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.

An Alternative Technique to Evaluate Crack Propagation Path in Hyperelastic Materials

2012 ◽  
Vol 40 (1) ◽  
pp. 42-58 ◽  
Author(s):  
R. R. M. Ozelo ◽  
P. Sollero ◽  
A. L. A. Costa
Keyword(s):  
Constitutive Model ◽  
Crack Propagation ◽  
Experimental Tests ◽  
Growth Direction ◽  
Propagation Path ◽  
J Integral ◽  
Alternative Technique ◽  
Crack Propagation Path ◽  
Hyperelastic Materials ◽  
Material Constitutive Model

Abstract REFERENCE: R. R. M. Ozelo, P. Sollero, and A. L. A. Costa, “An Alternative Technique to Evaluate Crack Propagation Path in Hyperelastic Materials,” Tire Science and Technology, TSTCA, Vol. 40, No. 1, January–March 2012, pp. 42–58. ABSTRACT: The analysis of crack propagation in tires aims to provide safety and reliable life prediction. Tire materials are usually nonlinear and present a hyperelastic behavior. Therefore, the use of nonlinear fracture mechanics theory and a hyperelastic material constitutive model are necessary. The material constitutive model used in this work is the Mooney–Rivlin. There are many techniques available to evaluate the crack propagation path in linear elastic materials and estimate the growth direction. However, most of these techniques are not applicable to hyperelastic materials. This paper presents an alternative technique for modeling crack propagation in hyperelastic materials, based in the J-Integral, to evaluate the crack path. The J-Integral is an energy-based parameter and is applicable to nonlinear materials. The technique was applied using abaqus software and compared to experimental tests.

An octahedral stress-based fracture criterion for hyperelastic materials

2020 ◽  
pp. 095440622097213
Author(s):  
A Hamdi ◽  
A Boulenouar ◽  
N Benseddiq
Keyword(s):  
Pure Shear ◽  
Thermoplastic Elastomer ◽  
Styrene Butadiene Rubber ◽  
Butadiene Rubber ◽  
Principal Stresses ◽  
Hyperelastic Materials ◽  
Biaxial Tests ◽  
Set Up ◽  
Loading Modes ◽  
Styrene Butadiene

No unified stress-based criterion exists, in the literature, for predicting the rupture of hyperelastic materials subjected to mutiaxial loading paths. This paper aims to establish a generalized rupture criterion under plane stress loading for elastomers. First, the experimental set up, at breaking, including various loading modes, is briefly described and commented. It consists of uniaxial tests, biaxial tests and pure shear tests, performed on different rubbers. The used vulcanizate and thermoplastic rubber materials are a Natural Rubber (NR), a Styrene Butadiene Rubber (SBR), a Polyurethane (PU) and a Thermoplastic elastomer (TPE). Then, we have investigated a new theoretical approach, based upon the principal stresses, to establish a failure criterion under quasi-static loadings. Thus, we have proposed a new analytical model expressed as a function of octahedral stresses. Quite good agreement is highlighted when comparing the ultimate stresses, at break, between the experimental data and the prediction of the proposed criteria using our rubber-like materials.

Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues—With Examples

1987 ◽  
Vol 40 (12) ◽  
pp. 1699-1734 ◽  
Author(s):  
Millard F. Beatty
Keyword(s):  
Constitutive Equations ◽  
Mechanical Energy ◽  
Finite Elasticity ◽  
Physical Nature ◽  
Decomposition Theorem ◽  
Elastic Stability ◽  
Biological Tissues ◽  
Field Equations ◽  
Energy Principle ◽  
Hyperelastic Materials

This is an introductory survey of some selected topics in finite elasticity. Virtually no previous experience with the subject is assumed. The kinematics of finite deformation is characterized by the polar decomposition theorem. Euler’s laws of balance and the local field equations of continuum mechanics are described. The general constitutive equation of hyperelasticity theory is deduced from a mechanical energy principle; and the implications of frame invariance and of material symmetry are presented. This leads to constitutive equations for compressible and incompressible, isotropic hyperelastic materials. Constitutive equations studied in experiments by Rivlin and Saunders (1951) for incompressible rubber materials and by Blatz and Ko (1962) for certain compressible elastomers are derived; and an equation characteristic of a class of biological tissues studied in primary experiments by Fung (1967) is discussed. Sample applications are presented for these materials. A balloon inflation experiment is described, and the physical nature of the inflation phenomenon is examined analytically in detail. Results for the different materials are compared. Two major problems of finite elasticity theory are discussed. Some results concerning Ericksen’s problem on controllable deformations possible in every isotropic hyperelastic material are outlined; and examples are presented in illustration of Truesdell’s problem concerning analytical restrictions imposed on constitutive equations. Universal relations valid for all compressible and incompressible, isotropic materials are discussed. Some examples of non-uniqueness, including that of a neo-Hookean cube subject to uniform loads over its faces, are described. Elastic stability criteria and their connection with uniqueness in the theory of small deformations superimposed on large deformations are introduced, and a few applications are mentioned. Some previously unpublished results are presented throughout.

scholarly journals Meshless Radial Point Interpolation Method for Hyperelastic Materials

10.29007/r7sp ◽  
2020 ◽  
Author(s):  
Trong Khiem Bui ◽  
Vu Tuong Nguyen ◽  
Thanh Nha Nguyen ◽  
Tich Thien Truong
Keyword(s):  
Large Deformation ◽  
Interpolation Method ◽  
Biological Tissues ◽  
Material Behavior ◽  
Nonlinear Material ◽  
Radial Point Interpolation Method ◽  
Hyperelastic Materials ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation

Hyperelastic materials are special types of material that tends to behavior elastically when they are subjected to very large strains. These materials show not only the nonlinear material behavior but also the large deformation and stress-strain relationship is derived from a strain energy density function. Hyperelastic materials are widely used in many applications such as biological tissues, polymeric foams, and moreover. Neo - Hookean is a material model for hyperelastic solid which contains only two material parameters: bulk modulus and shear modulus. In the field of numerical analysis, the radial point interpolation method (RPIM) is a well-known meshfree method based on Garlekin's weak form. With the property of “free of mesh”, the RPIM approach shows its advantage for large deformation problems. In this study, a meshless radial point interpolation method is applied to demonstrate the elastic response of rubber-like materials based on the Mooney- Rivlin model. The obtained results are compared with the reference solutions given by other methods to verify the accuracy of the proposed method.

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