Page 164 - The Combined Finite-Discrete Element Method

STRESS 147

This means that, for instance, vector

ˆ
˜
˜
˜
k is parallel to (i × j) (4.76)
i.e. it is orthogonal to the surface formed by these two vectors The volume of the deformed
element of material is given by
˜
˜
(det F) = (i × j) · k (4.77)
˜
Since

˜
ˆ ˆ
˜ ˜
k · k k
(4.78)
=
˜
(i × j) · k ˜ ˜ ˜
˜
(i × j)
and
ˆ ˜ ˜ ˜ ˜ ˜
k · k = 1; (i × j) · k = det F (4.79)
it follows that

˜
ˆ ˜ ˜ (4.80)
k (det F) = (i × j)
Since vector
ˆ
(i × j) is parallel to the vector k (4.81)
˜
˜
˜
it follows that
ˆ
(det F)k = (i × j) (4.82)
˜
˜
˜
By analogy, the following expression for the other two vectors of the triad is obtained:
ˆ
˜
(det F)i = (j × k) (4.83)
˜
˜
ˆ
˜
˜
˜
(det F)j = (k × i)
In other words, these vectors simply represent the surface normals of the deformed mate-
rial element. By substituting these into the defining formula for the first Piola-Kirchhoff
stress, the following expressions are obtained:
S = (det F)TF −T
= T[(det F)F −T ] (4.84)
ˆ ˆ ˆ
˜ ˜ ˜
= T(det F)[i, j, k]
ˆ
˜
ˆ
= T[i(det F), j(det F), k(det F)]
˜
˜
˜
ˆ
ˆ
ˆ
˜
= T[i(det F)) + Tj(det F) + Tk(det F)
˜
˜