scholarly journals Minimal Functional Bases for Elasticity Tensor Symmetry Classes

2022 ◽  
Author(s):  
R. Desmorat ◽  
N. Auffray ◽  
B. Desmorat ◽  
M. Olive ◽  
B. Kolev
Keyword(s):  
Elasticity Tensor ◽  
Symmetry Classes

scholarly journals Classification of first strain-gradient elasticity tensors by symmetry planes

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210165
Author(s):  
Hung Le Quang ◽  
Qi-Chang He ◽  
Nicolas Auffray
Keyword(s):  
Strain Gradient ◽  
Gradient Elasticity ◽  
Elasticity Tensor ◽  
Constitutive Law ◽  
Sixth Order ◽  
Strain Gradient Elasticity ◽  
Present Contribution ◽  
Symmetry Classes ◽  
Elasticity Tensors

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.

scholarly journals Characterization of the symmetry class of an elasticity tensor using polynomial covariants

2021 ◽  
pp. 108128652110108
Author(s):  
Marc Olive ◽  
Boris Kolev ◽  
Rodrigue Desmorat ◽  
Boris Desmorat
Keyword(s):  
Fourth Order ◽  
Sufficient Conditions ◽  
Cross Product ◽  
Elasticity Tensor ◽  
Theory Of Elasticity ◽  
Symmetry Class ◽  
Symmetric Tensors ◽  
Minimal Set ◽  
Algebraic Sets ◽  
Symmetry Classes

We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in physics. The novelty is that these conditions are written using polynomial covariants. As a corollary, we deduce that the symmetry classes are affine algebraic sets, a result which seems to be new. Meanwhile, we have been lead to produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor and introduce an original generalized cross-product on totally symmetric tensors. Finally, using these tensorial covariants, we produce a new minimal set of 294 generators for the invariant algebra of the elasticity tensor.

Irreducible matrix resolution for symmetry classes of elasticity tensors

2020 ◽  
Vol 25 (10) ◽  
pp. 1873-1895
Author(s):  
Yakov Itin
Keyword(s):  
Matrix Representation ◽  
Symmetric Matrix ◽  
Elasticity Tensor ◽  
Matrix Representations ◽  
Irreducible Decomposition ◽  
Symmetry Classes ◽  
Artificial Materials ◽  
Deformation Properties ◽  
Algebraic Properties ◽  
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.

scholarly journals A study of curvature theory for different symmetry classes of Hamiltonian

Pramana ◽  
2021 ◽  
Vol 95 (3) ◽  
Author(s):  
Y R Kartik ◽  
Ranjith R Kumar ◽  
S Rahul ◽  
Sujit Sarkar
Keyword(s):  
Symmetry Classes ◽  
Curvature Theory

scholarly journals Random Material Property Fields of 3D Concrete Microstructures Based on CT Image Reconstruction

Materials ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 1423
Author(s):  
George Stefanou ◽  
Dimitrios Savvas ◽  
Panagiotis Metsis
Keyword(s):  
Random Fields ◽  
Volume Fraction ◽  
Concrete Specimen ◽  
Elasticity Tensor ◽  
Statistical Characteristics ◽  
Moving Window ◽  
Window Method ◽  
Ct Image Reconstruction ◽  
Random Material Property ◽  
Spatially Varying

The purpose of this paper is to determine the random spatially varying elastic properties of concrete at various scales taking into account its highly heterogeneous microstructure. The reconstruction of concrete microstructure is based on computed tomography (CT) images of a cubic concrete specimen. The variability of the local volume fraction of the constituents (pores, cement paste and aggregates) is quantified and mesoscale random fields of the elasticity tensor are computed from a number of statistical volume elements obtained by applying the moving window method on the specimen along with computational homogenization. Based on the statistical characteristics of the mesoscale random fields, it is possible to assess the effect of randomness in microstructure on the mechanical behavior of concrete.

scholarly journals Structure–function relationships of the plant cuticle and cuticular waxes — a smart material?

2006 ◽  
Vol 33 (10) ◽  
pp. 893 ◽  
Author(s):  
Hendrik Bargel ◽  
Kerstin Koch ◽  
Zdenek Cerman ◽  
Christoph Neinhuis
Keyword(s):  
Structure Function ◽  
Smart Materials ◽  
Self Assembly ◽  
Water Repellency ◽  
Cuticular Waxes ◽  
Form Complex ◽  
Symmetry Classes ◽  
Research Activities ◽  
Terrestrial Habitats ◽  
Surface Waxes

The cuticle is the main interface between plants and their environment. It covers the epidermis of all aerial primary parts of plant organs as a continuous extracellular matrix. This hydrophobic natural composite consists mainly of the biopolymer, cutin, and cuticular lipids collectively called waxes, with a high degree of variability in composition and structure. The cuticle and cuticular waxes exhibit a multitude of functions that enable plant life in many different terrestrial habitats and play important roles in interfacial interactions. This review highlights structure–function relationships that are the subjects of current research activities. The surface waxes often form complex crystalline microstructures that originate from self-assembly processes. The concepts and results of the analysis of model structures and the influence of template effects are critically discussed. Recent investigations of surface waxes by electron and X-ray diffraction revealed that these could be assigned to three crystal symmetry classes, while the background layer is not amorphous, but has an orthorhombic order. In addition, advantages of the characterisation of formation of model wax types on a molecular scale are presented. Epicuticular wax crystals may cause extreme water repellency and, in addition, a striking self-cleaning property. The principles of wetting and up-to-date concepts of the transfer of plant surface properties to biomimetic technical applications are reviewed. Finally, biomechanical studies have demonstrated that the cuticle is a mechanically important structure, whose properties are dynamically modified by the plant in response to internal and external stimuli. Thus, the cuticle combines many aspects attributed to smart materials.

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

2021 ◽  
Author(s):  
V. Calisti ◽  
A. Lebée ◽  
A. A. Novotny ◽  
J. Sokolowski
Keyword(s):  
Circular Inclusion ◽  
Elasticity Tensor ◽  
Second Order ◽  
Topological Derivative ◽  
Microstructural Changes ◽  
Topological Derivatives ◽  
Spatial Dimensions ◽  
Elasticity Tensors ◽  
Derivatives Of ◽  
Optimization Of Microstructures

AbstractThe 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.

A Study on a Grade-One Type of Hypo-Elastic Models

2014 ◽  
Author(s):  
S. Alireza Momeni ◽  
Mohsen Asghari
Keyword(s):  
Constitutive Models ◽  
Specific Model ◽  
Elasticity Tensor ◽  
Elastic Model ◽  
Cauchy Stress ◽  
Spin Tensor ◽  
Positive Real ◽  
Corotational Rate ◽  
Deformation Rate Tensor ◽  
Grade One

In Hypo-elastic constitutive models an objective rate of the Cauchy stress tensor is expressed in terms of the current state of the stress and the deformation rate tensor D in a way that the dependency on the latter is a homogeneously linear one. In this work, a type of grade-one hypo-elastic models (i.e. models with linear dependency of the hypo-elasticity tensor on the stress) is considered for isotropic materials based on the objective corotational rates of stress. A positive real parameter denoted by n is involved in the considered type. Different values can be selected for this parameter, each selection leads to a specific model within the class of grade-one hypo-elasticity. The spin of the associated corotational rate is also dependent on the parameter n. In the special case of n=0, the corresponding hypo-elastic model reduces to a grade-zero one with the logarithmic rate of stress; noting that this rate is a corotational rate associated with the logarithmic spin tensor. Moreover, by choosing n=2, the model reduces to a grade-one hypo-elastic model with the Jaumann rate, i.e. the corotational rate associated with the vorticity spin tensor. As case studies, the simple shear problem is investigated with utilizing the considered type of hypo-elastic models with various values for parameter n, and the curves for the stress-shear response are depicted.

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