New S-Type Bounds of M-Eigenvalues for Elasticity Tensors with Applications

In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, S-type upper bounds for the M-eigenvalues of elasticity tensors are established. Finally, S-type sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the S-type M-eigenvalue inclusion sets.

Classification of first strain-gradient elasticity tensors by symmetry planes

First strain-gradient elasticity is a generalized continuum theory capable of modelling size effects in materials. This extended capability comes from the inclusion in the mechanical energy density of terms related to the strain-gradient. In its linear formulation, the constitutive law is defined by three elasticity tensors whose orders range from four to six. In the present contribution, the symmetry properties of the sixth-order elasticity tensors involved in this model are investigated. If their classification with respect to the orthogonal symmetry group is known, their classification with respect to symmetry planes is still missing. This last classification is important since it is deeply connected with some identification procedures. The classification of sixth-order elasticity tensors in terms of invariance properties with respect to symmetry planes is given in the present contribution. Precisely, it is demonstrated that there exist 11 reflection symmetry classes. This classification is distinct from the one obtained with respect to the orthogonal group, according to which there exist 17 different symmetry classes. These results for the sixth-order elasticity tensor are very different from those obtained for the classical fourth-order elasticity tensor, since in the latter case the two classifications coincide. A few numerical examples are provided to illustrate how some different orthogonal classes merge into one reflection class.

Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

FINITE STRAINS OF NONLINEAR ELASTIC ANISOTROPIC MATERIALS

Anisotropic materials with the symmetry of elastic properties inherent in crystals of cubic syngony are considered. Cubic materials are close to isotropic ones by their mechanical properties. For a cubic material, the elasticity tensor written in an arbitrary (laboratory) coordinate system, in the general case, has 21 non-zero components that are not independent. An experimental method is proposed for determining such a coordinate system, called canonical, in which a tensor of elastic properties includes only three nonzero independent constants. The nonlinear model of the mechanical behavior of cubic materials is developed, taking into account geometric and physical nonlinearities. The specific potential strain energy for a hyperelastic cubic material is written as a function of the tensor invariants, which are projections of the Cauchy-Green strain tensor into eigensubspaces of the cubic material. Expansions of elasticity tensors of the fourth and sixth ranks in tensor bases in eigensubspaces are determined for the cubic material. Relations between stresses and finite strains containing the second degree of deformations are obtained. The expressions for the stress tensor reflect the mutual influence of the processes occurring in various eigensubspaces of the material under consideration.

Irreducible matrix resolution for symmetry classes of elasticity tensors

In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

PSEUDOTENSOR FORMULATION OF THE MECHANICS OF HEMITROPIC MICROPOLAR MEDIA

The possibility of applications of relative tensors concepts to the mechanics of micropolar continuum and, in particular, for the hemitropic micropolar continua is considered. The fundamental tensors and orienting relative scalars in three-dimensional space are introduced. Permutation symbols and absolute Levi-Civita tensors are investigated in further details. Algebraic and differential properties of the relative tensors are discussed. The weights of the fundamental kinematic tensors are determined. The wryness tensor and the asymmetric strain tensor are constructed in terms of the vectors of micro-rotation and displacements. Notions of force and couple traction vectors, associated force and associated couple stress vector, force and couple stresses tensors are discussed in the frameworks of relative tensors algebra. The weights of the basic micropolar elasticity tensors are determined and discussed. The constitutive form of the micropolar elastic potential is introduced as an absolute scalar in order to obtain micropolar constitutive equations. In the linear case, the elastic potential is a quadratic form whose coefficients are pseudoscalars. The weights of the constitutive pseudoscalars are calculated. The dimensionless constitutive micropolar constants and constitutive constants with physical dimensions are discriminated. Statics and dynamics of micropolar elastic continua are developed in terms of relative tensors. Dynamic equations involving displacements and microrotations in the case of semi-isotropic (hemitropic) symmetry are derived and represented by the pseudotensor technique. The paper can be considered as a script of fundamental formulas and concepts related to the algebra and differentiation of relative tensors of arbitrary rank.