DIRECT COMPUTATION OF STRESSES IN PLANAR LINEARIZED ELASTICITY

Abstract

Given a simply-connected domain Ω in ℝ2, consider a linearly elastic body with [Formula: see text] as its reference configuration, and define the Hilbert space [Formula: see text] Then we recently showed that the associated pure traction problem is equivalent to finding a 2 × 2 matrix field ∊ = (∊αβ) ∈ 𝔼(Ω) that satisfies [Formula: see text] where [Formula: see text] where (Aαβστ) is the elasticity tensor, and ℓ is a continuous linear form over 𝔼(Ω) that takes into account the applied forces. Since the unknown stresses (σαβ) inside the elastic body are then given by σαβ = Aαβστ eστ, this minimization problem thus directly provides the stresses. We show here how the above Saint Venant compatibility condition ∂11 e22 - 2∂12e12 + ∂22e11 = 0 in H-2(Ω) can be exactly implemented in a finite element space 𝔼h, which uses "edge" finite elements in the sense of J. C. Nédélec. We then establish that the unique solution ϵh of the associated discrete problem, viz., find ϵh ∈ 𝔼h such that [Formula: see text] converges to ϵ in the space [Formula: see text]. We emphasize that, by contrast with a mixed method, only the approximate stresses are computed in this approach.