chung & ryou - viscoel const law fiber reinforced composites

COMPOSITES

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Composites Science and Technology 69 (2009) 284–291

SCIENCE ANDTECHNOLOGY

Development of viscoelastic/rate-sensitive-plastic constitutive lawfor fiber-reinforced composites and its applications. Part I: Theory

and material characterization

Kwansoo Chung *, Hansun Ryou

Department of Materials Science and Engineering, Intelligent Textile System Research Center, Seoul National University, Shillim-dong,

Gwanak-gu, Seoul 151-744, Republic of Korea

Received 18 February 2007; received in revised form 30 May 2007; accepted 1 June 2007Available online 14 June 2007

Abstract

The viscoelastic/rate-sensitive plastic constitutive law to describe the nonlinear, anisotropic/asymmetric and time/rate-dependentmechanical behavior of fiber-reinforced (sheet) composites was developed under the plane stress condition. In addition to the theoreticalaspect of the developed constitutive law, experiments to obtain the material parameters were also carried out for the woven fabric com-posite based on uni-axial tension and compression tests as well as stress relaxation tests, while the numerical formulation and verifica-tions with experiments are discussed in Part II.� 2007 Elsevier Ltd. All rights reserved.

Keywords: B. Stress/strain curves; B. Mechanical properties; B. Nonlinear behavior; C. Anisotropy; C. Stress relaxation

1. Introduction

Many attempts have been made to characterize themechanical properties of fiber-reinforced composites, butmainly based on the linear anisotropic elasticity. Experi-mental studies, however, confirm that fiber-reinforced com-posites show nonlinear hardening behavior and permanentdeformation after unloading in addition to the elasticbehavior [1,2]. Also time/rate-dependent behavior isobserved for fiber-reinforced composites [3,4]. Therefore,the constitutive law with time/rate-dependency both inthe elastic and plastic ranges was developed for fiber-rein-forced (sheet) composites in this work as schematicallyshown in Fig. 1: viscoelastic/rate-sensitive-plastic constitu-tive law. In general, fiber-reinforced composites showstrong directional difference (anisotropy) and also the differ-ent constitutive behavior between tension and compression,called the bi-modular property or asymmetry [5,6]. The con-

0266-3538/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2007.06.003

* Corresponding author. Tel.: +82 2 880 7189; fax: +82 2 885 1748.E-mail address: [email protected] (K. Chung).

stitutive equation developed in this paper accounts for thetime/rate-dependent behavior as well as anisotropic/asym-metric properties under the plane stress condition.

In particular, in addition to the anisotropic property ofviscoelasticity, researchers have observed the asymmetricviscoelastic property of fiber-reinforced composites [7,8].Besides, composite materials show the nonlinear unloadingbehavior, especially under the pre-strained condition [2,3].Therefore, nonlinear asymmetric/anisotropic viscoelastic-ity is considered in this work.

There have been continuous efforts to include the time/rate effect in plasticity. To describe the rate-dependentbehavior of composite materials, Weeks and Sun [9] devel-oped a rate-dependent nonlinear constitutive model forfibrous composites, while Thiruppukuzhi and Sun [10]modified the model to properly account for anisotropy.As for the time effect, Chaboche and Rousselier [11] mod-ified the conventional plasticity by adding the viscous term.Sun and Chen [12] developed the one-parameter viscoplas-tic model for creep analysis. As for the asymmetric/aniso-tropic rate-sensitive plastic property in this work, both

0

21

1 2

μ

μ μ

λ λ

Fig. 2. A 5-element viscoelastic Maxwell solid model.

Fig. 1. A schematic view of the stress–strain curve of fiber-reinforcedcomposites.

1 The particular stress relaxation expression is valid for the 5-elementlinear Maxwell solid model. However, the same expression is also used forthe nonlinear case here, assuming that the material parameters arepartially not constants.

K. Chung, H. Ryou / Composites Science and Technology 69 (2009) 284–291 285

the initial anisotropic yielding and anisotropic hardeningbehaviors were considered. Regarding initial anisotropic/asymmetric yielding, the Drucker–Prager yield criterionhas been modified for composite materials [13]. To accountfor anisotropic hardening, the anisotropic isotropic-kine-matic hardening law has been utilized [14].

The polymer matrix used in composite materials (here,the epoxy) shows the highly nonlinear behavior and ratesensitivity, therefore, providing significant effects on thenonlinear and rate sensitive properties of composites. Theaddition of textile fibers may affect the directionality ofthe composite property as well as nonlinearity and rate sen-sitivity, depending on the arrangement of fibers. Therefore,many researches have been carried out for the matrix prop-erties. Drozdov and Christiansen [15,16] developed theviscoplastic model to describe the rate dependent behavioras a phenomenological approach. Oshmyan et al. [17] con-sidered the structural–mechanical model to account for thenonlinear behavior in both loading and unloading. Roguetet al. [18] performed tension and torsion tests to character-ize the rate-dependent behavior of a semi-crystalline ther-moplastic far above its glassy transition temperature.

In this work, the proposed viscoelastic/rate-dependentplastic constitutive law is summarized. Besides, experi-ments to obtain the material parameters of the developedconstitutive law are carried out for the woven fabric com-posite based on uni-axial tension and compression testsas well as stress relaxation tests. The composite made ofthe seven layers of the plain glass fiber woven fabric withthe same directional alignment in the epoxy resin is usedfor the sample material. In Part II [19], the numerical for-mulation and verifications with experiments performed forthe time-dependent springback in rate-dependent threepoint bending tests are discussed.

2. Constitutive law

2.1. Viscoelasticity

In the viscoelastic/rate-sensitive plasticity theory, thetotal strain increment is assumed to be decoupled into

viscoelastic, dee, and rate-sensitive plastic parts, dep, i.e.,de = dee + dep, where the Cauchy stress is related to the vis-coelastic strain history by the stress relaxation modulus, C,as r ¼

R t0

Cðt � sÞ dee

ds ds. Here, t is the real (current) timeand s is the dummy time variable to represent the momentof straining. For orthotropic sheet materials under theplane stress condition,

rxðtÞ ¼Z t

0

C11ðt � sÞ deex

dsdsþ

Z t

0

C12ðt � sÞdee

y

dsds

ryðtÞ ¼Z t

0

C21ðt � sÞ deex

dsdsþ

Z t

0

C22ðt � sÞdee

y

dsds

rxyðtÞ ¼Z t

0

2C66ðt � sÞdee

xy

dsds

ð1Þ

where the subscripts ‘x’ and ‘y’ stand for the two in-planeorthogonal symmetric axes of the fiber-reinforced sheetcomposite (the axial and transverse directions). The stressrelaxation behavior was assumed to follow the 5-element

Maxwell solid model as shown in Fig. 2: CðtÞ ¼ l0þP2j¼1lj exp � ljt

kj

� �� l0 þ

P2j¼1lj exp � t

pj

� �.1 Therefore,

the component of the stress relaxation modulus becomes,

C11ðtÞ ¼ ðlxÞ0 þX2

j¼1

ðlxÞj exp � tðpxÞj

!;

C22ðtÞ ¼ ðlyÞ0 þX2

j¼1

ðlyÞj exp � tðpyÞj

!

C12ðtÞð¼ C21ðtÞÞ ¼ myxC11ðtÞ ¼ mxyC22ðtÞ;

C66ðtÞ ¼ ðlxyÞ0 þX2

j¼1

ðlxyÞj exp � tðpxyÞj

!:

ð2Þ

286 K. Chung, H. Ryou / Composites Science and Technology 69 (2009) 284–291

The material parameters to be measured from the relax-ation test are: (px/y/xy)j, (lx/y/xy)j, (lx/y/xy)0 = C11/22/66

(t =1) and Poisson’s ratios, mxy and myx.To account for the asymmetric viscoelastic behavior of

fiber-reinforced composites, the different values of thematerial parameters are used for tensile and compressivemodes, in particular during initial loading (when the accu-mulative effective plastic strain is zero: �e ¼ 0Þ and when theplastic deformation takes place along with the viscoelasticdeformation (plastic hardening: �e–0 and d�e–0Þ, i.e.,

ðliÞ0; li; mij ¼

l0Ti

� �0lT

i ; mTij for

deei P 0;

l0Ci

� �0; lC

i ; mCij for

deei < 0;

8>>><>>>:

where i; j ¼ x; y;

ð3Þwhere superscripts ‘T’ and ‘C’ stand for tensile and compres-sive values, respectively. Note here that pj and the shear com-ponent parameters are assumed to be not asymmetric.

In order to describe the nonlinear unloading behavior,the material parameters were further modified for unload-ing (when �e–0 and d�e ¼ 0 with the sign change of dee

i forunloading):

ðlkÞ0 ¼ ðl0kÞ0 1:0� gk

r�k � rk

r�k

� �dk !

where k ¼ x; y; xy;

ð4Þwhere r�k is the stress states before unloading and rk is thevalues at the moment of straining, s. Note that gx, gy andgxy vanish during loading (�e ¼ 0; initial loading or�e–0=d�e–0 plastic loading or �e–0=d�e ¼ 0 having the samesign of dee

i with that of previous plastic loading). Note thatg ¼ gð�eÞ in general with the initial condition,g ¼ gð�e ¼ 0Þ ¼ 0. With the nonvanishing g, viscoelasticitybecomes nonlinear. Also, lx/y and mxy/yx are determinedfrom Eq. (3) by the deformation mode during unloading,while px/y/xy and lxy are always constants.

Considering Eq. (1),

r ¼Z t

0

Cðt � sÞ dee

dsds ¼

Z t

0

DðsÞ þX2

j¼1

Fjðt � sÞ !

dee

dsds

ð5Þwhere

DðsÞ ¼ðlxÞ0 mxyðlyÞ0 0

myxðlxÞ0 ðlyÞ0 0

0 0 ðlxyÞ0

264

375 ð6Þ

and

Fjðt�sÞ¼

ðlxÞj exp � t�sðpxÞj

� �mxyðlyÞj exp � t�s

ðpy Þj

� �0

myxðlxÞj exp � t�sðpxÞj

� �ðlyÞj exp � t�s

ðpy Þj

� �0

0 0 ðlxyÞj exp � t�sðpxy Þj

� �

266664

377775:

ð7Þ

In Eq. (6), the components of D are determined fromEqs. (3) and (4) considering the direction and the modeof de at the moment of straining s, while lx/y and mxy/yx

in Eq. (7) are constants determined from Eq. (3) by thedeformation mode at s.

2.2. Plastic yield criterion

The modified Drucker–Prager yield criterion [13] wasused to describe the anisotropy and asymmetry of compos-ite materials

U ¼ p r_2

x � b22r_

xr_

y þ b222r_2

y þ 3b233r_2

xy

h i1=2

þ q r_

x þ jr_

y

� �� riso ¼ 0: ð8Þ

Here, r_ ¼ r� a, a is the back stress, �riso is the size of the

yield surface, p, q, b22, b33 and j are material constantscharacterizing the anisotropic and asymmetric behavior.Note that the back stress components are initially set tovanish (when �e ¼ 0Þ. The modified Drucker–Prager yieldcriterion can describe different values of tensile yield stressesin two directions (anisotropy) and different values of tensileand compressive yield stresses (asymmetry). Also, the shearyield stress can be given independently. The five materialparameters therefore can be determined from two tensileyield stresses rT

x , rTy , two compressive yield stresses rC

x , rCy

in the axial and transverse directions and the shear yieldstress rY

xy or the tensile yield stress along the 45� direction(or 1-direction) rY

1 . Here, the tensile yield stress in the x

direction is the reference value: �risoð�e ¼ 0Þ ¼ rTx .

2.3. Plastic hardening rules

In order to represent anisotropic hardening and ratesensitive behavior, the combined isotropic-kinematic hard-ening law was utilized. As for isotropic hardening, the fol-lowing power law type hardening law was utilized:

�riso ¼ K �eþ e0ð ÞN_�e_e0

� �M

; ð9Þ

where K, N, M, e0, and _e0 are material parameters. For theback stress evolution, the following anisotropic kinematichardening rule to represent anisotropic hardening wasdeveloped based on the Chaboche type back stress evolu-tion rule [14,20]:

da ¼ C1 �ðr� aÞ

�riso

d�e� C2 � ad�e; ð10Þ

where C1 and C2 are the fourth order tensors containingparameters to be experimentally determined and d�e is theeffective strain increment. For the plane stress condition,Eq. (10) becomes

dax

day

daxy

264

375¼

g11 g12 g13

g21 g22 g23

g31 g32 g33

264

375

nx

ny

nxy

264

375�

h11 h12 h13

h21 h22 h23

h31 h32 h33

264

375

ax

ay

axy

264

375

0B@

1CAd�e;

ð11Þ

15mm

75mm

30mm

160mm84mm

38mm 2mm

3.1mm

25mm

40mm

Fig. 3. Dimensions of test specimens for (a) tension and (b) compression.


where n ¼ ðr� aÞ=�riso, gij and hij are the components of C1

and C2 in the matrix form, respectively.In the classical plasticity, the combined isotropic-kine-

matic hardening rules are used to account for the Bausch-inger effect as well as the transient behavior of re-loading.In this work, however, the combination type hardeningrules are proposed to describe the anisotropic hardeningand rate-sensitive behaviors of fiber-reinforced compos-ites. The anisotropic back stress evolution rule accountsfor anisotropic hardening as shown in Eq. (11) andisotropic hardening describes the rate-dependent behavioras shown in Eq. (9). The proposed viscoelastic/rate-sensi-tive plastic constitutive law was implemented into thegeneral purpose finite element program ABAQUS/Stan-dard using the material user subroutine. The numericalformulation is discussed in Part II [19] along withverification.

3. Materials

3.1. Measurement of material properties

The preform of the plain woven fabric composite wasmade by laminating seven layers of the plain glass fiberwoven fabric with the same directional alignment. To fab-ricate the composite perform, the hot press method wasused with the epoxy as a resin. The hot press machinewas set at a pressure at 7.35 MPa and 3.5 h for the wholepressing process. After the curing process, the final samplewith 3.1 mm thickness was obtained. The basic mechanicalproperties of the glass fiber and the epoxy resin used in thiswork are shown in Table 1.

Simple tension and compression tests were carried outin the x and y directions for the woven fabric composite(supplied by Hankuk Fiber Glass CO,. LTD) by standardprocedures, ASTM D3039-76 and ASTM D3410-87 usingthe Instron 8516 system. The specimen geometries areshown in Fig. 3. The stress–strain curves of tension andcompression are measured at three different strain ratesas shown in Fig. 4(a)–(b). Note that because the wovenfabric composite have the same structure for the x andy directions, tension and compression have the samematerial properties for the two directions. Fig. 4(a) showsthat the tensile stress–strain curve is almost linear initiallyand becomes slightly nonlinear after the ‘‘yield points.’’ InFig. 4(b), the compressive stress–strain curve is alsoalmost linear initially but showed failure at smaller straindue to micro-buckling of fibers. Simple tension and com-

Table 1Basic mechanical properties of the glass fiber and the epoxy resin

Property Fibera Epoxy

Density (g/cm3) 2.52 1.12Young’s modulus (GPa) 73.0 2.13Poisson’s ratio 0.2 0.37

a Hancox et al. [21]

pression tests along the 45� direction were also performedat three different strain rates and the tensile curves werequite nonlinear as shown in Fig. 4(c). In the 45� compres-sion curves, buckling of samples followed soon after thelinear region and the measured strain range was signifi-cantly smaller than other cases because of very earlybuckling of samples as shown in Fig. 4(d). It was experi-mentally confirmed that the ‘‘yield point’’ is approxi-mately the limit of linearity here.

Note that the nonlinear behavior is related to fiber rota-tions and physical damages (e.g., matrix cracking, interfa-cial debonding and delaminations), therefore, thenonlinearity differs for various loading directions. Also,the amount of the matrix involvement in the deformationmode also would affect the nonlinearity. Since the tensionin the 45� direction involves more significant fiber rotationsand matrix contribution, the nonlinearity in the 45� direc-tion is more conspicuous compared to that in the x direc-tion. Also, the compression test data were obtained untilsamples buckle and the x direction test shows larger buck-ling load since fibers are aligned along this direction so thatthe sample is stronger. In the 45� test, the sample is weakerwith less resistance by fibers so that the sample buckleswith smaller loading.

Poisson’s ratios for tension and compression were alsomeasured by the standard procedure, ASTM E132-04.Because of the same structure for the x and y directions,mT

xy ¼ mTyx and mC

xy ¼ mCyx for the particular sample used in this

work. Their measured values were mTxy ¼ mT

yx ¼ 0:35 andmC

xy ¼ mCyx ¼ 0:33.

Unloading tests of simple tension and compression inthe x(=y) and 45� directions were also carried out for thewoven fabric composite at different strain rates as shownin Fig. 5. The unloading results of tension and compression

Fig. 4. Comparisons of measured and calculated stress–strain curves: (a) x(=y) direction tension (b) x(=y) direction compression (c) 45� direction tension(d) 45� direction compression.


in the x(=y) direction show that the specimen is almost lin-early unloaded, while tension in the 45� direction isunloaded with changing stiffness: nonlinear unloading.Compression in the 45� direction is linearly loaded and alsolinearly unloaded.

Materials with poly-crystal structures show almost thesame mechanical behavior in tension and compression(therefore, the same stiffness, yielding and hardeningbehavior) as well as in loading and unloading sincemicro-structural damage development is similar for variousdeformation modes. However, the woven fabric compositeshows the different behavior not only between tension andcompression but also between loading and unloadingbecause its micro-structural evolution associated with thematrix cracking and fiber rotations significantly differsfor various deformation modes.

Stress relaxation tests of tension and compression in thex(=y) and 45� directions were also performed. Tests wereperformed with two constant strains, below and aboveyielding, for all cases except compression in the 45� direc-tion (with strain below yielding), during 40000 s as shownin Fig. 6.

3.2. Characterization of material parameters

The viscoelastic properties were obtained from the stressrelaxation test results shown in Fig. 6, using the data mea-sured below yielding. The shear property was indirectlydetermined from the stress relaxation test of the 45� tension

based on the tensor transformation formula. The resultsare shown in Table 2(a).

The five material parameters for the proposed yield cri-terion were determined from the two tensile yield stressesrT

x , rTy , two compressive yield stresses rC

x , rCy in the axial

and transverse directions and the tensile yield stress rY1 in

the 45� direction. These yield stresses were determined asthe limits of linearity as marked in Fig. 4 and listed in Table2(b). The resulting parameters for the yield criterion arelisted in Table 2(c). The predicted yield surface with initialasymmetry and anisotropy is shown in Fig. 7.

To determine the hardening parameters, the simpletension true stress–true strain curves in the x, y and 45�directions were considered after the yield points. For theuni-axial tension tests in the x and y directions, the follow-ing differential equations are obtained from Eq. (11):dax ¼ðg11nx � h11axÞd�e and day ¼ ðg22ny � h22ayÞd�e whose solu-tions are ax ¼ g11nx

h11ð1� e�h11�eÞ and ay ¼ g22ny

h22ð1� e�h22�eÞ,

respectively. Now, Eq. (9) along with the definitionof n ¼ ðr� aÞ=�riso gives the following stress–strainrelations:

rx ¼ nxK �eþ e0ð ÞN_�e_e0

� �M

þ g11nx

h11

ð1� e�h11�eÞ and

ry ¼ nyKð�eþ e0ÞN_�e_e0

� �M

þ g22ny

h22

ð1� e�h22�eÞ: ð12Þ

Note that from the parallelism of a and r for proportionalsimple tension loading in the x and y directions, g21 =

Fig. 5. Comparisons of measured and calculated stress–strain curves including unloading: (a) x(=y) direction tension (b) x(=y) direction compression (c)45� direction tension (d) 45� direction compression.

Fig. 6. Comparisons of measured and calculated time–stress curves from stress relaxation: (a) x(=y) direction tension (b) x(=y) direction compression (c)45� direction tension (d) 45� direction compression.


Table 2Material Parameters

(a) Viscoelastic constants

ðlxÞT0 ¼ ðlyÞT0 ðlxÞT1 ¼ ðlyÞT1 ðlxÞT2 ¼ ðlyÞT225.19 GPa 3.29 GPa 3.16 GPaðlxÞC0 ¼ ðlyÞC0 ðlxÞC1 ¼ ðlyÞC1 ðlxÞC2 ¼ ðlyÞC237.6 GPa 39.07 GPa 4.82 GPa(lxy)0 (lxy)1 (lxy)2

6.68 GPa 1.76 GPa 3.70 GPa(px)1 = (py)1 (px)2 = (py)2 (pxy)1 (pxy)2

296.0 s 14578.2 s 3528.6 s 23.14 s

(b) Measured initial yield stresses

rTx ¼ rT

y rCx ¼ rC

y rY1

40.8 MPa 49.5 MPa 45.4 MPa

(c) The material constants of the modified Drucker–Prager yield criterion

p q b22 j b33

0.91 0.09 1.0 1.0 0.85

(d) Material parameters for the combined isotropic-kinematic hardening

law

g11 = g22 g13 = g23 g33 K

66100.8 MPa �57580.8 MPa 8520.0 MPa 318.8 MPah11 = h22 h13 = h23 h33 N M e0 _e0

151.1 189.2 340.3 0.98 0.30 0.00678 0.00005


h21 = g31 = h31 = 0 and g12 = h12 = g32 = h32 = 0. In thesimple tension test in 45� direction, rx = ry = rxy = r1/2where r1 is the stress in the simple tension test along the45� direction. Then, the following stress–strain relation isobtained:

rxy ¼r1

2¼ nxyKð�eþ e0ÞN

_�e_e0

� �M

þ g33nxy

h33

ð1� e�h33�eÞ; ð13Þ

after considering daxy ¼ ðg33nxy � h33axyÞd�e. The hardeningparameters shown in Eqs. (12) and (13), which are g11,g22, g33, h11, h22, h33, e0, _e0, K, N and M, are now obtainedfrom the curve fitting of simple tension stress–strain curvesin the x, y and 45� directions at various strain rates. Also,the parallelism of a, r and n for proportional simple ten-

contours every 10MPa

(MPa)-90 -60 -30 0 30 60

-90

-60

-30

0

30

60

x

y(MPa)σ

σ

σ

xy

Fig. 7. Yield criterion of the woven fabric composites.

sion loading in the 45� direction leads to g13 = g33 � g11,h13 = h33 � h11, g23 = g33 � g22 and h23 = h33 � h22. Theresulting material parameters for the combined isotropic-kinematic hardening law are listed in Table 2(d).

Using the material parameters obtained from the mea-sured test data, true stress–true strain curves for uni-axialtests and stress relaxation curves were re-calculated asshown in Figs. 4 and 6. Even though the hardening param-eters were obtained from the tensile hardening curves, thecompressive behavior also shows good agreement withexperiments as shown in Fig. 4(b). Note that the materialparameters for stress relaxation curves in Table 2(a) werecalculated below yielding. However, results above yieldingalso show good agreement with experiments as shown inFig. 6.

The true stress–true strain curves for unloading werealso compared with calculation results obtained using theproposed constitutive law as shown in Fig. 5. Especiallyfor the simple tension in the 45� direction, even thoughthe nonlinear unloading behavior was observed, the pro-posed constitutive law considering the following relation-ship of Eq. (4) properly described the nonlinearunloading behavior:

ðlxyÞ0 ¼ ðl0xyÞ0ð1:0� 38:0�e�dð

r�xy � rxy

r�xy

Þ0:7Þ; ð14Þ

where �e� is the accumulative effective plastic strain beforeunloading and d = 0 when the material is loaded. Note thatthe nonlinearity of (lx)0 and (ly)0 was ignored for this par-ticular woven sample used in this work.

Note that the compression results in the 45� directionwere not utilized for material characterization. However,comparisons in Figs. 4(d), 5(d) and 6(d) show goodagreement.

4. Summary

In order to describe the nonlinear, anisotropic/asymmet-ric and time-dependent mechanical behavior of fiber-rein-forced composites, the viscoelastic/rate-sensitive plasticconstitutive law was developed based on the modifiedDrucker–Prager yield criterion and the anisotropic isotro-pic-kinematic hardening law. Experiments to obtain thematerial parameters of the developed constitutive law werealso carried out for the woven fabric composite based onuni-axial tension and compression tests as well as stressrelaxation tests. The newly developed constitutive law isuseful to analyze the rate/time-dependent performance offiber-reinforced composites, as confirmed in the verificationbeing documented in Part II.

Acknowledgements

The authors of this paper would like to thank the KoreaScience and Engineering Foundation (KOSEF) for spon-soring this research through the SRC/ERC Program ofMOST/KOSEF (R11-2005-065).


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chung & ryou - viscoel const law fiber reinforced composites

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