The Finite Element Method for the Analysis of Non-Linear and ...

Example
B. Jaumann stress rate formulation
This formulation uses the following constitutive relationship:
t˜τ ij = t C ijrs t D rs (7)
where the velocity strain tensor t D is computed using the velocity gradient L (see
Lecture 4) :
[ ]
0 2
L = ẊX −1 =
0 0
L can be decomposed to s symmetric part D = D T (the velocity strain tensor) and a
skew symmetric part W = −W T (the spin tensor):
L = D + W
[
0 1
D =
1 0
which in this case yields
] [ ]
0 1
, W =
−1 0
Now we use the same constitutive matrix C and Eqn (7) to obtain the Jaumann stress at
time t:
⎡ ⎤ ⎡ ⎤
˜τ 11 0
⎢ ⎥ ⎢ ⎥
⎣ ˜τ 22 ⎦ = ⎣ 0 ⎦
˜τ 12
3846
Institute of Structural Engineering Method of Finite Elements II 16