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**Deformation Gradient**

The gradient of motion is generally called the deformation gradient and is denoted by F. Thus

Since, Ï‡ is a function of both X and t we have used a partial derivative in the definition of the deformation gradient. Also, we havenâ€™t defined it as Grad(x) because for the Grad operator, as defined in chapter 2, the range of the function for which gradient is sought is any vector; not just position vectors. The difference becomes evident in curvilinear coordinate systems like the cylindrical polar coordinates.

Let **{E _{i}} **be the three Cartesian basis vectors in the reference configuration and

F = F

In general, the basis vectors e

where (X, Y, Z) and (x, y, z) are the Cartesian coordinates of a typical material particle, P in the reference and current configuration respectively. Similarly, the matrix components of the deformation gradient in cylindrical polar coordinate system is:

where (R, Î˜, Z) and (r, Î¸, z) are the cylindrical polar coordinates of a typical material particle, P in the reference and current configuration respectively. Substituting

in (3.13) we obtain

where

from which we obtain (3.14) recognizing that

where {E

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