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.

1

In associated flow rule plastic strains evolve normal to the yield surface ˙ εp = ˙ λ ∂f ∂σ Associated flow rule can be derived from the maximum dissipation principle.

2

In non-associated flow rule the plastic strain rate is obtained from a separate plastic potential ˙ εp = ˙ λ ∂g ∂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) =

• 3J2 − σy = 0

where √3J2 = σeff is the effective stress. ˙ εp = ˙ λ ∂f ∂σ = ˙ λ ∂f ∂J2 ∂J2 ∂σ = ˙ λ 3 2√3J2 s = ˙ λ 3 2σeff s Notice that tr(˙ εp) = 0 which means that for von Mises plasticity the plastic flow is isochoric, i.e. volume preserving. Defining monotonously increasing effective plastic strain εp

eff =

• ˙

εp

eff dt,

where ˙ εp

eff =

• 2

3 ˙ εp : ˙ εp

• 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) =

• 3J2 + αI1 − β = 0.

For associated plasticity, the flow rule is ˙ εp = ˙ λ ∂f ∂σ = ˙ λ ∂f ∂J2 ∂J2 ∂σ + α∂I1 ∂σ

• = ˙

λ

• 3

2√3J2 s + αI

• = ˙

λ

• 3

2σeff s + αI

• The relative plastic volume change is

tr(˙ εp) = 3α 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) =

• 3J2 + α∗I1

and ˙ εp = ˙ λ ∂g ∂σ = ˙ λ

• 3

2σeff s + α∗I

• Now the relative plastic volume change is

tr(˙ εp) = 3α∗ Requirement from experiments α∗ < α.

• R. Kouhia (Tampere University, Structural Mechanics)

Introduction to materials modelling November 13, 2019 5 / 9

Hardening rules

1

Isotropic hardening

2

Kinematic hardening

3

Anisotropic or distortional hardening The yield function (and the plastic potential) is written as f(σ, Kα), α = 1, 2, ... where Kα = Kα(κβ) denote hardening parameters, depending on internal variables κβ, 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) =

• 3J2 − (σy0 + K(κ)) = 0

where K is a hardening parameter and κ is an internal variable. ˙ εp = ˙ λ ∂f ∂σ = ˙ λ ∂f ∂J2 ∂J2 ∂σ = ˙ λ 3 2√3J2 s = ˙ λ 3 2σeff s ˙ κ = − ˙ λ ∂f ∂K = ˙ λ Effective plastic strain rate ˙ ¯ εp =

• 2

3 ˙ εp : ˙ εp = | ˙ 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 K(κ) = K(¯ εp) = H¯ εp More realistic behaviour can be modelled with K = K∞(1 − exp(−h¯ εp/K∞)), thus σy = σy0 + K∞(1 − exp(−h¯ εp/K∞)) and H = dσy d¯ εp = h exp(−h¯ εp/K∞)

• 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 f(σ, α) =

• 3

2(s − α) : (s − α) − σy0 The Melan (1938) and Prager (1955) hardening rule ˙ α = c ˙ κ = c˙ εp The Ziegler (1955) hardening rule ˙ α = ˙ λ¯ c(σ − α) Non-linear Armstrong-Frederick (1966) model ˙ α = h 2 3 ˙ εp − α α∞ ˙ ¯ εp

• R. Kouhia (Tampere University, Structural Mechanics)

Introduction to materials modelling November 13, 2019 9 / 9