),was used for shear test. Specimens were tested for threedifferent strain rates: 0.001, 0.01 and 0.1 s–1. a Universalmaterial testing machine Zwick-Z20, with a video camerasystem was used. For the shear test a punch tool accordingto ASTM D 732-02 standard was used. On each specimenfor tensile test white ink marks for the video camera weremade. The material used in the experiments was PP TD20.

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The choice of an appropriate material model for simulationis an important decision in which many factors have to beaccounted for.Generally a material model should preciselydescribe material behaviour and at the same time be of sim-ple form. For industrial needs these two factors have to bebalanced. For the simulation of a drop test presented inthis paper rate dependent elastic-plastic model with linearDrucker-Prager yield criterion was used. This choicewas reasoned by sufficient accuracy of this model and sim-plicity of use due to implementation available in AbaqusFEA system.

Data registered during material testing has been analysedand values needed to define material model in Abaqus havebeen obtained. Using pictures created by a vision systemand data from standard test machine stress-strain plots weremade for each specimen. Figure 6 presents measuringpoints and approximation curves obtained for three testspecimens with three different strain rates. Measuringpoints have been fitted using function:

1 21 2( ) T T

T T e eε ε ε εσ ε = σ + σ (6)

where:σT – true tensile stress,εT – true tensile strain,

ε1�ε2,�σ1�σ2 – function parameters.

Fitted functions have been used to approximate mate-rial behaviour under bigger strain-rates. The least squaremethod was used to curve fitting. All hardening curves havebeen presented in the Figure 7. It is characteristic for poly-mer materials that yield strength increases with strain rate.This behaviour is pictured in the Figure 7, which allowthe assumption that material tests and approximations arecorrect. Tensile data has been used *)� +�-+�-�'��. '����-�1)��-��- ��� ',� �-��'�+ �)���)�>� ��'�)υe. �(��)�'�'�+�'���� �����'�&�'( 0���1�'�� μ ,�&� /��� +�-+�-�'�� ����.�,��� �nd tensile data (equation (3) has been used).Flowparameter μ' have been calculated using the plastic compo-nent of Poisson’s ratio (equation (5) has been used).

Page 4: Prediction of Plastic

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Page 5: Prediction of Plastic

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The 3D model of thehousing of car electrical elements usedfor simulation was presented in the Figure 8a.Because notall features of the geometry have an impact into analysisresults, the 3D model was simplified and then imported toAbaqus/CAE. Geometry after importing was presented inthe Figure 8b.

Parameters determined from test data were used to createa material model within the FE system. In Abaqus system,Drucker-Prager material model is defined using the angle offriction β, dilatation angle ψ, flow stress ratio K and har-dening curves for different strain rates. The angle of frictioncan be derived using equation (7).

arctanβ = μ (7)

where:β – angle of friction,μ – hydrostatic stress sensitivity parameter.

Dilatation angle can be derived using equation (8).

arctan ′ψ = μ (8)

where:ψ A ��-�'�'�)���.-��

μ' – flow parameter.

The third parameter required by Abaqus is Flow stressratio which describes differences in material behaviourunder tension and compression. This parameter used for thematerial model has been set to 1 which implicates com-parable behaviour under tension and compression. Pointscoordinates describing hardening curves were exported toa text file and then used in a created material model. Abaqus

uses these curves to approximate material behaviour underall strain rates, which occur in simulation. Other simulationparts including rigid contact surface, boundary conditionsand meshing were defined, and simulations at differentfalling directions were performed. The strike surface hasbeen modelled as a rigid surface with all degrees of freedomset to zero. The model of the housing has been positionedwith regard to created rigid surface so, that the distancebetween this surface and the nearest point of housing was0.01 mm. Initial velocity has been specified for the housing.This velocity has been calculated using the measured massof housing and the initial distance to the strike surface. Fi-gures 9 and 10 present the results of simulation and a realdrop test in which the housing drops at one of the handles.Because the failure model was not implemented in simula-tion, the handle is not snapped off. Plastic strain analysisshowed that the handle should break at places pointed to bythe arrows. Comparison with real test results has shown thatthe simulation gives acceptable results.

Figures 11 and 12 present results of a simulation anda real drop test at one of the housings corners. In this situa-tion in a real test only elastic deformation occurs. Charac-teristic shape deformations were pointed out by arrows ineach of the presented figures. Comparison of this regionshows that the simulation described clearly the componentdeformation under impact loading.

Fast FEA result achievement allows us to calculate se-veral iterations to optimize part geometry without expen-sive mould changes. A presented comparison yields theconclusion that the Drucker-Prager material model, pre-viously created for metal parts analysis, is good for fastand effective drop test modelling, although it should becomplemented by the influence of void formation andfailure which play preponderant roles in the plastic partsdestruction.

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Page 6: Prediction of Plastic

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Page 7: Prediction of Plastic

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Performed simulations have shown, that a real drop test ofplastic parts can be easily replaced with a numerical simula-tion. A presented computation method can be performedwithout problems in industrial conditions and can be used ina relatively short time. Simple tensile tests with differentstrain rates and shear tests are sufficient to create a rate de-pendent elastic-plastic material model, which can be used inthe modelling process of arbitrary shape impact analysis.The presented material model can be complemented by fail-ure criteria, which should further increase the accuracy atcalculations.

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Authors would like to acknowledge the Delphi TCK E/EAteam for sample providing and all their technical assistance.

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