NASA TECHNICAL MEMORANDUM 102665

RATE DEPENDENT CONSTITUTIVE

MODELS FOR FIBER REINFORCED

POLYMER COMPOSITES

Thomas S. Gates

May 1990

"I

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23665

(NASA-TM-10?6&5) RATF DEPENDENT

CON_TITUITVF MODELS FnR FIBER RrINFORCED

P_LYM_R C_MPUSITES (NASA) 30 D CSCL ]In

G3/Z4

N90-23492
44,
Rate Dependent Constitutive Models

for Fiber Reinforced

Polymer Matrix Composites

Thomas S. Gates

Mechanics of Materials Branch

NASA Langley Research Center

Summary

A literature survey has been conducted to assess the state-of-the-art in

rate dependent constitutive models for continuous fiber reinforced polymer

matrix composite (PMC) materials. Several recent models which include

formulations for describing plasticity, viscoelasticity, viscoplasticity and ratedependent phenomenon such as creep and stress relaxation are outlined and

compared. When appropriate, these comparisons include brief descriptions of

the mathematical formulations, the test procedures required for generating

material constants and details of available data comparing test results to

analytical predictions.

1 Introduction

Constitutive models to describe the rate dependent behavior of isotropic

metallic materials have taken many forms over the last 4 decades. Examination of published constitutive models reveal that several different forms

of viscoelasticity and viscoplasticity have been proposed and utilized by the

mechanics community. During 1966, in an extensive article on viscoplasticity, Perzyna [1] provided descriptions of several rate dependent constitutive

models and their capabilities. Further descriptions of rate dependent constitutive models for metallic materials can be found in Cristescu and Suliciti's
book on viscoplasticity [2]which was publishedin 1982. Thesetwo sources

provide a good background on many of the acceptedmodels used to describerate dependentbehaviorin isotropic metallic materials. In addition to

thesesources,Lindholm et.al. [3] recently reviewedconstitutive modelsfor

advancedmetallic alloysusedin high temperature applications.

The presentsurveypaperis focusedonadvancesin the application of rate

dependentconstitutive modelsto continuousfiber reinforcedorganicmatrix

composites.In orderto reviewthe state-of-the-art, emphasiswill begiven to

recent viscoelasticand viscoplasticconstitutive modelsfor PMC materials.

In general,all of the modelsoutlined in the referencesmentioned above

were developedfor usein describing rate dependent behavior in isotropic

materials. In recent years,many researchershaverelied on someof the models describedin thesesourcesto provide a foundation for developing rate

dependentconstitutive modelsfor PMC materials. In many cases,this is an

acceptablestarting point dueto the factthat PMC's mayexhibit phenomenological behavior which appearssimilar to that seenin advancedmetallics.

Despitesomeapparentsimilarities betweenobservedbehaviorin metallics

and PMC's, oneof the primary grossphysical differencesbetweenisotropic

directionally independent metals and PMC's is that continuous fiber reinforced compositesare anisotropic with well defineddirectional dependency.

As a result of this material anisotropy and the laminated construction common to most PMC's, the mechanicsof the material which contribute to the

observedrate dependentbehavior in PMC's may be quite different when

comparedto the mechanicsof isotropic metallics. Consequently,the constitutive modelsusedto describerate dependentelastic and inelastic behavior

may incorporate more material constantsand include more terms than the

general isotropic case. This increasedcomplexity implies that test requirements for generatingthe necessarymaterial constantswill be different than

thoseutilized in the isotropic modelsand that experimentalprocedureswill

be more involved. From an analytical viewpoint, the inclusion of material

anisotropy implies that the constitutive modelsfor PMC's will usually include the specificcaseof material isotropy asthe most generalapplication of

the compositematerial rate dependentconstitutive model.

It is generallyrecognizedthat most commercialfibersusedin composites

will exhibit linear elastic behavior up to very close to the ultimate fai!ure

point when loaded along their longitudinal axis. Conversely,many composite matrix materials may exhibit material nonlinearity and rate depend6nt
behavior when under load. The degreeof this type of behavior will depend

upon the molecular make-upof the matrix material, the operating environment, the load level and the matrix interaction with the fibers while under

a generalstate of stress.

For many years,the primary typesof compositematerials of interestwere

basedupon a thermosetepoxy matrix system. Generally,thesecomposites

were characterizedas brittle and elastic. Therefore,many of the early rate

dependent constitutive models for compositesconcentratedupon defining

viscoelasticmaterial behavior.

One early model of this type was formed by assumingtime independent properties along the fiber direction for compositesunder isothermal

conditions. Dillard et.al. [4,5]utilized this viscoelastic concept to describe

the failure of graphite/epoxy laminates while undergoing creep. In other

work, Schafferand Adams [6] performed nonlinear viscoelastic analysis of

unidirectional composites by using a form of Schapery's [7] integral constitutive equation. As part of this work, they formed a micromechanics model

which allowed them to study the effects of the individual constituents on

the laminate behavior. Their model was implemented via a finite element

code. Material constants for the model were found by using uniaxial test

specimens under constant temperature conditions. Additional information

on measurement of viscoelastic parameters can be found in work performed

by Kibler and Carter [9]. More recently, Mohan and Adams [8] extended

this work to include investigations of the nonlinear viscoelastic behavior of

neat epoxy resin and unidirectional graphite/epoxy specimens. The effects

of temperature and moisture absorption were also investigated. Numerical

procedures were developed to construct the viscoelastic coefficients needed

to characterize creep-recovery behavior.

Within the last decade, many new types of polymer matrix composites,

such as graphite/thermoplastics, have been developed for use in aircraft and

spacecraft structures. Many of these new materials have been shown to

display aspects of both material nonlinearity and strain rate dependency.

Several new PMC materials such as APC-2 and RADEL have also been developed for extended use at elevated temperatures under conditions of static

and cyclic mechanical loads. This condition of high operating temperature

along with the increased toughness of the new PMC's has made it apparent

that elastic, linear, rate-independent analysis procedures will not be sufficient

for a complete characterization of the material behavior.

3
Viewing the PMC as a tough material has required constitutive models

to be developedwhich utilize conceptsof rate independent plasticity and

rate dependentviscoplasticity in addition to rate dependentviscoelasticity.

Thesetypesof modelsmay includeformulations for describingthe behaviorof

laminatesunder monotonicand cyclic loadsfor both uniaxial and multi axial

states of stress. Various formulations have also beenproposed to describe

phenomenonsuchas creepand stressrelaxation.

The remainderof this paperwill outline someof the morecurrent constitutive modelsfor advancedPMC's and providea comparisonof their analytical formulations and associatedtest methods. The intent of this review is

to provide a starting point for literature searchesof current researchin rate

dependentconstitutive modelsfor PMC's.

2 Constitutive Model Description

Three types of constitutive models will be discussed in the following sections. Rate independent elastic/plastic models, rate dependent viscoelastic

models, and rate dependent viscoplastic models. In general, two approaches

have been used in the development of mechanical material properties for the

constitutive models. The first approach assumes that the fiber and matrix

material can be viewed as separate constituents and their individual contributions to composite behavior can be addressed. The second approach

uses a lamina level model to describe the phenomenological behavior of the

individual plys in the composite. Both types of approaches are used in the

constitutive models given in the following paragraphs.

2.1 Rate Independent Elastic/Plastic Behavior

Providing an accurate description of the rate independent elastic/plastic behavior can be an important aspect in the complete description of the rate

dependent viscoelastic or viscoplastic behavior. With this in mind, two rate

independent elastic/plastic models are described below.

In order to describe the observed plasticity in continuous fiber reinforced

metal matrix composites, Bahei-E1-Din and Dvorak [10,11] developed a micromechanics constitutive model based upon a continuum reinforced by cylindrical fibers of vanishingly small diameter. Their model relied upon a mi-
cromechanicsdescription of the material and usedvolumefractions of the individual constituents to assignrelative contributions of the fiber and matrix.

This model may be consideredan extensionof Hill's[12,13,14]micromechanics constitutive model for composites.In Bahei-E1-Din and Dvorak's model,

the stress and strain increments were written as

do" = C ldo. ! + Cm dtrm (1)

de. = Cfde I + C,,,de.,,,, (2)

where the volume fractions of the fiber and matrix are given as C! and Cm

respectively. The constituent relations in the elastic and plastic range for the

fiber and matrix constituents were given as

do', = L_de_ r = f(fiber), re(matrix) (3)

with L being the matrix of instantaneous local moduli.

It was assumed that the fibers behaved elastically while the matrix exhibited plasticity and obeyed a yield condition given by

f(B,,,eo') =0 (4)

where B,,,_ is a stress concentration factor indicating the contribution of the

individual constituent. The yield surface itself is of a VonMises type and

is allowed to harden kinematicaJly. The flow rule is also specified and is

associated with the hardening rule.

Comparison with experimental data in reference [11] indicated that the

model provided a reasonable approximation of the observed behavior for

a [0/-l- 45], boron reinforced aluminum laminate subjected to two loading

cycles. This is shown in figure 1. It was noted that the model should be

used on materials with low to moderate fiber densities and stress states with

low isotropic components. In addition, it was found that applications of

the model to thermal problems may require further theoretical development.

Although the model was applied to metal matrix composites, it appears likely

that it may also be applied with some confidence to PMC's.

In a series of articles on orthotropic plasticity, Sun and his co workers[14-

17] constructed a constitutive model for metal matrix and polymer matrix

composites. The original form of the inodel was formulated by Kenaga

et.al.[15] for the characterization of the nonlinear behavior of boron/aluminum

5
compositesunder a state of uniaxial tension. This model relied upon a

yield function which required three material parametersand wasquadratic

in stresses.Analytical results from the model were comparedto test data

from off-axis tension tests and found to correlate well within the range of

loading casesinvestigated.

This three parameter model was investigated further by Rizzi et. al.

[16] for cases involving laminated composites in a nonuniform stress field.

The constitutive model was incorporated into a finite element program by

Leewood et. al. [17] and verified by comparison to test data from tapered

specimens and rectangular specimens with a center hole. It was shown that

the selection of the proper parameters in the uniaxial work hardening law

resulted in a good comparison between test data and finite element predictions.

In the most recent form of the model by Sun and Chen [18], the model was

simplified further by using a one parameter flow law. This model assumed

that the total incremental strain can be decomposed into elastic and plastic

tensor components given by

Each of these components can be used in the constitutive relationships

d_ = S_jda_ i,j = 1 - 6 elastic (6)

d¢_ = S_daj i,j = 1 - 6 plastic (7)

where S0 represents the compliance matrix and da is the increment in total

stress.

The potential function used to describe the yield surface was found by

assuming the composite exhibits linear elastic behavior in the fiber direction.

In general, this function was given by

f(aij)--0 (8)

with the Hill-type yield function for orthotropic materials [19] as a special

case. The specific form of the yield function (f) used for fiber reinforced

composites was given by

2 f - a_2 + 2a_a_2 "(9)
The use of one parameter, (a66), in this function is significant in that only one

type of test, uniaxial tension of an off-axis composite specimen, was required

to quantify the parameter. The stress terms involved in this function are the

ply transverse and shear stresses.

By using the associated flow rule, the incremental plastic strains were

written in terms of the yield function. In addition, the increment in plastic

work was defined using effective stress/strain quantities. Specifically, this

was written as

dW p = _d_

where the effective stress was given by

(10)

_:_ (11)

and d_ is the effective plastic strain increment. Use of these relations allowed

for construction of complete constitutive relations to describe elastic/plastic

behavior of laminates. Verification of the model by comparison to test data

from polymer and metal matrix composites indicated that the chosen methods provided a reasonable approach to describing the observed nonlinear material behavior. Figures 2 and 3 show these results for off-axis stress/strain

curves of boron/aluminum and graphite/epoxy composites at various fiber

angles.

2.2 Viscoelastic Behavior

For describing rate dependent behavior, Schapery[20] has developed viscoelastic constitutive models for application to composites. In a recent report, Dan Jumbo, Harbert and Schapery[21] analyzed the strain rate effects

and creep behavior of graphite/thermoplastic composites. In their analysis

of this PMC, they examined both linear and nonlinear viscoelasticity solutions. For the linear viscoelastic solution, creep and recovery strains and the

constant rate behavior are interrelated through a convolution integral

D(t- (12)

where D(t) is the creep compliance. For a constant strain rate test, the

time-averaged relaxation modulus is given by

E, = 7 E(r)dr (t3)
and the recoverystrain following the application of constant stress is

= [t"- (t- tl)"] (14)

where tl is the time under load and

D = Dot" (15)

with Do and n being positive constants.

The use of linear viscoelastic theory to analyze a laminate allowed for

the ability to predict the creep modulus, 1/D, and the exponent n in the

creep strain. It was assumed that the modulus in the fiber direction and

the principal Poisson's ratio were independent of rate and time. The relaxation modulus, in-plane shear modulus and the transverse modulus were all

considered time dependent.

In dealing with the nonlinear viscoelastic behavior of laminates, a ply

constitutive theory was used which was based upon an elastic/plastic model

with constant elastic moduli modified for time dependence. Lamination theory was then employed to predict overall laminate response to uniaxial loads.

Uniaxial tension tests were run on rectangular specimens with varying

amounts of 0", -l-45 °, and 90 ° plys. Constant rate and creep-recovery tests

were performed. A comparison between test and theory showed a good agreement for most cases. However, some errors were observed in the prediction

of creep in the nonlinear range during the creep-recovery tests.

For constant rate tension loading, figure 4 shows Dan Jumbo's comparison

of test to theory for longitudinal stress/strain and Poisson's ration of a [0/4-

45/90] laminate. Results from both the linear and nonlinear theory are shown

in this figure. Examination of the results by the authors revealed that the

proposed method was inaccurate at high stresses when dealing with the creep

strains which occur just after loading. Despite this, the nonlinear theory

appears to give an accurate representation of the complete axial stress/strain

behavior.

In an analytical study of the thermo-viscoelastic behavior of composite

materials, Lin and Hwang[22] developed a finite element model using a variational formulation. A finite difference scheme was also developed to solve

the integral equations.

The constitutive model used by Lin an'd Hwang is for a linear viscoelast, ic

8
orthotropic material and is provided by the relation

ai(T,t) = Cij(T,t - r) (cj(r) - e;(r))dr (16)

oo

This is a form of the hereditary integral given by Schapery [23] which is

similar to equation 12 above. For this integral, ai is the stress, Q is the total

strain and e_ is the free thermal strain. The matrix Cij is composed of the

relaxation moduli which are functions of time (t) and temperature (T).

Lin and Hwang assumed that the composite material was thermo-rheologically

simple and that the relaxation moduli could be found from a master curve by

knowing a reference temperature and the temperature shift factor[24]. For

use in the computational scheme, the relaxation moduli were given in terms

of an exponential series

NT

Cij(t) = Cij.0 + _ Ci/,_e -'/x'i'_' (17)

where NT is the number of terms in the series and the relaxation times are

given by the constants Aij,_

Using their finite element model, the viscoelastic behavior of both notched

and unnotched laminates were obtained. The effects of environmental load

spectrums were also investigated. Material constants needed by the model

were taken from the literature using test data from orthotropic lamina. The

case of a [5=45/0/90], layup under isothermal loads was compared to lamination theory results predicted by Tuttle and Brinson[25]. A comparison of

the two methods showed good agreement. This comparison is given in figure

5. Additional test cases were run but not compared to test data or other

analytical results.

2.3 Viscoplastic Behavior

In order to describe the observed rate dependent behavior of graphite fiber

reinforced thermoplastic composites, Gates and Sun[26,27] developed an elastic/viscoplastic constitutive model for an orthotropic material. For their

model, it was assumed that the rate dependent strain could be decomposed

into an elastic and plastic component

_,j = _,_ + _,_ (_8)

9
where_implies a derivativewith respectto time and e and p indicate elastic

and plastic components, respectively.

The constitutive relation for the elastic terms was given by

_ = S_j_'j i,j = 1 - 6 (19)

where S_j is the compliance matrix of elastic constants. Similarly, the viscoplastic constitutive relationship was written as

e"Pi =S_jaj _" i,j=l-6 (20)

where ,.q_ is the viscoplastic compliance matrix found by using a form of the

associated flow rule

i.v._ A' (21)

'3

The quantity (f) in this expression is the potential function which describes

the rate dependent loading surface, while A' is a proportionality factor used

to bring in the time rate of change of stress. The potential function, given

by equation 9, accounts for material anisotropy and is formed by assuming

elastic behavior along the fiber direction and plane stress conditions.

The dynamic loading surface was assumed to be everywhere parallel to

the static yield surface with the plastic strain rate vector acting in a direction

normal to it. Consequently, the time independent elastic/plastic yield surface

and the rate dependent dynamic loading surface were represented by the same

potential function.

The proportionality factor A' in equation 21 was given more explicitly by

the expressions

2 (22)

3

"7= (23)

where _ is the effective stress and ¢(H) was given by

(q_(H)) = {_(H) 0 ifif HH >_< 00 (loading) (unloading) (24)

The scalar quantity H is the "overstress" and was defined by

H=

10
where _ and _* are the effective rate dependentand effective quasistatic

stress,respectively. The useof effectivestressquantities allows for the dependenceof the state of stresson the anglebetweenthe load and fiber directions and can be representedin a general senseby equation 11. The

quasistatic stressis assumedto be rate independent and is found by using

the incremental elastic/plastic expressionsof Sun and Chen[18].

The concept of overstressand its relationship to viscoplastic strain in

isotropic metalficshasbeenattributed to Malvern[28] and his work on high

strain rate conditions during wave propagation. Additional referencesto

overstressand its usein constructing viscoplasticmodelsin metalfics canbe

found in the work of EisenbergandYen]29,30].

Gates and Sun [26,27]adopted the concept of overstressbut changed

it to relate to effective stressin PMC's. In turn, the effective stress was

related to the combinedstate of stresswhich existed in orthotropic lamina

under a state of off-axis uniaxial loading. Useof effective stressand strain

along with the appropriate potential function allowedfor the generation of

master curvesfor elastic/plastic and elastic/viscoplastic material constants.

Overstress,and hence,the viscoplasticparameters,werefound by performing

multiple relaxation eventsduring a singletest and measuring the changein

stresslevelsovertime.

Material constantsand relatedfunctions neededby this model werefound

by performing uniaxial tension tests on off-axis laminates under isothermal

conditions. By collapsingthis data into master curves,five experimentally

derivedparameterscould be incorporatedinto the constitutive model. Comparison of the predicted behavior to experimental data showeda good correspondencefor both variable strain rate loading and stress relaxation of

off-axis laminates. The typical measuredand predicted stress/strain time

dependentbehavior of an off-axis laminate is shownin figure 6. Additional

information on the useof this type of model and its comparisonto test data

is given in reference[31].

Another analytical modeldevelopedfor describingthe rate dependentbehavior of PMC's wasdevelopedby HaandSpringer]32].Their model utilized

both viscoplasticand viscoelastic constitutive relationships to calculate the

rate dependentresponseof compositelaminates. Theserelationships were

developedfor individual unidirectional lamina and then used in a modified

form of lamination theory to predict the/responseof a multidirectional laminate.

11
For their model, it was assumed that the mechanical strain was composed

of three components

mech ne ve

q =q +q +e_' (26)

where the three terms represent nonlinear elastic, viscoelastic and viscoplastic

strains, respectively, and the subscript i indicates material direction. Addltional assumptions were that the total strain was to consist of the sum of the

mechanical and thermal strains, and the ply properties in the fiber direction

were to be independent of time.

The nonlinear elastic strain e_", as given by Ha and Springer[33], for a

ply is

,_ ai 1 + (27)

q = E!

where 1 indicates the type of loading and E and _ are temperature dependent

parameters.

The viscoelastic strain component e_' was assumed to behave nonlinearly

and exhibit temperature dependency. A form of the Schapery-Lou [34] integral expression, similar to equation 12, was given by

ei = ga dr (28)

where Ki is the time dependent creep compliance function which was given as

the summation of an arbitrary number of exponential terms. The additional

parameters 9il and .qi2 depend upon stress and temperature. The total number of experimentally derived material constants needed for this viscoelastic

component is 23 for shear and transverse loading combined.

For a single ply, an associated flow law was written to relate viscoplastic

strain rate and stress. The assumed form was

1, 0/

C' = (29)

71i vui

where r/is a constant and the potential function f was given by

(30)

12
The stress term in equation 29 is dimensionless and was given by

o"i

= (31)

where X_ is the ply strength in the i direction.

The function ¢_ in equation 29 is related to plastic strain rate and was

assumed to be given by a form of overstress function written in the form of

equation 24 with the overstress given by a power law

H = (& _ _.)b (32)

where the quantity _" was allowed to vary with temperature and strain. In

all, nine experimentally derived constants were required for the viscoplastic

component assuming combined transverse and shear loads.

To predict the rate dependent behavior in laminated composites, a numerical solution routine for a modified laminated plate theory was developed.

Experimental tests were run on graphite/epoxy specimens under four point

bending. These tests were used to obtain the necessary material constants

and to validate the model. Several layups, including multiple ply unidirectional, [4-451 and [4-60/4- 30] laminates were used. A comparison between

test and theory indicated that the analytical model gave good predictions for

creep behavior under various thermal and load histories. Figure 7 shows representative results for creep recovery tests for a [4-45]4, layup under constant

applied load and time varying temperature.

Sutcu and Krempl [35] and Krempl and Hong [36] also utilized the overstress concept to predict viscoplastic behavior in composites. Their model

was applied to metal matrix composites and was formulated assuming that

the total strain rate was composed of an elastic and inelastic component

similar to equation 18 above. The formulation required no definition of yield

surfaces and, consequently, loading and unloading conditions were not specified.

A modified form of laminate theory by Krempl and Hong assumed that

total strain and strain rates were constant through the thickness and the

stress rate was given by an average value. Material constants for the model

were found from off-axis tests of unidirectional material. It was found that

the most complete form of the model required 17 material constants and

two experimentally derived functions. Stress relaxation and creep were investigated for both tension and compression loading; however, comparisons

13
to test data were given only for off-axis tension loading of unidirectional

laminates. These comparisons indicated a reasonably accurate predictive capability. All of the cases explored in the analytical investigations were for

isothermal conditions.

Two additional viscoplastic models which were developed for use on rate

dependent anisotropic materials should be mentioned. These models were

applied to metal matrix composites; however, they may have some application to PMC materials.

The first model was developed by Robinson and Duffy [37] and assumes

the composite acts as a continuum with local transverse anisotropy. A macroscopic viewpoint of the material relationships was taken and a potential function, flow law and evolutionary law were developed and related to the material

parameters. It was noted that some of the required parameters were dependent upon the volume ratio of the constituents. Material constants required

by the model were found by performing tension/torsion tests on longitudinal and circumferential reinforced tubes. For the model, two parameters

and one function were needed to relate the degree of anisotropy and threshold strength while four parameters and one function were needed for the

viscoplastic components. Analytical predictions were made for specimens

under several different states of stress. No comparison was given between

test data and predicted values.

The other viscoplastic constitutive model which was applied to metal matrix composites was formulated by Aboudi [38] and assumes a microstructure

based continuum. The analysis was performed by looking at a representative

cell and assuming both the fiber and matrix to behave as elastic-plastic work

hardening materials. In his derivations, Aboudi provided formulations for the

displacement continuity conditions, the equations of motion for the continuum and the relevant boundary conditions. The constitutive equations used

to describe the stress/strain relationships were based upon work by Bodner

and Partom [39]. These relations have been described as a unified type of

model which do not consider loading and unloading conditions separately.

A comparison between test and theory was not made; however, analytical

results for several different cases were considered and compared.

14
3 Model Capabilities

A summary comparison of four rate dependent constitutive models for PMC's

is given in table 1. As was outlined above and is shown in table 1, all of the

models share some similar capabilities and exhibit some unique differences.

A comparison of the type of model indicates that selection of the analytical

foundation is dependent upon which type of material behavior is of interest to

the researcher. To account for rate-dependent behavior such as creep, creeprecovery, stress relaxation and aging, a combination of both viscoelasticity

and viscoplasticity may be needed. In addition, use of concepts such as

overstress to develop the viscoplastic strains may require a firm definition of

rate independent elastic/plastic behavior.

Important considerations when examining a model are; assessment of

the accuracy of the model when compared to actual experimental data, and

complete explanations of methods needed to arrive at the necessary material

constants. Figures 4-7 indicate the degree of accuracy of the selected models when compared to specific test cases. With respect to these figures, two

criteria need to be kept in mind when comparing predicted values to experimental data. One, what is the relative error associated with the predictions

for any point of interest? Two, does the model capture the correct overall

form of the data during complex events such as loading and unloading, creep

and relaxation? The first criteria of relative error may be of less important

since sources of errors can usually be attributed to inadequate procedures in

testing, data reduction or numerical implementation of the solution. This

implies that the second criteria, capturing the correct form of the data, is

of primary importance when assessing predictive capabilities. In particular,

this criteria may be applied when it becomes necessary to choose a model to

predict material behavior in actual structures. In any case, whatever criteria

is selected, it is recommended that a complete investigation of the data presented in the references be made before judging the overall accuracy of the

analytical model.

Development of experimental methods to generate test data and provision of a scheme for reducing it to the required material constants and

empirical functions is also of importance when judging the applicability of

any particular model. Table 1 indicates which models address these issues.

Specific information on the simplicity and reproducibility of their respective

test methods and data reduction schemes should be obtained directly from

15
Source

of Model

Dan Jumbo 21

Lin 22

Gates 26

Ha 32

Type of

Model

VE

VE

VP

VE/VP

Applied to

Elev. Temp.

No

Yes

Yes

[Creep

Yes

Yes

No

Yes

]Relaxation

No

Yes

Yes

No

Comparison

Experimental

[Methods

Yes

No

Yes

Yes Yes

Yes

No

Yes

Yes

Table 1: Comparison of Selected Rate Dependent Constitutive Models,

Key; VE =_ Viscoelastic, VP =_ Viscoplastic

the references themselves. The number of material constants required by any

given model gives some indication of the complexity of the associated test

techniques. However, a direct comparison of required constants for different

models is not available since several of the models have constants which may

be assumed or ignored for special case applications.

4 Further Research

This report compares some of the most recent constitutive models for rate

dependent behavior in fiber reinforced composites. Particular emphasis has

been placed upon those models which were applied to polymer matrix composites. These PMC constitutive models incorporate concepts of elasticity,

plasticity, viscoelasticity and viscoplasticity. Important aspects of each model

have been outlined and comparisons were made between analytical formulations, experimental methods and predictive capabilities.

Allowing for the fact that these models represent the state-of-the-art in

rate dependent constitutive behavior for PMC's, several areas which require

further research can be identified. Some of the more important future research topics are cyclic mechanical and thermal loading and associated frequency effects, behavior under repeated tension/compression loading, effects

of material hybridization, material selection and processing, combined environmental effects, response to biaxial or triaxial stress states, fracture and

damage growth. Many of these issues may need to be addressed and scrutinized before the proposed constitutive models can be fully accepted and

utilized. Despite this, it is felt that as the uses of new advanced PMC's increase and the operating conditions for these materials become more severe,

16
analytical and experimentalinvestigationson the types of constitutive models outlined in this report will be found to be invaluable in furthering our

knowledge and capabilities.

17
References

[1] Perzyna, Piotr, "Fundamental Problems in Viscoplasticity," Advances

in Applied Mechanics, Vol. 9, 1966, pp. 244-377.

[21

[3]

Cristescu, N. and Suliciu, I., Viscoplasticity, Martinus Nijhoff Publishers

Hague\ Boston\ London, 1982.

Lindholm, U.S., Chan, K.S., Bodner, S.R., Weber, R.M. and Walker,

K.P., "A Survey of Unified Constitutive Theories," Nonlinear Constitutive Relations for High Temperature Application -1984, NASA Conference Publication 2369, NASA Lewis Research Center, Cleveland, Ohio,

June 15-17, 1984.

[4] Dillard, D.A., Morris, D.H. and Brinson, H.F., "Creep and Creep Rupture of Laminated Graphite/Epoxy Composites," Virginia Polytechnic

and State University Report, VPI-E-81-3, Blacksburg, Virginia, 1981.

[5] Dillard, D.A., Morris, D.H. and Brinson, H.F., "Predicting Viscoelastic Response and Delayed Failures in General Laminated Composites,"

ASTM STP 787, I.M. Daniel, ed., 1982, pp. 357-370.

[6] Schaffer, B.G., and Adams, D.P., "Nonlinear Viscoelastic Analysis of

Unidirectional Composite Materials," ASME Journal of Applied Mechanics, Vol. 48, 1981, pp. 859-865.

[7] Schapery, R.A., "Application of Thermodynamics to Thermomechanicai, Fracture, and Birefringent Phenomenon in Viscoelastic Media,"

Journal of Applied Physics, Vol. 35, No. 5, May 1964, pp. 1451-1465.

[8] Mohan, R., and Adams, D.F., "Nonlinear Creep-Recovery Response of

a Polymer Matrix and Its Composites," Ezperimental Mechanics, Vol.

25, 1985, pp. 262-271.

[9] Kibler, K.G. and Carter, H.G., "Viscoelastic Parameters of Epoxy Resin

from Thermomechanical and Electrical Conductivity Measurements,"

Composite Materials: Testing and Design (Fifth Conf.), ASTM STP

674, American Society for Testing _nd Materials, 1979, pp. 282-288.

18
[lO]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

Dvorak, G.J. and Bahei-E1-Din, Y.A., "Plasticity Analysis of Fibrous

Composites," ASME Journal of Applied Mechanics, Vol. 49, June 1982,

pp. 327-335.

Bahei-E1-Din, Y.A., and Dvorak, G.J., "Plasticity Analysis of Laminated Composite Plates," ASME Journal of Applied Mechanics, Vol.

49, June 1982, pp. 740-746.

Hill, R., "Theory of Mechanical Properties of Fiber-Strengthened Materials, I: Elastic Behavior," Journal of Mechanics and Physics of Solids,

Vol. 12, 1964, pp.189-212.

Hill, R., "Theory of Mechanical Properties of Fiber-Strengthened Materials, II: Inelastic Behavior," Journal of Mechanics and Physics of Solids,

Vol. 12, 1964, pp.213-218.

Hill, R., "Theory of Mechanical Properties of Fiber-Strengthened Materials, III: Self-Consistent Model," Journal of Mechanics and Physics of

Solids, Vol. 13, 1965, pp.189-198.

Kenaga, D., Doyle, J.F., and Sun, C.T., "The Characterization of

Boron/Aluminum Composite in the Nonlinear Range as an Orthotropic

Elastic-Plastic Material," Journal of Composite Materials, Vol. 21, June

1987, pp. 516-531.

Rizzi, S.A., Leewood, A.R., Doyle, J.F., and Sun, C.T., "Elastic-Plastic

Analysis of Boron/Aluminum Composite Under Constrained Plasticity

Conditions," Journal of Composite Materials, Vol. 21, August 1987, pp.

734-749.

Leewood, A.R., Doyle, J.F., and Sun, C.T., "Finite Element Analysis of Laminated Anisotropic Elastoplastic Materials," Computers and

Structures, Vol. 25, No. 5, 1987, pp. 749-758.

Sun, C.T. and Chen, J.L., "A Simple Flow Rule for Characterizing Nonlinear Behavior of Fiber Composites," Journal of Composite Materials,

Vol. 23, October 1989, pp.1009-1020.

Hill, R., The Mathematical Theoi'y of Plasticity, Oxford University

Press, London, 1950.

19
[2o] Schapery, R.A., "Viscoelastic Behavior and Analysis of Composite Materials," Composite Materials, Vol. 2, G.P. Sendenckyj, Ed., Academic

Press, New York, 1974.

[21] Dan Jumbo, E.A., Harbert, B.C. and Schapery, R.A., " Constant

Rate, Creep Behavior and the Analysis of Thermoplastic Composite

Laminates," Composite Materials: Testing and Design (Ninth Volume),

ASTM STP 1059, S.P. Garbo, Ed., American Society for Testing and

Materials, Philadelphia, 1990.

[22] Lin, K.Y. and Hwang, LH., 'Thermo-Viscoelastlc Response of Graphite

Epoxy Composites", Journal of Engineering Materials and Technology,

Vol. 110, 1988, pp. 113-116.

[23] Schapery, R.A., "Stress Analysis of Viscoelastic Composite Materials,"

Journal of Composite Materials, Vol. 1, 1967, pp. 228-267.

[24] Williams, M.L., Landel, R.F. and Ferry, J.D., "The Temperature Dependence of Relaxation Mechanism in Amorphous Polymers and Other

Glass-Liquids," Journal of American Chemical Society, Vol. 77, 1955,

p.370.

[25] Tuttle, M.E., and Brinson, H.F., "Prediction of the Long-Term Creep

Compliance Of General Cornposite Laminates", Ezperimental Mechanics, Vol. 26, pp. 89-102, 1986.

[26] Gates, T.S. and Sun, C.T., "An Eiastic/Viscoplastic Constitutive Theory for Fiber Reinforced Thermoplastic Composites",

AIAA/ASME/ASCE/AHS/ASC 30th Structures, Structural Dynamics,

and Materials Conference, Mobile, Alabama, April 3-5, 1989.

[27] Gates, T.S. and Sun, C.T., "An Elastic/Viscoplastic Constitutive Model

for Fiber Reinforced Thermoplastic Composites," AIAA Journal, Ac-

..................... z : ::

cepted for publication February 1, 1990.

[28] Malvern, L:E., "The Propagation Of Longitu_n_:ffave_ o_?iastic Deformation in a Bar of Material Exhibiting a Strain-Rate Effect," ASME

Journal of Applied Mechanics, Vol. 18, 1951, pp. 203-208.

2O
[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

Eisenberg, M.A. and Yen, C.F., "A Theory of Multi axial Anisotropic

Viscoplasticity," ASME Journal of Applied Mechanics, Vol. 48, June

1981, pp. 276-284.

Eisenberg, M.A. and Yen, C.F., "The Anisotropic Deformation of Yield

Surfaces," American Society of Mechanical Engineers, Winter Meeting,

New Orleans, Louisiana, December 9-14, 1984.

Wang, C., Sun, C.T., and Gates, T.S., "Elastic/Viscoplastic Behavior

of Fiber-Reinforced Thermoplastic Composites",

AIAA/ASME/ASCE/AHS/ASC 31st Structures, Structural Dynamics,

and Materials Conference, Long Beach, California, April 2-4, 1990.

Ha, S.K., and Springer, G.S., "Time Dependent Behavior of Laminated

Composites at Elevated Temperatures," Journal of Composite Materials, Vol. 23, November, 1989, pp. 1159-1197.

Ha, S.K., and Springer, G.S., "Nonlinear Mechanical Properties of a

Thermoset Matrix Composite at Elevated Temperatures," Journal of

Composite Materials, Vol. 23, November, 1989, pp. 1130-1158.

Lou, Y.C., and Schapery, R.A., "Viscoelastic Characterization of Nonlinear Viscoelastic Materials," Polymer Engineerin 9 and Science, Vol. 9,

1969, pp. 295-310.

Sutcu, M. and Krempl, E., "A Simplified Orthotropic Formulation of

the Viscoplasticity Theory Based on Overstress," NASA Conference on

Nonlinear Constitutive Relations for High Temperature Applications,

Publication 10010, Akron, Ohio, June 11-13, 1986, pp. 89-95.

Krempl, E. and Hong, B.Z., "A Simple Laminate Theory Using the

Orthotropic Viscoplasticity theory based on Overstress. Part I: In-plane

Stress-Strain Relationships for Metal Matrix Composites," Composites

Science and Technology, Vol. 35, 1989, pp. 53-74.

Robinson, D.N., Duffy, S.F., and Ellis, J.R., "A Viscoplastic Constitutive Theory for Metal Matrix Composites at High Temperature," Thermal Stress, Material Deformation and Thermo-Mechanical Fatigue, Ed.

by Sehitoglu H. and Zamrik S.Y.,' ASME, PVP, Vol. 123, 1987.

21
[38] Aboudi, J., "A Continuum Theory for Fiber-Reinforced ElasticViscoplasticComposites,"International Journal of Engineering and Science, Vol. 20, 1982, pp. 605-621.

[39] Bodner, S.R. and Partom, Y., "Constitutive Equations for ElasticViscoplastic Strain Hardening Materials," Journal of Applied Mechanics,

Vol. 42, June,. 1975 pp. 385-389.

22
2nd cycle 0

QO

0 o

0

0

0

0

0

ol st

0

0

Trensverse

Stress

500

0

Model

Test Data

2nd cycle

200

c

cycle

0

0 0

0

0

0

100

0

0

0

0

0

0

0

0

0

0

0

Strain Long itudinal

Figure 1: Comparison of calculated and measured stress-strain curves for a

[0/4- 45], B-AL plate subjected to two loading cycles[ill.

23
Stress (MPa)

350 /

t / o e io ° o_ [] _<> • T..t Ooto

300 -_ /0 = --Mo.e,

250 I /0 ° v -_ 20°

200 t #__ ---- 0=3o °

150 4 _ _ _-------EJ- o

1 O0 e-45

500 ,., , ,-

0.000 0.002 0.004 0.006 0.008 0.01 0

Total Strain

Figure 2: Comparison of calculated and measured off-axis stress-strain curves

for B-AL material[18].

Stress (MPa)

1000

8OO

600

4O0

200

0

0

8=6

_D_ 0 Test Data

---Model

! g

0.000 0.005 0.010 0.015

Total Sfrain

Figure 3: Comparison of calculated and measured off-axis stress-strain curves

for graphite/epoxy material[18].

24
Stress Poisson's

(MPa) Ratio

800

700 t

600

500

4OO

300

20O

100

1.0

/

/

/

/

0

/

/

/

/

/

/

t

Nonlinear Model

Test Data

.... Linear Model

0.8

0.6

0.4

0.2

0 0.0

0.000 0.005 0.010 0.015 0.020

Longifudinal Strain

Figure 4: Comparison of calculated and measured stress-strain and Poisson's

ratio curves for a [0/:1: 45/90] graphite-PEEK material[21].

25
Strain

-3.0e-004

-3.5e-004

-4.0e-004

-4.5e-004

-5.0e-004

AT= -171°C

0

FEM

CLT

Solu'i'fon

Solution

0

' ' ' I ' ' ' I ' ' ' ! ' ' ' I ' '

50 100 150 200

!

I'''1'''

250 300 350

Time (hours)

Figure 5: Comparison of calculated and classical lamination theory stressstrain curves for a [+45/0/90], graphite/epoxy material under thermal

load[22].

26
Stress

(MPa)

25O

200

150

100

50

0

0

8=15

0

95 C

0

0 Test Data

S Rate Dependent

Model

...... Ouasistatic

IIIIIIIIIIIIIIIIII

0.000 0.004 0.008 0.012

Strain

Figure 6: Comparison of calculated and measured stress-strain curves for an

15 ° off-axis graphite/PEEK specimen[27].

27
Strain

0.004

0.003

Mr= 1.601bf-in/in

0.002

0.001

T =24°C T A- 149°C

0

0.000

0 200 400

0 Test Data

Model

600 800 1000 1200

Time (hours)

1400

Figure 7: Comparison of calculatecFand measured strain-time curves for a

[+45]4° graphite/epoxy material[32].

28
Report Documentation Page _lO",al t-_ecor_ufC S ar'_3

_£e t_3lr_t_ sir afOn

1. Report No.

NASA TM- 102665

2. Government Accession No.

4. Title and Subtitle

Rate Dependent Constitutive Models for Fiber

Reinforced Polymer Composites

7, Author(s)

Thomas S. Gates

9. Performing Organization Name and Address

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5225

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, DC 20546-0001

3. Recipient's Catalog No.

5. Report Date

May 1990

6. Performing Organization Code

8. Performing Organization Report No.

10, Work Unit No.

505-63-01-05

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14, Sponsoring/_gency Code

15. Supplementary Notes

16. Abstract

A literature survey._n conducted to assess the state-of-the-art in rate dependent constitutive models

for continuous fiber reinforced polymer matrix composite (PMC) materials. Several recent models which

include formulations for describing plasticity, viscoelasticity, viscoplasticity, and rate-dependent

phenomenon such as creep and stress relaxtion are outlined and compared. When appropriate, these

comparisons include brief descriptions of the mathematical formulations, the test procedures required for

generating material constants, and details of available data comparing test results to analytical predictions.

17. Key Words (Suggested by Author(s))

Constitutive models

Polymer matrix composites

Plasticity

Viscoplasticity

Vi_oelasticity

19, Security Classif. (of this report)

Unclassified

18. Distribution Statement

Unclassified - Unlimited

Subject Category - 24

20. Security Classif. (of this page)

Unclassified

21, No. of pages 22. Price

29 AO 3

NASA FORM 1E28 OCT 86
_I _ _ I_ I_z_ _ _