Tensor Analysis of the Crystallographic Problem

The problem above can also be solved analytically using tensor methods—the preferred technique when higher accuracy is required. In general, any homogeneous deformation can be represented by a second-rank tensor that operates on any vector in the initial material and transforms it into a corresponding vector in the deformed material. For example, in the lattice deformation, each vector, Ffcc, in the initial f.c.c. structure is transformed into a corresponding vector in the b.c.t. structure, Vbct, by 

If S is the lattice-invariant deformation tensor and R the rigid-body rotation tensor, the total shape deformation tensor, E, producing the invariant plane can be expressed as 

Wayman describes in detail how the tensor formalism can be used to solve the crystallographic problem [5]. A simple graphical demonstration, in two dimensions, of how an invariant line (habit plane) can be produced by the deformations B, S, and R is given in Exercise 24.6. 

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