1 Introduction

In recent decades,many advanced functional materials (such as metal/ceramic bonded materials and composite materials) have been expanded to the engineering application. Because of the difference of mechanical properties on different sides of the interface,the stress concentration in the vicinity of the interface could lead to the initiation and propagation of the micro-crack. In addition,the growth or arrest of the crack is attributed to the crack-tip field. Williams^[1] found that there was oscillatory singularity for stress field near the crack-tip in the isotropic bi-material interface crack.

Many scholars have done extensive work on interface cracks combining the different practical problems^{[2, 3, 4, 5, 6, 7]}. Wang et al.^[8] studied the anti-plane interface crack problem in composites under the impact loading. Viggo^[9] numerically analyzed the crack growth on an interface joining two elastic-plastic solids for small scale yielding. It can be found that the plastic flow near the crack-tip for the mode II crack has a much larger resistance to the crack propagation. L¨u et al.^[10] developed an analytical solution for the asymmetrical mode III interface propagation crack under point loading and unit-step loading. Yang et al.^[11] derived theoretical solutions of stress intensity factor and stress field basing on the limit uniqueness theorem for interface crack of dissimilar composites. Sun and Chen^[12] studied the viscoelastic interface crack for sandwich beams under dynamic and quasi-static loading. It is found that with shorter loading rise time, the oscillating characteristic for dynamic J-integral is more evident. Liang et al.^[13] studied the crack-tip fields on the elastic-viscoplastic/rigid interface of mode III. Their numerical results indicated that the material viscosity is an important factor for the crack-tip stress field. Wei et al.^[14] investigated the dynamic stress fields near the crack-tip for the interface of linear viscoelatic materials using the singular integral equation method. The results showed that the singularity index and the oscillation index were concerned with the material parameters and the load frequencies. Liu et al.^[15] studied the influence on generation of screw dislocations with the blunting Kelvin-type viscoelastic interface near the crack-tip under concentrated loads. Li et al.^[16] investigated the crack-tip field of mixed-mode quasi-static propagation crack with the compression load and shear load. Under the situation of frictional contact in crack surfaces,the crack-tip facture toughness increased with the friction enhancing between crack surfaces. Tang and Xu^[17] replaced the material parameters in elastic solutions with viscoelastic parameters by correspondence principle and obtained the approximate classical solution of viscoelasticity fields interface crack. Liang et al.^[18] established the governing equations of crack-tip field of dynamic propagating crack with the compression and shear load by considering the material viscosity and friction effect in crack surfaces. Their paper presented numerical solutions near the crack-tip field without the intermittence of stress or strain. Viggo and Brian^[19] analyzed the anisotropy of plastic material and the resistance of crack growing. Based on the numerical result,they showed that the fracture toughness is very sensitive to the mixed-loads level.

In this paper,based on the singularity near the crack-tip,the frictional contact on the crack surface and the parameter of mixed-loads level,the mechanical model of dynamic propagation interface crack of the compression-shear mixed mode is established using the material elasticviscoplastic constitutive model. The governing equations of propagation crack interface at cracktip are achieved. The numerical analysis is accomplished for the interface crack of compressionshear mixed mode. The distributed regularities of stress-strain fields near the interface crack-tip of compression-shear mixed mode are discussed when several special parameters are altered.

2 Formulation

Figure 1 shows the geometrical model of the steady state growth interface crack for the compression-shear mixed mode. (X1,X2) denote the fixed coordinate system. The systems (x,y) and (r,θ) move with the crack-tip,and the crack propagates along the x-direction with the constant velocity V between the elasto-viscoplastic material I and the rigid material II. For the steady state growth crack,the time derivative of one physical quantity in the crack-tip field is

The constitutive model of elastic-viscoplastic material is shown in Fig. 2,and the simultaneous equation system can be obtained as

where C denotes the fourth-order compliance tensor," and σ denote the strain and the stress tensor,e,p,and v denote the elastic property,the plastic property,and the viscose property, respectively,S,Sv,and Sp denote the partial tensor of stress σ,σv,and σp,respectively,σm p denotes the mean plastic stress,"˙p denotes the plastic strain rate tensor,λ denotes the plastic flow factor,and η denotes the material viscous property.

The constitutive equation in the three-dimensional situation can be yielded as

where ν is Poisson’s ratio,E is Young’s modulus,δ is the Kronecker delta,H is the Heaviside step function,J2 is the second invariant of stress tensor,and K is the yield strength. Moreover, the equivalent stress σ,the equivalent plastic stress σp,and the equivalent plastic strain "p are defined as follows: Then,introducing the Mises yield condition,we can obtain Since the viscosity effect is related to the equivalent plastic strain rate,the viscosity coefficient η can be shown as follows: where C is a non-negative constant. Substitute Eqs. (4),(5),and (6) into Eq. (3). After arrangement,for the perfect plastic material,the elastic visco-plastic constitutive equation can be expressed as follows: where the Heaviside step function is Further,we consider that the stress has the power singularity According to the magnitude coordination,we get Since 0 < δ < 1/2,only the case of 2/3 < β < 1 will be considered.

3 Governing equations of crack-tip field of compression-shear mixed mode interface crack

The geometry equations for polar coordinates are

The equations of motion are

Considering the incompressible condition,introducing a displacement function U of cracktip field of interface crack with compression-shear mixed mode,and matching the magnitude, the strain function is assumed as

where g(θ) is the angular distribution function in the crack-tip field. The displacement components are The singularity r−δ exists in the stress tensor,σij ,in the crack-tip field. The stress tensor can be expressed as where P(θ),R(θ),and T (θ) are the angular distribution functions of the crack-tip field. Therefore, the equivalent stress is signified as

Further introduce the following dimensionless quantities:

Substituting the relevant quantities into the motion equations and the constitutive equations, after dimensionless arrangement,we can obtain the governing equations of the crack-tip field for dynamic propagating interface crack with compression-shear mixed mode,i.e.,

4 Characterization parameters and boundary conditions of mixed mode crack 4.1 Characterization parameters

The mixed-load parameter (MP),supported first by Shih^[20],is introduced for defining the stress state of crack-tip

)

Substituting Eq. (15) into Eq. (22) yields

When MP = 1,it denotes the mode I crack with the tension (compression) load; when MP = 0, it denotes the mode II crack with the shear load; and when 0 < MP < 1,it denotes the mixed mode crack with tension (compression) load and shear load at the same time. As an exceptional case,the closed crack surface would generate friction along the surface with the compression load.

4.2 Boundary conditions

The boundary conditions of compression-shear mixed mode interface crack include both the continuity conditions at the front of crack (θ = 0) and the friction conditions in the crack surface (θ = π).

The continuity conditions in the front of crack (θ = 0) are

The friction at the crack surface is presumed to be compatible with the Coulomb friction law. Thus,the friction conditions at the crack surface (θ = π) are expressed as follows:

5 Numerical solution and analysis

In order to solve governing equations of the crack-tip field,there are five parameters needed to be selected,i.e.,the crack propagating velocityM,the crack the propagating velocity D∗,the index parameter β,the friction coefficient μ,and the mixed-load parameter MP. Because of the singularity of equations near the crack-tip,the start and end points of the numerical caloalation are,respectively,θ = 10−4 and θ = π − 10−4. The solution to the governing equation belongs to the two-point boundary value problem. Owing to the absence of definite condition,g′′(0) and P(0) are designated as adjustable parameters. By commissioning the two parameters,the boundary value problem can be changed into an initial value problem. The variation curves according to the special parameters in the crack-tip field are plotted (see Fig. 3,Fig. 4,Fig. 5,Fig. 6).

In Fig. 3,Fig. 4,Fig. 5,Fig. 6,near the crack-tip,when MP = 0.7 and MP = 0.3,with M increasing,that is,the stress components,the strain components,the hydrostatic stress,and the plastic flow factor decrease monotonously with the increase in the crack propagating velocity. At the same time,the strain component "rθ does not change significantly,but the opposite holds for the other solutions. When θ < 110◦,the stress component σr increases monotonically,σθ decreases monotonically,and σrθ increases monotonically first and then it decreases monotonically. When θ > 110◦,the stress component σr increases monotonically first and then decreases monotonically, σθ increases monotonically,and σrθ decreases monotonically. When θ < 90◦,the strain component "r increases monotonically first and then decreases monotonically,"θ decreases monotonically first and then increases monotonically,and "rθ and λ decrease monotonically. When θ > 90◦,the strain component "r decreases monotonically first and then increases monotonically, "θ increases monotonically first and then decreases monotonically,and "rθ and λ increase monotonically.

Based on the numerical results of the crack-tip field,von Mises yield criterion is adopted in the plastic zone of crack-tip,i.e.,the equivalent stress in the plastic zone of crack-tip is equal to the yield stress of the material^[21]. In order to verify the correctness of the results,we use the finite element package ABAQUS to analyze the crack propagation. The Mises stress of crack-tip fields is shown in Fig. 7(a),and the equivalent stress obtained from Eq. (16) under the same conditions is shown in Fig. 7(b). It can be seen that the contours of the equivalent stress are consistent.

The asymptotic analysis of compression-shear mixed mode interface propagating crack shows that "rθ will increase up to the maximum as the crack-tip is approached. Therefore,from the perspective of establishing strain fracture criterion,the fracture criterion is assumed as

where "crθ and r0 are the material constants,which can be obtained from micromechanics and experiment.

6 Conclusions

(i) When these individual parameters are altered,numerical results satisfy all continuity conditions and friction conditions in the front of compression-shear mixed mode interface crack.

(ii) The numerical results of the crack-tip field are completely continuous because of the presence of viscosity effect of crack-tip field. This fact emphasizes that the viscous effect is an important factor in studying the propagating crack-tip field.

(iii) The crack-tip singularity fields of compression-shear mixed mode interface propagating crack are governed by the material viscosity coefficient,the Mach number,and the singularity exponent. The mixed-load parameter can influence the distribution of crack-tip field.