68
3 Elasticity
Imposing, alternatively, a constant stress step σ(t) = σ 0 H(t) renders also a
Heaviside-type solution for the strain history
E (t) = H(t) σ 0 =⇒ (t) =: C(t) σ 0 .
(3.11)
Here C(t), i.e. the normalized strain history as response to an imposed constant unit
stress step σ(t) = H(t), has been introduced as the Heaviside-type “creep” function
that is illustrated in Fig. 3.3 (right)
C(t) := C H(t).
(3.12)
Based on the Boltzmann superposition process, the strain history (t) for t ≥ 0 as
response to an arbitrary stress history σ(t) ≡ H(t) σ(t) is formally retrieved from
the convolution integral
(t) = C σ(t) =
t
0
C(t − t
) ˙
σ(t
) dt
=: C(t) ) ˙
σ(t).
(3.13)
Thereby the convolution of the “creep” function with the stress rate history is abbreviated symbolically as C(t) ) ˙
σ(t).
Laplace Transformation Representation
Upon Laplace transformation, the convolution integral of the “relaxation” function
E(t) with a prescribed strain (rate) history, a causal signal (t) = H(t) (t) with
(0) = 0, results in
L{σ(t)} = L{E(t) ) ˙
(t)} = s L{E(t)} L{(t)} = E L{(t)}.
(3.14)
Choosing, as a particular example, a causal harmonic strain history with (t) =
H(t) a sin(ω t) and thus L{(t)} = a ω/[ω
2
+ s
2
] renders, after inverse Laplace
transformation,
2 also a causal in-phase harmonic signal for the resulting stress history,
whereby the initial condition σ(0) = E (0) is captured.
σ(t) = H(t) E
sin(ω t) a .
(3.15)
Thereby E
:= E is here formally introduced as abbreviation, however as will
become transparent in the sequel, it denotes the so-called storage stiffness modulus.
The stress history resulting from a sinusoidal strain history is shown in Fig. 3.4 (left).
2 The inverse Laplace transformation for the stress history follows from the following step by step
computation:
L{σ(t)}
a
= E
ω
ω 2 + s 2 = E
L{H(t) sin(ω t)}.