Introduction to materials modelling Lecture 8 - Plasticity Reijo - PowerPoint PPT Presentation

Introduction to materials modelling Lecture 8 - Plasticity Reijo Kouhia Tampere University, Structural Mechanics October 31, 2019 R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 1 / 10

Plasticity - basic incredients Characteristic feature in plastic deformation is the formation of permanent deformations. In a closed loading process energy is dissipated into structural changes of a material and into heat. To discribe plasticity three type of equations are needed: yield condition which determines the boundary of the elastic domain, 1 flow rule which describes how the plastic strains evolve, 2 hardening rule which describes the evolution of the elastic domain, i.e. the evolution of the yield 3 surface. R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 2 / 10

Yield function (initial) Written usually as f ( σ , parameters ) = 0 . Separates the elastic domain from the plastic state: f ( σ , .. ) < 0 stresss in the elastic domain f ( σ , .. ) = 0 plastic state f ( σ , .. ) > 0 not possible For an isotropic solid the yield function has to be independent of coordinate orientation, i.e. f ( σ , .. ) = f ( βσβ T , .. ) ∀ orthogonal β Thus f ( I 1 , I 2 , I 3 ) or preferably f ( I 1 , J 2 , cos 3 θ ) R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 3 / 10

Yield function (cont’d) If an isotropic yield function is given in the form f ( I 1 , J 2 , cos 3 θ ) = 0 , it facilitates the investigation of its symmetry properties in the deviatoric plane. The yield function is 120 ◦ periodic, i.e. ρ = √ 2 J 2 has to have same values at θ and θ + 120 ◦ . Since cos is an even function, there has to be symmetry with respect to θ = 0 ◦ . Due to the 120 ◦ periodicity, f has to be symmetric also wrt θ = 120 ◦ and θ = 240 ◦ . If we set θ = 60 ◦ + ψ , then cos(3 θ ) = − cos(3 ψ ) and setting θ = 60 ◦ − ψ gives cos(3 θ ) = − cos(3 ψ ) , so they have the same ρ , thus the yield curve at deviatoric plane is symmetric about θ = 60 ◦ , thus it has to be symmetric also about θ = 180 ◦ and θ = 300 ◦ . As a conclusion the initial yield curve for isotropic solids in the deviatoric plane is completely characterized by its form in the sector 0 ◦ ≤ θ ≤ 60 ◦ . R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 4 / 10

Some well known yield functions Pressure independent yield functions f ( J 2 , cos 3 θ ) = 0 : 1 ◮ Tresca τ max = 1 2 ( σ 1 − σ 3 ) − τ y = 0 ◮ von Mises √ √ 3 J 2 − σ y = 0 , or J 2 − τ y = 0 . Pressure dependent yield functions f ( I 1 , J 2 , cos 3 θ ) = 0 : 2 ◮ Drucker-Prager √ 3 J 2 + αI 1 − β = 0 ◮ Mohr-Coulomb mσ 1 + σ 3 − σ c = 0 R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 5 / 10

Tresca vs. von Mises yield surfaces σ 1 30 ◦ σ e /σ y 2 1 σ m /σ y σ 2 σ 3 − 2 − 1 0 1 2 τ/σ y σ 2 /σ y 0 . 75 1 . 0 0 . 50 0 . 5 0 . 25 σ/σ y − 1 . 0 − 0 . 5 0 . 5 1 . 0 σ 1 /σ y 0 − 0 . 5 0 0 . 25 0 . 50 0 . 75 1 . 00 − 1 . 0 RK R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 6 / 10

Tresca vs. von Mises - experiments R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 7 / 10

Mohr-Coulomb yield criteria τ = c − µσ τ R = 1 2 ( σ 1 − σ 3 ) c σ 3 σ 1 φ σ P = 1 2 ( σ 1 + σ 3 ) c/µ R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 8 / 10

Mohr-Coulomb yield criteria (cont’d) σ e /f c 5 θ = 60 ◦ σ 2 /f c 4 0 . 5 σ 1 σ 1 /f c 3 θ = 0 ◦ − 1 . 0 − 0 . 5 0 . 5 2 − 0 . 5 1 σ 2 σ 3 σ m /f c − 1 . 0 − 3 − 2 − 1 0 1 Kuvissa f c is the uniaxial compressive strength. R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 9 / 10

Failure surfaces for concrete σ 2 /f c σ e /f c θ = 60 ◦ σ 1 /f c 7 − 1 . 4 − 1 . 2 − 1 . 0 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 6 − 0 . 2 5 − 0 . 4 θ = 0 ◦ θ = 0 ◦ 4 − 0 . 6 3 − 0 . 8 2 − 1 . 0 1 σ m /f c − 1 . 2 − 5 − 4 − 3 − 2 − 1 0 1 − 1 . 4 (a) (b) Mohr-Coulomb with tension cut-off (green), Barcelona model (red), Ottosen’s model (blue). R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 31, 2019 10 / 10