Mechanics of Materials 99 (2016) 37–52

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Mechanics of Materials

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

Research paper

Micromechanical modeling and characterization of damage evolution

in glass fiber epoxy matrix composites

Zhiye Li a , Somnath Ghosh

b , ∗, Nebiyou Getinet c , Daniel J. O’Brien

c

a Department of Civil Engineering, Johns Hopkins University, Baltimore, MD 21218, United States b Department of Civil, Mechanical and Materials Science & Engineering, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, United States c Composite and Hybrid Materials Branch, U.S. Army Research Laboratory, Aberdeen, MD 21001, United States

a r t i c l e i n f o

Article history:

Received 6 March 2016

Revised 8 May 2016

Available online 20 May 2016

Keywords:

Glass fiber

Epoxy matrix

Non-local continuum damage mechanics

Strain-rate dependent

Cohesive zone models

RVE

a b s t r a c t

This paper develops an experimentally calibrated and validated 3D finite element model for simulating

strain-rate dependent deformation and damage behavior in representative volume elements of S-glass

fiber reinforced epoxy-matrix composites. The fiber and matrix phases in the model are assumed to be

elastic with their interfaces represented by potential-based and non-potential, rate-dependent cohesive

zone models. Damage, leading to failure, in the fiber and matrix phases is modeled by a rate-dependent

non-local scalar CDM model. The interface and damage models are calibrated using experimental results

available in the literature, as well as from those conducted in this work. A limited number of tests are

conducted with a cruciform specimen that is fabricated to characterize interfacial damage behavior. Val-

idation studies are subsequently conducted by comparing results of FEM simulations with cruciform and

from micro-droplet experiments. Sensitivity analyses are conducted to investigate the effect of mesh, ma-

terial parameters and strain rate on the evolution of damage. Furthermore, their effect on partitions of

the overall energy are also explored. Finally the paper examines the effect of microstructural morphology

on the evolution of damage and its path.

© 2016 Elsevier Ltd. All rights reserved.

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. Introduction

The utilization of glass-fiber epoxy-matrix based composites

n a variety military and commercial applications, e.g. rotor-craft

okes and prop-rotor blades, is substantial. Noteworthy among

hese are the S-glass fiber reinforced composites, containing mag-

esium alumino-silicate or borosilicate fibers, that are known for

heir high stiffness and strength to weight ratio, impact resistance,

nd durability under extreme temperature or corrosive environ-

ents. Design of these materials for various structural applications,

ubject to dynamic loading conditions require considerations of

complex mix of properties contributing to weight, performance

nd reliability. Robust modeling, accounting for microstructural de-

ails, as well as material and interfacial properties, is an indispens-

ble ingredient of the material design process. These models are

rucial in unraveling the underpinnings of microstructure-property

elationships.

The mechanical and damage response of fiber-reinforced poly-

eric composites depend on the microstructural morphology, as

∗ Corresponding author. Fax: +1 410 516 7473.

E-mail address: sghosh20@jhu.edu (S. Ghosh).

t

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ttp://dx.doi.org/10.1016/j.mechmat.2016.05.006

167-6636/© 2016 Elsevier Ltd. All rights reserved.

ell the material and interfacial properties. Damage mechanisms

re particularly sensitive to the local morphology, e.g. spatial dis-

ribution, size and interfacial strength. For dynamic conditions,

train-rate dependent material properties govern both the mechan-

cal and damage behavior. While rate-dependent material prop-

rties have been extensively investigated for metals over a wide

ange of strain-rates, there is a paucity of information on experi-

entally observed strain-rate effects on mechanical and failure be-

avior of reinforced composites. For glass-fiber epoxy matrix com-

osites, studies at a range of strain-rates have been conducted (e,g,

n Davies and Magee, 1975; Lifshitz, 1976; Okoli and Smith, 20 0 0;

taab and Gilat, 1995; Shokrieh and Omidi, 2009 ). Some of these

tudies demonstrated that while the elastic stiffness and failure

train are less sensitive to the strain rate, the dynamic failure stress

ould be 20 − 30% higher than the static failure stress. Also, larger

amaged regions have been observed with increasing strain-rates.

A variety of micro-mechanical computational models, using e.g.,

he finite element method, have been developed to predict de-

ormation and failure in composite micro-structures. A number of

hese models define representative volume elements (RVEs) or sta-

istically equivalent RVEs (SERVEs) of the microstructure as com-

utationally tractable reductions of the actual microstructure. A

ajority of damage studies in composite materials use unit cell

38 Z. Li et al. / Mechanics of Materials 99 (2016) 37–52

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models implying periodic repetition of single cells. Incorporat-

ing the material constitutive relations and damage mechanisms

along with appropriate boundary conditions, the micro-mechanical

RVE/SERVE problems are solved for deformation and failure behav-

ior, (e.g. in Voyiadjis et al., 2002 , Ghosh et al. (20 0 0) ). Continuum

damage mechanics (CDM) models have been used to represent the

evolution of defect distribution, where an effective damage vari-

able (either scalar or tensor) is used to depict material degrada-

tion leading to crack nucleation and propagation ( Chaboche, 1981;

Kachanov and Krajcinovic, 1987; Lemaitre and Chaboche, 1990;

Nemat-Nasser and Hori, 1999 ). Micro-mechanics-based CDM ap-

proaches have been used to analyze RVE failure in Lene and Leguil-

lon, 1982 , Chaboche et al. (1998) Jain and Ghosh (2008) , Chen and

Ghosh (2012) . On the other hand, initiation and growth of fiber-

matrix interfacial de-bonding has been modeled with the cohesive

volumetric finite element methods (e.g. in Needleman, 1992; Tver-

gaard, 1991; Costanzo et al., 1996; Geubelle, 1995 ). Two classes

of cohesive zone traction-separation laws have emerged, viz. the

potential-based model (e.g. in Park and Paulino, 2012 ) and non-

potential model (e.g. in Needleman, 1992; Lin et al., 2001 ). In

the potential-based models, the traction-separation relationships

across the crack surface are obtained from a potential function

that characterizes the normal and tangential fracture energy. The

two-dimensional Voronoi cell finite element method (VCFEM) has

been developed in Ghosh et al., 20 0 0; Swaminathan and Ghosh,

2006; Ghosh, 2011 for modeling large micro-regions of reinforced

composites undergoing interfacial decohesion and concurrent ma-

trix cracking. The VCFEM models are able to provide a good un-

derstanding of the effect of different morphologies and constituent

material properties on the overall cracking and failure behavior of

complex multi-inclusion micro-structures.

The present paper develops a 3D micro-mechanical finite ele-

ment model for simulating deformation and damage behavior in

RVEs of S-glass fiber reinforced epoxy-matrix composites under-

going rate-dependent loading. The overall objective is to develop

a modeling framework that can delineate failure characteristics of

this class of materials at the micro and meso-scales. These mod-

els can be subsequently used in hierarchical and concurrent multi-

scale models developed (e.g. in Ghosh et al., 2007; Ghosh, 2011;

Massart et al., 2007; Fish, 2013 ).

The fiber and matrix phases in the model are assumed to be

elastic with their interfaces are represented by potential-based and

non-potential, rate-dependent cohesive zone models. Specifically

a non-potential bilinear model ( Ortiz and Pandolfi, 1999 ) and a

potential-based PPR model ( Park and Paulino, 2012 ) are imple-

mented for interface characterization. Damage, leading to failure,

in the fiber and matrix phases is modeled by a rate-dependent

non-local scalar CDM model. The interface and damage models are

calibrated using experimental results available in the literature, as

well as from those conducted in this work. A limited number of

tests in this study are conducted with a cruciform specimen that

is fabricated to characterize interfacial damage behavior. Validation

studies are subsequently conducted by comparing results of FEM

simulations with experiments. Sensitivity analyses are conducted

to investigate the effect of mesh, material parameters and strain

rate on the evolution of damage. Furthermore, their effect on par-

titions of the overall energy are also explored. Finally the paper

examines the effect of microstructural morphology on the evolu-

tion of damage and its path.

2. Micro-mechanical model of the Representative Volume

Element

The representative volume element or RVE is defined as a

micro-scale sub-domain, on which volume average of variables are

taken to yield macro-scale model variables ( Drugan and Willis,

996; Kouznetsova et al., 2002; Kanit et al., 2003; Jain and Ghosh,

008; Chen and Ghosh, 2012 ). The choice of the RVE depends on

he material property of interest and can vary from one class of

roperties to another. Statistically equivalent RVEs have been de-

eloped from detailed characterization studies of non-uniformly

istributed micro-structures in Swaminathan et al., 2006, Swami-

athan and Ghosh, 2006 . In this section, the RVE of a uniformly

istributed unidirectional composite, shown in Fig. 1 , is simulated

or calibrating and validating the material constitutive and damage

roperties of the fiber-reinforced composite that will be modeled

n this paper. In Fig. 1 (c) the RVE is represented by a cuboidal ma-

rix containing a cylindrical fiber with a cohesive fiber-matrix in-

erface that is characterized by cohesive zone models ( Lee and Mal,

998; Jain and Ghosh, 2008; Chen and Ghosh, 2012 ).

Assuming periodicity of the RVE, the FEM model in Fig. 1 (c) is

ubjected to incremental periodic boundary conditions. Following

rocedures in Pellegrino et al., 1999; Segurado and Llorca, 2002 ,

he macroscopically applied strain increment on the RVE �E ij , is

btained by decomposing the displacement increment �U i on the

VE boundary into an RVE-averaged term and a periodically per-

urbed term, expressed as:

U i = �E i j X j + �U i (1)

he periodic displacement component �U i is equal for correspond-

ng nodes on opposite faces of the RVE, e.g. nodes n 1 and n 2 in

ig. 1 (b). Accordingly, the total displacements at the corresponding

ode-pair ( n 1 , n 2 ) are related in terms of the macroscopic strain

ncrement as:

( �U i ) n 2 − ( �U i ) n 1 = �E i j

[ (X j

)n 2

−(X j

)n 1

] (2)

here (X j

)n 1

and

(X j

)n 2

are coordinates of the node-pair on the

VE boundary. Details of implementation in commercial software

or the dynamic simulations is given by Wu and Koishi, 2009 .

he validity of applying periodical boundary condition for moder-

te strain-rates has been tested in Chen and Ghosh, 2012 .

.1. Constitutive and damage models for the fiber and matrix phases

n the microstructure

Constitutive models for the matrix and fiber phases, and mod-

ls representing damage in the matrix and fiber phases including

ber-matrix interfacial de-cohesion, are briefly discussed in this

ection. The fiber material typically considered in this study is

lass while the matrix is an epoxy material, discussed later. Both

hases are represented by an infinitesimal strain, linear elastic-

amage constitutive framework for their deformation and dam-

ge behavior. A finite rotation framework is adopted to allow for

he relative motion between multiple phases. In an incremen-

al/rate formulation, the coupled constitutive-damage relations for

n isotropic material are expressed in an un-rotated configuration

s:

˙ i j = C i jkl (D ) ε kl stress-strain relation (3a)

˙ i j = R ki ˙ σsp

kl R l j , ˙ ε i j = R ki ˙ ε

sp

kl R l j un-rotated stress, strain (3b)

i j =

∂x i ∂X j

= R ik U k j RU decomposition (3c)

i jkl (D ) = 2 G (D ) δik δ jl +

(K(D ) − 2

3

G (D ) δi j δkl

)elastic stiffness

(3d)

= E/ [2(1 + ν)] and K = E/ [3(1 − 2 ν)]

E = ( 1 − D ) 2 E 0 Damage-stiffness relation (3e)

Z. Li et al. / Mechanics of Materials 99 (2016) 37–52 39

ZY

X

(a) Macroscopic level (b) Microscopic unit cell (c) FEM mesh of RVE

n1 = - n2

Epoxy Matrix

Glass Fiber

continuous

n2

n1CZM

Interface

Fig. 1. Representation of a unidirectional composite domain and the RVE simulation model: (a) the microstructure with a periodically repetitive of unidirectional cylindrical

fibers, (b) RVE represented by a microstructural unit cell, and (c) finite element model of the RVE.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

σ/C

εR

ε /εR

S* = 2

S* = 7

S* = 50

S* = 100

Fig. 2. Stress-strain curve showing the effect of varying CDM parameters.

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σ ij and ε ij in Eq. (3a) are respectively the un-rotated Cauchy

tresses and infinitesimal strain components. They are derived by

otation of the respective spatial components σ sp i j

and ε sp i j

by the

rthogonal tensor R , ( R

T R = I ) , where R is generated by polar de-

omposition of the deformation gradient tensor. The fourth-order

tiffness tensor C ijkl is a function of the evolving damage variable D

nd E, ν , G and K are respectively the Young’s modulus, Poisson’s

atio, shear and bulk moduli. The continuum damage mechanics

r CDM-based scalar variable D provides a measure of reduction

f the stiffness with increasing damage. The relation in Eq. (3e) is

aken from Lemaitre and Chaboche (1994) to describe brittle dam-

ge, where E and E 0 are the elastic modulus in the damaged and

ndamaged states respectively. In Lemaitre and Chaboche (1994) ,

he evolution of D has been established to be a nonlinear function

f the equivalent strain and strain-rates, expressed as:

˙ =

{( ε /ε 0 )

S ∗ ˙ ε when ε ≥ ε D and

˙ ε > 0

0 when ε < ε D or ˙ ε ≤ 0

(4)

ere ε is equivalent strain and εD is a threshold value for dam-

ge initialization. S ∗ and ε 0 are material constants, which are de-

ermined from experimentally obtained stress-strain curves of fail-

re in Chocron et al. (2009) ; Fard (2011) . As shown in Fig. 2 , the

arameter S ∗ controls the shape of the dimensionless stress-strain

urve. An analytical relation for the evolving damage parameter D

nd the stress σ can be derived for the one-dimensional case with

nitial conditions D = 0 and ε = ε D = 0 , by integrating Eq. (4) and

sing Eq. (3d) as:

=

(ε

ε R

)S ∗+1

and σ = C ( D = 0 ) ε

[

1 −(

ε

ε R

)S ∗+1 ] 2

(5)

ere ε R is the rupture strain, defined in terms of ε 0 and S ∗ as:

R =

[( S ∗ + 1 ) ε 0

S ∗] 1

S ∗+1 (6)

= 1 when ε / ε R = 1 .

Since the objective of this work is to model strain rate-

ependent material behavior, the elastic moduli of the different

hases are considered to be rate-dependent. Following experimen-

al observations in Brown et al. (2010) , the undamaged elastic

oduli E 0 of both glass and epoxy are assumed to depend on the

train-rate ˙ ε according to the relation:

0 = E re f

(1 + C ln

˙ ε

˙ ε 0

)(7)

ref is a material dependent reference stiffness at an equivalent

train-rate ˙ ε 0 . C is a dimensionless material parameter that is cali-

rated from experiment data.

.1.1. Non-local damage representation

The local CDM in Eq. (4) is known to project mesh sensitivity

n the FEM solutions. Consequently, a non-local damage variable˙ nl ( x ) is used for damage representation at a point x , as:

˙ nl ( x ) =

1

V r ( x )

∫ V

α( s − x ) D ( s ) dV ( s ) where (8a)

r ( x ) =

∫ V

α( s − x ) dV ( s ) and α( x ) = exp

[−( κ| x | /l c )

2 ]

(8b)

Here α( x ) is a Gaussian distribution function that represents

he weighting of the damage variable ˙ D in a volume V around

he point x . κ is a constant that is taken to be (6 √

π) 1

3 from

azant and Pijaudier-Cabot (1988) ; Eringen (1972) ; Bazant and Ji-

asek (2002) . The non-local damage rate at a point x is used to

pdate the damage value at that point that is used in Eq. (3e) . The

arameter l c in Eq. (8b) is a material characteristic length that is

f the same order as the size of microstructural inhomogeneities

Bazant and Pijaudier-Cabot, 1988 ). In this paper, half of an aver-

ged inter-fiber distance in the RVE is evaluated as:

c =

1

ND

ND ∑

i =1

IF D (i ) (9)

here IFD ( i ) is the inter-fiber distance and ND is the number of

ber-pairs in the RVE used for distance measurement. This rela-

ion can be used to derive characteristic lengths for different fiber

rrangements in the microstructural RVE as:

c =

1

8

[4 ( L RV E − d ) + 4

(√

2 L RV E − d )]

uniform: Fig. 1(c)

(10a)

c =

√

2

2

L RV E − d hexagonal: Fig. 10(a) (10b)

Here L RVE is the dimension of the RVE and d is the diameter of

bers. The validity of the relations in Eqs. (10) will be further es-

ablished through numerical parametric studies in Section 4.2 . The

40 Z. Li et al. / Mechanics of Materials 99 (2016) 37–52

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calibrated values of all damage parameters for borosilicate fiber in

EPON 828/PACM matrix are listed in Table 6 .

2.2. Fiber-matrix interfacial de-bonding using Cohesive Zone Models

Two types of 3D cohesive zone models or CZMs are explored

for their suitability with respect to the materials and interfaces be-

ing modeled for fiber-matrix interfacial de-bonding. These are dis-

cussed next.

2.2.1. Bilinear CZM

A bi-linear cohesive zone model has been proposed in Ortiz

and Pandolfi (1999) , for which the norm of the traction vector

T (=

√

T 2 n + T 2 t1

+ T 2 t2

) is relate d to the effective interfacial separa-

tion δ(=

√

δ2 n + β2 δ2

t ) , where δt =

√

δ2 t 1

+ δ2 t 2

. T n , T t 1 , T t 2 are re-

spectively the normal and tangential components of the interfa-

cial traction vector. The corresponding components of the interfa-

cial separation vector are the normal opening δn and the tangential

components δt 1 , δt 2 in the interfacial plane. β is a shear factor that

determines the contribution of the tangential component to the ef-

fective displacement jump, assumed to be β = 1 in this work. The

bi-linear traction-separation relation is expressed as:

T =

{ σmax

δc δ ∀ δ ≤ δc (hardening region)

σmax

δc −δe (δ − δe ) ∀ δc < δ ≤ δe (softening region)

0 ∀ δ > δe (complete de-bonding)

(11)

In the softening region, the unloading path is from the current po-

sition to the origin, with a reduced stiffness that different from the

hardening region stiffness.

T =

σmax

δmax

δmax − δe

δc − δe δ ∀ δc < δmax < δe and δ < δmax (12)

Reloading in the softening region proceeds along the unloading

path till it reaches the softening curve. The total traction van-

ishes at complete separation when δ ≥ δe . Beyond this separation,

a large penalty stiffness spring is introduced under compression

between interfacial node-pairs in the normal direction, to prevent

penetration.

2.2.2. Potential based PPR CZM

A generalized potential based cohesive zone model, termed as

the PPR model, has been developed in Park and Paulino (2012) .

The PPR model is suitable for mixed-mode cohesive fracture and

incorporates physical parameters such as fracture energy, cohe-

sive strength and shape of cohesive interactions. Path depen-

dence of work-of-separation with respect to proportional and non-

proportional paths have shown consistency of the cohesive con-

stitutive model for arbitrary mixed-mode loading conditions. This

model introduces a cohesive fracture potential as:

( �n , �t ) = min (φn , φt )

+

[�n

(1 − �n

δn

)α(m

α+

�n

δn

)m

+ 〈 φn − φt 〉 ]

·[

�t

(1 − | �t |

δt

)β(n

β+ | �t | δt

)n

+ 〈 φt − φn 〉 ]

(13)

where φn and φt are normal and tangential fracture energies, �n

ans �t are the normal and tangential separations along fracture

surface with �t =

√

�2 t1

+ �2 t2

, and δn and δt are normal and tan-

gential final crack openings. The energy constants �n and �t are

functions of the normal and tangential fracture energy. The non-

dimensional exponents m and n are associated with the initial

lope, α and β are shape parameters and 〈 · 〉 is the Macauley

racket.

The normal and tangential traction forces have been obtained

rom the derivatives of the potential function with respect to

he normal and tangential separation displacements in ( Park and

aulino, 2012 ) as:

T n ( �n , �t )

=

�n

δn

[ m

(1 − �n

δn

)α(m

α +

�n

δn

)m −1 − α(1 − �n

δn

)α−1 (m

α +

�n

δn

)m

] ·[ �t

(1 − | �t |

δt

)β(n β

+

| �t | δt

)n + 〈 φt − φn 〉 ]

(14)

T t ( �n , �t )

=

�t

δt

[ n

(1 − | �t |

δt

)β(n β

+

| �t | δt

)n −1 − β(1 − | �t |

δt

)β−1 (n β

+

| �t | δt

)n ]

·[ �n

(1 − �n

δn

)α(m

α +

�n

δn

)m + 〈 φn − φt 〉 ]

�t | �t | (15)

he traction values at the critical separation δnc and δtc , respec-

ively correspond to the maximum cohesive strength T n ( δnc , 0 ) =max and T t ( δtc , 0 ) = τmax . Parameters λn and λt are initial slope

ndicators. Several properties of the PPR formulation have been

iven in ( Park and Paulino, 2012 ).

.2.3. Rate effects on CZM parameters

The cohesive parameters and the fracture energy depend on the

train-rate in rate dependent cohesive zone models. In ( Wei et al.,

013 ) the maximum traction T max and the final crack opening δe

ave been expressed in terms of the loading rate as:

max = T re f max

(1 + η ln

(˙ ε

˙ ε re f

)); δe = δre f

e

(1 + η ln

(˙ ε

˙ ε re f

))(16)

here ˙ ε re f , T re f

max , δre f e , and η are the reference effective local cohe-

ive separation rate, initial cohesive strength, initial critical open-

ng, and a rate-sensitive parameter, respectively. This rate depen-

ence is implemented in the present work. In the PPR model,

he normal and tangential final crack openings δn and δt in

ection 2.2.2 are governed by the same rate-dependent form as δe

n the second equation of Eq. (16) .

. Computational model calibration and validation with

xperiments

In advance of predicting material damage evolution in the com-

utational model of the fiber reinforced composite RVEs, compar-

sons are made with experimental results to calibrate and validate

he constitutive and damage models. Two experimental studies,

iz. a cruciform test and a droplet test, are considered for validat-

ng the bilinear and PPR cohesive models.

.1. Cruciform test with a single embedded fiber

The cruciform test has been developed and used in Warrier

t al. (1996) ; Tandon et al. (1999) ; Tandon et al., 20 0 0 ; Ghosh et al.

20 0 0) to evaluate quasi-static failure of a fiber-matrix interface.

nder a finite stress field, the ‘arms’ of the cruciform promote in-

erfacial de-bonding due to high stresses in the central region of

he specimen. In contrast, for a rectangular specimen with a lon-

itudinal fiber, the interface cannot be reliably characterized, since

tress singularities at the intersection of the fiber-matrix interface

nd the free surface always cause de-bonding to initiate at this lo-

ation.

Z. Li et al. / Mechanics of Materials 99 (2016) 37–52 41

Fig. 3. Experiments with the cruciform specimen: (a) schematic of the specimen, (b,c,d) photo of the apparent crack height h for loading forces F = 890 N, F = 1560 N, and

F = 2220 N respectively, (e) schematic of the cross-section showing de-bonding angle evaluation scheme.

Table 1

Dimensions of the cruciform specimen

in Fig. 3 (a).

units H l w t d

( mm ) 70 55 20 10 5

Table 2

Material parameters for the cruciform experiment.

Young’s Modulus (GPa) Poissons ratio Density(g/cc)

EPON 828 epoxy 2 .5 0 .4 1 .17

Borosilicate glass 64 0 .2 2 .23

c

T

e

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C

A

a

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ε

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2

F

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a

�

o

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a

Cruciform specimens, as shown in Fig. 3 (a), are fabricated by

asting in a silicone rubber mold with dimensions given in Table 1 .

he fillets at the cross junctions simplify the molding process and

ase stress concentration in the matrix at corners. The model fiber

s a borosilicate glass rod that is suspended in the mold via inte-

rated supports on either side of the cruciform. The resin used to

ast the specimen is a bisphenol A epoxide (EPON 828, Hexion Inc.,

olumbus, OH) cured with a cycloaliphatic amine (Amicure PACM,

ir Products Inc, Allentown, PA). This epoxy is optically clear and

llows for visualization of the interfacial de-bonding process. The

lass rod is coated with polysiloxane-based release agent prior to

lacing it in the mold, which results in a weaker bond between

he glass fiber and the epoxy matrix, promoting interfacial fail-

re. A camera records the stable growth of the de-bond crack. The

anufacturer provided elastic material properties for the fiber and

poxy matrix are listed in Table 2 .

In the experimental set up shown in Fig. 3 (a), the cruciform

pecimen is fixed at the bottom, while the top end is stretched in

ension at speed v = 0 . 381 mm/min. The force F on the specimen

s recorded for every displacement increment on the loading end

top). The average applied stress is obtained by dividing the ap-

lied force by the area of the top surface, i.e. σa v e = F / ( wt ) , where

and t are the cross-sectional dimensions shown in Fig. 3 (a).

urface displacements are measured by the digital image correla-

ion (DIC) technique and used to calculate the engineering strain

eng A

in the central location ‘A’ on the surface of the specimen in

ig. 3 (a). Additionally, the extent of de-bonding failure is recorded

y illuminating the interface at an oblique angle and observing

he reflection upon the appearance of a crack ( Bechel and Tandon,

002 ). The apparent crack height h is measured for loading forces

= 890 N, F = 1560 N and F = 2220 N respectively, as shown in

ig. 3 (b–d). The crack height is used to estimate the de-bonding

ngle in Fig. 3 (e), using the formula:

= 2 · arccos ( 1 − 2 h/d ) (17)

Initiation of fiber-matrix de-bonding is identified from images

f the illuminated interface, as well as change in the slope of the

tress-strain curve from the DIC readings in the central portion

f the cruciform. Only a thin outline is observed for F = 890 N

ith no obvious sign of damage in Fig. 3 (b). For F = 1560 N, a

lear crack is observed with easy-to-measure apparent crack height

. This indicates that de-bonding has started. h is slightly higher

or F = 2220 N in Fig. 3 (d). Fig. 5 (a) shows a plot of the aver-

ged stress σ ave as a function of the local engineering strain ε eng A

ecorded by the DIC system on the specimen surface. The corre-

ponding strains are also computed by a FE model of the cruciform

hown in Fig. 4 . The cruciform experiment is repeated four times

ith the same applied strain-rate, and Fig. 5 (a) shows the variabil-

ty in the experimentally calculated force values resulting from the

ncertainty of DIC measurements. The smooth change in the slope

orresponds to a relatively stable de-bonding process and a steady

ecline in the specimen stiffness. When the interfacial crack ceases

o grow, the stiffness stabilizes and the loading is terminated be-

ore matrix cracking.

.2. Calibration and validation of the FE model with cruciform test

esults

Cohesive zone parameters in Sections 2.2.1 and 2.2.2 are cal-

brated by a genetic algorithm or GA based scheme ( Deb et al.,

002 ), minimizing the difference between the FE and experimen-

al results. The GA implementation follows steps given in Carroll

1996) . A detailed FE model of the single-fiber cruciform specimen

s developed, as shown in Fig. 4 (a) for the calibration process. The

A-based calibration process evaluates the set of CZM parameters

y minimizing the difference in the data extracted from the σ ave -

eng A

plots obtained from the cruciform experiments and those gen-

rated from FE simulations. The resulting calibrated CZM parame-

ers for the bilinear and PPR models are given in Table 3 .

Results of the cruciform FE simulations with calibrated CZM pa-

ameters are shown in the contour plots of Figs. 4 (b) and 4 (c), and

ompared with experiments in Fig. 5 . Fig. 5 (a) plots the macro-

copic stress-strain results, while the local de-bonding angle � as a

unction of loading stress σ ave is depicted in Fig. 5 (b). The abscissa

n Fig. 5 (a) is the local engineering strain ε eng A

recorded by the DIC

ystem on the specimen surface in Fig. 3 . The corresponding FEM

trains are calculated in the element at the location A. The stress-

train plots in Fig. 5 (a) show good agreement with the experimen-

al behavior. The de-bonding angle � in Fig. 5 (b) increases rapidly

fter initiation with little additional loading, and subsequently

42 Z. Li et al. / Mechanics of Materials 99 (2016) 37–52

(a)

(Avg: 100%)S, Mises

+0.000e+00+6.376e+00+1.275e+01+1.913e+01+2.550e+01+3.188e+01+3.826e+01+4.463e+01+5.101e+01+5.739e+01+6.376e+01+7.014e+01+7.651e+01

(b)

(Avg: 100%)S, Mises

+0.000e+00+1.237e+00+2.473e+00+3.710e+00+4.947e+00+6.184e+00+7.420e+00+8.657e+00+9.894e+00+1.113e+01+1.237e+01+1.360e+01+1.484e+01

(c)

Fig. 4. (a) FE model and mesh of cruciform experiment in tension; contour plot of the von-Mises stress: (b) through the mid-section of the cruciform, and (c) on the

interfacial cohesive elements showing the de-bonded region, at de-bonding angle � = 120 ◦ .

0

4

8

12

16

20

0 0.25 0.5 0.75 1

σ ave

(M

Pa)

εengA (%)

ExperimentBilinear

PPR

(a)

0

40

80

120

0 3 6 9 12

Θ (

°)

σave (MPa)

ExperimentBilinear

PPR

(b)

Fig. 5. Comparison of the FE simulation results with experimental data for the cruciform specimen: (a) macroscopic stress-strain plots, and (b) evolution of de-bonding

angle as a function of applied load.

6

f

a

a

s

s

n

tapers off asymptotically to a stable value. A good agreement in the

evolution of � is seen for both the cohesive zone models. The cru-

ciform tests primarily invoke mode I type interfacial de-bonding in

the specimen, and both the bilinear and PPR type CZMs are found

to yield satisfactory results.

The corresponding simulated normal and tangential interfacial

stresses σ n and σ t for the bilinear CZM are plotted Figs. 6 (a) and

(b). The plot shows the distribution of stresses along the inter-

ace ( φ = 0 ◦ to φ = 180 ◦) for increasing values of the de-bonding

ngle � in Fig. 3 (e). The normal stress is symmetric about the

ngular position φ = 90 ◦, while the tangential stresses are anti-

ymmetric about this position. Prior to de-bonding, the normal

tress is high near φ = 0 ◦, while the shear stress is maximum

ear at φ = 45 ◦. Once de-bonding initiates, the peaks shift to the

Z. Li et al. / Mechanics of Materials 99 (2016) 37–52 43

−6

−3

0

3

6

0 45 90 135 180

σ n (

MPa

)

φ (°)

θ=0°θ=45°θ=90°

θ=120°θ=135°

(a)

−8

−4

0

4

8

0 45 90 135 180

σ t (

MPa

)

φ (°)

θ=0°θ=45°θ=90°

θ=120°θ=135°

(b)

Fig. 6. Distribution of stresses along the interface at angle φ in Fig. 3 (e) around the circular fiber for different values of θ : (a) normal stress σ n , (b) tangential stress σ t .

Table 3

Calibrated CZM parameters from cruciform tests.

Bilinear σ n max σ t

max σ n

max

δn c

σ t max

δt c

(GPa) (GPa) (GPa/mm) (GPa/mm)

7 . 104 × 10 −3 11 . 0016 × 10 −3 1 . 0 × 10 −3 1 . 0 × 10 −3

δn e − δn

c (mm) δt e − δy

c (mm)

1.0e-6 1.0e-6

PPR φn φt α β

(Gpa · mm) (Gpa · mm)

2 . 8 × 10 −8 6 . 6 × 10 −8 1 .0 0 01 1 .0 0 01

σ n max (GPa) σ t

max (GPa) λn λt

7 . 104 × 10 −3 11 . 0016 × 10 −3 0 .99999 0 .99999

c

t

f

m

q

a

t

t

i

a

F

f

c

d

s

r

m

w

f

p

C

f

F

i

e

b

a

u

s

F

T

p

t

g

r

f

t

c

c

fi

r

T

b

i

3

h

fi

b

o

t

i

w

B

m

t

5

d

A

o

l

d

p

o

s

T

i

r

p

h

t

m

w

s

i

t

t

t

m

o

rack tip, from which the normal stress decreases monotonically

ill φ = 45 ◦ where σ n is compressive due to Poisson’s effect. With

urther de-bonding, the peak tensile stress initially increases in

agnitude but subsequently decreases. This behavior is a conse-

uence of two competing phenomena, viz. (i) an increase in aver-

ge stress due to increasing de-bond length and (ii) a decrease in

he normal component of stress with increasing angular orienta-

ion, especially as φ → 90 °. The region of compressive stresses also

ncreases with increasing de-cohesion. Similar observations have

lso been made in Achenbach et al., 1990 Ghosh et al. (20 0 0) .

or the tangential stress σ t , the peak value at the crack tip is also

ound to first increase slightly and then decline with progressive

racking. These results validate the computational model in pre-

icting interfacial de-cohesion phenomena.

Tensile experiments are next conducted with the cruciform

pecimens at different loading speeds to study the effect of strain-

ate. Three loading rates, viz. (i) v = 1 . 27 mm/min, (ii) v = 12 . 7

m/min and (iii) v = 127 mm/min are considered. The load F f at

hich de-bonding is first observed in the DIC system is recorded

or each loading speed. For each speed, inverse calculations are

erformed using GA to calibrate T max and δe in Eq. 16 for the two

ZM’s. The objective function to be minimized in GA is the dif-

erence between the experimental and simulated failure load i.e.

exp

f − F sim

f . All other parameters in the CZM are kept the same as

n the previous examples. After calibrating T max and δe , the local

ngineering strain-rate ˙ ε is extracted from FEM simulations. Cali-

rated values for the case with loading speed v = 0 . 381 mm/min

re set to T re f

max and δre f e in Eq. 16 .

Fig. 7 (a) compares the experimental and calibration-based sim-

lation results of the failure load F f as a function of the local

train-rate in the central region A of Fig. 3 (a). The error bars in

ig. 7 (a) represent the variability in repeated experimental results.

he rate-dependent interfacial failure behavior in the cruciform ex-

eriment, as illustrated in Fig. 7 , clearly demonstrates the need for

he rate-dependent cohesive zone model. With no epoxy matrix or

lass fiber damage, an increase in the failure load is seen with a

ise in the local engineering strain-rate. The figure also shows the

ailure load for the rate-independent case, which cannot represent

he behavior with increasing rates. From this figure, the maximum

ohesive traction T max and the local engineering strain-rate ˙ ε are

alibrated and plotted as discrete data in Fig. 7 . A best-fit curve

tting is shown with the solid line in the figure. Parameters in the

ate-dependent model in Eq. (16) are evaluated as: η = 0.104805,

re f max = 7.0 (MPa), δre f

e = 1.0e-6 (mm), ˙ ε re f = 0.005443 ( s −1 ). The cali-

ration process is applied to both bilinear and PPR CZM’s. The cal-

brated T max , δe and η in Eq. 16 are the same for both the models.

.3. Micro-droplet test

Micro-droplet experiments, developed in Miller et al. (1987) ,

ave been conducted in Sockalingam et al. (2014) for characterizing

ber-matrix interfacial properties. The test specimens are prepared

y depositing resin micro-droplets of diameter is 90 μm on a fiber

f nominal diameter 9 μm and cooling down at prescribed rates

o form desirable interfaces. The fiber is S-glass, while the matrix

s DER 353 epoxy with PACM curing agent. The interface develops

ith matrix resin cure and interaction with a coating on the fiber.

oth the fiber and matrix have been qualified as isotropic elastic

aterials with properties listed in Sockalingam et al. (2014) . In

he experimental process, a quasi-static shear displacement-rate of

μm/s is applied by using a knife-edge to peel off the resin micro-

roplet from the fiber. This is shown in the FE model of Fig. 8 (a).

t quasi-static loading speeds, interfacial crack initiates at the top

f the droplet that is followed by progressive de-bonding in the

oading direction until a maximum load is reached. Upon complete

e-cohesion, the droplet undergoes a quasi-static frictional sliding

rocess due to the force caused by thermal shrinkage of the epoxy

n the fiber ( Table 4 ). The de-cohesion zone, as well as the con-

tant force in the Coulomb friction regime, are depicted in Fig. 8 (b)

he post-de-bond friction force is observed to be F s = 26 × 10 −3 N.

A detailed FE model of the micro-droplet test is set up as shown

n Fig. 8 (a). The interfacial de-cohesion behavior, in Fig. 8 (b). is

epresented by the bilinear and PPR cohesive zone models. In the

ost-de-bonding phase, all cohesive zone elements are assumed to

ave completely failed and no longer generate transverse forces in

he interface. This phase incorporates frictional sliding, which is

odeled with a Coulomb friction model (see Chaboche et al., 1997 )

ith a constant coefficient of friction. Considerable residual

tresses are found to be generated at the interface when the spec-

men is cooled from a maximum curing temperature of 150 °C to

he ambient temperature, due to mismatch in the coefficients of

hermal expansion (CTE) between the glass fiber and epoxy ma-

rix. To obtain the residual stresses in the FE model, a thermo-

echanical analysis is first conducted for a temperature change

f �θ = −150 ◦C , assuming that the composite is stress free at

44 Z. Li et al. / Mechanics of Materials 99 (2016) 37–52

1.2

1.6

2

2.4

0.01 0.1 1 10

F f (

kN)

Local strain rate (s−1)

ExperimentRate dependent model

Rate independent model

(a)

6

8

10

12

0.01 0.1 1 10

Tm

ax (

MPa

)

Local strain rate (s−1)

CalibrationCurve fitting

(b)

Fig. 7. (a) Experimental and simulated failure loads F f as a function of the local engineering strain-rate ˙ ε; (b) relation between T max and ˙ ε from the calibration process.

von-Mises

stress σ (GPa)

0.000.200.140.620.821.031.231.441.641.852.052.262.46

vM

(a)

von-Mises

stress σ (GPa)

0.0000.0170.0340.0510.0680.0850.1020.1190.1360.1530.1700.1870.205

vM

(b)

Fig. 8. Contour plots of von Mises stress at Force = 80 . 342 × 10 −3 N in: (a) the droplet and the fiber, and (b) interfacial cohesive elements showing de-bonding.

Table 4

Material parameters for micro-droplet experiment ( Sockalingam et al.,

2014 ).

Property Youngs Modulus (GPa) Poisson’s Ratio CTE (/ °C )

S-glass fiber 90 .0 0 .17 3 .4

Epoxy DER353 3 .2 0 .36 70 .0

s

m

t

b

i

p

f

r

a

e

d

θ = 150 ◦C . The resulting residual von Mises stress along the fully

bonded interface after the cooling is plotted in Fig. 9 (a). Subse-

quent mechanical loading, corresponding to a quasi-static shear

displacement-rate of 5 μm/s of the blade, is applied to the droplet

0

100

200

300

0 20 40 60 80

vo

n M

ises

str

ess

(MP

a)

Distance along fiber ( μm)

(a)

−3

Fig. 9. (a) Stress distribution along the interface at the end of thermal pre-loading, a

Sockalingam et al. (2014) and the FEM model with different CZMs.

pecimen with residual stresses. The CZM parameters in the two

odels, as well as the coefficient of friction μ, are calibrated from

he experimental data obtained from Fig. 9 (b). The GA-based cali-

ration process is used to minimize the difference between exper-

mental data and the simulated load-displacement data. The CZM

arameters are delineated in Table 5 and the friction coefficient is

ound to be μ = 0.60. For this mode II dominated problem, pa-

ameters for normal and transverse traction-separation laws are

ssumed to be the same. Both the bilinear and PPR cohesive mod-

ls show good agreement with the experiment.

Contour plots of the von Mises stress on a quarter of the

roplet model and on the partially de-bonded interface are shown

0

45

90

135

180

0 3 6 9 12

Forc

e x

10 N

Displacement (μm)

ExperimentBilinear

PPR

(b)

nd (b) force-displacement curve for the micro-droplet test from experiments in

Z. Li et al. / Mechanics of Materials 99 (2016) 37–52 45

Table 5

CZM parameters calibrated from the micro-droplet test by the GA algorithm.

Bilinear σ t max (GPa)

σ t max

δt c

(GPa/ μm) δt e − δy

c ( μm)

0 .12 1 .02910 2 .58

PPR φt (Gpa · μm) σ t max (GPa) β λt

0 .159 0 .12 25 .5 0 .037453

Table 6

Material parameters for nonlocal CDM model.

Material Fiber Borosilicate glass Matrix EPON 828 epoxy

CDM Parameters εD f

ε0 f

S ∗f

εD m ε0

m S ∗m 0 .0164 0 .014 7 .0 0 .0562 0 .070 7 .0

κ C ˙ ε 0 κ C ˙ ε 0 (6 √

π) 1

3 0 .029 1 s −1 (6 √

π) 1

3 0 .015 1 s −1

i

F

r

t

b

t

I

t

l

2

s

F

t

t

c

T

w

m

4

e

t

e

c

f

i

g

m

f

r

d

r

e

c

(

i

t

A

4

f

h

e

s

1

fi

c

R

E

l

1

r

s

d

E

w

s

r

c

a

r

i

o

d

4

fi

a

m

i

p

o

F

b

n Figs. 8 (a) and 8 (b), respectively. The plots are for a force of

= 80 . 342 × 10 −3 N in Fig. 9 (b), corresponding to the middle of

apid de-cohesion process. The figure shows that some elements in

he middle of the interface, close to the blade, have completely de-

onded. The blue region in Fig. 8 (b) represents the region where

he stress is reduced to near zero on the verge of de-bonding.

n both the experiments and FEM simulations, it is observed that

he initial damage position is located at a distance d = 10 μm be-

ow the blade. This is attributed to the fact that there is a gap of

μm between the blade tip and the fiber that leads to an initial

tress concentration away from the top of the interface as seen in

ig. 8 (a). The stress concentration zone gradually moves to the bot-

om of the interface ( d = 90 μm) with progressive blade peeling.

Comparison of the CZM parameters in Tables 3 and 5 indicates

hat the interface in the droplet specimen is stronger than that in

ruciform. This is attributed to the different interface treatments.

he treatment for the cruciform specimen leads to a weaker bond,

hile the sizing yields a stronger interface for the droplet speci-

en.

. Effects of Mesh, Model and loading parameters on damage

volution

In this section, 3D micro-mechanical analyses of unidirectional

wo-fiber RVEs are conducted to study mesh sensitivity and the

ffect of model parameters on matrix damage and interfacial de-

ohesion leading to material failure. The material modeled is that

or the cruciform specimen in Section 3.2 , i.e. borosilicate fiber

n EPON 828/PACM matrix. The material and CZM parameters are

iven in Tables 2 and 3 . Parameters in the CDM model for fiber and

atrix damage in Section 2.1 are calibrated from experimental ef-

ective stress-strain plots in Chocron et al. (2009) and ( Fard, 2011 )

espectively. They are given in Table 6 . All simulations are con-

Scaler damage

variable D

0.00.10.20.30.40.50.61.0

h

(a)

Err

or (

%)

ig. 10. (a) Mesh sensitivity study showing the contour plot in x 3 = 0 mm plane of the s

and-width as a function of increasing mesh density.

ucted using the trilinear 8-noded brick elements with selective

educed integration (C3D8R) in ABAQUS-Explicit with a lumped el-

ment mass matrix. Selection of stable time steps in the explicit

entral difference time-integration method is described in Bathe

1982) . The non-local continuum damage model is implemented

n the user subroutine VUMAT. The PPR CZM is implemented in

he user subroutine VUEL, while the bilinear CZM is included in

BAQUS-Explicit.

.1. Mesh sensitivity of the RVE model with damage modes

To examine the convergence characteristics of the FE mesh

or problems involving matrix damage, a unidirectional RVE with

exagonal arrangement of two fibers is analyzed using four differ-

nt mesh densities of 8-node brick elements. The RVE has dimen-

ions 2.0 mm × 2.0 mm × 1.0 mm with a fiber volume fraction of

0%. The coarsest mesh is composed of 9568 elements, while the

nest mesh consists of 220,160 elements. A uniaxial tensile load

orresponding to a global strain rate of 10 2 s −1 is applied to the

VE. Fig. 10 (a) shows a contour plot of the damage variable D in

q. (3e) (on a 0 − 1 scale) for the RVE with 58,0 0 0 elements. The

ocal strain-rates in the regions of higher damage are in the range

0 3 s −1 to 10 7 s −1 , which are much higher than the overall loading

ate.

Convergence is measured in terms of a parameter h that repre-

ents the local width of the damage zone (band). For each mesh

ensity, an error in the damage band-width is defined as:

rror =

∣∣(h a v g − h

re f a v g )

∣∣[0 . 5(h

re f max + h

re f min

) ] (18)

here h avg is the average width of the damage bands, obtained by

ampling at various locations on the bands shown in Fig. 10 (a). The

eference value h re f a v g is chosen for a very high density mesh, in this

ase with 220,160 elements. The denominator refers to the aver-

ge of the maximum and minimum sampled band-widths for the

eference mesh. Fig. 10 (b) plots the error as a function of increas-

ng mesh density. The stabilized results suggest that the solution

f the non-local damage model converges with the higher mesh

ensities.

.2. Effect of non-local damage parameter

A parametric study is conducted with the RVE of hexagonal

ber arrangement to understand the effect of the non-local dam-

ge characteristic length l c in Eq. (8b) on the overall response. The

esh of 58,0 0 0 elements ( ≤ 3% error in Fig. 10 (b)) is considered

n this study. The loading and boundary conditions, and material

arameters other than l c are kept the same as for the simulation

f Fig. 10 (a) in Section 4.1 .

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5

Number of elements (× 105)

(b)

calar-valued damage parameter D (0 ≤ D ≤ 1) in Eq. (3e) , (b) error in the damage

46 Z. Li et al. / Mechanics of Materials 99 (2016) 37–52

0.09

0.12

0.15

0.18

0 0.4 0.8 1.2 1.6

Ons

et Σ

vM

(G

Pa)

lc (mm)

(a)

0.09

0.1

0.11

0.12

0.13

0 0.2 0.4 0.6 0.8

Ons

et Σ

vM

(G

Pa)

lc (mm)

(b)

Fig. 11. Onset of damage in terms of the peak macroscopic stress in uniaxial tension as a function of the characteristic length l c for: (a) the hexagonal arrangement RVE in

Figs. 10 (a), and (b) the uniform arrangement RVE in Fig. 1 (c).

s

a

a

b

a

t

l

i

w

t

s

d

s

s

s

t

t

P

m

n

4

i

m

s

t

l

g

c

t

p

v

s

p

a

u

v

t

p

e

e

The volume-averaged von-Mises stress and effective strain are

respectively defined as:

�v M

=

√

3

2

�de v : �de v and e e f f =

√

2

3

e de v : e de v (19)

where �dev and e dev are the deviatoric components of the volume-

averaged stresses and strains respectively, expressed as:

� =

1

V

∫ V

σdV and e =

1

V

∫ V

ε dV +

1

2 V

∫ ∂V coh

( u � n + n � u ) dS

(20)

σ and ε are the microscopic stresses and strains in the RVE with

volume V . When interfacial de-cohesion is present, ∂V coh corre-

sponds to the fiber-matrix interface domain and u denotes the

jump in displacement components across the interface with out-

ward normal n . Prior to the onset of damage in the RVE, the plot

of volume-averaged von-Mises stress vs. effective strain shows ex-

hibits a linear regime, after which the stress drops with evolving

damage, e.g. as shown in Fig. 12 (a). The maximum stress, corre-

sponding to the onset of damage is termed as the onset von-Mises

stress.

For the hexagonal RVE, the onset von Mises stress is plotted

as a function of l c in Fig. 11 (a). The peak stress gradually in-

creases with l c and eventually stabilizes at a value l stab c ≈ 0 . 9 mm .

Beyond this stabilized value, the onset of damage is independent

of the characteristic length and any value can be chosen in nu-

merical studies. The stabilized characteristic length l stab c depends

on the microstructure. The onset stress- l c plot in Fig. 11 (b) shows

the trend for a single fiber uniform RVE, with a stabilized length

l stab c ≈ 0 . 5 mm . Eqs. (10a) and (10b) in Section 2.1.1 for these RVEs

yield the values: l uni f orm

c = 0 . 707 mm and l hexagonal c = 0 . 914 mm re-

spectively. The values derived from the analytical forms agree well

with results of the numerical parametric study.

4.3. Effect of loading directions and interface model

The effects of loading directions and the interface model on the

overall behavior of unidirectional composites are examined in this

section. The RVE considered has the hexagonal architecture and the

same material and damage properties as in Section 4.2 . The sta-

bilized characteristic length is taken as l c = 1 . 0 mm . For compari-

son, simulations are also conducted with a local CDM model i.e.

l c = 0 mm . The RVE is subjected to periodicity boundary conditions

and three different loading cases are applied to the RVE with the

macroscopic rates of deformation tensor d i j , given as:

1. Uniaxial tension test, d 11 = 100 s −1 , all other components are

zero.

2. Uniaxial compression test , d 11 = −100 s −1 , other components

are zero.

3. Shear test, d 12 = d 21 = 100 s −1 , all other components are zero.

The volume-averaged von-Mises stress - effective strain re-

ponse for the three loading cases are plotted in Figs. 12 (a), 12 (c)

nd 12 (e), respectively. The corresponding true strain contour plots

re given in Figs. 12 (b), 12 (d) and 12 (f).

The stress-strain responses display stiffness softening induced

y cumulative damage with increasing deformation. Prior to dam-

ge, the behavior with local and non-local CDM models are indis-

inguishable, but the subsequent deviation depends on the type of

oading. The maximum stress for the compressive loading is signif-

cantly higher than that for tension and shear, which is consistent

ith experimental observations on tension-compression asymme-

ry ( e.g. in Chocron et al., 2009; Fard, 2011 ). While for the ten-

ile loading, the post peak stress continues to drop with increasing

amage, the stress rises a little after initial drop for the compres-

ive and shear loading cases due to reduced damage evolution. The

tress-strain plots in Figs. 12 (a), 12 (c) and 12 (e) also compare re-

ponses with the bilinear and PPR cohesive zone models for in-

erfacial de-bonding. The responses are very similar in general. For

he shear load however, the stress drop is slightly faster with the

PR CZM due to the coupling of normal and tangential displace-

ents, causing the tangential deformation energy to contribute to

ormal damage evolution.

.4. Strain rate effects on damage evolution

The effect of applied strain-rate on tensile failure is studied

n this section and compared with experimental data. The RVE

odel in Section 4.3 is analyzed by subjecting it to a uniaxial ten-

ile strain-rate (or rate of deformation) in the horizontal direction,

hat is varied from d 11 = 10 s −1 to 300 s −1 . The results of simu-

ations are compared with experimental data for a unidirectional

lass/ML-506 epoxy composite in Shokrieh and Omidi (2009) . The

omposite in the experiments is subjected to tensile strain-rate

hat is varied from d 11 = 0 . 0017 s −1 to 85 s −1 . The variables com-

ared are those at the onset of damage, as defined in Section 4.3 ,

iz. the peak macroscopic von Mises stress, effective strain at peak

tress, and recoverable elastic energy density. Since the material

roperties in experiments and simulations are different, the vari-

bles are normalized with respect to their respective reference val-

es, i.e. �norm

v M

=

�v M �

re f v M

, e norm

e f f =

e e f f

e re f e f f

and W

norm

re =

W re

W

re f re

. The reference

alues for the experimental and simulation variables are taken as

heir respective values, evaluated for d re f = 46 s −1 . Fig. 13 com-

ares the normalized von Mises stress, effective strain strain and

lastic energy density for different applied strain-rates. While the

ffect of strain-rate is relatively low at the lower rates, there is

Z. Li et al. / Mechanics of Materials 99 (2016) 37–52 47

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

Σ vM (

GP

a)

eeff (%)

Nonlocal CDM, Bilnear CZMNonlocal CDM, PPR CZMLocal CDM, Bilnear CZM

(a)

Local logarithmic

strain e

−6.402e−04+9.027e−03+1.869e−02+2.836e−02+3.803e−02+4.769e−02+5.736e−02+6.703e−02+7.670e−02+8.636e−02+9.603e−02

11

(b)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10

Σ vM (

GP

a)

eeff (%)

Nonlocal CDM, Bilnear CZMNonlocal CDM, PPR CZMLocal CDM, Bilnear CZM

(c)

−9.791e−02−8.812e−02−7.833e−02−6.855e−02−5.876e−02−4.897e−02−3.918e−02−2.939e−02−1.961e−02−9.817e−03−2.883e−05

Local logarithmic

strain e 11

(d)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10

Σ vM (

GP

a)

eeff (%)

Nonlocal CDM, Bilnear CZMNonlocal CDM, PPR CZMLocal CDM, Bilnear CZM

(e)

−5.422e−04+2.101e−02+4.257e−02+6.413e−02+8.568e−02+1.072e−01+1.288e−01+1.504e−01+1.719e−01+1.935e−01+2.150e−01

Local logarithmic

strain e12

(f)

Fig. 12. Volume-averaged von Mises stress- effective strain response and contour plot of local logarithmic strain component at a section x 3 = 0 mm for: (a,b) uniaxial tension

(contour plot corresponding to e e f f = 3 . 07% ); (c,d) uniaxial compression (contour plot corresponding to e e f f = 4 . 01% ); (e,f) shear (contour plot corresponding to e e f f = 4 . 03% ).

a

e

t

5

p

T

a

a

d

W

W

w

fi

r

s

S

s

t

e

t

W

significant rise at higher rates close to d 11 = 10 2 s −1 . The rise is

specially significant for W

norm

re . The experimental trends are cap-

ured well in the simulations in these plots.

. Energy partitions with evolving damage

The volume-averaged energy in a damaging composite RVE is

artitioned into different components and studied in this section.

he total internal energy density W T is comprised of an irrecover-

ble, dissipative energy density W D due to damage and a recover-

ble energy density W R , i.e. W T = W D + W R . The partitioned energy

ensities are expressed as:

D =

1

V

(

N f ∑

k =1

W

e D f ( k )

V

e f ( k )

+

N m ∑

k =1

W

e Dm ( k )

V

e m ( k )

+

N coh ∑

k =1

�e Dcoh ( k )

S e coh ( k )

)

(21a)

R =

1

V

(

N f ∑

k =1

W

e R f ( k )

V

e f ( k )

+

N m ∑

k =1

W

e Rm ( k )

V

e m ( k )

+

N coh ∑

k =1

�e Rcoh ( k )

S e coh ( k )

)

(21b)

here N f , N m

and N coh are the total number of elements in the

ber, matrix and interface respectively. V e f ( k )

and V e m ( k )

are the cur-

ent volume of the k th element in the fiber and matrix phase re-

pectively, while V 0 e f ( k )

and V 0 e m ( k )

are their respective initial volumes.

imilarly, σe f ( k )

and σe m ( k )

, and ε e f ( k )

and ε e m ( k )

are the stresses and

trains in the k th element of the fiber and matrix phase respec-

ively. S e coh ( k )

and S 0 e coh ( k )

are the current and initial areas of the k th

lement on the cohesive zone interface.

Following ( Carol et al., 1994 ), the dissipative energy density of

he k th element in fiber and matrix phases are defined as:

e D f ( k )

= −1

2

∫ ε i j ( k ) ε i j ( k ) dE ( k ) (22a)

48 Z. Li et al. / Mechanics of Materials 99 (2016) 37–52

0

0.5

1

1.5

2

0.1 1 10 100 1000

Σonse

tvM

/ Σ

ref

vM

d_

11 (s−1)

RVE simulationExperiment

(a)

0

0.5

1

1.5

2

0.1 1 10 100 1000

eon

set

eff

/ eno

rmef

f

d_

11 (s−1)

RVE simulationExperiment

(b)

0 0.5

1 1.5

2 2.5

3

0.1 1 10 100 1000

Won

set

R

/

Wno

rmR

d_

11 (s−1)

RVE simulationExperiment

(c)

Fig. 13. Comparison of simulations with experiments in plots showing the effect of strain-rate on variables at the onset of damage: (a) normalized peak von Mises stress,

(b) normalized effective strain at peak stress, and (c) normalized recoverable energy density at peak stress. The plots are for uniaxial tension tests of the RVE in Fig. 10 (a).

W

W

W

w

e

f

m

a

f

a

s

i

f

d

(

i

e

l

l

d

t

u

6

a

s

a

a

1

M

m

i

m

e Dm ( k )

= −1

2

∫ ε i j ( k ) ε i j ( k ) dE ( k ) (22b)

The recoverable energy density of the kth element are conse-

quently expressed as:

e R f ( k )

=

∫ σi j ( k ) dε i j ( k ) − W

e D f ( k )

(23a)

e Rm ( k )

=

∫ σi j ( k ) dε i j ( k ) − W

e Dm ( k )

(23b)

On the cohesive interface, the fracture energy per unit area

( �e Dcoh ( k )

) and the recoverable energy per unit area ( �e Rcoh ( k )

) for

the k th element are defined as:

�e Dcoh ( k )

=

∫ T n d δn +

∫ T t d δt − �e

Rcoh ( k ) (24a)

�e Rcoh ( k )

=

1

2

T n δn +

1

2

T t δt (24b)

A critical value is assumed for the scalar damage variable for

an element to fail. Specifically D

e ( k )

= 0 . 9 is assumed and the cor-

responding damage volume fraction in the fiber and matrix phases

are expressed as:

V

0 f D

V

0 f

=

1

V

0 f

N f ∑

k =1

H

(D

e ( k )

− 0 . 9

)V

0 e f ( k )

(25a)

V

0 mD

V

0 m

=

1

V

0 m

N m ∑

k =1

H

(D

e ( k )

− 0 . 9

)V

0 e m ( k )

(25b)

where H ( x ) is the Heaviside function. On the cohesive interface, the

damage area fraction is assumed to be:

S 0 cohD

S 0 coh

=

1

S 0 coh

N coh ∑

k =1

H ( δ − δe ) S 0 e coh ( k )

(26)

The RVE in Section 4.3 is solved for the different loading cases

ith the bilinear CZM model. Fig. 14 depicts the corresponding

nergy densities, as well as the evolution of the damage volume

raction. The CZM parameters calibrated from the cruciform experi-

ent correspond to a weak interface. Hence, for all cases, the dam-

ged interface volume fraction rapidly approaches 100%, even be-

ore the effective strain reaches 0.5%. De-bonding in tension grows

t the fastest rate, while de-cohesion speed is the slowest for

hear. The initial volume fraction of the RVE is low. This causes the

nterface to de-bond fast and there is no fiber damage, i.e. V 0

f D

V 0 f

≈ 0

or all cases. Before matrix damage, there is no obvious energy

issipation. The damaged matrix volume fraction and dissipation

irrecoverable) energy density evolve slowly in the beginning that

s followed by rapid growth. During the rapid change of irrecov-

rable energy, the recoverable energy density either changes very

ittle (under tensile or compressive load) or decreases (under shear

oad). After a sharp increase of matrix damage, the rate of energy

issipation suddenly drops followed by slow subsequent growth. At

he same time, the matrix damage volume fraction remains nearly

nchanged at around 7 − 11% .

. Effect of microstructural morphology on damage evolution

The effect of microstructural morphology on damage nucleation

nd growth is examined in this section. A multi-fiber RVE, con-

isting of 172 randomly distributed unidirectional cylindrical fibers

s shown in Fig. 15 (a), is analyzed. Identical fibers of 4 μm di-

meter are considered. The size of the RVE analyzed is 100 μm ×00 μm × 10 μm with an average fiber volume fraction of 21.6%.

aterial properties for the borosilicate fiber and EPON 828/PACM

atrix, as well as the non-local CDM model properties are given

n Section 4 . Both the bilinear and PPR cohesive zone interfacial

odels are used with parameters calibrated from the cruciform

Z. Li et al. / Mechanics of Materials 99 (2016) 37–52 49

0

2

4

6

8

10

2 4 6 8 0

50

100

150

200E

nerg

y de

nsit

y (m

J/m

m3 )

Dam

aged

vol

ume

frac

tion

(%

)

Effective strain (%)

Total internal energyRecoverable energyIrrecoverable energyDamaged fiber volume fractionDamaged matrix volume fractionDamaged interface volume fraction

(a)

0

2

4

6

8

10

0 2 4 6 8 10 0

0.5

1.0

Ene

rgy

dens

ity

(mJ/

mm

3 )

Dam

aged

vol

ume

frac

tion

Effective strain (%)

Total internal energyRecoverable energyIrrecoverable energyDamaged fiber volume fractio nDamaged matrix volume fractio nDamaged interface volume fractio n

(b)

0

2

4

6

8

10

2 4 6 8 0

50

100

150

200

Ene

rgy

dens

ity

(mJ/

mm

3 )

Dam

aged

vol

ume

frac

tion

(%

)

Effective strain (%)

Total internal energyRecoverable energyIrrecoverable energyDamaged fiber volume fractionDamaged matrix volume fractionDamaged interface volume fraction

(c)

Fig. 14. Energy density and damage evolution plots for uni-directional single fiber

RVE in Fig. 1 (c) for (a) uni-axial tension, (b) uni-axial compression, and (c) shear

loading. Note that range of damaged volume fraction is [ 0, 1.0 ].

t

s

a

s

v

s

t

c

l

E

i

r

a

p

a

h

m

o

t

l

o

F

m

s

i

t

a

m

T

w

c

d

n

6

d

l

c

m

c

l

c

n

t

a

n

d

t

e

p

t

ests (weak interface). The RVE in Fig. 15 (a) is subjected to a tensile

train-rate d 11 = 10 s −1 in the horizontal direction.

Figs. 15 (a) and 15 (b) show the contour plots of the scalar dam-

ge variable D in Eq. (3e) with the interfacial de-bonding repre-

ented by the bilinear and PPR CZMs respectively. For determining

arious morphology-based characterization parameters, the cross-

ection of the microstructure is tessellated using Voronoi tessella-

ion method ( Ghosh, 2011 ). It is based on the subdivision of an Eu-

lidean space into a set of bounded convex polytopes, where any

ocation within each polytope is associated with a heterogeneity.

ach Voronoi cell, enclosing a single fiber at most, represents the

mmediate neighborhood of the fiber and the edges of the cell rep-

esent the neighbors of the fiber. Fig. 15 (c) shows the Voronoi cells,

long with the apparent crack path corresponding to the damage

arameter D .

Damage predominantly nucleates with interfacial decohesion

nd grows as localized matrix damage ahead of the interfacial co-

esive zone crack tip, where stresses concentrate. In Fig. 15 (a), al-

ost all fiber-matrix interfaces have initiated de-cohesion, while

nly about 12% of all fibers show neighboring matrix damage. Prior

o damage nucleation, the principal tensile stresses are large in the

oading direction. However with damage evolution, the directions

f principal stresses can change. This has been demonstrated in

ig. 6 (a). The damage contours generated by the two cohesive zone

odels in Figs. 15 (a) and 15 (b) are quite similar. Matrix cracks are

een to initiate and grow near the middle of the RVE with little

nfluence from the boundary. Matrix cracking normally starts from

he interfacial region where de-bonding has halted with saturation,

t anding angle ϕ as described in Fig. 15 (a). For this example with

ultiple fibers, the de-bonding angles range from ϕ = 0 ◦ → 180 ◦.

he matrix cracks grow, bridging interfacial de-bonds and merging

ith other cracks in the vicinity, to form longer cracks. The longer

rack paths are generally orthogonal to the direction of loading ( x -

irection in this case), thus making transverse cracking the domi-

ant mode.

.1. Characterization of the damaged microstructure

Quantitative characterization of the RVE microstructure is con-

ucted with the cell structure created by Voronoi tessellation. Fol-

owing some of the methods in Ghosh et al. (20 0 0) ; Ghosh (2011) ,

haracterization is used to understand effect of the non-uniform

ulti-fiber morphology on damage initiation and growth. Three lo-

alized damage zones in Fig. 15 (a) are marked as the dominant

ong cracks in the tessellated diagram of Fig. 15 (c). Along each

rack, the de-bonded fibers contributing to the cracked zone are

umbered. Various morphological parameters are measured along

he crack paths of Fig. 15 (c). These are:

1. Local volume fraction (LVF) obtained by dividing the associated

Voronoi cell area by the fiber cross-sectional area;

2. Number of neighborhood fibers (ND) that correspond to the num-

ber of edges of the associated Voronoi cell. For a given fiber,

there are ND distances between the fiber and its neighbors;

3. Nearest neighbor distance (NND) that corresponds to the small-

est of all ND distances between the fiber and its neighbors;

4. Second nearest neighbor distance (SNND) that corresponds to the

second smallest of all ND distances;

5. Third nearest neighbor distance (TNND) that corresponds to the

third smallest of all ND distances;

6. Distance between fibers along a crack (NCFD) , which is measured

as the distance between two contiguous fibers along a crack

path.

For comprehending the statistics of the crack path, these vari-

bles are plotted with respect to the fiber number (IDs) and crack

umber (IDs) in Fig. 16 . Figs. 16 (a, d and g) demonstrate that the

istance between fibers along a crack (NCFD) is mostly equal to

he nearest neighbor distance (NND). Cracks propagate to the near-

st fibers, then to the second nearest neighbor, and so on. The

lots of the local volume fraction (LVF) in Figs. 16 (c, f and i) show

hat fibers along the crack path have higher LVF than the average

50 Z. Li et al. / Mechanics of Materials 99 (2016) 37–52

Scaler damage

variable D

0.00.10.20.30.40.50.61.0

(a)

Scaler damage

variable D

0.00.10.20.30.40.50.61.0

(b)

10 30 50 70 90

10

30

50

70

9012

345

678910

1112

1

234

123

45

crack 3

crack2 crack 1

(c)

Fig. 15. Contour plot of the scalar-valued damage parameter D (0 ≤ D ≤ 1) for the RVE with: (a) bilinear interfacial CZM at t = 7 . 7 × 10 −3 s and e e f f = 3 . 1565% , (b) PPR

interfacial CZM at t = 7 . 7 × 10 −3 s and e e f f = 3 . 8203% ; and (c) Voronoi tessellated microstructure showing the crack path from FEM results of Fig. 15(b) .

d

b

d

c

e

T

i

i

c

c

v

a

c

t

o

d

e

a

c

o

p

d

g

e

T

l

s

w

o

l

over the volume fraction, which is 21.6%. The number of neigh-

boring fibers (ND) are shown in the histograms of Fig. 16 (b, e and

h). While the above conjectures generally hold, exceptions are ob-

served as well. For example, fiber 3 along crack 1 in Fig. 16 (a) does

not meet the nearest fiber criterion. The nearest fiber has a low

LVF and it is located in a direction that deviates too much from

the transverse direction. Likewise, fiber 1 along the crack 1 and

fiber 5 along crack3 contradict the high LVF criterion. This discrep-

ancy is attributed to the intersection of the tessellation with the

RVE boundary that results in a Voronoi cell with a lower LVF.

In summary, cracks in the RVE subjected to uniaxial tension,

initiate in form of interfacial de-bonding and evolve into matrix

damage for cells with high LVF. Subsequently, they propagate to

nearest cells with higher LVF in directions that are close to orthog-

onal to the loading. These are consistent with observations (3) and

(4) in Section 6 , in that cracks tends to grow towards the nearest

adjacent de-bond tips. In general, crack progression is governed by

various factors such as loading directions and microstructural mor-

phology.

7. Conclusions

This paper develops an experimentally validated computational

model for polymer-matrix composites to predict damage nucle-

ation and propagation in the microstructure, leading to overall fail-

ure. Specifically, the strain-rate dependent models are calibrated

and validated for S-glass fiber reinforced epoxy-matrix composites.

The computational tool is developed in the FEM code ABAQUS-

Explicit, in which the constitutive and damage models are incor-

porated using the user-subroutines VUMAT and VUEL. The mi-

crostructural representative volume elements or RVEs consist of

various arrangements of unidirectional, brittle glass fibers embed-

ed in an epoxy resin matrix. The RVEs are simulated with periodic

oundary conditions.

Damage in the fiber and matrix are modeled by a strain-rate

ependent, non-local isotropic CDM model. Fiber-matrix interfa-

ial de-bonding is described by two alternate CZMs, viz. a bilin-

ar traction-displacement model and a potential-based PPR model.

he CZ models of the fiber-matrix interface are calibrated and val-

dated using results from two experimental studies. The first study

nvolves a cruciform experiment for the borosilicate glass, epoxy

omposite that is fabricated and conducted by the authors. Besides

alibration and validation, the cruciform experiment results also

erify strain-rate dependency of the fracture energy. The second is

micro-droplet test from Sockalingam et al., 2014 , that is used to

alibrate and validate de-bonding as well as post de-bonding fric-

ional sliding process with a Coulomb friction model. Comparison

f the two experimental results indicates that the interface in the

roplet specimen is stronger than that in the cruciform.

The computational model is implemented to investigate the

ffect of morphological and loading characteristics on crack nucle-

tion and damage propagation in the microstructure. To build this

apability, a two-fiber RVE model is subjected to investigations

f mesh sensitivity, effect of non-local CDM parameters, material

roperty asymmetry, energy absorption properties, and strain-rate

ependence. The mesh density that is necessary to achieve conver-

ence is deciphered. A parametric study validates the equation for

stimating characteristic length in the non-local damage model.

he material property asymmetry study examines the effect of

oading type and interfacial CZM models on the stress-strain re-

ponse. In the post-damage response, the volume-averaged stress

ith the PPR CZM drops faster for simple shear due to coupling

f the normal and tangential fracture energies. Under compressive

oading, the RVE shows a higher energy absorption capacity than

Z. Li et al. / Mechanics of Materials 99 (2016) 37–52 51

6

8

10

2 4 6 8 10 12

Dis

tanc

e (

µm

)

Fiber number

NNDNCFDSNNDTNND

0

2

4

6

8

2 4 6 8 10 12

No.

of

NF

Fiber No. along crack 1

0

0.2

0.4

2 4 6 8 10 12

LV

F

Fiber number

6

8

10

2 4

Dis

tanc

e (

µm

)

Fiber number

NNDNCFDSNNDTNND

(a) )c()b(

0

2

4

6

8

1 2 3 4

No.

of

NF

Fiber No. along crack 2

0

0.2

0.4

2 4

LV

F

Fiber number

(d) )f()e(

6

8

10

2 4

Dis

tanc

e (

m)

Fiber number

NNDNCFDSNNDTNNDµ

0

2

4

6

8

1 2 3 4 5

No.

of

NF

Fiber No. along crack 3

0

0.2

0.4

2 4

LV

F

Fiber number

(g) (h) (i)

Fig. 16. Characterization plots of morphological parameters along crack paths in Fig. 15 (c): (a,b,c) for crack path 1, (d,e,f) for crack path 2, and (g,h,i) for crack path 3.

t

p

o

r

s

d

V

v

C

d

e

e

p

t

e

c

t

p

s

A

D

t

W

C

H

H

R

A

B

B

B

B

hose under shear and tension loadings. Finally, the macroscopic

eak stress, strain and recoverable energy density at the onset

f damage shows a increasing trend with increase in the strain-

ate.

The validated computational model is used to study micro-

copic crack nucleation and growth in a multi-fiber, randomly

istributed RVE subject to tension loading. The microstructure is

oronoi tessellated for quantitative characterization, which pro-

ides parameters that can be correlated to damage variables.

racks initiate with interfacial de-bonding and grow as matrix

amage ahead of the interfacial cohesive zone crack tip. With the

mergence of damage, cracks tend to propagate towards the near-

st neighborhood fibers along transverse directions. Also, cracks

rogress in directions of high fiber volume fraction. In summary,

he present paper lays a foundation for characterization and mod-

ling of damage evolution leading to failure of polymer-matrix

omposites and their consequent design. Stress wave propagation

hat is observed in these simulations will be discussed in a future

aper. These models can also be important in hierarchical multi-

cale modeling to develop higher-scale constitutive models.

cknowledgments

This work is sponsored by the Center for Materials in Extreme

ynamic Environments (CMEDE) of the Army Research Labora-

ory and was accomplished under Cooperative Agreement Number

911NF-12-2-0022. The sponsorship is gratefully acknowledged.

omputational support of this work has been provided by the

omewood High Performance Compute Cluster (HHPC) at Johns

opkins University.

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