A boundary-field equation method for a nonlinear exterior elasticity problem in the plane

Abstract

This paper presents a new method for solving a nonlinear exterior boundary value problem arising in two-dimensional elasto-plasticity. The procedure is based on the introduction of a sufficiently large circle that divides the exterior domain into a bounded region and an unbounded one. This allows us to consider the Dirichlet-Neumann mapping on the circle, which provides an expficit formula for the stress in terms of the displacement by using an appropriate infinite Fourier series. In this way we can reduce the original problem to an equivalent nonlinear boundary value problem on the bounded domain with a natural boundary condition on the circle. Hence, the resulting weak formulation includes boundary and field terms, which yields the so called boundary-field equation method. Next, we employ the finite Fourier series to obtain a sequence of approximating nonlinear problems from which the actual Galerkin schemes are derived. Finally, we apply some tools from monotone operators to prove existence, uniqueness and approximation results, including Cea type error estimates for the corresponding discrete solutions.

A FEM–DtN formulation for a non‐linear exterior problem in incompressible elasticity

“…Section 2) one can prove (cf. Lemmas 4.1-4.3 in Reference [17]) that a ij (·) and its ÿrst-order partial derivatives are continuous in R 2×2 , and that there exist C 1 ; C 2 ; C 3 ¿0 such that (19) for all :=( ij ), ÿ :=(ÿ ij )∈R 2×2 (see, also, Reference [28]). The rest of the proof proceeds similarly as for Theorems 5.1 and 5.2 in Reference [17] (see also Reference [28]).…”

A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate

“…(2.5)) one can show (cf. [25] or Lemmata 4.1 and 4.2 in [6]) that a ij (·) is continuous and has first order partial derivatives in R 2×2 , and there exist C 1 , C 2 > 0 such that…”

“…On the other hand, we recall from [25] (see also Lemma 4.3 in [6]) that the functions ∂ ∂r kl a ij (·) are continuous in R 2×2 , and there existsC > 0 such that…”