Direct computation of stresses in planar linearized elasticity

Given a simply-connected domain Ω in ℝ, consider a linearly elastic body with Ω as its reference configuration, and define the Hilbert space E(Ω)={e(eαβ) ε L2s (Ω) ∂11e22- 2∂12e12}+∂22e11 = 0 in H-2(Ω)}. Then we recently showed that the associated pure traction problem is equivalent to finding a 2 × 2 matrix field = (_αβ) ∈ () that satisfies j(ε)= infeεE(Ω) j(e), where j(e) = 1/2∫Ω A αβσ τeστeαβ dx - ℓ(e), where (A _αβστ) is the elasticity tensor, and ℓ is a continuous linear form over (Ω) that takes into account the applied forces. Since the unknown stresses (σ_αβ) inside the elastic body are then given by σ_αβ = A _αβστ e_στ, 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 ^h, which uses "edge" finite elements in the sense of J. C. Nédélec. We then establish that the unique solution ^h of the associated discrete problem, viz., find ^h ∈ ^h such that j(εh)=inf ehεEh j(eh) converges to in the space L2 s(Ω). We emphasize that, by contrast with a mixed method, only the approximate stresses are computed in this approach.