Transient dynamic analysis of 3-D gradient elastic solids by BEM

A boundary element methodology is presented for the frequency domain elastodynamic analysis of three-dimensional solids characterized by a linear elastic material behaviour coupled with microstructural effects taken into account with the aid of a simple gradient elastic theory obtained as a special case of the general one due to Mindlin. A variational statement to determine the equation of motion as well as all the possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem is provided. The gradient frequency domain elastodynamic fundamental solution is explicitly derived and used to construct the boundary integral representation of the problem with the aid of a reciprocal integral identity. In addition to a boundary integral representation for the displacement, a boundary integral representation for its normal derivative is also necessary for the complete formulation of a well-posed problem. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. The problem is solved in the frequency domain for a sequence of values of the frequency parameter and the transient response is obtained by a numerical inversion of the frequency domain solution through the fast Fourier transform algorithm. Two numerical examples serve to illustrate the method and demonstrate its accuracy.