Deformation Gradient - Civil Engineering (CE) PDF Download

Deformation Gradient

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



Since, χ is a function of both X and t we have used a partial derivative in the definition of the deformation gradient. Also, we haven’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 {e_i} the basis vectors in the current configuration. Then, the deformation gradient is written as

F = F_ije_i ⊗ E_j .                                                     (3.12)

In general, the basis vectors e_i and E_j 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, Θ, Z) and (r, θ, 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_R, E_Θ, E_Z} and {e_r, e_θ, e_z} 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