6.3 Isotropic Hyperelasticity with a von Mises–type Yield Surface

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6.3.1 Uncoupled isotropic hyperelastic model

Next, the attention is focused on an isotropic hyperelastic model of the form (Simo 1988a)

_(g, be−1, F) = 12

μ(J

−2/3I1be − 3) + U(J) (6.50)

where Ibe is the first invariant of the elastic left Cauchy–Green tensor. The first term on

the right-hand side of Equation (6.50) is isochoric, the second is volumetric. The choice of

U(J) is not unique. Here, we will take

U(J) = 12

K _12

(J 2 − 1) − ln J _ (6.51)

which satisfies the asymptotic requirements

lim

J→+∞

U(J) = +∞ (6.52)

lim

J→0

U(J) = +∞ (6.53)

discussed in Chapter 4. The parameter K is the bulk modulus.

For C = G, Equation (6.50) leads to the classical isotropic Hooke law. To prove this,

the derivative with respect to C is taken. Using relationships derived in Section 4.4 and

noting that I1b = I1C = I1, one obtains

∂_

∂CKL

= 12

μJ

−2/3 _−13

I1C

−1KL + GKL_ + 14

K _J 2 − 1_C

−1KL (6.54)

and

∂2_

∂CKL∂CMN

= −16

μJ

−2/3C

−1MN _GKL − 13I1C

−1KL_

+ 12

μJ

−2/3 _−13

GMNC

−1KL + 16I1(C

−1KM

C

−1LN + C

−1KN

C

−1LM

)_

+ 14

KJ2C

−1MN

C

−1KL − 18

K(J2 − 1)(C

−1KM

C

−1LN + C

−1KN

C

−1LM

). (6.55)

For C = G, the first and last term drop out and one obtains

4

∂2_

∂CKL∂CMN

 

C=G

= _K − 23

μ_GKLGMN + μ(GKMGLN + GKNGLM) (6.56)

where K − 23

μ = λ. This is the classical Hooke law for linear isotropic materials.

Using the appropriate push-forward/pull-back pairs, Equation (6.50) can be reformulated

as

_(C,Cp) = 12

μ(J

−2/3TRCp−1 − 3) + U(J) (6.57)

280 FINITE STRAIN ELASTOPLASTICITY

where

TR[·] = [·] : C (6.58)

is the pull-back of the trace operator in spatial coordinates:

tr(be) = be : g = Cp : C =: TR(Cp−1

). (6.59)

Equation (6.57) can also be written as

_(C,Cp) = 12

μ(C : Cp−1 − 3) + U(J). (6.60)

Applying Equations (6.19),(6.42) and (6.44), one obtains for the second Piola–Kirchhoff

stress

S = pJC

−1 + μDEV(C

p−1

) (6.61)

where

p = dU

dJ

= 12

K

            J 2 − 1

J

 (6.62)

and

C

p−1

:= J

−2/3Cp−1

. (6.63)

Accordingly (Equation (6.38)),

∂2_(C,Cp)

∂C∂Cp

= 1

2

∂S

∂Cp

= μ

2

DEV

_∂C

p−1

∂Cp

_

(6.64)

and

∂2_(C,Cp)

∂C∂Cp : ˙C

p = −μJ

−2/3DEV

_∂Cp−1

∂t

_

. (6.65)

6.3.2 Yield surface and derivation of the flow rule

One of the frequently used forms of the yield surface is due to von Mises:

_(S,C,Q) = DEVS + _23

q1 (6.66)

where q1 is a scalar internal plastic variable satisfying q1 = −h1(α1). The variables q1 and

α1 are spatial quantities. Since C

−1 is a symmetric matrix, one finds

DEV(C

−1) = C

−1 − 13

(C : C

−1)C

−1 = 0 (6.67)

leading to (Equation (6.61))

DEVS = μDEV(C

p−1

). (6.68)

FINITE STRAIN ELASTOPLASTICITY 281

Hence, Equation (6.66) can be rewritten as

_(C,Cp,Q) = μDEV(C

p−1

) − _23

h1(α1). (6.69)

The unit of _ is stress, and in the present chapter we will assume that this is the Kirchhoff

stress. Since (S, τ ) and (C, g) are push-forward/pull-back pairs, one can write

DEVS = _(DEVS)IJ (DEVS)KLCIKCJL (6.70)

= _(devτ)ij (devτ)klgikgjl . (6.71)

For the flow rule (Equation (6.38)), we need the derivative of _ with respect to C. One

can write

∂_

∂C

= ∂DEVS

∂C

(6.72)

= 1

2DEVS

∂

∂C

DEVS2. (6.73)

Furthermore (Equation (6.44)),

∂

∂C

DEVS2 = J

−2/3DEV_ ∂

∂C

DEVS2_

. (6.74)

Now (Equation (6.68)),

∂

∂C

DEVS2 = μ2 ∂

∂C

DEVC

p−12 (6.75)

= μ2 ∂

∂C

_(DEVCp−1

)IJ (DEVCp−1

)KLCIKCJL_ (6.76)

= 2μ2  ∂

∂C

(DEVCp−1

)IJ

(DEVCp−1

)KLCIKCJL

+ 2μ2(DEVCp−1

)IJ (DEVCp−1

)KLCJL. (6.77)

Since

∂

∂C

(DEVCp−1

) = ∂

∂C

_Cp−1 − 13

(C : Cp−1

)C

−1_ (6.78)

= ∂

∂C

_Cp−1 − 13

(C : Cp−1

)C

−1_ (6.79)

= −13

(I : Cp−1

)C

−1 + 13

(C : Cp−1

)I

C

−1 (6.80)

= −13

C

−1 ⊗ Cp−1 + 13

(Cp−1 : C)I

C

−1 (6.81)

where

_I

C

−1_IJKL

:= 12

_(C

−1

)IK(C

−1

)JL + (C

−1

)IL(C

−1

)JK_ (6.82)

282 FINITE STRAIN ELASTOPLASTICITY

and

I := II (6.83)

one obtains

             ∂

∂C

DEVS2AB

= 23

μ2[

(i)

_ __ _ (Cp−1 : C)I

C

−1 −

(ii)

_ __ _

C

−1 ⊗ Cp−1]IJAB(DEVCp−1

)KLCIKCJL

+ 2μ2(DEVCp−1

)AJ (DEVCp−1

)BLCJL

_ (_ii_i) _

. (6.84)

Substituting Equation (6.84) into

∂_

∂C

= J

−2/3

2DEVSDEV_ ∂

∂C

DEVS2_ (6.85)

one notices that ∂_/∂C consists of three additive terms, each of which will be treated

separately:

1.

DEV    J

−2/3μ2

3DEVS(Cp−1 : C)IIJAB

C

−1 CIKCJL(DEVCp−1

)KL

= DEV J

−4/3μ2

3DEVS

12

(Cp−1 : C)(δA

KδB

L

+ δB

KδA

L)(DEVCp−1

)KL (6.86)

= DEV J

−2/3μ

3DEVSTRCp−1

(DEVS)AB (6.87)

= μN (6.88)

where

μ := 13

μJ

−2/3TRCp−1 (6.89)

N := DEVS

DEVS. (6.90)

2.

− DEV J

−2/3

2DEVS

23

μ2 _C

−1 ⊗ Cp−1_IJAB

(DEVCp−1

)KLCIKCJL

 

= −DEV          μ2J

−4/3

DEVS(C

−1)IJ (Cp−1

)ABCIKCJL(DEVCp−1

)KL (6.91)

= −DEV          μ2J

−4/3

DEVS(Cp−1

)ABCKL(DEVCp−1

)KL = 0 (6.92)

FINITE STRAIN ELASTOPLASTICITY 283

since

C : (DEVCp−1

) = 0. (6.93)

To obtain Equation (6.93), recall that C

−1 : C = 3 since C is symmetric.

3.

DEV    J

−2/3

DEVSμ2(DEVCp−1

)AJ (DEVCp−1

)BLCJL

 

= DEV μ2

DEVS(DEVC

p−1

)AJ (DEVC

p−1

)BLCJL

 (6.94)

= DEV (DEVS)AJ (DEVS)BLCJL

DEVS

 (6.95)

= DEVSDEV(N2). (6.96)

Accordingly, Equation (6.38) yields

− ˙ γ

∂_

∂C

= −˙ γμ

            N +

DEVS

μ

DEV(N2)

 

. (6.97)

Equating Equations (6.65) and (6.97) yields the flow rule:

−J

−2/3μDEV

_∂Cp−1

∂t

_

= 2γ˙μ

            N +

DEVS

μ

DEV(N2)

 

. (6.98)

The second term on the right-hand side is much smaller than the first one and is usually

dropped.

For infinitesimal strains and rotations, Equation (6.98) reduces to Equation (5.95). Indeed,

substituting μ yields (neglecting the term with N2)

DEV( ˙C

p−1

) = −2γ˙

3

TR(Cp−1

)N. (6.99)

In the infinitesimal theory, one can write

Cp ≈ I + 2_p (6.100)

Cp−1 ≈ I − 2_p (6.101)

˙C

p−1 ≈ −2˙_p. (6.102)

The field _p is deviatoric, hence,

DEV( ˙C

p−1

) ≈ −2˙_ p (6.103)

TR(Cp−1

) ≈ 3 (6.104)

and Equation (6.99) reduces to

˙_

p = ˙γ n (6.105)

which coincides with Equation (5.95).

284 FINITE STRAIN ELASTOPLASTICITY

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