2.2 Computational Tools
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2.2 Computational Tools
2.2.1 Constitutive Integrator
2.2.1.1 General Setting
Material models are typically formulated as functional and/or differential relations
between the state functions, here the free energy ψ = ψ((, α) and the stress σ =
σ ((, α), and the (external and internal) state variables, here the strain and the set
of internal (state) variables α.
For a discrete representation of these state quantities the time arrow within the time
interval of interest T = [0, T ] is discretized, see Fig. 2.4, into discrete time instants
t
n with n = 0, 1, . . . n mx , whereby the starting and the terminating time instants are
t
0
= 0 and t
n mx = T , resulting in n mx time steps with time step size
n
:= t
n
− t
n−1 ,
i.e.
T =
n mx
n=1
[t
n
− t
n−1
] =
n mx
n=1
n
.
(2.29)
Then, for known (external and internal) state variables at the discrete time instant
t
n−1 , i.e. for known
n−1 and α
n−1 and thus for known state functions at the discrete
time instant t
n−1 , i.e. for known ψ
n−1
= ψ((
n−1
, α
n−1
) and σ
n−1
= σ (
n−1
, α
n−1
),
either
n or σ
n are prescribed at the discrete time instant t
n and the remaining state
quantities are sought to be updated algorithmically by a constitutive integrator.
Moreover, the derivative of σ
n with respect to
n , i.e. the algorithmic tangent E
n
a ,
is often needed.
Algorithmic Strain Control
The algorithm for strain control is denoted the basic constitutive integrator:
For known
n−1
, α
n−1 (and σ
n−1 ) and prescribed strain
n it computes the corresponding σ
n
, α
n along with the algorithmic tangent stiffness E
n
a , see Fig. 2.5.
T
t
0 = 0
t 1
· · ·
t
n−1
t
n
· · · t
nmx = T
Δt
1
Δt
2
· · ·
Δt
n
· · ·
Δt
nmx
t ∈
= [0 , T ]
Fig. 2.4 Discretization of time interval T = [0, T ] into discrete time instants t n with n =
0, 1, . . . n mx , whereby t 0 = 0 and t nmx = T , resulting in n mx time steps n := t n − t n−1