scholarly journals Stretching of the mechanical-geometric model and two common nonlinear elastic solids

2021 ◽  
Vol 273 ◽  
pp. 07019
Author(s):  
Daniil Azarov
Keyword(s):  
Geometric Modeling ◽  
Strain Energy ◽  
Geometric Model ◽  
Constitutive Relations ◽  
Theory Of Elasticity ◽  
Elastic Solids ◽  
Hyperelastic Materials ◽  
Deformation Properties ◽  
Nonlinear Elastic ◽  
Nonlinear Theory Of Elasticity

The variety of hyperelastic materials and the design of new modifications and technical applications requires the development of a description of nonlinear deformation properties. The most commonly used constitutive relations of the Mooney-Rivlin and Yeoh models are based on polynomial decompositions. Mechanical-geometric modeling (hereinafter - MGM) is a new way of constructing constitutive relations and strain energy densities within the nonlinear theory of elasticity. In this paper, a comparison of the deformation behavior of MGM with the traditional Mooney-Rivlin and Yeoh models was carried out. Comparative analysis is accompanied by diagrams for uniaxial and biaxial stretching. The effectiveness of the new model was proved.

scholarly journals Mechanical-Geometrical Modeling Of The Hyperelastic Materials At Uniaxial Stretching

2021 ◽  
Vol 2131 (5) ◽  
pp. 052017
Author(s):  
Daniil Azarov
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Geometric Modeling ◽  
Strain Energy ◽  
Theory Of Elasticity ◽  
Density Functions ◽  
Hyperelastic Materials ◽  
Weakly Compressible ◽  
Nonlinear Elastic ◽  
Nonlinear Theory Of Elasticity

Abstract Hyperelastic materials, such as rubber, occupy an important place in the design and operation of various technological equipment and machines. The article analyzed the deformation behavior of hyperelastic materials using a mechanical-geometric model. The method of mechanical-geometric modeling is a new method for obtaining constitutive relations and strain energy density functions for nonlinear elastic solids. It is based on physically and geometrically consistent prerequisites. The resulting models can describe broad classes of nonlinear elastic materials (both isotropic and anisotropic) depending on the mechanical and geometric properties “embedded” in them at the first stages of design. This paper discusses two basic types of models based on different initial geometry. The mechanical parameters of the models are constants, and the models themselves are considered in a statement corresponding to isotropic hyperelastic materials. The article presents the most common diagrams of deformation of artificial and natural rubbers, as well as steel. Hyperelastic materials, depending on the task, can be described in the nonlinear theory of elasticity as ideal incompressible, or as weakly compressible. Parameters of expressions of strain energy density functions of mechanical-geometric models obtained for cases of incompressible and weakly compressible continuous solids were identified. Stretch diagrams and diagrams of the transverse deformation function of the obtained mechanical-geometric models for the two cases mentioned above are plotted. The extension diagram for the model with parameters corresponding to the classic structural material of the steel type is also shown. Comments are given on the possibility of further paths of developing the method of mechanical-geometric modeling to obtain results not only in the field of nonlinear theory of elasticity, but also viscoelasticity.

Nonlinear Elastic Constitutive Relations of Auxetic Honeycombs

2009 ◽  
Author(s):  
Jaehyung Ju ◽  
Joshua D. Summers ◽  
John Ziegert ◽  
Georges Fadel
Keyword(s):  
Cell Walls ◽  
Strain Energy ◽  
Constitutive Relations ◽  
Effective Strain ◽  
Strain Energy Function ◽  
Periodic Boundary Conditions ◽  
Plane Shear ◽  
Local Cell ◽  
Nonlinear Elastic ◽  
Nonlinear Constitutive Relation

When designing a flexible structure consisting of cellular materials, it is important to find the maximum effective strain of the cellular material resulting from the deformed cellular geometry and not leading to local cell wall failure. In this paper, a finite in-plane shear deformation of auxtic honeycombs having effective negative Poisson’s ratio is investigated over the base material’s elastic range. An analytical model of the inplane plastic failure of the cell walls is refined with finite element (FE) micromechanical analysis using periodic boundary conditions. A nonlinear constitutive relation of honeycombs is obtained from the FE micromechanics simulation and is used to define the coefficients of a hyperelastic strain energy function. Auxetic honeycombs show high shear flexibility without a severe geometric nonlinearity when compared to their regular counterparts.

Universal deformations in inhomogeneous isotropic nonlinear elastic solids

2021 ◽  
Vol 477 (2253) ◽  
Author(s):  
Arash Yavari
Keyword(s):  
Strain Energy ◽  
Elastic Solid ◽  
Necessary Condition ◽  
Density Functions ◽  
Elastic Solids ◽  
Universal Deformations ◽  
Green Strain ◽  
The Right ◽  
Nonlinear Elastic ◽  
Isotropic Solids

Universal (controllable) deformations of an elastic solid are those deformations that can be maintained for all possible strain-energy density functions and suitable boundary tractions. Universal deformations have played a central role in nonlinear elasticity and anelasticity. However, their classification has been mostly established for homogeneous isotropic solids following the seminal works of Ericksen. In this article, we extend Ericksen’s analysis of universal deformations to inhomogeneous compressible and incompressible isotropic solids. We show that a necessary condition for the known universal deformations of homogeneous isotropic solids to be universal for inhomogeneous solids is that inhomogeneities respect the symmetries of the deformations. Symmetries of a deformation are encoded in the symmetries of its pulled-back metric (the right Cauchy–Green strain). We show that this necessary condition is sufficient as well for all the known families of universal deformations except for Family 5.

Equilibrium of a Thick-Walled Sphere of Inhomogeneous Nonlinear-Elastic Material

2013 ◽  
Vol 423-426 ◽  
pp. 1670-1674 ◽  
Author(s):  
Vladimir I. Andreev
Keyword(s):  
Stress Distribution ◽  
Nonlinear Theory ◽  
Spherical Cavity ◽  
Soil Mass ◽  
Theory Of Elasticity ◽  
Nonlinear Dependence ◽  
Nonlinear Elastic Material ◽  
Centrally Symmetric ◽  
Nonlinear Elastic ◽  
Nonlinear Theory Of Elasticity

Paper considers the problem of nonlinear theory of elasticity for inhomogeneous thick-walled sphere. The problem is solved in centrally symmetric statement. In general, all the parameters of the nonlinear dependence between the intensities of the stresses and strains are functions of the radius. As an example, solve the problem of stress distribution in the soil mass with a spherical cavity.

scholarly journals APPLICATION OF THE NONLINEAR THEORY OF ELASTICITY AND SIMILARITY METHOD FOR ASSESSMENT OF THE METALLIC JERSEY DEFORMATION PROPERTIES

2017 ◽  
pp. 105-113
Author(s):  
Vladimir Ivanovich KHALIMANOVICH ◽  
◽  
Lev Aleksadrovich KUDRYAVIN ◽  
Oleg Fedorovich BELYAEV ◽  
Vladimir Andreevich ZAVARUEV ◽  
...  
Keyword(s):  
Nonlinear Theory ◽  
Theory Of Elasticity ◽  
Deformation Properties ◽  
Similarity Method ◽  
Nonlinear Theory Of Elasticity

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

scholarly journals Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Energy Density ◽  
Constitutive Model ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
General Method

AbstractIn this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

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