using digital image corr - Core

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Procedia Engineering 10 (2011) 2110–2116

ICM11

Parameters identification of fatigue damage model for short glass fiber reinforced polyamide (PA6-GF30) using digital image correlation F. Meraghnia*, H. Nouria, N. Bourgeoisb, C. Czarnotac and P. Loryd a

LEM3 – Arts et Métiers ParisTech - Metz 4 Rue Augustin Fresnel, 57078 Metz, FRANCE b LEM3– UPVM - Metz Ile du Saulcy - 57045 Metz, FRANCE c LEMTA - GIP-InSIC, 27 rue d’Hellieule, 88100 Saint-Dié-des-Vosges. FRANCE d Technocentre Renault - TCR LAB 136 1, avenue du Golf - 78288 Guyancourt, FRANCE

Abstract The work deals with the parameters identification and the experimental validation of a phenomenological model for fatigue anisotropic damage in short glass fiber reinforced polyamide (PA6-GF30). The damage fatigue model has been formulated in terms of strain energy and was implemented into the finite element code ABAQUS/Standard through a user defined material subroutine UMAT. The present paper focuses mainly on the identification strategy based on homogeneous and heterogeneous fatigue tests. Damage parameters governing longitudinal and transversal damage (dLL and dTT) are identified from homogeneous tension-tension fatigue tests performed in both material directions. The damage parameters governing the shear induced damage (dLT) are identified using heterogeneous fatigue tests carried out on a specific configuration. This heterogeneous fatigue test gives rise to non-uniform distributions of the in-plane strain components: İLL, İTT and İLT. Heterogeneous strain fields are measured by the digital image correlation technique (DIC) during the fatigue test. The in-plane strain components are coupled to numerical computations through an inverse method to determine the parameters set governing the shear induced damage. The comparison between the damage model predictions and the damage curves obtained from fatigue tests shows a good agreement in both material directions. © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of ICM11

Keywords: Fatigue; Damage mechanics; Polymer matrix composite; Short fiber; Inverse method; Digital image correlation.

a

* Corresponding author. Tel.: +33-387-375-459; fax: +33-387-374-284. E-mail address: [email protected]

1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of ICM11 doi:10.1016/j.proeng.2011.04.349

F. Meraghni et al. / Procedia Engineering 10 (2011) 2110–2116

1. Introduction The proposed model, detailed elsewhere [1], predicts finely the anisotropic damage evolution observed for reinforced polyamide composites under cyclic loading. Damage kinetic evolution exhibited by reinforced polyamide under fatigue loading shows three stages, namely: i) material softening and damage initiation, ii) coalescence and propagation of micro-cracks, iii) macroscopic cracks propagation prior the material failure. The developed fatigue damage model is built in the framework of the continuum damage mechanics (CDM). It aims at capturing and predicting the three stages of damage kinetic notably the first stage due to the material softening specific to this kind of reinforced thermoplastics. Based on the mesomodel proposed by Ladevèze [2], the developed model has been formulated in terms of strain energy, so that makes easy its numerical implementation into the finite element code ABAQUS/Standard through a user defined material subroutine UMAT. The anisotropic in-plane damage is introduced through three internal state variables associated to the in-plane material stiffness reduction [3, 4]. The damage kinetic, including the first stage due to the material softening specific to short glass fiber reinforced thermoplastics, is modeled by a new incremental formulation of the dissipation energy yielding to the three damage rates. This formulation is very convenient to predict the anisotropic in-plane damage evolution under proportional or non-proportional multiaxial cyclic load even with variable amplitude. The present paper focuses mainly on the identification strategy based on heterogeneous fatigue tests. The latter allows determining the whole set of model parameters characterizing shear induced damage. The phenomenological model and the damage parameters identification have been experimentally validated on displacement and load controlled fatigue tests performed for several strain or applied load levels. The comparison between the model damage predictions and the damage curves obtained from fatigue tests shows a good correlation in both material directions. 2. Material description and Formulation of the fatigue damage model 2.1. Short glass fiber reinforced polyamide 6. The studied composite material is noted PA6-GF30. It is a polyamide matrix reinforced by 30% (weight) of glass fibre. The reinforcement makes the material highly resistant to abrasion, compression, tension or bending. Glass fibres have are discontinuous and have a length (L = 0.3 to 1.2 mm) with a nominal fibre diameter of 10 Pm. Due to the injection process, fibres in the shell layer of a plate are expected to be oriented parallel to the mold flow direction (MFD), whereas fibres in the core are more oriented perpendicular to the MFD. Therefore, the composite material will exhibit to main directions: longitudinal and transversal directions [5]. 2.2. Fatigue damage model To take into account the observed three damage stages in reinforced polyamide, a new polycylic fatigue damage model is formulated on the basis of the meso model proposed by Ladevèze and Le Dantec [2]. According to the continuum damage mechanics, the damage is introduced as an internal state variable coupled to elastic behaviour. The model has been formulated through five damage variables [1] but in the present paper, one will give the formulation for in-plane damage involving three variables: dLL, dTT and dLT. Relationships between longitudinal, transversal and shear moduli and associated damage variables are given by equation 1.

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ELL

0 ELL 1  d LL

ETT

0 ETT 1  dTT

GLT

0 GLT 1  d LT

(1)

ELL (resp. ETT) is the Young’s modulus in the longitudinal (resp. transversal) direction. GLT is the inplane shear modulus. dij are the damage variables associated to the corresponding moduli. The superscript 0 indicates initial non damaged moduli. For a damaged material, the elastic strain energy becomes dependent on the state variables dij: 1 1 0 ª ELL (1  d LL )H LL H LL  Q TLH TT 2 1 Q LTQ TL ¬

Wd 

1 1 0 ª ETT (1  dTT )H TT H TT Q LT H LL 2 1 Q LTQ TL ¬



0 H LL H LL Q TLH TT  ELL



º¼



0 H TT H TT Q LT H LL  ETT



0 2 º¼  GLT (1  d LT )J LT

(2)

Symbols (+) and (-) stand for the positive and negative parts of A, respectively. The thermodynamic dual variables Yij associated to the damage variables dij are deduced from the elastic strain energy Wd of the damaged material: YLL

w Wd 1 1 E 0 H H Q H w d LL 2 1 Q LTQ TL LL LL LL TL TT w Wd 1 1  E 0 H H Q H w dTT 2 1 Q LTQ TL TT TT TT LT LL w Wd 1 0 2  G J w d LT 2 LT LT



YTT YLT



(3) 

In the proposed modelling, the damage rate is assumed to be the sum of two components [1]: d( d LL ) D LLE LL E 1 YLL LL  O LL YLL e GLL N d( N ) 1  E LL d( dTT ) DTT ETT E 1 YTT TT  OTT YTT e GTT N d( N ) 1  ETT





d( d LT ) d( N )

(4)



D LT E LT E 1 YLT LT  O LT YLT e GLT N 1  E LT



(5)





N stands for the number of cycles. The first contribution, in Equations (4) to (6), is derived from a Norton-like function describing the dissipation potential. Note that in a previous work from Sedrakian et al [6], the damage evolution is only described by this first term. The second component is introduced by an exponential term and it describes the rapid stiffness reduction occurring during cyclic loading of reinforced thermoplastics and assigned to the material softening. The developed model is then a complete model in the sense that the entire damage process (3 stages) could be described. The instantaneous state variables dij(N) are obtained by numerical integration of Equations (4) to (6), with the initial conditions dij(N=0)=dqsij. These initial values are function of the imposed displacement. In the case where the applied strain Hҏappij is below a threshold value associated to the beginning of the damage process in the longitudinal, transversal or shear direction, dqsij=0. From Equations (4) to (6), it should be mentioned that the model is essentially developed for thin composite structures. In that particular case, relationships governing the damage evolution require 12 parameters to be identified namely, 4 parameters per damage variable. In the next section, an identification procedure is adopted to reach this goal.

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3. Experimental identification of the damage parameters 3.1. Longitudinal and transversal damage parameters identification 3.1.1.

Homogeneous fatigue tests and identification procedure

A first identification strategy involving homogeneous tensile fatigue tests have been applied to determine the longitudinal and transversal damage parameters. Displacement controlled fatigue tests have been performed in the longitudinal and transversal direction on a tension specimen [7]. Five strain controlled fatigue tests at different levels have been carried-out: İmax =20% of İrup, İmax =30% of İrup, İmax =35% of İrup, İmax =40% of İrup and İmax=45% of İrup. The applied strain ratio R= İmin/İmax remains constant for the different configuration (R=0.3) and the frequency was fixed up to 2Hz to avoid material heating during the fatigue test. Hrup denotes the maximum strain reached corresponding to the material failure under quasi-static loading [7]. The goal is to identify the parameters involved in the damage evolution, for the longitudinal direction (DLL, ELL, OLL and GLL governing dLL) and in the transverse direction (DTT, ETT, OTT and GTT governing dTT) for the PA6-GF30. The identification procedure is given for the longitudinal direction. The same strategy has been adopted to identify the transverse behaviour. For convenience, in the description, the subscript LL is omitted. An objective function representing, in the least squares sense, the difference between experimental and numerical damage values has been built and minimised using the Levenberg-Marquart algorithm. The cost function is written as: a

2 exp i

¦ ª¬d  P  d

F  P

i 1

a

num i

º¼

(7)

exp 2 i

¦ d i 1

a is the number of experimental data. num

and di

diexp denotes the experimental damage in the longitudinal direction

is the computed value obtained from the modelling. P is the unknown parameter vector.

3.1.2. Results and discussion Damage parameters have been identified exploiting experimental data from 2 different tests performed at two of the previously defined strain levels: İmax =30% İrup and İmax=45% İrup. The identified parameters governing the longitudinal parameters are given in table 1. It should be noticed that the inverse identification is robust since the determined damage parameters are insensitive to the initial values (global minimum). Indeed, using two different initial values, the algorithm converges to identical identified results which are the global minimum values. Table 1. Longitudinal damage parameters identified from experimental fatigue tests using two different initial values. Identification 1 (8 iterations)

Identification 2 (13 iterations)

D LL

Initial values 1 10-5

Identified values 1.8473 10-7

Initial values 1 10-3

E LL

1

5.5206 -4

O LL

1 10

G LL

1 10-3

Cost function

Identified values 1.8473 10-7

1 10-2

5.5205

-5

1 10-2

4.9487 10-5

9.6111 10-4

1 10-1

9.6117 10-4

4.9484 10

0.0047

0.0047

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Experimental validation of the damage model is achieved by comparing damage evolution dLL (figure 1) and dTT as a function of the cycle numbers. It is worth noting that the experimental validation relies upon experimental results obtained from displacement controlled fatigue tests performed at the three others strain levels. 0,6

Longitudinal damage (dLL)

LongitudinalFatigueTest:PA6Ͳ LongitudinalFatigueTest:PA6ͲGFL30,f=2Hz,straincontrolledtests 0,5

45 % HRup

Identification

0,4

simulation 40 % HRup

0,3

35 % HRup 0,2

Identification 30 % HRup 0,1

20 % HRup

0 0

20000

40000

60000

80000

100000

120000

Number of cycles

140000

160000

180000

Fig. 1. Comparison between experimental and simulated longitudinal damage (dLL) evolutions versus number of cycles (N) for PA6GF30. Displacement controlled fatigue tests.

3.2. Shear damage parameters identification 3.2.1. Heterogeneous fatigue tests and identification procedure The second identification strategy is based on the use of optical whole-field displacement/strain measurements by digital image correlation coupled to an inverse method. Strain fields are measured during fatigue tests performed on a modified Meuwissen configuration [8]. It is worth noting that this experimental configuration gives rise to heterogeneous stress/strain fields. The aim is to determine the whole shear damage parameters simultaneously from a single coupon. The Meuwissen sample geometry was modified to optimising the strain gradient temporal distribution (figure 2). The optimised configuration test seems to be suitable to enable the damage parameters identification of PA6-GF30. By incrementing the applied load, one obtains hence a spatio-temporal distribution of the strain. This may allow estimating simultaneously the damage parameters of the composite material in two directions (LL, TT) and the in-plane shear (LT). However, one limits this first application to the four shear damage parameters identification. Digital Image Correlation (DIC) technique has been utilized to determine the strain fields [9]. The spatial resolution of the method depends on the quality of the applied speckle. In this study, the pattern size was chosen as 32X32 pixels. It must be noticed that the larger the pattern size, the smaller the uncertainty. During the fatigue test, several pictures were taken by a 2 megapixel cooled camera (4096 grey level) placed in front of the specimen surface. The heterogeneous displacement/strain field was determined using the VIC 2D software. Fatigue tests performed using the specific configuration were load controlled. They were carried-out at frequency of f=2 Hz and for a load ratio R=Fmin/Fmax of 0.3. This value may enable avoiding the compression process during the fatigue test. T

Fig. 2. The modified Meuwissen specimen used for heterogeneous fatigue tests

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The inverse problem will be solved iteratively by minimizing the least squared difference between measured and computed strain fields by updating the four shear damage parameters [10-11]. The cost function is constructed on the basis of experimental/numerical strain components (İLL, İTT and İLT). By adding the squared difference of measured and computed displacement at the moving extremity of the specimen (Uexp and Unum), one may regularise the inverse problem resolution (Equation 8). ª ncycles ª ndata exp num « ¦ « ¦ H LL  H LL i j 1 1 ¬ 1 «« ncycles n data 2 2« H exp ¦ ¦ LL « i 1 i 1 « ¬



F P





2 j

º » ¼i

n cycles

ª ndata

¦ « ¦ H



i 1

¬

exp TT

j 1

ncycles n data



num  H TT

¦ ¦ H i 1

exp TT



2 j

º » ¼i

2

n cycles

ª ndata

¦ « ¦ H



i 1

¬

exp LT

 H num LT

j 1

ncycles n da ta

¦ ¦ H i 1

i 1

exp LT

2



2 j

º » ¼i

ncycles



¦

U

i 1

exp LL

num  U LL

U exp LL

i 1

2



2

º » » » » » » ¼

3.2.2. Results and discussion The resolution of the inverse problem led to the shear damage parameters given in table 2. The analysis of strain fields experimentally measured and those numerically computed shows a qualitative agreement regarding the distribution and gradient of the whole components (figure 3). Table 2. Shear induced damage parameters identified from Meuwissen fatigue tests using two different initial values. Identification 1 (10 iterations)

Identification 2 (15 iterations)

D LT

Initial values 1 10-5

Identified values 6.0112 E-4

Initial values 1 10-3

Identified values 5.88 E-4

ELT

1

3.6011

1 10-2

3.57

0.00475

1 10-2

0.00438

1.43

1 10-1

1.32

-4

O LT

1 10

G LT

1 10-3

T

H

exp TT

Hexp LL

Hexp LT

L

num HTT

H num LL

H num LT

(a) (b) Fig. 3. Comparison between experimental (a) and FE simulated (b) distribution of strain fields obtained at 5000 cycles of the Meuwissen configuration fatigue test.

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6. Conclusion In this work, a new cumulative fatigue damage model in an injection moulded polyamide reinforced short glass fibre (PA6-GF30) is presented, identified and experimentally validated. The identification of the material damage parameters was performed using a mixed experimentalnumerical inverse method. First, parameters governing longitudinal and transversal damage (dLL and dTT) were identified on the basis of homogeneous tensile fatigue tests. A second identification strategy, based on the use of optical full-field displacement/strain measurements by Digital Image Correlation (DIC) coupled to an inverse method, has been developed to determine the damage parameters governing the shear induced damage (dLT). Heterogeneous fatigue tests using a modified Meuwissen configuration was carried-out to give a rise to non-uniform distributions of the in-plane strain components. The spatial-temporal distributions of the in-plane strain were exploited through an inverse method to extract the parameters set governing the shear induced damage evolution. Using the identified damage parameters, one obtains a qualitative agreement regarding the distribution and gradient of the whole components. References [1]

Nouri H., Meraghni F. and Lory P., Fatigue damage model for injection-molded short glass fibre reinforced thermoplastics; International Journal of Fatigue Vol. 31 (2009) 934-942 [2] Ladevèze P., Le Dantec E., Damage modelling of the elementary ply for laminated composites; Composites Science and Technology; 43:257-67, 1992. [3] Van Paepegem W, and Degrieck J. A new coupled approach of residual stiffness and strength for fatigue of fibre-reinforced composites. International Journal of Fatigue 24 (2002) 747–762. [4] Degrieck J., Van Paepegem W. Fatigue damage modeling of fibre-reinforced composite materials: Review; Applied Mechanics Review 54 (2001) 279-299. [5] Bernasconi B., Davoli P., Rossini D., Armanni C., Effect of reprocessing on the fatigue strength of a fibre glass reinforced polyamide; Composites Part A: Applied Science and Manufacturing, Volume 38, Issue 3, March 2007, Pages 710-718. [6] Sedrakian A., Ben Zineb T., and Billoet JL, Contribution of industrial composite parts to fatigue behaviour simulation; International Journal of Fatigue, 307-318, 2002. [7] Nouri H., Czarnota C., Meraghni F., Lory P. Fatigue Damage Model for Short Glass Fibre Reinforced Thermoplastics; 17th Int. Conf. Composite Materials, Fatigue session ICCM-17, Edinburgh, July 2009. [8] Meuwissen M.H.H., Oomens C.W.J., Baaijens F.P.T., Petterson R., Janssen J.D. Determination of the elasto-plastic properties of aluminium using a mixed numerical–experimental method; Journal of Materials Processing Technology 75 (1998) 204–211 [9] Lecompte D., Smits A., Sven Bossuyt, Sol H., Quality assessment of speckle patterns for digital image correlation; Optics and Lasers in Engineering 44 (2006) 1132-1145. [10] Cooreman S., Lecompte S., Sol D., Vantomme H., Debruyne J., Elasto-plastic material parameter identification by inverse methods: Calculation of the sensitivity matrix; International journal of solids and structures, Vol. 44-13, 2007: 4329-4341. [11] Gavrus A., Massoni E. and Chenot JL. An inverse analysing a finite element model for identification of rheological parameters; Journal of Materials Processing Technology 60 (1996) 447-454.