A reexamination of some puzzling results in linearized elasticity

In this paper, we analyse three commonly discussed `flaws' of linearized elasticity theory and attempt to resolve them. The first `flaw' concerns cylindrically orthotropic material models. Since the work of Lekhnitskii (1968), there has been a growing body of work that continues to this day, that shows that infinite stresses arise with the use of a cylindrically orthotropic material model even in the case of linearized elasticity. Besides infinite stresses, interpenetration of matter is also shown to occur. These infinite stresses and interpenetration occur when the ratio of the circumferential Young modulus to the radial Young modulus is less than one. If the ratio is greater than one, then the stresses at the center of a spinning disk are found to be zero (recall that for an isotropic material model, the stresses are maximum at the center). Thus, the stresses go abruptly from a maximum value to a value of zero as the ratio is increased to a value even slightly above one! One of the explanations provided for this extremely anomalous behaviour is the failure of linearized elasticity to satisfy material frame-indifference. However, if this is the true cause, then the anomalous behaviour should also occur with the use of an isotropic material model, where, no such anomalies are observed. We show that the real cause of the problem is elsewhere and also show how these anomalies can be resolved. We also discuss how the formulation of linearized elastodynamics in the case of small deformations superposed on a rigid motion can be given in a succinct manner. Finally, we show how the long-standing problem of devising three compatibility relations instead of six can be resolved.

Anisotropic thermopiezoelectric solids with an elliptic inclusion or a hole under uniform heat flow

The two-dimensional problem of a thermopiezoelectric material containing an elliptic inclusion or a hole subjected to a remote uniform heat flow is studied. Based on the extended Lekhnitskii formulation for thermopiezoelectricity, conformal mapping and Laurent series expansion, the explicit and closed-form solutions are obtained both inside and outside the inclusion (or hole). For a hole problem, the exact electric boundary conditions on the hole surface are used. The results show that the electroelastic fields inside the inclusion or the electric field inside the hole are linear functions of the coordinates. When the elliptic hole degenerates into a slit crack, the electroelastic fields and the intensity factors are obtained. The effect of the heat how direction and the dielectric constant of air inside the crack on the thermal electroelastic fields are discussed. Comparison is made with two special cases of which the closed solutions exist and it is shown that our results are valid.

压电材料平面问题的尖端场和应力强度因子的求解

利用Lekhnitskii理论和Stroh理论的相互联系,把已知的基于Lekhnitskii理论平面应变结果转化为Stroh理论形式的结果,直接获得Stroh公式中A,B的显式表达式,此方法可扩展到平面应力情况,然后导出压电材料平面应变问题的尖端场Williams形式的展开式,采用半权函数法计算有限大压电体平面问题应力和电位移强度因子.对无穷大板含中心裂纹的情况下本文结果和已有结果进行了比较,表明本文方法得到的结果精度可靠.本文方法的最大优点是可以求解有限压电体的应力强度因子,并且需要的单元少,精度高,实用性好.

含孔洞和裂纹压电板问题的研究

压电陶瓷因具有良好的压电性而被广泛应用于智能结构。在工作状态下，由力或电载荷引起的应力或电位移的集中会导致压电元件的力电失效。本论文对压电陶瓷板的力电集中和断裂问题进行探讨。首次应用Rei ssner理论对含孔、裂纹压电板的弯曲问题进行了研究，得出了孔边和裂纹尖端场的力电分布形式；应用半权函数方法分析计算含裂纹压电板应力强度因子。全文包括以下几部分：（一）含孔压电板的弯曲基于Reissner板理论，结合有限元法和解析法研究含孔压电板的力电集中问题。给出了上下表面电源短路、含孔压电板方程的通解。计算了含圆孔无穷大压电板受纯弯曲作用的力电集中问题。结果表明：1）压电效应对面内弯曲正应力的影响很小，可忽略不计。2）压电效应对剪应力影响较大，切向电场强度和电位移的集中程度随径厚比变化趋势和切向剪力的变化趋势相同。3）中面上孔边电势和垂直于板中面的电场强度的集中程度随径厚比的减小而减弱，垂直于中面的电位移的集中程度随着径厚比的减小而增强。（二）含裂纹压电板的拉伸和弯曲（极化轴垂直于板的中面）用双重级数展开方法给出了基于Reissner板理论压电板的裂纹尖端奇异场。研究得出：垂直于板面的电场强度不奇异，但是电位移是奇异的；在裂纹尖端的奇异场里，板面内弯曲应力与电场无关，同时面内电位移奇异且只与剪力有关，垂直于板面的电位移与弯曲应力有关。计算了受纯弯矩作用含中心裂纹的压电矩形板，并与非压电材料板情况进行了比较，结果表明：压电效应对应力强度因子影响不大，可以忽略不计，因此可以用含裂纹的各向同性非压电板计算应力强度因子，并得出垂直于板面的电位移。（三）含裂纹压电板的拉伸和弯曲（极化轴平行于板的中面）基于Reissner板理论，分析了含裂纹且极化轴平行于板中面的压电板拉伸和弯曲问题。研究表明，当板的上下表面没有剪力和电荷载，只在板边加电荷载和力的情况下，此问题可以分解为压电材料的平面应力断裂和各向异性板的弯曲断裂问题。对于含裂纹压电材料的平面应力问题，利用Stroh理论导出压电材料平面问题尖端场Wiiliams形式的展开式。计算了有限压电板含中心裂纹受纯弯曲和板边加电荷载作用的断裂问题。得出以下结论：各向异性板受纯弯曲作用时，由线性压电材料的本构关系可以得出电位移的奇异性；整个问题电位移奇异性是由受电荷载作用的平面应力问题和受纯弯曲作用的各向异性板弯曲问题各自引起的电位移奇异性的叠加。（四）计算含裂纹压电板应力强度因子的半权函数法本文对含裂纹压电板的应力强度因子计算都采用了半权函数法。由功能互等定理导出用半权函数和积分围道裂纹尖端的参考场所表示的所研究情况的应力强度因子的表达式。对多种情况进行了计算，并与相关结果进行了比较，结果表明：此方法需要的单元少，精度高，实用性好。利用本文方法对实际应用中出现的有限尺寸压电介质断裂问题进行分析，为压电元件的力电祸合性态研究及其可靠性预测提供理论依据。

Mixed Mode Fracture Behaviour of Piezoelectric Materials

Piezoelectrics present an interactive electromechanical behaviour that, especially in recent years, has generated much interest since it renders these materials adapt for use in a variety of electronic and industrial applications like sensors, actuators, transducers, smart structures. Both mechanical and electric loads are generally applied on these devices and can cause high concentrations of stress, particularly in proximity of defects or inhomogeneities, such as flaws, cavities or included particles. A thorough understanding of their fracture behaviour is crucial in order to improve their performances and avoid unexpected failures. Therefore, a considerable number of research works have addressed this topic in the last decades. Most of the theoretical studies on this subject find their analytical background in the complex variable formulation of plane anisotropic elasticity. This theoretical approach bases its main origins in the pioneering works of Muskelishvili and Lekhnitskii who obtained the solution of the elastic problem in terms of independent analytic functions of complex variables. In the present work, the expressions of stresses and elastic and electric displacements are obtained as functions of complex potentials through an analytical formulation which is the application to the piezoelectric static case of an approach introduced for orthotropic materials to solve elastodynamics problems. This method can be considered an alternative to other formalisms currently used, like the Stroh’s formalism. The equilibrium equations are reduced to a first order system involving a six-dimensional vector field. After that, a similarity transformation is induced to reach three independent Cauchy-Riemann systems, so justifying the introduction of the complex variable notation. Closed form expressions of near tip stress and displacement fields are therefore obtained. In the theoretical study of cracked piezoelectric bodies, the issue of assigning consistent electric boundary conditions on the crack faces is of central importance and has been addressed by many researchers. Three different boundary conditions are commonly accepted in literature: the permeable, the impermeable and the semipermeable (“exact”) crack model. This thesis takes into considerations all the three models, comparing the results obtained and analysing the effects of the boundary condition choice on the solution. The influence of load biaxiality and of the application of a remote electric field has been studied, pointing out that both can affect to a various extent the stress fields and the angle of initial crack extension, especially when non-singular terms are retained in the expressions of the electro-elastic solution. Furthermore, two different fracture criteria are applied to the piezoelectric case, and their outcomes are compared and discussed. The work is organized as follows: Chapter 1 briefly introduces the fundamental concepts of Fracture Mechanics. Chapter 2 describes plane elasticity formalisms for an anisotropic continuum (Eshelby-Read-Shockley and Stroh) and introduces for the simplified orthotropic case the alternative formalism we want to propose. Chapter 3 outlines the Linear Theory of Piezoelectricity, its basic relations and electro-elastic equations. Chapter 4 introduces the proposed method for obtaining the expressions of stresses and elastic and electric displacements, given as functions of complex potentials. The solution is obtained in close form and non-singular terms are retained as well. Chapter 5 presents several numerical applications aimed at estimating the effect of load biaxiality, electric field, considered permittivity of the crack. Through the application of fracture criteria the influence of the above listed conditions on the response of the system and in particular on the direction of crack branching is thoroughly discussed.