6.2 Deriving the Flow Rule

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6.2.1 Arguments of the free-energy function and yield condition

Concentrating on mechanical applications, we start from a general energy function of

mechanical grade 1(cf Equation (1.377))

_ = _(F, Fp,X). (6.11)

Objectivity in the spatial configuration requires (cf Chapter 1)

_ = _(C, Fp,X). (6.12)

Now, in addition, invariance under arbitrary rigid motions in the intermediate configuration

is postulated. This is only satisfied if _ is a function of the inner product of any two vectors

in the intermediate configuration:

_ = _(C, FpT · Fp,X) (6.13)

= _(C,Cp,X) (6.14)

or, dropping X for convenience,

_ = _(C,Cp). (6.15)

In the theory of plasticity, additional internal variables are frequently defined, which we

will denote by A in their kinematic form. Accordingly,

_ = _(C,Cp,A) (6.16)

and

˙_

= ∂_

∂C

: ˙C + ∂_

∂Cp : ˙C

p + ∂_

∂A

: ˙A. (6.17)

276 FINITE STRAIN ELASTOPLASTICITY

In Equation (6.16), Cp and A represent the time–history dependence of plastic deformation.

Substitution of Equation (6.17) into the Clausius–Duhem inequality yields

1

θ

_−∂_

∂C

+ 1

2

S_ : ˙C − 1

θ

_ ∂_

∂Cp : ˙C

p + ∂_

∂A

: ˙A_ − ρ0

θ

˙ θη − 1

θ2Qθ · ∇0θ ≥ 0 (6.18)

where Qθ is the heat flux. Assuming similar relationships as in Equation (6.16) for η and

Qθ , Equation (6.18) is satisfied if

S = 2

∂_

∂C

(C,Cp,A) (6.19)

η = 0 (6.20)

Qθ = 0 (6.21)

− ∂_

∂Cp : ˙C

p − ∂_

∂A

: ˙A ≥ 0. (6.22)

Equation (6.19) is the classical expression for the second Piola–Kirchhoff stress S. Equations

(6.20) and (6.21) result from the fact that no temperature dependence is assumed.

The crucial equation left to be satisfied is the dissipation inequality (Equation (6.22)). It

suggests the definition of the dynamic form Q of the internal variables by

Q := −∂_

∂A

:= −h(A) (6.23)

reducing Equation (6.22) to

− ∂_

∂Cp : ˙C

p +Q : ˙A ≥ 0. (6.24)

The main goal is to derive expressions for the evolution of Cp and A, that is, expressions

for ˙C

p and ˙ A.

From the previous chapter, we know that an additional equation in the form of a yield

condition is required to describe plasticity. The yield condition is usually written in terms

of the stress S and the dynamic internal variables Q:

_(S,Q) ≤ 0. (6.25)

Because of Equations (6.19) and (6.23), this is equivalent to

_(C,Cp,Q) ≤ 0. (6.26)

6.2.2 Principle of maximum plastic dissipation

The previous section has shown that the thermodynamic state is characterized by the variables

{C,Cp,A}. Now, an uncoupled free energy in the internal state variables A is assumed

of the form

_ = _(C,Cp) + _(A). (6.27)

FINITE STRAIN ELASTOPLASTICITY 277

The plastic dissipation Dp amounts to Equation (6.24):

Dp(C,Cp,A; ˙C p; ˙A) := − ∂_

∂Cp : ˙C

p − ∂_

∂A

: ˙A. (6.28)

To derive the flow rule, the principle of maximum dissipation is invoked. It states that, for

fixed {Cp,A}, the field C will take such a value that for all other fields C satisfying the

yield condition, the plastic dissipation is smaller. Hence, defining the cone

Kφ := {˜C ∈ R6|_(˜C,Cp,Q) ≤ 0} (6.29)

of all the states satisfying the yield condition, we have

Dp(C,Cp,A; ˙C

p

, ˙A) ≥ Dp(˜C ,Cp,A; ˙C

p

, ˙A) ∀˜C ∈ Kφ (6.30)

or, by use of Equation (6.28),

−∂_(C,Cp)

∂Cp : ˙C

p ≥ −∂_(˜C,Cp)

∂Cp : ˙C

p

, ∀˜C ∈ Kφ. (6.31)

Accordingly, C satisfies

C = arg

_

max

˜C

∈Kφ

_

−∂_(˜C ,Cp)

∂Cp : ˙C

p__

(6.32)

or

C = arg

_

min

˜C

∈Kφ

_∂_(˜C ,Cp)

∂Cp : ˙C

p__

(6.33)

where “arg” denotes the argument of the function. This is a constrained minimization

problem amenable to mathematical analysis. Indeed, one can prove (Luenberger 1989) that

the solution of Equation (6.33) is equivalent to the minimization of the functional

Lp := ∂_(C,Cp)

∂Cp : ˙C

p + ˙γ_(C,Cp,Q) (6.34)

subject to

γ˙ ≥ 0 (6.35)

γ˙_(C,Cp,Q) = 0. (6.36)

The minimization of Lp is equivalent to

∂Lp

∂C

= 0 (6.37)

or

∂2_(C,Cp)

∂C∂Cp : ˙C

p = −˙ γ

∂_(C,Cp,Q)

∂C

. (6.38)

Equation (6.38) is the flow rule! The principle of maximum plastic dissipation leads to a

flow rule, which is a function of the hyperelastic free-energy potential and the yield surface

only. Accordingly, as soon as the hyperelastic free energy and yield surface are known, the

flow rule is uniquely defined.

278 FINITE STRAIN ELASTOPLASTICITY

6.2.3 Uncoupled volumetric/deviatoric response

The elastoplastic theory can be further simplified if one assumes a completely uncoupled

volumetric/deviatoric response throughout the entire range of deformation. It is obtained

through a multiplicative decomposition of the deformation gradient:

F = J 1/3F (6.39)

and accordingly

det(F) = 1. (6.40)

The associated right Cauchy–Green tensor takes the form

C = F

T · F = J

−2/3C. (6.41)

By using the chain rule and taking into account that

∂J

∂C

= J

2

C

−1 (6.42)

one obtains

∂C

∂C

= J

−2/3 _I − 13

C ⊗ C

−1_ (6.43)

and in general

∂(·)

∂C

= J

−2/3DEV         ∂(·)

∂C

 (6.44)

where

DEV[·] := (·) − 13

[C : (·)]C

−1 (6.45)

is the pull-back of the deviator in spatial coordinates. For example, we know that Cp−1 is

the pull-back of be:

Cp−1 = F

−1 · be · F

−T. (6.46)

Accordingly,

DEVCp−1 = F

−1 · devbe · F

−T, (6.47)

which leads to

DEVCp−1 = F

−1 · _be − 13

(be : g)_ · F

−T

= Cp−1 − 13

(be : g)C

−1

= Cp−1 − 13

(Cp−1 : C)C

−1. (6.48)

For metals, the plastic deformation is considered to be isochoric, and consequently the

volumetric response is purely elastic. Hence,

J = J e, Jp = 1. (6.49)

FINITE STRAIN ELASTOPLASTICITY 279

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