4.5 Generalized-Maxwell Model
169
π
∗
(σ v ) =
1
2
1
η m
|σ v |
2
.
(4.193)
The evolution law for the viscous strain of the Maxwell element then follows
as partial derivative of the dual dissipation potential with respect to its conjugated
variable
˙
v (σ v ) = ∂ σ v π
∗
(σ v ) =
1
η m
σ v .
(4.194)
Obviously the expressions in Eqs. 4.190 and 4.194 are inverse relations. The
smooth dissipation and dual dissipation potentials π = π(˙ v ) and π
∗
= π
∗
(σ v )
together with the resulting smooth constitutive relations σ v = σ v (˙ v ) and ˙
v = ˙
v (σ v )
are similar to those displayed in Fig. 4.2.
The Standard-Linear-Solid Maxwell model is summarized in Table 4.13.
Convolution Integral Representation
The rate form of the constitutive relation ˙
σ v = E m [˙ − ˙
v ] for the viscous stress and
the evolution equation for the viscous strain ˙
v = σ v /η m in the Maxwell element
together with the rate form of the constitutive relation ˙
σ ∞ = E ∞ ˙
in the Hooke
element may be arranged in a differential equation relating the total stress and strain
as
σ(t) + τ m ˙
σ(t) = E ∞ (t) + E 0 τ m ˙
(t).
(4.195)
Here E 0 := E m + E ∞ and τ m := η m /E m denote the instantaneous elastic stiffness
of the Standard-Linear-Solid Maxwell model and the relaxation time of the Maxwell
element. Moreover the stiffness ratio e := E 0 /E ∞ and the compliance ratio c :=
C 0 /C ∞ are introduced with C 0 := E
−1
0 = C ∞ C m /[C ∞ + C m ] and C ∞ := E
−1
∞ .
Imposing a constant strain step (t) = 0 H(t) (and thus ˙
(t) = 0 δ(t)) renders
an exponential stress relaxation in time
σ(t) + τ m ˙
σ(t) = [E ∞ H(t) + E 0 τ m δ(t)] 0 =⇒ σ(t) =: E(t) 0 .
(4.196)
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. 4.49 (left)
E(t) := E ∞ H(t)
1 + [e − 1] e
−t/τ m
= E ∞ H(t) + E m H(t) e
−t/τ m . (4.197)
The latter expansion clearly highlights the parallel arrangement of a Hooke and a
Maxwell element in the Standard-Linear-Solid Maxwell model. 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
169
π
∗
(σ v ) =
1
2
1
η m
|σ v |
2
.
(4.193)
The evolution law for the viscous strain of the Maxwell element then follows
as partial derivative of the dual dissipation potential with respect to its conjugated
variable
˙
v (σ v ) = ∂ σ v π
∗
(σ v ) =
1
η m
σ v .
(4.194)
Obviously the expressions in Eqs. 4.190 and 4.194 are inverse relations. The
smooth dissipation and dual dissipation potentials π = π(˙ v ) and π
∗
= π
∗
(σ v )
together with the resulting smooth constitutive relations σ v = σ v (˙ v ) and ˙
v = ˙
v (σ v )
are similar to those displayed in Fig. 4.2.
The Standard-Linear-Solid Maxwell model is summarized in Table 4.13.
Convolution Integral Representation
The rate form of the constitutive relation ˙
σ v = E m [˙ − ˙
v ] for the viscous stress and
the evolution equation for the viscous strain ˙
v = σ v /η m in the Maxwell element
together with the rate form of the constitutive relation ˙
σ ∞ = E ∞ ˙
in the Hooke
element may be arranged in a differential equation relating the total stress and strain
as
σ(t) + τ m ˙
σ(t) = E ∞ (t) + E 0 τ m ˙
(t).
(4.195)
Here E 0 := E m + E ∞ and τ m := η m /E m denote the instantaneous elastic stiffness
of the Standard-Linear-Solid Maxwell model and the relaxation time of the Maxwell
element. Moreover the stiffness ratio e := E 0 /E ∞ and the compliance ratio c :=
C 0 /C ∞ are introduced with C 0 := E
−1
0 = C ∞ C m /[C ∞ + C m ] and C ∞ := E
−1
∞ .
Imposing a constant strain step (t) = 0 H(t) (and thus ˙
(t) = 0 δ(t)) renders
an exponential stress relaxation in time
σ(t) + τ m ˙
σ(t) = [E ∞ H(t) + E 0 τ m δ(t)] 0 =⇒ σ(t) =: E(t) 0 .
(4.196)
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. 4.49 (left)
E(t) := E ∞ H(t)
1 + [e − 1] e
−t/τ m
= E ∞ H(t) + E m H(t) e
−t/τ m . (4.197)
The latter expansion clearly highlights the parallel arrangement of a Hooke and a
Maxwell element in the Standard-Linear-Solid Maxwell model. 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