86
4 Visco-Elasticity
Table 4.2 Algorithmic update for the specific Newton model
Input
n n−1
Trial Strain = n − n−1
Update Stress σ n =
η
t n
Tangent
E n
a =
η
t n
Output
σ n E n
a
σ
n
=
η
t n
.
(4.30)
The trial strain
is computable exclusively from known quantities at the beginning and at the end of the time step and follows as
:=
n
.
(4.31)
The sensitivity of σ
n with respect to
n is denoted the algorithmic tangent E a (thus
dσ = E a d) and is straightforwardly computed as
∂ σ
n
=
η
t n .
(4.32)
Note that, consequently, the algorithmic tangent degenerates to E a → ∞ for
t
n
→ 0, i.e. for very fast processes (as compared to the viscosity) the response
is rigid.
The algorithmic step-by-step update for the specific Newton model is summarized
in Table 4.2.
4.1.3 Specific Newton Model: Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Newton model to a prescribed Zig-Zag strain history is
displayed in Fig. 4.6a–f.
Figure 4.6a, c depict the prescribed Zig-Zag (viscous) strain history (t) with
amplitude a = 5 and period T = 4 in the time interval t ∈ [0, t max = 10], whereby
N = 100 time steps with t = 0.1 are computed.