EN222: Mechanics of Solids

 

 

 

 

 

  

 Division of Engineering

    Brown University

 

 

2.4 Viscoelasticity

 

Amorphous polymers show complex time-dependent behavior when subjected to a history of stress or strain.  Viscoelasticity theory was developed to approximate the main features of this behavior.

 

Features of the rate dependent response of polymers

 

Like plastically deforming metals, and rubbers, most polymers strongly resist volume changes, but show much more compliant shear response.  In addition, the response in shear is virtually independent of hydrostatic loading.   Consequently, most experiments focus on the shear response of the material – a typical test is to take a specimen that is free of stress at time , apply a constant shear stress  for  and measure the resulting shear strain  as a function of time.  The results are generally presented by plotting either the `creep compliance’

or the `relaxation modulus’

as a function of time.

 

The results of such a test depend on the degree of cross-linking in the polymer.  Heavily cross-linked materials show `retarded elastic’ behavior, while un-cross linked materials show steady-state creep.  In both cases, the results are strongly temperature dependent.

 

Typical `retarded elastic’ response is illustrated below, by plotting creep compliance as a function of time for various temperatures

The notable features of this behavior are:

(1)   There is always an instantaneous strain in response to a step change in stress.  The instantaneous compliance  is low, and only weakly dependent on temperature.

(2)   At low temperatures (well below the glass transition temperature) the solid is essentially elastic (there may be a very slow rate of creep), and has a very low compliance, comparable to .

(3)   At temperatures above the glass transition temperature, the solid is very compliant, and the compliance is a function of temperature.  The specimen will show an initial transient response, but will quite quickly settle to a constant strain (the strain may increase very slowly with time).  The time taken to reach steady state decreases with increasing temperature, and for some materials may be short enough that the transient response can be neglected.  In this case the material can be modeled using the hyperelastic constitutive law described in the preceding section.

(4)   For a range of temperatures both above and below the glass transition temperature, the solid shows a slow transient response.

(5)   The deformation is reversible – if the loading is removed, the specimen will eventually return to its original configuration, although in the transition regime this may take a very long time.

 

`Steady-state creep’ behavior is illustrated below. 

The notable features of this behavior are:

(1)   There is always an instantaneous strain in response to a step change in stress, exactly as in crosslinked polymers..

(2)   At low temperatures (well below the glass transition temperature) the solid is essentially elastic (there may be a very slow rate of creep), and has a very low compliance, comparable to .

(3)   At temperatures above the glass transition temperature, the solid is very compliant.  It may show rubbery behavior for very low stresses, but for most practical ranges of loading the compliance will increase more-or-less linearly with time (especially for short time intervals).  The rate of change of compliance is strongly temperature dependent

(4)   Above the glass transition temperature, the deformation is irrreversible – if the loading is removed, the specimen will not return to its original shape.

 

In addition to measuring the response of a material to a step change in load, one can subject it to cyclic stress, e.g. with stresses that vary sinusoidally with time .  The strain history will also be harmonic , and one can define a harmonic modulus by taking the ratio of stress amplitude to strain amplitude as

One can plot the modulus as a function of temperature or inverse frequency, with almost identical results – the typical trend is illustrated below

Actually, the connection between temperature and loading rate is more than just a qualitative trend.  This can be demonstrated clearly by measuring the (time dependent) relaxation modulus for the material, and showing that results for different temperatures can be collapsed onto a single master-curve.  The relaxation modulus is measured by subjecting a specimen to a step increase in strain, and measuring the resulting stress.  The stress progressively relaxes as the material’s compliance increases with time – in an uncrosslinked polymer it will relax to zero, whereas in a crosslinked polymer it will asymptote to a constant value.  The relaxation modulus is defined as .

 

One can measure the relaxation modulus at various temperatures.  Empirically, it is found that results for two temperatures  and  will be related by

where A is given by

and is called the Williams-Landell-Ferry (WLF) shift function.  The scaling holds for any two temperatures, but of course  and  must depend on the choice of .  It is convenient to use the glass transition temperature as the reference, in which case

provides the necessary scaling.  The values of  and  vary slightly (but surprisingly little) from one polymer to another.  The expression works (again surprisingly) for  both above and below  - but of course the whole thing blows up if .  For temperatures below the critical value, the response is elastic.

 

Given the variation of relaxation modulus with time at glass transition temperature, one can then deduce the variation at any arbitrary temperature, as

 

 

1-D models

 

Qualitatively, the behavior outlined in the preceding subsection can be modeled using simple linear spring-dashpot models.  Three common idealizations are shown below

 

In each case the force applied to the spring-dashpot system represents `stress’, while the extension represents `strain.’  It is straightforward to show that they are related by

Qualitatively, these models describe the behavior of a typical polymer.  The Kelvin-Voigt model gives retarded elastic behavior, and represents a crosslinked polymer.  The Maxwell model gives steady state creep, and would represent an uncrosslinked polymer.  With an appropriate choice of  and , the 3 parameter model can describe both types of behavior.  The models are too simple to give a good quantitative fit to any polymer over an extended period of time, but a generalized version of this relationship could be expressed as

which would provide enough constants to fit just about anything.

 

In practice, constitutive relations for polymers are usually expressed in integral, rather than differential form.  For example, note the the response of the Kelvin-Voigt material to a step increase in strain is

where  is a time constant.  The response to an arbitrary strain history can be found by summing an appropriate distribution of step functions

Again, this model is too simple to give a good fit to experiment.  But we can make a more versatile model by connecting a bunch of Maxwell elements in series, and then sticking a spring in parallel with the whole shebang.  The response of this stuff to a step increase in strain would be

where  is the steady-state stiffness (the parallel spring), and  are the time constants and stiffnesses of the Maxwell elements.  This expression contains enough constants to describe very complex relaxation behavior accurately.  The response to an arbitrary strain history can be computed by defining the relaxation modulus function

(This sum of exponentials is known as the `Prony series’), whence

This last expression is known as a `hereditary integral’ formulation of the constitutive response.  The general class of materials in which stress depends on strain history are known as materials with memory, and linear hereditary materials are a special class of such behavior.

 

Multi-axial constitutive relation

 

It is straightforward to extend the simple 1-D models outlined in the preceding section to more general loading.  To proceed, we assume

1.      Infinitesimal deformations;

2.      The solid is isotropic;

3.      The shear and hydrostatic response of the solid may be decoupled;

4.      Both shear and hydrostatic response can be described using a linear hereditary integral formulation.

 

Volumetric and deviatoric strains are defined in the usual way

and the stress-strain relation is then expressed as

where

are the shear and bulk relaxation modulus, and  is related to t through the WLF shift function

The constitutive relation must be calibrated by fitting values of  and  to experimental data.  Generally rate dependence of the bulk modulus can be neglected; and often the incompressible limit can be taken by setting  to a suitably large value.  The terms in the Prony series for shear modulus must be determined through an appropriate curve fit.   ABAQUS contains routines that will do this for you.