Relative deformation gradient tensor

Now consider two neighboring material points X, and X,+dX, which were located at neighboring places X z) and Jr(r)-l-dAr(r) at time r. The relative deformation gradient tensor F is defined by... 

Thus, the relative deformation gradient tensor describes the change of the local configuration that occurs between times z and t in the neighborhood of P. Since F depends on both r and /, it is a relative tensor and we write F,(r) to denote this. From Eqs. 2 and 4, we note that the relative deformation gradient tensor is the gradient of the motion, i.e. 

KINEMATICS AND STRESSES OF DEFORMABLE BODIES 23 In terms of the relative deformation gradient tensor t ), Eq. (2.29) may be written as... 

The velocity gradient tensor L(r) can be determined from the relative deformation gradient tensor F (r ) also, since we have... 

Calculate the deformation gradient tensor F(t) and the relative deformation gradient tensor F,(t ) for the motion given in Problem 2.1. 

The components of the relative deformation gradient Fy and the symmetrical tensor Jy can be calculated, respectively, from Eqs. (13.7) and (13.8), giving... 

Consequently, E has components relative to the reference configuration, and is a referential strain tensor. A complementary strain tensor may be defined from the inverse deformation gradient F ... 

From Example 2.2. Ic we see that the velocity gradient tensor is not zero for a solid body rotation. From the statement of Newton s viscosity law, we would not expect stress to be generated as a result of the flow in solid body rotation because there is no relative separation of points. Like the deformation gradient F, the velocity gradient tensor L contains rate of rotation as well as stretching. We need a way to remove this rotation. 

To describe deformation, we will examine the relative displacement of two neighboring fluid particles. At time tg, these particles are separated by the vector dx(fo) and at time by the vector dx(ti). A quantity that provides complete information about the relative displacement of any two such particles in a very small volume of the fluid is the displacement gradient tensor F whose components are given by ... 

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