Page 135 - Introduction to Continuum Mechanics

t20 Deformation Gradient

3.18 Deformation Gradient

We recall that the general motion of a continuum is described by

where x is the spatial position at time t, of a material particle with a material coordinate X.
A material element dX at the reference configuration is transformed, through motion, into a
material element dx. at time t. The relation between dX and dx is given by

i.e.,

where the tensor

is called the deformation gradient at X. The notation Vx is an abbreviation for the notation
V xx where the subscript X indicates that the gradient is with respect to X for the function
x(X, t). We note that with x = X 4- u, where u is the displacement vector,

Example 3.18.1

Given the following motion:

where both jc/ and X- t are rectangular Cartesian coordinates. Find the deformation gradient
au = 0 and at? = 1.
Solution. For rectangular Cartesian coordinates,

Thus, from Eq. (i) and (ii),