## Analysis of materials with strain-gradient effects: a meshless local Petrov-Galerkin (MLPG) approach, with nodal displacements only.(English)Zbl 1148.74346

Summary: A meshless numerical implementation is reported of the 2-D Fleck-Hutchinson phenomenological strain-gradient theory, which fits within the framework of the Toupin-Mindlin theories and deals with first-order strain gradients and the associated work-conjugate higher-order stresses. From a mathematical point of view, the two-dimensional Toupin-Mindlin strain gradient theory is a generalization of the Poisson-Kirchhoff plate theories, involving, in addition to the fourth-order derivatives of the displacements, also a second-order derivative. In the conventional displacement-based approaches in FEM, the interpolation of displacement requires $$C^1$$ -continuity (in order to ensure convergence of the finite element procedure for 4th order theories), which inevitably involves very complicated shape functions. These shape functions involve large numbers of degrees of freedom in every element, including nodal displacements, nodal rotations (i.e. first order gradients of displacement), and even higher order derivatives. $$C^1$$-continuous methods are mostly feasible only for one-dimensional problems. The standard approach for solving Bernoulli-Euler beam problems is by employing $$C^1$$-continuous Hermite cubic shape functions, interpolating both displacements and rotations (i.e., slopes). For two-dimensional problems, such as involving plate and shell analysis, $$C^1$$-continuous methods are very complicated, and formulations for three-dimensional problems as they arise from strain gradient theories become more or less intractable. The high computational cost and large number of degrees of freedom soon place such formulations beyond the realm of practicality. Recently some mixed and hybrid formulations, requiring only $$C^0$$-continuity, have also been developed and applied to strain gradient plasticity or elasticity problems. But they are even more problematic. While some of the developed elements have been subjected to the Patch test and other benchmark problems, a rigorous numerical analysis is missing: mathematical proofs of consistency and stability have not been demonstrated, and the rate of convergence has not been established. The large number of nodal degrees of freedom is still inevitable in such mixed methods. It is evident that currently, no efficient finite element methods are available for strain gradient theory formulations.
In this paper, a truly meshless approach, the Meshless Local Petrov-Galerkin Method(MLPG), is introduced for higher-order gradient theories that trace their roots to the work of Toupin. The degrees of freedom consist of only nodal displacements, i.e. nodal rotational degrees of freedom are not used. A numerical analysis of the method is presented, covering the usual ground of consistency, stability and hence, convergence in several examples. All the results show that, when solving fourth-order elliptic problems such as those arising in gradient theories of material behavior, the MLPG method is superior to primal or mixed finite element procedures.

### MSC:

 74S30 Other numerical methods in solid mechanics (MSC2010) 74C99 Plastic materials, materials of stress-rate and internal-variable type