Expression of Dirichlet boundary conditions in terms of the Cauchy–Green tensor field

Abstract

In a previous work, it was shown how the Cauchy–Green tensor field C:=∇ΦT∇Φ∈W2,s(Ω;S>3), s>3/2, can be considered as the sole unknown in the homogeneous Dirichlet problem of nonlinear elasticity posed over a domain Ω⊂R3, instead of the deformation Φ∈W3,s(Ω;R3) in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition Φ=Φ0 on a portion Γ0 of the boundary of Ω can be recast, again as boundary conditions on Γ0, but this time expressed only in terms of the new unknown C∈W2,s(Ω;S>3).