Direct computation of stresses in planar linearized elasticity

Philippe G. Ciarlet 1 Patrick Ciarlet 2
2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : Given a simply-connected domain Ω in ℝ2, consider a linearly elastic body with Ω as its reference configuration, and define the Hilbert space E(Ω)={e(eαβ) ∈ 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) ∈E(Ω) that satisfies j(∈)= inf e∈E(Ω) j(e), where j(e) = 1/2 ∫Ω Aαβστ eστ eαβ dx - l(e), where (A αβστ ) 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 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 ∈ Eh such that j(∈h)=inf eh∈Eh 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. © 2009 World Scientific Publishing Company.
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Submitted on : Wednesday, October 16, 2013 - 4:22:40 PM
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Philippe G. Ciarlet, Patrick Ciarlet. Direct computation of stresses in planar linearized elasticity. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2009, 19 (7), pp.1043-1064. ⟨10.1142/s0218202509003711⟩. ⟨hal-00873070⟩

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