An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain‐gradient theory of elasticity

Abstract

SUMMARYAn advanced boundary element method (BEM) for solving two-(2D) and three-dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form-II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems.

Boundary element formulation for plane problems in couple stress elasticity

“… and Karlis et al . concentrate on second‐gradient theories. However, all of these inherit the weaknesses of their underlying formulations, as noted above and with more detail in .…”

“…It should be mentioned that there have been a number of articles employing boundary element methods to solve elastic problems within the framework of the previous theories. For example, Das and Chaudhuri [11], Liang and Huang [12], and Shmoylova et al [13] focus on micropolar formulations, while Polyzos et al [14], Tsepoura et al [15] and Karlis et al [16][17][18] concentrate on second-gradient theories. However, all of these inherit the weaknesses of their underlying formulations, as noted above and with more detail in [9].…”

The Effect of Rotation and Inhomogeneity on the Transient Magneto-Thermoviscoelastic Stresses in an Anisotropic Solid

“…Fahmy [9] used the boundary element method to obtain thermal stresses in a nonhomogeneous anisotropie solid. More extensive applications of boundary element methods may be found in Canelas and Sensale [10], Davi and Müazzo [11], Fahmy and ElShahat [12], Hou et al [13], Karlis et al [14], Matsumoto et al [15], Mohammadi et al [16] and Zhang et al [17]. A completely different approach to address dynamic problems utilizing static fundamental solutions is the so-called dual reciprocity boundary element method (DRBBM).…”

A Time-Stepping Drbem for Magneto-Thermo-Viscoelastic Interactions in a Rotating Nonhomogeneous Anisotropic Solid

“…Fahmy [2007] used boundary element method to obtain thermal stresses in a nonhomogeneous anisotropic solid. A more extensive applications of boundary element methods may be found in Matsumoto et al [2005]; Fahmy and El-Shahat [2008]; Canelas and Sensale [2010]; Karlis et al [2010]; Hou et al [2011]; Davi and Milazzo [2011]. A completely different approach to handle dynamic problems utilizing static fundamental solutions is the so-called dual reciprocity boundary element method (DRBEM).…”