# Injective weak solutions in second-gradient nonlinear elasticity

Timothy J. Healey; Stefan Krömer

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 15, Issue: 4, page 863-871
- ISSN: 1292-8119

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topHealey, Timothy J., and Krömer, Stefan. "Injective weak solutions in second-gradient nonlinear elasticity." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2008): 863-871. <http://eudml.org/doc/90941>.

@article{Healey2008,

abstract = {
We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question here.
In particular, we demonstrate that the determinant of the gradient of any admissible deformation with finite energy is strictly positive on the closure of the domain. With this in hand,
Gâteaux differentiability of the potential energy at a minimizer is automatic, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including “mixed” boundary conditions. For each of the two possible pure displacement formulations (in second-gradient problems), we show that the resulting deformation is an injective mapping, whenever the imposed placement on the boundary is itself the trace of an injective map.
},

author = {Healey, Timothy J., Krömer, Stefan},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Gradient estimate; injective deformations; Euler-Lagrange equation; nonlinear elasticity; gradient estimate},

language = {eng},

month = {7},

number = {4},

pages = {863-871},

publisher = {EDP Sciences},

title = {Injective weak solutions in second-gradient nonlinear elasticity},

url = {http://eudml.org/doc/90941},

volume = {15},

year = {2008},

}

TY - JOUR

AU - Healey, Timothy J.

AU - Krömer, Stefan

TI - Injective weak solutions in second-gradient nonlinear elasticity

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/7//

PB - EDP Sciences

VL - 15

IS - 4

SP - 863

EP - 871

AB -
We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question here.
In particular, we demonstrate that the determinant of the gradient of any admissible deformation with finite energy is strictly positive on the closure of the domain. With this in hand,
Gâteaux differentiability of the potential energy at a minimizer is automatic, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including “mixed” boundary conditions. For each of the two possible pure displacement formulations (in second-gradient problems), we show that the resulting deformation is an injective mapping, whenever the imposed placement on the boundary is itself the trace of an injective map.

LA - eng

KW - Gradient estimate; injective deformations; Euler-Lagrange equation; nonlinear elasticity; gradient estimate

UR - http://eudml.org/doc/90941

ER -

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