122
4 Visco-Elasticity
0
1
2
3
t
cτ
E(t)/E ∞ = H(t) [1 + ¯ e e
−et/τ ]
(t) 0 = H(t)
0
1
2
3
0
1
2
3
0
1
2
3
t
C(t)/C ∞ = H(t) [1 + ¯ c e
−t/τ ]
σ(t)/σ0 = clH(t)
Fig. 4.25 Standard-Linear-Solid Kelvin model: Normalized relaxation function E(t)/E ∞ (left)
and normalized creep function C(t)/C ∞ (right) for a relaxation time τ k = 2 and E 0 = E k (thus
E ∞ = 1/2 E 0 , e = 2, ¯
e := e − 1 = 1, c = 1/2 and ¯
c := c − 1 = −1/2). Observe the jumps in both
functions at t = 0. The dotted lines depict the normalized step functions for the prescribed strain
and stress, respectively. The dashed line illustrates the meaning of the modified relaxation time c τ k
superposition process, the strain history for t ≥ 0 as response to an arbitrary
stress history σ(t) ≡ H(t) σ(t) follows from the convolution integral
=
t
0
C(t − t
) ˙
σ(t
) dt
=: C(t) ) ˙
σ(t).
(4.97)
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 = H(t) with = 0,
results in
L{σ(t)} = L{E(t) ) ˙
= s L{E(t)} L{
(4.98)
= E ∞
1 + τ k s
1 + c τ k s
L{
Choosing, as a particular example, a causal harmonic strain history with =
H(t) a sin(ω t) and thus L{ = a ω/[ω
2
+ s
2
] renders, after inverse Laplace
transformation,
6 a phase shifted causal harmonic signal for the resulting stress history
superposed by an exponentially decaying signal that is needed to enforce the initial
condition σ(0) = 0
6 The inverse Laplace transformation for the stress history follows from the following step by step
computation:
4 Visco-Elasticity
0
1
2
3
t
cτ
E(t)/E ∞ = H(t) [1 + ¯ e e
−et/τ ]
(t) 0 = H(t)
0
1
2
3
0
1
2
3
0
1
2
3
t
C(t)/C ∞ = H(t) [1 + ¯ c e
−t/τ ]
σ(t)/σ0 = clH(t)
Fig. 4.25 Standard-Linear-Solid Kelvin model: Normalized relaxation function E(t)/E ∞ (left)
and normalized creep function C(t)/C ∞ (right) for a relaxation time τ k = 2 and E 0 = E k (thus
E ∞ = 1/2 E 0 , e = 2, ¯
e := e − 1 = 1, c = 1/2 and ¯
c := c − 1 = −1/2). Observe the jumps in both
functions at t = 0. The dotted lines depict the normalized step functions for the prescribed strain
and stress, respectively. The dashed line illustrates the meaning of the modified relaxation time c τ k
superposition process, the strain history for t ≥ 0 as response to an arbitrary
stress history σ(t) ≡ H(t) σ(t) follows from the convolution integral
=
t
0
C(t − t
) ˙
σ(t
) dt
=: C(t) ) ˙
σ(t).
(4.97)
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 = H(t) with = 0,
results in
L{σ(t)} = L{E(t) ) ˙
= s L{E(t)} L{
(4.98)
= E ∞
1 + τ k s
1 + c τ k s
L{
Choosing, as a particular example, a causal harmonic strain history with =
H(t) a sin(ω t) and thus L{ = a ω/[ω
2
+ s
2
] renders, after inverse Laplace
transformation,
6 a phase shifted causal harmonic signal for the resulting stress history
superposed by an exponentially decaying signal that is needed to enforce the initial
condition σ(0) = 0
6 The inverse Laplace transformation for the stress history follows from the following step by step
computation: