6.1 Multiplicative Decomposition of the Deformation Gradient

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The multiplicative decomposition states that the deformation in an elastoplastic material

consists of a purely plastic part due to dislocation motion, leading to an intermediate

stress-free configuration, followed by a purely elastic deformation rotating and distorting

the crystal lattice (Figures 6.1 and 6.2).

F = Fe · Fp. (6.1)

The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt

2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-8

274 FINITE STRAIN ELASTOPLASTICITY

X

dX

x

dx

dx

∗

F

Fp

Fe

V0 V

Figure 6.1 Multiplicative decomposition of the deformation gradient

F

Fp

Fe

l1

l1

l1

l2

l2

l2

Figure 6.2 Deformation of the crystal lattice

The total left and right Cauchy–Green tensors satisfy (Chapter 1)

b = F · FT (6.2)

C = FT · F (6.3)

whereas the elastic left Cauchy–Green tensor be and the plastic right Cauchy–Green tensor

Cp are defined as follows:

be = Fe · FeT (6.4)

Cp = FpT · Fp. (6.5)

The inverse left elastic Cauchy–Green tensor or elastic Finger tensor be−1 and the right

plastic Cauchy–Green tensor Cp are a push-forward/pull-back pair:

Cp = FpT · Fp = FT · Fe−T · Fe−1 · F = FT · be−1 · F. (6.6)

FINITE STRAIN ELASTOPLASTICITY 275

Accordingly, Cp is the pull-back of be−1 and be−1 is the push-forward of Cp. Recall that

C and the spatial metric tensor g are push-forward/pull-back pairs, as well as C

−1 and

g

−1. From Figure 6.1, the following relationships prevail:

dx = F · dX (6.7)

dx

∗ = Fp · dX = Fe−1 · dx (6.8)

and consequently,

ds2 = CKL dXK dXL (6.9)

ds

∗2 = C

p

KL dXK dXL = b

e−1

kl dxk dxl . (6.10)

Hence, the plastic right Cauchy–Green tensor plays the role of a metric tensor in the

intermediate configuration with respect to the material frame of reference.

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