The exponentiated Hencky-logarithmic strain energy: Part II: Coercivity, planar polyconvexity and existence of minimizers

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autorschaft

  • Patrizio Neff
  • Johannes Lankeit
  • Ionel Dumitrel Ghiba
  • Robert Martin
  • David Steigmann

Externe Organisationen

  • Universität Duisburg-Essen
  • Universität Paderborn
  • Al. I. Cuza University
  • Romanian Academy
  • University of California at Berkeley
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Details

OriginalspracheEnglisch
Seiten (von - bis)1671-1693
Seitenumfang23
FachzeitschriftZeitschrift fur Angewandte Mathematik und Physik
Jahrgang66
Frühes Online-Datum4 Feb. 2015
PublikationsstatusVeröffentlicht - Aug. 2015
Extern publiziertJa

Abstract

We consider a family of isotropic volumetric–isochoric decoupled strain energies (Formula Presented.) based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, κ=2μ+3λ3>0 is the infinitesimal bulk modulus with $${\lambda}$$λ the first Lamé constant, (Formula Presented.) are dimensionless parameters, F=∇φ is the gradient of deformation, (Formula Presented.) is the right stretch tensor and (Formula Presented.) is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy (Formula Presented.) which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family (Formula Presented.) are polyconvex for (Formula Presented.) extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann’s polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations.

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The exponentiated Hencky-logarithmic strain energy: Part II: Coercivity, planar polyconvexity and existence of minimizers. / Neff, Patrizio; Lankeit, Johannes; Ghiba, Ionel Dumitrel et al.
in: Zeitschrift fur Angewandte Mathematik und Physik, Jahrgang 66, 08.2015, S. 1671-1693.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Neff P, Lankeit J, Ghiba ID, Martin R, Steigmann D. The exponentiated Hencky-logarithmic strain energy: Part II: Coercivity, planar polyconvexity and existence of minimizers. Zeitschrift fur Angewandte Mathematik und Physik. 2015 Aug;66:1671-1693. Epub 2015 Feb 4. doi: 10.48550/arXiv.1408.4430, 10.1007/s00033-015-0495-0
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abstract = "We consider a family of isotropic volumetric–isochoric decoupled strain energies (Formula Presented.) based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, κ=2μ+3λ3>0 is the infinitesimal bulk modulus with $${\lambda}$$λ the first Lam{\'e} constant, (Formula Presented.) are dimensionless parameters, F=∇φ is the gradient of deformation, (Formula Presented.) is the right stretch tensor and (Formula Presented.) is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy (Formula Presented.) which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family (Formula Presented.) are polyconvex for (Formula Presented.) extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann{\textquoteright}s polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations.",
keywords = "Coercivity, Existence of minimizers, Finite isotropic elasticity, Logarithmic strain, Plane elastostatics, Polyconvexity",
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T2 - Part II: Coercivity, planar polyconvexity and existence of minimizers

AU - Neff, Patrizio

AU - Lankeit, Johannes

AU - Ghiba, Ionel Dumitrel

AU - Martin, Robert

AU - Steigmann, David

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N2 - We consider a family of isotropic volumetric–isochoric decoupled strain energies (Formula Presented.) based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, κ=2μ+3λ3>0 is the infinitesimal bulk modulus with $${\lambda}$$λ the first Lamé constant, (Formula Presented.) are dimensionless parameters, F=∇φ is the gradient of deformation, (Formula Presented.) is the right stretch tensor and (Formula Presented.) is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy (Formula Presented.) which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family (Formula Presented.) are polyconvex for (Formula Presented.) extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann’s polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations.

AB - We consider a family of isotropic volumetric–isochoric decoupled strain energies (Formula Presented.) based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, κ=2μ+3λ3>0 is the infinitesimal bulk modulus with $${\lambda}$$λ the first Lamé constant, (Formula Presented.) are dimensionless parameters, F=∇φ is the gradient of deformation, (Formula Presented.) is the right stretch tensor and (Formula Presented.) is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy (Formula Presented.) which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family (Formula Presented.) are polyconvex for (Formula Presented.) extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann’s polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations.

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