Viscoelasticity - Orthopaedic Biomaterials in Research and Practice - page 85

t = t 0 (σ = σ 0 )

σ 0

L 0

σ

t = t 1 (σ = σ 0 )

ε

L

ε 0

+ ∞

t = t 1 (ε = ε 1 )

E ( t )

L

t 0 t 1

Time

FIGUre 4.8 the Kelvin unit: spring/hooke body attached in parallel
to a dashpot/newton body.

the externally applied force, reducing the net stress on the dashpot and
thus reducing the rate of deformation of the dashpot at a constant rate.
At t = t 1 , the external strain is fixed (made constant), and the stress in
the dashpot becomes zero but the stress in the spring does not, since it is
strained from its rest position. Thus, there is an instantaneous reduction
in stress, but not to zero, as in the case of a single dashpot. In this case,
however, the strain is unrecoverable and there are no further changes in
stress or strain.

PROBLEM 4.3
In Figure 4.8, what feature regulates the ratio σ 1 /σ 0 ?

ANSWER:
Since σ 1 is the stress in the spring at time t 1 , the ratio increases with
increasing t 1 − t 0 , asymptotically approaching a value of 1.

Standard
linear (three-
body) model

The Maxwell and the Kelvin units combine viscous and elastic behavior
but are not sufficiently complex to be able to represent the stress-strain
relationships in real solids. The simplest model that can do that is the
standard linear (or three-body) model. (There are other three-body mod-
els that can represent the properties of real materials. The simplest of
these is a Maxwell unit in parallel with a spring. This is equivalent to the
standard linear model considered here.) This consists of a Kelvin unit
in series with another spring, as shown in Figure 4.9. In this case, the
springs are shown as having different stiffnesses.
The results of the standard test are shown in Figure 4.10. There is an
initial elastic strain, ε 0 , corresponding to a deformation of spring 2, since
spring 1 is constrained by the dashpot. The dashpot begins to deform,
which produces a restraining stress in spring 1, reducing the stress on
the dashpot below σ 0 and thus reducing the strain rate. At t = t 1 , when