Page 159 - The Combined Finite-Discrete Element Method

142 DEFORMABILITY OF DISCRETE ELEMENTS

Tensor U is called the right stretch tensor. Tensor V is called the left stretch tensor.
Tensors U and V are symmetric and positive definite tensors with

det U = det V =| det F| > 0 (4.50)

representing the ratio between the volume of the deformed material element and the initial
material element. Both right and left stretch tensors can be decomposed into a succession
of three extensions in three mutually orthogonal directions:

3

U = λ i e i ⊗ e i (4.51)
i=1
= U 1 U 2 U 3
U i = I + (s i − 1)e i ⊗ e i
3

V = λ i ˜ e i ⊗ ˜ e i (4.52)
i=1
= V 1 V 2 V 3
V i = I + (s i − 1)˜ e i ⊗ ˜ e i

where scalars s 1 ,s 2 and s 3 represent principal stretches. Principal stretches are in essence
elongation in the principal directions, i.e. the ratio between the deformed length and initial
length. Principal stretches are the same for both right and left stretch tensor.
The right stretch tensor U therefore represents successive stretching of the material
element in three mutually orthogonal directions. This stretching is applied before any
rotation. In contrast, left stretch tensor V represents successive stretching of the material
element in three mutually orthogonal directions applied after rotation. Thus, the principal
directions of left stretch tensor V are obtained by simply rotating the principal directions
associated with the right stretch tensor U:

˜ e 1 = Re 1
(4.53)
˜ e 2 = Re 2
˜ e 3 = Re 3

4.4 STRAIN

Using stretch tensors U and V, different strain tensors can be defined. For instance,

T
T
T
T
T
C = F F = (RU) (RU) = U R RU = U U
(4.54)
T
T
T
T
T
B = FF = (VR)(VR) = VRR V = VV = V 2
are the right and left Cauchy–Green strain tensor, respectively. In the case of the right
Cauchy–Green strain tensor C, rotation occurs after stretch. Thus, the left Cauchy–Green