On strong ellipticity for isotropic hyperelastic materials based upon logarithmic strain

The logarithmic strain measures log U 2 , where log U is the principal matrix logarithm of the stretch tensor U = √ F T F corresponding to the deformation gradient F and . denotes the Frobenius matrix norm, arises naturally via the geodesic distance of F to the special orthogonal group SO(n). This purely geometric characterization of this strain measure suggests that a viable constitutive law of nonlinear elasticity may be derived from an elastic energy potential which depends solely on this intrinsic property of the deformation, i.e. that an energy function W : GL + (n) → R of the formwith a suitable function Ψ : [0, ∞) → R should be used to describe finite elastic deformations. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function Ψ such that W of the form (1) is Legendre-Hadamard elliptic.Similarly, we consider the related isochoric strain measure devn log U 2 , where devn log U is the deviatoric part of log U . Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case n = 2, we show that for n ≥ 3, no strictly monotone function Ψ : [0, ∞) → R exists such that F → Ψ( devn log U 2 ) is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form F → Ψ( devn log U 2 ) + W vol (det F ) cannot be rank-one convex for any function W vol : (0, ∞) → R if Ψ is strictly monotone.

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“…A similar implication was found by Sendova and Walton [47], who considered energy functions of the form…”

Section: Necessary Conditions For Rank-one Convexitysupporting

confidence: 82%

A non-ellipticity result, or the impossible taming of the logarithmic strain measure

Martín

Ghiba

Neff

2018

International Journal of Non-Linear Mechanics

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“…Without resorting to the Taylor expansion of the logarithm [11] or of the exponential [8] of a symmetric tensor, nor to its spectral decomposition [5], we have given an intrinsic proof of the existence of the potential α • exp between σ ρ and ln B. Numerous isotropic hyperelastic constitutive laws expressing directly σ in term of ln B have been proposed ( [2], [6], [7], [9], [12]) and numerically implemented [4]. When the potential α • exp is convex, the consideration of its Legendre-Fenchel-Moreau transform is a tool to perform the inversion of the constitutive law ( [1], [13], [14]), ie to express the Hencky logarithmic strain tensor ln B in term of the Cauchy stress tensor σ.…”

Section: Resultsmentioning

confidence: 99%

On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity

Vallée

Fortuné

Lerintiu

2008

Comptes Rendus. Mécanique

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Elastic materials are governed by a constitutive law linking the second Piola-Kirchhoff stress tensor Σ and the right Cauchy-Green strain tensor C = F T F . Isotropic elastic materials are the special ones for which the Cauchy stress tensor σ depends solely of the left Cauchy-Green strain tensor B = F F T . In this paper we revisit the following property of isotropic hyperelastic materials: if the constitutive law linking Σ and C derives from a potential α, then σ and ln B are linked by a constitutive law deriving from the potential α • exp. We give a new and concise proof which is based on an explicit formula expressing the derivative of the exponential of a tensor.

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“…In [222] some necessary conditions for the LH-ellipticity versus exponential-growth are discussed for energies depending on the Hencky strain log U . In fact, Sendova and Walton [222] have considered the energy W to be a function of K 2 = dev 3 log U and proved that W has to grow at least exponentially as a function of K 2 . They note, however, that "constructing conditions that are both necessary and sufficient for strong ellipticity to hold for all deformations still seem[s to be] a daunting task".…”

Section: Previous Work In the Spirit Of Our Investigationmentioning

confidence: 99%

The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity

Neff

Ghiba

Lankeit

2015

J Elast

126

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We investigate a family of isotropic volumetric-isochoric decoupled strain energiesbased on the Hencky-logarithmic (true, natural) strain tensor log U , where µ > 0 is the infinitesimal shear modulus, κ = 2µ+3λ 3 > 0 is the infinitesimal bulk modulus with λ the first Lamé constant, k, k are dimensionless parameters, F = ∇ϕ is the gradient of deformation, U = √ F T F is the right stretch tensor and devn log U = log U − 1 n tr(log U ) • 1 1 is the deviatoric part of the strain tensor log U . For small elastic strains, W eH approximates the classical quadratic Hencky strain energywhich is not everywhere rank-one convex. In plane elastostatics, i.e. n = 2, we prove the everywhere rankone convexity of the proposed family W eH , for k ≥ 1 4 and k ≥ 1 8 . Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for n = 2, 3 and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family W eH is not preserved in dimension n = 3 and that the energiesare not rank-one convex.

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On strong ellipticity for isotropic hyperelastic materials based upon logarithmic strain

Cited by 19 publications

References 15 publications

A non-ellipticity result, or the impossible taming of the logarithmic strain measure

A non-ellipticity result, or the impossible taming of the logarithmic strain measure

On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity

The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity

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