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.

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 [

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 [

The paper is organized as follows. In Section

Following [

The static and kinematic boundary conditions have the following form:

In what follows we are restricting ourselves by isotropic case. For a linear isotropic micropolar solid the constitutive equations are

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

From the experimental point of view it is better to use another set of material parameters [

Micropolar material parameters.

Description | Symbol | Formula |
---|---|---|

Shear modulus [MPa] | | |

Poisson’s ratio [—] | | |

Coupling number [—] | | |

Characteristic length (torsion) [m] | | |

Characteristic length (bending) [m] | | |

Polar ratio [—] | | |

The boundary value problems of the micropolar isotropic elasticity contain also the boundary value problems of classical elasticity as a special case. This coincidence may be used for verification of the developed code. The first way to reduce the problem is to assume _{1} transforms to the classic equilibrium equation of the linear elasticity which is independent of rotations whereas (_{2} includes only rotations. Let us also note that (_{2} has similar form to (_{1} but with different elastic moduli. So in this case the problem is decoupled and translations can be determined independently of microrotations.

The second reduction is possible if one assumes the microrotations to be fixed, _{1} again coincides with classic equilibrium equation with Lame’s moduli

Efficient solution of large boundary value problems requires application of an advanced software. In this research ABAQUS commercial program has been extended by the implementation of the user element (UEL) to solve the micropolar elasticity problems. Special code written in Fortran is linked with ABAQUS software allowing the user to practice all the ABAQUS features without paying an attention to their numerical implementation. From the practical point of view the most important features are creation of sophisticated geometry, application of loads and boundary conditions, applications of constrains and contact conditions, generation of 3D meshes, using the material library, and, the most important feature, being very effective solver.

During the solver execution UEL procedure is called twice for each Gaussian point in every element. In the first call the element stiffness should be provided by UEL procedure. Very often user element procedure requires calling UELMAT code (user material procedure) necessary to obtain the relation between stress and strain increments. Another call of UEL procedure is necessary to compute residual forces, element nodal forces resulting from element stresses, which is essential in the convergence monitoring during solving nonlinear problems. Designing of own finite elements is an ambiguous task recommended to advanced users only. It should be mentioned that UEL procedures should be very carefully tested and validated. There are some important disadvantages of using user elements in ABAQUS program. First of all, ABAQUS does not recognize the shape of the element; the element is represented by the set of nodes only. For example, 2D four nodes’ element can be defined as a truss structure, as a frame made up of beams, or as a quadrilateral flat element. In each of the mentioned cases different type of loads can be applied; for example, the pressure can be applied to quadrilateral element only, the bending moment can be applied to beam, and so forth. The visualization of obtained results is another matter. The visualization is not possible without detailed description of an element topology; the set of nodes does not provide this information. That is why only nodal displacements of the user elements can be displayed in ABAQUS postprocessor. Displaying strains and stresses requires developing own graphical programs or using uncomfortable techniques with “ghost” meshes made of ordinary ABAQUS elements constrained to meshes consisting of user elements.

The typical call of UEL procedure is presented below:

The most important parameters are

The whole procedure is typical for the isoparametric finite element formulation in 3D problems with the exception of considered degrees of freedom (in micropolar elasticity there are three displacement components and additionally three microrotations) and strain and stress measures (strain and stress tensors contain more components and are not symmetric). In UEL implementation of micropolar elasticity the same shape functions are used for displacement and microrotations

In the isoparametric element the shape functions (

Calculation of derivatives

The stiffness matrix of developed element is

Short description of micropolar elasticity 8-node isoparametric element implementation consists of several steps. The user element procedure based on (

Calculation of stiffness matrix of 8-node isoparametric element for micropolar elasticity. In the loop over the Gaussian points (the are 8 Gaussian points)

Find the shape functions at each Gaussian point and their derivatives with respect to natural coordinates

Find the Jacobian matrix (

Find the shape functions derivatives (

Find the matrix of shape functions derivatives

Find the constitutive matric

Compute

Additional computations

Find the element nodal forces resulting from element stresses and subtract them from the external nodal forces in order to compute residuals (necessary to check the rate of the solution convergence)

Make other computations; for example, update the strain energy

Using the presented above finite element we analyzed few 3D static problems for solids with certain singularities such as notch, hole, or small contact area for the contact problem of two solids [

Similar results obtained by the commercial software using the classical theory of elasticity and the results acquired by use of user element while reduced number of material data is used prove that the solution obtained by the UEL procedure is reliable. In this research the Hertzian contact between a parabolic stamp and a half space is considered. For classical Hertz theory of an elastic contact we refer, for example, to [

Fine element mesh in the contact problem.

The contact of a parabolic stamp made of polystyrene foam with the flat elastic plate is considered. The foam is modelled as a micropolar material. For the foam we use following material data [

In Figure

Von Mises stress [Pa]: commercial software, classical theory of elasticity.

In Figure

The distribution of couple stress

We discussed finite element approach adopted to the linear micropolar elasticity in order to model microstructured solids such as porous materials and beam lattices. The new 8-node hybrid micropolar isoparametric element and its implementation in ABAQUS are presented. Here we analyzed the contact problem between two elastic solids. Comparison of solutions based on classical and micropolar elasticity is carefully discussed. Numerical tests have shown that couple stress appears almost in the vicinity of contact zone.

The publication of the paper does not lead to any conflict of interests.

Authors acknowledge the support by the People Program (Marie Curie ITN transfer) of the European Union’s Seventh Framework Programme for research, technological development, and demonstration under Grant Agreement no. PITN-GA-2013-606878.

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