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.

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.

Rubber Composition–Properties Relationships during Tire Numerical Simulation and Design Optimization

2017 ◽  
Vol 45 (1) ◽  
pp. 71-84 ◽  
Author(s):  
Alexey Mazin ◽  
Alexander Kapustin ◽  
Mikhail Soloviev ◽  
Alexander Karanets
Keyword(s):  
Numerical Simulation ◽  
Design Optimization ◽  
Strain Tensor ◽  
General Purpose ◽  
Orthogonal Functions ◽  
Element Analysis ◽  
Time Investment ◽  
Footprint Analysis ◽  
Composition Properties ◽  
Nonlinear Elastic

ABSTRACT Numerical simulation based on finite element analysis is now widely used during the design optimization of tires, thereby drastically reducing the time investment in the design process and improving tire performance because it is obtained from the optimized solution. Rubber material models that are used in numerical calculations of stress–strain distributions are nonlinear and may include several parameters. The relations of these parameters with rubber formulations are usually unknown, so the designer has no information on whether the optimal set of parameters is reachable by the rubber technological possibilities. The aim of this work was to develop such relations. The most common approach to derive the equation of the state of rubber is based on the expansion of the strain energy in a series of invariants of the strain tensor. Here, we show that this approach has several drawbacks, one of which is problems that arise when trying to build on its basis the quantitative relations between the rubber composition and its properties. An alternative is to use a series expansion in orthogonal functions, thereby ensuring the linear independence of the coefficients of elasticity in evaluation of the experimental data and the possibility of constructing continuous maps of “the composition to the property.” In the case of orthogonal Legendre polynomials, the technique for constructing such maps is considered, and a set of empirical functions is proposed to adequately describe the dependence of the parameters of nonlinear elastic properties of general-purpose rubbers on the content of the main ingredients. The calculated sets of parameters were used in numerical tire simulations including static loading, footprint analysis, braking/acceleration, and cornering and also in design optimization procedures.

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 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.

scholarly journals Seismic wave propagation in anisotropic ice – Part 1: Elasticity tensor and derived quantities from ice-core properties

2015 ◽  
Vol 9 (1) ◽  
pp. 367-384 ◽  
Author(s):  
A. Diez ◽  
O. Eisen
Keyword(s):  
Seismic Waves ◽  
Dynamic Behaviour ◽  
Ice Core ◽  
Ice Cores ◽  
Seismic Wave Propagation ◽  
Elasticity Tensor ◽  
Reflection Coefficients ◽  
Seismic Velocities ◽  
Crystal Anisotropy ◽  
Elasticity Tensors

Abstract. A preferred orientation of the anisotropic ice crystals influences the viscosity of the ice bulk and the dynamic behaviour of glaciers and ice sheets. Knowledge about the distribution of crystal anisotropy is mainly provided by crystal orientation fabric (COF) data from ice cores. However, the developed anisotropic fabric influences not only the flow behaviour of ice but also the propagation of seismic waves. Two effects are important: (i) sudden changes in COF lead to englacial reflections, and (ii) the anisotropic fabric induces an angle dependency on the seismic velocities and, thus, recorded travel times. A framework is presented here to connect COF data from ice cores with the elasticity tensor to determine seismic velocities and reflection coefficients for cone and girdle fabrics. We connect the microscopic anisotropy of the crystals with the macroscopic anisotropy of the ice mass, observable with seismic methods. Elasticity tensors for different fabrics are calculated and used to investigate the influence of the anisotropic ice fabric on seismic velocities and reflection coefficients, englacially as well as for the ice–bed contact. Hence, it is possible to remotely determine the bulk ice anisotropy.

scholarly journals Computation of electrostatic fields in anisotropic human tissues using the Finite Integration Technique (FIT)

2005 ◽  
Vol 2 ◽  
pp. 309-313 ◽  
Author(s):  
V. C. Motresc ◽  
U. van Rienen
Keyword(s):  
Coordinate System ◽  
Electromagnetic Fields ◽  
Human Body ◽  
Local Coordinate System ◽  
Anisotropic Materials ◽  
The Body ◽  
Integration Technique ◽  
Data Set ◽  
Computer Data ◽  
Finite Integration Technique

Abstract. The exposure of human body to electromagnetic fields has in the recent years become a matter of great interest for scientists working in the area of biology and biomedicine. Due to the difficulty of performing measurements, accurate models of the human body, in the form of a computer data set, are used for computations of the fields inside the body by employing numerical methods such as the method used for our calculations, namely the Finite Integration Technique (FIT). A fact that has to be taken into account when computing electromagnetic fields in the human body is that some tissue classes, i.e. cardiac and skeletal muscles, have higher electrical conductivity and permittivity along fibers rather than across them. This property leads to diagonal conductivity and permittivity tensors only when expressing them in a local coordinate system while in a global coordinate system they become full tensors. The Finite Integration Technique (FIT) in its classical form can handle diagonally anisotropic materials quite effectively but it needed an extension for handling fully anisotropic materials. New electric voltages were placed on the grid and a new averaging method of conductivity and permittivity on the grid was found. In this paper, we present results from electrostatic computations performed with the extended version of FIT for fully anisotropic 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