Resolution of a nonlinear elasticity problem subject to friction laws

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 26; no. 4; p. 51
• Main Author: Boukrouche, Mahdi
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.12.2024; Springer Verlag
• Subjects: Computer Science; Coulomb friction; Elasticity; Fixed points (mathematics); Fourier law; Friction; Nonlinear Sciences; Partial differential equations; Schauder fixpoint theorem; Sobolev space; Strain; Tensors; Variational problem; Coulomb friction law; 70F40; 46E30 Nonlinear elasticity problem; Sobolev spaces with variable exponent; Secondary 74B20 70F40 46E30; existence; Secondary 74B20; Mathematics Subject Classification . Primary 35R35; Fourier law
• ISSN: 1661-7738; 1661-7746; 2730-5422
• 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 σij=∑k,h=13aijkhEhk(∇u) where u is a displacement of a substance, (aijkh)1≤i,j,k,h≤3 are the coefficients of elasticity and Ehk are the components of the nonlinear deformation tensor of St Venant E(∇u)=12T∇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.