7. VISCOELASTIC SOLIDS − FUNG’S QUASILINEAR THEORY
Viscoelastic solids and fluids
We have already recalled that a material has a viscoelastic behaviour if its response to external
actions is time dependent: the relationship between stresses and strains within the material is
not the same at all times, but it depends on their time history.
We already recalled that a creep experiment consists of applying at time t = 0 a step stress, i.e.
a stress increment ∆σ(0), which is kept constant for t > 0, and measuring the corresponding
strain response ε(t) for t ≥ 0.
Real fluid
σ
Ideal fluid
ε
Real solid
∆σ(0)
t=0
Ideal solid
t
t=0
t
An ideal (elastic) solid has a constant strain for t > 0 and t = 0+: ε(0+) = ε(t) ≠ 0 ∀t > 0.
An ideal (viscous) fluid does not have an instantaneous deformation response ε(0+) = 0 and it
creeps indefinitely under constant stress: ε(t) → ∞ as t → ∞.
A real fluid has a (small) instantaneous elastic deformation, ε(0+) ≠ 0, but it creeps indefinitely
under constant stress: ε(t) → ∞ as t → ∞.
A real solid has an instantaneous deformation ε(0+) ≠ 0, it creeps under constant stress ε̇(t) > 0,
for t > 0, but the deformation remains bounded for large times (ε(t) → ε∞ < ∞, as t → ∞).
In the case of linear viscoelasticity the Creep Compliance C(t) relates the creep strain ε(t) at
time t with the stress increment ∆σ applied at time 0:
ε(t) = C(t) ∆σ(0)
In other words, C(t) is the strain ε(t) at time t due to a unit stress increment at time 0: C(t)=ε(t)
for ∆σ(0) = 1
On the other hand, a relaxation experiment consists of applying at time t = 0 a step strain, i.e.
a strain increment ∆ε(0), which is kept constant for t > 0, and measuring the corresponding
stress response σ(t) for t≥0.
+∞
Ideal fluid
σ
ε
Ideal solid
∆ε(0)
Real fluid
t=0
t
t=0
Ideal fluid
Real solid
t
An ideal (elastic) solid has a constant stress for t > 0 and t = 0+: σ(0+) = σ(t) ≠ 0 ∀t > 0.
An ideal (viscous) fluid has an infinite stress σ(0) → ∞ at the initial strain jump ∆ε(0) (an
infinite strain rate at t = 0, ε̇(0) → ∞) and has a null stress for the subsequent constant
deformation: σ(t) = 0 ∀t > 0.
A real fluid has a (large) stress σ(0) at the initial strain jump, but, at constant strain, that stress
is completely relaxed after sufficiently large time: σ(t) → 0 as t → ∞.
A real solid has an instantaneous stress response σ(0+) for the initial strain jump ∆ε(0), it
relaxes part of that stress under constant stress (σ̇(t) < 0, for t > 0), but it keeps a residual stress
for large times (σ(t) → σ∞ as t → ∞, 0 < σ∞ < σ(0+)).
In the case of linear viscoelasticity the relaxation modulus G(t) relates the relaxation stress σ(t)
at time t with the strain increment ∆ε applied at time 0:
σ(t) = G(t) ∆ε(0)
In other words, G(t) is the stress σ(t) at time t due to a unit strain increment at time 0: G(t)=σ(t)
for ∆ε(0) = 1
Superposition of effects in linear viscoelasticity
Suppose that various strain increments ∆ε(τi) occur at the successive time instants τi, i = 1, 2, …
and that we wish to know the stress response at the later time t (t ≥ max τi). Superposing the
effects of the successive strain increments, we get:
σ(t) = ∑ G(t − τi) ∆ε(τi)
(S)
i
ε
σ
∆ε(τ3)
∆ε(τ2)
∆ε(τ1)
τ1
τ2
τ3
t
τ1
τ2
τ3
t
In case the strain varies smoothly with time the superposition of the effects of the elementary
∂ε
increments of strain at the previous times τ, dε(τ) =
(τ) dτ, leads to the hereditary integral
∂τ
∂ε
⌠ G(t − τ) (τ) dτ
σ(t) = ⌡
∂τ
t
−∞
(I)
Note that this means that the elementary increment of strain dε(τ) at the previous time τ leads to
an elementary increment of stress dσ(t) at the current time t given by:
dσ(t) = G(t − τ) dε(τ)
The integration starting at τ = −∞ means that the effect of all relevant previous strain changes
must be taken into account.
Assuming that the deformation starts at τ = 0, with ε(τ) = 0 for τ < 0, and that a possibly nonvanishing jump exists at τ = 0 [∆ε(0) = ε(0+) − 0 = ε(0+)], it follows from (S) and (I) that:
t
⌠ G(t − τ)
σ(t) = G(t) ε(0 ) + ⌡
+
0
∂ε
(τ) dτ
∂τ
(SI)
All the above developments show that in linear viscoelasticity the stresses depend linearly on
the strains and the stress at the current time t results from the linear superposition of the effects
of the strain changes in all relevant previous times τ.
Fung’s quasilinear theory for bio-viscoelastic solids (1D)
Fung’s theory is also based on the idea of linearly superposing at the current time the effects
from past changes in the system, and leads also to a hereditary integral. But the theory cannot
preserve the linear dependence of the stresses on the strains, since, as already observed, the
elastic behaviour of the soft tissues typically is highly nonlinear.
Let T be the nominal stress, λ the stretch ratio, and let T(e)(λ) denote the instantaneous elastic
response of the material, i.e. the stress instantaneously generated in the tissue when a step
stretch ratio λ is imposed on the tissue. In other words, T(e)(λ) characterizes the elastic
constitutive law that holds when relaxation effects are not taken into account. T(e)(λ) is typically
a nonlinear function that frequently involves exponential contributions.
In the figure below, λ(t) = 1, T(e)(t) = 0 for t < 0. At t = 0 a step stretch ratio is imposed on the
tissue such that, for all t > 0 λ(t) = λ(0+) and, by definition of T(e),
T(e)(λ(0+)) = T(λ(0+)).
(T0)
In order to take into account the relaxation effects in the quasilinear viscoelastic theory of Fung
a reduced relaxation modulus G(t) is introduced which relaxes the instantaneous elastic
response for t > 0,
T(t) = G(t) T(e)(λ(0+)),
with, from (T0) and (TG),
G(0) = 1 .
(TG)
λ
T
λ(0+)
T(λ(0+)) = T (e)(λ(0+))
T(t) = G(t) T(e)(λ(0+))
λ=1
t=0
t
t=0
t
Comparing these results with the linear viscoelasticity, it should be clear that:
• in linear viscoelasticity G(t) has dimensions of stress, but it is non-dimensional in
quasilinear viscoelasticity,
• in quasilinear viscoelasticity G(t) is normalized to be 1 at t = 0.
Assuming that λ(τ) = 1, T(e)(τ) = 0 for τ < 0, the proposed 1D quasilinear constitutive equation
for soft tissues has the form
∂T(e)
T(t) = G(t) T (λ(0 )) + ⌠
⌡ G(t − τ) ∂τ (τ) dτ
0
(e)
+
t
Consequently:
• This constitutive law involves linear superposition of the effects of the instantaneous elastic
stresses at all previous times. Linear viscoelasticity involves linear superposition of strains,
which is not valid in quasilinear viscoelasticity.