engleski

Two Different Mixed Meshless Operator-split Procedures for Gradient Elasticity

In recent years, scientific research has been aimed to improve the modeling accuracy of various effects that cannot be properly described using classical continuum theories. Among others, these phaenomena include material softening and strain localization effects. In order to solve the differential equations attached to the use of non-classical continuum theories by standard displacement approach, high-order approximation functions are necessity. Therefore, the use of standard Finite Element Method (FEM) for solving of gradient elasticity problems leads to computationally burdensome element formulations with complex shape functions that also require a large number of degrees of freedom. On the other hand, mixed FEM approaches require the satisfaction of complicated inf-sup condition in order to achieve stability. Hence, such FEM formulations are considered inefficient, especially in case of 3-D problems. When compared to FEM, the shape functions in meshless methods can be constructed with arbitrary order of continuity and fewer number of nodal unknowns in a simpler manner. Therefore, the meshless approaches seem to be more appropriate for modeling of deformation responses using gradient theories. In this contribution, the fourth-order equilibrium equations of gradient elasticity are solved using two different operator-split procedures, the displacement- and strain-based, in a staggered manner. The mixed strong-form meshless approach based on the Aifantis gradient theory is employed. The original equilibrium equations are solved as an uncoupled sequence of two sets of second-order differential equations. Several numerical examples are chosen and utilized to compare the attained results. The application of the operator-split procedure and the mixed meshless approach should result in stable numerical formulations. To further alleviate the well- known stability problems of collocation methods associated with the appearance of natural boundary conditions in the numerical model, the solution of the gradient elasticity problem using weak-strong formulation will also be considered.

meshless method, gradient elasticity, operator-split procedures

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