Composites: Part A 106 (2018) 52–58

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Composites: Part A

journal homepage: www.elsevier .com/locate /composi tesa

Cavitation in epoxies under composite-like stress states

https://doi.org/10.1016/j.compositesa.2017.12.0031359-835X/Published by Elsevier Ltd.

⇑ Corresponding author.E-mail addresses: anupamneogi@gmail.com (A. Neogi), nilanjan@civil.iitkgp.

ernet.in (N. Mitra), talreja@tamu.edu (R. Talreja).

Anupam Neogi a, Nilanjan Mitra b, Ramesh Talreja c,⇑aAdvanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur 721302, IndiabDepartment of Civil Engineering and Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, IndiacDepartment of Aerospace Engineering and Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA

a r t i c l e i n f o

Article history:Received 12 August 2017Received in revised form 20 November 2017Accepted 1 December 2017Available online 5 December 2017

Keywords:A. Composite failureB. Molecular dynamics simulation

a b s t r a c t

In a previous study (Asp et al., 1995) the experimentally observed low strains to transverse tensile failureof unidirectional (UD) polymer matrix composites were explained as an effect of triaxial (composite- like)stress state in the epoxy matrix. Assuming cavitation as an underlying mechanism for brittle crackingunder triaxial stress states, a dilatation energy density based criterion was put forth (Asp et al., 1996)and was shown to predict well the transverse failure of epoxy based UD composites (Asp et al., 1996).The assumption of cavitation in the epoxy matrix has hitherto not been supported by a mechanism study.The current study attempts to provide a systematic clarification of the cavitation mechanism by molec-ular dynamic simulation. By imposing uniaxial, equi-biaxial and equi-triaxial tension on a simulation cellof a crosslinked epoxy, the degrees of cavitation at various stages of the stress- strain response arerevealed. The results show that triaxiality of the stress states is a governing factor in cavitation of epoxies.

Published by Elsevier Ltd.

1. Introduction

It is commonly recognized that fiber/matrix interfaces in poly-mer matrix composites play critical roles in determining how fail-ure initiates and progresses under thermomechanical loading. Forinstance, the extent of debonding at the fiber/matrix interface onfiber breakage under axial tension is attributed to the strengthand fracture toughness of the interface [4]. Similarly, initiation ofdebonding and its growth at the fiber/matrix interface on imposedtensile loading transverse to the fiber axis is also assumed to begoverned by the same properties of the interface [5]. These assump-tions are nearly always based on studies involving a single fiberembedded in the polymermatrix [4–7]. One of these studies [6] sta-ted, ‘‘..we conclude that one cannot accurately predict the interfa-cial properties of a composite based solely upon conventionalsingle fiber and bulk matrix properties”. As one reason for this, theyconjectured, ‘‘a change in the state of stress in the epoxy near thefiber due merely to the presence of the fiber”. Motivated by theobservation that strain to failure under transverse tension of unidi-rectional (UD) glass-epoxy composites was found to be much lowerthan the strain at which pure epoxy failed in tension, Asp et al. [1]measured the strain to failure for four epoxies in the glassy stateunder what they called ‘‘composite-like stress state”, i.e. the triaxial

stress state experienced by epoxy within the UD composite loadedin transverse tension. They used a test called ‘‘poker-chip” test,developed previously for rubbers [8,9], to induce triaxial stressstates in epoxies and demonstrated that the strain to failure inthe primary direction reduced significantly compared to that in uni-axial stress state. Based on these results, and the previouslyreported observation that yielding in polymers is sensitive tohydrostatic stress [10], Asp et al. [2] posed the question: ‘‘What ifthe distortional effects are small and the dilatational effects aredominant?” In other words, if the triaxial stress state in the epoxymatrix approaches hydrostatic tension, what would the conse-quence be? In the study [1] where triaxial stress states wereinduced in epoxies, the fracture surfaces suggested (not conclu-sively) that the fracture may have nucleated from a point in theregion of the specimen where hydrostatic tension existed. How-ever, because of the unstable growth of the nucleated crack in thepurely elastic region, the fracture surface did not clearly provideevidence of a nucleation site and a probable cause of nucleation.

In the absence of clear physical evidence of how failure initiatesin epoxies under hydrostatic tension, Asp et al. [2] conducted a sys-tematic study of yielding and failure in three epoxies subjected todifferent triaxial stress states. They found that the initiation ofyielding could be predicted by the critical energy density of distor-tion, or by this criterion modified by the mean stress, as long as theenergy density of distortion was sufficiently high. At sufficientlylow energy density of distortion, Asp et al. [2] postulated thatcavitation occurs at a critical value of the energy density of

A. Neogi et al. / Composites: Part A 106 (2018) 52–58 53

dilatation. From the poker-chip tests, they calculated this criticalvalue for three different epoxies. In another study, Asp et al. [3]used this criterion to successfully predict the experimentallyobserved trends in transverse failure of a glass-epoxy UD compos-ite at different fiber volume fractions.

The calculated values [2] of the critical dilatational energy den-sity of three epoxies fall in the range 0.13–0.20 MPa. These low val-ues suggest the size of cavities at unstable growth to be small,likely in the sub-micron range. Direct in situ observation of cavitiesof this size does not seem feasible at this time. Therefore, theauthors of this paper decided to conduct a molecular dynamic sim-ulation instead. It is recognized upfront that such study may at bestreveal trends in cavitation as well as different types of energies butnot provide quantitative values of energies associated with thephenomenon.

2. Molecular Dynamic (MD) simulations

MD simulation has increasingly become a tool for studyingmaterials phenomena at atomistic levels. For epoxies, in particular,this tool has been applied to study various thermodynamic andstructural properties [11–21]. For instance, in the most recentstudy [21] MD simulation was conducted on diglycidyl etherbisphenol A (DGEBA), also known as EPON 828, and isophoronediamine (IPD) hardener to predict glass transition temperature,thermal expansion coefficient, curing-induced shrinkage, elasticmodulus, yield stress, and viscosity. In comparison, the failurebehavior of epoxies has not been studied much with MD simula-tions. The current study will address this aspect with focus onthe imposed stress states.

In the current study, individual molecules of EPON-862 (digly-cidyl ether bisphenol F) and DETDA (diethylene toluene diamine)hardener are generated by utilizing the ‘‘polymer builder moduleof the commercial simulation package, MedeA of Material Design,Inc [22]. Static cross-linking method [11] is adopted to obtain astoichiometric mixture (4:1 ratio) of 4 molecules of EPON-862and 1 molecule of DETDA. An iterative MD/molecular minimization(MM) procedure is used to crosslink the epoxy resin, with onecrosslink established per iteration. Energy of the system is moni-tored during the iterative process to ensure attainment of a stableequilibrium structure. The reaction followed along with the finalstructure of the resin mix is presented in Fig. 1. For more detailsof the process of static cross-linking, see [11–21].

The energy minimized equilibrated molecular structure is thenpacked into a structure (resulting in a cubic box with side lengthof 111.456 Å) consisting of 648 molecular clusters (crosslinkedEPON-862 + DETDA system) by using the Monte-Carlo (MC) based

Fig. 1. (a) Molecular structure of one cluster of crosslinked EPON-862 + DETDA model sand nitrogen atoms, respectively. (b) Chemical reaction of four epoxy resins with a curingis referred to the web version of this article.)

lengthy mixing simulation (utilizing ‘‘Amorphous builder” moduleof MedeA [22]). The amorphous cell is then equilibrated using theconstant volume/constant temperature (i.e. NVT ensemble, at ambi-ent temperature and pressure up to 100 pswith timestep size of 0.5fs) MD simulations. To emulate the curing regime, the condensedphase amorphous crosslinkedEPON-862 + DETDAsystem is initiallyheated up to 700 K for 200 ps and then equilibrated at 300 K tem-perature and 1 atm of pressure for 200 ps ofMD stage. The resultingstructure is subjected to subsequent energy minimization and thenMD simulation (utilizing NVT ensemble) to ensure that there is noresidual stress in the equilibrated structure. Prior to simulations,all the N-C bonds formed initially to obtain 100% crosslinking areexamined as to whether these bonds are still within the cut-off dis-tance [16] (RMS value of the bonds has been scanned within a 10 Ådistance) of physical zone of activationor reaction. Themass densityof the 100% crosslinked epoxy system is obtained as 1.243 gm/cc atambient temperature and pressure conditions.

Following previous studies [17,21], the polymer consistentforce field (PCFF) [23], in its modified version (PCFF+) [22] isadopted in this study. The expression of total PCCF + energy isreproduced for completeness:

Etotal ¼P Eb þ Ea þ Et þ Eo þ Ebb0 þ Eba þ Ebt

þEaa0 þ Eat þ Eaat þ Evdw þ Eelec

" #

Eb ¼X4n¼2

kbn b� b0ð Þn

Ea ¼X4n¼2

kan h� h0ð Þn

Et ¼X3n¼1

ktn 1� cos n/ð Þ

Eo ¼ ko v� v0

� �2Ebb0 ¼ kbb

0b� b0ð Þ b0 � b0

0

� �Eba ¼ kba b� b0ð Þ h� h0ð ÞEaa0 ¼ kaa

0h� h0ð Þ h0 � h00

� �Ebt ¼ b� b0ð Þ

X3n¼1

kbtn cos n/

Eat ¼ h� h0ð ÞX3n¼1

katn cos n/

Eaat ¼ kaat h� h0ð Þ h0 � h00� �

cos /

Evdw ¼ e 2 r�=rð Þ9 � 3 r�=rð Þ6h i

Eelec ¼ QiQj=rij

ystem. Black, red, white and blue colored balls represent carbon, oxygen, hydrogenagent. (For interpretation of the references to color in this figure legend, the reader

54 A. Neogi et al. / Composites: Part A 106 (2018) 52–58

The total energy has been divided into contributions from eachof the internal valence coordinates, cross-coupling terms betweeninternal coordinates and non-bonded interactions. The valence

energies consist of terms from distortions of bond lengths Eb, bondangles Ea, out of plane bending angles Eo, and torsion angles Et .Bond and angle terms contain up to quartic terms to characterizeanharmonic features, and torsion function is represented by a sym-metric Fourier expansion. The out of plane function is a simple har-monic function. The cross-coupling terms used in this force field:

Ebb0 ; Eba; Eaa0 ; Ebt ; Eat ; Eaat represent bond-bond, bond-angle,angle-angle, bond-torsion, angle-torsion, angle-angle-torsion cou-pling terms respectively. The non-bonded terms include the van

der Walls interaction Evdw represented by a 9–6 Lennard Jones

function and electrostatic interaction Eelec is written in the formof standard Columbic interaction with partial atomic charges.

Non-bonded cut-off distance has been taken as 10 Å. For long-range Columbic electrostatics interactions, particle-particleparticle-mesh (PPPM) solver [24] (which maps atomic pointcharges to a 3d grid and computes electrostatic energy with suffi-cient accuracy) is used with convergence criterion of 10�5.

3. MD cell deformation and failure

The equilibrated cross-linked epoxy system is subjected to uni-axial, equi-biaxial and equi-triaxial tension to study the mechani-cal response at a constant strain rate of 1010 s�1 with a time stepof 0.5 fs. The temperature in the simulations was kept at ambient(300 K). For uniaxial loading, a uniform displacement of atoms isimposed in one direction keeping other two boundaries periodic,while for biaxial loading, displacement is imposed along two per-pendicular directions keeping the third direction boundaries peri-odic. In order to maintain equi-biaxial loading, the ratio ofdisplacements along the two directions is kept the same as thatof the ratio of initial dimensions along those directions. For equi-triaxial loading, displacements along all the three directions areimposed such that they are in the ratios of the corresponding initialdimensions. The simulations were performed using the LAMMPSsoftware [25], where the deformation algorithm decouples theboundary in the loading direction from the NPT equations ofmotion [26] for uniaxial and equi-biaxial loading and NVT equa-tions of motion for the equi-triaxial loading situations. Tempera-ture of the sample is kept constant at ambient temperature byemploying the Nosé–Hoover thermostat.

The per-atom stress tensor at site k is calculated by using thefollowing expression [27].

rij ¼ �

mv iv j þ 12

PNpn¼1 r1iF1j þ r2iF2j

� �þ 1

2

PNbn¼1 r1iF1j þ r2iF2j

� �þ 1

3

PNan¼1 r1iF1j þ r2iF2j þ r3iF3j

� �þ 1

4

PNdn¼1 r1iF1j þ r2iF2j þ r3iF3j þ r4iF4j

� �þ 1

4

PNin¼1 r1iF1j þ r2iF2j þ r3iF3j þ r4iF4j

� �þKspace ria; Fib

� �

2666666666666664

3777777777777775

where m and v denote the atomic mass and velocity, respectively,F1; F2, etc., are the forces on atoms involved in an interaction,r1; r2, etc. are the corresponding atomic positions. Np and Nb, arethe number of pairs of atoms and number of bonds, respectively,associated with the given atom.

The terms Na;Nd and Ni are similar numbers for angle, dihe-dral and improper interactions involved. The Kspace indicateslong term Columbic interactions i.e. long range electrostatics

between atoms. In the present simulations particle-particle-particle mesh (PPPM) solver, which maps atom charges to 3Dmesh (by using 3d FFTs to solve Poisson’s equation on the meshand then interpolates electric fields on the mesh point back toatoms) has been used.

To determine yielding within the MD cell, the von Mises stressbased equivalent strain is calculated from the per-atom strain ten-sor, which in turn is computed from the per-atom deformation gra-dient tensor. The algorithm used for these calculations wasdeveloped by [28,29] and has been implemented in [30]. To deter-mine the deformation induced cavitation (void nucleation andgrowth) within the MD cell, a Delaunay tessellation of the atomis-tic solid is generated to subdivide the space into tetrahedral ele-ments (based on implementation in [30,31]) and each tetrahedralDelaunay element is subsequently classified as either solid orempty by computing the alpha complex [27]. For probing theempty surface during the polyhedral mesh construction, the radiusof the probe sphere was chosen as 5 Å, resulting in non-bondeddistance between atoms as 10 Å (which is greater than the non-bonded cut-off distance reported in literature [15]).

4. Results and discussion

The results will be described according to the imposed loadingconditions: uniaxial, equi-biaxial and equi-triaxial at strain rates of1010 s�1 and 300 K temperature. To determine strain rate and tem-perature effects, additional results were obtained at the strain rateof 108 s�1 and at 10 K temperature. It should be noted that theglass transition temperature of epoxy resin mix is approximately378 K and the temperatures chosen for the simulations are 300 Kand 10 K, which are below the glass transition temperature inwhich the resin is in workable condition.

4.1. Uniaxial loading

The stress-strain response of a MD epoxy cell under uniaxialloading has been reported in the literature [16–21]. Our resultsare shown in Fig. 2. The strain plotted is the nominal strain calcu-lated by dividing the imposed uniaxial displacement by the initialcell length and the stress is the axial virial stress averaged over theMD cell. The general features displayed by the stress-strain curvein the figure agree with the previous results. The new featuresare the contours of the von Mises equivalent strain and the voidformation (with void volume fraction indicated) within the MD cellat different stages of deformation labeled in the figure as (a)–(h).As seen in the figure, voids are found to appear near the beginningof the strain softening region, labelled (f), at a strain of 125%. Thesevoids are observed to nucleate in the regions of high von Misesstrain. The void volume fraction increases in the strain softeningregion due to enlargement of already nucleated voids. No newvoids are found to nucleate in the cell. The enlargement of thevoids takes place along approximately 45� to the loading direction(region of von Mises strain concentration). The void growth behav-ior observed is ductile in nature. The complex interplay of voidnucleation and evolution with inelasticity has been studied at anatomic/molecular level earlier for metallic materials (typicallypolycrystalline FCC and BCC materials) but has not been reportedbefore in the literature for polymeric materials. Computationswere also performed at a lower strain rate (108 s�1) and a lowertemperature (10 K). The stress-strain behavior was found to besimilar, except the maximum stress reached increased slightly atthe lower strain rate, while it increased significantly at the lowertemperature. The void nucleation and growth displayed essentiallythe same behavior qualitatively.

(a)

(b)

(c) (d)

(e)

(f)

(g)

(h)ε=5%

ε=10%

ε=12.9%

ε=25%

ε=50%

ε=125%

ε=135%

ε=150%

ab

cd

e fg

h

Fig. 2. Uniaxial tensile stress-strain response of the sample. The von Mises strain contours (with associated color bar), void-volume fraction and applied uniaxial strain (�) areshown at different instances in the stress-strain plots. The inset figures shows surfaces of the voids (indicated as green colored surface) in the samples as obtained from theDelaunay tessellation. The yellow colored arrow in the inset figure represents the position of the voids. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

A. Neogi et al. / Composites: Part A 106 (2018) 52–58 55

4.2. Equi-biaxial loading

The stress-strain response in equi-biaxial tension is shown inFig. 3. In comparison to the uniaxial response (Fig. 2), a striking fea-ture here is that the strain hardening is not present. Instead, thesoftening region follows immediately after the elastic range. Com-pared to the uniaxial loading case, voids in this case appear earlierin the softening regime. The voids are shown in the insertedimages in the figure at strains of 60% (d) and 100% (e). While inthe uniaxial loading the voids that initiated grew with strainingof the MD cell and no further void nucleation was found, in thebiaxial loading case, the increase in the void volume fraction iscaused by enlargement of already nucleated voids as well as bynew void formation. The evolution of voids does not follow anypreferred angle like 450 in the uniaxial loading, and can be viewedas random. However, it should be pointed out that the voids startsnucleation and eventual evolution in the softening region only.

4.3. Equi-triaxial loading

Finally, the stress-strain response in equi-triaxial tensile load-ing is shown in Fig. 4. Here, the strain hardening is absent, andthe strain softening is more prominent than in the previous twocases. Quite similar to the behavior observed in the previous twocases, the voids in this case are also observed to nucleate andevolve in the softening region. No voids are formed prior to strainlevels of around 7.5% (labelled (c) in Fig. 4). In this case, the voidvolume fraction at similar strains compared to that of the uniaxialand equi-biaxial cases are significantly large. Furthermore, therapid increase in the void volume fraction in this case is by enlarge-ment, multiplication and coalescence of voids. The brittle behavioras observed in this case is significantly different from the ductilebehavior observed for the uniaxial case.

4.4. Effect of strain rate and initial system temperature

The results reported above are for a strain rate of 1010 s�1 inwhich the initial system temperature is the ambient temperatureof 300 K. MD simulations have also been performed at lower strainrate of 108 s�1 and lower temperature of 10 K. The peak stresswas found to decrease slightly with reduction of strain rate(from 1010 s�1 to 108 s�1) whereas it increased significantly withdecrease in temperature (from 300 K to 10 K). Qualitatively thepattern of void nucleation and evolution was observed to be simi-lar for simulations at other strain rates and also at lower tempera-tures. Quantitatively, the void volume fraction reached at a givenstrain was slightly less at lower strain rate and at lowertemperature.

4.5. Effect of crosslinking

There has been interest in the effect of the degree of crosslink-ing in epoxies on the thermodynamic and structural properties[11–21]. While significant effects have been found in the citedstudies on glass transition temperature, thermal expansion coeffi-cient, curing-induced shrinkage, elastic modulus, etc.; not muchhas been reported on the failure characteristics and mechanismsof epoxies. The results reported here are all at 100% crosslinkingof the epoxy considered. We carried out simulations at othercrosslinking percentages and found little differences in softeningand cavitation behavior at the three stress states. A note is in orderconcerning the 100% crosslinking used here. While completecrosslinking in epoxies is not achievable in practical processing,the precise degree of crosslinking cannot be measured with cer-tainty. The common method of measuring the degree of crosslink-ing is infrared spectroscopy where the presence of other bondsaffects the crosslinking measurement. In any case, our simulations

a

ɛ=2.5%

ɛ=7.17%

bc

ɛ=30.0%ɛ=60.0%

d

(a)

(b) (c) (d)

ɛ=100.0%

e

(e)

void vol. frac.= 22.14%

void vol. frac.= 7.11%

Fig. 3. Equi-biaxial tensile stress-strain response of the sample. von Mises strain contours and void volume fractions are shown at different points in the stress-strain plots.The inset figures shows surfaces of the voids in the sample. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

Fig. 4. Triaxial tensile stress-strain response of the sample. von Mises strain contours and void volume fractions are shown at different points in the stress-strain plots. Theinset figures shows surfaces of the voids in the sample. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

56 A. Neogi et al. / Composites: Part A 106 (2018) 52–58

with 100% crosslinking are for comparing the cavitation andassociated characteristics in different stress states and for this pur-pose using this crosslinking state seems adequate.

It should also be noted that the results presented in this paper arespecific to the epoxy resin (EPON-862+DETDA)which is a thermosettypeof polymer.Usually for a thermoset polymer a single realization

of the cross-linked structure is fine for the purpose of decipheringresults from simulations. This is not true for a thermoplastic poly-mer inwhich one needs to use different realizations of the structuresince there may be differences in results due to chain atacticity,chain length, chain orientation, radius of gyration and so on. A wordof caution should also bementioned in here: the results presented is

0.015

0.030

0.045(a)

atom

)

Pairwise energy (evdwl+ecoul+elong)

A. Neogi et al. / Composites: Part A 106 (2018) 52–58 57

for epoxy resin system (EPON-862+DETDA) and thereby similarresults should not be generalized for other thermoset and/or differ-ent thermoplastic polymers without proper simulations and/orexperiments. Generalization of results to other thermoplastic andthermoset polymers is outside the scope of this work.

0 20 40 60 80 100 120 140

0.000

0.002

0.004

0.0060.000

E-E 0

(Kca

l/

Strain (%)

Bond energyAngle energyDihedral energy

0 20 40 60 80 100 120-0.006-0.0030.0000.0030.006

0 20 40 60 80 100 1200.00

0.03

0.06

0.09

0.12(b)

E-E 0

(Kca

l/ato

m)

Strain (%)

Bond energyAngle energyDihedral energy

Pairwise energy (evdwl+ecoul+elong)

E-E 0

)mota/lac

K(

Strain(%)

Bond energyDihedral energyAngle energy

Pairwise energy (evdwl+ecoul+elong)

-0.009

0.009

0.018

0.00

0.00

0.05

0.10

0.15

0.20

0 20 40 60 80 100

(C)

Fig. 5. Evolution of energy components with applied levels of strain for (a) uniaxialtensile (b) equi-biaxial tensile, and (c) equi-triaxial, tensile loading of the sample.The vertical dashed-dot lines in (a) depict the elastic limit and onset of failure. In (b)and (c), the vertical lines represent the elastic limit. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

4.6. Evolution of energy distribution

The difference in behavior as observed between the uniaxial(demonstrating a ductile response) and the equi-triaxial case(demonstrating a brittle response) can be correlated with the distri-bution of energies associated with bond length stretching (Ebond),bond angle bending (Eangle), dihedral angle torsion (Edihedral) andnon-bonded pairwise interaction energy (EvanDerWall þ EcolumbicþElong�range) shown in Fig. 5. The vertical lines (parallel to E� E0 axisand perpendicular to Strain axis) in all the subfigures denotes elas-tic regime and initiation of the softening regime.

The case for uniaxial tensile loading is shown in Fig. 5(a). It canbe observed that at all strains irrespective of the elastic, hardeningand the softening regions, the energy associated with bond angle ishigher than the other two bonded energies. Within the elasticregime, the energy associated with the bond stretching is highercompared to the dihedral energy; however in the strain hardeningregime, the dihedral energy is significantly higher compared to thebond stretching energy. This in fact indicates that torsion plays amajor role in the strain hardening region compared to the exten-sion of the bonds. However, amongst the bonded energies, at allregimes of behavior the change in angle associated with the bondplays a major role in the deformation response of the polymerchain. The non-bonded part of the energy (pairwise energy) isobserved to significantly higher compared to the bonded compo-nents of energy at all the regimes of response. In the softeningregion where void nucleation is also observed, the energy contribu-tion is primarily from the pairwise and the bond angle.

The case of equi-biaxial tension is shown in Fig. 5(b). The figureshows that the pairwise energy is significantly higher than the otherenergies which indicates that there is significant amount of chainslippage and/or interaction between chains in comparison to bondstretching, changes in bond angle bending and bond angle torsionalchanges. Within the elastic regime, the energies associated withbond stretching, bond angle and torsion are almost the same. Priorto formation of voids (which starts after the elastic regime), notmuch of difference can be obtained between the bond angle anddihedral energy. The bond length stretching has been observed tosteadily decrease (similar to that as observed in the uniaxial casehowever not to initial energy as observed in the uniaxial case) asthe bond lengthsmove towards new equilibrium bond length.Withthe development of voids, the dihedral energy is observed to followadecreasing trend (in comparison to previous increasing trend) sug-gesting a possibility of transformations from the gauche conforma-tion to that of lower energy trans conformations.

Fig. 5(c) shows the case for equi-triaxial loading. With regardsto energy distribution (Fig. 5(c)) it is observed that amongst thebonded energies, the bond length stretching and angle energy con-tributes almost equally and much greater than the dihedral energy.In other words, the bonds undergo stretching and changes in anglemore than that compared to the torsion of the bonds. This observa-tion is in fact significantly different from that observed for the uni-axial and the equi-biaxial case where dihedral rotations play asignificant role in energy dissipation (after the elastic regime). Sim-ilar to the uniaxial and equi-biaxial loading case the non-bondedpairwise energies are significantly greater than the bonded ener-gies thereby indicating that mechanisms of slippage between thechains may act as the main contributor to the energy dissipationunder applied loading. It is also interesting to note that the dihe-

dral and the angle energies almost approaches that of the initialequilibrated energy at higher strain rates and the bond angleenergy plays a significant role in energy dissipation. This observa-tion is quite different compared to the uniaxial and equi-biaxialcase in which the bond angle plays a nominal role in comparisonto the other two bonded energies. This observation of bond angleenergy playing a major role may be a possible reason for explaininghigh void nucleation and coalescence for softening regime in theequi-triaxial case compared to the other two loading cases.

5. Implications on failure in composites

As was stated at the outset, MD simulation is an effective toolfor understanding the behavior of epoxies (and other materials)

58 A. Neogi et al. / Composites: Part A 106 (2018) 52–58

via atomic interactions, but because of the demands placed oncomputational resources, it is not yet a fully quantitative predic-tion tool. We have used this tool here to get some clarificationon the effect of imposing stress states that exist within a UD fiberreinforced composite subjected to tension normal to the fibers.Inferences from stress and failure analysis in previous studies havesuggested a low-energy requiring failure mechanism driven bylocal hydrostatic tension. While in rubbers cavitation can beobserved in triaxial stress states due to the size of the voidsformed, in crosslinked epoxies this has not been possible dueapparently to the very small size of the voids. The MD simulationshere have shown a clear trend in the increase of the cavitation pro-cess as we go from the uniaxial to equi-biaxial to equi-triaxialstress states imposed on a MD cell.

If the void nucleation and growth mechanisms are as assumed,then the next sequence in the failure process of a UD composite intransverse tension could be the fiber/matrix bond breakage as aconsequence of the energy released by unstable growth of a voidin the matrix near the fiber surface. In fact, the local hydrostatictension near the fiber/matrix interface has been confirmed bystress analysis [1–3]. The fiber/matrix debonding has beenobserved in UD composites under transverse tension by directmicroscopic observations [32]. The future challenge for MD simu-lation would be to clarify the transition from cavitation in theepoxy matrix to debonding of the fiber/matrix interface.

6. Conclusions

In this paper, we have demonstrated by MD simulations thatthe deformation response and the underlying mechanisms inepoxy resin (EPON-862+DETDA) depend significantly on themacroscopic stress states. By studying the cases of imposed uniax-ial, equi-biaxial and equi-triaxial tension at different strain ratesand at different temperatures, we have shown that apart fromthe general response behavior of these cases the post-elasticresponse displays significant differences in the three cases. Moreimportantly, the void formation and evolution also shows highdependence on the imposed stress states. Most significant is thefinding that voids of up to 50% volume fraction can form in thematerial when equi-triaxial tension is imposed thus supportingessentially brittle fracture condition under equi-triaxial tension.The study provides a plausible explanation of the cavitation-induced brittle fracture proposed by Asp et al. [2,3] for unidirec-tional fiber-reinforced composites subjected to tension normal tofibers. The local stress states near the fiber-matrix interface in thiscase are found to be nearly equi-triaxial tension and therefore caninduce brittle cracking of the interface as a result of unstable voidgrowth. By investigating different energy components the studyalso provided plausible reason for activation of cavitation-induced brittle fracture for equi-triaxial tensile case and a ductilestrain hardening type response for the uniaxial tensile loadingsituation.

Acknowledgment

A Neogi is grateful to Indian Institute of Technology Kharagpurfor providing doctoral fellowship and Center for Theoretical Stud-ies for computational resources. A Neogi would also like to thankseveral friends at IIT Kharagpur who had helped him in this worksuch as S. Thamaraikannan and Dipak Prasad.

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