The effects of thermal residual stresses and interfacial properties on the transverse behaviors of fiber composites with different microstructures

1 Introduction

Metal matrix composites (MMCs) have been widely used in aerospace and other industries as structural materials for decades because of their superior mechanical performances and lightweight properties. In the manufacturing progress, because of the mismatch between the thermal expansion coefficients of the fiber and the matrix, thermal residual stresses develop at the fiber-matrix interface during the cooling, which seriously reduces the tensile strength of the composites [1]. However, generally speaking, a full experimental approach requires a considerable amount of human, financial and time resources. Therefore, it is necessary to develop a kind of accurate and cost-effective model that could consider the influence of thermal residual stress and interfacial properties.

The effect of thermal residual stresses on the transverse behaviors of the composites has been studied by many researchers using either finite element analysis or the analytical micromechanical approach [2–4]. Hobbiebrunken et al. [5, 6] used a micromechanical finite element to investigate the influence of matrix plasticity and thermal residual stresses on the behavior of unidirectional carbon fiber epoxy composites. Taking into account thermal residual stresses, Huang [7] conducted a micromechanical analysis using an isothermal strength formula [8] to study the strength of the SiC/Ti-24-11 composites. Tsai and Chi [9] investigated the effect of thermal residual stresses on the constitutive behaviors of the two-phase composites with three different fiber arrays using the generalized method of cells (GMC). However, the interfacial properties were not considered in their analysis.

Many attempts have been made in the past to investigate the effect of the interfacial properties on the transverse behavior of the composites [10–13]. Experimental studies have shown that the dominant damage mechanism involved in the transverse ply cracking of composites is the debonding occurring at the fiber-matrix interface [14]. A micromechanics damage model, which was presented by Vaughan and McCarthy [1, 15], was used to examine the effect of fiber-matrix debonding and thermal residual stresses on the transverse damage behavior of a unidirectional carbon fiber-reinforced epoxy composite. It was found that when a weak fiber-matrix interface was considered, the presence of thermal residual stress may induce damage before mechanical loading. Williamson et al. [16, 17] developed an elastic-plastic finite element numerical model for predicting thermal residual stresses at graded ceramic-metal interfaces. Goldberg et al. [18] investigated the effect of thermal residual stresses on the tensile stress-strain response of a representative titanium matrix composite with weak interfacial bonding using the GMC model. The fiber-matrix interface was modeled by a constant compliant interface (CCI) model [19], which is limited by the fact that the degree of debonding cannot increase once the interfacial debonding occurs. The modified Needleman interface (NI) model was then proposed by Lissenden [20], which overcomes the disadvantage of the CCI model and allows the degree of debonding to progress via the unloading of the interfacial stress. However, because of the complexity and nonlinearity introduced by the NI model into the micromechanical model, a significant computational cost is needed, and closed-form constitutive equations for the weakly bonded composite cannot be available. The evolving compliant interface (ECI) model, which combines the simplicity and physical accuracy of the CCI model and the NI model, was used to model the transverse tensile of SiC/Ti composites with imperfect bonding by Bednarcyk and Arnold [21] and Ye et al. [22]. Because the ECI model is an exponential form model and sensitive to the parameters, it cannot provide the explicit constitutive behavior of the interface, and the non-convergence problem may occur in the simulated composite transverse tensile response. The bilinear cohesive zone model (CZM) [23, 24], which was simpler and more suitable to simulate the interface failure in MMCs compared with the exponential form [25], was introduced into finite element method (FEM) model by Shao et al. [26] to investigate the particle size-dependent flow strengthening and interface damage in the particle-reinforced MMCs. However, the bilinear CZM has not been introduced into the GMC, and few studies concerning the effect of the residual stresses and interfacial properties on the transverse behaviors of the fiber composites with different fiber microstructures have been reported. At the same time, the stress concentration factor (SCF) in the matrix due to the introduction of a fiber has been calculated [27, 28], which is used to predicted the tensile strength of the SiC/Ti MMC subjected to off-axis loading.

In this paper, the effects of the residual stresses and interfacial properties on the stress and strain curves of composites with different fiber microstructures were investigated. By introducing the CZM model into the GMC model, a new micromechanical model was proposed to study the transverse response of the composites. During the analysis, the fiber was considered to be linear elastic, and the matrix was considered to be a viscoplastic material. The thermal residual stresses in the composites were calculated, and their effects on the composites were also investigated. Then a parametrical study was conducted to assess the influence of the interfacial properties and the fiber microstructures on the stress-strain curves of the composites. At last, the SCF is calculated, and the tensile strength of the SiC/Ti MMC subjected to off-axis loading is simulated.

2 Computational model

2.1 GMC model

To model the transverse mechanical responses of fiber composites using a micromechanical approach, the GMC proposed by Paley and Aboudi [29] is adopted. In the GMC model, the fiber-reinforced composites can be considered to be double periodic, as seen in Figure 1A. The repeating volume element (RVE) of the GMC is shown in Figure 1B.

Figure 1:

(A) Composite with double periodic array of fibers in x₁ direction. (B) Repeating volume element of GMC.

The RVE is usually divided into N_β×N_γ subcells. Each subcell may contain a distinct homogeneous material. The constitutive equation of the subcell (βγ) is given by

(1)σ¯(βγ)=C(βγ)(ε¯(βγ)-ε¯p(βγ)-ε¯T(βγ)),

where σ¯(βγ) is the vector of subcell stress, C^(βγ) is the subcell elastic stiffness matrix, and ε¯(βγ),ε¯p(βγ), and ε¯T(βγ) are the vectors of subcell total strain, plastic strain, and thermal strain, respectively. ε¯T(βγ)=α(βγ)ΔT,α^(βγ) is the subcell thermal expansion coefficient, and ΔT is the temperature change in the RVE.

On the basis of the continuity of displacement and traction on the adjacent subcells in conjunction with the periodicity condition of RVE, the relationship between subcell strain and global (RVE) strain can be expressed by

(2)A˜ε¯s=K˜ε¯+D˜(ε¯sp+ε¯sT),

where ε¯s=(ε¯(11), ⋯, ε¯(NβNγ)),ε¯sp=(ε¯p(11), ⋯, ε¯p(NβNγ)), and ε¯sT=(ε¯T(11), ⋯, ε¯T(NβNγ)) are the vectors of total strain, plastic strain, and thermal strain for all the subcells, respectively. ε¯=(ε¯11, ε¯22, ε¯33, ε¯23, ε¯13, ε¯12) indicates the global strain. Α˜,K˜, and D˜ are the strain concentration matrices.

Combining Equations (1) and (2), the relationship between subcell stress and global strain can be determined in the form

(3)σ¯(βγ)=Gε¯+Φ+ΓΔT,

where σ¯(βγ)=(σ¯11(βγ), σ¯22(βγ), σ¯33(βγ), σ¯23(βγ), σ¯13(βγ), σ¯12(βγ)) is the subcell stress vector, G is the mixed concentration matrix, and Φ and Γ are the vectors wherein the inelastic and thermal terms are included, respectively.

On the basis of the average sense, the global stress of the RVE can be written as

(4)σ¯=1hl∑β=1Nβ∑γ=1Nγhβlγσ¯(βγ).

Using Equations (3) and (4), the global constitutive equations for the composite are established as

(5)σ¯=C∗(ε¯-ε¯p-ε¯T),

where C∗=1hl∑β=1Nβ∑γ=1NγhβlγG is the global elastic stiffness matrix, ε¯p and ε¯T are the global plastic strain and thermal strain, respectively, and ε¯T=α∗ΔT,α^* is the global thermal expansion coefficient vector.

With the fiber and matrix material properties as well as the RVE geometry, Equation (5) can be used to predict the composites behaviors.

2.3 Cohesive zone model

Interfaces play an important role in the determination of the composite mechanical and thermal properties [32]. Therefore, it is very important to model the interface behavior accurately. There are two major ways recently [33]. The first approach is to model an actual interface region with its own constitutive behavior. In this way, the interface is regarded as the third material, which has its own elastic, plastic properties and its own thickness. The second way to define the effect of imperfect bonding on the composites is to assume that a jump in the displacement field at the interface may occur at certain given conditions, while the tractions still have continuity.

In this study, the interface constitutive response is defined in terms of a bilinear traction-separation law [34, 35], as shown in Figure 2. The relationship between the interfacial traction and the interfacial displacement discontinuities can be written as follows:

Figure 2:

Traction-separation law of the CZM.

(9.1){σnI=knδn, when 0≤δn≤δn0σnI=tn0-tn0δnf-δn0, when δn0<δn≤δnfσnI=0, when δn>δnf

(9.2){σsI=ksδs, when 0≤δs≤δs0σsI=ts0-ts0δsf-δs0, when δs0<δs≤δsfσsI=0, when δs>δsf

where subscripts n and s refer to the normal (n) and shear (s) directions. σnI and σsI are the interfacial traction, tn0 and ts0 are the interfacial strength, k_n and k_s are the stiffness of interface, δ_n and δ_s are the interfacial displacement, and δn0 and δs0 are the interfacial displacement when the interfacial traction is equal to interfacial strength. The interfacial failure displacement (δnf or δsf) is determined by the fracture energy G, which corresponds to the area under the traction-separation curve. In Figure 2, k is the interfacial stiffness before the debonding occurs, and k₁ is the interfacial stiffness after the interface occurs debonding.

2.4 Stress concentration factor

As mentioned by Huang et al. [27, 28], an SCF should be influenced by three kinds of factors. First, an SCF should be related to the loading direction. Second, it should also be closely related to the fiber, matrix, and interface properties. Third, an SCF should depend on the fiber volume fraction of the composite.

In the study of Yao and Huang [27], the fiber is isolated by using a coaxial cylinder (Figure 3). Obviously, the relation can be obtained as follows:

Figure 3:

A matrix domain embedded with a fiber cylinder subjected to a uniaxial tension.

(10)b=a/Vf.

Then an averaged dimensionless stress of σ22m along a straight line parallel to the x₂-axis in the matrix domain can be calculated as follows:

(11)KφI=1(x2b-x2a)∫x2ax2bσ22mσ220|x3=x3φdx2,

where x2a=acosφ,x3φ=asinφ, and x2b=b2-(x3φ)2.

The first portion of the SCF can be defined as follows:

(12)KI=max{KφI, 00≤φ≤900}.

Through mathematical calculation, it can be known that KφI reaches its maximum value at φ=0⁰ when the fiber is stiffer than the matrix. Therefore, if the fiber is stiffer than the matrix, K^I can be expressed as follows:

(13)KI=1(b-a)∫abσ22mσ220|φ=00dx2=1+Vf2A′+Vf2(3-Vf-Vf)B′,

where A′=[1-vm-2(vm)2]Ef-[1-vf-2(vf)2]EmEf(1+vm)+Em[1-vf-2(vf)2],

B′=Em(1+vf)-Ef(1+vm)-Em(1+vf)+Ef[-3+vm+4(vm)2].

In fact, the stress component (σ22m)GMC obtained by the GMC model is an averaged quantity with respect to the entire matrix domain. However, K^I is a dimensionless stress averaged only along a straight line in the matrix domain. Therefore, the second portion of the SCF should be defined as follows:

(14)KII=(σ22m)avg(σ22m)GMC=1π(b2-a2)(σ22m)GMC∫ab∫02πσ22mρdρdφ.

It can be seen that (σ22m)avg=σ220. At the same time, in the GMC model, the (σ22m)avg and the (σ22m)GMC can be obtained as follows:

(15)(σ22m)avg=1hl∑β=1Nβ∑γ=1Nγhβlγσ¯(βγ),

(16)(σ22m)GMC=1(1-Vf)∑(βγ)∈matrixhβlγσ¯(βγ).

Therefore, the SCF of the composite due to a transverse load can be calculated by

(17)K=KI×KII

3 Model verification

To verify the validity of the proposed model, it is used to predict the transverse tensile response of SiC/Ti composites. The fiber volume fraction is 0.35. The properties of SiC-6 fibers are listed in Table 1 [20]. The fibers are considered to be isotropic and linear elastic. The Ti-15-3 matrix is regarded as an elastic-viscoplastic material, whose viscoplastic properties are characterized by the Bodner-Partom model [37]. In the Bodner-Partom model, there are six material parameters, which can be obtained through a uniaxial tensile test at constant strain rate and a set of shear stress-shear strain curves with different strain rates. The specific values are presented by Lissenden [20] and listed in Table 1. The cure temperature of SiC/Ti composite is 650°C, and the initial stress state of simulated composites is at room temperature. Therefore, in the following predictions, thermal residual stresses are calculated by applying an effective temperature drop of 625°C. The thermal residual stresses in the fiber phase and in the matrix phase are calculated and listed in Table 2.

In recent years, single fiber push-out and pull-out tests are being increasingly used to study the interfacial behavior of the composites [38]. The values of the interfacial properties are listed in Table 3, as discussed by Chandra et al. [38].

The predicted results of the transverse tensile response of SiC/Ti composite is shown in Figure 4. Meanwhile, the CCI interface model has also been used for comparison. The curves denoted by (CCI, R) and (Proposed model) are the transverse tensile response curves of composites when thermal residual stresses (R) are considered and the interfacial properties are simulated by the CCI and the CZM interface models, respectively. The transverse tensile response of composites with perfect interface is shown by the curve denoted by (Perfect). It can be seen that the simulated results using the proposed model can agree well with the experimental results. Compared with the perfect bonding, the weak interface bonding produces a much lower stiffness. The results also show that the transverse response predicted by the CZM model are more closer to the experimental results than the response simulated by the CCI model. That is because the interfacial stresses will decrease when they exceed the interfacial strength and eventually reach zero in the CZM model. In other words, the interface will unload and cannot bear any load finally. However, in the CCI model, the interface stresses do not unload and increase with the increase of strain all the time. Therefore, when the interface stresses go beyond the interfacial strength, the response predicted by the CCI model becomes larger than the results predicted by the CZM model and reaches a steady-state value of 31% eventually.

Figure 4:

The transverse tensile response of SiC/Ti composite.

Moreover, regardless of using any one of the two interface models, the values of the debonding stress (dot A in Figure 4) when the residual stresses are not considered will be 32.4% smaller than the values (dot B in Figure 4) when the residual stresses are considered. The reason is that the composites must overcome thermal residual stresses first when the residual stresses are considered. It can be noted that the predicted results when the residual stresses are considered are closer to the experimental results. However, because debonding occurs at all the interface eventually during the transverse tension, the matrix will bear all the loading. Therefore, whether considering the residual stresses, the area of the remaining matrix material that sustains the load will be the same, so the stresses will converge to the same value.

4 Results and discussion

In the following simulation, the previously proposed model has been used. When the effects of a parameter on the transverse behavior of the composite are studied, all the other parameters remain unchanged, which are the same with the values used in Figure 4.

4.1 Effect of interfacial properties

As has been shown earlier, the interface plays an important role in the transverse behavior of composites. However, the equipment to perform interfacial tests is not commonly available, and the determination of interfacial properties by experiments is very difficult [39, 40]. Mostly, the obtained results are often doubtable [14]. Therefore, it is necessary to study the effect of interfacial properties on the transverse response of the composites.

4.1.1 Effect of interfacial stiffness

In fact, because interfacial stiffness is not a pivotal parameter in the simulation [14], the determination of its value is just for numerical consideration. To examine the effect of interfacial stiffness on the transverse behavior of the composites, two kinds of interfacial stiffness values have been studied, as shown in Figure 5. Because smaller interfacial stiffness means smaller connection strength between the matrix and the fiber, the stiffness of the composites decreases with the decrease of interfacial stiffness. When the interfacial strength (t⁰) and the fracture energy (G) remain unchanged, the value of k₁ increases with the decrease of interfacial stiffness k. This means that the speed of the stress unloading increases, so the stiffness of the composites will decrease. It should also be noted that the stress will converge to the same value (saturation stress).

Figure 5:

Effect of the interfacial stiffness on the transverse behavior of the composites.

4.1.2 Effect of interfacial strength

The effects of interfacial strength on the transverse behaviors of the composites are shown in Figure 6. For comparison, three kinds of interfacial strengths (tn0=ts0=t0) are investigated. Whether or not we consider the thermal residual stresses in the composites, the debonding stress (dot A in Figure 6) increases with the increase of the interfacial strength because higher interfacial strength requires higher stress to overcome the interfacial adhesion stress. Results show that the interfacial strength do not influence the initial stiffness of the composites before debonding occurs. When the interfacial strength is 100 MPa, the value of k₁ is smallest because the interfacial fracture energy is the same for the three interfacial strengths. In this circumstance, the speed of the interfacial stress unloading is small. It can be noted that the speed of stress increases because of the remaining intact material that is bigger than the speed of stress, which decreases as a result of the interfacial stress unloading. Therefore, the stiffness of the composites is still positive after the debonding, and the overall response of the composite continues to increase as the composite strain increases. When the interfacial strength increases to 200 MPa, the speed of the interfacial stress unloading increases accordingly; hence, although the stiffness of the composites is still positive, it decreases. When the interfacial strength reaches 300 MPa, the speed of the interfacial stress unloading is larger than the speed of stress increase because of the remaining intact material. The result is that the composites start to unload after the debonding, and the stiffness of the composites is negative. Of course, when the interfacial stress is zero because of unloading, the composites start to bear the load all by the matrix; hence, the stiffness of the composites becomes positive (dot B in Figure 6). Because the interfacial stress is zero at last for all the three interfacial strengths, the saturation stress will converge to the same value.

Figure 6:

Effect of the interfacial strength on the transverse behavior of the composites.

4.1.3 Effect of interfacial fracture energy

Figure 7 illustrates the effect of the interfacial fracture energy on the transverse behaviors of the composites. In terms of unchanged values of interfacial strength and interfacial strength, when the interfacial fracture energy G is bigger, the value of k₁ is smaller. Therefore, among the three kinds of interfacial fracture energy, the stiffness of the composites is the biggest when G is 500 J/m². The stiffness of the composite decreases with the decrease of interfacial fracture energy, which is contrary to the case of interfacial strength. When the interfacial fracture energy decreases from 50 to 5 J/m², the stiffness of the composites becomes negative from positive, which is because the speed of the interfacial stress unloading is larger than the speed of stress increases because of the remaining intact material. For all the three kinds of interfacial fracture energy, when thermal residual stresses are considered, the debonding stresses are much bigger because the interface needs to overcome thermal residual stress before it starts to debond. However, the debonding stresses do not change with the interfacial fracture energy. Likewise, the saturation stress will converge to the same stress value for different interfacial fracture energy.

Figure 7:

Effect of the fracture energy on the transverse behavior of the composites.

4.2 Effect of the fiber microstructure

4.2.1 Effect of the fiber volume fraction

The effect of the fiber volume fraction on the transverse behavior of the composites is shown in Figure 8. Because the transverse elasticity modulus of the SiC fiber is much larger than the transverse elasticity modulus of the Ti matrix, the transverse elasticity modulus of the composites before debonding occurs will increase when the fiber volume fraction increases, as shown in Table 4. When the fiber volume fraction is 0.1, the contact area between the fiber and the matrix is smallest, that is, the area of interface is smallest. Therefore, when the interface stress reaches interfacial strength, the area that starts to unload is smallest, which indicates that the decrease of the stiffness of the composites is smallest. With the increase of the fiber volume fraction, the area of the interface increases; hence, the stiffness of the composites decreases after debonding occurs. When the fiber volume fraction reaches 0.3, the speed of the interfacial stress unloading is larger than the speed of stress increases because of the remaining intact material. As a result, the stiffness of the composites is negative and the overall response is unloading. For the three kinds of fiber volume fraction, when the interfacial stress reaches zero, the matrix bears all the load; hence, stress will continue to increase and the stiffness of the composites is positive. Because the area of the remaining matrix material that sustains the load is different for various fiber volume fractions, the saturation stress will converge to different stress values. With the fiber volume fraction decreasing, the area of the remaining matrix material increases; hence, the saturation stress (σ_s) will increase, and the values can be seen in Table 4.

Figure 8:

Effect of the fiber volume fraction on the transverse behavior of the composites.

Table 4

Transverse elasticity modulus and saturation stress vary with the fiber volume fraction.

Vf	E22 (GPa)	σs (MPa)
0.1	103.8	678
0.3	138.4	18
0.5	187.0	231

4.2.2 Effect of the fiber section shape

To investigate the effect of the fiber section shapes on the transverse behaviors of the composites, three kinds of fiber section shapes have been studied: square, circular, and elliptical. However, for the elliptical fiber section shape with the same aspect ratio, the transverse response will be different when the force direction is different. Hence, the transverse response of the two normal stress direction is investigated, as shown in Figure 9C and D, which are denoted as a/b=1.33 and a/b=0.75, respectively.

The effect of the fiber section shapes on the transverse behavior of the composites in the x₂ direction is shown in Figure 10. The results show that the fiber section shape primarily affects the stiffness of the composites after the composites get into the debonding state. The stiffness of composites using elliptical fiber model with an aspect ratio of 0.75 is biggest, and that using elliptical fiber model with an aspect ratio of 1.33 is smallest. The simulation results using the square fiber model is higher than that of the circular fiber model. This is because the square fiber can provide a higher magnitude of hydrostatic stress in the matrix phase relative to the circular fiber, which can delay localized yielding and provide constraint on the expansion of the plastic zone throughout the matrix phase. In Figure 4, the circular fiber model has been used, and the results can match the experimental results very well. This shows that the predicted results using the circular fiber model and considering the thermal residual stress can provide a much better comparison with the experimental values in all the four fiber shapes. The four kinds of fiber shapes show great difference in the remaining matrix material area, which means that the difference in the saturation stress is very big. The saturation stress of the composites using the elliptical fiber model with an aspect ratio of 0.75 is a litter bigger than that of the square fiber model, but is 20.6% bigger than that of the circular fiber model and 58% bigger than that of the elliptical fiber model with an aspect ratio of 1.33. Overall, the results show that for modeling the transverse tensile behavior of the composites when the interface effect is considered, it is very important to correctly and accurately simulate the fiber shape.

Figure 9:

Four kinds of fiber section shapes.

Figure 10:

Effect of the fiber section shape on the transverse behavior of the composites.

4.3 Ultimate tension strength

The tension strengths of the fiber and matrix are 3240 MPa [41] and 870 MPa [42], respectively. The predictions of the off-axis tensile strength of SiC-6/Ti-15-3 can be seen in Figure 11. It can be seen that the strength predictions based on perfect interface are far from the experimental data [43]. However, when the SCF is considered, the strength predictions based on CZM interface model agree with the experimental data.

Figure 11:

Comparison of tensile strength for the SiC/Ti MMC subjected to off-axis loading.

5 Conclusions

By introducing the bilinear CZM into the GMC, a new micromechanical model is proposed to study the effects of thermal residual stresses and interfacial properties on the transverse behaviors of SiC/Ti composites with different microstructures. The simulated results can be exactly consistent with the experimental results when thermal residual stresses and weak interfacial bonding are considered. From the micromechanical analysis, it is indicated that the weak interfacial bonding can very seriously decrease the transverse response of the composites. On the contrary, thermal residual stress can increase the debonding stress of the composites. Moreover, when the SCF is considered and the interface is considered as imperfect, the strength predictions agree well with the experimental results, which are much better than the results using the perfect bonding,

The effects of the interfacial properties on the transverse behaviors of the composites have been studied, including interfacial stiffness, interfacial strength, and interfacial fracture energy. It can be found that interfacial stiffness can affect the stiffness before debonding occurs whereas other two cannot. However, only the interfacial strength can affect the debonding stress among them. With the increase of interfacial strength, debonding stress increases, and unloading may occur followed by loading. With the increase of interfacial fracture energy, debonding stress does not vary, and the transverse behavior tendency is contrary to the condition considering the interfacial strength.

The present study also discusses the effects of the fiber microstructure on the transverse behaviors of the composites. With the increase of the fiber volume fraction, the stiffness of the composites increases before debonding occurs and the saturation stress decreases. Among the four kinds of fiber section shapes, the stiffness and the saturation stress of the composites using the elliptical fiber model with an aspect ratio of 0.75 is biggest and that using the elliptical fiber model with an aspect ratio of 1.33 is smallest. The stiffness and the saturation stress of the composites using the square fiber model is bigger than that using the circular fiber model. The simulated results using the circular fiber model and considering thermal residual stresses agree better with the experimental results compared with the results using the square or elliptical fiber model.