Abstract:
Relationship of stress and strain for a carbon nanotube was obtained on the basis of micropolar elasticity. The principle of Hamilton was utilized to derive the dynamic differential equation. Dispersion relationship (the relationship between frequency and wavenumber) of wave propagating through a carbon nanotube was obtained from the dynamic differential equation. In addition, group velocities and characteristic wave surfaces were also investigated. The obtained results are discussed.

Differential equations to describe elasticity are derived without the use of stress or strain. The points within the body are the independent parameters instead of strain and surface forces replace stress tensors. These differential equations are a continuous analytical model that can then be solved using any of the standard techniques of differential equations. Although the equations do not require the definition stress or strain, these quantities can be calculated as dependent parameters. This approach to elasticity is simple, which avoids the need for multiple definitions of stress and strain, and provides a simple experimental procedure to find scalar representations of material properties in terms of the energy of deformation. The derived differential equations describe both infinitesimal and finite deformations.

Abstract:
The aim of this paper is to study the elastic stress and strain fields of dislocations and disclinations in the framework of Mindlin's gradient elasticity. We consider simple but rigorous versions of Mindlin's first gradient elasticity with one material length (gradient coefficient). Using the stress function method, we find modified stress functions for all six types of Volterra defects (dislocations and disclinations) situated in an isotropic and infinitely extended medium. By means of these stress functions, we obtain exact analytical solutions for the stress and strain fields of dislocations and disclinations. An advantage of these solutions for the elastic strain and stress is that they have no singularities at the defect line. They are finite and have maxima or minima in the defect core region. The stresses and strains are either zero or have a finite maximum value at the defect line. The maximum value of stresses may serve as a measure of the critical stress level when fracture and failure may occur. Thus, both the stress and elastic strain singularities are removed in such a simple gradient theory. In addition, we give the relation to the nonlocal stresses in Eringen's nonlocal elasticity for the nonsingular stresses.

Abstract:
Usual introductions of the concept of motion are not well adapted to a subsequent, strictly tensorial, theory of elasticity. The consideration of arbitrary coordinate systems for the representation of both, the points in the laboratory, and the material points (comoving coordinates), allows to develop a simple, old fashioned theory, where only measurable quantities -like the Cauchy stress- need be introduced. The theory accounts for the possibility of asymmetric stress (Cosserat elastic media), but, contrary to usual developments of the theory, the basic variable is not a micro-rotation, but the more fundamental micro-rotation velocity. The deformation tensor here introduced is the proper tensorial equivalent of the poorly defined deformation "tensors" of the usual theory. It is related to the deformation velocity tensor via the matricant. The strain is the logarithm of the deformation tensor. As the theory accounts for general Cosserat media, the strain is not necessarily symmetric. Hooke's law can be properly introduced in the material coordinates (as the stiffness is a function of the material point). A particularity of the theory is that the components of the stiffness tensor in the material (comoving) coordinates are not time-dependent. The configuration space is identified to the part of the Lie group GL(3)+, that is geodesically connected to the origin of the group.

Abstract:
Adequate mathematical modeling of selvage zone and natural fracturing as well as assessment of its impact on stress-strain state – urgent problems in calculation of hydraulic tunnels. Modern Russian regulations in fact give dependences only to solve the problems in plane deformation conditions. The specificity of work of the tunnel that crosses the discontinuity, as a space frame are not taken into account. This article presents influence of discontinuities and fracture characteristics on the rock mass stress-strain state around excavation. Fractured rock mass model was analyzed. Formula of modulus of elasticity for fractured rock mass at distance from the fault was deduced. Influence of discontinuities on the stress distribution was estimated with using experiment design method. On the basis of the conducted research it was established, that assessing rock stress-strain state around the fracture is necessary to consider rock mass fracture characteristics; and using principal stresses distribution in combination with Lode parameter we can clearly estimate the type of stress-strain state in each point, therefore, we can use different strength theories for different sections of the tunnel.

Abstract:
We say that there is a representation of the universal algebra B in the universal algebra A if the set of endomorphisms of the universal algebra A has the structure of universal algebra B. Therefore, the role of representation of the universal algebra is similar to the role of symmetry in geometry and physics. Morphism of the representation is the mapping that conserves the structure of the representation. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. The set of automorphisms of the representation of the universal algebra forms the group. Twin representations of this group in basis manifold of the representation are called active and passive representations. Passive representation in basis manifold is underlying of concept of geometric object and the theory of invariants of the representation of the universal algebra.

Abstract:
The purpose of this paper is to review some elasticity effects in epitaxial growth. We start by a description of the main ingredients needed to describe elasticity effects (elastic interactions, surface stress, bulk and surface elasticity, thermodynamics of stressed solids). Then we describe how bulk and surface elasticity affect growth mode and surface morphology by means of stress-driven instability. At last stress-strain evolution during crystal growth is reported.

Abstract:
In this paper, strain gradient elasticity formulation for analysis of FG (Functionally Graded) micro-cylinders is presented. The material properties are assumed to obey a power law in radial direction. The governing differential equation is derived as a fourth order ODE. A power series solution for stresses and displacements in FG micro-cylinders subjected to internal and external pressures is obtained. Numerical examples are presented to study the effect of the characteristic length parameter and FG power index on the displacement field and stress distribution in FG cylinders. It is shown that the characteristic length parameter has a considerable effect on the stress distribution of FG micro-cylinders. Also, increasing material length parameter leads to decrease of the maximum radial and tangential stresses in the cylinder. Furthermore, it is shown that the FG power index has a significant effect on the maximum radial and tangential stresses.

Abstract:
In this paper we consider and compare special classes of static theories of gradient elasticity, nonlocal elasticity, gradient micropolar elasticity and nonlocal micropolar elasticity with only one gradient coefficient. Equilibrium equations are discussed. The relationship between the gradient theory and the nonlocal theory is discussed for elasticity as well as for micropolar elasticity. Nonsingular solutions for the elastic fields of screw and edge dislocations are given. Both the elastic deformation (distortion, strain, bend-twist) and the force and couple stress tensors do not possess any singularity unlike `classical' theories.

Abstract:
Thermodynamic framework of finite strain viscoelasticity with second order weak nonlocality in the deformation gradient is investigated. The application of Liu procedure leads to a class of third grade elastic materials where the second gradient of the stress appears in the elastic constitutive relation. Finally the dispersion relation of longitudinal plane waves is calculated in isotropic materials.