Static and Dynamic BEM Analysis of Strain Gradient Elastic Solids and Structures

This paper reviews the theory and the numerical implementation of the direct boundary element method (BEM) as applied to static and dynamic problems of strain gradient elastic solids and structures under two- and three- dimensional conditions. A brief review of the linear strain gradient elastic theory of Mindlin and its simplifications, especially the theory with just one constant (internal length) in addition to the two classical elastic moduli, is provided. The importance of this theory in successfully modeling microstructural effects on the structural response under both static and dynamic conditions is clearly described. The boundary element formulation of static and frequency domain dynamic problems of strain gradient elasticity is accomplished with the aid of reciprocal theorems and corresponding fundamental solutions. Quadratic line and surface boundary elements are developed for two- and three- dimensional problems, respectively. Special crack tip or front, line and surface boundary elements of variable singularity are also developed for fracture mechanics problems. A variety of strain gradient elastic static and dynamic problems involving two- and three- dimensional solids and structures with or without cracks as solved by the BEM are presented in order to illustrate the method, demonstrate its advantages over the finite element method (FEM) and assess and discuss the influence of the microstructure on the response.