Viscoelasticity - Orthopaedic Biomaterials in Research and Practice - page 83

PROBLEM 4.2
Doesn't the statement “after removal of the external load, the internal
stress remains constant” violate translational equilibrium conditions?

ANSWER:
No. The external load is replaced by a constraint on strain. One can think
of it as replacing a weight on a spring with a pin or nail constraining
the end of the spring. The internal stress remaining will be balanced by
an equal and opposite resultant force exerted by the pin or nail, so that
equilibrium is maintained. If there is no internal stress, then there will
be no resultant force; this is different from the prior situation in which
the external force was fixed at a constant value. So in short, the external
force was simply replaced by a reaction force, which balances transla-
tional equilibrium conditions.
The utility of the standard test becomes more obvious when it is
applied to the Newton body. The results are shown in Figure 4.6. The
application of constant stress, σ 0 , produces no initial strain (ε 0 = 0), but
strain increases at a constant rate. The strain is then fixed at t 1 , and the
internal stress drops to zero.
In this case, the modulus, E ( t ), has an interesting behavior. At t 0 , it
is infinite (since ε 0 = 0), and it decreases asymptotically to σ 0 /ε 1 as time
approaches t 1 . At t = t 1 , the internal stress becomes zero and so does E ( t ).
The response of a St. Venant body to the standard test is identical to
that of the Newtonian body, so long as σ 0 ≥ σ c . (In Figures 4.6 through
4.9, stress vectors are omitted for simplicity.)
Using these three mechanical elements, it is possible to construct
more complex models whose stress and strain behaviors as a function of
time begin to resemble those found in real viscoelastic materials. These
models contain several of the fundamental bodies and are called units .

t = t 0

σ 0

σ

σ

L 0

ε

t = t 1

ε 0

+∞

L

E ( t )

ε 0 = 0 ε 1 = ( L - L 0 )/ L 0

E 0 = + ∞ E 1 = σ 0 /ε 1' E ( t > t 1 ) = 0

Time

t 0 t l

FIGUre 4.6 the standard test for the dashpot or newton body.