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# Deformation Gradient Civil Engineering (CE) Notes | EduRev

## Civil Engineering (CE) : Deformation Gradient Civil Engineering (CE) Notes | EduRev

The document Deformation Gradient Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Advanced Solid Mechanics - Notes, Videos, MCQs & PPTs.
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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 {Eibe the three Cartesian basis vectors in the reference configuration and {eithe basis vectors in the current configuration. Then, the deformation gradient is written as

F = Fijei âŠ— Ej .                                                     (3.12)

In general, the basis vectors ei and Ej need not be the same. Since the deformation gradient depends on two sets of basis vectors, it is called a twopoint tensor. It is pertinent here to point out that the grad operator as defined in chapter 2 (2.207), is not a two-point tensor either. The matrix components of the deformation gradient in Cartesian coordinate system is

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 {ER, EÎ˜, EZ} and {er, eÎ¸, ez} are the cylindrical polar coordinate basis vectors obtained from (2.242) using (3.15). Comparing equations (3.14) with (2.259) we see the difference between the Grad operator and  operator

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## Advanced Solid Mechanics - Notes, Videos, MCQs & PPTs

42 videos|61 docs

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