Advances in Materials Science and Engineering

Volume 2016, Article ID 9675604, 9 pages

http://dx.doi.org/10.1155/2016/9675604

## On Finite Element Computations of Contact Problems in Micropolar Elasticity

Rzeszów University of Technology, Al. Powstańców Warszawy 8, 35959 Rzeszów, Poland

Received 22 July 2016; Accepted 27 November 2016

Academic Editor: Ying Wang

Copyright © 2016 Victor A. Eremeyev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Within the linear micropolar elasticity we discuss the development of new finite element and its implementation in commercial software. Here we implement the developed 8-node hybrid isoparametric element into ABAQUS and perform solutions of contact problems. We consider the contact of polymeric stamp modelled within the micropolar elasticity with an elastic substrate. The peculiarities of modelling of contact problems with a user defined finite element in ABAQUS are discussed. The provided comparison of solutions obtained within the micropolar and classical elasticity shows the influence of micropolar properties on stress concentration in the vicinity of contact area.

#### 1. Introduction

Nowadays the interest grows to further development and application extended models of continuum mechanics in order to model micro- and nanostructured materials with complex inner structure. The basic idea of enhancement of classic Cauchy continuum model is to add additional fields describing additional degrees of freedom into constitutive equations or/and consider higher-order gradients of deformations. Among these generalized models there are the surface elasticity, micropolar or Cosserat continua, microstretched and micromorphic media, media with internal variables, gradient elasticity, and so forth. In particular, the micropolar model [1–3] proposed by Cosserat brothers more than hundred years ago found applications for modelling such materials as porous solids [4, 5], bones [6–8], masonries [9, 10], beam lattices [11], and other composite materials; see [1, 12–15] and reference therein. The micropolar elasticity possesses the description of size-effects and may be useful for description of the microstructured solids such as foam, bones, powders, and soils. In particular, the influence of micropolar properties may be important for the analysis of the stress concentration near holes and notches or in the vicinity of contact area. Within the Cosserat continuum model the translations and rotations determine the kinematics of the medium and the stress and couple stress tensors are introduced. The micropolar properties of material may be important near singularities or to describe observed experimentally size-effect [16–19]. Let us also note that for such complex media even more general models of continuum mechanics such as gradient elasticity may be useful; see, for example, [20–24]. Application of extended models of continuum for modelling of such structures as open-cell foams, beam lattices, and pantographic systems is motivated by their complex inner structure. For example, in the classic beam theory moments play an important role and their influence is inherited by the homogenized models. Let us also note that for structured and nonhomogeneous beams used as structural elements for foam or lattice description some extended models were proposed in [25–31]. For rods and beams there are some phenomena as warping of cross-section, instabilities, and sensibility to imperfections are also important; see, for example, [32–41]. Considering all of these phenomena may lead to rather complex models of homogenized media. From this point of view the micropolar elasticity may be treated as first step towards modelling of microstructured materials and their contact. Further extensions of the indentation problems can be performed based on the other enriched models of continuum; see, for example, strain gradients models [42, 43].

Let us note that the effective solution of boundary value problems of micropolar elasticity as well as of other enhanced models requires advanced numerical code such as the finite element method. The generalized models of continua require usually more computational efforts than the classical elasticity since there exist more degrees of freedom. For the micropolar continuum we use an isogeometric analysis [44, 45] as efficient FEM strategy which together with a hybrid mixed formulation was applied for generalized continua and structures; see, for example, [46–50]. For the moment commercial FEM software gives one the possibility to use user defined elements and user defined procedure for implementation of nonstandard material models. Here we developed new finite element and implement it in ABAQUS.

The paper is organized as follows. In Section 2 we present the basic equations of the linear micropolar elasticity. The equilibrium equations, static and kinematic boundary conditions, and constitutive equations are given. Here we are restricted by isotropic case using the Voigt notation. In Section 3 we discuss the finite element modelling for the micropolar solids. Finally, in Section 4 we present the solution of contact problem for micropolar parabolic stamp and an elastic thick plate.

#### 2. Basic Equations of the Micropolar Elasticity

Following [1–3] we recall here the basic equations of the linear micropolar elasticity of isotropic solids. The kinematic of a micropolar solid is described by two fields, that is, the field of translations and the field of rotations . The latter is responsible for the description of moment interactions of the material particles. Hereinafter the Latin indices take on value 1, 2, or 3 and we use the Einstein summation rule over repeating indices. The equilibrium equations take the form where and are the stress and couple stress tensors, respectively, is the Levi-Civita third-order tensor, and and are external forces and couples. Unlike in classical (Cauchy-type) continua, the tensors and are not symmetric. Equations (1) constitute the local balance of momentum and moment of momentum, respectively.

The static and kinematic boundary conditions have the following form: where is the unit vector of external normal to the boundary , and are external forces and couples, and and are given on surface fields of translations and rotations, respectively. Obviously, other mixed boundary conditions can be introduced.

In what follows we are restricting ourselves by isotropic case. For a linear isotropic micropolar solid the constitutive equations are where is the Kronecker symbol, and , , , , , and are the elastic moduli. In (3) we also introduced the stretch tensor and the wryness tensor given by formulas

Using the Voigt notation modified for the micropolar elasticity and introducing the stress and moment stress vectors with stretch and wryness vectors by the formulas we represent the constitutive equations in the following unified form:with symmetric stiffness matrix , where and are three-diagonal symmetric matrices given by Unlike classical elasticity where the stiffness matrix in the Voigt notations is a symmetric matrix here we have larger matrices. The values of used micropolar elastic moduli can be found from direct experiments provided in [4, 5, 8, 14] or using homogenization technique [6, 7, 9–11, 51]. Let us note that the homogeneous micropolar model can be derived on the base of the passage from heterogeneous classical (Cauchy) continuum and on using passage from inhomogeneous micropolar (Cosserat) material; see [52–55]. Analysis of general constitutive equations of anisotropic micropolar solids on the base of the material symmetry groups is performed in [2, 56, 57].

From the experimental point of view it is better to use another set of material parameters [4, 5, 8, 14] listed in Table 1.