70
3 Elasticity
Observe, furthermore, that direct application of the Laplace transformation to the
degenerated differential (i.e. algebraic) equation relating the total stress and strain
renders immediately a relation
L{σ(t)} = E L{(t)}
(3.19)
that is entirely conforming with the convolution integral representation.
Complex Harmonic Oscillation Representation
The degenerated differential (i.e. algebraic) equation relating the total stress and
strain reads in complex representation as
σ(t) = E (t).
(3.20)
Then, for a stationary harmonic oscillation of the total stress and strain with
σ(t) = σ
∗ e
i ω t and (t) =
∗ e
i ω t the relation between the corresponding complex
amplitudes
∗
= a e
i δ and σ
∗
= σ a e
i δ σ (where δ = −π/2 for sinusoidal strain
control and δ σ = −π/2 for sinusoidal stress control) follows as
σ
∗
= E
∗
=: E
∗
∗
.
(3.21)
Thereby the quantity relating the complex amplitudes of the total strain and stress
is denoted the complex stiffness modulus, its inverse is the complex compliance
modulus (so that E
∗ C
∗
= 1)
E
∗
(ω) := E
:= E and C
∗
(ω) := C
:= C.
(3.22)
Note that for the specific Hooke model, trivially, the complex moduli E
∗ and C
∗
have only real parts. Here, E
and C
denote the so-called storage stiffness modulus
and storage compliance modulus, respectively. The storage stiffness modulus and the
storage compliance modulus are plotted against the angular frequency ω for various
elasticity moduli in Fig. 3.5.
Finally, the (real) amplitudes E a and C a of the complex stiffness modulus and the
complex compliance modulus are defined via
E
∗
(ω) =: E
e
i0
=: E a e
i0 and C
∗
(ω) =: C
e
i0
=: C a e
i0
(3.23)
so that σ a = E a a (or a = C a σ a ) with δ σ = δ . Specifically, the angular frequency
independent amplitudes E a = E a (ω) and C a = C a (ω) follow as
E a := E
= E and C a := C
= C.
(3.24)
The amplitudes of the complex stiffness modulus and the complex compliance modulus, when plotted against the angular frequency ω for various elasticity moduli, are