Composite Structures 113 (2014) 1–11

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Study of localized damage in composite laminates using micro–macroapproach

http://dx.doi.org/10.1016/j.compstruct.2014.02.0160263-8223/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +91 22 25767507; fax: +91 22 25726875.E-mail addresses: ramesh@me.iitb.ac.in, rameshksingh@gmail.com (R. Singh).

Ganesh Soni a, Saurabh Gupta b, Ramesh Singh b,⇑, Mira Mitra c, Wenyi Yan d, Brian G. Falzon e

a Department of Mechanical Engineering, IITB-Monash Research Academy, Mumbai, Indiab Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, Indiac Department of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai, Indiad Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Australiae School of Mechanical and Aerospace Engineering, Queen’s University, Belfast, UK

a r t i c l e i n f o a b s t r a c t

Article history:Available online 21 February 2014

Keywords:Micro–macro methodPartial homogenizationMatrix damageInterface failure

A robust multiscale scheme referred to as micro–macro method has been developed for the prediction oflocalized damage in fiber reinforced composites and implemented in a finite element framework. Themicro–macro method is based on the idea of partial homogenization of a structure. In this method, themicrostructural details are included in a small region of interest in the structure and the rest is modeledas a homogeneous continuum. The solution to the microstructural fields is then obtained on solving thetwo different domains, simultaneously. This method accurately predicts local stress fields in stressconcentration regions and is computationally efficient as compared with the solution of a full scalemicrostructural model. This scheme has been applied to obtain localized damage at high and low stresszones of a V-notched rail shear specimen. The prominent damage mechanisms under shear loading,namely, matrix cracking and interfacial debonding, have been modeled using Mohr–Coulomb plasticityand traction separation law, respectively. The average stress at the notch has been found to be 44% higherthan the average stresses away from the notch for a 90 N shear load. This stress rise is a direct outcome ofthe geometry of the notch.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Failure in composites often originates from stress concentrationregions in the structure. The onset of such a failure is not onlydetermined by the material properties of the composite but alsodepends on the lay-up and geometry of the stress raisers, for exam-ple, at the edge of circular cut-outs, notches, etc. This makes thetask of developing a general theory for predicting the damageinitiation in composites even more challenging. It is important tonote that the present computational strategies for predicting dam-age initiation and evolution, in the proximity of stress raisers, areeither inapplicable in the regions of stress singularities or veryexpensive computationally [1–4].

A number of strength-based macro-mechanical models havebeen proposed in the literature over the years for the predictionof failure in composites. The macro-mechanical failure modelstypically used are Tsai–Hill [5], Hashin’s [6] and Puck’s damage cri-teria [7]. However, the macro-mechanical failure models do not

capture the failure mechanisms at the fiber and matrix level. Themost discernible approach of studying damage at the microscopiclevel is to carry out a full-scale microscopic analysis of thestructure by explicitly modeling all the heterogeneities. A directconsequence of inclusion of the microstructural details in thewhole structure is the inevitable complexity and massive compu-tational cost. In order to reduce computational cost and to studythe microscale behavior of composites, various multiscale/multi-level methods are used, as illustrated in Fig. 1.

Multiscale methods offer the efficiency of macroscopic modelsand the accuracy of microscopic models. The scope of multiscalemodeling is the designing of more efficient combined computa-tional methods than the solving of the full microscopic model pro-viding microscopic information to the desired accuracy, at thesame time. To study microscale behavior of composites,researchers like González and Llorca [8], Totry et al. [1,9] havedemonstrated the use of a Representative Volume Element (RVE)for the prediction of damage mechanisms at the microscale offiber-reinforced composites. An RVE is a volume element that rep-resents the composite statistically and includes a sample of all theheterogeneities that are present in the composite. These studies

Multiscale/multi-level modeling

methods

Using a representative

volume element (RVE) [1,8,9]

Voronoi cell finite element method [2]

FE2 [3]

Sub-modeling method [12]

Localized Langrangemultiplier method [4]

Embedded cell approach [10]

Inter-scale theory [11, 13]

Partial homogenization [14]

Fig. 1. Multiscale/multi-level methods.

2 G. Soni et al. / Composite Structures 113 (2014) 1–11

[1,8,9] were focused on deformation and failure mechanisms of fi-ber-reinforced composites under multi-axial stress states. Appro-priate damage mechanisms were incorporated in the constitutivebehavior of the microstructural elements such as fiber, matrixand interface. The properties of the RVE should be independentof its position in the structure. Therefore, models based on RVEs of-ten employ periodic boundary conditions [1,8,9]. Ghosh et al. [2]proposed a coupled multiscale two-dimensional model for com-posites. Homogenized material coefficients were derived by per-forming the Voronoi Cell Finite Element Method (VCFEM)analysis using periodic boundary conditions on the base cell. Abase cell is the microstructure cell containing all the inclusionsand voids modeled exclusively. A finite element based multiscalemodel was introduced by Feyel [3] called the FE2 method. In thismethod, macroscopic analysis was performed using the homogeni-zation of periodic media. An RVE was assigned to each integrationpoint at the macroscopic scale and a separate finite element com-putation was performed at the microscale, simultaneously. How-ever, the assumption of periodicity is lost at the points close tofree edges or points of high stress concentration, and thus, the ap-proaches presented in the literature [1–3,8,9], cannot be used inthese areas. To address the issues related to the unsuitability ofperiodic boundary conditions at stress raisers, different approacheshave been developed which are discussed later in the paper.

Markovic and Ibrahimbegovic [4] proposed a two-scale compu-tational strategy for modeling the inelastic behavior of composites.The finite element method was applied to the structural scale aswell as on the microscale. The localized Lagrange multipliers wereused to derive the interface conditions between the micro andmacroscales. Gonzalez and Llorca [10] simulated the fracturebehavior of a notched fiber-reinforced composite sample undertension, where the fibers were aligned in the direction of loadingand perpendicular to the notch. Their method was based on anembedded cell approach. In front of the notch tip, where the dam-age is concentrated, actual fiber/matrix topology was modeledwhile the rest of the beam was represented by a transversally iso-tropic homogeneous solid.

Sun and Wang [11] proposed a multi-level approach in whichmicroscale results were obtained in a small area of interest. Inthe initial macro-analysis, the effective properties were derivedusing a suitable homogenization method, e.g., laminate theorieswere utilized to predict the effective (macroscopic) stress andeffective (macroscopic) displacement fields. Subsequently, in themicroanalysis, microscale results were obtained in the area ofinterest by employing macro-stresses and/or macro-displacementsas boundary conditions on the microscale domain. All the micro-structural details like fibers, matrix, interface, etc., were explicitlymodeled in the area of interest (for example, stress concentrationarea). The size of the local domain was selected on the basis of a‘‘local domain test’’. The sub modeling methodology available in

ABAQUS Standard� [12] was used for analyzing a part of the struc-ture with more accuracy using a finer mesh density. Therefore, thismethod can potentially be used to implement the idea of homoge-nization and de-homogenization [11].

However, at present, the sub modeling method uses constanttime step increments which limit its application to highly non-lin-ear problems. Wang and Yan [13] proposed an inter-scale theory tostudy matrix failures in composite laminates. This theory utilizedthe macro-field of the laminate obtained after homogenization torecover the microfield at the fiber and matrix scale (de-homogeni-zation). Borokovand Sabadash [14] have proposed a partial homog-enization approach for obtaining microstresses in the region ofinterest. They proposed an approach in which the microscale andmacroscale are coupled as a single domain and analyzed simulta-neously. Both the models proposed by Sun and Wang [11] andWang and Yan [13] give accurate results at the microstructuralscale only when the region of interest lies away from the bound-aries of the microdomain. This limitation is addressed by Borokovand Sabadash [14] and the present work follows upon the idea ofpartial homogenization to develop and implement the proposedmicro–macro approach.

In this paper, an attempt has been made to formulate a multi-scale modeling approach which can address the limitations of theexisting modeling approaches mentioned earlier and can be uti-lized to predict damage mechanisms in the regions of high stressconcentrations. The ideas of homogenization and de-homogeniza-tion by Wang and Yan [13] have been extended and combined withthe idea of partial homogenization provided by Borokov and Saba-dash [14] for the formulation of the micro–macro approach. Themicro–macro method however extends this idea to inelastic defor-mation while it was restricted to only linear deformations in theprevious studies [14]. Following Borokov and Sabadash [14],microscale and macroscale are coupled as a single domain and ana-lyzed simultaneously in the micro–macro approach. The micro-structure is enriched with all the details like fibers, interface, etc.The use of partial homogenization leads to less computations ascompared to multiscale methods described in the previous studies[2–4,10]. The drawback of the works which present local analysisvia multiscale methods [11,13] are also addressed, since the micro-structural solution remains accurate even at the boundaries of themicrodomain. The details of the formulation of the scheme and itsapplication for investigating damage mechanisms in regions near anotch root are outlined in the following sections.

The next section presents the locality principle on which themicro–macro method has been formulated. The details of the finiteelement geometry, material models and the boundary conditionsused in the model are mentioned in this section. The micro–macromodel is validated by comparing the predicted shear response of across-ply laminate against the experimental shear response ob-tained from V-notched rail shear test [15]. After model validation,

Mat

rix d

amag

e (%

)

Shear strain (%)

Complete micro

Submodeling

Micro-macro

Fig. 3. Comparison between full scale microstructural response and responsepredicted by micro–macro scheme and sub-modeling method.

G. Soni et al. / Composite Structures 113 (2014) 1–11 3

the effects of phase and interface properties in the microdomainhave been characterized. Finally, two different micro–macro mod-els have been developed to distinguish the damage initiation andpropagation near the stress raisers and away from it. In the firstcase, damage evolution near a notch root is studied whereas thesecond case deals with damage evolution away from the notchroot.

2. Formulation of the micro–macro approach based on thelocality principle

The locality principle states that the effect of homogenization ofthe structure does not influence the homogenized part farther thanthe characteristic length of the structure. Typically, the character-istic length is of the order of the unit cell containing a single fiber[14]. This principle can be used to model the microstructure in theregion of interest and the remaining area could use homogenizedor effective properties.

In order to confirm that the proposed micro–macro methodol-ogy based locality principle is valid, a prediction of localized dam-age in the composite via a micro–macro analysis is compared witha full-scale microstructural analysis. The damage in the epoxymatrix is modeled using Mohr–Coulomb plasticity and the accu-mulated plastic strain is taken as a measure of the damage.Fig. 2(a) shows a randomly distributed fiber array consisting of25 fibers which are surrounded by the matrix material. It alsoshows the center fibers around which the accumulated damagein the matrix is captured. Fig. 2(b) shows the corresponding par-tially homogenized lamina as per the proposed micro–macroscheme. Fig. 2(c) shows the mesh geometry of the micro–macroanalysis. A shear load in X–Y plane is applied on the lamina asshown in Fig. 2(b). Fig. 3 shows the damage in the epoxy matrixversus the volume averaged shear strain in the localized regionof interest for three different approaches, viz., complete micro-structural modeling, sub-modeling and micro–macro. Thetwo-step sub modeling approach has also been used predict thelocalized damage and compared with the proposed micro–macromethod. The solution time for the sub-modeling approach is dou-ble that of the micro–macro method since it requires two separateanalyses (one at macroscale and another at microscale). It can beseen that the micro–macro scheme prediction is in good agree-ment with the full-scale microstructural analysis and the two-stepsub-modeling.

Note that the micro–macro method evaluates the distribution ofstresses in the macrostructural and the microstructural domains via

Center fibers

(a) (

Lamina under sh

Fig. 2. Geometry of the (a) lamina with complete microstructure, (b) partially homogenizlamina.

a coupled concurrent analysis of both the domains. This methoduses partial homogenization in which the region other than the areaof interest (where the microstructure is explicitly modeled) ishomogenized. The basic difference between conventional multi-level methods, i.e., inter-scale theory and the partial homogeniza-tion method is that the microstructural and macrostructuralanalyses are not performed simultaneously in the multi-levelmethods. On the other hand, in the partial homogenization method,the two length scales are analyzed simultaneously. This leads to therequired coupling of the two scales, eliminating the need for twoseparate analyses. In the proposed scheme, initially, a region ofinterest (or a local domain) is identified in the structure, which istypically the region of a stress concentration, free edges, crack tips,etc. This region is modeled with the microstructure of the compos-ite, that is, randomly distributed fibers in the matrix. The rest of thestructure is modeled as a homogeneous continuum with effectiveproperties. Strong kinematic coupling is incorporated betweenthe local domain and the homogeneous continuum. This modifiedstructure, that is, the original structure along with the modified lo-cal domain is then solved and the required micro-stresses in the re-gion of interest are obtained. Fig. 4 illustrates the micro–macromethod. Fig. 4(a) shows the combined micro–macro model,Fig. 4(b and c) show details of the microdomain in the combinedmodel.

Note here that the most of the multi-level models reported inthe literature assume elastic homogenized properties for the

b) (c)

ear loading

ed lamina with the applied shear loading, (c) meshing of the partially homogenized

Fig. 4. Proposed micro–macro approach.

4 G. Soni et al. / Composite Structures 113 (2014) 1–11

macro-domain [11,13]. The global response for non-homogenousstrain rate cannot be studied via this approach; ply-level failurecan be incorporated by using quadratic failure criteria, such asHashin’s and Tsai–Hill criteria. Consequently, some form ofmacro-level failure can be incorporated.

The micro–macro approach described here will be used in thefollowing sections for developing a model to analyze the damagebehavior of the specimen used in V-notched rail shear test [15].In-plane loading was chosen for developing the model, since thedamage mechanisms are fairly complex under shear. The modelutilizes the Mohr–Coulomb criteria for simulating damage in thematrix, and a traction separation law for simulating fiber–matrixinterface failure. Damage mechanisms in the stress concentrationarea near the notch root can be investigated and compared againstthe damage mechanisms away from the notch root.

3. Finite element modeling

3.1. Model geometry

This scheme requires the modeling of both the microstructureand the macrostructure. The geometry of the structure is same asthe geometry of the specimen used in the ASTM-D7078 standard[15], as shown in Fig. 5(a). The [0/90] laminate specimen contains

76.0

56.0 30.6

12.7

38.0

2

1.8

(a)Fig. 5. (a) Specimen geomet

a double V-notch and a significant portion to either sides of theV-shaped notches is clamped tightly during the experiment and doesnot undergo deformation. Consequently, these sections are notincluded in the finite element model geometry to reduce the com-putational load. This can be seen in Fig. 5(b) which provides the de-tails of the geometry of the structure used in the FE model. Themicrostructure consists of two cubical cells with fibers in one celloriented orthogonally with respect to fibers in the other cell to rep-resent a [0/90] laminate. Each cell contains randomly distributedglass fibers in the matrix with a fiber volume fraction of 28%. Thefiber volume was measured as 28% post fabrication. Hence, themodel used a volume fraction of 28% to corroborate the experi-mental response. However, this is not a limitation of the model.The fiber diameter is 24 lm and 24 fibers are modeled within acube of 0.2 mm edge length. The remainder of the cube iscomposed of matrix material. The cohesive layer between the fiberand the matrix is modeled as a layer of cohesive elements withzero initial thickness. 8-Noded 3D cohesive elements were usedfor meshing this layer in ABAQUS Standard�. The mesh is createdby off-setting a layer of zero thickness from the mesh containingfiber and the matrix. The size of the cube can be determined bythe volume required to capture all the relevant microstructuraldetails. The detailed geometry of the microstructure is as shownin Fig. 6.

56.0

27.4

13.71

(b)

2 x 90 0

X R 1.3

ry, (b) model geometry.

Fig. 6. Geometry of the microstructural domain.

Table 1Elastic properties of the lamina.

E11 (GPa) E22 (GPa) G12 (GPa) #12

23.8 6.368 2.2 0.2804

Table 2Elastic properties of fiber and matrix.

Material E (GPa) G (GPa) #

Epoxy resin EPOFINE-556/fine hardener-951 4.7 2.08 0.3E-rowing glass, qL = 4800 kg/m3, d = 24 lm 73 29.67 0.23

G. Soni et al. / Composite Structures 113 (2014) 1–11 5

Fig. 7 shows a magnified view of the section of the structurewhich is cut by the plane of symmetry PQRS and contains the lineAB. The position of the slot with reference to the structure is clearlyillustrated by the cross sectional view of the slot in the plane PQRS.The slot is symmetric about the line AB and the plane PQRS. Thedimensions of the slot are 0.2 mm � 0.2 mm � 0.4 mm. The micro-structure is placed in that region as shown in Fig. 7.

3.2. Material and damage models

Each layer of the macrostructure is considered to be a homoge-neous unidirectional elastic lamina under plane stress conditionsand the stiffness values are provided in Table 1, obtained by usinglaminate theory. E-glass (ER-459L) fibers were modeled as linearelastic isotropic solids and their constants are given in Table 2.The epoxy matrix (EPOFINE-556) with FINEHARD-951 hardenerwas assumed to behave as an isotropic, elasto-plastic materialand its elastic constants are also provided in Table 2.

The plastic deformation is governed by the multi-axialMohr–Coulomb criterion. The Mohr–Coulomb criterion describedin Jiang and Xie [16] considers that yielding takes place whenthe shear stress, s, acting on a specific plane reaches a criticalvalue, which is a function of the normal stress, rn acting on thatplane, as indicated in Eq. (1). The corresponding yield surface,

Fig. 7. Geometry of the unified sp

written in terms of the maximum and minimum principal stresses(r1 and r3) is given by Eq. (2):

jsj ¼ c þ rn tan u ð1Þ

Fðr1;r3Þ ¼ ðr1 � r3Þ þ ðr1 þ r3Þ sin u� 2c cos u ð2Þ

where c and u are matrix cohesion and the matrix friction angle,respectively. The matrix tensile strength, rmt = 75 MPa andrmc = 105 MPa have been measured experimentally and used forthe calculation of c and u using Eq. (3):

ecimen and microstructure.

Fig. 9. Boundary conditions on the finite element model for in-plane shear.

Mesh seeds Microstructure

6 G. Soni et al. / Composite Structures 113 (2014) 1–11

rmt ¼ 2ccos u

1þ sin uand rmc ¼ 2c

cos u1� sin u

ð3Þ

The fiber–matrix interfacial decohesion was simulated usingstandard cohesive surface elements in the ABAQUS Standard�

[12]. The mechanical behavior of the interface was simulated usinga traction–separation law which relates the displacement acrossthe interface to the force vector acting on it. In the absence ofany damage, the interface behavior was assumed to be linear witha high value of initial stiffness, K, to ensure displacement continu-ity at the interface. The linear behavior ends at the onset of dam-age, using a maximum stress criteria expressed as:

maxhtniN

;ts

S

� �¼ 1 ð4Þ

where tn and ts are normal and tangential tractions transferred bythe interface, respectively. N and S are the normal and tangentialinterfacial strengths, assumed to be equal, for simplicity. After theonset of damage, the damage evolution of the interface was dis-placement controlled and linear, as shown in Fig. 8.

Fracture energy, C0, is another parameter which controls theinterface behavior other than cohesive strength (N, S). It is definedas the total energy needed for complete decohesion of the inter-face. The interface failure model assumes that the energy con-sumed during the fracture of the interface is independent of theloading path. The fracture energy, C0 is described as

C0 ¼ 12

tDd ð5Þ

where t (tn or ts) is the cohesive strength of the interface and ‘Dd’ isthe total change in displacement across the interface. The interfaceproperties for the material system have not been explicitly reportedin the literature. The energy necessary for complete decohesion hasbeen chosen as 100 J/m2 for all the simulations in the present work,which is a reasonable range for glass fiber/epoxy matrix compositelaminate, as reported by Zhou et al. [17].

3.3. Boundary and contact conditions

The boundary conditions applied on the structure are illustratedin Fig. 9. Since the loading applied on the specimen is purelyin-plane shear, the face ABPQ is fixed and Ux is set to zero on theopposite face DCRS. The structure is loaded by giving some arbi-trary positive displacement d0 to face DCRS and setting Uy to d0.Displacement controlled loading has been used such that the loadcarried by the structure decreases as the structure fails whichallows for a slower rate of failure.

The connection between the microstructure and macrostruc-ture has been established through kinematic constraints. The con-straint type used in this work is a surface-based tie constraint [12],

K

1

δ δ

tn, ts

δmax

t

Fig. 8. Standard traction–separation law.

in which each node on the first surface (the slave surface) will havethe same values for its degrees of freedom as the point on the sec-ond surface (the master surface) to which it is closest. The mastersurfaces are selected from the macrostructure geometry, whereas acorresponding slave surface is selected from the microstructuregeometry.

3.4. Mesh sensitivity analysis

The mesh density around the microstructure affects the accu-racy of the results obtained at the microstructural level. Fig. 10shows the mesh seeds around the microstructure which can bevaried in number to achieve a finer mesh in the correspondingregions. The number of elements in the structural mesh was pro-gressively increased to establish the convergence in the results,as shown in Fig. 11.

It can be clearly seen from Fig. 11 that the stress–strain curvesconverge when the mesh density around the microstructure isincreased. The minimum number of mesh seeds required for thesolution to converge is 30. Consequently, the simulations in this

Fine mesh (macrostructure)

Coarse mesh (macrostructure)

Fig. 10. Mesh seeds in the microstructure.

Fig. 12. Comparison between experimental response and response predicted bymicro–macro scheme.

G. Soni et al. / Composite Structures 113 (2014) 1–11 7

work were carried out by keeping the number of mesh seeds fixedat 30, as no appreciable improvements in the results have beenobserved by increasing mesh density any further. SC8R continuumshell elements were used to discretize the macrostructure. C3D8R3D solid elements were used for discretizing the fiber and thematrix. COH3D8 cohesive elements were used for the modelingof the fiber–matrix interface. A total number of 142,526 elementsand 157,265 nodes have been used in the microscale model.

4. Model validation

Throughout this section, s12 and e12 refer to the in-plane shearstress and in-plane shear strain averaged over the volume of themicrostructure, unless mentioned otherwise. Volume averagedstresses and strains are determined by Eqs. (6) and (7).

Sij ¼1V

ZV

sijdV ð6Þ

Eij ¼1V

ZV

eijdV ð7Þ

where Sij, Eij are the volume averaged stresses and strains respec-tively and sij, eij are the microscopic strains in a microdomain whosevolume is V.

Since the strain gauges measure average strains over arelatively large area as compared with the micro-domain ofmicro–macro method, the experimental response depicts theaverage values of the strains averaged over a large area(5 mm � 5 mm). In order to compare this experimental responsewith the micro–macro prediction where the size of the micro-domain is 0.2 mm3, it is assumed that strain fields are uniform overa small macroscopic region near the center. Consequently, thepoint based value obtained in the model should be equal to theaverage value obtained from the strain gauges. Fig. 12 shows thatthe shear response predicted by the micro–macro scheme plottedas s12 versus e12 is in good agreement with the experimental re-sponse [18]. The curve showing experimental response illustratesthe in-plane shear response of the rail shear specimen testedaccording to ASTM Standard D7078 [15]. For doing micro–macrosimulations, the matrix friction angle, u, and matrix cohesivestrength, c, are determined as 10� and 44.7 MPa, respectively, fromEq. (3). The interface stiffness is chosen to be 10 GPa/mm, theinterface strength as 30 MPa and the interface energy as100 J/m2.As shown in Fig. 12, the linear portion of both the predicted as wellas experimental response extends up to 1.5% strain. The maximumprediction error is limited to�8%. The difference in the experimental

Fig. 11. Mesh convergence achieved by varying the number of mesh seeds.

response can be attributed to the assumptions in the model anderrors in measurement and the variation in the properties of theconstituents due to local defects and in-homogeneities. Once thematrix starts yielding, the response becomes non-linear in nature.The transition between the linear and the non-linear regimes is rel-atively smooth in the experimental data. However, the transition ismore pronounced in the model and the non-linear response com-mences at a strain of 1.5%. This sudden transition could be ex-plained by the initiation of matrix yielding. After modelvalidation, a parametric study was performed to study the effectof various phases and interface properties on the stress–strain re-sponse of the microstructural domain.

5. Effect of the phase and interface properties of themicrodomain

Interface and phase properties can affect the damage initiationand progression. This section is focused on quantifying the effectsof interface and phase properties which can affect the damage re-sponse of the microstructural domain. It has been observed inauthors’ previous work [18] that for the current material system(volume fraction and/or phase and interface properties) the effectof delamination is negligible on both local and global response.As a result, the effect of delamination was not explicitly consideredin the present model. However, it can be easily implemented in themodel for other loading conditions where delamination may besignificant.

5.1. Effect of interface fracture energy

The effect of the variation of fracture energy is discussed in thissection. The interface energy is a difficult quantity to measureexperimentally and the values can vary within an order of magni-tude depending upon the test method. The values for the fractureenergy reported in the literature lie anywhere between 1 J/m2

and 100 J/m2 [17]. The proposed approach can potentially capturethe effects of variations in the material properties which can bevery useful for design purposes. One can estimate the variationin the local response for the lower and upper bounds of the inter-facial energy and other parameters that can affect the interfacedamage and the matrix damage occurring in a local region. Thefracture energy is determined by the relationship given in Eq. (5),which indicates the amount of energy released during the failureof the interface. The strength and the stiffness of the interface

Fig. 13. Effect of changing fracture energy of the fiber–matrix interface on shearstress–strain response.

Fig. 15. Effect of changing matrix friction angle on shear stress–strain response.

8 G. Soni et al. / Composite Structures 113 (2014) 1–11

are kept constant with values equal to 30 MPa and 10 GPa/mm,respectively. The matrix friction angle u is kept as 10�. It can beseen from Fig. 13 that increasing the fracture energy reduces theamount of interfacial damage at a particular strain value. Atconstant strength and stiffness, increasing the fracture energy in-creases the separation, d, for the final fracture of the interface. Con-sequently, the completion of the debonding process (finalseparation) is delayed, which ensures high stress values at thesame strain for higher fracture energy. Fig. 14 shows the contourplots of interface damage at 8% shear strain. The legend indicatesthe value of the variable, which detects the final fracture of theinterface. The variable can take any value between zero and one.A value of less than one indicates that the damage has initiated,while a value of one indicates the completion of the debondingor decohesion process. It can be seen in Fig. 14 that the contourplot of the laminate with a fracture energy equal to 10 J/m2 showsa significant amount of completely damaged interfaces; whereas,the laminate with fracture energy equal to 100 J/m2 does not showcomplete fracture anywhere. This is an expected result (asdiscussed in this section) because the amount of interface damagedecreases with increasing fracture energy.

5.2. Effect of matrix friction angle

Other than the interface fracture energy, it is expected that thematrix properties may also affect the response of the microdomain.

Fig. 14. Contour plots of interfacial damage for fractu

The shear stress–strain responses of the microdomain predicted bythis micro–macro scheme are shown in Fig. 15. The interface prop-erties are kept fixed interface stiffness as 10 GPa/mm, interfacestrength as 30 MPa and interfacial energy as 100 J/m2. The matrixfriction angles have been changed from 5� to 15�, which are com-monly reported values of friction angle for the epoxy matrix [8].Since the matrix tensile strength was assumed to be constantand equal to 75 MPa, changes in the friction angles modifies thecohesive strength ‘c’ of the matrix as given in Eq. (3). The cohesivestrength of the matrix increases from 37.5 MPa (u = 0�) up to48.8 MPa (u = 15�). It can be seen from Fig. 15 that the onset ofnon-linear behavior is affected by changing the friction angle incomposites. A lower value of the friction angle leads to an earlieronset of matrix yielding because the cohesive strength reduceswhen the friction angle is decreased and, therefore, the resistanceto shear deformation decreases.

6. Application of micro–macro analysis for characterization ofstress raisers

The first step in performing the micro–macro analysis of anystructure is to identify the regions of interest based on the stressdistribution in a continuum model of the structure. Fig. 16 showsthe von-Mises stress distribution in a double V-notch cross-plylaminate under shear loading. The stress concentration regioncan be seen near the notches. Fig. 16 also shows the location of

re energies (a) 10 J/m2, (b) 100 J/m2 at 8% strain.

Fig. 16. von-Mises stress (MPa) distribution in a continuum model showing theregions of interest, i.e., regions near and away from the notch.

Fig. 17. Volume averaged shear stress as a function of applied shear load.

G. Soni et al. / Composite Structures 113 (2014) 1–11 9

the two microscale domains selected on the basis of the stressdistribution. One microscale domain is placed below the notch atthe top and the second one is placed away from the notch so thatthere is no stress concentration near the second microscale do-main. The idea behind placing two microscale domains is that

Fig. 18. Contour plot of von-Mises stress (MPa) in the matrix of cross-

the results from both cases will clearly elucidate the differencesin the damage initiation and propagation due to a stress raiser.

The effect of the stress-raiser in the microdomain is fairlypronounced if the shear stresses are plotted as a function of theapplied shear load as shown in Fig. 17. It can be seen that if themicrodomain lies in the stress raiser (near the notch root) the vol-ume averaged shear stress developed is 37.5% higher than the aver-age stresses developed in the region away from the stress raiserunder a 40 N shear load. The volume averaged stresses in the stressraiser increases multifold (�1.45 times the stresses developed inthe region away from the stress raiser) if the applied shear loadis increased to 90 N. Hence, it can be interpreted that the micro–macro approach is useful in characterizing damage initiation andpropagation in the hot zones, such as notches, holes, free edgesand geometric discontinuities.

It is important to note here that the difference between stressesat two hotspots, i.e., one close to the notch and another away fromthe notch, can be quantified through macro model analysis. How-ever, to quantify the damage initiation and evolution in terms ofaccumulated matrix damage accumulated and fiber–matrixdebonding the analysis of micro-domain is essential. These micro-scopic damage mechanisms distinctly elucidate the differences inthe behavior of regions close to stress singularities with thoseaway from such singularities. It may be noted that this knowledgecan be used for enhanced design/reinforcement at the hot spots.

6.1. Evolution of stresses and strains in microdomain

It may be noted that the accumulated plastic strains and stres-ses in the microdomains under the notch and away from it canpotentially provide information about damage initiation and prop-agation as shown in Totry et al. [19]. As seen previously in Fig. 17,at a given shear load, the microdomain near the notch has highervolume averaged shear stresses as compared to the microdomainaway from it. Consequently, it is expected that the stress evolutionin the microdomains in the stress raiser and away from it will bedifferent. To characterize the stresses in the two microdomains,the contour plots of von-Mises stress at a 90 N shear load areshown in Fig. 18. The maximum von-Mises stress in the regionnear the notch root is 82% higher than the maximum von-Misesstress induced in the region away from the notch root. This clearlyshows that the effect of stress raisers on the micro-stresses can becharacterized effectively via the micro–macro approach. Note thatFig. 19 shows the accumulated plastic strain in the matrix. It can beseen that the plastic strain accumulates in the form of shear bands

ply laminate at (a) notch (b) away from the notch at a 90 N load.

Fig. 19. Contour plot of accumulated plastic strain in the matrix of cross-ply laminate at (a) notch (b) away from the notch at a 90 N load.

Fig. 20. Comparison of evolution of the percentage of matrix elements damaged inthe microstructure between the notch and away from it.

10 G. Soni et al. / Composite Structures 113 (2014) 1–11

in the matrix and is not uniformly distributed in the matrix. Theseshear bands run parallel to the fibers and interact with each other,which leads to deformation in the matrix. It is seen that signifi-cantly high plastic strains are induced at the same load in the stress

Fig. 21. Contour plot of damage of the fiber–matrix interface (a) notch (b) away fro

raiser and the maximum accumulated plastic strain in the stressraiser is 160% higher than the maximum accumulated plastic straininduced in the microdomain of the region away from the notch.Note that stresses and strains at the boundaries of the micro-domain may be problematic, however as mentioned in Section 2,that homogenization of the structure does not influence the homoge-nized part farther than the characteristic length of the structure.The stresses and strains predictions away from the boundary arerelatively accurate.

6.2. Damage evolution at the microdomain

Besides evaluating the stresses and strains, the micro–macromethod can also be used for characterizing the damage evolutionin the matrix. Fig. 20 shows the evolution of the damage in the ma-trix as a function of the shear load for both the microdomains (inthe vicinity of the notch root and away from it). It is observed thatthe damage in the matrix is higher at the notch as compared toaway from the notch at every instance. This is expected becausewhen the von-Mises contour plots of Fig. 18 show that at any loca-tion, the magnitude of stresses at the notch are higher; this typi-cally gives rise to larger plastic deformation of the matrix.Damage in matrix material at notch is �10% higher as comparedto damage in matrix for away from the notch at a 90 N load.

The damage at the fiber–matrix interface is plotted in Fig. 21. Itcan be observed that the interface damage is more pronounced in

m the notch at a 90 N load (the value 1 indicates total failure of the interface).

G. Soni et al. / Composite Structures 113 (2014) 1–11 11

the microdomain at the notch root as compared to the microdo-main away from the notch. The maximum damage value increasesby 75% due to the presence of a stress raiser. It can also be observedthat the fiber–matrix decohesion progresses from the center to theedges of the fibers in both the cases.

7. Conclusions

A robust micro–macro multiscale scheme has been proposed inthis work which can be applied to predict local response andmicroscale damage in laminates under multi-axial loading. Thispaper is focused on using this scheme for predicting microscopicstress fields for shear loading of cross-ply laminate. It has beenfound that the response to in-plane shear loading is greatly influ-enced by the properties of the phase and interface. Furthermore,this work investigated the effect of the stress-raiser and the localresponse in the microdomain near notch root of a rail shear spec-imen used in ASTM Standard D7078. The response has been com-pared with the local response of the microdomain away from thenotch root and significant differences in stress evolution and dam-age behavior have been found. The key findings can be summa-rized as follows:

� The predicted response with proposed micro–macroscheme is in good agreement with the experimentalresponse and other established multiscale techniques, suchas sub-modeling.

� Interfacial fracture energy has significant effect on thematerial response. Increase in fracture energy enhancesthe plastic response of the composite laminate. The exper-imental results indicate that interfacial fracture energy of100 J/m2 is appropriate for capturing the constitutiveresponse in the plastic zone.

� A lower value of the matrix friction angle leads to an earlieronset of matrix yielding causing significant reduction instiffness of the material.

� It has been observed that 37.5% higher volume averagedshear stress is developed in the micro-domain near thenotch root as compared average stress developed in themicro-domain away from the notch at 40 N shear load.The volume averaged stress value in the stress raiser is�1.45 times the average stress developed in the regionaway from the stress raiser if the applied shear load isincreased to 90 N.

� The maximum von-Mises stress and accumulated plasticstrains in the region near the notch root are 82% and160% higher than the maximum von-Mises stress inducedin the region away from the notch root, respectively.

� Damage in matrix material at notch is �10% higher as com-pared to damage in matrix far away from the notch at a90 N load.

� It can also be observed that the fiber–matrix decohesionprogresses from the center to the edges of the fibers in boththe cases. Maximum interfacial damage is 75% higher in theregion near the notch as compared to the region away fromthe notch.

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