TY - JOUR
T1 - Direct computation of stresses in planar linearized elasticity
AU - Ciarlet, Philippe G.
AU - Ciarlet Jr., Patrick
PY - 2009/7
Y1 - 2009/7
N2 - 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 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. © 2009 World Scientific Publishing Company.
AB - 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 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. © 2009 World Scientific Publishing Company.
KW - Computation of stresses
KW - Edge finite element
KW - Finite element methods
KW - Linearized elasticity
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-64749095281&origin=recordpage
U2 - 10.1142/S0218202509003711
DO - 10.1142/S0218202509003711
M3 - RGC 21 - Publication in refereed journal
VL - 19
SP - 1043
EP - 1064
JO - Mathematical Models and Methods in Applied Sciences
JF - Mathematical Models and Methods in Applied Sciences
SN - 0218-2025
IS - 7
ER -