Introduction to materials modelling Lecture 9 - Plasticity, flow and - PowerPoint PPT Presentation

Introduction to materials modelling Lecture 9 - Plasticity, flow and hardening rules Reijo Kouhia Tampere University, Structural Mechanics November 13, 2019 R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling November 13, 2019 1 / 9

Flow rules Flow rule describes how the plastic strains evolve . In associated flow rule plastic strains evolve normal to the yield surface 1 λ ∂f ε p = ˙ ˙ ∂ σ Associated flow rule can be derived from the maximum dissipation principle . In non-associated flow rule the plastic strain rate is obtained from a separate plastic potential 2 λ ∂g ε p = ˙ ˙ ∂ σ The plastic multiplier ˙ λ is obtained from the consistency condition ˙ f = 0 which says the fact that when plastic deformation occurs the stress stays on the yield surface. Notice that in rate-independent plasticity the flow rule is homogeneous wrt time. R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling November 13, 2019 2 / 9

Associated von-Mises ideal-plasticity Yield condition � f ( σ ) = 3 J 2 − σ y = 0 where √ 3 J 2 = σ eff is the effective stress . λ ∂f λ ∂f ∂J 2 3 3 ε p = ˙ ∂ σ = ˙ ∂ σ = ˙ s = ˙ ˙ λ 2 √ 3 J 2 λ s ∂J 2 2 σ eff ε p ) = 0 which means that for von Mises plasticity the plastic flow is isochoric , i.e. Notice that tr(˙ volume preserving. Defining monotonously increasing effective plastic strain � � 2 ε p : ˙ ε p ε p ε p ε p eff = ˙ eff d t, where ˙ eff = 3 ˙ R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling November 13, 2019 3 / 9

Associated Drucker-Prager ideal plasticity Adding a linear pressure term to the von Mises criteria � f ( σ ) = 3 J 2 + αI 1 − β = 0 . For associated plasticity, the flow rule is � ∂f � � � � � λ ∂f ∂J 2 ∂ σ + α∂I 1 3 3 ε p = ˙ ∂ σ = ˙ = ˙ = ˙ ˙ λ λ 2 √ 3 J 2 s + α I λ s + α I ∂J 2 ∂ σ 2 σ eff The relative plastic volume change is ε p ) = 3 α tr(˙ Experiments have shown that for many materials it is too large, therefore non-associated flow rule has to be used. R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling November 13, 2019 4 / 9

Non-associated Drucker-Prager ideal plasticity Plastic potential � g ( σ ) = 3 J 2 + α ∗ I 1 and λ ∂g � 3 � ε p = ˙ ∂ σ = ˙ ˙ λ s + α ∗ I 2 σ eff Now the relative plastic volume change is ε p ) = 3 α ∗ tr(˙ Requirement from experiments α ∗ < α . R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling November 13, 2019 5 / 9

Hardening rules Isotropic hardening 1 Kinematic hardening 2 Anisotropic or distortional hardening 3 The yield function (and the plastic potential) is written as f ( σ , K α ) , α = 1 , 2 , ... where K α = K α ( κ β ) denote hardening parameters, depending on internal variables κ β , and can be scalars or second-order tensors. R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling November 13, 2019 6 / 9

Isotropically hardening von Mises plasticity Yield condition � f ( σ ) = 3 J 2 − ( σ y0 + K ( κ )) = 0 where K is a hardening parameter and κ is an internal variable. λ ∂f λ ∂f ∂J 2 3 3 ε p = ˙ ∂ σ = ˙ ∂ σ = ˙ s = ˙ ˙ λ 2 √ 3 J 2 λ s ∂J 2 2 σ eff λ ∂f κ = − ˙ ∂K = ˙ ˙ λ Effective plastic strain rate � 2 ε p = ε p : ˙ ε p = | ˙ ˙ ¯ 3 ˙ λ | R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling November 13, 2019 7 / 9

Isotropically hardening von Mises plasticity (cont’d) Simple linear hardening ε p ) = H ¯ ε p K ( κ ) = K (¯ More realistic behaviour can be modelled with ε p /K ∞ )) , ε p /K ∞ )) K = K ∞ (1 − exp( − h ¯ thus σ y = σ y0 + K ∞ (1 − exp( − h ¯ and H = d σ y ε p /K ∞ ) ε p = h exp( − h ¯ d¯ R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling November 13, 2019 8 / 9

Kinematic hardening von Mises plasticity Now the hardening parameter K is a deviatoric second order tensor α , which is called as the back stress � 3 f ( σ , α ) = 2( s − α ) : ( s − α ) − σ y0 The Melan (1938) and Prager (1955) hardening rule ε p α = c ˙ ˙ κ = c ˙ The Ziegler (1955) hardening rule α = ˙ ˙ λ ¯ c ( σ − α ) Non-linear Armstrong-Frederick (1966) model � 2 � ε p − α ε p ˙ α = h ˙ 3 ˙ ¯ α ∞ R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling November 13, 2019 9 / 9