Determining the intensity factors for stresses, electric-flux density, and electric-field strength in multiply connected electroelastic anisotropic media

Abstract

Formulas are derived to evaluate the intensity factors of stresses, electric-flux density, and electric-field strength in two-dimensional problems of elasticity, thermoelasticity, electroelasticity, magnetoelasticity, thermoelectroelasticity, and thermomagnetoelasticity for anisotropic bodies with holes and plane (rectilinear) cracks Keywords: intensity factors for stresses, electric-flux density, and electric-field strength, two-dimensional problem, anisotropic bodyIntroduction. Fracture strength analysis of structural members based on energy failure criteria necessarily involves calculation of stress intensity factors (SIFs), which requires a reliable method.Solution of two-dimensional problems of elasticity, thermoelasticity, electroelasticity, and magnetoelasticity is often based on complex potentials, which are determined from the boundary conditions on crack faces and smooth boundaries (external edge and holes).The boundary conditions on crack faces in isotropic plates are often reduced to the Riemann-Hilbert (linear conjugation) problem whose solution contains an isolated singularity at the crack tips. This allows deriving an analytic formula to calculate the SIFs [1-3, 10, 12, 14-18].In the case of two-dimensional problems of anisotropic elasticity, the boundary conditions on cracks lying along one straight line can also be reduced to the Riemann-Hilbert problem to derive analytic expressions for the 15,19]. The use of the linear conjugation method to solve problems of elasticity for multiply connected anisotropic bodies that have holes besides cracks (even if they lie along one straight line) involves severe mathematical and computational difficulties. Therefore, an approximate method was proposed in [7,9] to determine the SIFs for an arbitrary multiply connected domain with arbitrarily arranged rectilinear cuts. The method employs numerical limiting processes to find derivatives of complex potentials or even normal and tangential stresses in small neighborhood of crack tips. This method made it possible to determine the SIFs for arbitrary multiply connected bodies and to solve such problems for anisotropic media that could not be solved by other methods. Later, this method was extended to thermoelasticity [20], electroelasticity [8], magnetoelasticity [21], and thermoelectroelasticity [13]. However, this method may become unstable if a crack is close to a smooth boundary or concentrated forces act on the body. Note that problems for anisotropic inelastic and electroelastic cracked bodies were also addressed in [22,23].The present paper proposes a new approach to the determination of the intensity factors for stresses, electric-flux density, and electric-field strength that does not require passing to the limit.1. Two-Dimensional Problem of Anisotropic Elasticity. Consider an anisotropic body with elliptic cavities under external forces that induce a two-dimensional stress-strain state constant along the cavity generatrix, which coincides with the Oz-axis of a Cartesian coordinate system Oxyz. The ellipt...

Wedging of piezoceramic materials

“…The thickness of the wedges is limited and equal to 2h at infinity. Conditions (13) hold at the interface between the wedges and the cracks, as in the previous examples. The shape of the wedges is described by the equation…”

“…Hence, for the electric boundary conditions (13) or (14), it is necessary to determine the unknown harmonic function F to solve the problem. These boundary conditions differ by the constants A 1 Piezo or A 2 Piezo only.…”

Mathematical modeling and analysis of the stressed state in the orthotropic piezoelectric medium with a circle crack

“…В то же время решение пространственных трехмерных задач электроупругости является весьма сложной математической проблемой, поскольку исходная система уравнений для нахождения напряженного и электрического состояний пред-ставляет собой связанную систему дифференциальных уравнений в частных производных [1,3,4]. Поэтому до настоящего времени наиболее полно ис-следованы двумерные задачи электроупругости (с учетом связанности по-лей) для тел с концентраторами напряжений [8,10,11]. Для случая транс-версально-изотропных свойств электроупругого материала (представляют широкий класс пьезоэлектрических материалов) в работах [20,25] предло-жены подходы к построению общих решений системы связанных уравнений электроупругости, на основе которых получены решения ряда задач для пьезоэлектрического материала с полостями, включениями, трещинами, что специальным образом ориентированы относительно оси симметрии элек-троупругого материала.…”

Problems of electromagnetoelasticity for a half-plane and a strip with holes and cracks

“…Minimizing functional (12), we obtain a system of linear algebraic equations solving which yields the complex potentials (10), which can be used to determine the EMES at any point of the strip [7] and the intensity factors for stresses, electromagnetic-flux density, and electromagnetic-field strength at the crack tip [2,8].…”