Stress relaxation

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Stress relaxation describes how polymers relieve stress under constant strain. Because they are viscoelastic, polymers behave in a non-linear, non Hookean, fashion. This non-linearity is described by both stress relaxation and a phenomenon known as creep (deformation), which describes how polymers strain under constant stress.

Viscoelastic materials have the properties of both viscous and elastic materials and can be modeled by combining elements that represent these characteristics. Models like the Maxwell Model predict behavior akin to a spring (elastic element) being in series with a dashpot (viscous element), while the Kelvin-Voigt Model places these elements in parallel. Though Maxwell can predict stress relaxation it is fairly poor at predicting creep. Voigt, on the other hand, is good for predicting creep but rather poor at predicting stress relaxation. The most accurate of the viscoelastic models is the Standard Linear Solid Model, which combines the characteristics of both Maxwell and Voigt to display both creep and stress relaxation (See Viscoelasticity).

The following image shows the response of a Standard Linear Solid material to a constant stress, σ0, over time from t0 to a later time tf. For times greater than tf the load is removed. The curvature of the model represent the effects of both creep and stress relaxation.

E1 and E2 refer to the spring constants of the elastic elements of the model.

slsm image2

Stress relaxation calculations can differ for different materials:

To generalize, Obukhov uses power dependencies [2]:

\sigma(t)= \frac { \sigma_0 }{ 1-[1-(t/t*)(1^{1-n})]}

where σ0 is the maximum stress at the time the loading was removed (t*), and n is a material parameter.


Vegener et al. use a power series to describe stress relaxation in polyamides [2]:

\sigma(t)= \sum_{mn}^{} { A_{mn} [ln(1+t)]^m (\epsilon'_0)^n}


To model stress relaxation in glass materials Dowvalter uses the following [2]:

\sigma(t) = \frac { 1 }{ b }* log {\frac{10^{\alpha}(t-t_n)+1}{10^{\alpha}(t-t_n)-1}}

where α is a material constant and b and tn depend on processing conditions.


The following non-material parameters all affect stress relaxation in polymers [2]:

  • Magnitude of initial loading
  • Speed of loading
  • Temperature (isothermal vs non-isothermal conditions)
  • Loading medium
  • Friction and wear
  • Long-term storage


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