Intracrystalline deformation can be accommodated by the movement of disclocations through the crystal lattice, known as dislocation creep.  Within a given mineral, the movement of dislocations is along a certain number of lattice planes (slip planes) in a given direction (Burgers Vector).  Together, the orientation of the slip plane and the magnitude and direction of the Burgers vector define a slip system (Lister et al. 1977).  A given mineral will have a number of slip systems, each of which can accommodate a given amount of incremental strain.  Associated with each slip system is a Critical Resolved Shear Stress (CRSS), which is the minimum stress that needs to be exceeded in order for slip to occur.  The magnitude of the CRSS is strongly dependent on temperature, and lesser dependent on factors such as strain rate, pressure and chemical activity. 

Different slip systems can interact during deformation, causing dislocations to become tangled.  This entanglement of dislocations can block further dislocation movement and effectively stiffen the grain, a process known as strain hardening.

TBH recrystallization involves rigid body rotation of the crystallographic axis until a particular slip system is oriented to allow slip to occur, assuming the CRSS for that system is exceeded (Lister et al. 1977).  As dislocation glide continues, the crystallographic axis continues to rotate, and new slip systems may rotate into a preferred orientation, thus allowing slip to occur on a new slip system.  The TBH model assumes that there are five independent slip systems that can accommodate the deformation for any given strain increment (known as the Von Mises Criterion). 

The following assumptions are inherent in the TBH model (Lister et al. 1977):

1)    deformation is accommodated by dislocation glide only, which is treated as the simultaneous operation of several discreet glide systems, approximated as continuous simple shear relative to a glide plane;

2)    deformation is uniform through the polycrystalline material at all stages;

3)    the material has an elastic-plastic flow law

The six-component stress tensor represented in stress space. Each apex of the six-sided polygon is defined by a component of the stress tensor, and the distance from the origin in the centre of the polygon to the apex is the magnitude of the stress component. The apices are connected to form a yield surface, which separates zones of physically realistic stress conditions within the yield surface from physically unrealistic stress conditions outside the yield surface.

The resolved shear stress can be generalized as:

where t is the shear stress on system s, s is the applied stress in the ij direction, r^s is the component of the slip vector in the glide direction and v^s is the component of the slip vector normal to the glide plane.

If t is smaller than t_c, the slip system will be inactive, such that:

Stress space can be defined if each component of the stress tensor is represented by a vector:

Only six of these components are independent (to satisfy the condition that s_ij = s_ji), so the stress tensor can be reduced to:

 Thus, stress space can be represented as a six-sided hyperpolyhedron, with each stress vector represented as an apex. 

The stress space hyperpolyhedral separates stress space into a field of no deformation, a series of yield surfaces and a field of non-permisable stress states. 

There are two formulations of the TBH model: they Taylor formulation and the Bishop-Hill formulation (Lister et al. 1977).  The Taylor formulation assumes that of all the combinations of slip systems that can be active, the one that does the least amount of work is the one that actually operates.  Lister et al. (1977) expressed this principle in terms of modeling the following way:

 “Given the critical resolved shear stresses t_k for the glide system k = 1 … n, choose a set of g_k to minimize W_internal = t_kg_k, subject to the conditions 1/2 (n_il_j + n_jl_i)g_k = e_ij and g_k > 0 where e_ij is the imposed strain increment (i,j = 1,2,3)”  (Lister et al. 1977, p. 119)

The Bishop-Hill formulation assumes that the stress state existing within a grain during a strain increment is that which will do the most work compared with the virtual work done by other physically possible stress states (Lister et al. 1977).  If the stress state is known for the grain, then the active glide system is easily predicted.  Lister et al. (1977) expressed this principle in terms of modeling in the following way:

 “Given the strain increment e_ij, choose s_ij to maximize W_external = s_ije_ij, subject to the conditions ½(n_il_j + n_jl_i)_ks_ij < t_k and –infinity < s_ij < infinity, where s_ij is the state of stress in the grain (i,j = 1,2,3).” (Lister et al. 1977, p. 121)