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 σij = sij + σmδij, where σm is the mean stress and sij is the deviatoric stress tensor. Mean stress is σm = 1

3trσ

σ σ = 1

3σkk = σ11 + σ22 + σ33 = σx + σy + σz.

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Deviatoric invariants

Symmetric deviatoric tensor has five independent components. Eigenvalues of s: sn = λn (s − λI )n = 0. Characteristic equation −λ3 + J2λ + J3 = 0, where (notice J1 = trs = 0) J2 = 1

2tr(s2),

J3 = det s = 1

3tr(s3).

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Geometric interpretation

In the principal stress space: hydrostatic axis (blue line), deviatoric plane ⊥ hydrostatic axis. Red line NP lies on the deviatoric plane.

σ2 σ3 σ1 O ξ hydrostatic axis n

b

N ρ

bP(σ1, σ2, σ3)

σ1 σ2 σ3 θ ρ e1 N P

ξ = | ON|, ρ = | NP|, θ Heigh-Westergaard coordinates

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