Resolution of a nonlinear elasticity problem subject to friction laws Resolution of a nonlinear elasticity problem

• Published in: Journal of fixed point theory and applications Vol. 26; no. 4
• Main Author: Boukrouche, Mahdi
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2024
• Subjects: Analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Coulomb friction law; variational problem; 70F40; 46E30; Sobolev spaces with variable exponent; Nonlinear elasticity problem; existence; Secondary 74B20; Primary 35R35; Fourier law
• ISSN: 1661-7738; 1661-7746
• DOI: 10.1007/s11784-024-01144-5

We consider a problem in a bounded domain, with Dirichlet condition on one part of its boundary and on other parts non-linear slip conditions governed by Coulomb friction law and Fourier law. We assume that the problem is also governed by a particular constitutive law of elasticity system with a strongly nonlinear strain tensor given by σ ij = ∑ k , h = 1 3 a ijkh E hk ( ∇ u ) where u is a displacement of a substance, ( a ijkh ) 1 ≤ i , j , k , h ≤ 3 are the coefficients of elasticity and E hk are the components of the nonlinear deformation tensor of St Venant E ( ∇ u ) = 1 2 T ∇ u + ∇ u + T ∇ u ∇ u . The functional framework leads to use Sobolev spaces with variable exponent. The formulation of the problem leads to a variational inequality, for which we prove, by Schauder fixed point theorem, an existence solution.