Deformation gradient tensor simple shear

Note 3 The deformation gradient tensor for the simple shear of an elastic solid is... 

As an example, we shall consider simple shear when z/12 0, and find components of the tensor of the recoverable displacement gradients A12, An, A22, A33 the components of the tensor are calculated from the relaxation equations (9.49) or (9.58). In this case the matrix of the deformation tensor is determined as follows... 

In the case of a simple shear deformation, schematically indicated in Figure 4.6b, the only nonzero components of the displacement gradient and strain tensors are given by... 

In defining the material functions that describe responses to simple-shear deformations, a standard frame of reference has been adopted. This is shown in Fig. 10.4. The shear stress <7is the component < i (equal to <7i2 because of the symmetry of the stress tensor), and the three normal stresses are <7u, in the direction of flow (xj), Gjj in the direction of the gradient and <733, in the neutral (x ) direction. As this is by definition a two-dimensional flow, there is no velocity and no velocity gradient in the Xj direction. However, in describing shear flow behavior, we will follow the conventional practice of referring to the shear stress as <7, and the shear strain as y, where neither symbol is in bold or has subscripts. 

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