University:
Massachusetts Institute of Technology
Course:
16.225 | Computational Mechanics of Materials
Academic year:

2020
Views:

398

Pages:

10
Author:

Shyanne Johnson

Incompressible Elasticity Essentially the same equations as Stokes flow (viscous, incompressible, laminar Re→0). Background Hooke's Law for isotropic solid σ_ij = λ E_kk δ_ij + 2μ E_ij Volumetric and deviatoric components σ_ij^dev = σ_ij - p δ_ij, p = T_kk/3 σ_ij^vol = p δ_ij σ_ij = σ_ij^dev + σ_ij^vol σ_ij^dev σ_ij^vol = 0 ε_ij^dev = ε_ij - 1/3 δ_ij, θ = ε_kk = 1/3 u_k,k = ∇.u ε_ij^vol = 1/3 δ_ij ε_ij = ε_ij^vol + ε_ij^dev Hooke's Law σ_ij^dev = 2μ ε_ij^dev σ_ij^vol = 3K ε_ij^vol, K = bulk modulus = E / 3(1-2ν) ⇒ Isotropic Hooke's law decouples into deviatoric and volumetric parts. Incompressible limit ν → 0.5 K → ∞ For p = Kθ < ∞ as K→∞, p finite θ → 0 In limit θ → 0, p not determined from constitutive equations E_kk = ∇.u = 0 Governing equations: σ_ij,j + f_i = 0, σ_ij = σ_ij^dev + p δ_ij = 2μ ε_ij^dev + p δ_ij (2μ ε_ij^dev + p δ_ij),j + f_i = 0 ε_ij = 1/2 (u_i,j + u_j,i) u_k,k = 0 → ε_ij^dev = ε_ij ε_ij^dev = 1/2 (M_ij + M_ji) μ M_ij + p δ_ij + f_i = 0 in B incompressible elasticity u_i = u_i on S₁ (2μ M_ij + p δ_ij) n_j = t_i on S₂ M_ij = 0 Completely different variational structure (saddle-point problem)

16.225 Lecture 9 Notes

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