## Viscoelastic Response of Polymers under Stress

When viscoelastic materials are subjected to
stress they undergo deformations by molecular rearrangements and by
viscoelastic flow. To study these deformations, *creep experiments*
are often conducted. In these experiments, a sample is subjected to an instantaneous load
at
time *t*_{0} and the
strain (creep) is recorded as a function of time at constant
temperature.

**Crosslinked
elastomers (**ideal elastic materials) undergo only elastic
deformations. The creep under stress of these materials can be described with the
one-dimensional Voigt model. The
basic equation for the time dependence of strain during creep is
given by

*σ(t)* = *σ(t) _{Elastic}* +

*σ(t)*=

_{Viscous}*E*·

*ε*+

*η*· dε / d

*t*

or

*σ(t)* / *η* = dε / d*t* + *ε*(*t*) / *λ*

where *λ* is the *response or relaxation time* of the one-dimensional
Voigt element which is represented by the ratio of viscosity* η* to
Young's modulus
*E*. To solve this differential equaition we multiply both
sides with the integration factor exp(*t* / *λ*):

*σ(t)* e^{t/λ} / *η* = dε / d*t* e^{t/λ }+ *ε*(*t*) e^{t/λ} / *λ*

or

*σ(t)* e^{t/λ} / *η* = d[*ε*(*t*)
e^{t/λ}] / d*t*

Notice, the viscoelastic flow is restricted by the dash-pot in
parallel. Hence, there is not instantaneous strain when
stress is applied at the start of the experiment at *t*_{0}. Assuming *ε*(*t _{0}*) = 0, integration of
the equation above yields

Assuming constant stress, *σ*(*t* ≥ *t _{0}*) =

*σ*, the integral on the right-hand-side can be easily solved:

_{0}Hence, the strain *ε*(*t*) reaches the
asymptotic limit *σ _{0}* /

*E*when

*t*→

*∞*which is the equlibrium extension of the elastic element. The equilibrium time depends on the temperature; when the temperature increases the

*relaxation time*decreases and the asymptotic limit is reached faster.

**Plastic materials** undergo irreversible
deformation when exposed to stress. This situation can be
best described with the one-dimensional Maxwell model. If
a constant stress *σ*_{0} is applied then the Maxwell element reaches following strain:

*σ _{0 } *

_{}

*= η*(d

*ε*/ d

*t*) ⇒

*ε*(*t*) = *σ*_{0} {1 / *E *+ * t* /* η*}

The initial response is identical with the response of the spring, because the movement of the spring in the Maxwell element is not restricted by the dash-pot in series. Thus

*ε*(*t *= 0) =
*σ*_{0} / *E *

The dash-pot simulates the liquid-like behavior at large observation times. After the stress is removed, the spring of the Maxwell element recovers instantaneously but the dash-pot remains at its final position which represent the viscous flow of the material. This behavior is typical for amorphous polymers without chemical crosslinks which exhibit viscous flow (chain slippage) when exposed to stress.

The viscoelastic behavior of real polymeric materials is usually more complicated. To accurately describe the creep behavior of these materials, Maxwell and Voigt elements have to be combined to more complex arrangments.