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

2020
Views:

20

Pages:

14
Author:

ddifenwadbyx

Page 1 Boundary value problem ∂σ_ij/∂x_j + f_i = 0 in B σ_ij n_j = t_i on S₂ u_i = u_i on S₁ ε_ij = 1/2 (u_ij + u_ji) in B ε_e, ε_p incompatible Variational principle (minimum potential energy) Page 2 207 Assume: ∃ U(ε,y) internal energy potential y = (ε_p, q) σ_ij = ∂U(ε,y)/∂ε_ij Lubliner (1972): U(ε,y) = U^e(ε-ε_p) + U^p(q) (decoupled) Example: linear elasticity U = 1/2 G_ijkl ε_ij ε_kl → fully implicit algorithm is contractive (errors in initial conditions are reduced by algorithm) STRONG STATEMENT OF STABILITY Specific model: J2-Isotropic hardening - fully implicit ε̇_ij = λ.3 S_ij / 2 σ' = (3/2 S_ij S_ij)^1/2 ε̇_ij = 3/2 S_ij / σ' τ_ij = C_ijkl ε̇_kl λ = ε_0 [ (σ'/σ_0)^m - 1 ] / φ if φ ≠ 0 Ω = σ'/ε̇ φ = σ_y [ (σ'/σ_0)^m - 1 ] σ(ε) = σ_y (1 + ε/ε_0)^1/n ε = ∫_0^t λ dt σ_{n+1} = σ_n + C (Δε - Δλ Γ_{n+1}) Δλ = Δt/η Φ(σ_{n+1}, σ_{n+1m}) σ_{0,n+1} = σ_0(ε_n + Δλ) Isotropic elasticity [p_{n+1} = p_n + K Δε_kk] pressure update is elastic S_{n+1} = S_n + 2μ (Δε - Δλ 3/2 S_{n+1}/σ_{n+1}) Δε = Δε_kk/3 I + Δe S_{n+1}^{*} = S_n + 2μ Δe σ_{n+1}^{*} = (3/2 S_{n+1}^{*}:S_{n+1}^{*})^{1/2} S_{n+1} = S_{n+1}^{*} - (Δλ 3/2 S_{n+1}/σ_{n+1}) 2μ = S_{n+1}^{*} - 3μ Δλ S_{n+1}/σ_{n+1}

16.225 Lecture 26 Notes

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