On the correspondence between two- and three-dimensional Eshelby tensors

We consider both three-dimensional (3D) and two-dimensional (2D) Eshelby tensors known also as energy–momentum tensors or chemical potential tensors, which are introduced within the nonlinear elasticity and the resultant nonlinear shell theory, respectively. We demonstrate that 2D Eshelby tensor is introduced earlier directly using 2D constitutive equations of nonlinear shells and can be derived also using the through-the-thickness procedure applied to a 3D shell-like body

On dynamic extension of a local material symmetry group for micropolar media

For micropolar media we present a new definition of the local material symmetry group considering invariant properties of the both kinetic energy and strain energy density under changes of a reference placement. Unlike simple (Cauchy) materials, micropolar media can be characterized through two kinematically independent fields, that are translation vector and orthogonal microrotation tensor. In other words, in micropolar continua we have six degrees of freedom (DOF) that are three DOFs for translations and three DOFs for rotations. So the corresponding kinetic energy density nontrivially depends on linear and angular velocity. Here we define the local material symmetry group as a set of ordered triples of tensors which keep both kinetic energy density and strain energy density unchanged during the related change of a reference placement. The triples were obtained using transformation rules of strain measures and microinertia tensors under replacement of a reference placement. From the physical point of view, the local material symmetry group consists of such density-preserving transformations of a reference placement, that cannot be experimentally detected. So the constitutive relations become invariant under such transformations. Knowing a priori a material’s symmetry, one can establish a simplified form of constitutive relations. In particular, the number of independent arguments in constitutive relations could be significantly reduced

Flexoelectricity and apparent piezoelectricity of a pantographic micro-bar

We discuss a homogenized model of a pantographic bar considering flexoelectricity. A pantographic bar consists of relatively stiff small bars connected by small soft flexoelectric pivots. As a result, an elongation of the bar relates almost to the torsion of pivots. Taking into account their flexoelectric properties we find the corresponding electric polarization. As a results, the homogenized pantographic bar demonstrates piezoelectric properties inherited from the flexoelectric properties of pivots. The effective stiffness properties of the homogenized bars are determined by the geometry of the structural elements and shear stiffness whereas the piezoelectric properties follow from the flexoelectric moduli of the pivots

On the Bending of Multilayered Plates Considering Surface Viscoelasticity

We discuss the bending resistance of multilayered plates taking into account surface/interfacial viscoelasticity. Within the linear surface viscoelasticity we introduce the surface/interfacial stresses linearly dependent on the history of surface strains. In order to underline the surface viscoelasticity contribution to the bending response we restrict ourselves to the elastic behaviour in the bulk. Using the correspondence principle of the theory of viscoelasticity we present an e_ective bending relaxation function

On phase equilibrium of an elastic liquid shell with wedge disclination

Based on the six-parameter shell theory we consider the phase equilibrium of a two-phase liquid membrane containing a wedge disclination. The considered problems are related to modelling of phase transitions in biological or lipid membranes. In order to capture the membrane behavior we consider a special case of elastic shells which energy is invariant under major transformations of a reference configuration and can be treated as an two-dimensional elastic micropolar fluid which can resist bending deformations. Based on variational approach we formulate the equilibrium conditions at a phase interface. As an example, a two-phase state is analysed in the vicinity of a wedge disclination

On solvability of initial boundary-value problems of micropolar elastic shells with rigid inclusions

The problem of dynamics of a linear micropolar shell with a finite set of rigid inclusions is considered. The equations of motion consist of the system of partial differential equations (PDEs) describing small deformations of an elastic shell and ordinary differential equations (ODEs) describing the motions of inclusions. Few types of the contact of the shell with inclusions are considered. The weak setup of the problem is formulated and studied. It is proved a theorem of existence and uniqueness of a weak solution for the problem under consideration