0020-7683(94)00233-9

e Pergamon

Int. J. Solids Structures Vol. 32, No. 13, pp. 1859-1872, 1995Copyright © 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved0020-7683/95 $9.50 + .00

MICROMECHANICAL FORMULAS FOR THERELAXATION TENSOR OF LINEAR

VISCOELASTIC COMPOSITES WITH TRANSVERSELYISOTROPIC FIBERS

E. J. BARBEROWest Virginia University, Morgantown, WV 26506-6106, U.S.A.

and

R. LUCIANOtUniversity of Cassino, via Zamosh 43, Cassino, Italy

(Received 18 March 1994; in revisedform 15 September 1994)

Abstract-Explicit analytical expressions for the relaxation moduli in the Laplace domain ofcomposites with viscoelastic matrix and transversely isotropic fibers are developed. The cor­respondence principle in viscoelasticity is applied and the problem in the Laplace domain is studiedusing the solution of the elastic problem having periodic microstructure. Formulas for the L~place

transform of the relaxation functions of the composite are obtained in terms of the properties ofthe matrix and the fibers. The inversion to the time domain of the relaxation and the creep functionsof composites reinforced by transversely isotropic fibers is carried out numerically when a powerlaw model is applied to represent the viscoelastic behavior of the matrix. Finally, comparisons withexperimental results are presented.

1. INTRODUCTION

The creep response of polymer and metal matrix composites is one of the limiting designparameters for advanced composite structures expected to operate for long periods of timeon a variety of applications (Barbero, 1994). The widespread use of carbon fibers providesthe motivation for the development of a model capable of representing composite materialsreinforced with transversely isotropic fibers.

Many micromechanical models have been developed to estimate the elastic propertiesof composite materials (Christensen, 1990; Mura, 1987). The macroscopic viscoelasticproperties of fiber-reinforced materials were evaluated by the cylinder assemblage modelproposed by Hashin (1965, 1966), where the correspondence principle (Christensen, 1979)was applied. Christensen (1969) proposed an approximate formula for the effective complexshear modulus in the case of materials with two viscoelastic phases by using the compositesphere model. The self-consistent method and a numerical inversion method were used byLaws and McLaughlin (1978) to obtain the response in the time domain. Yancey andPindera (1990) estimated the creep response of unidirectional composites with linear vis­coelastic matrices and transversely isotropic elastic fibers by applying the micromechanicalmodel proposed by Aboudi (1991) to obtain the Laplace transform of the effective vis­coelastic moduli. Then, they used Bellman's numerical method for the inversion to the timedomain. Wang and Weng (1992) used the Eshelby-Mori-Tanaka method (Mori andTanaka, 1973) in order to obtain the overall linear viscoelastic properties.

While several micromechanical models initially developed for the analysis of the elasticbehavior of composites have been extended to the viscoelastic case, no model has beendeveloped for linear viscoelastic solids with periodic microstructure, even though manyr~sults are available for the elastic case (Nemat-Nasser and Hori, 1993). Furthermore, noattempt has been made to develop explicit analytical expressions in the time domain.

t Presently at Department of Mechanical and Aerospace Engineering, West Virginia University, Morgan­town, WV 26506-6106, U.S.A.

1859

1860 E. J. Barbero and R. Luciano

Interest in using the elastic solution for composites with periodic microstructure (Lucianoand Barbero, 1994a) is motivated by the following reasons. First, the periodic solution isthe best approximation available for advanced composites that have periodic micro­structure, taking into account not only the periodicity of the microstructure but also thegeometry of the inclusions. For composites without perfectly periodic microstructure, theperiodic solution provides a bound on the estimate of the overall properties, while othermethods (e.g. the self consistent method) provide a bound that represents the perfectlyrandom distribution of the inclusions (Nemat-Nasser and Hori, 1993).

In the present. paper, analytical expressions in the Laplace and time domain for thecoefficients of the creep and relaxation tensors of composite materials with periodicallydistributed transversely isotropic fibers and linear viscoelastic matrix are proposed. Theinversion in the time domain is carried out numerically because of the nonlinearity of theviscoelastic behavior of the matrix, which is represented by a power law. Comparisons withavailable experimental data obtained by Yancey and Pindera·(1990) and other results thatillustrate the capability of the model are presented.

2. VISCOELASTIC CONSTITUTIVE EQUATIONS

The constitutive equations of a linear viscoelastic isotropic material can be expressedin the time domain in the, following way:

(J(t) = /(2) Loo A(t-T)tre(T) dT+2 Loo /l(t-T)e(T) dT, (1)

where aCt) and e(t) are the stress and strain tensor, A(t) and /let) are the two stress-relaxationfunctions, the dot indicates the differentiation with respect to time and ]<2) denotes theidentity tensor of second order.

If the Laplace transform of a functionf(t) is denoted as:

](s) = f: J(t) exp ( -st) dt,

then eqn (1) can be expressed in the following way:

a(s) = sX(s)tre(s)/(2) +2sj1(s)e(s) = sC(s)e(s).

(2)

(3)

The Poisson ratio in the transformed domain VTD is written in terms of ~(s) and pes)

as:

VTD = ~(s)/2(X(s) +pes»~. (4)

In the following, only the set of linear viscoelastic materials with Poisson ratio thatremains constant in the course of the deformation [i.e. vet) = v = V

TD] will be analysed [s~e

Aboudi (1991) and Wang and Weng (1992)].

3. PERIODIC EIGENSTRAIN IN THE LAPLACE DOMAIN

Consider an infinitely extended linearly viscoelastic solid with periodic microstructurecharacterized by a unit cell D (Fig. 1). Let each cell D be a parallelepiped with dimensionsaj in the direction of the coordinate axes Xj, where j = 1, 2, 3 and let V be its volume. Theunit cell is divided into two parts: the fiber Q and the matrix D-Q and let Vf be the volumefraction of the fibers (Vr = Vn/V).

In the following a composite with viscoelastic isotropic matrix and elastic transverselyisotropic inclusions will be considered. The viscoelastic behavior of the matrix will be

Relaxation tensor of viscoelastic composites

II ,/

I Ie a3I

Fig. 1. Geometry of the unit cell D.

1861

represented by eqn (1) or (3), while the constitutive equation for the transversely isotropicfibers is written as :

u(s, x) = SCI (s)e(s, x) = Cle(s, x) in Q. (5)

In order to compute the viscoelastic properties of the composite, the fibers, periodicallydistributed in the body, can be simulated in the Laplace domain by an homogenizationeigenstrain 8*(S, x), periodic in xj . In this way, the analysis of the viscoelastic behavior of acomposite material with periodic microstructure is reduced to the viscoelastostatic problemof a solid subject to a periodic eigenstrain. The expression of the Laplace transform of theperiodic strain e(s, x) inside Q in terms of 8'*(s, x) can be derived (Luciano and Barbero,1994b) from tne elastic expression (Luciano and Barbero, 1994a). However, in order toobtain the relaxation moduli of the composite it is not necessary to have the exact expressionof 8'(s, x), but only its volume average in Q, 8'(s). In particular, the volume average of thestrain in the inclusion 8(S) can be written in terms of the volume average of the eigenstrain8'*(s) in the following way:

8'(s) = pes) : C(s) : 8'*(s) = 8(s) :8'*(s)

where 8(s) is the Eshelby tensor in the Laplace domain for solids with periodic micro­structure and depends only on the periodicity of the unit cell (al ) , the geometry of theinclusions and the viscoelastic properties of the matrix.

If the expression of 8(s) is known, the equivalent eigenstrain method can be appliedin order to obtain the exact volume-average homogenization eigenstrain which simulatesthe presence of the periodic fibers inside the body. To this end, let an applied average straintensor 80(S) be arbitrarily prescribed in the unit cell. Then, use the following averageconsistency condition (Mura, 1987) in the Laplace domain (i.e. the equiv~lence betweenthe stress in the homogeneous material and the heterogeneous one) :

C'(s): (eo(s)+P(s): C(s) :e*(s)) = C(s): (eo(s) + (P(s) : C(S)_/(4)) :e*(s)), (6)

where /(4) is the identity fourth order tensor. It is worth noting that the tensor pes) takesinto account the geometry of the inclusion and can be evaluated only once for every C I ofthe fibers. Then from eqn (6), the equivalent average volume eigenstrain e*(s) can beobtained in terms of the tensors C/(S), C(s), pes) and ~ for every s as:

8'*(s) = [«(C(s)-C/(s))-J -P(s)): C(S)]-l : eo (s).

The Laplace transform of the uniforlTI overall stress O"o(s) in the unit cell is:

(7)

1862 E. J. Barbero and R. Luciano

sC*(s) : 80(S) = sC(s) : (80 (s) ---- VrE*(S», (8)

where C*(s) is the overall relaxation tensor of the composite material. Using eqn (7) andnoting that so(s) is arbitrary, the following expression of C*(s) is obtained:

sC*(s) = sC(s)-sVr({C(s)-C/{S»-1 -P(S»-l. (9)

Taking into account the periodicity of the microstructure and using a Fourier series rep­resentation of the field variables, the tensor pes) for an isotropic matrix can be written as[see Nemat-Nasser and Hori (1993) and Luciano and Barbero (1994a) for the elastic case] :

pes) = ,ao~s) ~' t(~)(Sym (~® 1(2) ® ~)- 2(1 ~vo) (~® ~ ® ~ ®~»). (10)

where

(90{~») (90{-~»). i-t(~) = Vr --v;- Vn

With go(~) = n exp (l~X) dx (II)

and ';o{s) and vo are the Laplace transform of the shear modulus and the Poisson ratio ofthe matrix respectively, ~ = {~h ~2' ~3} with ~j = 2nnj /aj {nj = 0, ± 1, ±2 .. .j not summed,j = 1,2,3) and"~ = ~/I~I_ Then, from eqn (10), eqn (9) can be written as:

where 10{s) is the Laplace transform of the Lame constant A{t) of the matrix. Then, definingthe series 8f (with 1= 1-9) as in Luciano and Barbero (1994a), the final expressions of thecomponents of the tensor C*(s) can be obtained for any shape of inclusion. However, inthis paper we are interested in composite materials reinforced with long fibers which maybe transversely isotropic. In this case five series are different from zero and only three areindependent (Nemat-Nasser et al., 1982). If the fibers are aligned with the xl-axis, it holdsthat:

8 1 = 84 = 8 8 = 8 9 = 0,

8 2 = 8 3 , 8 5 = 8 6 -

Defining i o = slo(s) and flo = s/1o(s), the components of eqn (12) become:

SerI(S) = i o+2flo

T,T ( .2 2) (" (2flo +2io- C;3- C;3){a~ - a~) 2{a4 - a3). (io - C/l2 )2)-1-Yr -a4+ a3 - +-------

al ai

SCf2(S) = 20+ Vr ((20 -C/I:~(a4 -a3»)

x ((2#0 +220- C;3 - C;3)(a~ - an + 2(a4- a3)(20 - C'12)2)- 1

al ai

(13)

Relaxation tensor of viscoelastic composites

X(2PO+2lo-C;3-C~3)(a~-aD+ 2(a4-a3)(~O-C'12)2)-1

al al

x (2Po+2'xo-C;3-C~3)(a~-an + 2(a4-a3)('xo-C;2)2)-1

al at

where

8 "'3 8 "'2C' +12"21 4"2C' . 2" C,2 4" 1 C' 4" C' C'a2 = 110 - 110 33 110 AO - flo II - Jlo 23 + Jlo AO 23 + Jlo I I 33

- 8,aoi oC;3 -4,u0 C;~ +2,uoC;~ -4,uOC'II i o+8,uoi oC; 2

+2ioC /II C;3 +4C'I2 C;3io-4C/

I2 C;3io-2ioc'II C;3

. 86s--­

3 2- 2vo

,ao

1863

(14)

The series 8 3 , 8 6,87 were computed by Nemat-Nasser et al. (1982) for several valuesof the 'volume fraction of the inclusions. The numerical values can be fitted with thefollowing parabolic expressions using a least-square method (Luciano and Barbero, 1994a) :

8 3 = 0.49247 -0.47603 Vr-O.02748 vI

S6 = O.36844-0.14944Vr-O.27152Vl

8 7 = 0.12346-0.32035Vr+0.23517Vl. (16)

1864 E. J. Barbero and R. Luciano

4. RELAXATION TENSOR IN THE TIME DOMAIN

The C0111pOnents of the relaxation tensor in the Laplace domain are given in eqn (14)for composite nlaterials reinforced with transversely isotropic fibers. The lnatrix can berepresented by any linearly viscoelastic lnaterial. The inversion to the time domain can beperformed nU111ericaliy (Belhnan, 1966; Schapery, 1974; Laws and IvlcLaughlin, 1978;Yancey and Pindera, 1990) or analytically if the nlatrix is represented with the two- or four­paranleter Inodel (Barbero and Luciano, 1994). Experimental data for viscoelastic matrixare usually available in the [orIn of creep strain rate as a function of time t for variousvalues of applied stress (j. Pi. viscoelastic material can be represented by the power law

(17)

The power law provides very good representation of the data, and the data reductionis very simple, with the coefficients A, Band D obtained simply by multiple linear regression.However, the power law in eqn (17) leads to nonlinear models for which the correspondenceprinciple, used in this work, cannot be applied. Several simplified models can be derivedfrom eqn (17). If the coefficient D is assumed to be equal to one, the material is said to belinearly viscoelastic. Such nlaterial is amenable to a well developed viscoelastic theory,based on the correspondence principle (Christensen, 1979). In the following, we use alinearly viscoelastic material represented by a power law of time, for which the creepcompliance can be written

J(t) = Do + Ctl! (18)

where Do represents the elastic response, and C and n are determined by linear regressionof the logarithm of the creep data. Defining

A __ Cr(n+ 1)J(s) = sJ(s) = Do + ,

SI1

then

A_IE(s) = sEes) = J(s) ,

from which the Lanle functions of the matrix in the transformed domain are

flo = 2(1 +vo)'

(19)

(20)

(21)

(22)

Introducing these expressions in eqn (14), the coefficients of the relaxation tensor are.obtained as functions of the Laplace variable s. The inversion to the time domain is donenun1erically (Bellman, 1966) using J-Jegendre polynomials to find N discrete values of thesolution. Then, a po\ver lav,l is fitted to these N discrete points using a chi-square goodness­of-fit statistical test. The initial value theorem is used to obtain the elastic response. Forexalnple, we consider a graphite-epoxy cOlnposite Inaterial containing transversely isotropicfibers (T300 graphite) represented by the [oHowing properties (Yancey and Pindera,1990) :

Relaxation tensor of viscoelastic con1posites 1865

-Uj-.

200/0

'fiber·

F-1620/0

1E31E21E-2 1E-1 1EO 1E1

time [min]

34.5 .------------------.---.~---,

34 + ='--==.::: -- ·········1

w-I 33.5 + : ~ =*""~~';;;;;.:;::: ································1Ow.§. 33 -t-•..................................................................................................

:> 32.5 + ·····················1

.........y- 32 T····················································· ·······················1

~

() 31.5

31 .:r:;·;:.:;;;;;;;;;;:.;;;;;;;~;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;~~~~~='I""':·~Ei"!'!""~·:""""···· ··· :""""'"- 1

30.5 -j--r-++-1+H+t--l--+-t+H+tt--t-t-+++t-Ht---t-+-H-t+H-t--t-H-I+Htf-·--t-++++-++t1

1E-3

Fig. 2. Coefficient CTI (t) of the relaxation tensor.

EA == 29.42 X 106psi

ET == 3.6 X 106 psi

VA == 0.443

VT == 0.05

G == 6.40 X l06 psi

Vr == 0.62 (23)

and a viscoelastic Inatrix, represented by a power law with Do == 1.53 1/(106psi), C == 0.0931j(106psi-min), n==0.17, and vo==0.311. The experilnental data (y-rancey and Pindera,1990) are available for only 120 min. Plots of the components of the relaxation tensor areshown in Figs 2-7 for variolls values of fiber volume fraction. The cOlnponent Cft in Fig.2 is divided by the fiber volume fraction Vr to be able to sho'" all results in a single figure.The values of Cf~ 1 of the fiber are included for comparison. It can be noted thatCfl j Vr~ c~ 1 when Vr == 0.62, but as Vrdecreases, the contribution of the tnatrix beC0111eSmore important and the curves of CTI / Vr beco111e separated fro111 the fiber value Cf} l' Therelaxation of the matrix is included in Fig. 3 for c0111parison. The effect of the reinforcen1entresults in larger values of Ci2 for larger values of Vr. Ho~wever, the relaxation rate withtime is very similar to that of the Inatrix. Sin1ilar observations can be drawn for theremaining components of the relaxation tensor.

2 1Il===:=:::------------------.-.--------.--.----.-.---.--.---..----,

1.8 T- =,~~'S) 1

1.6 + ,

~1.4 ~ .....-..en E3--t3E:l3-'-El-S=~.E ~~

~1.2~.......................................•..•.......................

0 1 + ~

0.8 t .··1

0.6 lw--..----.---- .. --- .- - - - -~ ~~~~;~.- I

0.4 L-H-t+ttJt--+-H+ttttr.--+-H+Httl----H-H+lti+-+-t+++!HC..->-+++iJ

40%

200/01-

i11atrix

1E-3 1E-2 1E-1 1EO 1E1time [min]

riE2 1E3

Fig. 3. Coefficient C5.2 (1) of the relaxation tensor.

1866. E. J. Barbero and R. Luciano

0.9 + -..;::::..~'-'IIIIIIIII_liii ·..·..·· .. ·..·..· ···· ..1

....... 0.8 + · ·..· ······ ······..· ·· 1

"(jj

E...... O.7 + ':::~---~~ .:::: · · ·· ·..·..·..·1N~

() 0.6 + ·..· ·..· · · ·..1

62%

400/0

0.5 + ···:· ~ ..~~~~~~~;::::::::-· ..·· ··..·I

1E31E21E-1 1EO 1E1time [min]

0.4 -l--I-l-I--I+I+++--f--I-H-H+H--+-I-++HI+H---+-I-H+i+H--t-H-I-ttttt---+--H-H-ttI1

1E-3 1E-2

Fig. 4. Coefficient CT2 (1) of the relaxation tensor.

0.5 .,---------------------,

62%

20%

40%

1E31E21E-1 1EO 1E1time [min]

0.4 +--i-H-H+tti;--+-t-++t+ItI--+-H-I+I+H--+-I-H++++l----.-If-+++++H+=-+"--l-4-H1+I+l

1E-3 1E-2

('I)o0.44 -r ,"'-" =:I:~.~a: · •· ..·..·..· · ·..·· ·..1

o.42 -r ~ )

.--.

.~ 0.46

......

0.48 :t".~: _ · · ·• ·..·..· •···•..· ..··..1

Fig. 5. Coefficient C~3 (t) of the relaxation tensor.

20%

40%

1E3·1E21E-1 1EO 1E1time [min]

0.25 +--i-H-t+H+l;--+-t-+++HtI--+-H-I+I+H--+-I-+++H#--I-H-+H-Hf--+--H+1-H+l

1E-3 1E-2

0.55 + =~•..:::: · ········ ·..···..·..··..· 1

0.6 -r---------------------,

0.3 + ~ ~r:~~~= ··..·..· ·1

0.5 + ·..···..··· · · ·..·..· · · ·..·..·..· · ·1

......."(j)E 0.45......~ 0.4 + =-----.::::: ·..· · ·..·..··..·..1()

0.35 + =-==..· · ···· ·..·..· ··1

Fig. 6. Coefficient C:4 (t) of the relaxation tensor.

5. TRANSVERSELY ISOTROPIC COMPOSITE MATERIAL

Because of the periodicity of the microstructure, the relaxation tensor C*(t) for aunidirectional composite represents an orthotropic material with square symmetry. In thecase considered in Section 3, .the directions X2 and X3 are equivalent and the relaxation

Relaxation tensor of viscoelastic composites 1867

0.9 + ~'"-::.-: ···········..····················..·..·..···········..1

0.8 + _ _ - ···..· ······1

20%

40%

620/0

1E31E21E-2 1E-1 1EO 1E1time [min]

0.3 +--l-H-+t+ttt--+--+-++t+t+t----+-f-++H+tt--t-+-H-I+HT--l--+-HH+H+--+-++H+H-I

1E-3

0.4 + " " · ··_···..·············..1

.......

.~ 0.7 + ··········_··..···..··-···· ······1

COCO 0.6 + " _ ······· · ··.. ··..··· ·1

t)

0.5 +_ _ =====~~~~~a;;;.:::..····.. ········ ·· ·..···..1

Fig. 7. Coefficient C~6(t) of the relaxation tensor.

tensor is unchanged by a rotation about Xl of nn/2 (n = 0, ±1, ±2 . ~ .). This implies thatonly·six components are required to describe completely the tensor.

In order to obtain a transversely isotropic relaxation tensor C*TI(t), equivalent in anaverage sense to the relaxation tensor with square symmetry, the averagtng procedureproposed by Aboudi (1991) is used. Then, the following expressions are obtained explicitlyin terms of the coefficients of the tensor C*(t) described in Section 3 :

C I tT1(t) = Cfl (t)

ClfT1(t) = Cf2(t)

C2f T1 (t) = ~C~2(t) +~C~3 (t) +~C:6(t)

C2tT1 (t) = ~C~2(t) +~C~3(t)-~C:6(t)

C4tT1 (t) = ~ (C~2 (t) - C~3 (t) )

C6l T1 (t) = Ct6(t). (24)

6. CREEP COMPLIANCE

The creep compliance in the Laplace domain can be obtained from the relaxationmoduli [eqn (14)] by using:

(25)

Plots of the components of the creep compliance tensor are shown in Figs 8-13. Thecompliance of the matrix is included in Fig. 9 for comparison. It can be noted that the'reinforcement changes the creep behavior even at low values of the fiber volume fraction.The effect of the volume fraction is consistent among all the results presented in Figs 8-13.

7. COMPARISONS WITH EXPERIMENTAL RESULTS

Comparisons with experimental results are presented in this section for epoxy 934reinforced with a 62% fiber volume fraction of T300 graphite fibers. Yancey and Pindera(1990) measured the viscoelastic properties of 934 epoxy resin at two temperatures [NR:room temperature (72°F) ; NE: elevated temperature (250°F)], as shown in Table 1.

1868 E. J. Barbero and R. Luciano

O. 16 "l;:===::=~- v v V J ; hOwe' v....... .. 00 05

0.14 -/-H ··..····H ,

~ 0.12 + HH HHH H H H H.j

E........~1-1 0.1 -!-HH.. H HH.HH..HH H H .•HHH .. H.H.H H H H ·•·•· H H. H ···.H H···· HH ·HH ·H •.. H···H • · HH ·_._ •••.. ·1

~

(j) 0.08 ~::::::::~~~~~~~~~~~~~ H H H· I

620/0

400/0-....200/0

0.06 -/- H.H H.H H H •. H.H H HH HH H H HH H H H H H H HHH H H.HHH •.. H·.. • .. 1

1E31E21E-2 1E-1 1EO 1E1time [min]

0.04 -t--t-++-H+tfJI--I--+1-HI1-1+11H+-III --1-1-HI1+111-++111f---4-H-I+I-I+I--+--H-+-I-++l+----I--l-H-I-H+1

1E-3

Fig. 8. Coefficient Sfl (t) of the creep tensor.

1.8 -,----------------------,

1.6 t===:.::::;:;~-------"""""""= H H..H.j

1.4 -/- HH.. HH H H .. H H H.H .. H H.··H · H··· · ·H • • H • H H· ..• ·1,....-,"U).€ 1.2 jHH H.._ H HH H HH H·=:::::;;;;;~ __ ···_..• •.. ••• I~1-1

NNen

0.8 -l- _ HH_ H _ ==-"EI--EJr.EE~~~~~=_..H_._ _ _ ••1

620/0 \

400/0

20%

matrix0.6 ol.--'-'...._.H.. _._._...__....,...-.... _ .._......._...HH .•_... _ ..._._._. ..._....._ ..~•••••- ••_ ..... lif_'--'-'-

1E31E21E-2 1E-1 1EO 1E1time [min]

0.4 +-t-t-++ttttt-t---t-I-H-II ffillitt-li-IHII-+I++1I HttIIlI--+-1+1-11+ffitt-----I--le-t+HI-Ht--I-+-++++ttl

1E-3

Fig. 9. Coefficient S~2 (t) of the creep tensor.

-0.02 -,--------------'---------,11------------ 111_ .._ ..-0.025 -t- H H H H H H H HH ..HH·..H .. H•. H ·H H H H H ·H H·· H· ..•· .. · .. H H..H_HH H.H.H H·l

,....-,"U).€ -0.035 + HH .. H _ .. H HH H H H H H _ _ _._· • • ·_·..• .. • •.. •.. •.. • •.. •• • -1

~N -0.04 -!- H H H H H H HH ··H· ·.H H·H · ·.H HH H ..• •.. •.. ···· .. · .. ·• • • I

~

en -0.045 -l- H H HH H•.. ·······H· ···•· ..•· ..H.···H·.·· ··· H .•.. H H H H H H _ _ j

62%

400/0

20%

-0.05 + H HH H H HH _ HH H _ _ H j

1E31E2

v "VO"OM --

1E-2 1E-1 1EO 1E1time [min]

Fig. 10. Coefficient - Sf2 (t) of the creep tensor.

-0.055 ~f-H-H+ttI--t-++I+++tt--+-t-H+t+Ift--t--H-I-++tH--I-IH-.......tH~+H-+-H+++tH

1E-3

The fiber is T300 graphite with properties back calculated from composite propertiesso that the micromechanic predictions of the method of cells (Aboudi, 1991) used byYancey and Pindera (1990) fit the experimental data. Minor discrepancies between theexperimental results and the model results presented in this work can be attributed to thefact that the fiber properties were back calculated to fit exactly Aboudi's model (1991) and

Relaxation tensor of viscoelastic composites 1869

-0.1 -,-----.--------.--------,

-0.15 + ·..· · · ..1

-0.2 -1- I

620/0

40%

1E31E21E-1 1EO 1E1time [min]

Fig. 11. Coefficient - S~3 (t) of the creep tensor.

.......

.~ -0.25 +·..·..·....·......·..·..····....:==:::::::~~~~~~~~···· .. ····..·..····....I

.........~ -0.3 + ·· ..· ·..·..· · ·..···..·..······ · ·.. ·..··1

rt')

~ -0.35

-0.4 + ~It=:.: ·..· ·•.. ·· ·..··· 1

-0.45 + ~ ~ :::::~~= · ·..·..•· ·..·..·•······ 1

-0.5 +-H-++t+H1--+-H-t+Htt--t-H-t+H-H--+--t-'H-HtH---iI-++++Hif--l-+++H+l-l

1E-3 1E-2

4 ..,--------------------,

200/0

40%

620/0

.................. jjj=..2

3.5 -l- , 'W!:~~~~Y:::::~~~~~~~~J

~ 2.5 J¥ - ·..· ··..· · · ·..· · ·····..1

Y)

.......

.~ 3 + -.- ·..········ ·· ········..· ·..··..·· ··· · ·1

.........~

1201001.5 -+---+--+---l------1~-+--__+--I----+.----!---1--+----l

o 20 40 60 80time [min]

Fig. 12. Coefficient S t4 (t) of the creep tensor.

3.5 -,--------------------.,

200/0

40%

62%

12010040 60 80time [min]

20

~ 2en

3 1...~ ..sd~~.., =..:!..:!':~..~~~!'.~!~v~=!.·!~~~~.!~.. ~~-v-..~~~~I

.......-(jj.E 2.5 -t- (

.........~

Fig. 13. Coefficient St6 (t) of the creep tensor.

not the initial response of the periodic microstructure model presented in this work. Thefiber properties (Yancey and Pindera, 1990) are shown in Table 2.

The predictions obtained with the present model, using the properties given in Tables1 and 2, are compared with the experimental data in Figs 14-17. The experimental data,available only up to 120 min, were obtained from tests performed on coupons with fiber

SAS 32: 13-E

1870 E. J. Barbero and R. Luciano

Table 1

NR NE

Do(ljl06psi)C (ljl06psi-min)nVo

1.5300.0930.1700.311

2.050.1730.2000.317

Table 2

NR NE

EA(106psi) 29.42 31.09ET (106psi) 3.67 2.15VA 0.443 0.450VT 0.05 0.05G (106psi) 6.40 9.89

•

c

NE experiment

NE model

NRmodel

NR experiment

40 60 80 100 120time [min]

20

10 deg.

-~L: - - - - -I

oo

0.03

0.15

0.12

~. 0.06C/)

......"USE 0.09"'­~.......

Fig. 14. Comparison with experimental results of axial creep response in the global system ofaxes (10°).

100 12040 60 80time [min]

45 deg.

NR model

w L..J U U L.J NR experiment

• • • • •NE model

t:J

NE experiment.

1.2 -r---------------~1.1

10.9

1Wo.aE 0.7.........~0.6 ~

>< 0.5 .r··~ 0.4 l.-,....._~-.......--....----------........-----.....

0.30.20.10+--1--+--1-+--1--+--1---+--~-+--l---1

o 20

Fig. 15. Comparison with experimental results of axial creep response in the global system ofaxes (45°).

orientations of 10°, 45° and 90°, as indicated in Figs 14-17. Since experimental results for8 66 from tests at 10° and 45° are almost coincident, all the data are collapsed in Fig. 16.While the compliance of the composite at 10° from the fiber direction Sxx is fiber dominated,

Relaxation tensor of viscoelastic composites 1871

3

2.5

~ 2E.......~1.5

cocoen

10 and 45 deg.

0 CI 0 CI

;:]0 CICI

';i'

~.. . • • • •

NR model•

NR experiment

NE modelCJ

NE experiment

0.5

oo 20 40 60 80 100 120

time [min]

Fig. 16. Comparison with experimental results of shear creep response in the material system ofaxes (10° and 45°).

1.8 ,..----------------,

1.6 -t- _ c · •..•.. • · ..·1

1.4 + ··..· ·.._..·..·..· ·1

100 120

CJ

•

20

~1.2

~ 1~ [JCl CJ

~ 0.8 +r;--.._ _ · · · · · ·..· ·..1

N -=.en 0.6 1diiiiii~~~~-=--~~II=IF ~_ ~......

0.4 -J- · ·..· · ·1

0.2 -t- · · · ··..· .. ·..1

0+--1---l---+--I---I----l---+-+--+---!--+---t

o 40 60 80time [min]

Fig. 17. Comparison with experimental results of transverse creep response in the material systemof axes (90°).

the compliances 866 (inverse of the in plane shear modulus) and 822 are matrix dominatedproperties (Figs 16 and 17). The comparison presented demonstrates the ability of theproposed model to represent the viscoelastic behavior of fiber-reinforced compositematerials.

8. CONCLUSIONS

Analytical expressions for the Laplace transform of the relaxation and creep tensorsof composite materials reinforced with transversely isotropic fibers and linear viscoelasticmatrix are presented. Numerical inversion to the time domain was implemented along witha power law representation ofthe matrix creep. Good agreement with available experimentaldata is obtained. The interaction effects between the constituents and the geometry of theinclusions are fully accounted for by the micromechanical model used.

Acknowledgements-This work was supported by the Italian National Council of Research (CNR) and theConstructed Facilities Center (CFC) at West Virginia University. We wish to express our appreciation to Dr M.Pindera for providing the experimental data, and to the reviewers for a thorough review and valuable suggestions.

1872 E. J. Barbero and R. Luciano

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