Crystal Plasticity Finite Element Method

The elastic-plastic deformation of crystalline multiphase aggregates depends on the direction of loading, i.e. crystals are mechanically anisotropic.
The directionality or orientation dependence of the mechanical response of crystals under load is due to the anisotropy of the elastic tensor and
to the orientation dependence of the activation of certain crystallographic deformation mechanisms (dislocations, twins, martensite). 
A consequence of crystalline anisotropy is that the associated mechanical phenomena such as shape change, crystallographic texture
evolution, strength, strain hardening, deformation-induced surface roughening, damage, and fracture are also orientation dependent. This is not a
trivial statement since it implies that mechanical parameters of crystalline matter are tensor quantities.

An example is the uniaxial stress-strain curve, which is the most important mechanical measure in structural materials design. The statement above means that such flow curves represent an incomplete measure since they reduce a 6-dimensional yield surface and its change upon loading to a 1-dimensional (scalar) yield curve.
Another consequence of this statement is that crystallographic texture and its evolution during forming is a quantity that is inherently connected with plasticity theory. The texture describes the orientation distribution of a crystalline portion of matter. It can, therefore, be used to describe the integral
anisotropy of polycrystals in terms of the individual tensorial behavior of each grain and the orientation dependent boundary conditions among the crystals. 
Formally, the connection between crystallographic shear and texture evolution becomes also clear from the fact that any deformation gradient can be expressed as the product of its skew-symmetric portion, which represents a pure rotation leading to texture changes if not exactly matched by the rotation implied by plastic shear,  and a symmetric tensor that is a measure of pure stretching. This means that plastic shear creates, as a rule both shape changes and orientation changes, except for certain highly symmetric shears.  Hence, a theory of the mechanical properties of crystals must include first, the crystallographic and anisotropic nature of those mechanisms that create shear and second, the orientation(s) of the crystal(s) studied relative to the applied boundary conditions (e.g. loading axis, rolling plane).

Early approaches to describe anisotropic plasticity under simple boundary conditions have considered these aspects, such as for instance the Sachs, Taylor, Bishop-Hill, or Kröner formulations. 
However, these approaches were neither designed for considering explicitly the complex mechanical interactions among the crystals in a polycrystal  nor for responding to complex internal or external boundary conditions.
Instead, they are built on certain assumptions of strain or stress homogeneity to cope with the intricate interactions within a polycrystal. For that reason variational methods in the form of finite element approximations have gained enormous momentum in the field of crystal plasticity.  These methods, which are referred to as crystal plasticity finite element (CPFE) models, are based on the variational solution of the equilibrium of the forces and the compatibility of the displacements using a weak form of the principle of virtual work in a given finite

volume element. The entire sample volume under consideration is discretized into such elements.  The essential step which renders the deformation kinematics of this approach a crystal plasticity formulation is the fact that the velocity gradient is written in dyadic form. This reflects the tensorial crystallographic nature of the underlying defects that lead to shear and consequently,  to both shape changes (symmetric part) and lattice rotations
(skew-symmetric part).  This means that the CPFE method has evolved as an attempt to employ some of the extensive knowledge gained from experimental and theoretical studies of single crystal deformation and dislocations to inform the further development of continuum field theories of deformation. The
general framework supplied by variational crystal plasticity formulations provides an attractive vehicle for developing a comprehensive theory of plasticity that incorporates existing knowledge of the physics of deformation processes with the conceptual and computational tools of continuum mechanics to develop advanced and physically based design tools for engineering applications. 

One main advantage of CPFE models lies in their capability of solving crystal mechanical problems under complicated internal and/or external boundary conditions. This aspect is not a mere computational advantage but it is an inherent part of the physics of crystal mechanics since it enables one to tackle those boundary conditions that are imposed by inter- and intra-grain micro-mechanical interactions. 

However, the success of CPFE methods is not alone built on their efficiency in dealing with complicated boundary conditions. They also offer high flexibility with respect to including various constitutive formulations for the flow and hardening behavior at the elementary shear system level. The corresponding constitutive flow laws that were suggested during the last decades have gradually developed from empirical viscoplastic formulations into physics-based multiscale internal-variable models of plasticity including a variety of size-dependent and interface mechanical effects. 

In this context it should be emphasized that the finite element method is in itself not the actual model but the variational solver for the underlying constitutive equations mapping the anisotropy of elastic-plastic shears associated with the various types of lattice defects.