46
2 Preliminaries
x(t) ≡ H(t) x(t) =⇒ x(t) =
⎧
⎨
⎩
0
t < 0
for
x(t)
t ≥ 0
⎫
⎬
⎭
.
(2.77)
Moreover, it sometimes proves convenient to also define the modified Heaviside
function H 0 (t), see Fig. 2.12 (right), as
H 0 (t) :=
⎧
⎨
⎩
0
t ≤ 0
for
1
t > 0
⎫
⎬
⎭
.
(2.78)
The modified Heaviside function is particularly helpful in the context of representing
the algorithmic tangent of computational material models.
2.3.2 Laplace Transformation
The complex-valued Laplace transformed signal X (s) in complex frequency domain
of a real-valued causal signal x(t) in time domain is obtained by Laplace transformation L{x(t)} defined as
X (s) = L{x(t)} :=
∞
−0
x(t) exp(−s t) dt.
(2.79)
Thereby t and s denote the real-valued time variable and the complex-valued frequency or rather Laplace transformation variable, respectively. Likewise the realvalued inversely Laplace transformed causal signal x(t) of a complex-valued signal
X (s) is obtained by inverse Laplace transformation L
−1
{X (s)} defined by
x(t) = L
−1
{X (s)} :=
1
2π i
(s)+i ∞
(s)−i ∞
X (s) exp(s t) ds.
(2.80)
From these definitions the Laplace transformation inherits a number of useful properties, of which only a few that are relevant for the discussion in the sequel are
mentioned:
• The Laplace transformation of a linear combination a 1 x 1 (t) + a 2 x 2 (t) of two
real-valued causal signals x 1 (t) and x 2 (t) renders the same linear combination of
Laplace transformed signals X 1 (s) and X 2 (s), i.e.
L{a 1 x 1 (t) + a 2 x 2 (t)} = a 1 X 1 (s) + a 2 X 2 (s).
(2.81)
• The Laplace transformation of the rate ˙
x(t) of a real-valued causal signal x(t)
with initial condition x(0) = 0 computes as