Effects of thermal stress and imperfect interfacial bonding on the mechanical behavior of composites subjected to off-axis loading

Materials Science and Engineering A 527 (2010) 7530–7537

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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea

Effects of thermal stress and imperfect interfacial bonding on the mechanical behavior of composites subjected to off-axis loading Junjie Ye, Xuefeng Chen ∗ , Zhi Zhai, Bing Li, Yanyang Zi, Zhengjia He State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 3 June 2010 Received in revised form 15 July 2010 Accepted 29 July 2010

Keywords: Thermal residual stress Interfacial debonding Off-axis loading Stress–strain response

a b s t r a c t Metal matrix composites exhibit inelastic response due to the viscoplasticity of matrix, and imperfect interfacial bonding will decrease the flow stress sharply. The present study develops the generalized model of cells (GMC) to predict mechanical behavior of unidirectional metal matrix composites with imperfect interfacial bonding, which is subjected to off-axis loading. The model incorporates viscoplastic model for the matrix and interfacial debonding model for the fiber/matrix interface. The effects of fiber volume fraction and thermal residual stress on stress–strain response of composites are also discussed. Results show that stress–strain response influenced by fiber cross-section shape becomes more evident with the increase of fiber off-axis angle. Similar stress–strain response can be found in the early stage of loading regardless of the thermal residual stress. However, the effect of thermal residual stress on the stress–strain behavior of composites with imperfect interfacial bonding is closely dependent on fiber off-axis angle in the plastic stage. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Due to their low density, excellent mechanical strength, stiffness, creep resistance, as well as flexible design and manufacture according to service condition, composites have been new engineering materials widely used in aerospace, energy resources, transportation, machinery and biology. An unexpected failure may result in significant economic losses and safety problems. In order to minimize the accidents, many researchers have studied mechanical properties of composites [1,2,4,5]. However, they are restricted to study composites with perfect interfacial bonding. Many practical applications [6–8] show that structural difference between matrix and reinforced phase, as well as physical and chemical incompatibility lead to poor interfacial cohesive force. In other words, there is an imperfect interface between matrix and reinforced phase, which affects the mechanical properties of composites greatly. Finite element method [9–12] and analytical micromechanical method [13–16,18] have been widely used to describe the effects of interface. Kang et al. [9] discussed the effect of interfacial bonding state between particulate and metal matrix on the ratchetting of SiC particulate reinforced 6061Al alloy composites by using a finite element code ABAQUS. The simulated results showed that composites with imperfect bonding are closer to the correspond-

∗ Corresponding author. Tel.: +86 29 8266 7963; fax: +86 29 8266 3689. E-mail address: [email protected] (X. Chen). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.07.104

ing experiments than those obtained with assumption of perfect interface. Mondal et al. [10] modeled the interface in a metal matrix composites by using finite element method to study the deformation behavior of metal matrix composites as a function of interfacial characteristics. The interface is modeled as a thin layer of artificial material. The thickness and the modulus of the artificial material are varied as a means to vary the extent of coherency between the matrix and the particles. By implementing imperfect interfaces into finite element analysis, Nairn [11] proposed an approach to characterizing interfacial stiffness. Furthermore, some possible experiments for measuring the imperfect interfacial parameters needed for modeling were discussed. Caporale et al. [12] investigated the behavior of unidirectional fiber-reinforced composites with imperfect interfacial bonding through implementing an interfacial failure model by connecting the fibers and the matrix at the finite element nodes by normal and tangential brittle-elastic springs. Nie and Cemal [13] proposed a micromechanical model based on generalizations of Eshelby method to analyze effective elastic properties of particle filled acrylic composites with imperfect interfacial bonding. Based on a modified Needleman [14] type cohesive zone model, Lissenden [15] presented a threedimensional fiber–matrix debonding model for weakly bonded composites. Model predictions for transverse tensile and axial shear responses of silicon carbide/titanium showed good agreement with the experiments. Through application of a displacement discontinuity between the fiber and matrix, Aboudi [16] incorporated flexible interface (FI) model [17] into the method of cells to investigate damage in composites with imperfect bonding. Later, constant

J. Ye et al. / Materials Science and Engineering A 527 (2010) 7530–7537

7531

Fig. 1. Composite with periodic array.

compliant interface (CCI) [18] model, which was improved from FI model through adding a finite interfacial bond strength, was incorporated into GMC micromechanics model by Goldberg and Arnold [19] to investigate the tensile response of titanium matrix composite with imperfect interfacial bonding. However, both CCI and FI considered the interfacial debonding parameters to be constant. In other words, neither FI model nor CCI model can be used to describe progressive debonding of composites. To overcome this difficulty, Bednarcyk and Arnold [20] proposed a new interfacial debonding model named evolving compliant interface (ECI) model. The ECI model was implemented into GMC model to study the relationship between interface characteristic length and stress–strain response [21,22]. However, different fiber cross-section shapes of composites with imperfect interfacial bonding subjected to the influence of thermal residual stress arising from curing were not found in the studies above. The present study investigates the influence of thermal residual stress on elasto-plastic response of composites with imperfect interface bonding. Furthermore, due to the complexity of loading and boundary condition, off-axis loading, which received relatively little attention, is also discussed. The outline of this paper is organized as follows. A brief introduction of thermal residual stress calculation is given in Section 2. Section 3 presents the theory of incorporating interface debonding model into the GMC micromechanical model. In Section 4, the theory is used to investigate stress–strain response of composites subjected to different fiber off-axis angles. Meanwhile, both fiber cross-section shape and fiber volume fraction are also considered. Conclusions are given in Section 5. 2. Theoretical calculation of thermal residual stress

(ˇ)

= ε¯ 11

(ˇ = 1, . . . , Nˇ ;  = 1, . . . , N )

(ˇ)

hˇ ε¯ 22

= h¯ε22

( = 1, . . . , N )

(3)

ˇ=1 N 

(ˇ)

= lε¯ 33

l ε¯ 33

(ˇ = 1, . . . , Nˇ )

(4)

=1 N

ˇ 

(ˇ)

hˇ ε¯ 12

= h¯ε12

( = 1, . . . , N )

(5)

ˇ=1 N 

(ˇ)

l ε¯ 13

= lε¯ 13

(ˇ = 1, . . . , Nˇ )

(6)

=1 N

N ˇ  

(ˇ)

hˇ l ε¯ 23

= hlε¯ 23

(7)

=1 ˇ=1

where hˇ and l indicate the sub-cell sizes, respectively. h and l indicate the size of representative volume element, respectively. Superscript (ˇ) refers to the sub-cell. For a fixed column of sub-cells (1). . .(Nˇ ) of normal stress

(2)

(ˇ)

T22 and fixed row of sub-cells (ˇ1). . .(ˇN ) of normal stress T33 , stress continuity conditions can be expressed as: (1)

(Nˇ ) = · · · = ¯ 22 = T22

(ˇ1)

(ˇN ) = · · · = ¯ 22 = T33

¯ 22 ¯ 33

()

( = 1, . . . , N )

(8)

(ˇ)

(ˇ = 1, . . . , Nˇ )

(9)

Substituting Eq. (1) into Eqs. (3) and (4), respectively, that is,



(1)

In the composites, fibers are considered to be periodic distribution (see in Fig. 1). Based on the theory of generalized model of cells [26,27], the representative volume element (RVE) is usually divided into Nˇ × N sub-cells as shown in Fig. 2. The relationship between sub-cell strain and overall strain can be expressed as: ε¯ 11

N

ˇ 

()

Generally speaking, the thermal expansion coefficient between fiber and matrix is always different. Once processing temperature changes, inconsistent thermal expansion or compression between fiber and matrix will lead to thermal stress. Many researches show that mechanical properties of composites depend deeply on thermal residual stress [23–25]. Therefore, the influence of thermal residual stress of composites should be calculated accurately. The constitutive relationship of composites for each sub-cell can be written by: ε¯ (ˇ) = S (ˇ) ¯ (ˇ) + ε¯ p(ˇ) + ˛(ˇ) T

Fig. 2. Element division of RVE.

 hˇ

= h¯ε22 −

+



(ˇ)2

(ˇ) S22

 

hˇ

−



S12

() T22

(ˇ) S11 (ˇ)

hˇ

S12

(ˇ)

S11

ε¯ 11 +

+



p(ˇ) ε¯ (ˇ) 11 S11

p(ˇ) − ε¯ 22

 hˇ

 hˇ



(ˇ)

S12



(ˇ) S23

(ˇ) (ˇ)

−

S12 S13 (ˇ)2 S11

(ˇ) ˛ (ˇ) 11 S11

(ˇ)

T33



(ˇ)

S12



(ˇ) − ˛22

T

(10)

7532



J. Ye et al. / Materials Science and Engineering A 527 (2010) 7530–7537



(ˇ) (ˇ)

(ˇ) S23

l



= lε¯ 33 −

+





−

 hˇ



S12 S13

() T22

(ˇ)

S11

(ˇ)

hˇ

S12

(ˇ)

S11

ε¯ 11 +

(ˇ) S12 p(ˇ) ε¯ (ˇ) 11 S11



+



 

(ˇ)2

(ˇ) S33

l

(ˇ) ˛ (ˇ) 11 S11



(ˇ)

T33

(ˇ)

S11

1, 2, 3) are confirmed. Overall stress of composites can be obtained from a sum of sub-cell stresses, that is,



(ˇ)

S12

hˇ

−



S13

(ˇ) − ˛22

¯ 

T

11

¯ 22 ¯ 33

p(ˇ) − ε¯ 22

1 = hl

⎡

(11)

Eqs. (10) and (11) can be given in matrix form as follows: A B

B D

    T2 T3

H 0

=

  0 L

ε¯ 22 +

  c e

ε¯ 33 −

  d f

ε¯ 11 +

 

T +

P1 P2

(12)

Through manipulate calculation, sub-cell normal stresses in the 2 and 3 direction can be written as: ()

()

()

()

()

()

¯ 22 = a2 ε¯ 11 + b2 ε¯ 22 + c2 ε¯ 33 + ˚2 + 2 (ˇ)

(ˇ)

(ˇ)

(ˇ)

(ˇ)

(ˇ)

¯ 33 = a3 ε¯ 11 + b3 ε¯ 22 + c3 ε¯ 33 + ˚3 + 3

T T

(14)

where can be found in Ref. [28]. (ˇ) Meanwhile, sub-cell average stress of ¯ 11 is defined as: (ˇ)

=

1

(ˇ)

(ˇ,)

S11

(ˇ)

(ˇ)

(¯ε11 − S12 ¯ 22

(ˇ)

− S13 ¯ 33

p(ˇ)

− ε¯ 11

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(ˇ)

− ˛11

T )



(ˇ)

¯ 11

(ˇ)

hˇ l

¯ 22

(ˇ)

ˇ=1

1 hl

¯ 33

=1

Nˇ N  

ˇ=1

1 l

1 hl

(ˇ) hˇ l a1

Nˇ N  

=1 N

ˇ=1



1 l

()

l a2 =1

Nˇ  1

h ˇ=1

(13)

() () () () () (ˇ) (ˇ) (ˇ) (ˇ) (ˇ) a2 , b2 , c2 , a3 , b3 , c3 , ˚2 , ˚3 , 2 , 3

¯ 11

=



Nˇ N  

1 hl

(ˇ) hˇ l b1

=1 N

Nˇ N  

ˇ=1



1 l

()

l b2 =1

Nˇ  1

(ˇ)

hˇ a3

h

N  1

ˇ=1

1 l

(ˇ)

=1 N ()

l 2 =1

1 h

(15)

Nˇ 

(ˇ)

()

l c 2

h

⎤

hˇ l 1





Nˇ  1

hˇ b3

Nˇ

⎢ hl ⎢ ⎢

ε¯  ⎢ ⎢ 11 × ε¯ 22 +⎢ ⎢ ε¯ 33 ⎢ ⎢ ⎢ ⎣

=1 N

=1

ˇ=1

⎡

⎤ (ˇ) hˇ l c1

(ˇ)

hˇ 3

⎡

(ˇ)

hˇ c3

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

ˇ=1 N  1

⎤

Nˇ

⎥ ⎢ hl ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

ˇ=1

1 l

(ˇ)

hˇ l ˚1

=1 N



()

l ˚2 =1

1 h

ˇ=1

Nˇ 

(ˇ)

hˇ ˚3

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

ˇ=1

(18) Substituting Eqs. (13) and (14) into Eq. (15), the normal stress (ˇ) ¯ 11 for each sub-cell is written as: (ˇ)

=

¯ 11

1

(ˇ)

(ˇ,) S11

(a1

(ˇ)

ε¯ 11 + b1

(ˇ)

ε¯ 22 + c1

(ˇ)

ε¯ 33 + ˚1

(ˇ)

+ 1

T ) (16)

where (ˇ)

=

a1

1

(ˇ) ()

(ˇ) S11

(ˇ) (ˇ)

[1 − S12 a2 − S13 a3 ],

(ˇ)

b1

=−

1 (ˇ) S11

(ˇ) ()

[S12 b2

=

1

(ˇ) () − [S c2 (ˇ) 12 S11 (ˇ) () + S12 2

(ˇ)

˚1

=−

1

(ˇ) (ˇ) + S13 c3 ],

(ˇ) 1

=

⎢ ⎢ ⎢

0 ⎢ ⎢ = ⎢ 0 ⎢ 0 ⎢ ⎢ ⎢ ⎣

1

(ˇ) − [˛11 (ˇ) S11

(ˇ) (ˇ) + S13 3 ]

(ˇ)

p(ˇ)

(ˇ)

S11

()

(ˇ)

(ˇ)

(ˇ)

¯ 11

⎤

⎡

(ˇ)

(ˇ)

(ˇ)

a1

b1

c1

(ˇ) a3

(ˇ) b3

(ˇ) c3

⎤⎡

ε¯ 11

⎡

+

(ˇ) ⎤

1

⎡

⎤

ε¯ 33 (ˇ) ⎤

˚1

⎢ () ⎥ ⎢ ⎥ ⎣ 2 ⎦ T + ⎣ ˚() ⎦ 2 (ˇ)

3

Nˇ N  

ˇ=1

1 l

1 hl

(ˇ) hˇ l a1

=1 N



1 l

()

l a2 =1

Nˇ  1

h

Nˇ N  

ˇ=1

1 hl

=1 N

(ˇ)

hˇ a3

⎢ ⎢ ⎢

ε¯  ⎢ ⎢ 11 × ε¯ 22 + ⎢ ⎢ ε¯ 33 ⎢ ⎢ ⎢ ⎣

1 hl

 ˇ=1

(ˇ)

1 l

=1 N



()

l 2 =1

Nˇ  1

h

(ˇ)

hˇ 3

Nˇ  1

(ˇ)

hˇ b3

N

hˇ l 1

()

l c2 =1

ˇ=1 Nˇ

⎤ (ˇ) hˇ l c1

=1 N



1 l

()

l b2

h

Nˇ N  

ˇ=1



Nˇ  1

ˇ=1

(17)

(ˇ) hˇ l b1

=1

⎡

⎢ (ˇ) ⎥ ⎢ (ˇ) (ˇ) (ˇ) ⎥ ⎢ ⎥ ⎣ ¯ ⎦ = ⎣a b2 c2 ⎦ ⎣ ε¯ 22 ⎦ 22 2 (ˇ) ¯ 33

1 hl

ˇ=1

+ S12 ˚2 + S13 ˚3 ]

[¯ε11

Combining Eqs. (13), (14) and (16), the thermal stresses in each sub-cell are written as:

⎡

(ˇ)

to zero. However, local stresses ¯ ii are not zero which may seriously affect mechanical properties of composites. Therefore, Eq. (18) can be written as:

⎡

(ˇ) (ˇ)

+ S13 b3 ]

(ˇ) c1

It should be noted that during cooling from the processing temperature (high temperature) to room temperature (low temperature), the global stresses of composites ¯ ii (i = 1, 2, 3) are equal

⎤

h ˇ=1

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

(ˇ)

hˇ c3

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤

N  Nˇ

1 hl

(ˇ)

hˇ l ˚1 ˇ=1

1 l

=1 N



()

l ˚2 =1

Nˇ  1

h

(ˇ)

hˇ ˚3

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

ˇ=1

(19)

(ˇ)

˚3

From Eq. (17), we can see that thermal residual stresses in each sub-cell can be acquired when overall strain increment ε¯ ii (i =

Through transposing the matrix, the overall strain of composites during the temperature change can be written as:

J. Ye et al. / Materials Science and Engineering A 527 (2010) 7530–7537

7533

Table 1 Comparison between the prediction and experiment. Curing temperature = 120 ◦ C

Properties

Materials 1 (VF = 0.62)

E11 E22 G12

v12 ˛1 ˛2 f 11 m 22 ˛∗1 ˛∗2

Materials 2 (VF = 0.60)

Fiber

Matrix

Fiber

Matrix

80 80 33.3 0.2 4.9 4.9 −16.84 (−12.55) −5.90 (−6.86)

3.35 3.35 1.24 0.35 58 58 27.48 (20.47) 9.63 (11.2)

74 74 30.8 0.2 4.9 4.9 −18.07 (−13.55) −6.16 (−7.32)

3.35 3.35 1.24 0.35 58 58 27.11 (20.33) 9.24 (10.98)

6.40 (6.23) [8.6] 28.3 (20.62) [26.4]

6.65 (6.46) [8.6] 29.68 (21.67) [26.4]

⎧⎡ N ⎤ ⎡ N N ⎤⎫ N  ˇ ˇ     ⎪ ⎪ ⎪ ⎪ 1 1 (ˇ) (ˇ) ⎪ hˇ l 1 hˇ l ˚1 ⎥⎪ ⎪ ⎢ hl ⎥ ⎢ hl ⎪ ⎪ ⎪ ⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ˇ=1 =1 ˇ=1 =1 ⎪ ⎢ ⎥ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢  ⎥ ⎢ ⎥ ⎪ Nˇ Nˇ

ε¯  N N ⎬ ⎨    ⎢ ⎥ ⎢ ⎥ 11 1 1 () () −1 ⎢ hˇ l 2 ⎥ T + ⎢ hˇ l ˚2 ⎥ ε¯ 22 = −A ⎢ hl ⎥ ⎢ hl ⎥⎪ ⎪ ε¯ 33 ⎢ ˇ=1 =1 ⎥ ⎢ ˇ=1 =1 ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎢ ⎥⎪ ⎪ Nˇ Nˇ ⎪ N N ⎪ ⎢  ⎥ ⎢  ⎥⎪ ⎪   ⎪ ⎪ 1 1 ⎪ (ˇ) ⎦ (ˇ) ⎦⎪ ⎣ ⎣ ⎪ ⎪ hˇ l 3 hˇ l ˚3 ⎪ ⎪ hl ⎭ ⎩ hl ˇ=1

=1

ˇ=1

=1

(20) where

⎡

N

A=

ˇ N 1 

hl

(ˇ)

a1

hˇ l ⎣ a(ˇ) 2 (ˇ) a3

ˇ=1 =1

(ˇ)

(ˇ)

b1

c1

b2

c2

(ˇ)

(ˇ)

(ˇ) b3

⎤ ⎦

(ˇ)

c3

By substituting Eq. (20) into Eq. (17), residual thermal stresses in each sub-cell during cooling can be acquired, that is,

⎡

⎤ (ˇ)

⎡

¯ 11

⎣ ¯ (ˇ) ⎦ = ⎣ 22 (ˇ) ¯ 33

(ˇ) a1 () a2 (ˇ) a3

⎡

+

(ˇ) b1 () b2 (ˇ) b3

⎤ (ˇ)

⎤ (ˇ)

c1

()

c1

 ⎡ (ˇ) ⎤ 1 ε¯ 11 ⎦ ε¯ 22 + ⎣  () ⎦ T

˚1

(ˇ)

⎡ =

(ˇ)

(ˇ)

a1

⎣ a() 2

c1

b2

c2

()

(ˇ) a3

⎡

(ˇ) b3

(ˇ)

a1

+ ⎣ a() 2

⎡ +

(ˇ)

b1

(ˇ) a3

(ˇ)

˚1

(ˇ)

()

(ˇ)

c1

b2

c2

(ˇ) b3

⎤

⎣ ˚() ⎦ 2

⎦

(ˇ) c3

b1

()

⎤

()

(ˇ) c3

˛∗11 ˛∗22 ˛∗33

⎤ ⎦



ε¯ P11 ε¯ P22 ε¯ P33

According to interfacial debonding models, interfacial stress components in each direction are required continuously, while displacement components are discontinuous. Compliant interface (CCI) model and flexible interface (FI) model have been used to describe interfacial debonding between fiber and matrix. However, they are restricted to describing progressive debonding of composites due to the consideration of the interfacial debonding parameters as constant. Compared with FI model and CCI model, the ECI model allows interfacial debonding coefficients to evolve with time. It means that the ECI model can be used to describe the progression of the debonding by unloading the interfacial stress. According to Ref. [20], the normal and tangential displacement discontinuities can be written as:



I [u˙ n ] = Rn (t)˙ n |I + R˙ n (t)n |I I [u˙ n ] = 0 I [u˙ t ] = Rt (t)˙ t |I + R˙ t (t)t |I I [u˙ t ] = 0

n |I ≥ DB |I n |I < DB |I

(22)

t |I ≥ DB |I t |I < DB |I

(23)

where subscripts n and t refer to the normal and tangential components, respectively. Superscript I indicates the interface. ˙ i and  i refer to stress rate and stress, respectively.  DB is associated with the debonding stress at the interface. Once sub-cell stress in the fiber exceeds  DB , interfacial stress will unload. R and R˙ are interfacial debonding parameters which evolve with time. The proposed forms of debonding parameter time-dependence, which were given by Bednarcyk and Arnold, can be expressed as [20,31,32]:

⎣ ˚() ⎦ 2 ˚3

3. Imperfect interfacial bonding



2 (ˇ) 3

ε¯ 33

(ˇ) c3

the calculation results and experimental data of Huang [30] are shown in round brackets and square brackets, respectively. Obviously, higher accuracy of thermal expansion coefficients can be acquired by the GMC method.

T

 ⎡ (ˇ) ⎤ 1 + ⎣  () ⎦ T



2 (ˇ) 3

R(t) =  exp ˙ R(t) = (21)

(ˇ)

˚3

It should be noted that shearing stresses between fiber and matrix produced by thermal residual stresses are equal to zero [29]. Table 1 shows the results of thermal expansion coefficients of comf m posites and thermal residual stresses in the sub-cells. 11 and 22 ∗ indicate sub-cell stresses in fiber and matrix, respectively. ˛1 and ˛∗2 indicate longitudinal and transversal thermal expansion coefficients of composites, respectively. In order to verify the method,

t − t  DB

B



−1

(24)

 exp((t − tDB )/B) B

(25)

where  and B are empirical constants specific to the interface, and tDB is the time since debonding. In order to describe the effect of composites subjected to imperfect interfacial bonding, Eqs. (3) and (4) should be modified as follows: N

ˇ 

(ˇ) hˇ ε¯˙ 22 + n (Rn (t)˙ n |I + R˙ n (t)n |I ) = hε¯˙ 22

( = 1, . . . , N )

ˇ=1

(26)

7534

J. Ye et al. / Materials Science and Engineering A 527 (2010) 7530–7537

Fig. 3. Two different fiber cross-sections of composites with interfaces.

N 

(ˇ) l ε¯˙ 33 + nˇ (Rn (t)˙ n |I + R˙ n (t)n |I ) = lε¯˙ 33

(ˇ = 1, . . . , Nˇ )

=1

(27) n

nˇ

where and indicate the number of interfaces at th column and ˇth low in the sub-cells, respectively. [ui ]I refers to the displacement component at the interface, I. Substituting interfacial displacement function Eq. (23) into Eqs. (5)–(7), the rate relationship between sub-cell shear strains and macroscopic shear strains can be expressed as: N

ˇ 

(ˇ)

I

hˇ ε¯˙ 12 + n [u˙ 1 ] = hε¯˙ 12

( = 1, . . . , N )

(28)

ˇ=1 N 

(ˇ)

I

l ε¯˙ 13 + nˇ [u˙ 1 ] = lε¯˙ 13

(ˇ = 1, . . . , Nˇ )

(29)

=1

⎛

N

N ˇ  

(ˇ) 1 hˇ l ε¯˙ 23 + 2

=1 ˇ=1

⎝

N

ˇ 

N

ˇ

I

n hˇ [u˙ 2 ] +

n=1

ˇ 

Fig. 4. Stress–strain response of 90◦ fiber composites with and without thermal residual stress.

⎞ nˇ l [u˙ 3 ]

I

⎠ = hlε¯˙ 23

n=1

(30) Detailed information of incorporating the interfacial debonding models into the GMC method. 4. Results and discussion 4.1. Model validation As a first step to validate the presented model, the transverse tensile properties of composites with imperfect interfacial bonding are compared with experimental results [34]. Bodner–Partom viscoplastic model [35], one of the unified theories, is employed

to describe plastic behavior of composites. The constituents are assumed to be isotropic. And viscoplastic material parameters can be seen in Table 2. The manufacturing temperature of 850 ◦ C is assumed to be zero strain between fiber and matrix. It should be noted that the measured average aspect ratio (width (h2 ) divided by height (h1 ) of the cell) R = 0.88 and fiber volume fraction 0.32 are employed according to Ref. [34]. Meanwhile, for the presented SiC/Ti composites, interfacial strength  DB = 103 MPa is employed [8]. Fig. 3(b) shows the RVE with circular fiber contains 30 × 30 subcells. In the figure, the fiber, matrix and interface between the fiber and matrix are indicated by gray, white and black, respectively. For comparison purposes, composites with perfect interfacial bonding are also shown in Fig. 4. It can be seen that composites with perfect interfacial bonding provide much higher stiffness behavior than composites with imperfect interfacial bonding. However, the

Table 2 Material parameters of SiC fibers and Ti matrix. Material

E (Gpa)



n

m

Z0 (Mpa)

Z1 (Mpa)

D0−1

␣ (×10−6 )

Ti SiC

114.1 400

0.36 0.25

7 –

1700 –

965 –

1172 –

10−4 –

3.752 9.89

J. Ye et al. / Materials Science and Engineering A 527 (2010) 7530–7537

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Fig. 5. Stress–strain response of 90◦ fiber composites with different fiber volume fractions.

Fig. 7. Stress–strain response of 15◦ fiber composites with two different crosssection shapes.

theoretical prediction exhibits good agreement with experimental results only when both thermal residual stress and imperfect interfacial bonding are taken into consideration. And the method can be used to investigate the thermal residual stress effect on the mechanical behavior of fiber-reinforced composites subjected to different off-axis angles. Fig. 5 shows the effect of fiber volume fraction on the transverse tensile response of composites. Thermal residual stresses are included in all cases. As can be seen in the figure, with the increase of fiber volume fraction, the effect of interfacial condition on stress–strain response tends to increase evidently. Increasing fiber volume fraction will increase the stiffness behavior of composites with perfect interfacial bonding. This is in sharp contrast to the composites with imperfect interfacial bonding wherein increasing fiber volume fraction will decrease the stiffness behavior because it leads to the increase of interfacial region a (see in Fig. 3(b)). Once interfacial debonding occurs, less matrix region will undertake overall stress.

direction indicates the loading direction and the y direction is perpendicular to the x direction. Loading acts on composites should be divided into a series of incremental loadings. Meanwhile, stress increment in coordinate direction (d 1 , d 2 , d 12 ) can be acquired by stress increment in loading direction by using coordinate transformation formula as follows:

4.2. Stress–strain response of composites subjected to different off-axis angles In the case of off-axis loading, the loading should be decomposed into longitudinal, transversal and in-plane shear directions (see in Fig. 6). Two coordinate systems are defined as follows the (x1 , x2 ) system where x1 axis coincides with the fiber direction and x2 axis is perpendicular to the fiber direction and the (x, y) system where x

Fig. 6. Off-axis loading of directional fiber-reinforced composites.

T

{d11 , d22 , d12 } = dx {cos2 , sin2 , sin cos }

T

The elasto-plastic response corresponding to 0.32 fiber volume fraction of composites with imperfect bonding for 15◦ , 30◦ , 45◦ , 60◦ , 75◦ is shown in Figs. 7–11, respectively. Both square fiber and circular fiber employed the identical aspect ratio, namely R = 1 for comparsion. Compared with imperfect interfacial bonding, composites with perfect interfacial bonding always provide much more stiffness behavior. Moreover, this difference is closely dependent on off-axis angle. Increasing the fiber off-axis angle further enhances the difference in the stress–strain response of composites. In addition, the effect of the fiber cross-section shape on the stress–strain response tends to be more evident with the increase of fiber off-axis angle. No discernible differences can be found only when the fiber off-axis angle is more than 30◦ . As mentioned in Ref. [36], stress–strain response of composites had a characteris-

Fig. 8. Stress–strain response of 30◦ fiber composites with two different crosssections.

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tic three-stage deformation behavior, that is, stages I–III (see in Fig. 10). Similar stress–strain response can be found in the early stage of loading regardless of the thermal residual stress. However, stages II, and III differ greatly. The stiffness behavior of composites with imperfect bonding tends to be stiffer when thermal residual stress is taken into account in stage II. Meanwhile, the affected area “W” (see in Fig. 10) is increased with the increasing off-axis angle. However, the effect of thermal residual stress on stage III is closely dependent on fiber off-axis angle. For the 15◦ , 30◦ , 45◦ , 60◦ off-axis condition, thermal residual stress decreases the stiffness behavior of composites. On the contrary, the stiffness behavior will be increased when the off-axis angle increase to 75◦ . 5. Conclusion

Fig. 9. Stress–strain response of 45◦ fiber composites with two different crosssections.

The GMC micromechanical model is developed to investigate the effects of various parameters such as fiber cross-section shape, thermal residual stress and fiber off-axis angle on the stress–strain response of SiC/Ti composites with imperfect interfacial bonding. Theoretical prediction accords with experimental data only when thermal residual stress and interfacial debonding are considered simultaneously in the analysis. Results show that square fiber always provides more stiffening behavior than circular fiber in case of imperfect interfacial bonding regardless of thermal residual stress. Increasing fiber volume fraction of composites with imperfect interfacial bonding will decrease the stiffness behavior, forming a contrast with those of perfect interfacial bonding. Furthermore, the stiffness behavior of composites with imperfect bonding tends to increase when thermal residual stress is taken into account in stage II. However, the effect of thermal residual stress on stage III is closely dependent on fiber off-axis angle. Acknowledgements This work is supported by National Natural Science Foundation of China (No. 50875195), a Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 2007B33) and FOK YING TUNG Education Foundation (No. 121052). Special thanks should be expressed to Prof. J. Aboudi and Prof. Z.M. Huang for their instructions.

Fig. 10. Stress–strain response of 60◦ fiber composites with two different crosssections.

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Fig. 11. Stress–strain response of 75◦ fiber composites with two different crosssections.

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