A Method for the Numerical Solution of Boundary Value Problems Governed by Second-order Elliptic Systems

On an Antiplane Crack Problem for Functionally Graded Elastic Materials

“…For such materials, the c (0) i jkl in (4.1) may be conveniently expressed in terms of five constants A, N , F, C and L (see Clements [6]). If the x 3 axis is normal to the transverse planes then the nonzero c (0) i jkl are related to the constants A, N , F, C and L by the equations…”

“…Expressions for the c (0) i jkl referred to any Cartesian frame of reference in terms of the five constants A, N , F, C and L may be readily obtained from (7.1) by employing the transformation law for fourth-order Cartesian tensors (see Clements [6]). Thus, if the orientation of the Cartesian coordinate frame within the material is such that the x 2 at https://www.cambridge.org/core/terms.…”

“…The general solution of (3.7) and (3.8) may be written in terms of an analytic function in the form (Eshelby et al [12], Clements [6])…”

A note on the boundary integral equation method for the solutions of second order elliptic equations

“…In (2.1) the convention of summing over repeated Latin subscripts is employed. The constants a iJkl must satisfy certain symmetry and ellipticity conditions which are detailed in Clements and Rizzo [3]. The problem is to find a solution to (2.1) valid in a region ® in E 2 with boundary C. On C either the dependent variables <}> k are specified or the P i are specified where where rij is the unit (outer) normal to Si.…”

“…The method is essentially obtained by expressing the solution to a particular boundary value problem in terms of an integral equation with the integral taken round the boundary of the region under consideration. This integral equation is then solved numerically by employing approximate quadrature formulas and then solving the resulting system of linear algebraic equations (see for example Jaswon and Symm [4] and Clements [1]). The aim of the present work is to introduce an improvement to this procedure for the class of problems which are governed by a system of second order, linear partial differential equations.…”