A comparison between the performance of different Mindlin's theory based strain gradient models in local singularities

Abstract

AbstractThe classical continuum mechanics theories are originally supposed to determine deformations in ranges from millimeter to meter, the so‐called macroscopic scale. In fact, these theories are approximations of physical systems neglecting the underlying microstructure. For instance, a Cauchy continuum, i.e. a continuum with an elastic energy determined as a function of the gradients of its macroscopic displacement, can only approximate the behavior of a physical system sufficiently as long as the microstructure has a much smaller length‐scale than the macrostructure [1]. Although these models were exploited in studies for large and small scales, experiments have shown that the classical models are not able to properly cover the smaller scales; in particular, problems in micron‐ and nano‐dimensions are frequently observed. Size‐effects, which cannot be captured exploiting these theories, seem to be the source of this issue. On top of that, the appearance of local singularities at the crack tips (or more broadly, in the presence of point and line loads) is one of the known limitations of the classical continuum mechanics theories. Generalizing these models by introducing additional kinematic terms to consider the underlying microstructure effects at macroscopic levels is one way of overcoming the already mentioned problems. In this contribution, we will focus on the Mindlin's theory of elasticity with microstructures [2] and its different forms. Therein, it is shown that for the first strain gradient theory, five additional parameters must be introduced. However, in practice, due to the complexities of measuring the new parameters, various simplified versions of the theory are being exploited, among them we name Altan et al. [3] and Reiher et al. [4]. Our aim here is to compare the performance of these simplified theories in removing the local singularities of the conventional continuum mechanics theory.