Victor A. Eremeyev

Mechanics of Viscoelastic Plates Made of FGMs

Considering the viscoelastic behavior of polymer foams a new plate theory based on the direct approach is introduced and applied to plates composed of functionally graded materials (FGM). The governing two-dimensional equations are formulated for a deformable surface, the viscoelastic effective stiffness parameters are identified assuming linear viscoelastic material behavior. The material properties are changing in the thickness direction. Solving some problems of the global structural analysis it will be demonstrated that in some cases the results significantly differ from the results based on the Kirchhoff-type theory. The aim of this paper is to extend the results of the analysis given in (ZAMM 88:332–341, 2008; Acta Mech 204:137–154, 2009; Key Eng Mater 399:63–70, 2009) related to the case of general linear viscoelastic behaviour and to discuss how the effective viscoelastic properties reflect the properties in the thickness direction.

Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction

The application of the linear micropolar theory to the strength analysis of bioceramic materials for bone reconstruction is described. Micropolar elasticity allows better results to be obtained for microstructural and singular domains as compared to the classical theory of elasticity. The fundamental equations of the Cosserat continuum are cited. The description of FEM implementation of micropolar elasticity is given. The results of solving selected 3D test problems are presented. Comparison of classical and micropolar solutions is discussed.

On Finite Element Computations of Contact Problems in Micropolar Elasticity

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.

On Equilibrium of a Second-Gradient Fluid Near Edges and Corner Points

Within the framework of the model of second-gradient fluid we discuss the natural boundary conditions along edges and at corner points. As for any strain gradient model the model of second-gradient fluid demonstrates some peculiarities related with necessity of additional boundary conditions. Here using the Lagrange variational principle we derived the latter boundary conditions for various contact angles.

A Note on Reduced Strain Gradient Elasticity

We discuss the particular class of strain-gradient elastic material models which we called the reduced or degenerated strain-gradient elasticity. For this class the strain energy density depends on functions which have different differential properties in different spatial directions. As an example of such media we consider the continual models of pantographic beam lattices and smectic and columnar liquid crystals.

Some Introductory and Historical Remarks on Mechanics of Microstructured Materials

Here we present few remarks on the development of the models of microstuctured media and the generalized continua.

On Computational Evaluation of Stress Concentration Using Micropolar Elasticity

We discuss the implementation the finite element approach to the linear micropolar elasticity in order to perform the analysis of the stress concentration near holes and notches. Within the micropolar elasticity we analyze the behaviour of such microstructured solids as foams and bones. With developed new finite element few problems are analyzed where the influence of the microstructure may be important. The provided comparison of solutions obtained within the micropolar and classical elasticity show the influence of micropolar properties on stress concentration near notches and contact areas.

On the Variational Analysis of Vibrations of Prestressed Six-Parameter Shells

We discuss the variational statements of the theory of linear vibrations of prestressed six-parameter shells. Initial or residual stresses can significantly influence buckling and oscillations of thin-walled structures. Within the six-parameter theory of shells a shell is modeled as a deformed material each point of it has six degrees f freedom, that is three translational and three rotational ones. Starting with the governing equations of the six-parameter shell theory the constitutive equations are analyzed. The linearization of the boundary-value problem is realized. After a brief discussion of the eigen-vibrations of the prestressed six-parameter shells the Rayleigh principle is introduced and discussed.

Bending of a Three-Layered Plate with Surface Stresses

We discuss here the bending deformations of a three-layered plate taking into account surface and interfacial stresses. The first-order shear deformation plate theory and the Gurtin-Murdoch model of surface stresses will be considered and the formulae for stiffness parameters of the plate are derived. Their dependence on surface elastic moduli will be analyzed.

Comparison of anti-plane surface waves in strain-gradient materials and materials with surface stresses

Here we discuss the similarities and differences in anti-plane surface wave propagation in an elastic half-space within the framework of the theories of Gurtin–Murdoch surface elasticity and Toupin–Mindlin strain-gradient elasticity. The qualitative behaviour of the dispersion curves and the decay of the obtained solutions are quite similar. On the other hand, we show that the solutions relating to the surface elasticity model are more localised near the free surface. For the strain-gradient elasticity model there is a range of wavenumbers where the amplitude of displacements decays very slowly.