Experimental and Micromechanical Investigation ofT300/7901 Unidirectional Composite Strength

Y. Q. Zhao ,1,2 Y. Zhou,1 Z. M. Huang,1* R. C. Batra21School of Aerospace Engineering & Applied Mechanics, Key Laboratory of the Ministry of Education forAdvanced Civil Engineering Materials, Tongji University, Shanghai, 200092, China

2Department of Biomedical Engineering and Mechanics, Virginia Polytechnic Institute and State University,Blacksburg, Virginia

We experimentally find mechanical properties in uniaxialtension/compression and torsional deformations of the7901 epoxy resin used as a matrix to fabricate T300/7901unidirectional (UD) fiber-reinforced composite. A seriesof off-axis tensile tests on the composite were con-ducted, and micrographs of the fiber–matrix interface atdifferent load levels, using a Scanning Electron Micro-scope, taken to ascertain their ultimate tensile strengths.Values of the elasto-plastic parameters for the epoxydetermined from the tests, and the handbook values ofthe fiber parameters were used in a micromechanics-based bridging model to predict ultimate tensilestrengths of off-axis loaded composite by consideringstress concentration effects due to debonding at thefiber/matrix interface. For the 30� off-axis loaded lami-nate, the predicted ultimate strength is found to agreewell with that determined experimentally, and the inter-face debonding occurs before the ultimate failure. TheUD composite is assumed to fail when either the fiber orthe matrix failure criterion is satisfied. POLYM. COMPOS.,40:2639–2652, 2019. © 2018 Society of Plastics Engineers

INTRODUCTION

A major challenge in the fiber-reinforced compositecommunity is the prediction of failure initiation and of theultimate strength of a laminate subjected to an arbitraryload. Many important issues, such as matrix failure/crack-ing and fiber breakage, relevant to the failure analysis of alaminate cannot be well addressed without knowing stres-ses in the fiber and the matrix. Another important failure

mode of interface debonding requires accurately analyzingdeformations at and near the fiber/matrix interface. It iswell known that the load carrying capacity of a compositeis dominated to a large extent by the fiber/matrix interfacecharacteristics. For a given loading scenario, the load whenthe interface debonding occurs cannot be well understoodwithout knowing stresses and deformations at and near thefiber/matrix interface. Most current approaches to charac-terize composite failure are phenomenological, for exam-ple, see the world-wide failure exercises (WWFEs) [1,2].Furthermore, very few of the micromechanics approachesemployed in the exercises directly used as input parametersthe constituent properties and the matrix strength providedby the exercise organizers. These were subsequentlyadjusted using the measured composite data [3–5].

Recently, computational micromechanics including mul-tiscale modeling [6] has become a powerful tool to investi-gate failure mechanisms and predict ultimate strengths ofcomposites subjected to various loads. A UD laminateunder off-axis tension loading is frequently used as abenchmark case to assess the efficiency of a developed the-ory. Table 1 summarizes some such typical attempts.

Zhang et al. [7] used the maximum principal strain fail-ure criterion to predict crack initiation in the matrix, andfound that the predicted critical strain was not close to thatdetermined experimentally in uniaxial tensile deformationsof the resin. Asp et al. [8] used the dilatational energy den-sity criterion for matrix failure under multi-axial stressstates. They developed a Poker–Chip experimental methodto find the critical dilatational energy density of a resin.However, the dilatational energy density only considers thesum of the three principal stresses, which equals the sum ofthe normal stresses on three mutually perpendicular planes.Hence it may not well predict the ultimate strength for anoff-axis loaded UD composite because of neglecting shearstresses. Asp et al.’s simulations showed that the fiberarrangement patterns in a representative volume element(RVE), for example, square, hexagonal and square diago-nal, significantly influenced the predicted moduli and

Correspondence to: Z.M. Huang; e-mail: huangzm@tongji.edu.cn.Financial supports from the National Natural Science Foundation of China(Grant Nos. 11832014, 11472192) are acknowledged.Contract grant sponsor: US Office of Naval Research (ONR); contractgrant number: N00014-18-1-2548. contract grant sponsor: National Natu-ral Science Foundation of China; contract grant numbers: 11832014;11472192.DOI 10.1002/pc.25059Published online in Wiley Online Library (wileyonlinelibrary.com).© 2018 Society of Plastics Engineers

POLYMER COMPOSITES—2019

strengths [8]. Kumagai et al. [6] also employed the dilata-tional energy density criterion by discretizing a UD com-posite into a series of block elements, and considering eachas a localized square RVE. The position and the directionof a crack in a RVE were presumed by using a methoddeveloped in Ref. 9. The critical parameters in the criterionwere retrieved from the data of failure tests on the compos-ite, rather than from the experimental data of the resin.Govaert et al. [10] introduced a hybrid experimental/numerical technique to analyze off-axis tensile strengths ofa UD composite. They found critical shear strain when theaveraged Cauchy stress in a hexagonal RVE subjected to10� off-axis tension equaled the measured tensile strength.It is possible that the combined experimental/numericalmethod improves the model accuracy.

The aforementioned methods did not consider a weakfiber–matrix interface. By a weak interface we mean thatinitially there is perfect bonding but debonding occursbefore the ultimate failure. Experiments have shown thatthe debonding of a weak interface greatly affects the off-axis tensile properties of UD composites [11–15]. Theinterface properties depend not only upon those of the fiberand the matrix, but also on the surface treatment and chem-ical reactions occurring during the fabrication pro-cess [16,17].

The interface damage in UD composites has been stud-ied both experimentally [18] and theoretically [19–24].Kim et al. [19] and Okabe et al. [20] developed 3D shear-lag models to analyze effects of interface debonding on thestrength behavior of a UD composite. Okabe et al. [20]considered the size effect of a debonded interface arclength on the response of a UD composite. Using Sillinget al.’s [21] peridynamics approach, Kilic et al. [22] founda good agreement between the computed and the experi-mentally observed crack path in an off-axis loaded UD

composite. Vaughan et al. [23] developed a random RVEwith 80 fibers to realistically model fiber stacking in a UDcomposite. They used a traction-separation law to simulatedebonding at the interface, and a coupled experimental/ana-lytical technique to ascertain the constitutive and damageparameters for a cohesive zone element (CZE). They quali-tatively studied the effect of interface debonding on thetransverse strength of a UD composite, but did not reportthe ultimate strength of off-axis loaded composites. Agh-dam et al. [24] using CZEs showed that the off-axis tensilefailure was dominated by the interface failure, which wasdetected by using a criterion based on the radial, the cir-cumferential and the axial stresses at the interface. Theyselected parameters in the interface constitutive modelthrough a trial and error curve fitting to multi-axial loadingtest data [25], and showed good correlation between pre-dicted and experimental off-axis strengths for metal/matrixcomposites. However, it is generally difficult to obtain thethree strength parameters of the interface. Recently, Huanget al. [26] have developed an analytical method to identifythe initiation of interface debonding.

Here, we experimentally and numerically find tensilestrengths of the off-axis loaded T300/7901 UD compositeconsidering debonding at the fiber/matrix interface. Thecomputational model employs a micromechanics-basedbridging model, experimentally determined elasto-plasticmechanical properties of the 7901 matrix and the literaturevalues of the T300 fiber [27]. A series of off-axis tensiletests, and scanning electron microscope micro-graphs ofthe fiber–matrix interfaces at different load levels are usedto identify the load at which the interface debondingoccurred. It is found that the predicted and the measuredoff-axis tensile strengths as well as the load at debondinginitiation for the 30� off-axis specimen correlate reasonablywell with each other.

TABLE 1. A summary of studies on off-axis strengths

Interfaceassumption Stress calculation

Researchorientation

Presetcrack Input data source

Experimentalverification

Failurepart Reference

Perfectinterface

Finite element square RVE withtwo fibers

Off-axisstrength

Cracklocation

Mixed experiments/numerical analysis

Off-axistensile test

Matrixfailure

[6]

Perfectinterface

Finite element square RVE withone fiber

45� Off-axisstrength

None Original constituentsproperties

Other’s data Matrixfailure

[7]

Perfectinterface

Finite element RVE with twofibers in three kinds ofarrangements--square,hexagonal, square-diagonal

Transversestrength

None Original constituentsproperties

Transversetensile test

Matrixfailure

[8]

Perfectinterface

Finite element hexagonal RVEwith two fibers

Off-axisstrength

None Mixed experiments/numerical analysis

None Matrixfailure

[10]

Weakinterface

3D shear lag method Axial strength None Spectral experiments Other’s data Fiberfailure

[20]

Weakinterface

Peridynamics Off-axis crackpropagation

Initialcrack

Experiments deduceddata

Other’s data Interfacefailure

[22]

Weakinterface

Finite element random RVE with80 fibers

Transversestrength

None Mixed single fiberexperiments/analyticalmethod

Other’s data Interfacefailure

[23]

Weakinterface

Finite element RVE with 1/4fiber

Off-axisstrength

None Trial and error curvefitting method

Other’s data Interfacefailure

[24]

2640 POLYMER COMPOSITES—2019 DOI 10.1002/pc

EXPERIMENTAL

UD Composite Material and Specimen Preparation

Sixteen 0.125-mm thick T300/7901 prepregs, containinga nominal fiber volume fraction, of Vf= 0.62, of carbonfiber reinforced epoxy polymeric composite were hand laid,and cured for 2.5 h in an autoclave at 120�C temperatureand 750 kPa pressure. We calculated the fiber volume frac-tion from the mass density of the cured T300/7901 UDlaminate, constituents’ mass densities, and the rule of mix-tures. The fiber mass density listed on manufacturer’s web-site is 1.76 g/cm3, the measured mass densities of the curedand composites’, respectively, equaled 1.20 and1.547 g/cm3. The cured panels were cut with respect to(w.r.t.) fibers at off-axis angles of 0�, 15�, 30�, 45�, 60�,75�, 90�, and pasted on 1 mm thick protective glass plate

at each end of the 20 mm wide specimen, as shown inFig. 1, to give the gauge length of 100 mm.

Resin Material and Sample Preparation

The viscosity–temperature, viscosity–time curves at dif-ferent temperatures, and the heat flow–temperature curvesof the 7901 epoxy obtained through Differential ThermalAnalysis (DTA) using the Rheometer mars III, Type006-1385 with heating rate of 5�C/min and DifferentialScanning Calorimeter, TA instrument, 970501.901, areexhibited in Fig. 2a–c, respectively. It is clear from the plotof Fig. 2a that the epoxy viscosity decreases from morethan 104 Pa s at room temperature (RT) to less than 20 Pa sat temperatures above 80�C. The epoxy glass transitiontemperature read from the plot of Fig. 2c equals 120�C. Inorder to guarantee that the epoxy will not cure very fastduring the vacuuming and the pouring process, the temper-ature and the holding time were, respectively, set equal to80�C and 30 min.

The 7901 epoxy was cured by heating it to 120�C, hold-ing the temperature for 150 min, and annealing by keepingit at 120�C for 120 min. The epoxy samples lathed to the

FIG. 1. Dimensional size of off-axis UD composite specimens (mm).

FIG. 2. 7901 epoxy. (a)Viscosity versus temperature curve, (b) viscosity–temperature versus time curve, and(c) heat flow versus temperature curve.

DOI 10.1002/pc POLYMER COMPOSITES—2019 2641

geometry prescribed by the GBT 2567-2008 Standard areexhibited in Fig. 3.

Tensile Tests on Off-Axis UD Composite

Following ASTM Standard D3039, for each offsetangle, six specimens with two L-shaped strain gagesaffixed on them to measure the axial and the lateral strainswere tested on an Electromechanical Universal TestingMachine (WD-20A) at a constant displacement rate of1.25 mm/min at RT.

We have plotted in Fig. 4 the axial stress vs. the axialstrain curves in global coordinates for the composite. Thecurves are least squares fits to the data averaged for the speci-mens tested. Whereas the curves for the off-axis angles of 15�

and 30� are nonlinear, those for the off-axis angles of 45�,60�, 75�, and 90� are essentially linear before failure.

We have listed in Table 2 the measured ultimate axialstrengths and the axial Young’s moduli.

Epoxy Characterization

Following GBT 2567-2008 Standard, the epoxy speci-mens were tested in uniaxial tension and compression onan Electromechanical Universal Testing Machine (WD-20A) at a constant displacement rate of 1.25 mm/min atRT. Two strain gages bonded to the specimens measuredthe axial and the lateral strains. To measure shear proper-ties, the specimens were tested in torsion using a microcontrol torsion testing machine (NDW30500) at a relativeangular displacement between the two clamped ends of50�/min at RT. Test systems are shown in Fig. 5.

The stress–strain curves derived from the test data, aver-aged over the number of specimens tested, and photos offractured specimens are exhibited in Fig. 6.

FIG. 3. 7901 epoxy samples. (a) Tensile, (b) compression, and (c) torsion test. [Color figure can be viewed atwileyonlinelibrary.com]

FIG. 4. Averaged axial tensile stress–axial strain responses of the UD composite. (a) Zero degree axial ten-sion, (b) off-axis tension.

2642 POLYMER COMPOSITES—2019 DOI 10.1002/pc

The monolithic epoxy is assumed to be isotropic.Young’s modulus, Em, the shear modulus, the ultimatestrength, and the hardening parameter found from the testdata are listed in Tables 3 and 4. Both the strengths and theelastic moduli have low standard deviations. The slightlydifferent values of Young’s moduli in tension and compres-sion are within experimental errors. Poisson’s ratio com-puted from values of Young’s modulus and the shearmodulus equals 0.355. We note that Poisson’s ratio = 0.5for an incompressible plastic and isotropic material.

SEM Images

Samples were cut from the UD composite specimensand imaged using a SEM with an acceleration voltage of20 kV to observe topologies of fractured surfaces. In orderto counteract effects of polymer charging in the SEM underhigh voltage and essentially vacuum conditions, theobserved surfaces were sputter-coated with a conductivegold alloy at a deposition current of 15 mA for 3 min.

The micrographs exhibiting the fracture surface and thefiber–matrix interface damage morphology of the tensilespecimens are displayed in Fig. 7a. Those images indicatedmixed failure modes at the fiber–matrix interfaces andwithin the matrix. To observe the progressive fiber/matrixinterface damage developed in the UD composite duringtensile loading, SEM micrographs of the 30� specimensloaded to 30, 50, and 70 MPa axial stress are shown inFig. 7b–d. We conclude from Fig. 7b that the fiber/matrix

interface is perfectly bonded for axial stress = 30 MPa.However, the interface has debonded at the axialstress = 50 MPa, see Fig. 7c. This debonding became clearlydiscernable in the images taken at high magnification of5,000×. At the axial stress of 70 MPa, as shown in Fig. 7d,significant interface debonding and localized regions oftransverse matrix cracks had developed. Capturing preciselythe load when debonding initiated requires taking in situSEM images as the specimen is being loaded. However, thiscould not be done because of a lack of such facilities.

Using a smart cell phone, we recorded sound emittedduring off-axis tensile loading of a UD laminate. However,we could not capture the precise instant of fiber/matrixdebonding initiation.

STRENGTH PREDICTION

Mechanical Stresses in Constituents

We assume that a plane state of stress exists in the lami-nate. We use rectangular Cartesian coordinate axes withthe x1-axis along the fiber, and denote the in-plane axialstresses by σ1 and σ2, and the in-plane shear stress by σ3.

A composite is a heterogeneous material. We define astress increment, dσi, using its value averaged over a RVE by

dσi ¼ðV’

d~σidV

0@

1A=V 0 ¼Vf dσ

fi +Vmdσ

mi , i¼ 1,2,3, ð1Þ

TABLE 2. Tensile properties (with standard deviation or errors/%) of T300/7901 UD composite with fiber volume fraction of 62%.

Elasticity modulus (GPa) Strengths (MPa)

Experimentalresults

Predicted results withweak interface

Predicted results withideal interface

Experimentalresults

Predicted results with weakinterface and TRS

Predicted results with idealinterface and TRS

0� 137.7(0.18) 143.8(4.4) 143.8(4.4) 1784.9(4.2) 1560.2(13.0) 1560.2(13.0)15� 45.05(0.35) 47.39(5.2) 47.39(5.2) 223.5(1.51) 222.9(0.3) 241.6(8.1)30� 17.23(0.69) 18.33(6.4) 18.33(6.4) 104.0(6.30) 100.8(3.1) 123.9(19.1)45� 10.22(0.23) 11.37(11.2) 11.37(11.2) 64.50(5.84) 63.9(0.9) 87.4(35.5)60� 9.02(0.17) 9.34(3.5) 9.34(3.5) 57.30(2.68) 49.2(14.1) 71.5(24.8)75� 8.58(0.01) 8.92(3.9) 8.92(3.9) 42.8(2.9) 41.9(2.1) 64.4(50.5)90� 8.04(0.22) 8.93(11.0) 8.93(11.0) 40(3.26) 40.0(0.0) 62.3(55.8)

FIG. 5. Test systems. (a) Tensile, (b) compression, and (c) torsion test. [Color figure can be viewed atwileyonlinelibrary.com]

DOI 10.1002/pc POLYMER COMPOSITES—2019 2643

where, V0 is the RVE volume. Using the bridging model

equation, dσmi� �¼ Aij

� �dσ fj

n o, where [Aij] is the bridging

tensor [28], increments in the fiber and the matrix stressesare related to that in the macroscopic stress by

dσ fin o

¼ Vf I½ �+Vm Aij

� �� �−1dσj� �¼ Bij

� �dσj� � ð2aÞ

dσmi� �¼ Aij

� �Bij

� �dσj� � ð2bÞ

Superscripts f and m on a quantity signify, respectively,its value for the fiber and the matrix. Furthermore,

Aij

� �¼ a11 a12 a130 a22 a230 0 a33

24

35 ð3aÞ

a11 ¼Em=Ef11 a12 ¼

Sf12−Sm12

� a22−a11ð Þ

Sm11−Sf11

a13 ¼ d2β11−d1β21β11β22−β12β21

ð3bÞ

a22 ¼ 0:3 + 0:7Em

Ef22

a23 ¼ d1β22−d2β12β11β22−β12β21

a33 ¼ 0:3 + 0:7Gm

Gf22

ð3cÞ

Em ¼Em, if σme ≤ σmY

EmT , if σ

me > σmY

(ð3dÞ

Gm ¼ 0:5Em= 1 + νmð Þ, if σme ≤ σmYEmT =3, if σ

me ≤ σmY

ð3eÞ

d1 ¼ S1m3 −S

f13

� a11−a33ð Þ ð3fÞ

d2 ¼ S2m3 −S

f23

� Vf +Vma11� �

a22−a33ð Þ

+ Sm13−Sf13

� Vf +Vma33� �

a12ð3gÞ

β11 ¼ Sm12−Sf12, β12 ¼ Sm11−S

f11, β22 ¼ Vf +Vma22

� �Sm12−S

f12

� ð3hÞ

FIG. 6. (a) Stress–strain curves for the epoxy. (b–d) Fractured epoxy specimens. (b) Tensile, (c) compression,and (d) torsion test. [Color figure can be viewed at wileyonlinelibrary.com]

TABLE 3. Details of 7901 epoxy test results.

Elasticity modulus/GPa(Standard deviation) Ultimate strength/MPa(Standard deviation)

Tensile/Em Shear/Gm Compression Tensile/σmu, t Shear/σmu,s Compression/σmu,c

3.17(0.07) 1.17(0.24) 3.23(0.39) 85.1(1.25) 52.6(1.35) 106.4(0.98)

2644 POLYMER COMPOSITES—2019 DOI 10.1002/pc

β21 ¼Vm Sf12−Sm12

� a12− Vf +Vma11

� �Sf22−S

m22

� ð3iÞ

In Eq. 3 Vm= (1 − Vf) is volume fraction of the matrix,

[I] is a 3 × 3 unit tensor, Efij, ν

fij, G

fij and Em, νm, Gm are

Young’s modulus, Poisson’s ratio, and the shear modulusof the fiber (assumed to be a transversely isotropic materialwith the axis of transverse isotropy along the fiber) and theepoxy matrix, respectively. The yield stress σmY of thematrix is set equal to the axial stress at the axial strain of0.13%, and σme is the von Mises stress based on true stres-ses computed from Eq. 4 below. Furthermore, Em

T isthe tangent modulus of the matrix for its inelasticdeformations. Using the curve fitted by the R-Squaremethod to the measured axial stress-axial strain curve ofthe resin deformed in uniaxial tension, we get

EmT ¼ ln 19:695−0:2199σme

� �. The compliance matrices, Sfij

and Smij , are given by Hooke’s law for the fiber and arecomputed using the Prandtl–Reuss theory [28] for thematrix.

True Stresses in the Matrix

The homogenized stresses in the matrix computed usingEq. 2b are converted to true values by multiplying themwith respective stress concentration factors (SCFs), as indi-cated below.

σm11� �

l ¼ σm11� �

l−1 + dσm11 ð4aÞ

σm22� �

l ¼ σm22� �

l−1 +K22dσm22 ð4bÞ

σm12� �

l¼ σm12� �

l−1 +K12dσm12 ð4cÞ

K22 ¼Kt22, if dσ

m22 > 0 and before int erface debonding

Kt22, if dσ

m22 > 0 and after int erface debonding

Kc22, if dσ

m22 < 0

8>><>>: ð4dÞ

Here Kt22 and K

t22 are the transverse tensile SCFs of the

matrix before and after interface debonding. Even thoughthe initiation of fiber/matrix debonding is accounted for by

modifying the value of Kt22 to K

t22, the change in its value

due to crack propagation in the debonded region is not con-sidered. Kc

22 and K12 are the matrix transverse compressiveand in-plane shear SCFs, respectively. Kt

22, Kc22, and K12

are summarized below [26,29], whereas Kt22 is given in

Appendix A.

K22ðφÞ¼ 1þA

2

ffiffiffiffiffiVf

pcos2φþ B

2ð1− ffiffiffiffiffiVf

p Þ ½V2f cos4φ

(

þ4Vf ðcosφÞ2ð1−2cos2φÞþffiffiffiffiffiVf

p ð2cos2φþ cos4φÞ�o

ðVf þ0:3VmÞEf22þ0:7VmEm

0:3Ef22þ0:7Em

ð5aÞ

A¼ 2Ef22E

mðvf12Þ2þEf11fEmðvf23−1Þ−Ef

22½2ðvmÞ2þ vm−1�gEf11½Ef

22þEmð1−vf23ÞþEf22v

m�−2Ef22E

mðvf12Þ2ð5bÞ

B¼ Emð1þ vf23Þ−Ef22ð1þvmÞ

Ef22½vmþ4ðvmÞ2−3�−Emð1þ vf23Þ

ð5cÞ

Kt22 ¼K22 0ð Þ, Kc

22 ¼K22 φð Þ andφ¼ π

4+12arcsin

σmu,c−σmu, t

2σmu,c

ð5d and 5fÞ

K12 ¼ 1−VfGf

12−Gm

Gf12 +G

mW Vf

� �−13

�" #

Vf + 0:3Vm

� �Gf

12 + 0:7VmGm

0:3Gf12 + 0:7G

m

ð5gÞ

W Vf

� �¼ πffiffiffiffiffiVf

p 14Vf

−4

128−

2512

Vf −5

4096V2f

�ð5hÞ

In Eq. 5 σmu, t and σmu,c are, respectively, the original ten-sile and compressive strengths of the matrix. Fiber’smechanical stress concentration factor does not exist, dueto a uniform stress distribution on its cross section [30].

Thermal Analysis

The work reported in Refs. [31–34] has shown thatthe thermal residual stress (TRS) influences both the non-linear mechanical response and the failure strength ofcomposites. Ye et al. [31,32] have developed a microme-chanical constitutive model to study macroscopic defor-mations of composites by considering thermal residualstresses in the RVE constituents. However, they did notpresent the thermal residual stress influence due to curingat the RT of their material [31]. Their predicted resultsconsidering residual stresses well agreed with the experi-mental findings [32]. Ye et al. [33] have shown thatresidual stresses noticeably influence the composites fail-ure strength. Shah et al. [35] have proposed an optimizedcuring cycle to minimize residual stresses developed dur-ing the curing process.

TABLE 4. Tensile stress/strain response of 7901 epoxy.

i 1 2 3 4 5 6 7 8

σmY� �

i=MPa 17.82 25.55 39.13 50.54 60.13 68.19 74.97 80.66

EmT

� �i=GPa 3.17 2.97 2.50 2.10 1.77 1.48 1.24 1.05

DOI 10.1002/pc POLYMER COMPOSITES—2019 2645

In this work, we employ Benveniste and Dvorak’s [34]analytical stress equations for UD composites to evaluateconstituents’ thermal residual stresses. We set mechanicalstresses equal to zero before applying external loads,

assume that constituents deform elastically during the cur-ing process, and find the thermal residual stress in constitu-

ents, σrij

� Tð Þ, by using the relation

FIG. 7. (a) SEM images of the fracture surfaces showing fiber–matrix interface debonding. (b) Fiber–matrixinterface morphology at 30 MPa. (c). Fiber–matrix interface morphology at 50 MPa. (d) Fiber–matrix interfacemorphology at 70 MPa.

2646 POLYMER COMPOSITES—2019 DOI 10.1002/pc

σrij

� Tð Þ¼ brij

h iΔT ð6aÞ

Here r = f(m) for the fiber (matrix), the temperaturechange ΔT = −100�C corresponding to 120�C curing tem-perature and 20�C RT, and

brij

h i¼ I− Cr

ij

h ih iSfij−S

mij

� −1αmij −α

fij

� ð6bÞ

where, Crij

h iis the mechanical stress concentration factor

of phases found from Eqs. 2a and 2b for the fiber and the

matrix, respectively, and αmij and αfij are the thermal expan-

sion tensors whose values are taken from [36]. The compli-ance matric Smij in Eq. 6b is calculated by using Hooke’slaw since deformations are assumed to be linearly elastic.

Hence, the initial stresses in the matrix and the fiber inEqs. 4a–4c before loading equals the thermal residual stres-ses, that is,

σmij

� 0¼ σmij

� Tð Þ, σfij

� 0¼ σ fij

� Tð Þð6cÞ

We have listed in Table 6 the residual stresses in thefiber and the matrix.

Interface Debonding Criterion

Let σ022 be the critical transverse tensile stress in thecomposite at which the interface debonding occurs. Thecorresponding stresses in the matrix using the Bridgingmodel and neglecting plastic deformations, if any, in thematrix, are given by

FIG. 7. (Continued)

TABLE 5. Modulus and strength of T300 fiber.

Ef11/GPa Ef

22/GPa Gf12/GPa Gf

23/GPa νf12 νf23 σ fu, t/MPa Vf

230 15 15 7 0.2 0.2 2500 0.62

DOI 10.1002/pc POLYMER COMPOSITES—2019 2647

σm11 ¼Vf a12

Vf +Vma11� �

Vf +Vma22� � σ022 ð7aÞ

σm22 ¼0:3Ef

22 + 0:7Em

Vf + 0:3Vm

� �Ef22 + 0:7VmEm

σ022 ð7bÞ

Furthermore, we have σm,Y22 ¼ 0:3Ef22 + 0:7E

m

Vf + 0:3Vmð ÞEf22 + 0:7VmEm

Y ,

where Y is the transverse tensile strength of the composite.

From the condition Kt22 σm,Y22 − σ

m

22

� +Kt

22σm22= σmu, t at the

transverse tensile failure, the critical transverse tensilestress is obtained as

σ022 ¼K

t22Y

Kt22�Kt

22

−Vf + 0:3Vm

� �Ef22 + 0:7VmEm

0:3Ef22 + 0:7E

m�

Kt22−K

t22

� σmu, t ð7cÞ

For general loading, we postulate that the interfacedebonds when [26].

σme� �

l−1 > σme and σ1m

� �l−1 > 0 ð7dÞ

where, σ1m, σme , and σme are, respectively, the maximumprincipal tensile, the von Mises and the critical von Misesstresses in the matrix found using the true stresses given byEqs. 4a–4d, and

σme ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσm11� �2

+ Kt22σ

m22

� �2−Kt

22σm11σ

m22 + 3 K12σ

m12

� �2qð7eÞ

Fiber Failure Criterion

A fiber is assumed to fail when the maximum principal

stress in the fiber equals its ultimate tensile strength, σ fu, t.That is,

σ1f ¼σ f11 + σ

f22

2+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ f11−σ

f22

2

!2

+ σ f12

� 2vuut ≥ σ fu, t ð8Þ

Matrix Failure Criterion

We assume that a matrix point fails when the followingTsai-Wu criterion is satisfied there.

F1 σm11� �2

l + σm22� �2

l − σm11� �

l σm22

� �l

h i+F2 σm12

� �2l +F3 σm11

� �l + σm22� �

l

h i≥ 1,

F1 ¼ 1= σmu, tσmu,c

� , F2 ¼ 1= σmu,s

� 2, F3 ¼ 1=σmu, t −1=σ

mu,c: ð9Þ

Here σmij are the true stress components given byEqs. (4a)–(4c), and σmu,s is the shear strength of matrix. Forthe 7901 epoxy, the experimentally found values for σmu, t,σmu,c, and σmu,s are 85.1, 106.4, and 52.6 MPa, respectively.

FIG. 8. Comparison of predicted and experimental off-axis tensile strength of T300/7901.

TABLE 6. Thermal expansion coefficients (αrij, 10−6/�C) and residual

stresses [ σrij

� Tð Þ, MPa] in the fiber and the matrix.

α11 α22 α12 σr11� � Tð Þ

σr22� � Tð Þ

σr12� � Tð Þ

Fiber −0.7 12 0 −12.6 −5.3 0Matrix 55 55 0 20.2 8.6 0

TABLE 7. SCFs and critical von Mises stress for the matrix.

ψ Kt22 K

t22 Kc

22 K12 σme /MPa σ022/MPa

73.4� 2.37 5.54 1.81 1.17 35.89 24.18

2648 POLYMER COMPOSITES—2019 DOI 10.1002/pc

UD Composite Strength

We define the composite strength as the load at whichone of its two constituents first fails.

COMPARISON BETWEEN PEDCITED ANDEXPERIMENTAL RESULTS

For the fiber and the matrix properties listed inTables 3–6, computed values of the interface crack angle,

the matrix SCFs Kt22, K

c22, K12, and K

t22, and the critical

von Mises stress, σme are given in Table 7.The off-axis 0�, 15�, 30�, 45�, 60�, 75�, and 90� T300/7901

UD composite laminae were tested both experimentally and

numerically in tension under off-axis loading. We found thatthe interface debonding occurred prior to the load reaching itsultimate value at failure. The predicted tensile modulus and theultimate tensile strength considering constituents’ residual stres-ses are compared with the corresponding measured data inTable 2. The difference between the test data and a predictedvalue is listed in the table in parentheses.

It is seen the interface debonding does not influence thecomposite elastic modulus. This is reasonable since thestructural stiffness is an overall property. However, the fail-ure is localized, and the interface debonding significantlydecreases the composite ultimate strength.

We have compared in Fig. 8 experimental and predictedultimate tensile strengths for off-axis loading of the

FIG. 9. (a–f ) Comparison of computed and experimental axial stress versus axial strain of T300/7901 UDcomposites. [Color figure can be viewed at wileyonlinelibrary.com]

DOI 10.1002/pc POLYMER COMPOSITES—2019 2649

composite laminae considering thermal residual stressesdeveloped during the curing process by designating asfollows the curves for other factors included in theanalysis.

A: interface debonding, SCFs and elasto-plastic matrix.B: ideal interface, SCFs and elasto-plastic matrix.C: ideal interface, SCFs and elastic matrix.D: ideal interface and elasto-plastic matrix,E: load at which interface debonding initiated.For the ideal interface, our results exhibited in Fig. 8

agree with those of Ref. 33. The tensile failure strengthof 45� off-axis UD composite is lower than that of otheroff-axis specimens. From values reported in Table 7, wesee that the SCFs of the matrix in the transverse direc-tion before and after interface debonding equal 2.37 and5.54, respectively. By comparing Eqs. 4 and 8, it is clearthat without considering these SCFs, the transversestrength will be overestimated. The strengths predictedby considering the interface debonding and the matrixplasticity match very well with the experimental results.The predictions with the idea fiber–matrix interface seemreasonable for the fiber off-axis angle less than 15�, anderrors increase with an increase in the fiber off-axis angle.We observe that the ultimate off-axis ultimate strength sig-nificantly drops as the off-axis angle is increased from 0�

to 15�. For off-axis angles less than 15�, the debonding hasless effect on the ultimate strength as the differencebetween the results with the idea and the debonded inter-faces is relatively small. The reason for this is in that noSCF exists in the matrix when a composite is longitudinallyloaded. Furthermore, interface debonding only influencesthe transverse load carrying ability of the composite, asseen from Eq. (4d), and the transverse tensile stresses aresmall for off-axis angle less than 15�. As shown in Fig. 8,the predicted loads at the interface debonding equal approx-imately one-half of the off-axis tensile strengths, implyingthat an interface debonding may occur far before the ulti-mate failure load is reached.

Our experimental result of the decrease in the ultimatetensile strength with an increase in the off-axis angle ofUD composite qualitatively agrees with the experimentalobservations reported in Refs. 37–41.

The matrix plasticity seems not to affect much the pre-dicted failure load. However, by comparing predictedresults with the ideal interface and no SCFs in the matrixwith those for the ideal interface but with SCFs reveal thatthe importance of considering the SCFs.

We note that for the 30� off-axial tension specimen,the predicted interface debonding load, 47.1 MPa(Fig. 8, dashed line E), agrees well with the experimen-tal value read from Fig. 7. The closeness of the com-puted and the test results of the axial stress versus theaxial strain curves for off-axis tension loading, shown inFig. 9a–f, establish the accuracy of the current analysis(considering interface debonding, SCFs and thermalresidual stress) for damage and failure of UD compos-ites. In Fig. 9a–f, the longest curve is the computed

results considering ideal (perfect) interface and SCFs.The second longest curve is the predicted results consid-ering ideal interface, SCFs and thermal residual stress.By comparing the second longest curve with the longestcurve, we can see that the thermal residual stress cangreatly reduce the predicted results of ultimate strengthwhen related to smaller offset fiber angle (<30�) com-posite, and lead to better matches to the experimentalresults. But, significant errors are still found whenrelated to large offset fiber angle (45�, 60�, 75�, and90�) composite, as shown in Fig. 9c–f. The presentanalysis, represented by the third curve in addition tothe longest and the second longest ones, agrees bestwith the experiment data in Fig. 9c–f.

CONCLUSIONS

We have used Bridging Model that incorporatesdebonding-dependent stress concentration factors (SCFs)to find stresses in the fiber and the matrix of the compos-ite laminate and considering thermal residual stressesdeveloped during the cooling from the cure temperatureto the room temperature. Only the transverse tensilestrength of a unidirectional (UD) fiber reinforced com-posite is used to define the critical value of the vonMises stress for identifying debonding initiation betweenthe fiber and the matrix interface. Whereas the 7901 resinproperties are derived from our test data in uniaxial ten-sion/compression and torsional loading, material proper-ties for the T300 carbon fibers are taken from theliterature. Off-axis UD laminae were tested in tension tillfailure. For the 30� off-axis T300/7901 UD specimenloaded in tension, the computed load at debonding initia-tion is found to agree well with the measured one. Thework reveals the importance of considering the SCFs ina failure prediction and illustrates that the considerationof matrix plastic deformations has insignificant effect onthe predicted ultimate load at failure. The ultimatestrength noticeably drops as the fiber angle is increasedfrom 0 to 15� but gradually decreases with furtherincreases in the fiber angle.

ACKNOWLEDGMENTS

The research received financial supports from theNational Natural Science Foundations of China withgrant numbers of 11832014 and 11472192. We acknowl-edge the experimental facilities provided by Prof. XiaoYi from the School of Aerospace Engineering andApplied Mechanics at Tongji University. RCB’s workwas partially supported by the US Office of NavalResearch (ONR) grant N00014-18-1-2548 to VirginiaPolytechnic Institute and State University. YQZ has beenat VPI&SU since August 2016 as a visiting researcher inRCB’s group.

2650 POLYMER COMPOSITES—2019 DOI 10.1002/pc

APPENDIX

MATRIX TRANSVERSE TENSILE SCF WITH ANINTERFACE CRACK

For the problem schematically shown in Fig. A1, thecrack angle ψ is found from the following equations in theBridging model.

Kt22 ¼ K

t22 ψð Þ

¼ Re e−2iψM beiψ� �

a2=b−b� �

−e− iψ N2−N1a2

be− iψ

� �� �

+ e− iψ 2 + e−2iψ� �

N beiψ� �

−N3� �� Vf + 0:3Vm

� �Ef22 + 0:7VmEm

2 b−að Þ 0:3Ef22 + 0:7E

m�

where

N zð Þ¼Fz+a2k

z− z−aeiψ� �0:5 + iλ

z−ae− iψ� �0:5− iλ

F−0:5ð Þ− D

a2z

�

N1 zð Þ¼Fz+a2k

z+1ξ

z−aeiψ� �0:5 + iλ

z−ae− iψ� �0:5− iλ

F−0:5ð Þ− D

a2z

�

N2 ¼ aFe− iψ + akeiψ , N3 ¼Faeiψ + e− iψak,

M zð Þ¼F−a2k

z2− F−0:5ð Þz +H +

C

z+D

z2

�χ zð Þ

F¼ 1− cosψ + 2λsinψð Þexp 2λ π−ψð Þ½ �+ 1−kð Þ 1 + 4λ2� �

sin2ψ4k −2−2 cosψ + 2λsinψð Þexp 2λ π−ψð Þ½ �H¼ a cosψ + 2λsinψð Þ 0:5−Fð Þ

C¼ k−1ð Þ cosψ −2λsinψð Þa2 exp 2λ ψ −πð Þ½ �D¼ 1−kð Þa3 exp 2λ ψ −πð Þ½ �

χ zð Þ¼ z−aeiψ� �−0:5 + iλ

z−ae− iψ� �−0:5− iλ

k¼ μ1 1 + κ2ð Þ1 + ξð Þ μ1 + κ1μ2ð Þ

λ¼ − lnξð Þ= 2πð Þ, ξ¼ μ2 + κ2μ1ð Þ= μ1 + κ1μ2ð Þκ1 ¼ 3−4vm,

κ2 ¼ 3−νf23−4νf12ν

f21

1 + νf23,

μ1 ¼Em

2 1 + νmð Þ ,

μ2 ¼Ef22

2 1 + νf23

� ,b¼ a=

ffiffiffiffiffiVf

pThe crack angle ψ is found by solving the following

equation.

Re G0−1k−

2 1−kð Þkexp iφð Þ exp 2λ ψ −πð Þ½ �

� �R eiφ� � �

φ¼ψ −γ

¼ 0

where

R exp iφð Þð Þ¼ exp i φð Þð Þ−eiψ� �0:5 + iλexp i φð Þð Þ−e− iψ� �0:5− iλ

� exp − i φð Þð Þ,

G0 ¼1− cosψ + 2λsinψð Þexp 2λ π−ψð Þ½ �+ 1−kð Þ 1 + 4λ2

� �sin2ψ

2−k−k cosψ + 2λsinψð Þexp 2λ π−ψð Þ½ � ,

γ¼2λ J21 + J

22

� �J21 + J

22 −2J2J3

, if ξ< 1

−2λ J21 + J

22

� �J21 + J

22 −2J2J3

, if ξ> 1

8>>><>>>:

:

J1 ¼ kG0−1−2 1−kð Þξexp 2λψð Þcos ψð Þ,J2 ¼ 2 1−kð Þξexp 2λψð Þsin ψð Þ,

J3 ¼ 2 1−kð Þξexp 2λψð Þ J1 cos ψð Þ−J2 sin ψð Þ½ �=J2:

If ξ = 1, then there is no solution for ψ , and the corre-sponding interface crack is called a singular crack. How-ever, one can slightly vary either the fiber or the matrixproperty to have ξ 6¼ 1.

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