For a simple shear with respect to the perpendicular unit vectors and ,
and
If , , , and , then (in matrix notation)
The volume change is given by .
Properties of homogeneous displacement fields
If is a homogeneous displacement field, then , where is a rigid displacement and is a pure strain from an arbitrary point .
Every pure strain can be decomposed into the the sum of three simple extensions in mutually perpendicular directions, .
Every pure strain can be decomposed into a uniform dilatation and an isochoric pure strain, where , , and .
Every simple shear of amount with respect to the direction pair () can be decomposed into the sum of two simple extensions of the amount in the directions .
Every simple shear is isochoric. Every isochoric pure strain is the sum of simple shears.
Any displacement field that does not satisfy the condition of homogeneity is inhomogenous. Most deformations in engineering materials lead to inhomogeneous displacements.
Let be a displacement field, be the corresponding strain field. Let and be continuous on B. Then, the mean strain depends only on the boundary values of .
where is the unit normal to the infinitesimal surface area .