4.2 Kelvin Model
101
C(t) := C H(t) [1 − e
−t/τ
],
(4.52)
where C := E
−1 denotes the inverse of the elastic stiffness, i.e. the elastic compliance. 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) follows from the
convolution integral
(t) =
t
0
C(t − t
) ˙
σ(t
) dt
=: C(t) ) ˙
σ(t).
(4.53)
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 [1 + τ s] L{(t)}.
(4.54)
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,
4 a phase shifted causal harmonic signal for the resulting stress history,
whereby the initial condition σ(0) = η ˙
(0) is captured.
σ(t) = H(t)
E
sin(ω t) + E
cos(ω t)
a .
(4.55)
Thereby E
:= E and E
:= E τ ω are here formally introduced as abbreviations,
however as will become transparent in the sequel, they denote the so-called storage
and loss stiffness moduli (note that E
/E
= τ ω). The stress history resulting from
a sinusoidal strain history is shown in Fig. 4.14 (left).
Upon Laplace transformation, the convolution integral of the creep function C(t)
with a prescribed stress (rate) history, a causal signal σ(t) = H(t) σ(t) with σ(0) = 0,
results in
4 The inverse Laplace transformation for the stress history follows from the following step by step
computation:
L{σ(t)}
a
= E [1 + τ s]
ω
[ω 2 + s 2 ]
= E
[ω + τ ω s]
[ω 2 + s 2 ]
= E
L{H(t) sin(ω t)} + E
L{H(t) cos(ω t)}
.