The Construction of Nonlinear Elasticity Tensors for Crystals and Quasicrystals

2017 ◽  
Vol 09 (06) ◽  
pp. 1750080 ◽  
Author(s):  
Yuri Astapov ◽  
Dmitrii Khristich ◽  
Alexey Markin ◽  
Marina Sokolova
Keyword(s):  
Elastic Properties ◽  
Nonlinear Elasticity ◽  
Constitutive Relations ◽  
Symmetry Plane ◽  
Order Effects ◽  
Isotropic Materials ◽  
Icosahedral Quasicrystals ◽  
Elasticity Tensors ◽  
Nonlinear Elastic ◽  
Elastic Modules

The problem of construction of constitutive relations in nonlinear elasticity theory for crystals and quasicrystals is associated with finding the construction of the structure of tensors of elastic modules of 2nd and 3rd orders. In the case of linear constitutive relations, the problem is entirely solved for crystals and some quasicrystals. In the case of nonlinear relations, the quantity of nonzero elastic modules of the 3rd order is known only for some classes of crystals. The problem of construction of nonlinear constitutive relations for quasicrystal materials has not been considered. In this article, representations of linear and nonlinear elastic properties of crystals and quasicrystals in the form of decompositions by special invariant tensor bases are considered. This allows us to concretize constitutive relations for icosahedral and axial quasicrystals. It is shown that axial quasicrystals’ behavior towards elastic properties coincides with transversal-isotropic materials. Icosahedral quasicrystals’ behavior matches with the isotropic materials’ one. On the basis of the obtained decompositions of nonlinear elasticity tensors for axial quasicrystals having a symmetry plane and for graphene films with and without defects, the analysis of their mechanical behavior at some types of loading is fulfilled. It is shown that differences in the behavior of these materials appear only in the second order effects.

scholarly journals FINITE STRAINS OF NONLINEAR ELASTIC ANISOTROPIC MATERIALS

2021 ◽  
pp. 103-116
Author(s):  
M.Yu. Sokolova ◽  
◽  
D.V. Khristich ◽  
Keyword(s):  
Coordinate System ◽  
Elastic Properties ◽  
Mutual Influence ◽  
Strain Tensor ◽  
Anisotropic Materials ◽  
Elasticity Tensor ◽  
Finite Strains ◽  
Cubic Materials ◽  
Elasticity Tensors ◽  
Nonlinear Elastic

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.

A non-classical theory of elastic dielectrics incorporating couple stress and quadrupole effects: part I – reconsideration of curvature-based flexoelectricity theory

2021 ◽  
pp. 108128652110015
Author(s):  
YL Qu ◽  
GY Zhang ◽  
YM Fan ◽  
F Jin
Keyword(s):  
Classical Theory ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Constitutive Relations ◽  
Stress Theory ◽  
Isotropic Materials ◽  
Flexoelectric Effect ◽  
Beam Bending ◽  
Gradient Theories ◽  
Anisotropic And Isotropic Materials

A new non-classical theory of elastic dielectrics is developed using the couple stress and electric field gradient theories that incorporates the couple stress, quadrupole and curvature-based flexoelectric effects. The couple stress theory and an extended Gauss’s law for elastic dielectrics with quadrupole polarization are applied to derive the constitutive relations of this new theory through energy conservation. The governing equations and the complete boundary conditions are simultaneously obtained through a variational formulation based on the Gibbs-type variational principle. The constitutive relations of general anisotropic and isotropic materials with the corresponding independent material constants are also provided, respectively. To illustrate the newly proposed theory and to show the flexoelectric effect in isotropic materials, one pure bending problem of a simply supported beam is analytically solved by directly applying the formulas derived. The analytical results reveal that the flexoelectric effect is present in isotropic materials. In addition, the incorporation of both the couple stress and flexoelectric effects always leads to increased values of the beam bending stiffness.

scholarly journals Membranes from the Material Assuming Nonlinear Elastic Properties if Exposed to Aggressive Environment

2021 ◽  
Vol 1079 (6) ◽  
pp. 062013
Author(s):  
V V Erastov ◽  
A V Erastov ◽  
I V Erofeeva ◽  
A A Treshev ◽  
A A Bobryshev ◽  
...  
Keyword(s):  
Elastic Properties ◽  
Aggressive Environment ◽  
Nonlinear Elastic

scholarly journals Effective elastic properties of transversely isotropic materials with concave pores

2021 ◽  
Vol 153 ◽  
pp. 103665
Author(s):  
K. Du ◽  
L. Cheng ◽  
J.F. Barthélémy ◽  
I. Sevostianov ◽  
A. Giraud ◽  
...  
Keyword(s):  
Elastic Properties ◽  
Transversely Isotropic ◽  
Effective Elastic Properties ◽  
Isotropic Materials ◽  
Transversely Isotropic Materials

scholarly journals Structure–Elastic Properties Relationships in Gelling Carrageenans

Polymers ◽  
2021 ◽  
Vol 13 (23) ◽  
pp. 4120
Author(s):  
Loïc Hilliou
Keyword(s):  
Elastic Properties ◽  
Red Algae ◽  
Food Industry ◽  
Structural Features ◽  
The Self ◽  
Structural Units ◽  
Self Assembling ◽  
Linear And Nonlinear ◽  
Nonlinear Elastic ◽  
Elastic Characteristics

Gelling carrageenans are polysaccharides extracted from the Gigartinales order of red algae. These are additives used essentially in the food industry for texturizing, stabilizing or gelling various formulations. Although a consensual gel mechanism has been reached which encompasses a coil-to-helix transition followed by the self-assembling of helices in a network, the structure–elastic relationships in the network are still to be clearly established. This paper reviews the reports in which carrageenan gel structures have been systematically compared with gel elastic properties. The focus is on the sizes documented for structural units, such as strands, aggregates, voids or network meshes, as well as on the reported linear and nonlinear elastic characteristics. The insufficient rationalization of carrageenan gel elasticity by models which take on board mechanically relevant structural features is underlined. After introducing selected linear and nonlinear elastic models, preliminary results comparing such models to structural and rheological data are presented. In particular, the concentration scaling of the strain hardening exhibited by two types of carrageenan gels is discussed.

Sign in / Sign up

Export Citation Format

Share Document