State-space approach is developed to analyze the bending and free vibration of a simply supported, cross-ply laminated rectangular plate featuring interlaminar bonding imperfections, for which a general linear spring layer model is adopted. The analysis is directly based on the three-dimensional theory of orthotropic elasticity and is completely exact. Numerical comparison is made, showing that although the plate theory developed in the literature behaves well for moderately thick perfect laminates it can become inaccurate when bonding imperfections are present. The special problem of the laminate in cylindrical bending is also considered, and the validity of the assumption of cylindrical bending is investigated through numerical examples.
NomenclatureA = dimensionless coef cient matrix a; b; h = length, width and height of plate c i j = elastic constants E; G; ¹ = Young's modulus, shear modulus, and Poisson's ratio h k = thickness of the kth layer K i = stiffness constants of interface M k ; P k = transfer matrices of layer and interface m; n = positive integers N = number of layers in the laminate T; T i j = global transfer matrix and the elements u; v; w = displacement components in Cartesian coordinates V = dimensionless state vector z k = z coordinate of the kth interface »;´; = dimensionless coordinates ½ = mass density ratio ¾ i ; ¿ i j = normal and shear-stress components Ä, ! 0 , ! ¤ = dimensionless frequency parameters ! = circular frequency Subscripts L ; T = directions parallel and perpendicular to the bers k = kth layer or interface Superscript (k) = kth layer
This paper derives a general solution of the three-dimensional equations of transversely isotropic piezothermoelastic materials (crystal class, 6 mm). Two displacement functions are first introduced to simplify the basic equations and a general solution is then derived using the operator theory. For the static case, the proposed general solution is very simple in form and can be used easily in certain boundary value problems. An illustrative example is given in the paper by considering the symmetric crack problem of an arbitrary temperature applied over the faces of a flat crack in an infinite space. The governing integro-differential equations of the problem are derived. It is found that exact expressions for the piezothermoelastic field for a penny-shaped crack subject to a uniform temperature can be obtained in terms of elementary functions. [S0021-8936(00)01704-9]
By introduction of a special dependent variable and separation of variables technique, the electroelastic dynamic problem of a nonhomogeneous, spherically isotropic hollow sphere is transformed to a Volterra integral equation of the second kind about a function of time. The equation can be solved by means of the interpolation method, and the solutions for displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow sphere with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are presented at the end.
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