Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

Abstract

In a previous work, it was shown how the linearized strain tensor field e:=12(∇uT+∇u)∈L2(Ω) can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain Ω⊂R3, instead of the displacement vector field u∈H1(Ω) 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 u=0 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 e∈L2(Ω).