Studies On Two-Dimensional Problems Of Functionally Graded Materials With A Hole Or Inclusion

Holes or inclusions are common defects in materials, and they are one of the important factorsleading to local stress concentration. So how to study the stress concentration near the hole orinclusion is one of the classic problems in the field of solid mechanics, and a lot of studies have beendone on this topic. But, it should be pointed out that most of the previous works are mainly for thecase of homogeneous medium. In fact, the nonhomogeneous materials, such as functionally gradedmaterials (FGMs), have found wide application in engineering. But relatively little works are done forthe problem of stress concentration in FGMs, and they are mainly restricted to the case of thestructures with straight geometric boundary. In this paper we study the problems of functionallygraded materials with a hole or inclusion and functionally graded structures with curved boundary.The main contents are as follows:In the first Chapter, a brief introduction to FGMs is given and the problems needed to be solvedare outlined.In the second Chapter, the complex variable functions of stress and boundary conditions abouttwo-dimensional thermoelastic problems are presented based on Muskhelishvili’s theory. Then thebasic theory of elastic dynamics is introduced, and the complex variable method for solving elasticwave scattering problems is briefly presented.In the third Chapter, the stress concentrations around a circular hole in an infinite and finite FGMplate are analyzed, respectively. By using the method of piece-wise homogeneous layers, the FGMplate with a circular hole is reduced approximately to the case where the homogeneous plate containsN rings with equal thickness, and then it can be solved based on Muskhelishvili’s theory combinedwith the least squares boundary collocation technique. Finally, the numerical results are shown todiscuss the effects of the variations of Young’s modulus and finite boundary conditions on the stressdistribution in the FGM plate.In the fourth Chapter, the two-dimensional dynamic stress in a FGM plate with a circular holeunder plane compressional waves is investigated. By using the similar method as the previous Chapter,the problem is reduced to the case where the homogeneous plate contains N rings. The dynamic stressdistributions in the FGM plate having radial arbitrary material parameters are presented based on thecomplex variable method.In the fifth Chapter, the two-dimensional thermoelastic problems of an infinite matrix with afunctionally graded coated circular inclusion under uniform heat flow are analyzed. Firstly, by decomposing the coating to N rings with equal thickness, the complex potential functions oftemperature and stress field in the inclusion, matrix and each ring can be expressed by series. Then thetemperature and stress field can be derived based on the continuous conditions at the interfacebetween ecah rings.In the sixth Chapter, the two-dimensional thermal stresses in a finite matrix with a functionallygraded coated circular inclusion are studied. On the basis of the fifth Chapter, the least squaresboundary collocation technique is used here. That is, taking the collocation points along the finiteouter boundary and substituting the points into boundary equations, one can obtain additional linearequations, which can be used to determine the temperature and stress field in the finite matrix.In the seventh Chapter, the thermal stresses in a functionally graded coating system withparabolic boundary are analyzed. The closed-form solutions of thermal stress related to compositionalgradient, coating thickness and substrate curvature are obtained. The numerical results are shown todiscuss the effects of three parameters on the thermal stress distribution in the structure.Finally, in the last Chapter, the present work is summarized and some future works are proposedon the topic.