Effect of Constitutive Parameters on Formation of a Spherical Void in a Compressible Nonlinear Elastic Material

1994 ◽  
Vol 61 (2) ◽  
pp. 395-401 ◽  
Author(s):  
Shiro Biwa ◽  
Eiji Matsumoto ◽  
Toshinobu Shibata
Keyword(s):  
Strain Energy ◽  
Finite Elasticity ◽  
Void Formation ◽  
Strain Energy Function ◽  
Elastic Materials ◽  
Spherical Void ◽  
Nonlinear Elastic Material ◽  
Nonlinear Elastic ◽  
Constitutive Parameters ◽  
Bifurcation Behavior

Void formation in materials under isotropic tension is considered within the theory of finite elasticity. This phenomenon is described as the bifurcation of the solution containing a spherical cavity from the state of homogeneous deformation of a solid hyperelastic sphere. The equation giving the bifurcation point is derived, and the critical stretch and stress are numerically calculated for a special class of compressible nonlinear elastic materials for which the strain energy function was proposed by Hill. The effects of constitutive parameters on the post-bifurcation behavior as well as on the critical stretch and stress are discussed.

On the Theory of Elastic Undamageable Materials

2013 ◽  
Vol 135 (2) ◽  
Author(s):  
George Z. Voyiadjis ◽  
Peter I. Kattan
Keyword(s):  
Elastic Material ◽  
Deformation Process ◽  
Strain Energy ◽  
Damage Mechanics ◽  
Materials Science ◽  
Higher Order ◽  
Damage Variable ◽  
Elastic Materials ◽  
Nonlinear Elastic Material ◽  
Nonlinear Elastic

In this work, both the concepts of Voyiadjis–Kattan materials and undamageable materials are introduced. The Voyiadjis–Kattan material of order n is defined as a nonlinear elastic material that has a higher-order strain energy form in terms of n. The undamageable material is obtained as the limit of the Voyiadjis–Kattan material of order n as n goes to infinity. The relations of these types of materials to other nonlinear elastic materials from the literature are outlined. Also, comparisons of these types of materials with rubber materials are presented. Finally, a proof is given to show that the value of the damage variable remains zero in an undamageable material throughout the deformation process. It is hoped that these proposed new types of materials will open the way to new areas of research in both damage mechanics and materials science.

Mechanical power and hyperelasticity

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Mechanical Power ◽  
Elastic Materials ◽  
Cyclic Motion

This chapter covers the notion of hyperelasticity—the concept that stress is derived from a strain—energy function–by invoking an analogy between elastic materials and springs. Alternatively, it can be derived by invoking a work inequality; the notion that work is required to effect a cyclic motion of the material.

scholarly journals Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids

2012 ◽  
Vol 468 (2146) ◽  
pp. 2881-2903 ◽  
Author(s):  
A. N. Norris ◽  
W. J. Parnell
Keyword(s):  
Strain Energy ◽  
Wave Motion ◽  
Deformation Gradient ◽  
Strain Energy Function ◽  
Formal Equivalence ◽  
Isotropic Elasticity ◽  
Symmetry Constraints ◽  
Constitutive Parameters ◽  
Wave Properties ◽  
Transformation Acoustics

Transformation elasticity, by analogy with transformation acoustics and optics, converts material domains without altering wave properties, thereby enabling cloaking and related effects. By noting the similarity between transformation elasticity and the theory of incremental motion superimposed on finite pre-strain, it is shown that the constitutive parameters of transformation elasticity correspond to the density and moduli of small-on-large theory. The formal equivalence indicates that transformation elasticity can be achieved by selecting a particular finite (hyperelastic) strain energy function, which for isotropic elasticity is semilinear strain energy. The associated elastic transformation is restricted by the requirement of statically equilibrated pre-stress. This constraint can be cast as tr F = constant, where F is the deformation gradient, subject to symmetry constraints, and its consequences are explored both analytically and through numerical examples of cloaking of anti-plane and in-plane wave motion.

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.

Antiplane Wave Scattering from a Cylindrical Void in a Pre-Stressed Incompressible Neo-Hookean Material

2012 ◽  
Vol 11 (2) ◽  
pp. 367-382 ◽  
Author(s):  
William J. Parnell ◽  
I. David Abrahams
Keyword(s):  
Wave Scattering ◽  
Small Amplitude ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Incident Field ◽  
Constitutive Behaviour ◽  
Void Size ◽  
Scattering Coefficients ◽  
Nonlinear Elastic Medium ◽  
Nonlinear Elastic

AbstractAn isolated cylindrical void is located inside an incompressible nonlinear-elastic medium whose constitutive behaviour is governed by a neo-Hookean strain energy function. In-plane hydrostatic pressure is applied in the far-field so that the void changes its radius and an inhomogeneous region of deformation arises in the vicinity of the void. We consider scattering from the void in the deformed configuration due to an incident field (of small amplitude) generated by a horizontally polarized shear (SH) line source, a distance r0 (R0) away from the centre of the void in the deformed (undeformed) configuration. We show that the scattering coefficients of this scattered field are unaffected by the pre-stress (initial deformation). In particular, they depend not on the deformed void radius a or distance r0, but instead on the original void size A and original distance R0.

A Mathematical Model of Lung Parenchyma

1980 ◽  
Vol 102 (2) ◽  
pp. 124-136 ◽  
Author(s):  
A. D. Karakaplan ◽  
M. P. Bieniek ◽  
R. Skalak
Keyword(s):  
Strain Energy ◽  
Strain Energy Function ◽  
Strain Curve ◽  
Lung Parenchyma ◽  
The Finite Element Method ◽  
Stress Strain Curve ◽  
Proposed Model ◽  
Elastic Membranes ◽  
Mammalian Lung ◽  
Nonlinear Elastic

The geometry of the proposed model of the parenchyma of a mammalian lung reproduces a cluster of alveoli arranged around a lowest-level air duct. The alveolar walls are assumed to be nonlinear elastic membranes, whose properties are described in terms of a strain energy function which reflects the hardening character of the stress-strain curve. The effect of the surfactant is included in terms of a variable (area-dependent) surface tension. Analyses of various mechanical processes in the parenchyma are performed with the aid of the finite element method, with the geometric and physical nonlinearities of the problem taken into account.

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