Direct computation of stresses in planar linearized elasticity

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Original languageEnglish
Pages (from-to)1043-1064
Journal / PublicationMathematical Models and Methods in Applied Sciences
Issue number7
Publication statusPublished - Jul 2009


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 (σαβ) inside the elastic body are then given by σαβ = A αβστ 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 hh 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. © 2009 World Scientific Publishing Company.

Research Area(s)

  • Computation of stresses, Edge finite element, Finite element methods, Linearized elasticity