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C. A. Bronkhorst et al.
Fig. 6 Schematic of the kinematical structure used in the current single crystal model
Within a finite deformation kinematical framework, the deformation gradient is
decomposed as
F = F
e F
p
(28)
The second Piola-Kirchoff stress is given as a function of the energy conjugate
Green-Lagrange elastic strain and the thermal expansion term
T
∗
= C
E
e
− A (θ − θ 0 )
,
(29)
where in relation to the Cauchy stress, T ∗ in the deformed configuration
T
∗
=
det F
e
F
e −1 TF
e −T .
(30)
The elastic stiffness tensor as a function of temperature θ is given as a linear
expression with temperature
C ij kl = C ij kl 0 + m C ij kl θ.
(31)
The elastic Green-Lagrange strain is given by
E
e
=
1
2
F
e T F
e
− 1
.
(32)