Prediction of impact-induced damage accumulation in

a composite using a macromolecular polymer model

Xavier Poulain∗ and Ramesh Talreja† and A. Amine Benzerga‡

Texas A&M University, College Station, Tx, 77843, USA

Polymers and their composites are increasingly sought for applications where impactresistance constitutes an important design specification. One example of practical signif-icance is the use of polymer-based composites in novel designs of fan blade containmentcases (BCCs) of jet engines. During a blade-out event, a failed blade may penetrate theBCCs, but damage has to be contained within the composite. Here, we develop the ba-sic ingredients of a multiscale modeling methodology with focus on the scale of the basicstructural unit, where polymeric matrix and reinforcements are explicitly modeled. Thepolymer model accounts for nonlinear behavior, finite strains, intrinsic softening, tension-compression asymmetry, rate-sensitivity, thermal softening and kinematic-like hardeningassociated with macomolecular mechanisms of chain reorientations. In addition to thepolymer model, models of matrix cracking and fiber debonding are used. The materialparameters entering the macromolecular model are identified based on tests conducted onan untoughened epoxy resin for a wide range of temperatures and strain rates. Resultsfrom unit-cell calculations are presented to discuss damage initiation and progression aswell as competition between modes of failure.

I. Introduction

Polymer Matrix Composites (PMCs) have been progressively introduced in aeronautics and aerospaceindustries. The combination of the mechanical properties of their phases (low weight polymer, strong andstiff reinforcement) made them appropriate alternative to common materials, especially in an economicenvironment in which fuel efficiency is sought. As a consequence, the use of PMCs as a material for jet engineBlade Containment Cases (BCCs) has naturally emerged and a new design has been recently developed.These BCCs are required to exhibit excellent performance under impact. For instance, damage is requiredto be contained within the composite in case of a blade-out event.1 To reduce costly and extensive testing,the optimization of such designs may benefit from computational simulations which aim at reproducing thebehavior of the composite under impact. A fundamental requirement consists in capturing the complexpolymer response, which has been showed to be dependent on temperature, strain rate and loading mode.2

Numerical methodologies have been developed in the past decades to study the macroscopic behavior,the microscopic evolution of mechanical fields and the fracture mechanisms in metal-matrix composites(MMCs).3, 4 However, only a few of such studies have been recently extended to PMCs.5 Therefore, fun-damental investigation concerning the evolution of thermomechanical fields and damage is still lacking forPMCs. In this paper, we present a modeling methodology that incorporates fundamental ingredients tocapture damage accumulation in PMCs. It includes a macromolecular polymer model which describes thestructural evolution at micro-scale and accounts for temperature, pressure-sensitivity and strain-rate ef-fects.6, 7 It is enriched with a model which represents the onset and accumulation of damage, as in the caseof craze failure in thermoplastics.8 A debonding model is also included, which accounts for failure at thematrix-reinforcement interface.8

The paper is organized as follows. We first present experiments on a representative epoxy resin and weshow the effects of temperature, strain rate and loading mode on the polymer response. Then we detail

∗Ph.D Candidate, Department of Aerospace Engineering, 736C H.R. Bright Building†Professor, Department of Aerospace Engineering, 736A H.R. Bright Building‡Professor, Department of Aerospace Engineering, 736C H.R. Bright Building

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Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

the constitutive equations which model the polymer behavior and the accumulation of damage within thepolymer matrix. A unit cell structure will be used to investigate failure mechanisms in composites, such aspolymer-fiber debonding. After illustrating the capabilities of the model to capture the polymer responsefor both thermosetting and thermoplastic polymers, the influence of a void and of strain rate on the damageprogression in composite unit cells loaded transversally to the fiber will eventually be discussed.

II. Experiments

Extensive uniaxial tension and compression tests for a large range of temperatures (25C-80C) and strainrates (10−5-10−1/s) were conducted on Epon E862 specimens to characterize its behavior and failure.2

Contrary to traditional strain gauges which perturb the local thermo-mechanical fields and promote localizedfailure, an optical method was used to capture the full response of the tested specimen.2

0

50

100

150

200

0 0.1 0.2 0.3 0.4 0.5 0.6

Σ 22(

MP

a)

ε22

25C

50C

80C

0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5 0.6

Σ 22(

MP

a)

ε22

10−1/s

10−3/s

COMPRESSION

TENSION

(a) (b)

Figure 1. Stress-strain curves of Epon E862 Epoxy resin2 in uniaxial tension (open circles) and compression (full

circles) showing (a) temperature effects (at ε = 10−1/s) and (b) strain rate effects (at T=50C)

In Fig. 1 are plotted the commonly observed stress-strain curves in uniaxial tension and compressionat different temperatures (a) and strain rates (b). In compression, a glassy polymer exhibits a three-stageresponse : (i) a low strain hardening, which continously decreases until a peak stress is reached; (ii) a softeningregime which can spread over an extended range of deformation; (iii) and a more or less important progressivehardening at large strains until final failure. In tension, because of earlier fracture, the rehardening stageis not clearly observed. Besides this enhanced ductility observed in compression, some general traits areexperimentally observed in glassy polymers: (i) a high temperature clearly softens the response (Fig. 1(a));(ii) the behavior is rate-dependent, as observed by a stiffer and stronger response with higher strain rates(Fig. 1(b)); (iii) a compression–tension asymmetry is observed with a stronger response in compression thanin tension at fixed temperature and strain rate.

III. Polymer Constitutive Model

III.A. Behavior

In the polymer, the total rate of deformation D is decomposed into an elastic part De, given by a hypoelasticlaw, and a viscoplastic part Dp, which is specified by using a macromolecular model. The flow rule reads:

Dp = ˙εp, p =3

2σe

σ′

d (1)

where ˙ε is the effective strain rate defined as ˙ε =√

2/3Dp′

: Dp′

with X′ referring to the deviator of

second-rank tensor X, and σe is an effective stress defined by:

σe =

√

3

2σ

′

d : σ′

d, σd = σ − b (2)

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with σd the driving stress and b the back stress tensor that describes the orientation hardening of thematerial.9 It is evolved following:

∇

b= R : D (3)

R being a fourth-order tensor, which is specified here by using a non-Gaussian network model9 that combinesthe classical three-chain rubber elasticity model6 and the eight-chain model,7 so that

R = (1 − κ)R3-ch + κR8-ch (4)

where κ = 0.85λ/√

N , N is a material constant and λ is the maximum principal stretch, which is calculatedbased on the left Cauchy–Green tensor B = F · FT . For example, the contravariant components of theeight-chain back-stress moduli tensor, R8-ch, are given by:

Rijkl8-ch =

1

3CR

√N

[(

ξc√N

− βc

λc

)

BijBkl

trB+

βc

λc

(

gikBjl + Bikgjl)

]

(5)

where gij and Bij are the components of the metric tensor inverse and the left Cauchy–Green tensor,respectively, CR and N are material constants known as the rubbery modulus and average number of linksbetween entanglements, respectively, and

λ2c =

1

3trB, βc = L−1

(

λc√N

)

, ξc =β2

c

1 − β2c csch2βc

(6)

where L−1 is the inverse Langevin function defined as L(x) = coth x − 1x . Further details as well as the

components of R3-ch can be found in the paper of Chowdhury et al.10 When the value of either λ or λc

approaches the average limit stretch of a molecular chain, which is actually given by√

N , the network locksand no further viscoplastic flow is allowed.9

Strain rate effects are accounted for through a viscoplastic law of the form:6

˙ε = ε0 exp

[

−A (s − ασh)

T

(

1 −(

σe

s − ασh

)5

6

)]

(7)

where ε0 and A are material parameters, α is a factor describing pressure sensitivity, T is the absolutetemperature, σh = σk

k is the trace of Cauchy stress and s is a micro-scale athermal shear strength. Thefollowing evolution law is used for s:

s = h1(ε)

(

1 − s

s1

)

ε + h2(ε)

(

1 − s

s2

)

ε (8)

where s1 and s2 are adjustable parameters and h1(ε) and h2(ε) are smooth, Heaviside-like functions thatallow to model the pre-peak hardening and the post-peak softening independently:

h1(ε) = −h0

{

tanh

(

ε − εp

f εp

)

− 1

}

; h2(ε) = h0

{

tanh

(

ε − εp

f εp

)

+ 1

}

(9)

Note that (8) differs from its original form,6 which does not capture well the pre-peak behavior (Fig.3(a)).After computation of displacements and velocities, the deformation gradient, the strain rate and other

kinematic quantities are directly computed. The constitutive updating is based on the rate tangent modulusmethod11 giving the Jaumann rate of Cauchy stress and therefore the convected rate of Kirchhoff stress foruse in the principle of virtual power :

∫

V

τ ijδηijdV =

∫

S

T iδuidS −∫

V

ρ∂2ui

∂t2δuidV (10)

Here, τ ij are the contravariant components of Kirchhoff stress, ηij the covariant components of Green-Lagrange strain on the deformed, convected coordinate net, T i the contravariant surface tractions, ui thecovariant displacements and ρ the mass density. V and S respectively denote the volume and surface of thebody in the reference configuration.

Discretization of (10) results in equations of motion that are integrated using a Newmark algorithm.10

Also, a lumped mass matrix is used since this is preferable for explicit integrators. The updating of theback stress b in (3) is obtained using standard kinematic relations. The plane strain specialization is basedon linear displacement triangular elements arranged in quadrilaterals to avoid volumetric locking at largestrains.10

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III.B. Fracture

The constitutive relations (1)–(8) define the plastic flow prior to localization in the polymer matrix but areno longer valid once matrix cracking occurs. Built on physical arguments and remaining phenomenological innature, a new model for polymer matrix cracking in thermosets and thermoplastics was recently developed.12

The onset of natrix cracking is assumed to occur when the maximum principal stress σI attains or exceedsa (positive) pressure-dependent critical value, σc, while the mean normal stress σk

k/3 is positive. The pressuredependence of σc is specified by:

σc(σkk ) = c1(T ) + c2(T )/σk

k (11)

where the ci’s are temperature-dependent material constants: ci = ci1T + ci2.The general form (1)1 of the flow rule applies but a localized mode of deformation takes place with the

direction of plastic flow being set by p = eI ⊗ eI, and the magnitude of flow set through:

˙ε = ˙εcr0

(

σI

(1 − (χ/χc)2)scr

)1

m

(12)

Here, χc is typically about 0.8 and eI refers to the direction of maximum principal stress. In (12) χ is astate variable representing induced damage, and varies between 0, at the onset of cracking, and χf = 1 atzero stress. χ is meant to describe the volume fraction of active fibrils in the damaged structure. Also, ˙εcr0is a reference parameter chosen to ensure continuity of plastic stretching at the transition from shear flowto localization flow, and m and scr are additional material constants. The following evolution equation isadopted for χ:

χ = C (χf − χ) ˙ε (13)

with χf as above and C a material constant. The loss of stress bearing capacity is a natural outcome tothe damaging process. The finite element implementation essentially follows along the lines developed byChowdhury et al.10

IV. Results

IV.A. Polymer response modeling

Numerical simulations, based on the integration of the previously introduced constitutive equations on astress or strain path, are carried out for various temperatures and strain rates in tension and in compres-sion. The material parameters present in the model are identified by matching numerical simulations withexperimental data. These estimated material paramaters are summarized in Table 1 : the density ρ affectsthe response essentially at high strain rates; E defines the slope of the initial slope; s0 controls the initialyield; s1 controls the peak stress; s2 describes the amount of softening; h0 affects the slope of the softeningfor a better fit; ε0 determines the rate-sensitivity; A the temperature-sensitivity; α the pressure-sensitivity;and CR and N define the large strain hardening.

Some experimental and numerical stress-strain curves are plotted together in Figure 2, in which thematerial parameters corresponding to Epon E862 epoxy resin were used. The effects of the temperature andstrain rate are clearly captured by the model, especially at low strains. However, some discrepancies arevisible at higher strains : in compression, the model predicts a stronger response than what experimentsexhibit, while the model underestimates the response in tension. A simple explanation is the following: sincethe simulations were based on a single element, they did not capture the heteregeneous strain and stress fieldsthat are actually present in experiments. Therefore, full specimen simulations would give a more refinedestimate of material parameters and give insight for the identification of fracture parameters. Subsequentlya better match between experimental and simulations would be obtained.

Besides its capability to capture the response of thermosetting polymers such as EPON E862 epoxyresin, this macromolecular model can also predict the behavior of thermoplastic polymers such as PMMA.8

Indeed, the response is clearly predicted by the model (Figure 3). As a result, the behavior of glassypolymers, regardless of their class (thermoplastics or thermosets) can be satisfactorily predicted by ourmacromolecular model, for different loading modes (tension, compression) and for a large range of strainrates and temperatures.

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Material Units Description PMMA E832 Epoxy

parameter

ρ kg/m3 mass density 1300 1100

E GPa Young’s modulus 3.2 2.6

ν – Poisson’s ratio 0.33 0.36

s0 MPa initial shear strength 70 65

s1 MPa pre-peak strength 114 107

s2 MPa saturation strength 104 104.8

h0 MPa slope of yield drop 1300 3000

ε0 1012s−1 rate-sensitivity factor 20 5

A K−1 temperature-sensitivity factor 225 220

α – pressure sensitivity parameter 0.067 0.05

CR MPa rubbery modulus 9.5 3.5

N – number of links 5.1 5.1

between entanglements

Table 1. Material parameters representative of a thermoplastic polymer (PMMA) and a thermosetting polymer (EponE862 Epoxy resin).

0

50

100

150

200

0 0.1 0.2 0.3 0.4 0.5 0.6

Σ 22(

MP

a)

ε22

25C

50C

80C

experiment

simulation

-10

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5

Σ 22(

MP

a)

ε22

10−1/s

10−3/s

(a) (b)

Figure 2. Stress-strain curves of Epon E862 Epoxy resin2 and single element simulations in (a) uniaxial compression

(at ε = 10−1/s) and in (b) uniaxial tension (at T=50C)

IV.B. Fracture of a composite unit-cell

The “unit-cell” concept is used to examine the onset and propagation of damage and preliminary resultsare obtained for a PMMA matrix composite. The effects of manufacturing induced voids on the overallcomposite response and on the failure modes were recently investigated.8 In Figure 4 the failure mechanismsin a unit cell composed of a void and a fiber embedded in a PMMA matrix were investigated in uniaxialtension or uniaxial compression (vertical direction) and with zero average traction prescribed on the lateralside. The onset and propagation of damage in the composite are strongly dependent on conditions such asloading mode or strain rate : while in compression failure only initiates from matrix cracking and propagatesalong the loading direction, in tension low strain rates favor fiber–matrix debonding, as opposed to higherstrain rates which enhance matrix cracking induced failure. Cavitation induced debonding in the compositeis set to occur in an interphase region of prescribed thickness when the dilatational energy density reachesa critical value.13

Stresses at which localization occurs were recorded for various temperatures and used for the identificationof the fracture parameters.14 The obtained values are : c11 = −0.065 MPa/K, c21 = −0.065 MPa2/K,c12 = 65 MPa and c22 = 806 MPa2 (Eq. 11); scr = 200 MPa, m = 0.04 (Eq. 12); and C = 7.5 (Eq. 13).

The elastic properties of the fiber material are taken to be representative of glass with E = 72.4 GPaand ν = 0.2.

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0

50

100

150

200

250

0.00 0.15 0.30 0.45 0.60 0.75

ε

σ (M

Pa)

Experiments [Arruda et al, 1995]

Original macromolecular

Modified macromolecular

0

20

40

60

80

100

120

140

160

180

200

220

240

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

σ(M

Pa)

ε

0.1/s

0.01/s

0.001/s

Plane Strain

(a) (b)

Figure 3. (a) Experimental PMMA stress-strain curves along with matching simulations using original and modifiedmacromolecular model;(b) Effects of strain rate on the PMMA response.

E22 = 0.0105 E22 = 0.0107 E22 = 0.0109

E22 = 0.0158 E22 = 0.0160 E22 = 0.0163

T=90C

σI (MPa)0.001s−1

1s−1

Debonding Crazing

Crazing

Debonding

(a)

E22=0.05 E22=0.09 E22=0.14

crackpath

σmax

I

(MPa)

(b)

Figure 4. Contours of maximum principal stresses and crack path in a unit cell composed of a void (bottom left) and afiber (bottom right) embedded in a PMMA matrix : (a) in tension, strain-rate dependence on the competition between

fiber debonding and matrix craking; (b) in compression, at 25C and E22 = 1/s only matrix craking can occur

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V. Conclusion

A new modeling methodology to predict the behavior of thermoplastics and thermosets was developed. Itis based on constitutive equations that describe the evolution of the polymer structure at a macromolecularlevel. It captures the influence of temperature, strain rate and loading mode on the macroscopic responseand microscopic evolution of mechanical fields. With the addition of fracture models relevant to matrixcracking and fiber debonding, the onset and the propagation of damage in glassy polymers and PMCs canbe assessed. A precise calibration of material and fracture parameters is fundamental to accurately predictthe behavior and failure mechanisms of multi-scale PMC structures, as in traditional top-down methods.The incorporation of aging effects and thermal softening in the polymer model, the refinement of fractureparameters, and a better modeling of debonding mechanisms are suggested for future investigation.

Acknowledgments

We acknowledge partial support from NASA Glenn Research Center under cooperative agreement NNX07AV39A and a grant from the Supercomputing Center at Taxas A&M University.

References

1Federal Aviation Regulation Part 33 Section 94-blade containment and rotor unbalance tests.2Littell, J. D., Ruggeri, C. R., Goldberg, R. G., Roberts, G. R., Arnold, W. A., and Binienda, W. K., “Measurement of

epoxy resin tension, compression, and shear stress-strain curves over a wide range of strain rates using small test specimens,”

Journal of Aerospace Engineering , Vol. 21, 2008, pp. 162–173.3Llorca, J., Needleman, A., and Suresh, S., “An analysis of the effects of matrix void growth on deformation and ductility

in metal ceramic composites,” Acta metall. mater., Vol. 39, 1991, pp. 2317–2335.4Aboudi, J., Pindera, M.-J., and Arnold, S., “Higher-order theory for functionally graded materials,” Composites: Part

B , Vol. 30, 1999, pp. 777–832.5Gonzalez, C. and LLorca, J., “Mechanical behavior of unidirectional fiber-reinforced polymers under transverse compres-

sion: microsocopic mechanisms and modeling,” Comp. Sci. Tech., Vol. 67, 2007, pp. 2795–2806.6Boyce, M. C., Parks, D. M., and Argon, A. S., “Large inelastic deformation of glassy polymers, Part I: Rate dependent

constitutive model,” Mech. Mater., Vol. 7, 1988, pp. 15–33.7Arruda, E. M. and Boyce, M. C., “A three-dimensional constitutive model for large stretch behaviour of rubber materials,”

J. Mech. Phys. Solids, Vol. 41, 1993, pp. 389–412.8Chowdhury, K. A., Talreja, R., and Benzerga, A. A., “Effects of manufacturing-induced voids on local failure in polymer-

based composites,” J. Eng. Mat. Tech., Vol. 130, 2008, pp. 021010.9Wu, P. D. and Van der Giessen, E., “Computational aspects of localized deformations in amorphous glassy polymers,”

Eur. J. Mech., Vol. 15, 1996, pp. 799–823.10Chowdhury, K. A., Benzerga, A. A., and Talreja, R., “A computational framework for analyzing the dynamic response

of glassy polymers,” Comput. Methods Appl. Mech. Engrg , Vol. 197, 2008, pp. 4485–4502.11Peirce, D., Shih, C. F., and Needleman, A., “A tangent modulus method for rate dependent solids,” Comput. Struct.,

Vol. 18, 1984, pp. 875–887.12Chowdhury, K. A., Benzerga, A. A., and Talreja, R., “An analysis of impact-induced deformation and fracture modes in

amorphous glassy polymers,” Eng. Frac. Mech., Vol. 75, 2008, pp. 3328–3342.13Asp, L. E., Berglund, L. A., and Talreja, R., “Prediction of Matrix-Initiated Transverse Failure in Polymer Composites,”

Comp. Sci. Tech., Vol. 56, 1996, pp. 1089–1097.14Gearing, B. P. and Anand, L., “On modeling the deformation and fracture response of glassy polymers due to shear

yielding and crazing,” Int. J. Solids Structures, Vol. 41, 2004, pp. 3125–3150.

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