Linear Isotropic Relations in Finite Hyperelasticity: Some General Results

Murphy, Jeremiah G.

doi:10.1007/s10659-006-9088-7

Cited by 11 publications

(8 citation statements)

References 8 publications

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“…(1.4) Additional motivations of the logarithmic strain tensor were also given by Vallée [205,206], Rougée [182, p. 302] and Murphy [142]. An extensive overview of the properties of the logarithmic strain tensor and its applications can be found in [209] and [154].…”

Section: What's In a Strain?mentioning

confidence: 99%

Geometry of Logarithmic Strain Measures in Solid Mechanics

Neff

Eidel

Martín

2016

Arch Rational Mech Anal

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We consider the two logarithmic strain measures ω iso = dev n log U = dev n log F T F andwhich are isotropic invariants of the Hencky strain tensor log U , and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group GL(n). Here, F is the deformation gradient, U = √ F T F is the right Biot-stretch tensor, log denotes the principal matrix logarithm, . is the Frobenius matrix norm, tr is the trace operator and dev n X = X − 1 n tr(X ) · 1 is the n-dimensional deviator of X ∈ R n×n . This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor ε = sym ∇u, which is the symmetric part of the displacement gradient ∇u, and reveals a close geometric relation between the classical quadratic isotropic energy potentialin linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energywhere μ is the shear modulus and κ denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonalIn memory of Giuseppe Grioli

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Section: What's In a Strain?mentioning

confidence: 99%

Geometry of Logarithmic Strain Measures in Solid Mechanics

Neff

Eidel

Martín

2016

Arch Rational Mech Anal

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“…(32,34) to obtain T (m) , H (m) explicitly in terms of T (1) , H (1) and U in tensorial form. Connections between components of T (m) and T (1) , H (m) and H (1) on the Lagrangian principal axis are easier to obtain [11].…”

Section: Thermal and Mechanical Tensors Conjugate To A General Class mentioning

confidence: 99%

“…However, the pioneering work in this context is as early as 1915 by Armanni [29]. This problem is still of interest, for example, we can refer to the recent works of Batra [30,31], Nader [32], Chiskis and Parnes [33], Murphy [34], Darijani et al [25] and Gilchrist et al [35].…”

Section: Constitutive Modelingmentioning

confidence: 99%

Conjugated kinetic and kinematic measures for constitutive modeling of the thermoelastic continua

Darijani

2014

Continuum Mech. Thermodyn.

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In this paper, the energy-type terms such as the stress power, the rate of the heat transferred to the system and the rate of the specific internal energy are presented in the Lagrangian, Eulerian and twopoint descriptions for thermoelastic continua. In order to solve a problem based on the energy viewpoint, the mechanical, thermal and thermo-mechanical tensors conjugate to the Seth-Hill strains, and a general class of Lagrangian, Eulerian and two-point strain tensors are determined. Also, the energy pairs for thermoelastic continua are simplified for special cases of isentropic and isothermal deformation processes as well as the so-called entropic elastic materials (rubber-like materials and elastomers). At the end, a strain energy density function of the Saint Venant-Kirchhoff type in terms of different strain measures and temperature is considered for modeling the thermo-mechanical behavior of the rubber-like materials and elastomers. It is shown that this constitutive modeling can give results which are in good agreement with the experimental data.

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“…One focus of attention has been on obtaining non-linear equivalents of the classical stress-strain relation. Although considered as early as 1915 by Armanni [2], this problem is still of interest, as can be seen in the recent work of Batra [3,4], Nader [5], Chiskis and Parvis [6] and Murphy [7].…”

Section: Introductionmentioning

confidence: 99%

Generalisations of the strain-energy function of linear elasticity to model biological soft tissue

Gilchrist

Murphy²,

Rashid

2012

International Journal of Non-Linear Mechanics

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Publication informationInternational Journal of Non-Linear Mechanics, 47 (2): 268-272Publisher Elsevier Item record/more information http://hdl.handle.net/10197/4602 Publisher's statementThis is the author's version of a work that was accepted for publication in International Journal of Non-Linear Mechanics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in International Journal of Non-Linear Mechanics (47, 2, (2012) ABSTRACT Strain measures consistent with the classical, infinitesimal form of the strain-energy function are obtained within the context of isotropic, homogeneous, compressible, non-linear elasticity. It will be shown that there are two distinct families of such measures. One family has already been much studied in the literature, the most important member being the strains whose principal values are a function only of the corresponding principal stretches. The second family of strains appears new. The motivation for studying such strains is the intuitive expectation that, for at least moderate deformations, a good fit with experimental data from material characterisation tests will be obtained with the corresponding strain-energy functions. In particular, there is the expectation that such models could prove useful for the modelling of biological soft tissue, whose physiological response is characterised by moderate strains. It will be shown that this is indeed the case for simple tension tests on porcine brain tissue.

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Linear Isotropic Relations in Finite Hyperelasticity: Some General Results

Cited by 11 publications

References 8 publications

Geometry of Logarithmic Strain Measures in Solid Mechanics

Geometry of Logarithmic Strain Measures in Solid Mechanics

Conjugated kinetic and kinematic measures for constitutive modeling of the thermoelastic continua

Generalisations of the strain-energy function of linear elasticity to model biological soft tissue

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