Interaction between heat dipole and circular interfacial crack

The heat dipole consists of a heat source and a heat sink. The problem of an interracial crack of a composite containing a circular inclusion under a heat dipole is investigated by using the analytical extension technique, the generalized Liouville theorem, and the Muskhelishvili boundary value theory. Temperature and stress fields are formulated. The effects of the temperature field and the inhomogeneity on the interracial fracture axe analyzed. As a numerical illustration, the thermal stress intensity factors of the interfacial crack are presented for various material combinations and different positions of the heat dipole. The characteristics of the interfacial crack depend on the elasticity, the thermal property of the composite, and the condition of the dipole.     

Interaction between heat dipole and circular interracial crack

The heat dipole consists of a heat source and a heat sink. The problem of an interfacial crack of a composite containing a circular inclusion under a heat dipole is investigated by using the analytical extension technique, the generalized Liouville theo-rem, and the Muskhelishvili boundary value theory. Temperature and stress fields are formulated. The effects of the temperature field and the inhomogeneity on the interracial fracture are analyzed. As a numerical illustration, the thermal stress intensity factors of the interfacial crack are presented for various material combinations and different po-sitions of the heat dipole. The characteristics of the interfacial crack depend on the elasticity, the thermal property of the composite, and the condition of the dipole.     

Fracture analysis of circular-arc interface cracks in piezoelectric materials

The anti-plane problem of N arc-shaped interfacial cracks between a circular piezoelectric inhomogeneity and an infinite piezoelectric matrix is investigated by means of the complex variable method. Cracks are assumed to be permeable and then explicit expressions are presented, respectively, for the electric field on the crack faces, the complex potentials in media and the intensity factors near the crack-tips. As examples, the corresponding solutions are obtained for a piezoelectric bimaterial system with one or two permeable arc-shaped interfacial cracks, respectively. Additionally, the solutions for the cases of impermeable cracks also are given by treating an impermeable crack as a particular case of a permeable crack. It is shown that for the case of permeable interfacial cracks, the electric field is jumpy ahead of the crack tips, and its intensity factor is always dependent on that of stress. Moreover all the field singularities are dependent not only on the applied mechanical load, but also on the applied electric load. However, for the case of a homogeneous material with permeable cracks, all the singular factors are related only to the applied stresses and material constants.     

正交异性材料平面裂纹尖端应力场

根据正交各向异性材料力学性能确定出了用应力函数表示的弹性力学基本方程,利用坐标变换和复变函数方法求解了正交异性材料平面裂纹体的应力边值问题。借鉴一般断裂力学解法构造了I型和II型裂纹问题的应力函数,推导出了正交各向异性板裂纹尖端区的奇异应力场。通过数值计算说明了裂纹尖端应力表达式的正确性,验证了裂尖前沿应力变化规律,即σx与材料特征参数h2成正比,而σy和τxy不随材料特性变化。     

Thermal shock analyses for a strip containing an infinite row of periodically distributed cracks normal to its edge

The transient thermal stress problem of a semi-infinite plate containing an infinite row of periodically distributed cracks normal to its edge is investigated in this paper. The elastic medium is assumed to be cooled suddenly on the crack-containing edge. By the superposition principle, the formulation leads to a mixed boundary value problem, with the negating tractions arisen from the thermal stresses for a crack-free semi-infinite plate. The resulting singular integral equation is solved numerically. The effects on the stress intensity factors due to the presence of periodically distributed cracks in a semi-infinite plate are illustrated. For both the edge crack and the embedded crack arrays, the stress intensity factors increase, due to the reduction of the shielding effect, as the stacking cracks are more separated. For the case of embedded crack array, one has the further conclusion that the stress intensity factors decline as the crack array shifts from the plate edge.     

夹杂对两相材料界面裂纹的影响

根据界面上应力和位移的连续条件,得到了单向拉伸状态下,含有椭圆夹杂的无限大双材料组合板的复势解。进一步通过求解Hilbert问题,得到了含有夹杂和半无限界面裂纹的无限大板的应力场,并由此给出了裂尖的应力强度因子K。计算了夹杂的形状、夹杂的位置、夹杂的材料选取以及上、下半平面材料与夹杂材料的不同组合对裂尖应力强度的影响。计算结果表明夹杂到裂尖的距离和夹杂材料的性质对K影响较大,对于不同材料组合,该影响有较大差异。夹杂距裂尖较近时,会对K产生明显屏蔽作用,随着夹杂远离裂尖,对K的影响也逐渐减小。另外,软夹杂对K有屏蔽作用,硬夹杂对K有反屏蔽作用,而夹杂形状对K几乎没有影响。     

A conducting arc crack between a circular piezoelectric inclusion and an unbounded matrix

The present paper investigates the problem of a conducting arc crack between a circular piezoelectric inclusion and an unbounded piezoelectric matrix. The original boundary value problem is reduced to a standard Riemann–Hilbert problem of vector form by means of analytical continuation. Explicit solutions for the stress singularities δ=−(1/2)±iε are obtained, closed form solutions for the field potentials are then derived through adopting a decoupling procedure. In addition, explicit expressions for the field component distributions in the whole field and along the circular interface are also obtained. Different from the interface insulating crack, stresses, strains, electric displacements and electric fields at the crack tips all exhibit oscillatory singularities. We also define a complex electro-elastic field concentration vector to characterize the singular fields near the crack tips and derive a simple expression for the energy release rate, which is always positive, in terms of the field concentration vector. The condition for the disappearance of the index ε is also discussed. When the index ε is zero, we obtain conventionally defined electro-elastic intensity factors. The examples demonstrate the physical behavior and the correctness of the obtained solution.     

DECAGONAL QU ASICRYSTALLINE ELLIPTICAL INCLUSIONS UNDER THERMOMECHANICAL LOADING

In this work, an elegant method is proposed to derive the thermoelastic field in- duced by thermomechanical loadings in a decagonal quasicrystalline composite composed of an infinite matrix reinforced by an elliptical inclusion. The thermomechanical loadings include a uniform temperature change, remote uniform in-plane heat fluxes and remote uniform in-plane stresses. The corresponding boundary value problem is ultimately reduced to the solution of two independent sets of four coupled linear algebraic equations, each of which involves four complex constants characterizing the internal stress field. The solution demonstrates that a uniform tem- perature change and remote uniform stresses will induce an internal uniform stress field, and that uniform heat fluxes will result in a linearly distributed internal stress field within the elliptical inclusion. The induced uniform rigid body rotation within the inclusion is given explicitly.     

Interaction between curved crack and elastic inclusion in an infinite plate

Summary In this paper, the curved-crack problem for an infinite plate containing an elastic inclusion is considered. A fundamental solution is proposed, which corresponds to the stress field caused by a point dislocation in an infinite plate containing an elastic inclusion. By placing the distributed dislocation along the prospective site of the crack, and by using the resultant force function as the right-hand term in the equation, a weaker singular integral equation is obtainable. The equation is solved numerically, and the stress intensity factors at the crack tips are evaluated. Interaction between the curved crack and the elastic inclusion is analyzed. Received 8 October 1996; accepted for publication 27 March 1997     

Orthotropic semi-infinite medium with a crack under thermal shock

A cracked orthotropic semi-infinite plate under thermal shock is investigated. The thermal stresses are generated due to sudden cooling of the boundary by ramp function temperature change. The superposition technique is used to solve the problem. The crack problem is formulated by applying the thermal stresses obtained from the uncracked plate with opposite sign to be the only external loads on the crack surfaces as the crack surface tractions. The Fourier transform technique is used to solve the problem leading to a singular equation of the Cauchy type. The singular integral equation is solved numerically using the expansion method. The influence of the material orthotropy on the stress intensity factors is shown by comparing the results obtained for different orthotropic materials and isotropic materials in the case of plane stress. The numerical results of the stress intensity factors are demonstrated as a function of time, crack length, location of the crack and the duration of the cooling rate.     

Stress intensity factors around a cylindrical crack in an interfacial zone in composite materials

Axisymmetric stresses around a cylindrical crack in an interfacial cylindrical layer between an infinite elastic medium with a cylindrical cavity and a circular elastic cylinder made of another material have been determined. The material constants of the layer vary continuously from those of the infinite medium to those of the cylinder. Tension surrounding the cylinder and perpendicular to the axis of the cylinder is applied to the composite materials. To solve this problem, the interfacial layer is divided into several layers with different material properties. The boundary conditions are reduced to dual integral equations. The differences in the crack faces are expanded in a series so as to satisfy the conditions outside the crack. The unknown coefficients in the series are solved using the conditions inside the crack. Numerical calculations are performed for several thicknesses of the interfacial layer. Using these numerical results, the stress intensity factors are evaluated for infinitesimal thickness of the layer.     

Interaction between rigid-disc inclusion and penny-shaped crack under elastic time-harmonic wave incidence

Influence of a rigid-disc massive inclusion on a neighboring penny-shaped crack induced by the time-harmonic wave propagation in an infinite elastic matrix is investigated by the numerical solution of associated 3D elastodynamic problem. No restrictions on the mutual orientation of interacting objects and direction of wave incidence are assumed. The inclusion is perfectly bonded with a matrix and supposes the translations and rotations, the crack faces are load-free. Frequency-domain problem is reduced to a system of boundary integral equations (BIEs) relative to the interfacial stress jumps (ISJs) on the inclusion and the crack opening displacements (CODs). The subtraction technique in conjunction with mapping technique, under taking into account the structure of solution at the fronts of inclusion and crack, is applied for regularization of BIEs obtained. A discrete analogue of equations is constructed by using the collocation scheme. Numerical calculations are carried out for the grazing incidence of a plane P-wave on the crack, where the interacting inclusion is coplanar and perpendicular to the crack, and has the same radius. The shielding and amplification effects of inclusion are assessed by the analysis of mode-I stress intensity factor (SIF) in the crack vicinity depending on the wave number, incident wave direction, position of the crack front point, inclusion mass, crack-inclusion orientation and distance.     

Stress intensity factors for circular hole and inclusion using finite element alternating method

A centre cracked plate subjected to remote tensile and shear loading is considered for the analysis. Effect of circular hole and influence of shrunk fit inclusion on stress intensity factors are studied. Multiply connected domain boundary value problem is solved using finite element alternating method (FEAM). Parametric studies involving drilled hole/inclusion sizes and locations are investigated. Energy release rates evaluated using the stress field obtained by FEAM are in good agreement with other methods. The optimum location in reducing the stress intensity factor with hole/inclusion is obtained and located at a distance 20% of semi-crack length from crack tip on the side opposite the ligament for Mode-I loading and it is also observed that the location is almost invariant of hole sizes. For Mode-II loading, the optimum location for the hole is located at a distance about 23% of semi-crack length from the middle of the crack along the transverse direction.     

Green’s functions for anti-plane deformations of a circular arc-crack at the interface of piezoelectric materials

Summary The anti-plane deformation problem of an interfacial debounding crack between a circular piezoelectric inclusion and a piezoelectric matrix is investigated by means of the complex variables method. For a line load applied within the matrix or inside the inclusion, Greens functions are presented for the complex potentials, intensity factors and electric fields on the crack faces, respectively, in closed and explicit form. The solutions are valid for both permeable and impermeable crack models. It is shown that, in the general case of permeable cracks, the electric field singularity is always proportional to the stress singularity.The first author (C.F.Gao) would like to express his gratitude for the support of the Alexander von Humboldt Foundation (Germany).     

Analysis of the Singularity for the Vertical Contact Problem of Line Crack-Inclusion

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Solution of multiple crack problem in a finite plate using coupled integral equations

This paper studies a numerical solution of multiple crack problem in a finite plate using coupled integral equations. After using the principle of superposition, the multiple crack problem in a finite plate can be converted into two problems: (a) the multiple crack problem in an infinite plate and (b) a usual boundary value problem for the finite plate. For the former problem, the Fredholm integral equation is used. For the latter problem, a BIE based on complex variable is suggested in which a Cauchy singular kernel exists. For the proposed BIE, after using the inverse matrix technique, the dependence of the traction at a domain point from the boundary tractions is formulated indirectly. This is a particular advantage of the present study. Several numerical examples are provided and the computed results for stress intensity factor and T-stress at crack tips are given.     

Thermoelastic problem of steady-state heat flows disturbed by a crack at an arbitrary angle to the graded interfacial zone in bonded materials

Plane thermoelasticity solutions are presented for the problem of a crack in bonded materials with a graded interfacial zone. The interfacial zone is treated as a nonhomogeneous interlayer having spatially varying thermoelastic moduli between dissimilar, homogeneous half-planes. The crack is assumed to exist in one of the half-planes at an arbitrary angle to the graded interfacial zone, disturbing uniform steady-state heat flows. The Fourier integral transform method is employed in conjunction with the coordinate transformations of field variables in the basic thermoelasticity equations. Formulation of the current nonisothermal crack problem lends itself to the derivation of two sets of Cauchy-type singular integral equations for heat conduction and thermal stress analyses. The heat-flux intensity factors and the thermal-stress intensity factors are defined and evaluated in order to quantify the singular characters of temperature gradients and thermal stresses, respectively, in the near-tip region. Numerical results include the variations of such crack-tip field intensity factors versus the crack orientation angle for various combinations of material and geometric parameters of the dissimilar media bonded through the thermoelastically graded interfacial zone. The dependence of the near-tip thermoelastic singular field on the degree of crack-surface partial insulation is also addressed.     

A generalized screw dislocation interacting with interfacial cracks along a circular inhomogeneity in magnetoelectroelastic solids

The interaction of a generalized screw dislocation with circular arc interfacial cracks under remote antiplane shear stresses, in-plane electric and magnetic loads in transversely isotropic magnetoelectroelastic solids is dealt with. By using the complex variable method, the general solutions to the problem are presented. The closed-form expressions of complex potentials in both the inhomogeneity and the matrix are derived for a single circular-arc interfacial crack. The intensity factors of stress, electric displacement and magnetic induction are provided explicitly. The image forces acting on the dislocation are also calculated by using the generalized Peach–Koehler formula. For the case of piezoelectric matrix and piezomagnetic inclusion, the shielding and anti-shielding effect of the dislocation upon the stress intensity factors is evaluated in detail. The results indicate that if the distance between the dislocation and the crack tip remains constant, the dislocation in the interface will have a largest shielding effect which retards the crack propagation. In addition, the influence of the interfacial crack geometry and materials magnetoelectroelastic mismatch upon the image force is discussed. Numerical computations show that the perturbation effect of the above parameters upon the image force is significant. The main result shows that a stable or unstable equilibrium point may be found when a screw dislocation approaches the surface of the crack from infinity which differs from the perfect bonded case under the same conditions. The present solutions contain a number of previously known results which can be shown to be special cases.     

A misfitting elastic inclusion in an infinite plane containing a crack

The problem discussed in this paper is that of a misfitting circular inclusion in an infinite elastic medium which contains a straight crack. The crack is stress free. The stresses develop in the elastic medium because of the misfit. The point force method is used to solve the problem. The problem reduces to finding two sets of complex potential functions: {(z), (z)}: One for the infinite medium and the other for the misfitting inclusion. The solution has been obtained in closed form. Graphs are drawn for stress intensity at the crack tip and also for normal, shear and hoop stresses at the common interface of medium and misfitting inclusion.     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.