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.

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.

Constitutive Equations for Hyperelastic Materials Based on the Upper Triangular Decomposition of the Deformation Gradient

2018 ◽  
Vol 24 (6) ◽  
pp. 1785-1799 ◽  
Author(s):  
Y. Q. Li ◽  
X.-L. Gao
Keyword(s):  
Energy Density ◽  
Constitutive Equations ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Deformation Gradient ◽  
Density Functions ◽  
Triangular Decomposition ◽  
Deformation Gradient Tensor ◽  
Hyperelastic Materials ◽  
Distortion Tensor

The upper triangular decomposition has recently been proposed to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular tensor called the distortion tensor, whose six components can be directly related to pure stretch and simple shear deformations, which are physically measurable. In the current paper, constitutive equations for hyperelastic materials are derived using strain energy density functions in terms of the distortion tensor, which satisfy the principle of material frame indifference and the first and second laws of thermodynamics. Being expressed directly as derivatives of the strain energy density function with respect to the components of the distortion tensor, the Cauchy stress components have simpler expressions than those based on the invariants of the right Cauchy-Green deformation tensor. To illustrate the new constitutive equations, strain energy density functions in terms of the distortion tensor are provided for unconstrained and incompressible isotropic materials, incompressible transversely isotropic composite materials, and incompressible orthotropic composite materials with two families of fibers. For each type of material, example problems are solved using the newly proposed constitutive equations and strain energy density functions, both in terms of the distortion tensor. The solutions of these problems are found to be the same as those obtained by applying the polar decomposition-based invariants approach, thereby validating and supporting the newly developed, alternative method based on the upper triangular decomposition of the deformation gradient tensor.

A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part II: Applications

1996 ◽  
Vol 63 (4) ◽  
pp. 869-876 ◽  
Author(s):  
Jiun-Shyan Chen ◽  
Cheng-Tang Wu ◽  
Chunhui Pan
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Projection Method ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Finite Elasticity ◽  
Element Formulation ◽  
Density Functions ◽  
Hyperelastic Materials ◽  
Pressure Projection

In the first part of this paper a pressure projection method was presented for the nonlinear analysis of structures made of nearly incompressible hyperelastic materials. The main focus of the second part of the paper is to demonstrate the performance of the present method and to address some of the issues related to the analysis of engineering elastomers including the proper selection of strain energy density functions. The numerical procedures and the implementation to nonlinear finite element programs are presented. Mooney-Rivlin, Cubic, and Modified Cubic strain energy density functions are used in the numerical examples. Several classical finite elasticity problems as well as some practical engineering elastomer problems are analyzed. The need to account for the slight compressibility of rubber (finite bulk modulus) in the finite element formulation is demonstrated in the study of apparent Young’s modulus of bonded thin rubber units. The combined shear-bending deformation that commonly exists in rubber mounting systems is also analyzed and discussed.

Peridynamic Modeling of Hyperelastic Membrane Deformation

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
D. J. Bang ◽  
E. Madenci
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Equations Of Motion ◽  
Uniaxial Loading ◽  
Density Functions ◽  
Loading Conditions ◽  
Element Analysis ◽  
Material Parameters ◽  
Loading Case

This study concerns the development of peridynamic (PD) strain energy density functions for a Neo-Hookean type membrane under equibiaxial, planar, and uniaxial loading conditions. The material parameters for each loading case are determined by equating the PD strain energy density to that of the classical continuum mechanics. The PD equations of motion are derived based on the Neo-Hookean model under the assumption of incompressibility. Numerical results concern the deformation of a membrane with a defect in the form of a hole, a crack, and a rigid inclusion under equibiaxial, planar, and uniaxial loading conditions. The PD predictions are verified by comparison with those of finite element analysis.

Development of Hyperelastic Model for Weldlines Containing Natural Rubber Molded Part

2013 ◽  
Vol 747 ◽  
pp. 631-634
Author(s):  
Watcharapong Chookaew ◽  
Jirachai Mingbunjurdsuk ◽  
Pairote Jittham ◽  
Somjate Patcharaphun
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Mathematical Formulation ◽  
Constitutive Models ◽  
Hyperelastic Materials ◽  
Design Engineers ◽  
Strain Energy Density Function ◽  
Molded Part ◽  
Rubber Part

Several constitutive models of non-linear large elastic deformation based on strain-energy-density functions have been developed for hyperelastic materials. These models, coupled with the Finite Element Method (FEM), can effectively utilized by design engineers to analyze and design elastomeric products operating under the deformation states. However, due to the complexities of the mathematical formulation which can only obtained at the moderate strain and the assumption of material used for the analysis. Therefore it is formidable task for design engineer to make use of these constitutive relationships. In the present work, the strain-energy-density function of weldline containing rubber part was constructed by using the Neural Network (NN) model. The analytical results were compared to those obtained by Neo-Hookean, Mooney-Rivlin, Ogden models. Good agreement between developed NN model and the existing experimental data was found, especially at very low strain and at very high strain.

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