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Remarks on plastic material behavior

1) Yield surfaces (a surface in the stress space representing the condition/criterion whether the solid responds elastically or plastically to the applied load)

The von Mises and Tresca yield conditions are represented by the following yield surfaces in the 1 stress space (view long the (1,1,1 ) direction). 3 von Mises condition

σ III

σ I σ II

Tresca condition

2) The strain can be decomposed into elastic and plastic parts. In incremental form,

+= ddd εεε PE

The elastic part is related to stress via the usual linear elastic equations. The plastic part of strain in incremental form can be written as

⎧3 ' dε P ⎪ σ ij σσσ eYe ≥= 0d,if, 2 σ e P ⎪ dε ij = ⎨ ⎪ ,0 otherwise ⎪ ⎩

J 2 - flow theory:

1 '' ∂J 2 ' J 2 = σσ ijij , ' = σ ij 2 ∂σ ij

Based on the above relations, the plastic strain can be rewritten in term of J 2 as

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⎧3 ∂J dε 2 P loading, ⎪ ' 2 ∂σ σ eij P ⎪ dε ij = ⎨ ⎪ ,0 otherwise ⎪ ⎩

3) Normality rule

Yield stress:

3 = '' σσσσ ⇒= f ()' = σσ e 2 Yijij Yij

' where f (σ ij ) represents a general yield surface. 3 Differentiate the equation = σσσ 2'' gives 2 eijij ' ⎛ 3 ⎞ 3 ∂σ 3 σ ij '' 2 '' e d⎜ ijij ⎟ = d()σσσ e ⇒ = dd σσσσ eeijij ⇒ ' = ⎝ 2 ⎠ 2 ∂σ ij 2 σ e

' The plastic strain for a general yield condition f (σ ij ) can be written as

⎧ ∂f ε f εσ ≥= 0d,if,d ⎪ ' P Y P ∂σ ij P ⎪ dε ij = ⎨ ⎪ ,0 otherwise ⎪ ⎩ which also indicates dε P is normal to the yield surfaces.

4) Convexity rule

The yield surface must be convex for plastically stable solids. One cannot have a nonconvex yield surface such as the figure below.

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σ III

σ I σ II

Nonconvex

The convexity rule can be related to principle of maximum plastic resistance.

5) Limitations of von Mises law σ = σ Ye

Cyclic loading: σ

t

The σ = ε curve under a cyclic loading is illustrated below.

ε

σ

3 The isotropic hardening as implied by von Mises condition ( '' = σσσ ) is generally not 2 Yijij valid in case of cyclic loading. The so-called Bauschinger effect indicates that, if the solid first undergoes plastic deformation in tension and then loaded in compression, the yield stress in

3 EN0175 11 / 02 / 06 compression would generally become smaller. Alternatively, deforming the material in compression also tends to soften the material in tension. To account for this, kinematic hardening laws have been proposed to allow the yield surface to translate, without changing its shape, in stress space.

σ III

αv

σ I σ II

To account for the fact that yield surface translate without changing its shape in cyclic loading, the von Mises yield condition can be modified as

3 f ()' ()()' ' =−−= σασασσ ij 2 Yijijijij

6) J 2 - deformation theory

If there is no unloading and the stresses increase proportionally to each other. The J 2 - flow

theory can be integrated to give the so-called J 2 - deformation theory of plasticity.

ε

σ

'' 0 P ∂f ijij dd εεσσ ij =⇒= ' ε P ∂σ ij

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1+ν ν ε E = σ − σ ij E ij E kk

7) Single crystal plasticity (1970s-)

Modeling plastic deformation by considering slip along discrete crystal planes and orientations. e.g. Cu: FCC crystal, 12 slip systems: 4 (111) planes time 3 [110] directions.

Viscoelasticity

σ

σ

σ 0

t T

For a constant loading σ 0 acting from time 0 to T , the strain responses of elastic materials and viscoelastic materials are illustrated in the figures below.

5 EN0175 11 / 02 / 06 ε ε creep σ 0 E relaxation t t T T

elastic viscoelastic

Mathematical tools E σ Spring: E εεσ =⇒= E η σ Dash pot: εεησ && =⇒= η

Basic viscoelastic models:

Maxwell model: E η σ

σ& σ E εεε &&& η +=+= E η

The strain response of Maxwell model to a constant loading σ 0 acting from time 0 to T is ε

t T

Kevin model:

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E σ η

σ = σ E +ση = Eε +ηε&

The strain response of Kevin model to a constant loading σ 0 acting from time 0 to T is ε

t T

Standard linear solid model:

E2 E1 σ η

The strain response of standard linear solid model to a constant loading σ 0 acting from time 0 to

T is ε

t T

More sophisticated models (e.g., for biological systems)

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E1 E2 En L σ

η1 η2 ηn

η E1 1

E2 η2 σ L Lη En n

General mathematical structure of viscoelstic models: ⎛ ∂ ∂2 ∂n ⎞ ⎛ ∂ ∂2 ∂n ⎞ ⎜ + pp + p ++ p ⎟σ ⎜ += qq + q ++ q ⎟ Eε ⎜ 10 2 2 L n n ⎟ ⎜ 10 2 2 L n n ⎟() ⎝ ∂t ∂t ∂t ⎠ ⎝ ∂t ∂t ∂t ⎠

σ = (EQP ε )

∂ 21 Sometimes fractional derivatives (like ) have also been used to describe the stress-strain ∂t 21 relations of complex viscoelastic responses.

Generalize to 3D,

' ' Linear elastic: σ = Eε is generalized to ij = 2μεσ ij , σ kk = 3Kε kk

' ' Viscoelastic: σ = (EQP ε ) can be likewise generalized to ij = QP (2μεσ ij ) ,

σ kk = ()3KQP ε kk

Additional remarks on viscoelasticity:

σ

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1) Creep modulus: the strain response to a unit constant stress. σ ε creep 1 modulus

t t Relaxation modulus: the stress response to a unit constant strain. ε σ relaxation 1 modulus

t t

2) Storage & loss modulus

σ = σ 0 sinωt

ε = ε 0 sin(ωt −δ ) t

σ Elastic case: σ = σ sinωt , ε = 0 sin = sinωεω tt (no delay for strain response) 0 E 0

Viscoelastic case: ε = ε 0 sin()ωt −δ = ε 0 δ sincos ωt −ε 0 δ cossin ωt

ε cosδ 0 : storage compliance σ 0

ε sin δ 0 : loss compliance σ 0

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