Suppose that the region occupied by a body in a reference configuration is a unit cube...

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Suppose that the region occupied by a body in a reference configuration is a unit cube as shown in Fig. 1. The body undergoes the homogeneous deformation described by x = \X, x2 = X and x3 = A3X3, where the components have been taken with respect to the orthonormal basis {e,e2, e3} that is aligned with the axes of the cube. Derive relationships between the principal stretches in each of the following cases: e3 T S R Figure 1: P Ro 1. The body is composed of an incompressible material. 2. The length of the fiber OP remains unchanged by the deformation. 3. The angle between the fibers OP and QR remains unchanged by the deformation. 4. The area of the plane RSQT remains unchanged by the deformation. 5. The orientation of the plane RSQT remains unchanged by the deformation. Furthermore, if the body undergoes uniform extension in all the three directions, find the matrix representations of tensors F, R, U, E and e. Suppose that the region occupied by a body in a reference configuration is a unit cube as shown in Fig. 1. The body undergoes the homogeneous deformation described by x = \X, x2 = X and x3 = A3X3, where the components have been taken with respect to the orthonormal basis {e,e2, e3} that is aligned with the axes of the cube. Derive relationships between the principal stretches in each of the following cases: e3 T S R Figure 1: P Ro 1. The body is composed of an incompressible material. 2. The length of the fiber OP remains unchanged by the deformation. 3. The angle between the fibers OP and QR remains unchanged by the deformation. 4. The area of the plane RSQT remains unchanged by the deformation. 5. The orientation of the plane RSQT remains unchanged by the deformation. Furthermore, if the body undergoes uniform extension in all the three directions, find the matrix representations of tensors F, R, U, E and e.