Non-deterministic Calibration
173
3.1 Concepts
For the remainder of this chapter, consideration is given to CP as a model for the
homogenization of the underlying motion of dislocations on each slip system. As
such, the primary parts of a CP model are the kinematics of slip and a constitutive
model relating the external forces to slip rates through resolved shear stress. The
mathematical model of the kinematics of finite deformation relates the original
reference configuration of a continuum to a current configuration that is obtained
through the application of external loads and displacements. The total deformation
gradient, F , relates the reference and current configurations directly.
The decomposition of F into its elastic and plastic parts can be thought of
as a multiplicative transformation, Eq. 3. Therein, F e represents the reversible
component of deformation, while F p represents the deformation that remains upon
removal of the external forces and displacements. If irreversible deformation is
present, an intermediate configuration is obtained upon removal of external forces
and displacements. This intermediate configuration is related to the reference
configuration by F p . Furthermore, the lattice orientation remains unchanged in the
intermediate configuration, resulting in a stress-free configuration. Effectively, this
relies on an assumption that any dislocations formed must be passed beyond its local
neighborhood. The intermediate and current configurations are related by F e , where
lattice distortions lead to material stresses. This concept that the stress is induced
by the elastic portion of the deformation is fundamental both to the development
of the following constitutive equations and to the calibration method presented in
Sect. 4.4.
F = F e F p .
(2)
However, this decomposition does not yet have information regarding the
underlying crystallography essential for CP modeling. To capture crystallographic
kinematics, the plastic velocity gradient, L p , is defined as a tensor that transforms
the plastic deformation gradient, F p , to its time rate of change:
˙
F p = L p F p .
(3)
Since the consideration here is limited to dislocation slip as the only plastic
deformation process, L p is formulated as the sum of rates of slip on each system,
˙
γ α , along with the slip direction for each system, m α , and its corresponding plane
normal, n α :
L p =
n
α=1
˙
γ
α m
α
⊗ n
α .
(4)
It is with this definition that the crystallographic kinematics are modeled.
173
3.1 Concepts
For the remainder of this chapter, consideration is given to CP as a model for the
homogenization of the underlying motion of dislocations on each slip system. As
such, the primary parts of a CP model are the kinematics of slip and a constitutive
model relating the external forces to slip rates through resolved shear stress. The
mathematical model of the kinematics of finite deformation relates the original
reference configuration of a continuum to a current configuration that is obtained
through the application of external loads and displacements. The total deformation
gradient, F , relates the reference and current configurations directly.
The decomposition of F into its elastic and plastic parts can be thought of
as a multiplicative transformation, Eq. 3. Therein, F e represents the reversible
component of deformation, while F p represents the deformation that remains upon
removal of the external forces and displacements. If irreversible deformation is
present, an intermediate configuration is obtained upon removal of external forces
and displacements. This intermediate configuration is related to the reference
configuration by F p . Furthermore, the lattice orientation remains unchanged in the
intermediate configuration, resulting in a stress-free configuration. Effectively, this
relies on an assumption that any dislocations formed must be passed beyond its local
neighborhood. The intermediate and current configurations are related by F e , where
lattice distortions lead to material stresses. This concept that the stress is induced
by the elastic portion of the deformation is fundamental both to the development
of the following constitutive equations and to the calibration method presented in
Sect. 4.4.
F = F e F p .
(2)
However, this decomposition does not yet have information regarding the
underlying crystallography essential for CP modeling. To capture crystallographic
kinematics, the plastic velocity gradient, L p , is defined as a tensor that transforms
the plastic deformation gradient, F p , to its time rate of change:
˙
F p = L p F p .
(3)
Since the consideration here is limited to dislocation slip as the only plastic
deformation process, L p is formulated as the sum of rates of slip on each system,
˙
γ α , along with the slip direction for each system, m α , and its corresponding plane
normal, n α :
L p =
n
α=1
˙
γ
α m
α
⊗ n
α .
(4)
It is with this definition that the crystallographic kinematics are modeled.