Strain gradient elasticity with geometric nonlinearities and its computational evaluation

Wolfgang H Müller,B Emek Abali,Victor A Eremeyev

doi:10.1186/s40759-015-0004-3

Abstract

Abstract Background The theory of linear elasticity is insufficient at small length scales, e.g., when dealing with micro-devices. In particular, it cannot predict the “size effect” observed at the micro- and nanometer scales. In order to design at such small scales an improvement of the theory of elasticity is necessary, which is referred to as strain gradient elasticity. Methods There are various approaches in literature, especially for small deformations. In order to include geometric nonlinearities we start by discussing the necessary balance equations. Then we present a generic approach for obtaining adequate constitutive equations. By combining balance equations and constitutive relations nonlinear field equations result. We apply a variational formulation to the nonlinear field equations in order to find a weak form, which can be solved numerically by using open-source codes. Results By using balances of linear and angular momentum we obtain the so-called stress and couple stress as tensors of rank two and three, respectively. Since dealing with tensors an adequate representation theorem can be applied. We propose for an isotropic material a stress with two and a couple stress with three material parameters. For understanding their impact during deformation the numerical solution procedure is performed. By successfully simulating the size effect known from experiments, we verify the proposed theory and its numerical implementation. Conclusion Based on representation theorems a self consistent strain gradient theory is presented, discussed, and implemented into a computational reality.

Highlights

The theory of linear elasticity is insufficient at small length scales, e.g., when dealing with micro-devices
We propose to apply two balance equations of momenta in the current frame:
In order to analyze the effect of the material parameters, α, β, γ in the proposed constitutive equation for the couple stress μijk we construct a simple example to solve

Summary

Introduction

The theory of linear elasticity is insufficient at small length scales, e.g., when dealing with micro-devices. It cannot predict the “size effect” observed at the micro- and nanometer scales. We model the linear response at small deformations with HOOKE’s law, which has the same form for huge and small structures. Such a simple approach becomes inadequate at the micrometer scale. Sub-micrometer structures frequently show a stiffer response than predicted by traditional theory. This socalled size effect has been known experimentally for a long

Methods

Results

Conclusion