Details
| Original language | English |
|---|---|
| Pages (from-to) | 1671-1693 |
| Number of pages | 23 |
| Journal | Zeitschrift fur Angewandte Mathematik und Physik |
| Volume | 66 |
| Early online date | 4 Feb 2015 |
| Publication status | Published - Aug 2015 |
| Externally published | Yes |
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.
Keywords
- Coercivity, Existence of minimizers, Finite isotropic elasticity, Logarithmic strain, Plane elastostatics, Polyconvexity
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Physics and Astronomy(all)
- General Physics and Astronomy
- Mathematics(all)
- Applied Mathematics
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In: Zeitschrift fur Angewandte Mathematik und Physik, Vol. 66, 08.2015, p. 1671-1693.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The exponentiated Hencky-logarithmic strain energy
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
PY - 2015/8
Y1 - 2015/8
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.
KW - Coercivity
KW - Existence of minimizers
KW - Finite isotropic elasticity
KW - Logarithmic strain
KW - Plane elastostatics
KW - Polyconvexity
UR - http://www.scopus.com/inward/record.url?scp=84938803720&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1408.4430
DO - 10.48550/arXiv.1408.4430
M3 - Article
AN - SCOPUS:84938803720
VL - 66
SP - 1671
EP - 1693
JO - Zeitschrift fur Angewandte Mathematik und Physik
JF - Zeitschrift fur Angewandte Mathematik und Physik
SN - 0044-2275
ER -