The free-space Green function for a two-dimensional exponentially graded elastic medium is derived. The shear modulus µ is assumed to be an exponential function of the Cartesian coordinates (x, y), i.e. µ ≡ µ(x, y) = µ 0 e 2(β 1 x+β 2 y) , where µ 0 , β 1 , and β 2 are material constants, and the Poisson ratio is assumed constant. The Green function is shown to consist of a singular part, involving modified Bessel functions, and a non-singular term. The non-singular component is expressed in terms of one-dimensional Fourier-type integrals that can be computed by the fast Fourier transform.
A Mode-III crack problem in a functionally graded material modeled by anisotropic strain-gradient elasticity theory is solved by the integral equation method. The gradient elasticity theory has two material characteristic lengths and , which are responsible for volumetric and surface strain-gradient terms, respectively. The governing differential equation of the problem is derived assuming that the shear modulus G is a function of x, i.e., G Gx G 0 e x , where G 0 and are material constants. A hypersingular integro-differential equation is derived and discretized by means of the collocation method and a Chebyshev polynomial expansion. Numerical results are given in terms of the crack opening displacements, strains, and stresses with various combinations of the parameters , , and. Formulas for the stress intensity factors, K III , are derived and numerical results are provided.
This article provides a detail derivation of a singular Fredholm integral equation for the solution of a mixed mode crack problem in a nonhomogeneous medium. The integral equation derived here has already been addressed by F. Delale and F. Erdogan (Delale & Erdogan 1983), one of the most cited and pioneer papers in fracture mechanics that uses singulalr integral equation method (SIEM) to solve crack problems. However, probably due to its limit of paper length, some mathematical details are not provided to bring this powerful method, SIEM, to its full strength. In this paper we fill in the mathematical gaps, and both analytical and numerical parts are addressed in details. Some discussions from the view point of differential equations are given, and new numerical outcomes under different loading functions are provided.
We study the solitary waves and their interaction for a six-order generalized Boussinesq equation (SGBE) both numerically and analytically. A shooting method with appropriate initial conditions, based on the phase plane analysis around the equilibrium point, is used to construct the solitary-wave solutions for this nonintegrable equation. A symmetric three-level implicit finite difference scheme with a free parameter θ is proposed to study the propagation and interactions of solitary waves. Numerical simulations show the propagation of a single solitary wave of SGBE, and two solitary waves pass by each other without changing their shapes in the head-on collisions
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