4.1 Newton Model
81
Convolution Integral Representation
The constitutive relation for the total stress may be considered a differential equation
relating the total stress and strain as
σ(t) = η ˙
(4.10)
Imposing a constant strain step = 0 H(t) (and thus ˙
= 0 δ(t)) renders a
Dirac-delta-type solution for the stress history
σ(t) = η δ(t) 0 =⇒ σ(t) =: E(t) 0 .
(4.11)
Here E(t), i.e. the normalized stress history as response to an imposed constant
unit strain step = H(t), has been introduced as the Dirac-delta-type relaxation
function that is illustrated in Fig. 4.3 (left)
E(t) := η δ(t).
(4.12)
Based on the Boltzmann superposition process, the stress history σ(t) for t ≥ 0 as
response to an arbitrary strain history ≡ H(t) follows from the convolution
integral
σ(t) =
t
0
E(t − t
) ˙
) dt
=: E(t) ) ˙
(4.13)
Thereby, the convolution of the relaxation function with the strain rate history is
abbreviated symbolically as E(t) ) ˙
Imposing, alternatively, a constant stress step σ(t) = σ 0 H(t) (and thus ˙
σ(t) =
σ 0 δ(t)) renders linearly increasing creep strain in time
0
1
2
3
0
1
2
3
0
1
2
3
E(t)/η = δ(t)
t
∞
(t) 0 = H(t)
0
1
2
3
t
C(t) η = H(t) t
σ(t)/σ0 = H(t)
Fig. 4.3 Specific Newton model: Normalized relaxation function E(t)/η (left) and normalized creep
function C(t) η (right). The dotted lines depict the normalized step functions for the prescribed strain
and stress, respectively