3. Flow kinematics

This section presents a detailed discussion of the different distortion measures that form the geometrical backbone of the CPFE framework. The review is given in terms of finite deformation measures as the small distortion case can always be derived from it. The kinematics of isothermal finite deformation describes the process where a body originally in a reference state (or “configuration”), B ⊂ R³, is deformed to the current state, S ⊂ R³, by a combination of externally applied forces and displacements over a period of time, Δt. In this treatment we choose the perfect single crystal as reference state. Other possible choices of a reference state would be the state just before deformation, or in an incremental formulation, the state at any time t. The latter has the disadvantage of a constantly changing reference state while the former might contain an undefined amount of crystal defects.

In crystal plasticity one has to distinguish three coordinate systems. The shape coordinate system is a curvilinear system, based on the physical shape of the body, which deforms congruently with the total changes in shape occurring during deformation. In contrast, the lattice coordinate system has coordinate axes fixed locally parallel to the crystallographic directions, i.e. the nodes of the coordinate network maintain a one-to-one correspondence with crystal lattice points during deformation except where singularities in the atomic array occur due to the defect cores. Therefore, the space defined by the deformed lattice coordinate system is connected to ordinary Euclidean space by functions that depend on the defect content and distribution [235]. The distinction between shape and lattice distortion is critical when internal stresses are to be calculated [236]. Since these stresses arise from internal reaction forces generated when atoms experience a relative displacement from their equilibrium positions, they can only be related to lattice deformations because shape deformations do not necessarily follow the deformation of the underlying crystal lattice. Deformations in the two types of coordinates are coincident only when no motion of crystal defects occurs, which is the usual assumption made for calculating elastic stresses from measurements of the small deformations of a coordinate system deposited on the surface or embedded within a body. Finally, it is often convenient to describe the deformations of the overall shape of a body and its associated lattice in terms of a third reference system, laboratory coordinates, that does not deform with the body. Computations of the deformation of both shape and lattice coordinate systems can be made in terms of their components in laboratory coordinates. This has the practical advantage of providing a reference system that is fixed throughout the deformation of the body.

To cast this in a formal way we represent the positions of neighboring material points relative to an arbitrary origin in the reference configuration by the vector dx. As a result of deformation, this vector is mapped into its image in the current configuration, dy =dx+du, where du is the differential total displacement vector. These vectors are related by the total, or shape, deformation gradient, F:

where I is the second rank identity tensor. The second rank tensor formed from the partial derivatives of u with respect to x is known as the shape (also total) distortion tensor, β and is a perfect dfferential if the deformations that produce it do not introduce any discontinuities, i.e., gaps or cleavages, in the global body. That is, there exists a one-to-one mapping of material points from the current state to the reference state. The Lagrangian and Eulerian (Almansi) finite strain tensors, E and E*, associated with a deformation (defined by the deformation gradient F), respectively, are symmetric tensors defined by:

where the superscript (T and -T) indicates the transpose and its inverse, respectively, and the superscript (-1) indicates the inverse of the tensor. It is useful to note that any deformation gradient, F, can be expressed as the product of a pure rotation, R, and a symmetric tensor that is a measure of pure stretching. Two representations are possible, depending on which operation occurs first:

where the symmetric tensors U and V are, respectively, the right and left stretch tensors. The shape deformation can be decomposed into two components [237, 211, 7, 213, 238] (see also figure 5):

The “elastic” deformation, F_e, is the deformation component due to the reversible response of the lattice to external loads and displacements (as well as rigid-body rotations) while the “plastic” deformation, F_p, is an irreversible permanent deformation that persists when all external forces and displacements that produce the deformation are removed. In this sense, transformation of the reference state by F_p leads to an intermediate configuration which is free from external stresses and which is generally considered to maintain a perfect lattice. However, this state requires that none of the dislocations which produced the permanent shape change reside anymore within the material point neighborhood, but are located at its periphery. In reality, this assumption is typically not fulfilled, such that (balanced) internal stresses remain due to the (homogeneous) presence of dislocations within the neighborhood.

This decomposition dffers from that proposed by Bilby et al. [239] in that their model requires that the stress-free deformation producing the intermediate state does not leave residual deformation in the lattice, hence does not change the thermodynamic state of the material. In their formulation, no residual dislocations are present in the intermediate state regardless of the vanishing of the net Burgers vector. Making this distinction, Bilby, Bullough, and Smith refer to F_p as the ”dislocation deformation” and F_e as ”lattice correspondence functions”. In this case F_e also contains a component of lattice deformation due to sources of internal stress distributed throughout the body.

For example, consider the processes shown in figures 6 and 7 by which an initially perfect crystal undergoes the same change in shape by shear, however involving different contributions from dislocation slip. In figure 6 the crystal lattice is unchanged in the reference and current states, so all the work expended in the process is dissipated as heat and the material remains in the same thermodynamic state before and after the deformation. At the end of the process the external loads can be removed and no lattice deformation remains, F = F_p. In contrast, the crystal undergoes an purely elastic shape change in figure 7. 

In this case there are no dislocations, the lattice is distorted congruently with the external shape of the body, F = F_e, and the external cause for the deformation must be maintained in order to preserve the change in shape; removal of the external boundary conditions causes the body to revert to the reference state. Also the thermodynamic states of the reference and current states are different because of the stored elastic energy due to the lattice deformation.

As for the total shape deformation, F, given in equation (1), the elastic and plastic deformations can each be expressed as sums of the identity tensor and second-rank tensors, β_e and β_p, called the elastic distortion and plastic distortion, respectively. However these quantities need not be integrable derivatives of displacement vectors for the elastic and plastic deformations, since displacements associated with these deformations can be incompatible, i.e., may introduce discontinuities into the body.

When dislocation deformation occurs by slip on two or more systems, the spatio-temporal order in which deformation occurs seems important in the kinematic treatment. For example, figure 8 illustrates that a pure dislocation deformation of a volume element by slip on two slip systems that lead to the same F_p results in a different configuration of the reference volume. In figure 8(a) dislocation deformation initially causes the deformation component F_p21, followed by F_p12 resulting in a state of pure shear. In figure 8(b) the same final state is reached by reversing the order of the dislocation deformation. The resulting surface configuration of extra half planes results in two different types of surface dislocation configuration upon subsequent insertion into the original body. Denoting the dislocation deformation due to the k^th slip event as F_p^(k), the appropriate expression for the multiplicative decomposition of n successive events is

In equation (6) the index refers to the order in which the slip event occurs. Different index values can apply to slip on the same system occurring with intervening deformation on other systems. However, the distinction between different orders is generally not important, since the deformation carried by individual slip events is small unless atomic dimensions are concerned. This can be illustrated by expanding the plastic distortion into its components on the active slip systems:

where the trailing terms on the right hand side refers to terms of order higher than one. Since all components are small in the usual sense of linear approximations, these higher order terms can be neglected, leaving only the sum of first order terms in β^(k)_p . This sum of linear terms can be rearranged to group the summands according to slip system, leading to an expression for F_p that contains terms from all active slip systems regardless of the order in which slip on each occurs.

Development of the kinematics of finite deformations requires an expression for the time rate of change of F. The velocity of each material point of a body in motion forms a vector field measured in the current state, v = ú, where the superimposed dot refers to the time derivative of the quantity. The spatial gradient of the total velocity, L, is defined as

where the index, y, of the gradient operator refers to the fact that it is evaluated at the current location of the material point. The relationship of L to L_e and L_p can be obtained by combining equations (5) and (8). Application of the product rule of differentiation to the former gives

which, when applied to equation (8), yields the velocity gradient

The term in parentheses in equation (10) is the plastic velocity gradient, which is evaluated in the intermediate cofiguration and, therefore, must be mapped into the deformed configuration by F_e.