4.4 Maxwell Model
147
Obviously the expressions in Eqs. 4.145 and 4.149 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 specific Maxwell model is summarized in Table 4.10.
Convolution Integral Representation
The rate form of the constitutive relation ˙
σ v = E [˙ − ˙
v ] = ˙
σ for the viscous stress
(as well as for the total stress) together with the evolution equation for the viscous
strain ˙
v = σ v /η = σ/η may be arranged in a differential equation relating the total
stress and strain as
σ(t) + τ ˙
σ(t) = η ˙
(t).
(4.150)
Accordingly, for ˙
= 0 the stress satisfies σ = −τ ˙
σ at any t, thus τ := η/E has
been introduced as the relaxation time of the Maxwell model.
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 ˙
σ(t) = η δ(t) 0 =⇒ σ(t) =: E(t) 0 .
(4.151)
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. 4.37 (left)
E(t) := E H(t) e
−t/τ
.
(4.152)
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).
(4.153)
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 a linearly increasing creep strain in time
η ˙
(t) = [H(t) + τ δ(t)] σ 0 =⇒ (t) =: C(t) σ 0 .
(4.154)
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 linearly increasing creep
function that is illustrated in Fig. 4.37 (right)
147
Obviously the expressions in Eqs. 4.145 and 4.149 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 specific Maxwell model is summarized in Table 4.10.
Convolution Integral Representation
The rate form of the constitutive relation ˙
σ v = E [˙ − ˙
v ] = ˙
σ for the viscous stress
(as well as for the total stress) together with the evolution equation for the viscous
strain ˙
v = σ v /η = σ/η may be arranged in a differential equation relating the total
stress and strain as
σ(t) + τ ˙
σ(t) = η ˙
(t).
(4.150)
Accordingly, for ˙
= 0 the stress satisfies σ = −τ ˙
σ at any t, thus τ := η/E has
been introduced as the relaxation time of the Maxwell model.
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 ˙
σ(t) = η δ(t) 0 =⇒ σ(t) =: E(t) 0 .
(4.151)
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. 4.37 (left)
E(t) := E H(t) e
−t/τ
.
(4.152)
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).
(4.153)
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 a linearly increasing creep strain in time
η ˙
(t) = [H(t) + τ δ(t)] σ 0 =⇒ (t) =: C(t) σ 0 .
(4.154)
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 linearly increasing creep
function that is illustrated in Fig. 4.37 (right)