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.

Large Deformation Isotropic Elasticity—On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids

1973 ◽  
Vol 46 (2) ◽  
pp. 398-416 ◽  
Author(s):  
R. W. Ogden
Keyword(s):  
Mathematical Analysis ◽  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Strain Energy Function ◽  
Usual Practice ◽  
Principal Axes ◽  
Simple Tension ◽  
Isotropic Elasticity ◽  
Strain Invariants

Abstract Many attempts have been made to reproduce theoretically the stress-strain curves obtained from experiments on the isothermal deformation of highly elastic ‘rubberlike’ materials. The existence of a strain-energy function has usually been postulated, and the simplifications appropriate to the assumptions of isotropy and incompressibility have been exploited. However, the usual practice of writing the strain energy as a function of two independent strain invariants has, in general, the effect of complicating the associated mathematical analysis (this is particularly evident in relation to the calculation of instantaneous moduli of elasticity) and, consequently, the basic elegance and simplicity of isotropic elasticity is sacrificed. Furthermore, recently proposed special forms of the strain-energy function are rather complicated functions of two invariants. The purpose of this paper is, while making full use of the inherent simplicity of isotropic elasticity, to construct a strain-energy function which: (i) provides an adequate representation of the mechanical response of rubberlike solids, and (ii) is simple enough to be amenable to mathematical analysis. A strain-energy function which is a linear combination of strain invariants defined by ϕ(α)=(α1α+α2α+α3α)/α is proposed; and the principal stretches α1, α2, and α3 are used as independent variables subject to the incompressibility constraint α1α2α3=1. Principal axes techniques are used where appropriate. An excellent agreement between this theory and the experimental data from simple tension, pure shear and equibiaxial tension tests is demonstrated. It is also shown that the present theory has certain repercussions in respect of the constitutive inequality proposed by Hill.

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 H. Hencky’s Approximate Strain-Energy Function for Moderate Deformations

1979 ◽  
Vol 46 (1) ◽  
pp. 78-82 ◽  
Author(s):  
L. Anand
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Finite Strain ◽  
Large Deformations ◽  
Strain Energy Function ◽  
Strain Measure ◽  
Logarithmic Measure ◽  
Infinitesimal Strain ◽  
Isotropic Elasticity ◽  
Good Agreement

It is shown that the classical strain-energy function of infinitesimal isotropic elasticity is in good agreement with experiment for a wide class of materials for moderately large deformations, provided the infinitesimal strain measure occurring in the strain-energy function is replaced by the Hencky or logarithmic measure of finite strain.

scholarly journals A Bimodular Polyconvex Anisotropic Strain Energy Function for Articular Cartilage

2006 ◽  
Vol 129 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Stephen M. Klisch
Keyword(s):  
Articular Cartilage ◽  
Strong Interaction ◽  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Deformation Gradient ◽  
Strain Energy Function ◽  
Finite Deformations ◽  
Orthotropic Materials ◽  
Interaction Terms

A strain energy function for finite deformations is developed that has the capability to describe the nonlinear, anisotropic, and asymmetric mechanical response that is typical of articular cartilage. In particular, the bimodular feature is employed by including strain energy terms that are only mechanically active when the corresponding fiber directions are in tension. Furthermore, the strain energy function is a polyconvex function of the deformation gradient tensor so that it meets material stability criteria. A novel feature of the model is the use of bimodular and polyconvex “strong interaction terms” for the strain invariants of orthotropic materials. Several regression analyses are performed using a hypothetical experimental dataset that captures the anisotropic and asymmetric behavior of articular cartilage. The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson’s ratios, as well as axial stress–axial strain responses, in tension and compression for finite deformations.

Modeling of Rate Dependent Finite Deformation Viscoelastic Behavior of Foams

2008 ◽  
Author(s):  
Y. Anani ◽  
M. Asghari ◽  
R. Naghdabadi
Keyword(s):  
Energy Function ◽  
Finite Deformation ◽  
Strain Energy ◽  
Deformation Gradient ◽  
Strain Energy Function ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Gradient Tensor ◽  
Deformation Gradient Tensor ◽  
Rate Dependent

The behavior of foams is typically rate-dependent and viscoelastic. In this paper, multiplicative decomposition of the deformation gradient and the second law of thermodynamics are employed to develop the differential constitutive equations for isotropic viscoelastic foams experiencing finite deformations, from a phenomenological point of view, i.e. without referring to micro-structural viewpoint. A model containing an equilibrium hyperelastic spring which is parallel to a Maxwell model has been utilized for introducing constitutive formulation. The deformation gradient tensor is decomposed into two parts: elastic deformation gradient tensor and viscoelastic deformation gradient tensor. A strain energy function is presented for the equilibrium spring as a function of the invariants of the left Cauchy-Green stretch tensor to obtain equilibrium stress components. Also, a strain energy function is presented for the intermediate spring as a function of the invariants of elastic deformation gradient tensor to determine overstress components. The constants of the strain energies are calculated by using nonlinear regulation numerical methods and by comparing with the experimental data obtained from uniaxial tension tests. The developed finite deformation constitutive equations are derived such that for every admissible process, the second law of thermodynamics is satisfied.

Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids

1972 ◽  
Vol 328 (1575) ◽  
pp. 567-583 ◽  
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Pure Shear ◽  
Density Ratio ◽  
Strain Energy Function ◽  
Elastic Solids ◽  
Simple Tension ◽  
Isotropic Elasticity ◽  
Full Discussion ◽  
Theory And Experiment

A method of approach to the correlation of theory and experiment for incompressible isotropic elastic solids under finite strain was developed in a previous paper (Ogden 1972). Here, the results of that work are extended to incorporate the effects of compressibility (under isothermal conditions). The strain-energy function constructed for incompressible materials is augmented by a function of the density ratio with the result that experimental data on the compressibility of rubberlike materials are adequately accounted for. At the same time the good fit of the strain-energy function arising in the incompressibility theory to the data in simple tension, pure shear and equibiaxial tension is maintained in the compressible theory without any change in the values of the material constants. A full discussion of inequalities which may reasonably be imposed upon the material parameters occurring in the compressible theory is included.

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 Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

2021 ◽  
Author(s):  
Afshin Anssari-Benam ◽  
Andrea Bucchi ◽  
Giuseppe Saccomandi
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Limit Point ◽  
Strain Energy ◽  
Cylindrical Shells ◽  
Strain Energy Function ◽  
Model Parameters ◽  
Wide Range ◽  
Cylindrical Tubes ◽  
Gent Model

AbstractThe application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure ($P$ P ) – inflation ($\lambda $ λ or $v$ v ) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

Mechanical characterization of a woven multi-layered hyperelastic composite laminate under uniaxial loading

2021 ◽  
pp. 002199832110115
Author(s):  
Shaikbepari Mohmmed Khajamoinuddin ◽  
Aritra Chatterjee ◽  
MR Bhat ◽  
Dineshkumar Harursampath ◽  
Namrata Gundiah
Keyword(s):  
Composite Materials ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Mechanical Tests ◽  
Stress Strain ◽  
Aerospace Applications ◽  
Envelope Material ◽  
The Individual ◽  
High Dielectric

We characterize the material properties of a woven, multi-layered, hyperelastic composite that is useful as an envelope material for high-altitude stratospheric airships and in the design of other large structures. The composite was fabricated by sandwiching a polyaramid Nomex® core, with good tensile strength, between polyimide Kapton® films with high dielectric constant, and cured with epoxy using a vacuum bagging technique. Uniaxial mechanical tests were used to stretch the individual materials and the composite to failure in the longitudinal and transverse directions respectively. The experimental data for Kapton® were fit to a five-parameter Yeoh form of nonlinear, hyperelastic and isotropic constitutive model. Image analysis of the Nomex® sheets, obtained using scanning electron microscopy, demonstrate two families of symmetrically oriented fibers at 69.3°± 7.4° and 129°± 5.3°. Stress-strain results for Nomex® were fit to a nonlinear and orthotropic Holzapfel-Gasser-Ogden (HGO) hyperelastic model with two fiber families. We used a linear decomposition of the strain energy function for the composite, based on the individual strain energy functions for Kapton® and Nomex®, obtained using experimental results. A rule of mixtures approach, using volume fractions of individual constituents present in the composite during specimen fabrication, was used to formulate the strain energy function for the composite. Model results for the composite were in good agreement with experimental stress-strain data. Constitutive properties for woven composite materials, combining nonlinear elastic properties within a composite materials framework, are required in the design of laminated pretensioned structures for civil engineering and in aerospace applications.

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