Introduction to materials modelling Lecture 2 - Decomposition of - PowerPoint PPT Presentation

Introduction to materials modelling Lecture 2 - Decomposition of stress, geometric interpretation Reijo Kouhia Tampere University, Structural Mechanics October 4, 2019 R.Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 1 / 4

Deviatoric stress Additive decomposition of stress tensor σ ij = s ij + σ m δ ij , where σ m is the mean stress and s ij is the deviatoric stress tensor. Mean stress is σ m = 1 σ = 1 3 tr σ σ 3 σ kk = σ 11 + σ 22 + σ 33 = σ x + σ y + σ z . R.Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 2 / 4

Deviatoric invariants Symmetric deviatoric tensor has five independent components. Eigenvalues of s : sn = λ n ( s − λ I ) n = 0 . Characteristic equation − λ 3 + J 2 λ + J 3 = 0 , where (notice J 1 = tr s = 0 ) J 2 = 1 2 tr( s 2 ) , J 3 = det s = 1 3 tr( s 3 ) . R.Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 3 / 4

b Geometric interpretation In the principal stress space: hydrostatic axis (blue line), deviatoric plane ⊥ hydrostatic axis. Red line σ 1 σ 1 b P ( σ 1 , σ 2 , σ 3 ) P ρ θ hydrostatic axis e 1 NP lies on the deviatoric plane. n ρ N O ξ N σ 3 σ 2 σ 2 σ 3 ξ = | � ρ = | � ON | , NP | , θ Heigh-Westergaard coordinates R.Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 4 / 4