Research ArticleA Study of Failure Strength for Fiber-Reinforced CompositeLaminates with Consideration of Interface

Junjie Ye,1 Yuanying Qiu,1 Xuefeng Chen,2 Yumin He,3 and Zhi Zhai2

1Key Laboratory of Ministry of Education for Electronic Equipment Structure Design, Xidian University, Xi’an 710071, China2State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China3College of Mechanical and Electrical Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China

Correspondence should be addressed to Junjie Ye; ronkey6000@sina.com

Received 1 December 2014; Revised 9 March 2015; Accepted 9 March 2015

Academic Editor: Jainagesh A. Sekhar

Copyright © 2015 Junjie Ye et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Composite laminates can exhibit the nonlinear properties due to the fiber/matrix interface debonding and matrix plasticdeformation. In this paper, by incorporating the interface stress-displacement relations between fibers and matrix, as well as theviscoplastic constitutive model for describing plastic behaviors of matrix materials, a micromechanical model is used to investigatethe failure strength of the composites with imperfect interface bonding. Meanwhile, the classic laminate theory, which provides therelation between micro- and macroscale responses for composite laminates, is employed. Theory results show good consistencywith the experimental data under unidirectional tensile conditions at both 23∘C and 650∘C. On this basis, the interface debondinginfluences on the failure strength of the [0/90]s and [0/±45/90]s composite laminates are studied.The numerical results show that allof the unidirectional (UD) laminates with imperfect interface bonding provide a sharp decrease in failure strength in the 𝜎

𝑥𝑥

-𝜎𝑦𝑦

plane at 23∘C. However, the decreasing is restricted in some specific region. In addition, for [0/90]𝑠

and [0/±45/90]𝑠

compositelaminates, the debonding interface influences on the failure envelope can be ignored when the working temperature is increased to650∘C.

1. Introduction

For improving material properties and satisfying specialfunctional requirements, composites become one of themost important materials nowadays and are widely usedin aerospace, energy sources, and so forth [1–4]. Due totheir unique constructions and special designs, compositesshow obvious macroscopic and microscopic characteristics.In otherwords, the failures of composite structures are relatedto macroscopic and microscopic stress field. Therefore, itis important for researchers to investigate the microscopicstructure feature influence on the macroscopic mechanicalproperties of composites. At present, both analytical methodand finite element method are used to predict the nonlinearstress-strain responses of metal-matrix and polymer-matrixcomposites. However, most of the traditional analyticalmethods always rely on various experience parameters. Itis difficult for the method to calculate microscopic stress-strain field with a high accuracy. For the finite element

method, poor calculation efficiency, difficult modifying forthe structure model, results in a poor application in studyingmicrostructure variation effects on themacroscopic mechan-ical responses.

Due to its flexible structural framework and high cal-culation efficiency, the generalized method of cells (GMC)based on the uniform axial deformation constraint conditionand interface displacement continuous conditions, whichwas proposed by Aboudi, has been widely used in studyingelastoplastic [5–7] and failure problems [8]. Coupled with theGMC and classical lamination theory, the failure envelopesof the polymer-matrix composites were studied by Tangand Zhang [8]. For failure envelopes of [90∘/30∘/−30∘]

2s and[90∘/±45∘/0∘]

2s laminates, it can be concluded that all matrixfailure criteria produce similar results. Based on the GMCand laminate analogy approach, Sun et al. [9] predicted theelastic properties of short sisal fiber reinforced polypropylenecomposites. The results show that the fiber volume fractionhas very small effects on the transverse Young’s modulus,

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2015, Article ID 158578, 10 pageshttp://dx.doi.org/10.1155/2015/158578

2 Advances in Materials Science and Engineering

axial shear modulus, and axial Poisson’s ratio. It is no doubtthat the micromechanical problems can be easily solvedby using GMC model. However, uncoupled subcell normalstress and shear stress lead to the poor calculation accuracyfor GMC model. By employing the higher-order displace-ment expansion, the high-fidelity generalized method ofcells (HFGMC) was proposed for solving microscopic stress-strain field with a higher accuracy [10, 11]. By using theHFGMC theory, Aboudi and Ryvkin [11] studied the progres-sive damage effects on the behavior of elastic composites.Haj-Ali and Aboudi [12] proposed a new parametric formulationfor theHFGMC theory.The numerical results of spatial stressfields for circular and elliptical fibers show good consistencywith analytical and numerical solutions.

In the preparation of composite materials, the interfacephases between inclusions and matrix will be produced. Asa kind of connecting bridge, the interphase plays a criticalrole in the mechanical properties of composites [13, 14].At present, some representative interface theories, such aschemical bond theory, transition layer theory, and weakboundary layer theory, are used to describe the interfaceinfluences on themechanical responses of composites [15, 16].Similarly, the interface debonding model, which is one of theweak boundary layer theories, has been implemented into theGMC or HFGMC theory for investigating interface failureproblem. Once a critical debonding stress has been exceeded,the displacement discontinuity between fibers and matrix isconsidered. On this basis, Arnold et al. [17] studied the tensileresponse of composites with imperfect interfacial bonding.By implementing into the evolving compliant interface (ECI)model, Chen et al. [18] developed the GMC to predict themechanical behaviors of unidirectional metal matrix com-posites with imperfect interface bonding. The results showedthat the theoretical prediction exhibits good agreement withexperimental results only when both imperfect interfacebonding between fibers and matrix and thermal residualstresses in subcells are taken into consideration. Via fiber ofappropriate constitutive relation for fiber/matrix debonding,both the GMC andHFGMC have been applied by Bednarcyket al. [19] to investigate the transverse deformations of thefiber-reinforced composites. The results indicated that theHFGMC shows a higher accuracy than the GMC for per-fecting the transverse response of composites with imperfectinterface bonding. However, in spite of HFGMC effectivenessin the mechanical responses and other applications, it hasnot been applied for composite laminates with imperfectinterface bonding. Considering its ability in investigatingnonlinear failure problems, the HFGMC is used to studythe interface debonding influence on composite laminates bycombining with the classical laminate theory.

2. Modeling Framework ofFiber-Reinforced Composites withImperfect Interface Bonding

By establishing the quantitative relation betweenmicroscopicconstituent properties and macroscopic material properties,the micromechanical methods can be effectively used to

investigate the failuremechanisms andmechanical propertiesof composites. For calculating microscopic stress compo-nents with a higher solution precision, the HFGMC was pro-posed by introducing a higher order displacement compo-nent in subcells. For the fiber-reinforced composite laminatesas shown in Figure 1(a), a proper unit (Figure 1(b)) is selectedand the unit is defined as a representative volume element(RVE). The local constitutive relation between the subcellstress components𝜎(𝛽𝛾) and the subcell elastic stiffnessmatrixC(𝛽𝛾) can be written as follows:

𝜎(𝛽𝛾)

= C(𝛽𝛾) (𝜀(𝛽𝛾) − 𝜀𝑇(𝛽𝛾)) , (1)

where the superscripts 𝛽 and 𝛾 indicate the correspondingsubcells. The strain components 𝜀𝑇(𝛽𝛾) indicate the subcellthermal strains. The strain components 𝜀(𝛽𝛾) are subcellstrains. It should be noted that the subcell strain components𝜀(𝛽𝛾) contain the elastic and inelastic strain components.Moreover, the inelastic strain component can be described byusing the unified constitutive theory.

For the continuous fiber-reinforced composites, the sub-cell displacement field function ��

(𝛽,𝛾)

𝑖

with higher ordercomponents in the HFGMC, as well as the subcell averagestress variations 𝑆(𝛽𝛾)

𝑖𝑗(𝑚𝑛)

, can be written as follows [19, 20]:

��(𝛽,𝛾)

𝑖

= 𝜀𝑖𝑗𝑥𝑗+𝑊(𝛽,𝛾)

𝑖(00)

+ 𝑦(𝛽)

2

𝑊(𝛽,𝛾)

𝑖(10)

+ 𝑦(𝛾)

3

𝑊(𝛽,𝛾)

𝑖(01)

+1

2

(3𝑦(𝛽)2

2

−

ℎ2

𝛽

4

)𝑊(𝛽,𝛾)

𝑖(20)

+1

2

(3𝑦(𝛾)2

3

−

𝑙2

𝛾

4

)𝑊(𝛽,𝛾)

𝑖(02)

(𝑖 = 1, 2, 3) ,

(2)

𝑆(𝛽𝛾)

𝑖𝑗(𝑚𝑛)

=1

ℎ𝛽𝑙𝛾

∫

ℎ𝛽/2

−ℎ𝛽/2

∫

𝑙𝛾/2

−𝑙𝛾/2

𝜎(𝛽𝛾)

𝑖𝑗

(𝑦(𝛽)

2

) (𝑦(𝛾)

3

) 𝑑𝑦(𝛽)

2

𝑑𝑦(𝛾)

3

(𝑖, 𝑗 = 1, 2, 3) ,

(3)

where the parameters ℎ𝛽and 𝑙𝛾are the subcell height and

length, respectively. The strain components 𝜀𝑖𝑗indicate the

average macroscopic strain. The displacement components𝑊(𝛽,𝛾)

𝑖(00)

and 𝑊(𝛽,𝛾)𝑖(𝑚𝑛)

denote the subcell average displacementand high order average displacement components, respec-tively. The parameters 𝑦(𝛽)

2

and 𝑦(𝛾)3

are the local coordinatesystem in the subcell as shown in Figure 1(b).

The displace field function of composites with perfectinterfacial bonding can be acquired by employing (1)–(3).According to the displace and stress boundary conditionsin the local coordinate system, combining with the subcelldisplacement field function ��(𝛽,𝛾)

𝑖

, the subcell average stress

Advances in Materials Science and Engineering 3

The first UD laminateThe second UD laminate

x

y

z

The integration point

...

(a)

Interface

Matrix

Fibery3

y2

(b)

Figure 1: Composite laminate theory analysis with micromechanical model embedded to the proper unit. (a) Composite laminate theoryand (b) a proper unit with interface (RVE).

variations 𝑆(𝛽𝛾)𝑖𝑗(𝑚𝑛)

in the 2 and 3 directions can be acquired asfollows:

𝑆(1𝛾)

2𝑖(00)

−6

ℎ1

𝑆(1𝛾)

2𝑖(10)

= 𝑆(𝑁𝛽𝛾)

2𝑖(00)

−6

ℎ𝑁𝛽

𝑆(𝑁𝛽𝛾)

2𝑖(10)

(𝛾 = 1, . . . , 𝑁𝛾) ,

𝑆(𝛽1)

3𝑖(00)

−6

𝑙1

𝑆(𝛽1)

3𝑖(01)

= 𝑆(𝛽𝑁𝛾)

3𝑖(00)

−6

𝑙𝑁𝛾

𝑆(𝛽𝑁𝛾)

3𝑖(01)

(𝛽 = 1, . . . , 𝑁𝛽) ,

𝑆(𝛽𝛾)

2𝑖(00)

+6

ℎ𝛽

𝑆(𝛽𝛾)

2𝑖(10)

= 𝑆(𝛽+1𝛾)

2𝑖(00)

−6

ℎ𝛽+1

𝑆(𝛽+1𝛾)

2𝑖(10)

(𝛽 = 1, . . . , 𝑁𝛽− 1; 𝛾 = 1, . . . , 𝑁

𝛾) ,

𝑆(𝛽𝛾)

3𝑖(00)

+6

𝑙𝛾

𝑆(𝛽𝛾)

2𝑖(01)

= 𝑆(𝛽𝛾+1)

3𝑖(00)

−6

𝑙𝛾+1

𝑆(𝛽𝛾+1)

3𝑖(01)

(𝛽 = 1, . . . , 𝑁𝛽; 𝛾 = 1, . . . , 𝑁

𝛾− 1) .

(4)

By using the equilibrium equations and the surface stresscomponents per unit area, the subcell average stress variationrelations between 𝑆(𝛽𝛾)

2𝑖(10)

and 𝑆(𝛽𝛾)3𝑖(01)

can be expressed as fol-lows:

1

ℎ2

𝛽

𝑆(𝛽𝛾)

2𝑖(10)

+1

𝑙2

𝑟

𝑆(𝛽𝛾)

3𝑖(01)

= 0. (5)

According to [19], with consideration of the interfacialbonding influences on continuous fiber-reinforced compos-ites, the stress relation between the neighboring subcellsalong with 2 directions can be written as

𝑢(𝛽𝛾)

𝑖

(𝑦(𝛽)

2

=

ℎ𝛽

2

) − 𝑢(𝛽+1𝛾)

𝑖

(𝑦(𝛽)

2

= −

ℎ𝛽+1

2

) = − [𝑢2𝑖]𝐼

,

(6)

where the superscript 𝐼 indicates the interphase betweenthe neighboring subcells. The parameter 𝑢

2𝑖indicates the

interface displacement components along with 2 directions.

Similarly, the stress relation between the neighboringsubcells along with 3 directions for composites with theinterfacial debonding can be written as

𝑢(𝛽𝛾)

𝑖

(𝑦(𝛾)

3

=

𝑙𝛾

2

) − 𝑢(𝛽𝛾+1)

𝑖

(𝑦(𝛽)

3

= −

𝑙𝛾+1

2

) = − [𝑢3𝑖]𝐼

. (7)

The parameter [𝑢3𝑖]𝐼 is interface displacement components

along with 3 directions.Here, the interface displacement components can be

written as follows [21]:

[𝑢𝑗𝑖]

𝐼

= 𝑅(𝛽𝛾)

𝑗𝑖

𝜎𝐼(𝛽𝛾)

𝑗𝑖

, (8)

where the parameters 𝑅(𝛽𝛾)𝑗𝑖

and 𝜎𝐼(𝛽𝛾)𝑗𝑖

indicate the interfacedisplacement debonding coefficients and the interface stresscomponents between fiber and matrix, respectively. It shouldbe noted that the variations 𝑅(𝛽𝛾)

𝑗𝑖

and 𝜎𝐼(𝛽𝛾)𝑗𝑖

are both time-dependent variations.

By combining (2)with (3), the average sense results in (7)-(8) can be written as

𝑊(𝛽𝛾)

𝑖(00)

+

ℎ𝛽

2

𝑊(𝛽𝛾)

𝑖(10)

+

ℎ2

𝛽

4

𝑊(𝛽𝛾)

𝑖(20)

− (𝑊(𝛽+1𝛾)

𝑖(00)

−

ℎ𝛽+1

2

𝑊(𝛽+1𝛾)

𝑖(10)

+

ℎ2

𝛽+1

4

𝑊(𝛽+1𝛾)

𝑖(20)

)

= −𝑅(𝛽𝛾)

2𝑖

(𝑆(𝛽𝛾)

2𝑖(00)

+6

ℎ𝛽

𝑆(𝛽𝛾)

2𝑖(10)

) ,

𝑊(𝛽𝛾)

𝑖(00)

+

𝑙𝛾

2

𝑊(𝛽𝛾)

𝑖(01)

+

𝑙2

𝛾

4

𝑊(𝛽𝛾)

𝑖(02)

− (𝑊(𝛽𝛾+1)

𝑖(00)

−

𝑙𝛾+1

2

𝑊(𝛽𝛾+1)

𝑖(01)

+

𝑙2

𝛾+1

4

𝑊(𝛽𝛾+1)

𝑖(02)

)

= −𝑅(𝛽𝛾)

3𝑖

(𝑆(𝛽𝛾)

3𝑖(00)

+6

𝑙𝛾

𝑆(𝛽𝛾)

3𝑖(01)

) .

(9)

4 Advances in Materials Science and Engineering

By combining (4)-(5) with (9), the subcell average stresscomponents 𝑆(𝛽𝛾)

𝑖𝑗(𝑚𝑛)

can be calculated. Furthermore, the non-linear stress-strain behaviors of composites with imperfectinterface bonding can be acquired through implementingsubcell average stress components into (1). According to thehomogenization theory, the macroscopic stress components𝜎 of the composite can be obtained as follows:

𝜎 =1

ℎ𝑙

𝑁𝛽

∑

𝛽=1

𝑁𝛾

∑

𝛾=1

ℎ𝛽𝑙𝛾C(𝛽𝛾) (𝜀(𝛽𝛾) − 𝜀𝑇(𝛽𝛾)) . (10)

The thermal residual stress is defined as the mismatchstress between fibers and matrix due to the temperaturevariation from the manufacturing temperature to the roomtemperature. According to the orthogonalization character-istics of Legendre polynomial in [20], the subcell straincomponents 𝜀(𝛽𝛾) can be expressed with the macroscopicaverage strain components. It should be noted that thereis no mechanical loading applied during cooling. In otherwords, the macroscopic stress components 𝜎 should be equalto zero. From (10), it can be found the thermal residualstress components in the RVE can be acquired once themacroscopic average strain components are confirmed.

3. Unidirectional Tensile Response ofComposite Laminates

3.1. TheRelation between Internal Force and Strain of Compos-ite Laminates. In the design process of composite products, aunidirectional (UD) laminate is always economically unrea-sonable due to its especial engineering requirements or com-plex load environments. Typical composite structures in engi-neering field are multidirectional layer composite materials,such as composite laminates. According to their designableand strength requirement, the composite laminates can bemanufactured by a series of unidirectional laminates withdifferent layer orientations. Furthermore, the mechanicalproperties can be further improved by using optimizationalgorithm. The stress-stain relation of composite laminatescan be written as follows [22]:

[[

[

N⋅ ⋅ ⋅

M

]]

]

=

[[[[

[

A... B

⋅ ⋅ ⋅

B D

]]]]

]

[[

[

𝜀

⋅ ⋅ ⋅

k

]]

]

, (11)

where A andD are the in-plane stiffenss matrix and bendingstiffness matrix, respectively. The matrix B is the couplingstiffness matrix, which refers to the in-plane strain andinternal bending force. The vectors N and M indicate theinternal force components and internal moment compo-nents, respectively.The vectors 𝜀 and k are referred to as straincomponents.

3.2. Model Validation. In this study, the composites systeminclude SCS-6 fiber and titanium matrix alloy (Timetal 21S)matrix.The fiber volume fraction is 33%.The fiber andmatrix

are assumed to be strain free at 850∘C. The thermal residualstresses in the RVE are generated when the manufacturingtemperature cools down to room temperature at 20∘C. Thefibers are assumed to be of linear elasticity until they reach tothe ultimate strength. While the Timetal 21S matrix, whichis described by a generalized viscoplasticity with potentialstructure (GVIPS), is considered to be elastoplastic material,the relationship between the inelastic strain rate 𝜀

𝐼

𝑖𝑗

and thedeviatoric stress components 𝑠

𝑖𝑗in the GVIPS can be written

as follows [23]:

𝜀𝐼

𝑖𝑗

=3

2

𝜇𝐹𝑛

𝜅

𝑠𝑖𝑗

√(3/2) (𝑠𝑖𝑗− Λ𝑖𝑗) (𝑠𝑖𝑗− Λ𝑖𝑗)

,

𝐹 = ⟨

√𝐽2

𝜅

− ⟨1 −𝛽

𝜅0

√3

2

Λ𝑖𝑗Λ𝑖𝑗⟩⟩.

(12)

The differential equation of the components Λ𝑖𝑗with

respect to time can be expressed as follows:

Λ𝑖𝑗=

𝜅2

0

3𝐵0(1 + 𝐵

1𝑝𝐺𝑝−1

)

⋅ (𝛿𝑖𝑘𝛿𝑗𝑙−

3𝐵1𝑝 (𝑝 − 1)𝐺

𝑝−2

𝜅2

0

(1 + 𝑝𝐵1𝐺𝑝−1

(6𝑝 − 5))

𝐴𝑖𝑗𝐴𝑘𝑙) ��𝑘𝑙,

��𝑘𝑙= 𝜀𝑘𝑙− Λ𝑘𝑙(3𝛽𝜅

2𝜅2

0

√2

3

𝜀𝑘𝑙

𝜀𝑘𝑙

𝐻] [𝑌]

√𝐺

−3

𝜅2

0

𝑅𝛼𝛽𝑜𝐺𝑞

) ,

(13)

where the material parameters 𝜅, 𝛽, 𝐵𝑜, 𝑅𝛼, 𝑛, 𝑝, 𝑞, 𝐵

1are

acquired by experimental method.The material parameters of SCS-6/Timetal 21S are shown

in Table 1. From Table 1, it can be easily found that thefailure strength of the fiber and matrix are both temperature-dependent variations. To offer evidence for the predictivecapabilities of the modeling framework, the macroscopicstress-strain responses of cross-ply symmetric laminates([0/90]s) are displayed as shown in Figure 2. In Figure 2,the thermal residual stresses in subcells and the thermalvariation are both considered. However, the failure problemsin subcells are not considered. For comparisons, the exper-imental data acquired by Cervay [24] are also shown in thecorresponding figures. Obviously, the theoretical predictionsand experimental data at both 23∘C and 650∘C offer a certainevidence for the feasibility of using the modeling frameworkto analyze the failure strength of the composite laminates. Inaddition, by comparing Figure 2(a) with Figure 2(b), it canbe easily found that the stress-strain curves of compositesare temperature dependent for composite laminates due totheir property differences in constituent materials at differentworking temperatures.

4. Failure Envelope Investigation ofComposite Laminates

According to the actual engineering demanding, themechan-ical properties of composite structures can be improved

Advances in Materials Science and Engineering 5

Table 1: Material constant and viscoplastic parameters of Timetal 21S matrix.

Material Timetal 21S matrix SCS-6 fiber𝑇 (∘C) 23 300 650 21 650Elastic modulus 𝐸/Gpa 114.1 107.9 80.7 393 59.7Thermal expansion coefficient (10−6) 7.72 9.21 12.13 3.564 3.826Poisson’s ration 0.365 0.365 0.365 0.25 0.25𝜅 (Mpa) 1029 768.4 5.861 — —𝛽 0.001 0 0 — —𝐵0

(Pa) 69.08 103.5 587.0 — —𝑅𝛼

(1/sec) 0 0 1.0 ∗ 10−6 — —Failure strength (Mpa) 1190 — 460 2584 2380

𝑛 = 3.3, 𝐵1

= 0.05, 𝑝 = 1.8, 𝑞 = 1.35

0 0.002 0.004 0.006 0.008 0.010

200

400

600

800

1000

1200

Experimental data

Stre

ss (M

Pa)

Strain

Prediction results, 23∘C

(a)

0 0.002 0.004 0.006 0.008 0.01 0.0120

100

200

300

400

500

600

700

800

Experimental data

Stre

ss (M

Pa)

Strain

Prediction results, 650∘C

(b)

Figure 2: The predicted and experimental tensile stress-strain responses of [0/90]s laminate: (a) 23∘C and (b) 650∘C.

through optimizing stacking sequences. However, due to thelong experimental period and high cost, the experimentalmethod can hardly be used to acquire the failure strength ofcomposite laminates. Here, the failure envelopes of [0/90]sand [0/±45/90]s composite laminates with imperfect inter-face bonding are studied. The material parameters of SCS-6/Timetal 21S can be seen in Table 1.

In the composite laminates, the microscopic stress andstrain fields for each of the UD laminate can be acquiredby using the HFGMC theory at the integration point alongthickness direction as shown in Figure 1(a). Once the micro-scopic stress or strain distributions are acquired, the Tsai-Hillfailure criterion can be employed to investigate the failurestrength of composite laminates. Here, the subcell stiffness iszero when the subcell stress reaches to the failure strength.In addition, it should be noted that the failure strength ofcomposite laminates is defined as the complete failure forall subcells in the RVE along with the thickness direction asshown in Figure 1(a).

4.1. The Failure Envelope Predictions of [0/90]s CompositeLaminates. There is no doubt that the failure strength ofcomposites is closely related to the interfacial propertiesbetween fibers and matrix. For the composite laminates,a different arrangement of the UD laminate (as shown inFigure 1(a)) with imperfect interface bonding between fibersand matrix may lead to the variation of failure strength.In order to investigate the most unfavorable conditions, allof the UD laminates with imperfect interface bonding areinvestigated.Working temperatures, namely, 23∘Cand 650∘C,are both considered in the examples. The numerical resultsare also shown in Figures 3 and 4, respectively.

For the [0/90]s composite laminates with perfect interfacebonding, the failure envelopes that intersect the horizontaland vertical axes of Figures 3 and 4 are defined as A, B,C, and D. Similarly, the failure envelopes that intersect thehorizontal and vertical axes of Figures 3 and 4 are definedas A1, B1, C1, and D

1for the [0/90]s composite laminates

with imperfect interface bonding. The failure stresses that

6 Advances in Materials Science and Engineering

Table 2: The failure stresses of the [0/90]𝑠

composite laminates (Mpa).

23∘C, no-debonding interface 23∘C, debonding interface 650∘C, no-debonding interface 650∘C, debonding interfacePoints xx-yy xx-xy Points xx-yy xx-xy Points xx-yy xx-xy Points xx-yy xx-xyA 1496.1 1496.1 A1 1157.6 1157.6 A 663.2 663.2 A1 621.2 621.2B 1565.1 873.6 B1 1173.4 574.3 B 663.3 283.2 B1 619.8 255.6C −1605.1 −1605.1 C1 −1605.1 −1605.1 C −677.7 −677.7 C1 −677.7 −677.7D −1544.4 −873.5 D1 −1544.4 −567.4 D −669.4 −283.2 D1 −669.4 −253.5

−2000 −1000 0 1000 2000−2000−1500−1000

−5000

500100015002000

Imperfect interface bondingPerfect interface bonding

B1

B

C/C1

D/D1

A1 A

Stress (MPa)

Stre

ss (M

Pa)

(a)

−1500 −1000 −500 0 500 1000 1500

−1000

−500

0

500

1000

Imperfect interface bondingPerfect interface bonding

C/C1 A1 A

B1

B

Stress (MPa)

Stre

ss (M

Pa)

D

D1

(b)

Figure 3: The failure envelopes of [0//90]s laminate at 23∘C: (a) xx-yy and (b) xx-xy.

−800 −600 −400 −200 0 200 400 600 800−1000

−800−600−400−200

0200400600800

1000

Imperfect interface bondingPerfect interface bonding

B

B1

C/C1

D/D1

AA1

Stre

ss (M

Pa)

Stress (MPa)

(a)

−800 −600 −400 −200 0 200 400 600 800−400−300−200−100

0100200300400

Imperfect interface bondingPerfect interface bonding

B

B1

C/C1

D

D1

AA1

Stress (MPa)

Stre

ss (M

Pa)

(b)

Figure 4: The failure envelopes of [0//90]s laminate at 650∘C: (a) xx-yy and (b) xx-xy.

correspond to the horizontal and vertical axes of Figures 3and 4 can be found in Table 2. In a word, compared with nodebonding interfaces of composite laminates, all of the UDlaminates with debonding interfaces showdistinct decreasingin failure strength. Meanwhile, the failure strength of SCS-6/Timetal 21S composites is also temperature dependent.

From Figure 3(a), it can be concluded that the failurestrength differences between the no-debonding interface andall of the UD laminates with debonding interfaces can onlybe found in the first, second, and fourth quadrants. However,for compression-compression loading in the third quadrant,the decrease of failure strength can hardly be found in

the 𝜎𝑥𝑥-𝜎𝑦𝑦

stress plane. Moreover, the maximum differencesare presented in the 𝑥- and 𝑦-directional tensile loadings. Forthe 𝑥- and 𝑦-directional tensile loadings, the no-debondinginterface provides a 29% and 33.5% increase relative to thatof all of the UD laminates with debonding interfaces, respec-tively. Figure 3(b) shows the failure envelopes in the 𝜎

𝑥𝑥-𝜎𝑥𝑦

stress plane. Comparing Figure 3(a) with Figure 3(b), thefailure envelope shape shows distinct difference. Moreover,the pure shear loading provides the maximum difference forfailure strength between no debonding interfaces and all ofthe UD laminates with debonding interfaces in the 𝜎

𝑥𝑥-𝜎𝑥𝑦

stress plane.

Advances in Materials Science and Engineering 7

−2000 −1000 0 1000 2000−2000−1500−1000

−5000

500100015002000

All layers, debondingFirst layer, debondingSecond layer, debondingThird layer, debonding

Stre

ss (M

Pa)

Stress (MPa)

C1/C2/C3

B1/B2/B3

A

I

B

C

DEF

A1/A2/A3

D1/D2/D3

(a)

−1500 −1000 −500 0 500 1000 1500

−1000

−500

0

500

1000

All layers, debondingFirst layer, debondingSecond layer, debondingThird layer, debonding

Stre

ss (M

Pa)

Stress (MPa)

C1/C2/C3

B1/B2/B3

A

B

CD

D1/D2/D3

A1/A2/A3

(b)

Figure 5: The failure envelopes of [0/±45/90]s laminate at 23∘C: (a) xx-yy and (b) xx-xy.

−600 −400 −200 0 200 400 600−800−600−400−200

0200400600800

All layers, debondingFirst layer, debondingSecond layer, debondingThird layer, debonding

Stre

ss (M

Pa)

Stress (MPa)

C/C1/C2/C3

B/B1/B2/B3

A

D/D1/D2/D3

A1/A2/A3

(a)

−600 −400 −200 0 200 400 600−500−400−300−200−100

0100200300400500

All layers, debondingFirst layer, debondingSecond layer, debondingThird layer, debonding

Stress (MPa)

Stre

ss (M

Pa)

C/C1/C2/C3

B/B1/B2/B3

A

G

III

HD/D1/D2/D3

A1/A2/A3

(b)

Figure 6: The failure envelopes of [0/±45/90]s laminate at 650∘C: (a) xx-yy and (b) xx-xy.

Figure 4 shows the failure envelopes of the [0/90]s com-posite laminates when the working temperature is increasedto 650∘C. Compared with the failure envelope at 23∘C, itcan be clearly seen that the interface debonding influenceson failure envelope are relatively small. The relative errorof maximum difference value in the 𝜎

𝑥𝑥-𝜎𝑦𝑦

stress planeand 𝜎

𝑥𝑥-𝜎𝑥𝑦

stress plane is less than 10%. That is, theinterface debonding influences on the failure strength of[0/90]s composite laminates can be ignored in the 𝜎

𝑥𝑥-𝜎𝑦𝑦

and 𝜎𝑥𝑥-𝜎𝑥𝑦

stress planes when the working temperature isincreased to 650∘C.

4.2.The Failure Envelope Predictions of [0/±45/90]s CompositeLaminates. As one of the most typical layer structures, thefailure analysis of [0/±45/90]s composite laminates is alsoinvestigated. For the composite structures with imperfectinterface bonding in a specific unidirectional composite

laminate, it is hardly for researchers to study the imperfectinterface influences on failure envelopes. Therefore, a UDlaminate with imperfect interface bonding is designed indifferent layers. The definition of layer sequence is as shownin Figure 1. Clearly, the top of the layer is defined as the firstUD laminate. Due to its symmetrical features, the failureenvelopes of the first, second, and third UD laminate withimperfect interface bonding are discussed. For comparisons,all of the UD laminates with debonding interfaces are alsodiscussed as shown in the corresponding figures. Moreover,the working temperature variations influences on the failurestrength are also investigated in the 𝜎

𝑥𝑥-𝜎𝑦𝑦

and 𝜎𝑥𝑥-𝜎𝑥𝑦

stress planes. Figures 5 and 6 show the debonding interfaceinfluences on the failure envelopes of composite laminates at23∘C, 650∘C, respectively. For all of the UD laminates withdebonding interfaces, the failure envelopes that intersect thehorizontal and vertical axes of Figures 5 and 6 are defined asthe points A, B, C, and D. The corresponding results can be

8 Advances in Materials Science and Engineering

Table 3: The failure stresses of the [0/±45/90]𝑠

composite laminates with imperfect interface bonding at 23∘C (Mpa).

All of the UD laminates The first UD laminate The second UD laminate The third UD laminatePoints xx-yy xx-xy Points xx-yy xx-xy Points xx-yy xx-xy Points xx-yy xx-xyA 986.6 986.6 A1 1358.7 1358.7 A2 1306.5 1306.5 A3 1331.2 1331.2B 972.8 620.7 B1 1290.4 697.2 B2 1252.6 810.9 B3 1304.2 797.1C −1104.4 −1104.4 C1 −1374.5 −1374.5 C2 −1316.6 −1316.6 C3 −1354.5 −1354.5D −1162.4 −620.7 D1 −1428.2 −671.0 D2 −1328.3 −815.1 D3 −1342.1 −810.9

Table 4: The failure stresses of the [0/±45/90]𝑠

composite laminates with imperfect interface bonding at 650∘C (Mpa).

All of the UD laminates The first UD laminate The second UD laminate The third UD laminatePoints xx-yy xx-xy Points xx-yy xx-xy Points xx-yy xx-xy Points xx-yy xx-xyA 431.3 431.3 A1 467.1 467.1 A2 467.1 467.1 A3 467.1 467.1B 479.5 343.1 B1 479.5 343.1 B2 493.3 343.1 B3 493.3 343.1C −467.1 −467.1 C1 −467.1 −467.1 C2 −467.1 −467.1 C3 −467.1 −467.1D −493.3 −343.1 D1 −493.3 −343.1 D2 −493.3 −343.1 D3 −493.3 −343.1

found in Tables 3 and 4, respectively. The subscripts 1, 2, and3 in Figures 5 and 6 indicate the first, second, and third UDlaminate with imperfect interface bonding.

Figure 5 presents the interface debonding influences onthe failure strength of the composite laminates at 23∘C.Table 3 shows the failure stresses that correspond to thehorizontal and vertical axes at 23∘C. As can be seen in Figures5(a)-5(b), the imperfect interface bonding influences onfailure envelopes show substantial difference in the 𝜎

𝑥𝑥-𝜎𝑦𝑦

and 𝜎𝑥𝑥-𝜎𝑥𝑦

stress planes. Some specific UD laminate withimperfect interface bonding provides the higher resistantfailure ability than all of the UD laminates with imperfectinterface bonding. In addition, the failure envelopes in thefour different bonding conditions present small overlapregions between two characteristic points E and F in the𝜎𝑥𝑥-𝜎𝑦𝑦

stress plane as shown in Figure 5(a). The overlapregion is defined as region I. For four different interfacebonding conditions, the failure strength in region I showsgood consistency. However, compared with some specificUD laminate with imperfect interface bonding, the failurestrength tends to decrease sharply when all of the UD lami-nateswith imperfect interface bonding are considered outsidethe region I.Moreover, the failure envelope differences amongdifferent UD laminates with imperfect interface bonding canbe clearly distinguished. For the 𝑥-directional tensile andcompressive loadings, as well as the 𝑦-directional tensile andcompressive loadings, it can be seen fromTable 3 that the firstUD laminate with imperfect interface bonding provides thehighest failure strength. From Figure 5(b), it is clearly seenthat there is no overlap region in the 𝜎

𝑥𝑥-𝜎𝑥𝑦

stress plane.All of the UD laminates with imperfect interface bondingprovide the lowest failure strength. Meanwhile, the secondlayer UD laminate with imperfect interface bonding providesthe highest pure shear failure strength.

Figure 6 shows the imperfect interface bonding influ-ences on failure strength of composite laminates when theworking temperature is increased to 650∘C. Table 4 shows thefailure stresses that correspond to the horizontal and vertical

axes at 650∘C. By comparing Figure 5, the working tempera-ture effects on the failure strength show a great difference. Forthe 𝜎𝑥𝑥-𝜎𝑦𝑦

stress loading as shown in Figure 6(a), the failurestrength in four different interface bonding conditions showsgood consistency in the third quadrant. It means that theinterface bonding modes can be ignored in the compression-compression loading. However, only a little difference can befound in the second quadrant. In the 𝑥-directional tensileloading, the failure strength tends to be consistency forthe first, second, and third UD laminates with imperfectinterface bonding as shown in Table 4. In the 𝜎

𝑥𝑥-𝜎𝑥𝑦

stressplane as shown in Figure 6(b), the failure envelopes presentaxisymmetric characteristics. And two characteristic pointsare defined as G and H. On the left side of the line C-Dis defined as region I, and the other region is defined asregion II. For the four different debonding modes, it is hardlyto discern the interface debonding influences on failureenvelopes of composite laminates in region I. In addition,it should be noted that the imperfect interface bondinginfluences on failure envelopes can be ignored in the pureshear loading.

5. Conclusions

In this paper, a micromechanical model is used to investigatethe interface debonding influences on failure envelopes oftypical composite structures. The classic laminate theoryand the GVIPS viscoplastic model are both introduced fordescribing the nonlinear behaviors of the composite lami-nates. In addition, the evolving debondingmodel is presentedto study the interface debonding influences on [0//90]s and[0/±45/90]s composites laminates with imperfect interfacebonding. The failure envelopes are both considered at 23∘Cand 650∘C. Based on investigations, the following conclusionscan be drawn.

(1) The failure envelopes of SCS-6/Timetal 21S compos-ites are temperature dependent.

Advances in Materials Science and Engineering 9

(2) For [0/90]s composite laminates in the 𝜎𝑥𝑥-𝜎𝑦𝑦

stressplane, the failure strength between no-debondinginterface and all of the UD laminates with debondinginterfaces tend to be consistent in compression-compression loading when the working temperatureis 23∘C.Moreover, the interface debonding influenceson failure strength can be ignored in both𝜎

𝑥𝑥-𝜎𝑦𝑦

and𝜎𝑥𝑥-𝜎𝑥𝑦

stress planes when the temperature workingis increased to 650∘C.

(3) Some specific UD laminate with imperfect interfacebonding provides a higher resistant failure abilitythan all of the UD laminates with imperfect interfacebonding in the 𝜎

𝑥𝑥-𝜎𝑦𝑦

stress plane and 𝜎𝑥𝑥-𝜎𝑥𝑦stress

plane.

(4) For [0/±45/90]s composite laminates, the interfacialbonding modes can be ignored in the compression-compression loading when the working temperatureis increased to 650∘C.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (nos. 51305320 and 51225501) and theFundamental Research Funds for the Central Universities(K5051304003).

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