Direct Expression of Incompatibility in Curvilinear Systems

Abstract

We would like to present a method to compute the incompatibility operator in any system of curvilinear coordinates (components). The procedure is independent of the metric in the sense that the expression can be obtained by means of the basis vectors only, which are first defined as normal or tangential to the domain boundary, and then extended to the whole domain. It is an intrinsic method, to some extent, since the chosen curvilinear system depends solely on the geometry of the domain boundary. As an application, the in-extenso expression of incompatibility in a spherical system is given.

Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

“…The solution is immediately found as w = re r and K = I. Considering the form E = ϕ(r)I + ψ(r)e r ⊗ e r we have CE = ((3λ + 2µ)ϕ + λψ) I + 2µψe r ⊗ e r , and (see[40])inc E = ϕ + ϕ r − ψ r I + −ϕ + ϕ r + ψ r − 2ψ r 2 e r ⊗ e r .Hence CE + inc E = K if and only if  E N = 0 on ∂Ω entails ϕ (1) = ψ(1). The solution of the system is the classical elastic solution given by strain incompatibility in this case.…”