6.4 Extensions

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6.4.1 Kinematic hardening

Frequently, a more generalized form of the yield surface is used, in which the center of

the yield surface can move. This is accomplished by replacing Equation (6.69) by

_(C,Cp,Q) = DEV(μC

p−1 +Q2) − _23

h1(α1). (6.106)

Here, −Q2 represents the moving yield-surface center. Equation (6.68) still applies. Replacing

DEV(S) by DEV(S +Q2) and C

p−1

by C

p−1 +Q2/μ in the previous section, one

obtains for the flow rule

−J

−2/3μDEV

_∂Cp−1

∂t

_

= 2γ˙μ

_

N +

DEV(S +Q2)

μ

DEV(N2)

_

(6.107)

where

μ := 13

J

−2/3TR(μCp−1 +Q2) (6.108)

= μ + 13

J

−2/3TRQ2 (6.109)

N = DEV(μC

p−1 +Q2)

DEV(μC

p−1 +Q2)

= DEV(S +Q2)

DEV(S +Q2)

(6.110)

Q2 = J

−2/3Q2. (6.111)

The left-hand side of Equation (6.98) is not changed since it derives from the potential

function. Equation (6.107) is an evolution equation for Cp−1. The field Q2 is called the

back stress and represents an internal plastic variable for which an evolution equation

is needed as well. Since the fields Q2 and μCp−1 are related, an equation similar to

Equation (6.107) seems plausible:

−J

−2/3DEV_∂Q2

∂t

_ =

_h

eq’

2

3μ

_

2γ˙μ

_

N +

DEV(S +Q2)

μ

DEV(N2)

_

. (6.112)

The factor

1

3μ

h

eq’

2 := 1

3μ

∂h

eq

2

∂α

eq

2

(6.113)

was introduced to assure that Equation (6.112) reduces to its infinitesimal equivalent,

Equation (5.111). Indeed (q2 is deviatoric, )

DEV( ˙Q2) ≈ dev(˙q2) ≈ ˙q2 (6.114)

J ≈ 1 (6.115)

μ ≈ μ + 13

trq2 ≈ μ (6.116)

FINITE STRAIN ELASTOPLASTICITY 285

yielding

−˙q2 = 2

3

∂h

eq

2

∂α

eq

2

γ_________˙n. (6.117)

Here too, one defines

h

eq

2 := _32

h2 (6.118)

α

eq

2 := _23

α2. (6.119)

In the previous chapter, we derived for the infinitesimal theory

α1 = α

eq

2

= _peq (6.120)

and

˙_

peq = _23

γ˙ . (6.121)

These equations will also be used for the finite theory. Accordingly,

h

eq’

2 := ∂h

eq

2

∂_peq . (6.122)

The only curves to be provided by the user are h1(_peq) for isotropic hardening and h

eq

2 (_peq)

for kinematic hardening.

Equation (6.121) can be interpreted as the definition of _peq for finite strains. Combining

Equation (6.121) with the flow rule, Equation (6.99), yields the following kinematic

relationship for ˙_peq:

˙_

peq = _32

DEV( ˙C

p−1

)

TR(Cp−1

)

. (6.123)

Notice that Equations (6.107) and (6.112) determine only the deviatoric part of ˙C

p−1

and ˙Q2. To guarantee a unique definition of ˙Cp−1 and ˙Q2, the following additional constraints

can be defined:

TR( ˙C

p−1

) = 0. (6.124)

TR( ˙Q2) = 0. (6.125)

6.4.2 Viscoplastic behavior

For plastic behavior, it is assumed that the stress tensor cannot exceed the yield surface.

To illustrate this, the yield surface, as given by Equation (6.106), is shown in Figure 6.3

in deviatoric principal stress space (assuming that Q2 and S have the same eigenvectors).

The yield surface can also be written as (cf Equation (6.106))

DEV(S +Q2) = _23

h1(α1) (6.126)

286 FINITE STRAIN ELASTOPLASTICITY

(devτ )1

(devτ )2

(devτ )3

_23

h1(α1)

−FQ2FT

Figure 6.3 von Mises yield surface in deviatoric principal Kirchhoff stress space

which shows that the von Mises surface is indeed a sphere with its center at −Q2 and

radius _23

q1. In the classical plasticity theory, any physical stress state must lie on or

within the yield surface. However, both q1 and Q2 can change because of isotropic and

kinematic hardening respectively. In the viscoplastic theory, the stress state can momentarily

lie outside the yield surface; however, it tends asymptotically to the yield surface as time

goes by. The way in which the yield surface is approached is generally given by a creep

law of the form

τvm = f (˙_ peq). (6.127)

The quantity τvm is the von Mises equivalent stress of the Kirchhoff tensor satisfying

τvm := _32

devτ = _32

√

devτ : devτ . (6.128)

Accordingly, the plastic equality in Equation (6.126) is replaced in the viscoplastic case by

DEV(S +Q2) − _23

h1(_peq) = _23

f (˙_ peq) (6.129)

if DEV(S +Q2) − _23

h1(_peq) > 0, else the material remains elastic. A typical example

of a creep law in the infinitesimal theory is the Norton law

˙_

peq = Aτn

vm (6.130)

which can also be written as

τvm = _˙_peq

A

_(1/n)

. (6.131)

FINITE STRAIN ELASTOPLASTICITY 287

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