6.8 Isochoric Plastic Deformation

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In the previous derivation, the volume-preserving aspect of plastic deformation (Equation

(6.49)) has not been taken into account (Simo and Miehe 1992). Indeed, J p = 1 implies

detCp−1 = 1 (6.257)

and accordingly,

˙

detCp−1 = 0 (6.258)

or

∂ detCp−1

∂Cp−1 :

˙

Cp−1 = 0. (6.259)

Using Equation (1.509) for the derivative of the third invariant of a matrix, this yields

˙

Cp−1 : Cp = 0 (6.260)

which does not agree with the assumption in Equation (6.124):

TR(

˙

Cp−1

) = ˙

Cp−1 : C = 0. (6.261)

Accordingly, it looks as if Equation (6.177) does not hold and Equation (6.179) yields

DEVn+1Cp−1

n+1 and not Cp−1

n+1. However, we know that (Equation (6.48))

Cp−1 = DEVCp−1 + 13

TR(Cp−1

)C

−1 (6.262)

FINITE STRAIN ELASTOPLASTICITY 301

which implies that the knowledge of TRn+1(Cp−1

n+1) suffices to determine Cp−1

n+1. Defining

the invariants of Cp−1

n+1 by

J1Cp−1 := Cp−1 : C = trbe (6.263)

J2Cp−1 := (Cp−1 · C · Cp−1

) : C = trbe2 (6.264)

J3Cp−1 := (Cp−1 · C · Cp−1 · C · Cp−1

) : C = trbe3 (6.265)

one arrives at, Equations (4.304) to (4.306)

I1Cp−1 = J1Cp−1 = I1be (6.266)

I2Cp−1 = 12

(J 2

1Cp−1 − J2Cp−1 ) = I2be (6.267)

I3Cp−1 = DETCp−1 = 16

(2J3Cp−1 + J 3

1Cp−1 − 3J1Cp−1J2Cp−1 ) = I3be . (6.268)

Since

Cp−1 = F

−1 · be · F

−T (6.269)

one finds

detCp−1 = 1 ⇔ det be = DETCp−1 = J 2. (6.270)

Let us, for the simplicity of notation, denote Cp−1 by A in what follows. The eigenvalues

satisfy the characteristic equation:

_3

A

− I1A_2

A

+ I2A_A − I3A = 0. (6.271)

The same applies to the eigenvalues and invariants of DEVA:

_3

DEVA

+ I2DEVA_DEVA − I3DEVA = 0 (6.272)

since

I1A = TR(DEVA) = 0. (6.273)

The eigenvalues of A and DEVA are related by

_DEVA = _A − 13

I1A. (6.274)

Accordingly, Equation (6.272) reduces to

(_A − 13

I1A)3 + I2DEVA(_A − 13

I1A) − I3A = 0. (6.275)

Expanding Equation (6.275) and substituting Equation (6.272) yields

I3A − I2A_A + 13

_AI 2

1A

− (13

I1A)3 + I2DEVA_A − (13

I1A)I2DEVA − I3DEVA = 0.

(6.276)

302 FINITE STRAIN ELASTOPLASTICITY

This equation contains the unknowns I1A, I2A, I3A and _A and the known quantities

I2DEVA and I3DEVA. Ultimately, we are looking for I1A. From Equation (6.270), we know

that I3A = J 2. To eliminate I2A, the following equation is used:

TR[(DEVA)2] = TR(A2) − 13

[TR(A)]2 (6.277)

which can be obtained by simple expansion. Accordingly,

J2DEVA = J2A − 13

J 2

1A (6.278)

or, using Equations (6.266) to (6.268),

I2A = 13

I 2

1A

+ I2DEVA. (6.279)

Substitution of Equation (6.279) into Equation (6.276) yields

(13

I1A)3 + (13

I1A)I2DEVA + (I3DEVA − J 2) = 0. (6.280)

This is a cubic equation in I1A = TRCp−1, which can be solved explicitly (Abramowitz

and Stegun 1972).

The condition in Equation (6.261) was actually also used to determine μn+1 (see

Equations (6.149)–(6.152)). Since at this point TRCp−1 is not necessarily constant in time,

the result of Equation (6.280) allows for an update of μn+1 and an iterative procedure

ensues.

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