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SUMMARYThe present work is inspired by Anderson et al. (Phys. D Nonlinear Phenom. 2000; 135(1-2):175-194) and Noll (http://www.math.cmu.edu/wn0g/noll) and falls in the conceptual line of the Ginzburg-Landau class of first-order phase-transition models based on the concept of phase-field parameter. Trying to keep the exposition as much general as possible, we develop below a thermodynamically consistent rationalization of the physical process of (anisotropic) deposition of pyrolytic carbon from a gas phase. The derivation line we follow is well established in the field of the modern continuum physics. From the covariance of the first principle of thermodynamics, the second Newton's law and the Liouville's theorem with respect to the one-dimensional Lie groups of transformations, the balance laws for the temperature, linear momentum and density are formulated. This system of partial differential equations is comprehended further by the constitutive laws for the phase field, the stress, and the heat and entropy fluxes obtained in a form consistent with the Clausius-Duhem understanding of the second law. The result referred to as a local, strongly coupled initial boundary value problem of chemical vapor deposition (IBVP-CVD) constitutes the general mathematical description of the CVD process. The weak form of isotropic IBVP-CVD is then derived and discretized by means of the discontinuous Galerkin method. At the end of the paper, we also derive the weak formulations for the local lifting operators that provide the stabilization mechanism for the discontinuous Galerkin discretization scheme.

A B S T R A C T In this paper, transient thermoelastic crack analysis in two-dimensional, isotropic, continuously non-homogeneous and linear elastic functionally graded materials subjected to a thermal shock is presented. The Laplace transform technique is used to eliminate the time dependence of the governing equations of the linear coupled thermoelasticity. Fundamental solutions for isotropic, homogeneous and linear elastic solids in the Laplacetransformed domain are applied to derive boundary-domain integral equations for the mechanical and thermal fields. The radial integration method is employed to transform the domain integrals into the boundary integrals. A collocation-based boundary element method is implemented for the spatial discretization of the boundary-domain integral equations. The time-dependent numerical solutions are obtained by using Stehfest's inversion algorithm. Numerical results are presented and discussed to show the influences of the material gradation, the thermo-mechanical coupling, the crack orientation and the thermal shock loading on the dynamic stress intensity factors.Keywords boundary element method; dynamic stress intensity factors; functionally graded materials; Laplace transform; radial integration method; thermal shock. N O M E N C L A T U R Ea = half crack length a j i = unknown expansion coefficients b j = unknown expansion coefficients c (x) = specific heat at constant strain c 0 (x) = free-term coefficient c 0 jk (x) = free-term coefficients c i jkl (x) = elasticity tensor d A = support size for the application point A E(x) = Young's modulus h = semi-height of the FG plate H(t) = Heaviside step function I = identity matrix k(x) = thermal conductivitȳ K ± I (t) = normalized mode-I dynamic stress intensity factor at the crack-tips x 1 = ±ā K ± I I (t) = normalized mode-II dynamic stress intensity factor at the crack-tips x 1 = ±a N = total number of the unknown quantities N A = total number of the application points N s = total number of the approximation terms N q = total number of the boundary elements N d = total number of the internal points N w = total number of the boundary nodes N a (ξ ) = standard shape functions for quadratic elements n i = components of the outward unit normal vector Correspondence: A. Ekhlakov,

A boundary element method for the transient thermoelastic fracture analysis in isotropic, continuously non-homogeneous and linear elastic functionally graded materials subjected to a thermal shock is presented. The material parameters are assumed to be continuous functions of the Cartesian coordinates. Laplace-domain fundamental solutions of linear coupled thermoelasticity for infinite, isotropic, homogeneous and linear elastic solids are applied to derive the boundary-domain integral equation formulation. The numerical implementation is performed by using a collocation method for the spatial discretization. Numerical results for the dynamic stress intensity factors are presented and discussed.