An Inelastic Damage Model forFiber Reinforced Laminates

EVER J. BARBERO·Mechanical and Aerospace Engineering

< West Virginia UniversityMorgantown, WV 26506-6106, USA

'.

'.

PAOLO LONETTIDepartment of Structural Engineering

University of CalabriaItaly

(Received December 12, 2000)(Revised April 13, 2001)

ABSTRACT: A new model for damage behavior of polymer matrix compositelaminates is presented. The model is developed for an individual lamina, and thenassembled to describe the nonlinear behavior of the laminate. The model predicts theinelastic effects as reduction of stiffness and increments of damage and unrecoverabledeformation. The model is defined using Continuous Damage Mechanics coupledwith Classical Thermodynamic Theory. Unrecoverable deformations and Damageare coupled by the concept of effective stress. New expressions of damage andunrecoverable deformation domains are presented so that the number of modelparameters is small. Furthermore, model parameters are obtained from existing ·testdata for unidirectional laminae, supplemented by cyclic shear stress-strain data.Comparison with lamina and laminate test data are presented to demonstrate theability of the model to predict the observed behavior.

INTRODUCTION

T HE ADVANCING USE of polymer matrix composites (PMC) in applications with longlife cycles requires better analysis techniques to predict material degradation andfailure. Experimental results show marked nonlinear effects when a single lamina is loadedby shear [1,2]. Instead, lo~er nonlinearity appears when the loading is along the fiberdirection or transverse to 'it. The most common modes of failure of PMCs are fiberbreakage, fiber-matrix debonding, matrix cracks, and fiber kinking. Fabrication of PMCsinevitably creates body defects in the form of matrix cracks, voids, crazes, delaminat'ions,

*Author to whom correspondence should be addressed. E-mail: ebarbero@wvu.edu.

Journal o/COMPOSITE MATERIALS, Vol. 36, No. 08/2002

0021-9983/02/08 0941-22 $10.00/0 001: 10.1106/002199802023549,,© 2002 Sage Publications

941

EVER J. BARBERO AND PAOLO LONETTI

fiber-matrix debond, and so on. The evolution of such defects influence stress~str~inresponse of the laminae and the maximum load that a laminate can support. The presentpaper deals with Continuous Damage Mechanics of homogenized media where the effectof the constituents has beeD: averaged. Therefore, the model cannot identify individual"failure modes. The effect of the likely failure modes is felt in the damage" measures du, onan average sense only. .The proposed model is constructed coupling the damage and unrecov~rabledeforma­tions theories using a thermodynamic formulation. A mesoscale approach is used, thus the ,constitutive equations refer to a single lamina of orthotropic material. The global behavioris detertnined assembling the contributions of each lamina using the classical laminatetheory, modified for a continuously damaging material with plastic-like unrecoverabledeformations. Damage and. unrecoyerable deformations surfaces are developed toaccomplish good correlation with experimental observations while using a few adjustableparameters that" have clear physical interpfetation and are easy to determine from existingexperimental data. The evolution law~s are defined in standard and nonstandardformulation for unrecoverable deformation.s. and damage respectively. The principle ofmaximum entropy is used in order to define the evolution of internal variables.While several models are available to account damage and unrecoverable deformationseffects for a lamina [3,4,6], the proposed model is defined in terms of fewer adjustableparameters. Furthermore, the parameters in the proposed model are evaluated fromexisting experimental data rather than from specialized, nonstandard tests. Although themodel still contains five adjustable parameters, all of them can be determined fromavailable experimental data. Experimental results for unidirectional laminae and laminatesdemonstrate the qualities of the proposed model.

942

DAMAGE DEFINITION

The damage definition due to Kachanov-Rabotnov is used [7,8]. The local density ofdefects is assumed uniformly distributed in a representative volume element. Laminaexperimental results evidence different damage modes and evolution for longitudinal,transverse, and shear loading [1,2]. In addition, shear loading leads to longitudin~l andmostly transverse damage [9-11]. Therefore, the orientation of defects is described by asecond order tensor with principal directions coinciding with the material coordinate axesof the lamina.The damage variables are (a) a second order symmetric damage tensor D, defined toaccount for the anisotropic evolution of defects along matrix and fiber directions and (b) ascalar damage hardening parameter 8 that controls the size of the damage surface. Theevolution law is assumed to be isotropic for simplicity and due to the lack of experimentalobservations indicating anisotropic evolution of the damage surface.The unrecoverable deformation variables are (c) a second order symmetric tensor eP toaccount for the accumulated, unrecoverable deformations and (d) a scalar unrecoverabledeformation hardening para1nete~ p that controls the size of the unrecoverable deforma­tion surface. The evolution law is assumed to be isotropic for simplicity and due to tne lackof experimental observations ipdicating otherwise.A diagonal second order· tensor is chosen to describe the damage effects D =

[d. h d22, d33], because it yields simplicity to the constitutive equations. A refined formula­tion could be obtained using a fourth order damage tensor [3,12,13] but it would result in

An Inelastic Damage Model for Fiber Reinforced Laminates '943.

greater complexity. The use of damage and integrity tensors describ~~~~j_'fi~~~~~-tv~nthe effective C and damaged C configurations by a unique transfQ~~\j~I"~'itt;4P.~xth~ , ,square root theorem [14,15], a unique transformation connects tht%,4,~~~ge;;~#di~t~grity

tensor 0 = .JI - D. The eigenvalues of the damage tensor d; repJ,:es~!~~;~he;,n~t,;area .reduction along the principal axis, which coincide with the priI1.cipaJ;(id!r~ctions/oftheintegrity tensor Q. Experimental results on a single composite lamin.a sh6wthat during the 'deformation, microcracks, voids, and defects have preferential directions along.;;whichdamage and unrecoverable deformation growth occurs. Since they typically.coincide withmaterial axes, it is assumed that the principal axes of the damage tensor D are alignedwith the material 'axes and do not necessarily match the principal directions of stress. Theeffective stress is defined [16las

u = M-1(D)u = (0-1® O~I)CT == [(~rl ® (.JI -l?)-I]U (I)

where M is a fou~th order tensor' called the'-damage effect tensor. By the symmetry of the uand (j tensors, th~; damage effect tensor_M is doubly symmetric. An explicit expression isthen found in cOlltracted notation [17] as

'. n 2 0 '0 0 0 01Q2 0 0 0 02

Q2 0 0 03Q3 n 2

0 0Mijk/ = QikQ/j 2 (2)n 1n3

0sym2

nln2

2

In terms of Equation (2), Equation (I) reduces to the following explicit relationships

UII Ul20

(I - dl) J(l - d1)(1 - d2)

a= U220sym

(1 - d2)

0 0 0 (3)

811(I- dt) ( 812J(1 - dl)(l - d2) 0-I

822(1 - d2) 08 = sym

0 0 0

where prime indicates the elastic, recoverable part of the strain.

CONSTITUTIVE MODELI

General coupled elastoplastic and damag~ theory [18-20] allows us to write the total.strain tensor 8 as sum of unrecoverable and elastic strains. The inelastic process js modeled

944 EVER J. BARBERO AND PAOLO. LONETTI

introducing a set of internal variables that describes the unrecoverable and ,damagebehavior of the material. The Helmholtz free energy is a function of internal variables, t. .,

elastic strain, and temperature. The Helmholtz free energy is postulated to be sum of twoterms, the strain energy cp(e., eP, D) and the dissipation energy rr(p, 8) [5;6]

1/1 = Vi(e, eP,p, D,~) = epee, eP, D) + n(p, 8) (4)

.,

where eP, D are the unrecoverable strain and damage second order tensors and p, ~ are th~

hardening parameters. The strain energy can be assumed to be "

1q;(e,eP,D) =: "2(e - eP)E(D)(e - eP) (5)

where the E(D} is a fourth order tensor that expresses the damaged stiffness of thematerial. The ~amaged stiff1;less tensor i~ defined according to the principle of equivalent

(~--

elastic energy [21,22], which states that· the' elastic energy of the damaged material is thesame i~ form~as that of an effective materialexcept that the stress tensor is replaced by theeffective stress

'. 1 1 "I - 1 1 1 - 1 1ep(li,O) = -(1(E)- (1 = ~li(E)- li = -uM- (E)- M- u,2 2 2

(6)

The damage stiffness tensor is written in terms of the effective stiffness tensorE=MEM, which is a fourth order tensor, quadratic in the damage variable D. In terms ofEquation (2), we get the stiffness tensor explicitly as

- 2 E12(1 - dI)(1 - d2) 0 0 0 0Ell(l - dl)- 2 0 0 0 0E22(1 - d2)

0 0 0 0E(D) =

0 0 0(7)

sym 0 0

!E66(1 - d})(1 - d2)

and the compliance tensor as C = ~1. The dissipation energy n(p, 8) can be separated intotwo terms, which express the evolution of the damage and unrecoverable deformationsurfaces

Here, we postulate that

and

1T(p, 8) = R(p) + y(8)

1R(P) = _-cfp2

2

(8)

(9)

{10)

Using Equation (7), explicit expressions for·-t~he thermodynamic forces are found as

• '~. r, ,/,

where cf, Ct, C2 are material parameters. Since the function 1C(p,8) is assumed to be.,:convex, its second derivative must be positive. Therefore, the material constants~ mustsatisfy the following conditions Cl > 0; C2 < 0; cf > 0 [23]. The conjugate thermodynami9 '.,forces associated to the cinematic variables (8, eP,p, D, 8) are '

An Inelastic Damage Model for Fiber Reinforced Laminates

a1/lu = -- = E(D)(e - eP)ae

a1/l 1 a 'y =--= --(e - eP)-[ME(D)M](e - eP)aD 2 aD

a1/lR =--. = -c!?p.. op 1

. Y= _ 81/1 =c.1'exp ( 8d

).' a8,., c2

.Q45

(11)

and Y3 =0 for a state of plane stress. Considering the principle of maximum entropy [24]and using the Lagrange minimization method the evolution equations can be defined as

- · agPsP = AP_·

aCi '(13)

w·here -AP, i d are the damage and unrecoverable deformations multipliers, gP,.r' are thedissipation potentials. In addition, the unloading conditions follow directly from theKuhn-Tucker optimality conditions [24] as

'AP > o· i d ~ 0- ,gP ::s 0; gd ::s 0 (14)

gPip = 0; gdid =0.;

DAMAGE DOMAIN

Next, the damage surface and the dissipation potential are written in the thermo- .dynamic variables domain Yi, which in thermodynamic sense are the conjugate variablesof the damage tensor D [23]. The damage surface limits the state for which the material

946 EVER J. BARBERO AND PAOLO LONETTI

experiences no damage. A nonstandard formulation is then used to define ,.th~ damagefunction gd and potential surface I as .. ': ~"

gd = g(Y) - y(8) - Yo = Jy - J. Y + JIB· YI- (y(8)+ Yo),

fd =/(Y) - y(8) - Yo = Jy - J. Y - (y(8) + Yo)

:;"

(15)

where J and H are fourth and second order tensors respectively, Yo is the dainage energythreshold. The tensors J and H are material dependent. It is worth noting that the linea~

term in Equation (15) allows us to take into account different behavior of the compositelamina in tension and c,ompression. The coefficients in the tensors J and Hare univocallydetermined in terms of material ..properties as explained in section Internal MaterialConstants. The damage damain surface gd in Equation (15) has the shape of the Tsai-Wusurface when the first of Equation ,(15) is written in stress space [23]

'. g(a) =fla l +12(j2 +111(jY·+12201 + 2/12(j1 C12 + 166C1~ - (y(8) + yo)'e 1 1 . . 1 1 1

11 =---; 12 =---; III =-.- (16)Fit Fie . F2t F2e FltFl t

". 1" 1 0.5122 =F F ;' 166 = F2 ; 112 ~ - F2

21 21 6 It

The Tsai-Wu criterion is recovered when the last term on the right hand side is set toone. The magnitude of the last term affects the size of the damage surface but its shaperemains similar to that of a Tsai-Wu surface. Since the Tsai-Wu criterion predicts plyfailure, the damage surface coincides with ply failure when the last term is equal to one.Then y(8) = 0, the last terms reduces to Yo. Thus, the initial size of the surface Equation (16)is smaller and it represents the loci of stress at which damage starts. For the sake ofsimplicity, it is assumed that the shape of the damage domain remains the same while itssize grows until it reaches the Tsai-Wu criterion at ply failure.

The evolution of the damage is governed by a hardening function that involves twovariables Ch C2, which are material dependent. The proposed damage hardening function isgiven by Equation (10). The consistency condition (gel =g" = 0), and the normality ruleEquation (13), used in the CDM framework [25,26] allows us to derive the followingequation in terms of the damage and unrecoverable deformation multipliers i..d, i p , using

An Inelastic Damage Model for Fiber Reinforced Laminates

UNRECOVERABLE DEFORMATIONS DOMAIN

~947

Unrecoverable deformations of composite materials are more evident for shear '..loading than longitudinal and transverse loading [2]. The unrecoverable· strain 'evolutiortis realized by the classical plasticity formulation [20]. The loading surface' -and thedissipation potential are identical in the effective stress space, thus follow~.ng a standardformulation. A Tsai-Wu surface is chosen because of its ability to represent differentbehavior among the different load paths in stress space. Therefore, the loading surface"'used herein is '

gP =g(cf) - R(P) - Ro

where the unrecoverable deformation domain. is defined in effective stress space as

(18)

(19)

and Ro is the unrecoverable deformations threshold. The parameters Fj are materialdependent and represent the strength values in tension, compression and shear for a singlecomposite lamina. These values are tabulated in the literature [17,27], or they can be easilyobtained following standardized test methods [28]. Unrecoverable deformation hardeningis modeled by a simple function Equation (9), thus requiring the minimum number ofinternal parameters to describe it. As before, by the consistency condition (gP = gP = 0),the normality evolution rules, and the effective stress concept, it is possible to obtain thefollowing relations

R(p) - Kpp; 0: = M-1u; 8 = M-1e

agPa+ agP R= 0au aR

.. au . au r.p au· Ai 1 au. M 1 au.p a [ 1 •U = as s + asp s + aD D = - as s + - asp s + aD M- u]D

a: = M-l aue+ M-1 au M-1i.p agP +~[M-lu]id afas asp au aD aD (20)

t' = agP M- 1oue + M- 1 au M-1ipagP +~[M-lu]id af =0au as asp au aD aD

agP M-1aue+ agP M- 1 au M- 1i pagP + agP ~[M-lu]id af + agP aR agP = 0au as au asp · au au aD aD aR ap aR

-Ap[agP M-1 au M- 1agP + agPaRagP] +id[agP ~[M-lu] af] + agP M-1au. = 0au asp au aR ap aR au aD aD au a8 s

948 EVER J. BARBERO AND PAOLO LONETTI

in which the damage and unrecoverable deformation multipliers i p , i d are t:Q~: tpJ.known ,quantities. Solving the linear system defined by Equations (17) and (20), it is possibletQ ~,

determine the values of 'AP, i d which define the damage and unrecoverable deformationsevolution laws. "..

LAMINATE MODEL

The model' presented so far predicts the behavior of a single lamina. Classical laminatetheory is used ,next to introduce the damage and unrecoverable effects in the constitutiveequations for a laminate [~7]. Damage and unrecoverable effects are introduced so thatstress redistribution among' various la,yers in a laminate is properly accounted for. Thestress-strain relations in the ~aterial,c,oordinates for the top or the bottom surface of asingle lamina .;' are '

where the subscripts d, p, refer to damage and unrecoverable effects, respectively, andsuperscripts t, b refer to top and bottom, respectively. In the framework of the classicalplate theory (CPT) the kinematic varia~les are the midplane strain and curvatures. Theelastic-damaged terms of the reduced stiffness matrix are defined by a linear function ofthe material property of top and bottom surface of the kth lamina as

Q'!.( ) = Q'!.(Zb) +~(ZD - ~(Z~) ( _ Zb)lJ Z lJ k (Zl _ Z~) Z k

(22)

r

where an overbar (-) indicates quantities in the global coordinate system of the laminate.Such quantities are obtained by standard coordinate transformation [17] of Equation (21).The tension Uj over the lamina is a linear function of the top and bottom values of thestress, given by

_( ) ~ _(Zb) + IT;(Zk) - IT;(ZZ) ( Zb)(1, Z - (1, k (Zl _ Z~) Z - k (23)

To assemble the total stiffness of the material we use the definition of -force andmoment resultants (Equations (6.11) in [17]). Therefore, the laminate constitutive

An Inelastic Damage Modelfor Fiber Reinforced Laminates

equations become

N x All A 12 AI3 BII B12 B 13 SO - ePx x

Ny A I2 A2~ A23 B 12 B22 B23 sO - ePy y

Nxy AI3 A23 A33 BI3 B23 B33 o ,pYxy - Y:Xy

=M x BII B 12 BI3 DII D12 D13 Kx -K~

M' B12 B 22 B23 DI2 D22 D23 K -KPY Y Y

M xy ' BI3 B 23 B33 D13 D23 D33 Kxy - K~y

949

~' (24)

where the coeffic~ents are 'computed in terms 'Of the damaged values of the reducedstiffness coefficients in global coordinates Qr,j using the standard equations (Equation (6.16)in [17]). Noting th~t the damage and un~ecoverableeffects are piecewise linear functionsthrough the thickness of the laminate we obtain: the following explicit equations for thecoefficients of the la.minate stiffness matrices

'.·A.. =!(~(ZD +~(Z%»)(zt _Zb)2

l) tk 2 k k

Bij = -61 ! {(Zl)3[2Qi(Zi) +Qi(ZZ)] + (ZZ)3[Qi(Zl) + 2Qi(ZZ)]

tk

-3 · ZkZ~[Q~(Zk)Zk + Qi(ZZ)ZZ]}

Dij = 112 t~ {(Zl)4[3Qi(Zl) + Qi(ZZ)] + (ZZ)4[Qi(Z1) + 3Qt(Z%)]

- 4Z~Zk[Qi(Zk)(Zk)3 + Qi(ZZ)(Z~)3]}

INTERNAL MATERIAL CONSTANTS

(25)

The constitutive model'''developed in the previous sections involves a series of param­eters that are material dependent and remain constant during the analysis, as follows: Jand H are tensors that control shape of the damage and the potential surfaces, Ro and yare the damage and unrecoverable deformations thresholds Ct, C2, Cf are the damageand unrecoverable hardening constants. Since both J and H are diagonal tensors, onlyeleven coefficients need to be found (only nine for plane stress). While several fonnuiationsexist for the J tensor [29], they involve a large number of parameters that, altho~gh

increase the accuracy, significantly increase the complexity of parameter identification,and consequent testing. With the addition of the H tensor, the damage surface can bewritten in stress space as the Tsai-Wu surface. In this way the coefficients in the J and Htensors can be uniquely obtained in terms of the strength values Fi of the unidirectionallamina, which for the most part are available in the literature.

950 EVER J. BARBERO AND PA~LO LONETT

Writing the gd function for longitudinal uniaxial tension and compression·,·Equati.~~,{J5:becomes

CII -IB tl-6Ftt = (Y* + YO) = 1n1 .

Equation' (26) represents a linear system in which FIt and FIc are the longitudinalcompressive and tension stress of a ~ingle composite lamina, the values J}1 and HI are theunknown quantities, while the quantities Ole and nit are the critical tension andcompression values of the integrity tehsQrs. Comparing the Tsai-Wu criterion to the linearsystem Eq.uation (26) at failure, we set (y* + YO) equal to 1. Hence, there are two equationsthat allow'·us to solve for the two, parameters JII and HI.

In the transverse direction the damage function Equation (15) reduces to

'.(27)

In this way it is possible to express, J22 in function of H 2 by the following equation

J22 = (1 - (28)

Considering in-plane shear loading and Equation (15), we get the following equation

(29)

Here F6 value is the shear strength for a single composite lamina, while n1s and n2s

are critical values of the integrity tensor at failure for a state of in-plane shear stress. Sincethe shear response of a fiber-reinforced lamina in material directions is independent of thesign of the shear stress, the coefficient of the linear term in F6 of Equation (29) must bezero, leading to a relationship between HI and H 2, namely

/'

HI H2 Q~s-+-=0~H2=--Hl =-rHlQ2 n2' n 2 s

Is . 2s Is(30)

where rs = (n2s/ Q ls)2 is an. intermediate constant. Then H 2 can be written as a function ofthe r s , and Equation (29) becomes .

where Gi2 is the unloading damaged shear modulus prior to imminent failure (e~g-, lastunloading curve in Figure 1). Since J 11 is known from Equation (26)" and k; is a. Jcnown .constant (see section Critical Damage Parameters), using Equations (28) and (31)" it ispossible to determine J22 and H 2• Thus, all the coefficients in the tensors J and "~H arefound in terms of known parameters Fi and Qi-

An Inelastic Damage Model for Fiber Reinforced Laminates

Jll's h2 2C66 p2 - ( ) - 1-k +k k 6-Y+Yo-,

s srs s

951

(31) . ': '.'

CRITICAL DAMAGE PARAMETERS

The critical integrity parameters Q i introduced in the previous section represent thevalues of the integt:ity components when the·~tna~erial is near failure. They can be evaluatedby formulas based on experimental results and analytical procedures.

In a pure in-plmJ.~ shear test, and using Equation (3), the unloading shear stress-strainpath just prior to f~.ilure (Figure 1) can be described in terms of either the damaged or theundamaged (i.e., virgin) shear stiffness as·

- 2-Gn n I 2G* n-In-l IC16 = 12~'ls~'2s86 = 12~'ls ~'2s 86; (32)

which defines the known coefficient ks used in Equation (31) in terms of experimental data.In .this way, the product of Qys and Q~s represents the ratio between the damaged shearmodulus (unloading) at imminent failure and the virgin state value, both of which can bedetermined by experimental in-plane shear data for a unidirectional composite lamina.

On the longitudinal direction, the probability of failure of a fiber in a composite laminacan be expressed by the Weibull statistic distribution [30]. The probability for unbrokenfibers is

100 .....-~-------------..

3.53

II

I,I

I,I

II,,,

I,I

II

II

II

II

1.5 2 2.5....... strain (%J..'

0.5 f 1OIwJ-i.......................-.u..........t.t-................-t-...............-I

o

20

80

180II 40

Figure 1. Cyclic shear behavior of unidirectional T300/915 Carbon/Epxoy.

952 EVER J. BARBERO AND PAOLO LONETTI

'. (33)

wh~ere 0 < Pu < 1, CI is the stress in the unbroken fibers, 8 is the shear. iag len'gth, Lois"tltelength of the test fiber used to obtain the Weibull parameters Clo and m, which representthe dispersion of fiber strength. The bundle stress is Clb = PuCl'~ At fracture, the bundlestress Clb has a maximum. Solving for the fiber stress 0', the critical value of fiber stress is

(8m)-1/m

Clc = CIa Lo (34)

(35)'.

The bundle strength can be found by substituting Equation (34) back into O'b = Pu(1.The critical damageDlt at fracture is the ratio of broken to original fiber area, which isfound from Equations (33) and (34) as

, (-1).D lt = 1 - exp -;;

The Weib~ll dispersion parameter· ~s available from the literature for most types offibers [34,35,37]..

For longitudinal compression, the damage critical value can be evaluated according tothe microbuckling and misalignment of the fiber. Fiber misalignment, which is stronglyde.pendent on processing conditions and can be considered as an inherent defect, increasesthe possibility for the fiber to buckle. For each misalignment angle 9, the composite area­fraction with buckled fiber w(9) corresponding to fibers with misalignment angle greaterthan 9, can be taken as measure of damage. Since the fiber misalignment distribution isGaussian, the critical compressive damage D1c is

Dle = w«(J*) = 1 - erf(A~) (36)

where erf(-) is the error function, A the standard deviation of the actual Gaussiandistribution of fiber misalignment, and 9* is the critical misalignment angle at failure[31,32]. The value of A can be measured experimentally [33] or computed in terms ofexperimental values for F ic, F6 and 0 12 using Equations (4.74)-(4.75) in [17].

In the transverse direction, it is possible to think of the matrix as a periodic linkstructure enclosed between fibers. Assuming a constant probability distribution for thefailure of the matrix link it is possible to derive the maximum admissible strength for thebundle matrix [5]. The ratio between the broken links and the initial ones can beconsidered a measure of damage. Therefore, the critical damage value is

1Dlc =-

2(37)

MODEL IDENTIFICATION AND VALIDATIONI

Unidirectional lamina experimental data for two materials, Aramid/Epoxy and CarboniEpoxy are used to obtain all the parameters required in the model [2]. Laminate and

An Inelastic Damage Modelfor Fiber Reinforced Laminates 953

off-axis experimental data for the same materials is then used to demonstrate,: thepredictive capabilities of the proposed model.

In-plane shear experimental data of unidirectional composite in Figure 1 is used toadjust the model parameters for Carbon/Epoxy T300/915. The material properti~s are "given in Table 1. The intermediate parameters, determined completely in terms of thematerial properties are given in Table 2. The adjustable model parameters, determinedwith the aid of the shear stress-strain plot, are given in Table 3. The model is use,d, topredict the amount of unrecoverable strain, and the results are compared withexperimental data in Figure 2. The predicted damage evolution is shown in Figure 3.Failure is predicted'to occur when the transverse damage d2 equals the critical value D2t, asshown in Figure 3. In the actual test, the sample failed at about 3% shear strain, in goodagreement with the predicted val~e of str~in at failure. Next, the model is used to predict

Table 1. Material properties.

Propm-ty

E1(GPa). undamagedE2(GPa)e undamagedG12(GPa) undamagedG12(GPa) damagedV12 undamagedF1t(MPa)F1C(MPa)F2t(MPa)F2C(MPa)F6 (MPa)

Carbo,n/Epoxy T300/915

142.0'°10.3

'7.293.710.27

183910965757

86

Aramid/Epoxy

73.45.53.652.210.34

1137.1212.5

272747

Table 2. Intermediate parameters unlvocallv determined Interms of the material properties given in Table 1.

Property Carbon/Epoxy T300/915 Aramld/Epoxy

J11 0.2523910783 x 10-14 0.4647013345 X 10-13

J22 0.1166642654 X 10-12 0.1380462161 X 10-12

~ 0 0H1 0.1297694304 X 10-7 0.7518017147 X 10-6 ,

H2 ~0.75544663738 X 10-8 -0.3245140897 X 10-6

Hs 0 0ks 0.515 0.605's 0.581 0.416

/ Table 3. Model parameters.

Property Carb~n/Epoxy T300/915

0.17-8.9 x 105

-0.21.2 x 10-6

0.25

Aramld/Epoxy

0.1-4.0 x 105

-0.31.5 x 10-6

0.5

954 EVER J. BARBERO AND PAOLO LONETTI

the monotonic loading behavior of a [45/-45]2s laminate, which compares favorably with:cyclic experimental data in Figure 4. Comparison between predicted and meastlreaunrecoverable strain versus total shear strain is shown in Figure 5. Comparison betweenpredicted and measured axial stiffness versus total axial strain of a [45/-45]2s is shown inFigure 6.

Experimental data for unidirectional Aramid/Epoxy loaded with cyclic in-plane shear isused to adjust the model parameters, as shown in Figure 7. The model is then used topredict the amount of unrecoverable strain, and the results are compared )\lithexperimental data in Figure 8. The predicted damage evolution is shown in Figure 9.

3.531 1.5 2 2.5

tobalshear shin [%)

0.50 --. ,.......

o

2~5

2C experimerMI data e

i!I: -modet...

'. J

I1.5

'.JI 0.5

Figure 2. Accumulated unrecoverable strain of unidirectional T300/915 Carbon/Epoxy under shear loading.

0.6 ....-------------------......

3

D2t

2.5

-damage v~riabled2

.... -.Critical damage

1.5 2

shear strain r~]

O........- .............-....- .........- ...........~_._-.........- ......__I

o

0.1

0.5

SCI 0.4.!

1~ 0.3

•rE-I 0.2

Figure 3. Damage evolution up to failure of unidirectional T300/915 Carbon/Epoxy un~er shear loading.

An Inelastic Damage Modelfor Fiber Reinforced Laminates 955

3.532.5

.,­---- II

II

II

II

II

II

II

II

II

II

II

II

I/~ - - - --------

I -model resultsI

II

II - 0- -experimental data.. II ....._.. ._-_......

I

1.5' 2'axial strai!, [Ok]

200

175

150..Q;.

125!.

! 100•:i~ 15•

50

25

0'. 0 0.5

Figure 4. Comparison with experimental axial b.ehavior of [451-451~s T300/915 Carbon/Epoxy.'.

2

1.75

....~ 1.5I:gII 1.25u~..&I...5-a-I 0.75:;E::a8 0.5&I

0.25

o experimental data

-model results

3.531.5 2 2.5

total shear strain [%]'0.5

O..................- ........-..---I~....- ........- ...............-+--....--I

o

Figure 5. Comparison with experimental accumulated unrecoverable strain in [45/-45128 T300/915 Carbon/Epoxy.

The AramidjEpoxy failed in shear at about 3% shear strain, when the predictedlongitudinal damage d1 is 90% of the critical value D 1c= 0.114. Next, the model is usedto predict the monotonic loading behavior of a 10° off-axis lamina, which comparesfavorably with cyclic experimental data in Figure 10. Also, the model is able to accuratelypredict the accumulated unrecoverable strain of the 10° off-axis test, as shown in Figure 11.

956 EVER J. BARBERO AND PAOLO LONETII

225 ....---------------------....

3.53

o experimental

2.5

-model results

1.5 2

axial strain rAIl

o175

200

.. 150a.e.i 125

E 100ii 75.=

50

25

00 0.5 -1

Figure 6. eomparison with experimenta/·r~duced stiffness of [45/-45128 T300/915 Carbon/Epoxy.

60

~ ":~m~;~;~ata-model results

_ - , , _ _...

50

40

li!.

130

III

20

10

0.5 1.5 2

shear strain [%]

2.5 3 3.5

Figure 7. Cyclic shear behavior of unidirectional Aramid/Epoxy.

SUMMARY

The procedure used to adjust the model parameters is summarized in this sectipq. Themodel parameters describe. damage and unrecoverable deformation respectively.

The damage evolution parameters Cl, C2 control the damage evolution by Equation (10).The damage threshold Yo represents the initial size of the damage surface. No damage' canoccur until the thermodynamic forces Y reach the damage surface. These three parameters

An Inelastic Damage Mod~lfor Fiber Reinforced Laminates 957

1.5

1.25....~e;:.c

i..J•ui 0.75...eI 0.5'SE::Ic,)c,)II

0.25

'0e 0 0.5

o experimental

-model results

1.5 2

total shear strain rAt]2.5 3 3.5

Figure 8. Accumulated unrecoverable strain of u~idirectionalAramid/Epoxy under shear loading.

0.40.--------------------------....

3.5

d=D1c

3.02.5

-Transverse damage d2

- - 'Longitudinal damage d1

1.5 2.0

total shear strain [Ok]1.00.5

-----_ .... ---_ ..... -----

~ .. ",,,,,~

0.00 .....----.,....----.---....---.....---....----.---...0.0

0.30

0.10

.q,;ellaaIIEII

:: 0.20

~~

E:su~

Figure 9. Damage evolution up to failure of unidirectional AramidlEpoxy under shear loading

are adjusted with the monotonic loading part of the shear stress-strain diagram, such· asthe top curve in Figure I.

The unrecoverable deformatio~ parameter cf controls the evolution of unrecoverablestrain. The threshold parameter Ro represents the initial size of the unrecoverabledeformation surface (yield surface). These two parameters are adjusted with the experi-

958 EVER J. BARBERO AND PAOLO~ LONETTI

250 .....----------------------.,

200

'i'a.i! 150

•,J•CIt:; 100

j

50

0.2

[ -0-, ;perim~~taldat;---model results

0.4 0.6 0.8

shear deformation rAt]

1.2

'.Figure 10. Comparis~n with experimental axial behavior of 10° off-axis test of Aramid/Epoxy.

0.14------------------------.

0.12

C experimental data

l - model resultsc: 0.1

IUJ0.08

.1') 0.08

.!:IE80.04

•0.02

0.90.80.3 0.4 0.5 0.6 0.7

total shear strain (%]

0.20.1O[]oopoo.......~.....,:;;................_+...................~~..................~...................._.+.............

oI'

Figure 11. Comparison with experimental accumulated unrecoverable strain of 10° off-axis test of Aramid/Epoxy.

mental unrecoverable deformations resulting from unloading, such as those read fromunloading in Figure 1 and plotted in Figure 2.

The influence of these parameters on the predictions is discussed next, usingT300j915material as an example. The performance of the model is measureq by the x2

statistical measure of the difference between the predicted values Pi and the experim~ntal

values ei during an appropriate test over which the values have significant influence. Thestatistical measure is defined as follows

An Inelastic Damage Modelfor Fiber Reinforced Laminates

0.020 Error

959

0.015

-o-C1

~'&"C210**4

-o-yO 0.010

~,,,A ,\ ,~ ,4'~

~\ ~I~ I

I\.:I 4 i6 \ i

Q, \ i

~ \ 11;1 beAJ

-1.5 '-1 -0.5 O' 0.5 1.5

C1, ~X1-04, yO

Figure 12. Sensitivity of results to changes in the damage evolution parameters C1, C2, Yo.

'.(38)

First, the damage evolution parameters (c}, C2, Yo) are varied within one order ofmagnitude around the optimum values previously determined in the preceding section,.while keeping the unrecoverable deformation parameters fixed. Equation (38) is used toevaluate the error in predicting the loading portion of the shear stress-strain curve (e.g.,top curve in Figure 1). It can be seen in Fig~re 12 that the error of the prediction, asmeasured by Equation (38), has a minimum for the optimum values of CI and C2. Anydeviation from those values results in a clear loss of accuracy.

From Equations (10) and (15), setting the damage threshold Yo < - Cl starts damageimmediately (tS = 0). Larger values can be used to delay the onset of damage when it isnecessary, but for T300/915, damage must start immediately in order to yield curvature tothe stress-strain loading curve from the onset of loading, as shown in Figure 1.

Keeping the damage evolution parameters fixed, the unrecoverable deformationparameters (cf, Ro) were varied within one order of magnitude around the optimumvalues previously determil)ed in the preceding section. Equation (38) is used to evaluatethe error in predicting the unrecoverable deformations (e.g., Figure 2). It can be seen inFigure 13 that the error of the prediction, as measured by Equation (38), has a minimumfor the optimum value of cf The unrecoverable deformation threshold Ro controls whenunrecoverable deformations start accumulating (see Figure 2).

The internal material constants J II , J22, HI, H2, rs, and ks are used to write the ~odel

equations in a concise form. These are not adjustable model parameters since their valuesare univocally determined in terms of the available material constants Flh Fie, F2h F2c, F6,

Gi2 (see section Internal Material Constants) and the critical damage values D}h Die, D21 '

(see section Critical Damage Parameters). The latter are fixed values in terms of mat~rialconstants m and A as described by Equations (35)-(37). The Weibull dispersion of fiber .strength m and the misalignment angle A are avai~able from the literature [9,34,35,37] and

960

0.05 Error

0.04

EVER J. BARBERO AND PAOLO LONETTI

-6-C1p -o-RO

0.03

0.02

0.01

0.2 0.4 0.6, c1PX105• Ro

0.8

Figure 13. Sensitivity of results to changes in the-unrecoverable deformation evolution parameters ~, Ro•

'.well-established procedures exist to measure them for new materials [33,36]. Finally, thematerial constants E1, E2, G12, and V12 are readily available. The only information that isnot readily available is the shear stress-strain plot, as in Figure 1. The loading part of theplot is necessary to adjust the damage evolution parameters (Cl, C2, Yo). The unloading partof the plot is necessary to adjust the unrecoverable deformation evolution parameters(cf,.Ro). Such a plot can be determined experimentally using a shear test fixture such asdescribed in ASTM D5379.

CONCLUSIONS

The proposed model predicts the damage and unrecoverable deformations phenomenaof a laminated composite material under proportional loading. A limited number ofinternal variables are used in the model to represent the evolution of the damage andunrecoverable deformations phenomena. Comparisons between experimental data andmodel predictions are good in terms of damage and unrecoverable deformationsevolution. Some further development and improvements are envisioned to account forclosure during unloading. Also, further validation would be useful as additional low-cyclestress-strain data becomes available.

REFERENCES/'

1. Aboudi, J. (1991). Mechanics of Composite Materials: A Unified Approach. Elsevier.2. Herakovich, C.T. (1988). Mechanics' of Fibrous Composites. N.Y: John Wiley. .3. Chaboche, J.L. (1993). Development of CDM for elastic solids sustaining anisotropic, and

unilateral damage. International Journal of Damage Mechanics, 2(4): 311-317.4. Ladeveze, P. and LeDante~, E. (1992). Damage modelling of the elementary ply for laminated

composites. Composites Science and Technology, 43: 257-267.5. Krajcinovic, D. (1996). Damage Mechanics. Holland, Amsterdam; North Holland Pub.

An Inelastic Damage Model for Fiber Reinforced Laminates 961

6. Voyiadjis, G.Z. and Deliktas, B. (2000). A coupled anisotropic damage model for the i~elasticresponse of composite materials. Compo Methods in Appl. Mech. Eng., 183: 159-199.. '. ,;I

7. Kachanov, L.M. (1958). On the creep fracture time. Izv. Akad Nauk USSR Otd.Tekh., 8: ,26-31.

8. Kachanov, L.M. (1992). Effective elastic properties of cracked solids: critical review of somebasic concepts. Appl. Mech. Rev., 45(8): 304-335. ' . :

9. Wen, E. (1999). Compressive strength prediction for composite unmanned" aeriaf vehicles.Thesis, West Virginia University, Morgantown, WV. .

10. Barbero, E.J. and Wen, E. (2000). Compressive strength of production ..parts withoutcompression testing. Composite Structures: Theory and Practice. ASTM STP 1383, ASTM, .PA, 470-483.

11. Piggott, M.R., 'Liu, K. and Wang, J. (2000). New experiments suggest that all the shear andsome tensile failure processes are inappropriate subjects for ASTM standards. ASTM STPComposite Structures: Theory and Practice. ,

12. Chaboche, J.L. (1984). Anisotropic creep damage in the framework of continuum damagemechanics. Nuclear Eng. and Design, 79; 3097""319.

13. Lubarda, V.A~ and Krajcinovic,D. (1993):\ Damage tensors and cracks density distribution.Int. J. Solids Structures, 30: 2859-2877.

14. Ogden, R.W. (1984). Non-Linear Elastic Deformation. Mineola, NY: Dover Publications, Inc.15. Barbero, E.J. and De Viro, L. (2001). A Constitutive Model for Elastic Damage in Fiber

Reinforced PMC' Laminae, J. of Damage Mechanics, 10(1): 73-93.

16. Marsden, H. (198~). Mathematical Foundati~n ofElasticity. Englewood Cliffs, NJ: Prentice HallInc.. .

17. Barbero, E.J. (1999). Introduction to Composite Materials Design. Philadelphia, PA; Taylor &Francis.

18. Trusdell, C. (1966). The Elements of Continuum Mechanics. New York: Springer-Verlag.19. Coleman, B.D. and Gurtin, M.E. (1967). Thermodynamics with internal state variables. J. Chern.

Phys., 47: 597-613.20. Lubliner, J. (1972). On the thermodynamic foundations of nonlinear solids mechanics. Int. J.

Non-Linear Mech., 7: 237-254.21. Chow, C.L. and Wang, J. (1987). An anisotropy theory of elasticity for continuum damage

mechanics. Int. J. Fracture, 33: 3-16..

22. Cordebois, J.L. and Sirodoff, F. (1979). Damage Induced Elastic Anisotropy, ColloqueEuromech 115, Villard de Lans, France.

23. Barbero, E.J. and Lonetti, P. (2001). Damage Model for Composites Defined in Terms ofAvailable Data, J. Mechanics of Composite Materials and Structures, 8(4): 299-316.

24. Strang, G. (1986). Introduction to Applied Mechanics. Wellesley, MA: Wellesley-Cambridge.25. Chow, C.L. and Wei, Y. (1991). A model of continuum damage mechanics for fatigue failure.

Int. J. Fracture, 50: 301-316.

26. Hansen, N. and Schreyer, H.L. (1994). A thermodynamic consistent framework for theoriesof elastoplasticity coupled with damage. Int. J. Solids Structures, 31: 359-389.

27. Schwartz, M.M. (1997).' Composite Materials: Properties, Non-destructive Testing, and Repair.NJ: Prentice Hall.

28. ASTM Standards (2000). High Modulus Fibers and Composites, vI5.03.29. Chow, C.L. and Lu, T.J. (1989). On the evolution laws of anisotropic damage. Eng. Fracture

Mechanics, 34(3): 679-701.,-

30. Kelly, K. (1993). The effect of fiber damage on longitudinal creep of a CFMMC. Int. :/. SolidsStructures, 30: 3417-3429.

31. Barbero, E.J. (1998). Prediction of compression strength of unidirectional polymer matrix,composites. Journal of Composites Material, 32(5): 483-502.

32. Barbero, E.J. and Tomblin, J. (1996). A damage mechanics model for compression strengthof composites. Int. J. Solids Structures, 33: 4379-4393.

962 EVER J. BARBERO AND PAOLO LONETTI

33. Yurgartis, S.W. and Sternstein, S.S.(1992). Experiments to reveal the role of matrix properties ..and composite microstructure in longitudinal compression response of composite sttuctur.es.ASTM, 16-17 Nov. -', t ci

34. McDonough, W.O. and Clough,' R.B. (1996). The measurement of fiber strength 'Parametersin fragmentation tests by using ~coustic emissions. Composite Science' and Te~hnologJJ, 56~1119-1127. '.

35. Schultheiz, C.R., McDonough, W.O., Shrikant, K., Shutte, C.L., Mcturk, K.S., McAulife, M.and Hunston, D.L. 13th Symposium on Composites Testing and Design, ASTM, PA. pp. 257-286.

36. Kasai, Y. and Saito, M. (1979}.Weibull analysis of strengths of various reinforcing filaments.Fibre Science and Technology, 11: 21-29.

37. Haberle, J.G. (1991). Strength and failure mechanisms of unidirectional carbon fibre-reinforcedplastics under axial co~pression. PhD Thesis, Imperial College, London, U.K.

'.'..