International Journal of Solids and Structures 193–194 (2020) 172–191

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International Journal of Solids and Structures
journal homepage: www.elsevier.com/locate/ijsolstr

Multiscale analysis and experimental validation of crack initiation inquasi-isotropic laminates
Yuta Kumagai a, Sota Onodera a, Marco Salviato b, Tomonaga Okabe a,c,∗a

Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan
William E. Boeing Department of Aeronautics & Astronautics, University of Washington, 311D Guggenheim Hall, Seattle, WA 98195-2400, USAc
Department of Materials Science and Engineering, University of Washington, 302 Roberts Hall, Seattle, WA 98195, USAba r t i c l e

i n f oArticle history:
Received 10 November 2019
Revised 22 January 2020
Accepted 5 February 2020
Available online 13 February 2020
Keywords:
Composite materials
Crack
Fiber reinforced
Free edge
Laminate
Numerical methods

a b s t r a c t
A multiscale approach comprising laminate-scale finite-element (FE) analysis and fiber-diameter-scale periodic unit cell (PUC) analysis was developed to predict matrix microcracking in quasi-isotropic laminates;
further, this method was validated through comparison of our predictions with experimental results. In
the mesoscopic FE analysis, the nonlinear deformation in the unidirectional laminae was incorporated toreproduce the deformation behavior of the laminate and obtain deformation histories at the locations of
expected crack initiation in the laminate. In the microscopic analysis, the nonlinear behavior and crack
initiation in the matrix resin were simply modeled by an elasto-viscoplastic law and a stress-based failure criterion, respectively. To predict crack initiation considering both the macroscopic deformation fields
and the microscopic heterogeneity of the material, the mesoscopic FE analysis was conducted first. Subsequently, the microscopic PUC analysis was undertaken based on the strain histories obtained from themesoscopic analysis. Our multiscale approach was applied to quasi-isotropic laminates with several laminate configurations to predict the matrix cracks in the 90◦ ply of the laminates. In addition to referring toexperimental data cited in literature, initial and transverse cracks were observed when conducting tensile tests of quasi-isotropic laminates using the in situ replication technique and ex situ X-ray computed
tomography. Through comparison of the predicted values with experimental results quoted in literatureand obtained in this work, we validated the prediction capability of our multiscale analysis and evaluatedthe process of crack formation from the mesoscopic and microscopic points of view. Moreover, we examined the sensitivity of the predicted results to fiber arrangement and the influence of constitutive andfailure modeling of the two-scale analysis on the predicted cracking strains. The reported method canpredict initial and transverse cracks on quasi-isotropic laminates; further, it depicts the damage progress
wherein microcrack nucleation and coalescence occurring before the full-width transverse cracking inlaminated composites are observed under tensile loading conditions.
© 2020 Elsevier Ltd. All rights reserved.

1. Introduction
In recent years, fiber-reinforced plastics (FRPs) have been applied to a wide range of products, such as aircraft, automobiles,
and wind turbines, to achieve superior performance compared tothat offered by conventional metal-based materials (Lukaszewicz,
2013; Rana and Fangueiro, 2016; Song and Gupta, 2012; Stenzenberger, 1993). In the aerospace industry, carbon fiber reinforced
plastics (CFRPs) have been mainly used to fabricate structural components to reduce the weight of aircraft, which contributes to en∗
Corresponding author at: Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan.
E-mail address: okabe@plum.mech.tohoku.ac.jp (T. Okabe).

https://doi.org/10.1016/j.ijsolstr.2020.02.010
0020-7683/© 2020 Elsevier Ltd. All rights reserved.

hanced environmental friendliness and payload (Daniel and Ishai,
1994; Rana and Fangueiro, 2016). In aerostructures, CFRPs are primarily employed as laminates, manufactured from prepreg sheets
consisting of carbon fibers and matrix resin. These structures havea hierarchical nature that ranges from the fiber-diameter scale tothe structural component scale. The several length scales represent
some of the causes of complex failure mechanisms of composite
structures (Christensen, 2013; Talreja and Singh, 2012). To improvethe design quality, performance, and safety of structures fabricatedfrom laminated composites, accurate methods for the prediction of
failure events for composites are essential.
From the failure prediction point of view, seizing the influenceof the micro- and meso-structures on the damage evolution is key
to capturing the dominant failure mechanisms. On the microscopic

Y. Kumagai, S. Onodera and M. Salviato et al. / International Journal of Solids and Structures 193–194 (2020) 172–191

scale, the heterogeneous microstructure comprising carbon fibers
and matrix resin affects the microscopic stress/strain fields significantly. In fact, due to large material property differences between fibers and polymer matrix, large stress concentrations occur
in the matrix phase leading to microscopic damage initiation andsubsequent propagation (Asp et al., 1996a; Hobbiebrunken et al.,2006; Okabe et al., 2011). Since the foregoing microscopic damage can grow to mesoscopic damage e.g., in the form of transverse cracks and delaminations, it is important to predict damage initiation starting from the microscopic scale while accounting for material heterogeneity. Another aspect of utmost importance is the material anisotropy induced by the complex heterogeneous micro- and meso-structures. Although, if properly exploited,material anisotropy offers a great performance advantage over theisotropic nature of metals, it also causes complex deformation
fields around the free edges of laminates (Lecomte-Grosbras et al.,2009; 2013; Okabe et al., 2015). These singular deformation fields
can induce transverse cracks and subsequent delaminations, which
can eventually lead to complete failure.
Considering these phenomena, computational models for theaccurate evaluation of structural integrity must explicitly accountfor the nonlinear effects of material heterogeneity at the microscale and the laminate configuration at the mesoscale.
In the quest for an accurate failure prediction method for composite structures, numerous experimental and numerical studies
have been performed on the microscopic and mesoscopic scales.
In situ observations of transversely loaded composites by scanning electron microscopy (SEM) were performed to evaluate theonset of transverse cracks and the influence of voids on crack initiation (Hobbiebrunken et al., 2006; Aratama et al., 2016). X-ray
micro-computed tomography was also used for detailed experimental observation to gain a better understanding of microscopic
kink band formation, which is a fiber-dominant compressive failure mode, (Wang et al., 2017), mesoscopic transverse crack accumulation during tensile loading (Wright et al., 2008; Scott et al.,2012; Yang et al., 2015), and mesoscopic damage accumulation under fatigue conditions (Qiao et al., 2019; Qiao and Salviato, 2019).
The digital image correlation (DIC) technique (Bornert et al., 2009),
which enables measurement of the displacement and strain fieldsof the specimen, was employed to characterize free-edge strain
fields of cross-ply (Okabe et al., 2015) and angle-ply (LecomteGrosbras et al., 2009; 2013) laminates. To predict the experimentally observed failure behavior, numerical approaches involving thedominant mechanisms of damage evolution have been developed.
On the microscopic scale, representative volume elements (RVEs)
consisting of carbon fibers and matrix resin were developed to predict the onset of matrix cracks and fiber/matrix debonding (Aspet al., 1996b; Melro et al., 2013; Elnekhaily and Talreja, 2018; 2019;
Sudhir and Talreja, 2019), and kink band formation (Bai et al., 2015;
Naya et al., 2017). On the mesoscopic scale, intralaminar and interlaminar cracks were predicted for several laminate configurations, using laminae having homogenized orthotropic properties
with cohesive zone and/or continuum damage mechanics models (Su et al., 2015; Yang et al., 2015; Higuchi et al., 2017; Lu
et al., 2018). In addition to numerical approaches, theoretical modeling, which contributes to establishing laminate behavior description, was performed to predict intralaminar crack accumulation(Ogihara et al., 20 0 0; Singh and Talreja, 2010), stiffness degradation
caused by intralaminar cracks (Carraro and Quaresimin, 2015; Onodera and Okabe, 2019), and delamination onset due to transverse
cracks (Nairn and Hu, 1992; Carraro et al., 2017). These numerical and theoretical studies have contributed to establishing a better description of failure initiation and propagation and provided
good predictions of failure strength and damage patterns in composite laminates. However, several limitations need to be overcome
in order to improve the accuracy of the established models. In fact,

173

microscopic analysis alone cannot reflect the complex deformation
fields, due to the lack of upper scale data. On the other hand,due to homogenization of the fiber and matrix phases, mesoscopic
analysis cannot address thermal-load-induced matrix deformation
appropriately, which is known to affect the initial matrix-dominantfailure significantly (Hart-Smith, 2014).
To address these challenges, multiscale formulations that account for both the microscopic and mesoscopic characteristics havebeen developed for problems related to damage prediction ranging from static to dynamic loadings (Souza et al., 2008; Sato et al.,2014; Okabe et al., 2015). These multiscale approaches employ
two finite-element (FE) analyses on the different length scales:a microscopic analysis considering the heterogeneity of materials
and a macroscopic analysis treating each lamina as homogeneous.
By applying these methods to composite laminates, thicknessdependent cracking strains in cross-ply laminates (Okabe et al.,2015) and fiber-direction-dependent initial cracking strains in unidirectional laminates (Sato et al., 2014) have been successfully predicted. However, validation of a more practical configuration, suchas quasi-isotropic laminates, has not been demonstrated in detail.
The transverse crack is one of the dominant damages in tensileloaded composite laminates and was investigated in this study. According to extensive research undertaken on transverse cracking(Pagano et al., 1998; Okabe et al., 2015; Herráez et al., 2015; Yanget al., 2015; Zhuang et al., 2018; Kohler et al., 2019; Maragoni andTalreja, 2019), the process of a full-width transverse crack formation can be classified into three stages, as summarized in Fig. 1.
At stage 1, nucleation of microcracks is induced near the free edgeof the laminate by the free-edge stress concentration. At stage 2,
coalescence of microcracks occurs owing to microcracks generated
at stage 1 being connecting, which leads to free-edge crack formation. Finally, at stage 3, steady-state crack growth is initiated after energy criteria are satisfied, and then a full-width transverse
crack is formed. The onset and propagation of such transverse
cracks have been discussed for cross-ply laminates using a twodimensional model (Okabe et al., 2008; Arteiro et al., 2014; Herráez et al., 2015). However, the prediction capability for the damage progress depending on laminate configurations has not beenvalidated for quasi-isotropic laminates using a three-dimensionalmodel.
In this study, a multiscale approach consisting of a laminatescale FE analysis and a microscopic periodic unit cell (PUC) analysis was developed to predict the onset of matrix cracks in the90◦ layer of quasi-isotropic laminates. A mesoscopic FE analysis
that reproduced the nonlinear off-axis deformation of each ply
was performed to obtain the deformation histories at the locations where matrix crack onset was most likely to occur. Then, amicroscopic PUC analysis was conducted to predict matrix crack
initiation based on the strain histories obtained from the mesoscopic analysis. In the microscopic analysis, material nonlinearity and crack initiation in the matrix phase were simply modeledby an elasto-viscoplastic law and a multiaxial stress-based failure
criterion. Our multiscale analysis was applied to experiments on
quasi-isotropic laminates reported in literature (Kobayashi et al.,
20 0 0; Ogihara et al., 2001) and those undertaken in this work tovalidate the prediction capability of the developed multiscale approach and discuss the cracking events in laminated composites
with practical laminate configurations. For comparison with experimental data reported previously, nine laminate configurations
including cross-ply and quasi-isotropic laminates were simulated
to demonstrate the prediction capabilities for mesoscopic stress–
strain behavior and laminate-configuration-dependent transverse
cracking strains in the 90◦ layer of laminates. For further discussion of transverse crack prediction, detailed microscopic observation of initial and transverse cracking on three laminate configurations were conducted by the authors using the in situ replication

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Y. Kumagai, S. Onodera and M. Salviato et al. / International Journal of Solids and Structures 193–194 (2020) 172–191

Fig. 1. Schematic illustration of transverse cracking process in 90◦ layer of composite laminate.

2.1. Mesoscopic FE analysis of laminated composites
The laminae in a unidirectional CFRP composite under offaxis loading exhibit a nonlinear stress–strain behavior due to irreversible, nonlinear deformation in the matrix. The mesoscopic
analysis was conducted leveraging an anisotropic elasto-plastic
constitutive law proposed by Yokozeki et al. (Sun and Chen, 1989;
Yokozeki et al., 2007) to capture the nonlinear behavior of theresin. In this constitutive law, the effective stress σ̄eff and yieldfunction f are defined by the stress components associated withthe principal material axes as follows:σ̄eff =!

# 2$% 2 23"
2 + 2a2+ a1 σ11(σ22 − σ33 )2 + 2a44 σ23
66 σ12 + σ13
2

(1)

3f,

(2)

+ a1 (σ11 + σ22 + σ33 ),σ̄eff =

Fig. 2. Computational procedure for decoupled micro–macro multiscale analysis.

technique and ex situ X-ray computed tomography. The validity ofour developed approach and cracking sequence in the 90◦ layer ofquasi-isotropic laminates were discussed through the comparisonof the predicted values with experimental observations.

2. Simulation methods
The strains at microcrack initiation in laminated composites
were predicted by a multiscale approach that consists of laminatescale finite-element (FE) analysis and fiber-diameter-scale periodic
unit cell (PUC) analysis. Fig. 2 shows a flowchart summarizing thecomputational procedure of our multiscale analysis which featuresa micro–macro decoupling approach (Terada et al., 2014) to perform the two-scale analysis eﬃciently. The following subsections
describe the mesoscopic and microscopic analyses and the simulation procedure.

&

where the subscript 1 represents the fiber axis direction, the subscript 2 represents the in-plane transverse direction, and the subscript 3 represents the out-of-plane transverse direction. The yieldfunction f given by Eq. (2) was used as the plastic potential forthe flow rule (Sun and Chen, 1989) with the material parameters a44 , a66 , and a1 governing the plastic behavior. Following Satoet al. (2014), the values a44 = 2.0, a66 = 1.6, and a1 = 0.01 wereadopted in this work.
The relationship between the effective stress σ̄eff and the efp
fective plastic strain ε̄eff , which follows a Ramberg–Osgood model(Ramberg and Osgood, 1943), can be expressed as follows:' pε̄eff = A1 (σ̄eff )n1
pε̄eff= A2 (σ̄eff )n2

thresholdσ̄eff < σ̄effthresholdfor σ̄eff ≥ σ̄eff

for

(3)

where A1 , n1 , A2 , and n2 are parameters to be fitted against
the experimental uniaxial stress–strain curve. Two sets of parameters were used in the same way as reported in literature(Weeks and Sun, 1998) for accurate reproduction of nonlinear behavior. In this study, we assumed A1 = 3.2 × 10−11 , n1 = 3.8, A2 =threshold = 138 MPa, as reported by
4.5 × 10−18 , n2 = 7.0, and σ̄eff
Sato et al. (2014). A list of all the material properties used in themesoscopic analysis is provided in Table 1.

Y. Kumagai, S. Onodera and M. Salviato et al. / International Journal of Solids and Structures 193–194 (2020) 172–191

175

Fig. 3. Finite-element model used for macroscopic analysis showing the applied boundary conditions. The insert shows a detail of the three-dimensional mesh adopted inthis model.
Table 1
Material properties used in mesoscopic FE analysis.
Material constant

a44
a66
a1
A1
n1
A2
n2
threshold
σ̄eff

2.0
1.6
0.01
3.2 × 10−11
3.8
4.5 × 10−18
7.0
138 MPa

A typical FE model used in the mesoscopic analysis is shownin Fig. 3, which shows that each lamina was modeled as a homogeneous anisotropic body with fiber angles assigned based on thestacking sequence of the laminate. A three-dimensional mesh of
43,200 eight-node full-integration hexahedral elements was used
for each ply. The temperature change #T was applied to the analysis model to reproduce the thermal residual strains due to thetemperature difference between the fabrication temperature androom temperature. Then, incremental uniaxial displacement wasapplied to the edge of the analysis model with a displacement rateof 0.5 mm/min and a time increment of 0.96 s. A static explicit
FE code employing the direct sparse solver PARDISO provided by
the Intel Math Kernel Library (ASTM International, 2012) was developed and used for the mesoscopic analysis.

2.2. Microscopic periodic unit cell analysis
To predict the crack initiation under several loading conditions,a three-dimensional unit cell model comprising five carbon fibers
and a polymer matrix was developed. Fig. 4 shows the mesh of
the FE model used for the microscopic analysis along with theapplied periodic boundary conditions (PBCs). The carbon fibers
were modeled as an orthotropic elastic body whereas the mechanical behavior of the matrix was described by an isotropic elastoviscoplastic formulation. In this study, the fiber and the matrix
were assumed to be perfectly bonded in order to focus only on
the matrix cracking. Further, PBCs were imposed on the unit cell
model (Fig. 4). A three-dimensional mesh of approximately 55,0 0 0
ten-node second-order tetrahedral elements was used for the unit
cell. A static-explicit FE code employing the direct sparse solver
PARDISO (ASTM International, 2012) was developed and used forthe microscopic PUC analysis.
To predict the initiation of matrix cracking, nonlinear material
behavior was considered in the matrix phase in the PUC analysis.
The nonlinear stress–strain response of the epoxy resin was reproduced by the following an elasto-viscoplastic model:

σ˙ = C em : ε˙ −3µε̄˙ p ′
σ

σ̄

(4)

where σ is the stress tensor, C em is the elastic stiffness tensor forthe matrix resin, ε is the strain tensor, µ is the Lamé constant, ε̄ pFig. 4. FE model for microscopic PUC analysis.

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Y. Kumagai, S. Onodera and M. Salviato et al. / International Journal of Solids and Structures 193–194 (2020) 172–191
Table 2
Material properties of matrix resin used in PUC analysis.
Reference strain rate ε˙ rStrain-rate sensitivity parameter mHydrostatic stress sensitivity βHardening rule g1
g2
g3
Tensile strength of resin T
Compressive strength of resin C
Dimensionality for nonlocalization k
Reference length for nonlocalization lHere, f is the external force vector acting on St , u is the displacement vector, and δ represents the virtual component. Now
we employ quasi-static formulation to solve unknown states attime t ′ = t + #t using known physical quantities at time t. The linearization of Eq. (11) assuming the limit #t → 0 gives the following equation:

1.0 × 10−5
1/35
0.2
90 MPa0.08
20 MPa120 MPa
200 MPa3
0.15 µm),t
σ˙ : δεdV +σ˙ : δεdV
Vf
Vm
(,
,
,
t′
t
=
f · δ ud S −σ : δεdV +

#t

(,

t

St

is the equivalent plastic strain, σ̄ represents the von Mises stress,
σ ′ the deviatoric stress tensor, and · indicates time differentiation.
Following Matsuda et al. (2002), the equivalent plastic strain rateε̄˙ p was determined leveraging the following equations, including
the hydrostatic stress dependence (Okabe et al., 2011):ε̄˙ p = ε˙ r(

σ̄ + βσmg(ε̄ p )

) m1

(5)

,

g

g(ε̄ p ) = g1 (ε̄ p ) 2 + g3 .

*T
F =3 1−C+Tσˆ m + σ̄ˆ 2 ≤(7)CHere, T and C are the tensile and compressive strength of the resin,and σˆ m and σ̄ˆ are the hydrostatic and equivalent stresses normalized by C, respectively. In this study, we assumed T = 120 MPa andC = 200 MPa, as proposed in Kumagai et al. (2017). To avoid spurious mesh dependence in the presence of matrix damage, nonlocalization of the variable F was conducted according to literature(Bažant and Pijaudier-Cabot, 1988):

F̄ (x ) =1
Vr (x )

'

,

,

V

-

(8)

k|x|2
,
l2

(9)

h(s − x )dV (s ).

(10)h(x ) = exp −Vr (x ) =h(s − x )F (s )dV (s ),

V

,

Vf

σ : δεdV +,Vm

σ : δεdV =,St

f · δ udS.

(11)

(12)

σ˙ = C ef : ε˙ .

(13)

C ef

Here,
is the elastic stiffness tensor of the fiber. Substituting
Eqs. (4) and (13) into Eq. (12) gives the following virtual work
equation:

,

Vf

# e

$

C f : #ε : δεdV +

=

, t′
St

+

f · δ ud S −

,

Vm

, t

3 µ #ε p #

σ

,

Vm

# e

$

C m : #ε : δεdV

σ : δεdV

Vf +Vm$σ t ′ : δε dV.

(14)

In the PUC analysis, the displacement vector u and the strain increment #ε should be decomposed into a global component that
represents the average deformation of a unit cell and a local component that corresponds to a local deviation in the unit cell.

u = uG + uL

#ε = #εG + #εL(15)Here, the subscripts G and L represent the global component andlocal component, respectively. The global component is a constantregardless of the local coordinate system in the unit cell. Applying
this decomposition to Eq. (14), the following equation is derived:

,

Vf

# e

$

C f : #εL : δεdV +

=−

,

t

V +Vm

,

σ : δεdV +

Vm

,

# e

$

C m : #εL : δεdV

Vm$
3µ#ε p #t ′σ : δε dV

, f#
,
$
−
C ef : #εG : δεdV −
Vf

σ

# e

$

C m : #εG : δεdV.

Vm

(16)

In Eq. (16), the mechanical boundary St disappears because of theperiodic boundary condition. Finally, discretization of Eq. (16) gives
the following simultaneous linear equation:

#t

$

#

$

BeT Def Be dV ,

Km =

#

K f + t K m #U L = − t Q f + t Q m + t Q v − #Q f,G + #Q m,G

where

Here, F̄ is the nonlocalized damage parameter, V is the reference
volume, k is the dimensionality (k = 3 in this study), and l is a reference length for nonlocalization. We considered that l = 0.15 µm,
as reported previously (Okabe et al., 2011).
The material properties used in the PUC analysis are listed inTable 2.
Finally, the FE formulation for PUC analysis is briefly explained.
The fiber region, the matrix region, and the mechanical boundary
are expressed by Vf , Vm , and St , respectively. The principle of virtual work is written as,Vm)σ : δεdV .

The constitutive law for a fiber is given by(6)

Here, ε˙ r is the reference strain rate; σ m is hydrostatic stress; mis an exponent controlling the strain rate sensitivity; β defines
the hydrostatic stress sensitivity; and g1 , g2 and g3 are material
constants. We assumed ε˙ r = 1.0 × 10−5 , m = 1/35, β = 0.2, in accordance with a previous report (Okabe et al., 2011). The values
g1 = 90 MPa, g2 = 0.08, and g3 = 20 MPa were determined basedon the neat resin experiment performed by Fiedler et al. (2001).
The process of matrix crack onset was predicted by Christensen’s failure criterion (Christensen, 2013), which account for
both brittle failure under elastic deformation and ductile failure
under plastic deformation:

Vf

t

Kf =
Qf =
Qv =

#Q f , G =

.,
e

Vfe

.,
e

Vfe

.,
e

Vme

.,
e

BeT σˆ dV ,Vfe

3 µ #ε p

σ

Qm =

.,
e

.,
e

Vme

$

(17)

BeT Dem Be dV ,BeT σˆ dV ,Vme′

BeT σˆ dV,

BeT Def #εG dV,#Q m , G =

.,
e

Vme

BeT Dem #εG dV.
(18)

Here, #U is the nodal displacement increment, Kf is the stiffness
matrix of the fiber elements, Km is the stiffness matrix of the matrix elements, Q is the internal force vector, Qv is the internal force

Y. Kumagai, S. Onodera and M. Salviato et al. / International Journal of Solids and Structures 193–194 (2020) 172–191

177

Fig. 6. Strain histories of the 90◦ layer at the −45◦ /90◦ interface of the[0/ ± 45/90]s laminate obtained by mesoscopic analysis: (a) strain against temperature change and (b) strain against applied macroscopic axial strain.

strain history at the locations of expected crack initiation in
the laminate was stored at each step.
(2) The thermal stresses in the unit cell model were calculated,decreasing the temperature from the curing to room temperature. Here, the incremental global strain was adjusted by asimple feedback approach so that the global stress components were zero (Okabe et al., 2015). The stress/strain fieldobtained from this calculation was assumed as the initialstate.
(3) The strain history obtained in step (1) was applied as theglobal strain increment of the microscopic 3D PUC model.
The initial cracking strain was defined as the applied strainof the laminate at which the failure criterion described in
the previous section was first satisfied for an element.

Fig. 5. Comparison of stress–strain curves obtained by mesoscopic laminate analysis and experiments (experimental data up to first transverse cracking taken from
Kobayashi et al. (20 0 0) and Ogihara et al. (2001)).

vector derived from viscous components in Eq. (4), B is the compatibility matrix between the strain and displacement, D is theconstitutive matrix, and σˆ is a vector form of the stress tensor. Qf,Gand Qm,G are the internal force vectors driven by the macroscopicstrain increment #εG . The PUC analysis was conducted based on
the macroscopic strain increment #εG that was determined frommesoscopic analysis described in Section 2.1.
2.3. Computational procedure

3. Multiscale analysis of tensile tests on quasi-isotropic andcross-ply laminates of T80 0H/390 0-2

To account for the microscopic structure of composites andtheir macroscopic deformation simultaneously, the multiscale analysis described in the previous sections was carried out. The approach can be summarized as follows.

Multiscale crack prediction was conducted for experimental
data available in literature to demonstrate the crack prediction capability. The following subsections describe the simulation settingsof the multiscale analysis and discuss the results.

(1) The thermal residual strain for each location in the laminate
was calculated by the macroscopic 3D FEA, for the curing
temperature down to room temperature. After calculating
the thermal residual strain, incremental external displacement was applied to the FE model. During the analysis, the3.1. Problem descriptionon

Our multiscale analysis was applied to the tensile tests
quasi-isotropic and cross-ply laminates conducted by178

Y. Kumagai, S. Onodera and M. Salviato et al. / International Journal of Solids and Structures 193–194 (2020) 172–191

Fig. 7. Comparison of predicted initial cracking strains with experimental results. The blue dashed lines show experimentally characterized strains of transverse crack
initiation (Kobayashi et al., 20 0 0; Ogihara et al., 20 01). Results of simulations are represented by solid symbols. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

Kobayashi et al. (20 0 0) and Ogihara et al. (2001). In the experiments, tensile loading was interrupted at certain strain levels,and then the number of transverse cracks was counted using
optical microscopy and soft X-ray tomography. The transverse
crack was defined as a crack that completely propagated over thewidth of the laminate. In our analysis, the first transverse cracks

in the 90◦ ply were predicted for nine laminate configurations
listed in Table 3. Because of the symmetrical stacking sequenceof the laminates, only half of the laminate was discretized forthe mesoscopic FE analysis, as shown in Fig. 3. Expected cracking
points on the mesoscopic analysis were determined based on theexperimental and computational studies of the initial cracks inY. Kumagai, S. Onodera and M. Salviato et al. / International Journal of Solids and Structures 193–194 (2020) 172–191
Table 3Laminate configurations and specimen dimensions used for multiscale
analysis of T80 0H/390 0-2 laminated composites.
Laminate configuration[0/90]s[ ± 45/90]s[0/ ± 45/90]s[45/0/ − 45/90]s[ ± 45/0/90]s[0/90/ ± 45]s
[0/45/90/ − 45]s
[45/0/90/ − 45]s
[90/45/0/ − 45]s

Gauge length
(mm)

Width
(mm)

Ply thickness
(mm)

80

25

0.18

cross-ply laminates reported by Okabe et al. (2015). According totheir work, the first crack in the 90◦ ply of cross-ply laminates
was initiated at the free-edge interlaminar area. Subsequently, thecracks are generated on the midplane of the 90◦ ply. Based on
the results, strain histories used for subsequent PUC analysis wereextracted from the free-edge interlaminar area between 90◦ andneighboring plies and from the center of the thickness and widthof the 90◦ ply. The longitudinal positions for both strain-extracted
points were at the center of the length of the model, i.e. x = L/2.
Material constants used for mesoscopic and microscopic FE analyses are summarized in Table 4. These material constants werecollected from literature (Shigemori et al., 2014; Toray Composite
Materials America, 2018b; Yoshioka et al., 2016) to avoid material
property fitting. Consistency between mesoscopic analysis andmicroscopic PUC analysis was discussed in Appendix A.
3.2. Results and discussion
First, the mesoscale FE analysis was performed to obtain the strain data for the subsequent microscopic analysis. Fig. 5 compares the stress–strain curves obtained by ourmesoscopic FE analysis to the experimental data reported by
Kobayashi et al. (20 0 0) and Ogihara et al. (2001). Solid and dashedlines show our predictions whereas the symbols represent the experimental data up to the transverse crack initiation. As can benoted, our mesoscopic analysis, based on the anisotropic elastoplastic constitutive model described by Eqs. (1)–(3), successfully
reproduced the stress–strain response up to failure initiation forevery laminate configuration investigated in this work. On theother hand, Fig. 6(a) and (b) show the strain histories obtained
from the 90◦ layer at the −45◦ /90◦ interface of the [0/ ± 45/90]slaminate. In Fig. 6(a), the mesoscale strain components are plotted
as a function of the temperature change from curing temperature
to room temperature and, in Fig. 6(b), the strain components areplotted against the applied macroscopic uniaxial strain. These nontrivial strains were applied to the unit cell used in the microscopic
analysis to predict the strains at the onset of the first cracks in the90◦ layer of the laminated composites.
After imposing the macroscopic strain to the PUC, the microscopic model was solved by FEA to calculate the related stress distribution in the matrix according to the elasto-viscoplastic model
described by Eqs. (4)–(6). Then, employing Christensen’s failure criterion (Christensen, 2013) in the form summarized by Eq. (7), initiation of the first matrix cracks could be identified for the 90◦
layer. Figs. 7(a)–(i) summarize the comparison between the simulated initial cracking strains obtained by our multiscale analysis
and suggested by the experimental values as reported in literature(Kobayashi et al., 20 0 0; Ogihara et al., 20 01). In each figure, thehorizontal axes indicate the mesoscopic strain applied to the laminates, calculated by subtracting the thermal residual strain fromthe total strain of the unit cell, whereas the vertical axes schematically represent the locations in the laminate investigated in the179

PUC analysis. Colored and uncolored plots represent the simulated
results obtained from free edges and internal sections respectively,
whereas the experimental results by Kobayashi et al. (20 0 0) andOgihara et al. (2001) are represented in the figure by blue dashedlines. In all cases, the first crack of the ply initiated at the interlaminar area of the ply on the free edge, followed by a crack at theinternal section. This implies that an initial microcrack occurs at
the interlaminar on the free edge, and then grows to a transverse
crack. As seen in Fig. 7(a)–(i), internal cracks obtained by FE analysis agreed well with experimental initial transverse cracking strains
except for the [0/90/ ± 45]s laminate. To understand the transverse
crack initiation of the 90◦ ply in further detail, cracking strains of
the laminates were predicted by the following energy-based criterion, and compared with the multiscale analysis and experimental
results:&(ρ ) > &c ,&(ρ ) = −U (ρ /2 ) − 2U (ρ )
,
t

(19)
(20)

where & c is the critical energy release rate, U is the strain energy of the laminate, t is the ply thickness, and ρ is the ply crack
density. The critical energy release rate &c = 200 J/m2 was determined through comparison of predicted and experimental valuesof the [0/90]s laminate. Details on the energy-based criterion canbe found in Appendix B. The predicted cracking strains of the energy criterion are shown in Fig. 7 by red dashed dotted lines. As
seen in Fig. 7(a)–(e) and (i), the predicted values of the energy criterion agreed well with the numerical and experimental results.
On the other hand, the energy criterion overestimated the cracking
strains, as shown in Fig. 7(f)–(h). For these cases, predicted values with &c = 150 J/m2 are also shown in the figures by orange
dashed dotted lines. In the case of Fig. 7(f), the predicted value
with &c = 150 J/m2 agreed well with experimental one, which isthe case that the multiscale simulation underestimated the value.
This can be explained based on the multiscale prediction that thecritical energy release rate is decreased by the failure process zone
near the crack tip generated at the early loading stage. As seenin Fig. 7(g) and (h), the energy criterion with &c = 150 J/m2 stilloverestimated the experimental transverse cracking strain. In thesecases, the neighboring plies of the 90◦ layer had different fiber angles, e.g. 45◦ and −45◦ . The energy criterion assumed both ply interfaces to have been deformed uniformly determined by the applied strain and thus did not account for nonuniform deformation
due to different angles of neighboring plies. This discrepancy canbe one of the reasons for the overestimation of the energy criterion
as seen in Fig. 7(g) and (h).
As can be noted in Fig. 7(a)–(i), in all of the 90◦ layers, the first
crack initiated near the interlaminar area on the free edge of thelaminate. Subsequent cracks occurred at the internal section of theplies. Fig. 8 shows the stress–strain curves obtained by the PUC
analysis of the 90◦ ply of the [0/45/90/ − 45]s laminate. In thisfigure, the axial stress is the volume-averaged stress over the unit
cell whereas the strain in the x-axis represents the mesoscale axial strain composed of strains due to both thermal and mechanical loadings. The open circles correspond to the cracking strainspresented in Fig. 7(g). In the free-edge cases, the axial stress washigher than that of the internal case, and the cracking strain was
smaller than that of the internal one. This is because a multiaxial stress state was induced near the free edge by the interaction
between the 90◦ and neighboring plies. Figs. 9 and 10 present thehydrostatic stress, equivalent stress, and matrix crack distribution
obtained by PUC analysis of the [0/45/90/ − 45]s laminate at eachloading step. In the stress distribution, only the matrix elements
are visualized for clarity. In the matrix crack initiation patterns, thefiber and cracked matrix elements are plotted by blue and red elements, respectively. Based on comparison of Figs. 9(a) and 10(a),180

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Table 4
Mechanical properties used in mesoscopic FE analysis and PUC analysis of T80 0H/390 0-2 laminates
Mesoscopic FE analysisa
Longitudinal Young’s modulus E1
Transverse Young’s modulus E2 , E3
Shear modulus G12 , G13
Shear modulus G23
Poisson’s ratio ν 12 , ν 13
Poisson’s ratio ν 23
Coeﬃcient of thermal expansion for longitudinal direction α 1
Coeﬃcient of thermal expansion for transverse direction α 2 , α 3Temperature change #T
Microscopic PUC analysis
Fiber longitudinal Young’s modulus EL
Fiber transverse Young’s modulus ET
Fiber longitudinal Poisson’s ratio ν L
Fiber transverse Poisson’s ratio ν TFiber’s coeﬃcient of thermal expansion for longitudinal direction α L
Fiber’s coeﬃcient of thermal expansion for transverse direction α T
Matrix Young’s modulus Em
Matrix Poisson’s ratio ν mMatrix’s coeﬃcient of thermal expansion α mFiber volume fraction Vf
Fiber diameter df

151 GPa9.16 GPa
4.62 GPa
2.55 GPa
0.302
0.589
0 × 10−6 /K
33 × 10−6 /K−150 K294 GPab
19.5 GPac
0.17c
0.46c
−1.1 × 10−6 /Kc
10 × 10−6 /Kc
3.2 GPac
0.38c
60 × 10−6 /Kc
56%
5 µm ba

All mechanical properties for mesoscopic analysis were taken from Shigemori et al. (2014).
Properties were taken from Datasheet (Toray Composite Materials America, 2018b).
c
Properties were taken from Yoshioka et al. (2016).
b

Fig. 8. Comparison of stress–strain curves obtained by PUC analysis of [0/45/90/ − 45]s laminates.

the hydrostatic stress at the free edge was higher than that at theinternal section due to the free-edge effect. As shown in Fig. 9(b),this higher hydrostatic stress induced crack initiation at the interfiber matrix where fibers were aligned parallel to the loading direction. In the case of the internal section, the matrix crack initiated at the inter-fiber matrix at a higher axial strain because of thelower hydrostatic stress concentration than in the case of the freeedge, as indicated in Fig. 10(b). This cracking sequence in the 90◦
ply was consistent with the experimental and numerical results reported for cross-ply laminates by Okabe et al. (2015). These results
indicate that the dominant crack formation mechanism of the 90◦
ply in cross-ply and quasi-isotropic laminates is identical.
Next, to investigate the initial cracking on the 90◦ ply in quasiisotropic laminates in further detail, detailed microscopic observation of crack initiation and multiscale analysis were conducted.
4. Experimental observation and multiscale analysis of tensile
tests on quasi-isotropic and cross-ply laminates of T700S/2592
Experimental observation of microcracking and multiscale analysis were undertaken to investigate the onset of matrix cracking on
the 90◦ ply in quasi-isotropic laminates in greater detail. The ex-periment procedure, simulation conditions, and experimental andsimulated results are presented in the following subsections.
4.1. Experiments
Tensile tests on laminated composites were performed following procedures reported in the literature (Kobayashi et al., 20 0 0;
Ogihara et al., 2001) with a different material system to validate
our multiscale approach. Laminate configurations and specimen dimensions used for the experiment are presented in Table 5.
Laminate specimens were made from unidirectional prepreg
sheets of T700S/2592 (Toray Industries). The prepreg sheets werestacked with the laminate configuration shown in Table 5. The
laminated sheets were then cured in an autoclave (ASHIDA MFG
Co.) under a gauge pressure of 0.3 MPa and a holding temperatureof 130 ◦ C for 2 h. Cured laminates were cut into specimens withthe dimensions specified in Table 5. Unidirectional laminates wereused for elastic modulus measurements, and cross-ply and quasiisotropic laminates were employed for initial crack observation.
For tensile testing, GFRP tabs with a thickness of 1.5 mm werebonded on a specimen. Monotonic tensile loading with a displacement rate of 0.5 mm/min was applied to a specimen by a servohy-

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181

Fig. 9. Hydrostatic stress, equivalent stress, and matrix crack distribution obtained by PUC analysis at 45/90 interface on free edge of 90◦ ply of [0/45/90/ − 45]s laminate
at each loading step.

Table 5Laminate configurations and specimen dimensions used for experiments
and multiscale analysis of T700S/2592 laminates.
Laminateconfiguration[0]8[90]16[45]16[0/90]s[ ± 45/0/90]s[0/90/ ± 45]s
[0/45/90/ − 45]s

Length
(mm)

Width
(mm)

Ply thickness(mm)

End-tab size
(mm)

250

15

0.10

56 × 15
25 × 15
25 × 15

150

25

0.10

35 × 25

draulic testing machine (MTS Systems Corporation), and specimen
deformation was measured by strain gauges (Kyowa Electronic Instruments Co.) attached on the laminate. An edge face of cross-ply
and quasi-isotropic laminates was polished to observe transverse
cracks by the replication technique (International, 2012). For the
replication technique, the tensile loading was interrupted at intervals of 0.1% strain, and a replica of a specimen edge face was obtained using RepliSet (Struers), as shown in Fig. 11. After an initialtransverse crack was observed by microscopy on the replica films,
X-ray computed tomography was conducted to confirm that generated cracks propagated over the width of a specimen.

4.2. Simulation conditions
In the multiscale analysis, the first transverse cracks in the 90◦
ply were predicted for cross-ply and quasi-isotropic laminates aslisted in Table 5. In the same way as in the case of T80 0H/290 02, only half of the laminate was discretized for the mesoscopic
FE analysis because of the symmetrical configuration of the laminates. Mechanical properties used for mesoscopic and microscopic
FE analyses are presented in Table 6. Mechanical properties formesoscopic analysis were determined from tensile tests of unidirectional laminates listed in Table 5 and the other properties weretaken from literature (Okabe et al., 2015; Toray Composite Materials America, 2018a). In the next subsection, the simulated resultsof the two-scale analysis are compared to the experimental results
to confirm the dominant crack formation mechanism on the 90◦
ply in quasi-isotropic laminates.
4.3. Results and discussion
Fig. 12 summarizes the experimental and simulated results obtained by our experiment and multiscale FE analysis. In each figure,colored and uncolored symbols represent the simulated results obtained for free edges and internal sections, respectively. The rangesof experimental initial and transverse cracking strains on the freeedge are denoted by red and blue rectangles, respectively. The predicted cracking strains of the energy criterion are also shown in182

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Fig. 10. Hydrostatic stress, equivalent stress, and matrix crack distribution obtained by PUC analysis at center of ply thickness on internal section of 90◦ ply of [0/45/90/ −45]s laminate at each loading step.

Table 6
Mechanical properties used in mesoscopic FE analysis and PUC analysis of T700S/2592 laminates.
Mesoscopic FE analysis
Longitudinal Young’s modulus E1
Transverse Young’s modulus E2 , E3
Shear modulus G12 , G13
Shear modulus G23
Poisson’s ratio ν 12 , ν 13
Poisson’s ratio ν 23
Coeﬃcient of thermal expansion for longitudinal direction α 1
Coeﬃcient of thermal expansion for transverse direction α 2 , α 3Temperature change #T
Microscopic PUC analysis
Fiber longitudinal Young’s modulus EL
Fiber transverse Young’s modulus ET
Fiber longitudinal Poisson’s ratio ν L
Fiber transverse Poisson’s ratio ν TFiber’s coeﬃcient of thermal expansion for longitudinal direction α L
Fiber’s coeﬃcient of thermal expansion for transverse direction α T
Matrix Young’s modulus Em
Matrix Poisson’s ratio ν mMatrix’s coeﬃcient of thermal expansion α mFiber volume fraction Vf
Fiber diameter df
a

115 GPaa
7.74 GPaa
3.54 GPaa
2.78 GPaa
0.315a0.394a0.4 × 10−6 /Kb
36 × 10−6 /K b−100 K
230 GPac
16.5 GPab
0.20b
0.45b
−1.1 × 10−6 /Kb
10 × 10−6 /Kb
3.4 GPab
0.31b
60 × 10−6 /Kb
51%a
7 µmc

Properties were determined from tensile tests of unidirectional laminates listed in Table 5.
Properties were taken from Okabe et al. (2015).
c
Properties were taken from Datasheet (Toray Composite Materials America, 2018a).
b

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183

Fig. 11. Interrupted tensile tests of laminated composites with the replication technique.

Fig. 12 by red dashed dotted lines. In the microscopic replica observation in the experiment, all initial cracks occurred at the interlaminar area on the free edge, as seen in Fig. 13(a), and then acrack encompassing the entire thickness of the ply on the free edge
was generated. It should be noted that we observed both the fullthickness crack generated from the initial interface crack and thatgenerated independently from the initial interface crack. In addition, no evident delamination was seen near the initial cracks andthe subsequent free-edge cracks. In the X-ray CT experiments, as
shown in Fig. 13(b), the generated free-edge crack did not growto a full-width crack at the free-edge cracking strain. This cracking behavior agrees with previous experimental reports, whichstated that thin-ply quasi-isotropic laminates can suppress immediate full-width transverse crack formation after free-edge crack
initiation (Kohler et al., 2019). This experimental observation was
supported by the analytical prediction using the energy criterion.
Because a full-width transverse crack was not observed in the experiments, identifying the critical energy release rate by comparison of prediction of the energy criterion with experimental values was inappropriate. Hence, the typical value of an energy release rate of 200 J/m2 was employed for the discussion below. Theenergy criterion using the typical energy release rate of 200 J/m2
overestimated the free-edge cracking strains of Fig. 12(a) and (b)Fig. 13. Microscopic crack observations by replication technique and X-ray CT in
quasi-isotropic laminate.

and predicted no transverse crack formation for the applied strain
of 2%, as shown in Fig. 12(c). These results indicate that the ply
thickness of 0.10 mm was thin enough to suppress immediate fullwidth transverse crack formation in this case. Note that when afull-width transverse crack is formed in the present case with large
applied strains shown in Fig. 12, delamination can also be formed
because of the large stress in the laminate. This case can be predicted by analytical models proposed in literature (Nairn and Hu,
1992; Carraro et al., 2017).
In the case of multiscale analysis, the predicted initial cracking strains on the free edge agreed with the experimental results.

Fig. 12. Comparison of predicted initial cracking strains with experiment results obtained for T700S/2592 laminates.

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Fig. 14. Cracking strain distribution along the midplane of the 90◦ layer of quasi-isotropic laminates.

In addition, although transverse cracks did not propagate over thewidth of the laminates, the simulated cracking strains at the internal section agreed reasonably with the experimental free-edge
cracking strains.
To discuss the free-edge cracking event in further detail, additional FE analysis was conducted on the midplane of the 90◦ ply.
Fig. 14 shows the cracking strain distribution obtained by two-scale
analysis on the midplane of the 90◦ layer of the quasi-isotropic
laminates. The horizontal axis represents the depth of analysis
points from the free edge, and the vertical axis illustrates crack
onset strains at each analysis point. The cracking strain at the internal section is depicted by red dashed lines. As shown in the figure, the cracking strains near the free edge converged on the internal cracking strain, and the convergence distance from the freeedge was comparable with the thickness of several plies. Assumingthat a free-edge crack is formed as a semicircular crack, the crack
initiation at the distance equal to the thickness of the 90◦ layer
can be considered as a simplified criterion of the free-edge crack
initiation. In Fig. 14, a depth equal to the 90◦ layer thickness andthe experimental free-edge cracking strains, which are also shownin Fig. 12, are illustrated by black dashed lines and blue bands,
respectively. Comparison of the results of the two-scale analysis
and the experiments showed that the predicted cracking strains at
the depth equal to the thickness of the 90◦ layer reasonably reproduced the experimental onset strains despite our simplified assumption.
To validate the developed two-scale approach in more detail,the influence of fiber arrangement in PUC analysis on cracking
strains was examined. Fig. 15 shows the unit cell models used forthe validation of the fiber arrangement effect. The unit cells withrandomly distributed fibers were generated by following literature(D’Mello et al., 2016). Note that Model I is identical to the unit cell
model shown in Fig. 4, hence, two additional models were analyzed for the validation. Fig. 16 compares predicted cracking strainsobtained by the two-scale analysis using three different unit cells.
The circular, square, and diamond symbols represent the predictedcracking strains of Model I, II, and III, respectively. Although thepredicted cracking strains were slightly affected by the fiber arrangement, the cracking sequence, which is crack initiation at thefree-edge interlaminar area followed by cracking at internal section, was identical among the three cases. This result indicates thatthe prediction capability of the developed two-scale analysis wasinsensitive to the fiber arrangement of unit cells.
Moreover, we evaluated the influence of the constitutive modelof mesoscopic analysis and the constitutive and failure models of PUC analysis on crack prediction of quasi-isotropic laminates. In mesoscopic analysis, each lamina was modeled as an orthotropic elastic body. In PUC analysis, based on the discussion ofOkabe et al. (2015), the matrix phase of a unit cell was assumed
to be an elastic body and crack initiation was modeled by thefollowing dilatational energy density (DED) criterion proposed by
Asp et al. (1996a):Uv =1 − 2ν
(σ1 + σ2 + σ3 )2 .
6E

(21)

Here, Uv is the dilatational energy density, E is Young’s modulus, νis Poisson’s ratio, and σ i is the principal stress. A matrix crack was
assumed to occur when Uv reached its critical value Uvcrit . We employed Uvcrit = 1.1 MPa, referring to literature (Okabe et al., 2015).
Fig. 17 compares the predicted cracking strains obtained by theelastic modeling case with the nonlinear modeling case and experiments presented in Fig. 12. The modeling strategies of cases

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185

Fig. 15. Unit cell models for the validation of fiber arrangement effect.

Fig. 16. Comparison of predicted cracking strains obtained by PUC analysis using three fiber configurations.

Fig. 17. Effect of constitutive and failure modeling of two-scale analysis on predicted cracking strains. The difference between the cases EP and E is constitutive and failure
models in two-scale analysis and summarized in Table 7.

EP and E, and detailed computational results are summarized inTable 7. As shown in the results, the predicted results of case Emodeling were practically identical to those of case EP modeling.
These results indicate that, if the transverse cracks in the 90◦ layerof quasi-isotropic laminates are analyzed, the effect of plastic deformation on the mesoscopic and microscopic scales on crack prediction is negligible, and hydrostatic stress-induced cracking has
the dominant effect on crack prediction.

Finally, the results of our two-scale analysis were interpreted
along with the transverse cracking process described in Fig. 1. Our
microscopic PUC analysis only modeled the nucleation of microcracks in stage 1 and successfully reproduced the initial cracking
strains near the free-edge interlaminar area. In addition, two-scale
analysis of the internal section of laminates reasonably captured
the free-edge cracking, which can be caused by coalescence of microcracks in stage 2, despite the simplified modeling in PUC analy-

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Table 7Comparison of predicted cracking strains on different constitutive and failure modeling strategies of two-scale analysis.
Laminate configuration

Location

Case EP
Mesoscale: elasto-plastic
PUC: elasto-viscoplastic
Failure: Christensen

Case E
Mesoscale: elastic
PUC: elastic
Failure: DED[0/90]s0/90 interface at free edge (%)
Internal section (%)
0/90 interface at free edge (%)
Internal section (%)
0/90 interface at free edge (%)
90/45 interface at free edge (%)
Internal section (%)

0.542
0.792
0.412
0.862
0.502
0.522
0.862

0.552
0.782
0.422
0.872
0.513
0.563
0.873

[ ± 45/0/90]s[0/90/ ± 45]s

sis. As shown in Fig. 14, the cracking strains on the midplane of 90◦
plies converged near the free edge of the laminates where the constraining effect by neighboring plies is weaker than that offered by
the internal section. Crack initiation at the convergence point canbe considered as a suﬃcient condition for a free-edge crack formation assuming the free-edge crack formed as a semicircular shapecrack. Hence, the prediction at the internal section acted as a representative point for the free-edge crack prediction. The full-widthtransverse crack formation in stage 3 could be predicted for thecase of T80 0H/390 0-2 only by PUC analysis. In this case, the 90◦
layer was thick enough to satisfy the energy criterion immediately
after stress-based crack nucleation.
One possible approach to predict the entire process of transverse cracking is micro-mechanical FE modeling, in which a unit
cell consisting of fibers and matrix is embedded between homogenized plies (Arteiro et al., 2014; Herráez et al., 2015). In this kindof modeling, matrix damage is modeled by continuum damage
mechanics and crack band (Bažant and Oh, 1983) models and all
cracking stages presented in Fig. 1 can be captured. However, application of this strategy is still limited to two-dimensional cases
because of the appreciable computational costs involved. Future
development of a high-performance computing infrastructure will
help us establish a detailed multiscale FE approach for transverse
crack prediction. Our future work will involve detailed discussionof the full transverse cracking process obtained computationally,
although the major cracking process for tensile loading cases presented here will not be affected by the investigation.

modeling of the two-scale analysis on predicted results was examined. It showed that the prediction of the developed two-scale approach was insensitive to the fiber configuration of unit cells. If thetransverse cracks in the 90◦ layer of quasi-isotropic laminates are
predicted, the results indicated that inelastic deformation on themesoscopic and microscopic scales did not affect the crack prediction and that the hydrostatic stress-induced cracking was influential in crack prediction. The reported method shows the damage
progress wherein microcrack nucleation and coalescence followedby the full-width transverse cracking in laminated composites thatcan be seen under tensile loading conditions.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared toinfluence the work reported in this paper.
CRediT authorship contribution statement
Yuta Kumagai: Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review &editing, Visualization. Sota Onodera: Validation, Formal analysis,
Investigation, Writing - review & editing. Marco Salviato: Writing review & editing. Tomonaga Okabe: Conceptualization, Resources,
Writing - review & editing, Supervision.
Acknowledgements

5. Conclusions
In this work, a multiscale study on matrix crack initiation in
the 90◦ ply of cross-ply and quasi-isotropic laminates was undertaken. The laminate-scale deformation behavior was reproduced bya mesoscopic FE analysis that assumed each lamina to be a homogeneous anisotropic elasto-plastic body. The material nonlinearity and the process of crack formation on the fiber-diameter scale
were simply modeled by an elasto-viscoplastic constitutive modeland a two-parameter stress-based failure criterion in a microscalelevel FEA in a PUC. Our multiscale formulation, combining mesoscopic and microscopic FE analyses, indicated that the cracking
sequence in the 90◦ ply in quasi-isotropic laminates was identical to the one in cross-ply laminates. Cracks in the 90◦ layer
initiated at the interlaminar area on the free edge, followed by
microcracks at the internal section. In addition to the computational work, detailed microscopic observation of crack initiation
was conducted on cross-ply and quasi-isotropic laminates, using
the in situ replication technique and ex situ X-ray computed tomography. Comparison of simulated results with microscopic observations demonstrated that our multiscale analysis can reasonably predict the initial microcrack induced near the free-edge interlaminar area and the free-edge crack. In addition, the sensitivity
to fiber arrangement and the influence of constitutive and failureThis work was supported by Council for Science, Technology
and Innovation (CSTI), the Cross-ministerial Strategic Innovation
Promotion Program (SIP), “Materials Integration” for Revolutionary
Design System of Structural Materials (Funding agency: JST). The
authors would like to acknowledge the crucial encouragement andsupport received from the University of Washington–Tohoku University: Academic Open Space (UW–TU: AOS). This work was also
supported by Japan Society for the Promotion of Science (JSPS)
Grant-in-Aid for JSPS Research Fellow Numbers JP17J02004 andJP18J20899. The authors thank Toray Industries, Inc. and Dr. Nobuo
Takeda, Dr. Shu Minakuchi, and Mr. Shinsaku Hisada of the University of Tokyo for their support in specimen fabrication. The authors would like to thank Editage (www.editage.com) for English
language editing.
Appendix A
In this section, the compatibility between mesoscopic analysis
and microscopic PUC analysis was examined, and the influence of
their mechanical properties on the prediction of crack initiation
was evaluated. First, the two-scale analysis was applied to tensile
tests of unidirectional laminates under off-axis loading to present
compatibility between mesoscopic and microscopic analysis, using

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187

Fig. A1. Comparison of stress–strain curves obtained by mesoscopic laminate analysis, PUC analysis, and experiment of unidirectional laminates (experimental data takenfrom Yoshioka et al. (2016)).

Table A1
Mechanical properties used in mesoscopic FE analysis and PUC analysis of T80 0H/390 0-2B laminates. Mechanical properties of fiber and matrix were modified from Table 4 to improve compatibility between mesoscopic and PUC analyses.
Mesoscopic FE analysisa
Longitudinal Young’s modulus E1
Transverse Young’s modulus E2 , E3
Shear modulus G12 , G13
Shear modulus G23
Poisson’s ratio ν 12 , ν 13
Poisson’s ratio ν 23
Coeﬃcient of thermal expansion for longitudinal direction α 1
Coeﬃcient of thermal expansion for transverse direction α 2 , α 3Temperature change #T
Microscopic PUC analysis
Fiber longitudinal Young’s modulus EL
Fiber transverse Young’s modulusET
Fiber longitudinal Poisson’s ratio ν L
Fiber transverse Poisson’s ratio ν TFiber’s coeﬃcient of thermal expansion for longitudinal direction α L
Fiber’s coeﬃcient of thermal expansion for transverse direction α T
Matrix Young’s modulus Em
Matrix Poisson’s ratio ν mMatrix’s coeﬃcient of thermal expansion α mFiber volume fraction Vf
Fiber diameter df
a

151 GPa9.16 GPa
4.62 GPa
2.55 GPa
0.302
0.589
0 × 10−6 /K
33 × 10−6 /K−150 K260 GPab
19.5 GPac
0.17c
0.46c
−1.1 × 10−6 /Kc
10 × 10−6 /Kc
3.8 GPad
0.38c
60 × 10−6 /Kc
56%
5 µme

All mechanical properties for mesoscopic analysis were taken from Shigemori et al. (2014).
Properties were taken from Tane et al. (2019).
c
Properties were taken from Yoshioka et al. (2016).
d
Assumed value.
e
Properties were taken from Datasheet (Toray Composite Materials America, 2018b).
b

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Fig. A2. Comparison of stress–strain curves obtained by mesoscopic laminate analysis, PUC analysis, and experiment of unidirectional laminates after material constantfitting (experimental data taken from Yoshioka et al. (2016)).

Fig. A3. Influence of mechanical properties used for PUC analysis on the predicted cracking strains. Two-scale analysis was carried out for [ ± 45/0/90]s laminate with
T80 0H/390 0-2 case as shown in Fig. 7(e). The blue dashed lines show experimentally characterized strains of transverse cracking (Kobayashi et al., 20 0 0; Ogihara et al.,2001). Simulated results using properties from Tables 4 and A.1 are represented by circular and square symbols, respectively. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

the parameters listed in Table 4. Fig. A.1 shows the comparison ofstress–strain curves obtained by mesoscopic analysis and microscopic PUC analysis. In the figure, the dashed lines represent theresults of mesoscopic analysis and the solid lines illustrate the results of PUC analysis. Moreover, experimental data available in literature (Yoshioka et al., 2016) is depicted by symbols for the purpose of validation. The compatibility between mesoscopic and PUC
analyses can be improved by fitting the mechanical properties.
Subsequently, the fiber longitudinal Young’s modulus EL and thematrix Young’s modulus Em were adjusted to improve the compat-

ibility of the two FE analyses. The value EL = 260 GPa was determined based on experimental measurement of elastic constants ofcarbon fibers reported in literature (Tane et al., 2019) and the value
Em = 3.8 GPa was identified through the comparison of stress–
strain curves between mesoscopic and PUC analyses. The mechanical properties obtained after stress–strain curve fitting are summarized in Table A.1. Fig. A.2 compares the stress–strain curvesof the off-axis tensile tests of unidirectional laminate obtained bymesoscopic and PUC analyses. By employing mechanical properties

Y. Kumagai, S. Onodera and M. Salviato et al. / International Journal of Solids and Structures 193–194 (2020) 172–191

listed in Table A.1, the compatibility between mesoscopic and PUC
analyses was improved.
Finally, the influence of the compatibility between mesoscopic
and PUC analyses on the prediction of the crack initiation was
evaluated by predicting cracking strains on the 90◦ layer of quasiisotropic laminate. Fig. A.3 compares the predicted cracking strainsof [ ± 45/0/90]s laminate in the case of T80 0H/390 0-2 obtained using the mechanical properties listed in Tables 4 and A.1. Although
the predicted cracking strains were marginally affected by the mechanical properties, the cracking sequence, which is the crack initiation at the free-edge interlaminar area followed by cracking at
the internal section, was identical between the two cases. This result indicates that the compatibility between mesoscopic analysis
and microscopic PUC analysis was suﬃcient for the developed twoscale analysis in the present case.
Appendix B
An analytical model (Onodera and Okabe, 2019) used to predict
steady-state cracking strain in the 90◦ ply of quasi-isotropic laminates is briefly described here. Steady-state ply cracking is a fracture mode wherein a ply crack propagates over the width of thespecimen under constant thermo-mechanical loading. The model
used here assumes that a new ply crack is generated between two
pre-existing cracks with a crack spacing 4l: thus, a unit cell of atarget ply shown in Fig. B.1 was used for the prediction. The energy release rate associated with new crack formation is expressedas:

U (ρ /2 ) − 2Uk (ρ ),tk&k (ρ ) = − k

(B.1)

189

where Uk (ρ ) is the strain energy of the kth ply of N-ply laminate
calculated on the unit cell with crack spacing 2l and ply thickness
tk and ρ is the ply crack density and can be defined as ρ = 1/(2l ).
The strain energy Uk (ρ ) is given by the following equation:

Uk (ρ ) =tL{σ̄ L − σ̄ th }T S̄L {σ̄ L − σ̄ th }.
2ρ

(B.2)

Here, σ̄ L is the applied stress of the laminate, σ̄ th is the laminate stress that removes the contribution of the thermal residual stress of the kth ply from Uk (ρ ), tL is the laminate thickness,and S̄L is the effective compliance matrix of the laminate with ply
cracks. S̄L can be calculated using the three-dimensional laminate
theory (Gudmundson and Zang, 1993) and the continuum damage
mechanics based on the effective stiffness matrix C as describedin literature (Onodera and Okabe, 2019). Detailed formulation of
the analytical crack prediction model can be found in literature(Onodera and Okabe, 2019).
Steady-state ply cracking analysis of the laminate with arbitrarylay-ups was conducted by the following procedure. First, the energy release rate of each ply, & k , according to Eq. (B.1) was calculated for the applied laminate stress σ̄ L as a function of the ply
crack density ρ , assuming that cracks are formed only in the kth
ply. Then, the critical applied laminate stress associated with kth
ply cracking, σ̄ cL,k , was determined as the applied laminate stresswhere max [&k (ρ )] is equal to the critical energy release rate & c .
ρFinally, the initial steady-state cracking strain of laminate was determined as the strain at which a ply crack was formed in a ply
subjected to the minimum ply cracking applied stress min[σ̄ cL,k ].

Fig. B1. Representative unit cell used for analytical prediction of steady-state cracking in kth ply of laminate.

k

190

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