A simplified second gradient model for dilatant materials: Theory and numerical implementation

Abstract

a b s t r a c tThis paper deals with the development of a new second gradient model, its numerical implementation and its validation. In order to remedy to the spurious mesh dependency of the post localized computation enhanced models incorporating some internal length are necessary. These models are very time consuming. In this paper we present a simplified theory within the framework of constrained micromorphic models involving only the micro volumetric strain. Provided the use of an additional penalty term in the numerical treatment, this model is quite efficient to regularize problems modelling behaviors exhibiting plastic volumetric strain such as the ones of geomaterials. More over this model is notably less time consuming than the more general ones.

Using a penalty term to deal with spurious oscillations in second gradient finite elements

“…A similar approach was proposed by 39 in the context of plasticity. Noting C the penalty factor, the weak formulation of the problem becomes: …”

An identification method to calibrate higher-order parameters in local second-gradient models

“…In the same way that the micromorphic model was transformed to the second-gradient model, a kinematic constraint can be introduced between the macro volume change * V and the microdilation χ * . This modification produces the second-gradient dilation model [23]. We thus obtain the following equation for a kinematically admissible displacement field u * i :…”

“…In our experience, however, we have found that a minimum of four or five elements are required to model the width of the shear bands in order to obtain mesh-independent results [23,29]. This number of elements ensures an accurate description of localisation for the various applications considered in this study.…”

“…It is therefore necessary to consider simplified enhancements. The strong volume changes that occur in most geomaterials during the development of shear bands (e.g., [19][20][21]) have led to the development of models that involve volumetric quantities (e.g., [22,23]). In these models, the enhanced kinematical variable is a scalar (instead of a complete second-order tensor).…”

“…Note, however, that this type of theory is unsuitable for materials that do not exhibit volume changes during plastic loading. In this paper, we work within the framework of the second-gradient dilation model developed by Fernandes et al [23].…”