4.3 Generalized-Kelvin Model
121
Convolution Integral Representation
The constitutive relations v = − σ/E 0 for the total stress in the Hooke element
(and its rate form) and σ v = η k ˙
v = σ − σ e for the viscous stress in the Kelvin
element together with the constitutive relation σ e = E k v = σ − σ v for the elastic
stress in the Kelvin element may be arranged in a differential equation relating the
total stress and strain as
σ(t) + c τ k ˙
σ(t) = E ∞ (t) + E ∞ τ k ˙
(t).
(4.91)
Here E ∞ := E 0 E k /[E 0 + E k ] and τ k := η k /E k denote the equilibrium elastic
stiffness of the Standard-Linear-Solid Kelvin model and the relaxation time of the
Kelvin element. Moreover the stiffness ratio e := E 0 /E ∞ and the compliance ratio
c := C 0 /C ∞ are introduced with C ∞ := E
−1
∞ = C k + C 0 and C 0 := E
−1
0 .
Imposing a constant strain step (t) = 0 H(t) (and thus ˙
(t) = 0 δ(t)) renders
an exponential stress relaxation in time
σ(t) + c τ k ˙
σ(t) = [E ∞ H(t) + E ∞ τ k δ(t)] 0 =⇒ σ(t) =: E(t) 0 .
(4.92)
Here E(t), i.e. the normalized stress history as response to an imposed constant unit
strain step (t) = H(t), has been introduced as the exponentially decaying relaxation
function that is illustrated in Fig. 4.25 (left)
E(t) := E ∞ H(t)
1 + [e − 1] e
−e t/τ k
.
(4.93)
Based on the Boltzmann superposition process, the stress history σ(t) for t ≥ 0 as
response to an arbitrary strain history (t) ≡ H(t) (t) follows from the convolution
integral
σ(t) =
t
0
E(t − t
) ˙
(t
) dt
=: E(t) ) ˙
(t).
(4.94)
Thereby the convolution of the relaxation function with the strain rate history is
abbreviated symbolically as E(t) ) ˙
(t).
Imposing, alternatively, a constant stress step σ(t) = σ 0 H(t) (and thus ˙
σ(t) =
σ 0 δ(t)) renders an exponentially saturating creep strain in time
E ∞ (t) + E ∞ τ k ˙
(t) = [H(t) + c τ k δ(t)] σ 0 =⇒ (t) =: C(t) σ 0 .
(4.95)
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 exponentially saturating
creep function that is illustrated in Fig. 4.25 (right)
C(t) := C ∞ H(t)
1 + [c − 1] e
−t/τ k
= C k H(t)
1 − e
−t/τ k
+ C 0 H(t). (4.96)
The latter expansion clearly highlights the serial arrangement of a Kelvin and a
Hooke element in the Standard-Linear-Solid Kelvin model. Based on the Boltzmann
121
Convolution Integral Representation
The constitutive relations v = − σ/E 0 for the total stress in the Hooke element
(and its rate form) and σ v = η k ˙
v = σ − σ e for the viscous stress in the Kelvin
element together with the constitutive relation σ e = E k v = σ − σ v for the elastic
stress in the Kelvin element may be arranged in a differential equation relating the
total stress and strain as
σ(t) + c τ k ˙
σ(t) = E ∞ (t) + E ∞ τ k ˙
(t).
(4.91)
Here E ∞ := E 0 E k /[E 0 + E k ] and τ k := η k /E k denote the equilibrium elastic
stiffness of the Standard-Linear-Solid Kelvin model and the relaxation time of the
Kelvin element. Moreover the stiffness ratio e := E 0 /E ∞ and the compliance ratio
c := C 0 /C ∞ are introduced with C ∞ := E
−1
∞ = C k + C 0 and C 0 := E
−1
0 .
Imposing a constant strain step (t) = 0 H(t) (and thus ˙
(t) = 0 δ(t)) renders
an exponential stress relaxation in time
σ(t) + c τ k ˙
σ(t) = [E ∞ H(t) + E ∞ τ k δ(t)] 0 =⇒ σ(t) =: E(t) 0 .
(4.92)
Here E(t), i.e. the normalized stress history as response to an imposed constant unit
strain step (t) = H(t), has been introduced as the exponentially decaying relaxation
function that is illustrated in Fig. 4.25 (left)
E(t) := E ∞ H(t)
1 + [e − 1] e
−e t/τ k
.
(4.93)
Based on the Boltzmann superposition process, the stress history σ(t) for t ≥ 0 as
response to an arbitrary strain history (t) ≡ H(t) (t) follows from the convolution
integral
σ(t) =
t
0
E(t − t
) ˙
(t
) dt
=: E(t) ) ˙
(t).
(4.94)
Thereby the convolution of the relaxation function with the strain rate history is
abbreviated symbolically as E(t) ) ˙
(t).
Imposing, alternatively, a constant stress step σ(t) = σ 0 H(t) (and thus ˙
σ(t) =
σ 0 δ(t)) renders an exponentially saturating creep strain in time
E ∞ (t) + E ∞ τ k ˙
(t) = [H(t) + c τ k δ(t)] σ 0 =⇒ (t) =: C(t) σ 0 .
(4.95)
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 exponentially saturating
creep function that is illustrated in Fig. 4.25 (right)
C(t) := C ∞ H(t)
1 + [c − 1] e
−t/τ k
= C k H(t)
1 − e
−t/τ k
+ C 0 H(t). (4.96)
The latter expansion clearly highlights the serial arrangement of a Kelvin and a
Hooke element in the Standard-Linear-Solid Kelvin model. Based on the Boltzmann