Uniformly moving screw dislocation interacting with interface cracks in anisotropic bimaterials

This paper attempts to investigate the problem for the interaction between a uniformly subsonic moving screw dislocation and interface cracks in two dissimilar anisotropic materials. Using Riemann–Schwarz’s symmetry principle integrated with the analysis singularity of complex functions, we present the general elastic solutions of this problem and the closed form solutions for interface containing one and two cracks. The expressions of stress intensity factors at the crack tips and image force acting on moving dislocation are derived explicitly. The results show that the stress intensity factors at the crack tips decrease with increasing velocity of dislocation, and larger dislocation velocity leads to the equilibrium position of dislocation leaving from crack tips. The presented solutions contain previously known results as the special cases.     

Elastic fields due to dislocation arrays in anisotropic bimaterials

Based on the single-dislocation Green’s function, analytical solutions of the elastic fields due to dislocation arrays in an anisotropic bimaterial system are derived by virtue of the Cottrell summation formula. The singularity in the Peach–Koehler (P–K) force is removed by both rigorous mathematical approach and physical energy consideration. Numerical results for dislocation arrays in the Cu/Nb bimaterial with Kurdjumov–Sachs (K–S) orientation show that: (1) the traction continuity and periodic condition are both satisfied; (2) the maximum magnitude of the traction at the interface due to a mixed dislocation array is smaller than that due to a single mixed dislocation. In other words, the traction at the interface could be suppressed by the corresponding array with a relatively high density (L < 10 nm); however, the shear stress on the glide plane increases with increasing dislocation density; (3) the Cu/Nb interface attracts the mixed dislocation array in copper and repels the screw one there. This implies that the P–K force depends not only on the material properties, but also on the crystal orientation and the type of Burgers vector, among others.     

Transient dynamic analysis of interface cracks in anisotropic bimaterials by the scaled boundary finite-element method

The transient response of finite bimaterial plates with interface cracks is analyzed directly in the time domain by using the scaled boundary finite-element method. A bimaterial plate is divided into a few subdomains. Only the boundaries of the subdomains are discretized with line elements leading to great flexibility in mesh generation. The displacement and stress fields are expressed as a series solution which separates the singular stress term from other high-order terms. The oscillatory stress singularity in the radial direction emanating from the scaling center is represented analytically. The complex dynamic stress intensity factors are evaluated directly from either the stresses or the crack opening displacements of the singular stress term. Numerical examples of cracked anisotropic bimaterial plates are presented to verify the accuracy of the present technique and to provide additions to the very limited number of reference solutions in the literature.     

A cracked sliding interface between anisotropic bimaterials

A general solution for the stresses and displacements of a cracked sliding interface between anisotropic bimaterials subjected to uniform tensile stress at infinity is given by using the Stroh’s formulation. Horizontal and vertical opening displacements on the interface, stress intensity factors, and energy release rate are expressed in real form, which are valid for any kind of anisotropic materials including the degenerate materials such as isotropic materials. It is observed that stresses exhibit the traditional inverse square root singularities near the crack tips, and the vertical opening displacement and energy release rate are intimately related to a real parameter λ determined by the elastic constants of the anisotropic bimaterials.     

Elastic fields of dislocation loops in three-dimensional anisotropic bimaterials

By applying semi-analytical point-force Green's functions obtained via the Stroh formulism, we derive simple line integrals to calculate the elastic displacement and stress fields for a three-dimensional dislocation loop in an anisotropic bimaterial system. The solutions for the case of anisotropy are more convenient for treating an arbitrary dislocation loop compared with traditional area integration. With this new formulation, we numerically examine the displacement, stress, and energy due to the interaction between a dislocation loop and the bimaterial interface in an Al–Cu system. The interactive image energy due to the elastic moduli mismatch across the interface is then numerically evaluated. The result shows that a dislocation loop is subjected to an attractive force by the interface when it lies in the stiff material, and a repulsive force when it lies in the soft material. Moreover, the dependence of the interactive image energy of a dislocation loop on the position and size of the dislocation loop are also demonstrated and discussed. Significantly, it is found that the interactive image energy for a dislocation loop depends only on the ratio d/a, where a is the loop diameter and d is its distance to the interface. The examples studied provide benchmark solutions for anisotropic bimaterial dislocation problems.     

Some basic problems in anisotropic bimaterials with a viscoelastic interface

This research is devoted to the study of anisotropic bimaterials with Kelvin-type viscoelastic interface under antiplane deformations. First we derive the Green’s function for a bimaterial with a Kelvin-type viscoelastic interface subjected to an antiplane force and a screw dislocation by means of the complex variable method. Explicit expressions are derived for the time-dependent stress field induced by the antiplane force and screw dislocation. Also presented is the time-dependent image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. Second we investigate a rectangular inclusion with uniform antiplane eigenstrains embedded in one of the two bonded anisotropic half-planes by virtue of the derived Green’s function for a line force. The explicit expressions for the time-dependent stress field induced by the rectangular inclusion are obtained in terms of the simple logarithmic and exponential integral functions. It is observed that in general the stresses exhibit the logarithmic singularity at the four corners of the rectangular inclusion. Our results also show that when one side of the rectangular inclusion lies on the viscoelastic interface, the interfacial tractions are still regular at the two corners of the inclusion which are located on the interface. Last we address a finite Griffith crack normal to the viscoelastic interface by means of the obtained Green’s function for a screw dislocation. The crack problem is formulated in terms of a resulting singular integral equation which is solved numerically. The time-dependent stress intensity factors at the two crack tips are obtained and some interesting features are discussed.     

Two-dimensional Green's functions in anisotropic multiferroic bimaterials with a viscous interface

We derive, by virtue of the unified Stroh formalism, the extremely concise and elegant solutions for two-dimensional and (quasi-static) time-dependent Green's functions in anisotropic magnetoelectroelastic multiferroic bimaterials with a viscous interface subjected to an extended line force and an extended line dislocation located in the upper half-plane. It is found for the first time that, in the multiferroic bimaterial Green's functions, there are 25 static image singularities and 50 moving image singularities in the form of the extended line force and extended line dislocation in the upper or lower half-plane. It is further observed that, as time evolves, the moving image singularities, which originate from the locations of the static image singularities, will move further away from the viscous interface with explicit time-dependent locations. Moreover, explicit expression of the time-dependent image force on the extended line dislocation due to its interaction with the viscous interface is derived, which is also valid for mathematically degenerate materials. Several special cases are discussed in detail for the image force expression to illustrate the influence of the viscous interface on the mobility of the extended line dislocation, and various interesting features are observed. These Green's functions can not only be directly applied to the study of dislocation mobility in the novel multiferroic bimaterials, they can also be utilized as kernel functions in a boundary integral formulation to investigate more complicated boundary value problems where multiferroic materials/composites are involved.     

Green's functions for anisotropic piezoelectric bimaterials with a slipping interface

An explicit full-field expression of the Green's functions for anisotropic piezoelectric bimaterials with a slipping interface is derived. When the electro-elastic singularity reduces to a pure dislocation in displacement and electric potential, interaction energy between the dislocation and the bimaterials is obtained explicitly while the generalized force on the dislocation is given in a real form which is also valid for degenerate materials. The investigation demonstrates that the boundary conditions at the slipping interface between two piezoelectric materials will exert a prominent influence on the mobility of the dislocation. Project supported by the National Natural Science Foundation of China (No. 59635140).     

Modeling core-spreading of interface dislocation and its elastic response in anisotropic bimaterial

Interfacial dislocation may have a spreading core corresponding to a weak shear resistance of interfaces. In this paper, a conic model is proposed to mimic the spreading core of interfacial dislocation in anisotropic bimaterials. By the Stroh formalism and Green's function, the analytical expressions of the elastic fields are deduced for such a dislocation. Taking Cu/Nb bimaterial as an example, it is demonstrated that the accuracy and efficiency of the method are well validated by the interface conditions, a spreading core can greatly reduce the stress intensity near the interfacial dislocation compared with the compact core, and the elastic fields near the spreading core region are significantly different from the condensed core, while they are less sensitive to a field point that is 1.5times the core width away from the center of the spreading core.     

Multiple cracks in thermoelectroelastic bimaterials

The formulation for thermal stress and electric displacement in an infinite thermopiezoelectric plate with an interface and multiple cracks is presented. Using Green's function approach and the principle of superposition, a system of singular integral equations for the unknown temperature discontinuity defined on each crack face is developed and solved numerically. The formulation can then be used to calculate some fracture parameters such as the stress–electric displacement and strain energy density factor. The direction of crack growth for many cracks in thermopiezoelectric bimaterials is predicted by way of the strain energy density theory. Numerical results for stress–electric displacement factors and crack growth direction at a particular crack tip in two crack system of bimaterials are presented to illustrate the application of the proposed formulation.     

Fracture mechanical assessment of interface cracks with contact zones in piezoelectric bimaterials under thermoelectromechanical loadings II. Electrically impermeable interface cracks

This paper constitutes the second part of a study of interface cracks with contact zones in thermopiezoelectrical bimaterials, and it is concerned with the case of an electrically impermeable interface crack. The principal physical peculiarity of this case in comparison with an impermeable interface crack is connected with the dependencies of the contact zone length and the fracture mechanical parameters on the prescribed electrical flux, and in a mathematical sense the main peculiarity is concerned with the reduction of the problem in question to the joint solution of inhomogeneous combined Dirichlet–Riemann and Hilbert boundary value problems. The exact analytical solutions of the mentioned problems have been found for an arbitrary contact zone length, and the required thermal, mechanical and electrical characteristics at the interface as well as the associated fracture mechanical parameters at the corresponding crack tips are presented. The transcendental equations for the determination of the real contact zone length have been obtained for a general case and for a small contact zone length in an especially simple form. Using the admissible directions of the heat and the electrical fluxes defined in this paper as well, the dependencies of the real contact zone length and the associated fracture and electrical intensity factors on the intensities of the thermal and electrical fluxes are presented in tables and associated diagrams.     

Analytical solutions for anisotropic bimaterials under several boundary conditions on the interface

Summary Analytical solutions are proposed for the stress and displacement fields induced by in-plane loading of a bimaterial under several boundary conditions. The two joined orthotropic layers forming the bimaterial are allowed to possess different types of anisotropy with their planes of elastic symmetry arbitrarily inclined with respect to the horizontal.     

Fracture mechanical assessment of interface cracks with contact zones in piezoelectric bimaterials under thermoelectromechanical loadings: I. Electrically permeable interface cracks

An electrically permeable interface crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric spaces under the action of a remote electromechanical loading and a temperature flux is considered. Assuming that all fields are independent on the coordinate x₂ co-directed with the crack front, the stresses, the electrical and the temperature fluxes as well as the derivatives of the jumps of the displacements, the electrical potential and the temperature at the interface are presented via a set of analytic functions in the (x₁,x₃)-plane with a cut along the crack. Due to this representation firstly an auxiliary problem concerning the direction of the heat flux permitting a transition from a perfect thermal contact to a separation has been solved for a piezoelectric bimaterial. Besides, an inhomogeneous combined Dirichlet–Riemann boundary value problem has been formulated and solved exactly for the above mentioned interface crack. Stress and electrical displacements intensity factors are found in a clear analytical form which is especially easier for a small contact zone length. A simple equation and a closed form analytical formula for the determination of the real contact zone length have been derived and compared with the associated equation of the classical (oscillating) interface crack model defining the zone of crack faces interpenetration. For a numerical illustration of the obtained results a bimaterial cadmium selenium/glass has been used, and the influence of the heat flux upon the contact zone length and the associated stress intensity factor has been shown.     

Weight functions for interface cracks in dissimilar anisotropic materials

Bueckner‘s work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials. The difficulties in separating Stroh‘s six complex arguments involved in the integral for the dissimilar materials are overcome and then the explicit function representations of the integral are given and studied in detail. It is found that the pseudo-orthogonal properties of the eigenfunction expansion form (EEF) for a crack presented previously in isotropic elastic cases, in isotopic bimaterial cases, and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases. The relation between Bueckner‘s work conjugate integral and the J-integral in these cases is obtained by introducing a complementary stressdisplacement state. Finally, some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.     

Analysis method of planar interface cracks of arbitrary shape in three-dimensional transversely isotropic magnetoelectroelastic bimaterials

Using the fundamental solutions for three-dimensional transversely isotropic magnetoelectroelastic bimaterials, the extended displacements at any point for an internal crack parallel to the interface in a magnetoelectroelastic bimaterial are expressed in terms of the extended displacement discontinuities across the crack surfaces. The hyper-singular boundary integral–differential equations of the extended displacement discontinuities are obtained for planar interface cracks of arbitrary shape under impermeable and permeable boundary conditions in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. An analysis method is proposed based on the analogy between the obtained boundary integral–differential equations and those for interface cracks in purely elastic media. The singular indexes and the singular behaviors of near crack-tip fields are studied. Three new extended stress intensity factors at crack tip related to the extended stresses are defined for interface cracks in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. A penny-shaped interface crack in magnetoelectroelastic bimaterials is studied by using the proposed method.The results show that the extended stresses near the border of an impermeable interface crack possess the well-known oscillating singularity r^?1/2±iε or the non-oscillating singularity r^?1/2±κ. Three-dimensional transversely isotropic magnetoelectroelastic bimaterials are categorized into two groups, i.e., ε-group with non-zero value of ε and κ-group with non-zero value of κ. The two indexes ε and κ do not coexist for one bimaterial. However, the extended stresses near the border of a permeable interface crack have only oscillating singularity and depend only on the mechanical loadings.     

Asymptotic fields in the finite element analysis of electrically permeable interface cracks in piezoelectric bimaterials

Summary The interface crack problem for a piezoelectric bimaterial based on permeable conditions is studied numerically. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical and electrical loads, the complex stress intensity factor for an interface crack is obtained. The influence of the applied loads on the electromechanical fields near the crack tip is also studied. For a particular case of a short crack with respect to the bimaterial size, the numerical results are compared with the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.One author (V.G.) gratefully acknowledges the support provided by the Alexander von Humboldt Foundation of Germany.accepted for publication 7 June 2004     

基体裂纹与界面刚性线的弹性干涉

研究两种材料界面上的刚性线与其它任意位置处直线裂纹弹性干涉的反平面问题。基于界面上刚性线与任意位置处螺型位错干涉的基本解,运用连续位错密度模型法将问题转化为奇异积分方程。用半开型积分法求解奇异积分方程,得到位错密度函数的离散值,计算裂纹尖端处的应力强度因子。算例说明该方法可用于工程实际问题。     

On cracks and screw dislocation pile ups crossing a bimaterial interface

The situation where a screw dislocation pile up (or a crack in anti plane strain) crosses a bimaterial interface is considered. It is shown that a closed form solution can be obtained when the pile up crosses the interface in a symmetric fashion.

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Resume On discute le problème d'un empilement de dislocations-vis (ou d'une fissure sous déformation anti-plan) qui traverse l'interface entre deux matériaux aux propriétés différentes. On démontre l'existence d'une solution explicite dans le cas où l'empilement traverse l'interface de façon symétrique.

     

Out-of-plane loading of anisotropic bimaterials and semi-infinite materials

Summary Analytical closed-form solutions are proposed in a rather compact form for the stress and displacement fields induced by out-of-plane loading of a semi-infinite anisotropic material with inclined strata. The solutions are then extended to include the case of a bimaterial with a planar interface. Several boundary conditions are considered for the interface which may be between two anisotropic half-planes with different elastic properties, or two different orientations of the strata in the same material.     

Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by consideration of various contact zone models

Summary An interface crack with an artificial contact zone at the right-hand side crack tip between two piezoelectric semi-infinite half-planes is considered under remote mixed-mode loading. Assuming the stresses, strains and displacements are independent of the coordinate x ₂, the expression for the displacement jumps and stresses along the interface are found via a sectionally holomorphic vector function. For piezoceramics of the symmetry class 6 mm and for electrically permeable crack faces, the problem is reduced to a combined Dirichlet-Riemann boundary value problem which can be solved analytically. Further, analytical expressions for the stresses, electrical displacements, derivatives of elastic displacement jumps, stress and electrical intensity factors are found at the interface. Real contact zone lengths and the well-known oscillating solution are derived from the obtained solution as well. Analytical relationships between the fracture-mechanical parameters of various models are found, and recommendations are suggested concerning the application of numerical methods to the problem of an interface crack in the discontinuity area of a piezoelectric bimaterial. Received 16 March 1999; accepted for publication 31 May 1999