An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

Integration of Singularities in FE/BE Analyses of Soil-foundation Interaction with Non-homogeneous Elastic Soils

This paper addresses the derivation of the boundary integral equations for a non-homogeneous elastic half-space subjected to constant surface tractions on an arbitrarily shaped area on the basis of the respective Green's functions. The type of non-homogeneity considered is a power law variation of Young's modulus with depth below the surface of the half-space. Two different methods, a contour integral and an integration-free approach are presented, applicable for arbitrarily and rectangular shaped boundary elements, respectively. In the first one the divergence theorem is used in order to reduce the integration of a two-dimensional surface element to an integration over the element's confining boundary only. In the second approach no integration at all is needed since the solution is found simply by evaluating functions to be determined at the boundaries of the loaded rectangle.     

Lamb Problem for a Half-Space Covered with a Two-Axially Prestretched Layer

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, an approach for investigating the Lamb problem for a half-space covered with a layer is developed. It is assumed that the half-space and the cover layer have two-axial initial stresses operating parallel to the layer plane. To the free face of the layer, a normal point force changing harmonically with time is applied. For solving the corresponding boundary value problems, the double-exponential Fourier transformation is employed. An algorithm for obtaining numerical results is proposed, which is examined in particular cases. Numerical results for the influence of prestretching the cover layer on the interfacial stresses are presented.     

半平面压电体的Green函数及其应用

本文研究半平面压电体在线力、电荷和位错作用下的弹性场和电场，即Green函数．基于各向异性弹性力学中的Stroh方法和解析延拓理论，推导了Green函数的封闭形式的解．作为解的应用，分析了含半无限裂纹的无限大压电介质的机电耦合场，给出了应力和电位移强度因子的解析表达式．     

Reflection of plasma waves from a plane boundary

We state and analytically solve the linearized problem of the reflection of plasma waves from the half-space boundary in gas plasma for the first time. We consider mirror and diffusion boundary conditions and find the reflection coefficient as a function of the original problem parameters. We analyze the long-wave limit. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 498–510, March, 2007.     

Green functions for an incompressible linearly nonhomogeneous half-space

Summary Time-harmonic vibrations of an incompressible half-space having shear modulus linearly increasing with depth are studied. The half-space is subjected to a surface load which has vertical or hovizontal direction. The general solution of the time-harmonic, in the vertical direction nonhomogeneous problem is constructed for arbitrary angular distribution in the horizontal plane. Numerical results concerning surface displacements due to a point force are given for the case of nonzero shear modulus at the surface. These results show that nonhomogeneity can considerably increase amplitudes at large distances from the applied force.     

On the relationship between the solutions of static contact problems of elasticity and electroelasticity for a half-space

The paper establishes the relationship between the static contact problems of elasticity and electroelasticity (in the absence of friction) for a transversely isotropic half-space whose surface is the isotropy plane. This makes it possible to avoid solving the electroelastic problem by finding all the characteristics of electroelastic contact from known cases of purely elastic interaction. Moreover, the electroelastic state of the half-space can be fully described using a known harmonic function, which is a solution of the purely elastic problem. The approach is exemplified by solving contact problems of electroelasticity for flat, elliptic, two circular, conical, and paraboloidal (circular and elliptic in plan) punches __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 69–84, November 2006.     

Asymptotic Formulas for Thermography Based Recovery of Anomalies

We start from a realistic half space model for thermal imaging, which we then use to develop a mathematical asymptotic analysis well suited for the design of reconstruction algorithms. We seek to reconstruct thermal anomalies only through their rough features. With this way our proposed algorithms are stable against measurement noise and geometry perturbations. Based on rigorous asymptotic estimates, we first obtain an approximation for the temperature profile which we then use to design nonit-erative detection algorithms. We show on numerical simulations evidence that they are accurate and robust. Moreover, we provide a mathematical model for ultrasonic temperature imaging, which is an important technique in cancerous tissue ablation therapy.AMS subject classifications: 35R20, 35B30     

分层半空间表面非线性瑞利波的激发*

针对非线性瑞利波在均匀分层半空间结构中的激发和传播规律进行研究。根据摄动理论和模态分解将分层半空间结构中瑞利波的二次谐波声场表示为二倍频瑞利波模式的线性组合,经由互易关系得到各模式的展开系数表达式。对不同分层半空间结构中瑞利波二次谐波的激发和传播特性进行讨论,结果表明相速度匹配的瑞利波模式其二次谐波分量随传播距离线性增长,非匹配模式的二次谐波分量则沿传播方向周期震荡传播。此外,文中定义非线性参数表征瑞利波模式产生的非线性程度,这有利于选择出具有明显非线性效应的匹配点,为后续检测工作提供理论依据,具有指导意义。     

非均匀切向荷载下弹性半空间二阶效应的实例计算*

继文[1],本文详细讨论了一个实例,对二阶弹性材料在Z方向的位移和应力作了数值计算,我们发现,二阶效应使Z方向位移减小而使应力t_rz的值在荷载作用圆内减小而在圆外却增加。