Creep and Relaxation

In this section, we discuss in more detail the physical content of the constitutive relations introduced in Sect. 1.2. Consider the behaviour of a material suddenly subjected to a constant stress a at time having been unstrained before then. From (1.2.30) we have that 

In other words, the resultant strain is not instantaneous but develops over time to a final value 

It is observed experimentally, and is anyway intuitively reasonable, that the suddenly imposed stress causes a certain instantaneous strain. Subsequently, the strain increases either to some final value or indefinitely. 

This behaviour is termed creep and, as noted previously, J(t) is termed the creep function. If creep ceases eventually, this corresponds to saying that /(oo) is finite. Equation (1.4.2) suggests that /(oo) may be regarded as a natural generalization of the inverse elastic modulus. We will sometimes denote it by J. For certain materials, creep continues indefinitely. Such behaviour is akin to that of a liquid. Certain classes of viscoelastic materials are in fact referred to as viscoelastic liquids. There are materials with the property that the limit 

Inserting (1.4.4) into (1.4.1) and differentiating with respect to time shows that rj is the effective coefficient of viscosity at large times. Note that J(t) may diverge for large /, in the case of a viscoelastic solid, as a 1, for example. 

---

DavkJ Ford Sims, Viscoelastic Creep and Relaxation Behavior of Laminated Composite Plates, Ph.O. dissertation. Department of Mechanical Engineering and Solid Mechanics Center, Institute of Technology, Southern Methodist University, Dallas, Texas, 1972. (Also available from Xerox University Microfilms as Order 72-27,298.)... 

The behavior of materials under dynamic load is of considerable importance and interest in most mechanical analyses of design problems where these loads exist. The complex workings of the dynamic behavior problem can best be appreciated by summarizing the range of interactions of dynamic loads that exist for all the different types of materials. Dynamic loads involve the interactions of creep and relaxation loads, vibratory and transient fatigue loads, low-velocity impacts measurable sometimes in milliseconds, high-velocity impacts measurable in microseconds, and hypervelocity impacts as summarized in Fig. 2-4. 

Material behavior have many classifications. Examples are (1) creep, and relaxation behavior with a primary load environment of high or moderate temperatures (2) fatigue, viscoelastic, and elastic range vibration or impact (3) fluidlike flow, as a solid to a gas, which is a very high velocity or hypervelocity impact and (4) crack propagation and environmental embrittlement, as well as ductile and brittle fractures. 

Predictions can be made on creep behavior based on creep and relaxation data. 

There is generally a less-pronounced curvature when creep and relaxation data are plotted log-log. Tliis facilitates extrapolation and is commonly practiced, particularly with creep modulus and creep-rupture data. 

Viscoelastic and rate theory To aid the designer the viscoelastic and rate theories can be used to predict long-term mechanical behavior from short-term creep and relaxation data. Plastic properties are generally affected by relatively small temperature changes or changes in the rate of loading application. 

Creep and relaxation AMS, styrene -acrylonitrile copolymer, poly vinyl chloride), polyfrnclliyl methacrylate)... 

Star-shaped polymer molecules with long branches not only increase the viscosity in the molten state and the steady-state compliance, but the star polymers also decrease the rate of stress relaxation (and creep) compared to a linear polymer (169). The decrease in creep and relaxation rate of star-shaped molecules can be due to extra entanglements because of the many long branches, or the effect can be due to the suppression of reptation of the branches. Linear polymers can reptate, but the bulky center of the star and the different directions of the branch chains from the center make reptation difficult. 

The relationship between creep and relaxation experiments is more complex. The complexity of the transforms tends to increase when stress and strain lead experiments are transformed in the time domain. This can be tackled in a number of ways. One mathematical form relating the two is known as the Volterra integral equation which is notoriously difficult to evaluate. Another, and perhaps the conceptually simplest form of the mathematical transform, treats the problem as a functional. Put simply, a functional is a rule which gives a set of functions when another set has been specified. The details are not important for this discussion, it is the result which is most useful ... 

These two equations enable creep and relaxation to be related and complete our simple combination of interrelations. The interrelations are summarised in Figure 4.18. The question we would really like to answer is how these interrelations apply to real systems. We can get an idea by looking at simple models. This is considered in the following section. 

For the same reason, the creep and relaxation behaviours are not as good as for the thermosets. 

K. Onaram, W.H. Findley, Creep and Relaxation of Nonlinear Viscoelastic Materials , Dover Publications, New York (1989). 

Creep and relaxation experiments are carried out on time scales ranging from several... 

In combination with creep and relaxation measurements a total time span of l(f8 sec to 108 sec (3 years) can be covered.The usual techniques to measure the response of a polymer over this time span are schematically represented in Figure 6.12. 

Deformation ( + A ) and recovery ( — AE) along dissimilar pathways beget hysteresis. The elastic segment of creep and relaxation can occur at the same rate only when there is no hysteresis. Accordingly, in the absence of hysteresis, t1 is the time required for a viscoelastic fluid to reach 63% of the maximum deformation under stress. 

Measurement of the linear viscoelastic properties is the basic rheological characterization of polymer melts. These properties may he evaluated in the time domain (mainly creep and relaxation experiments) or in the frequency domain in this case we will talk about mechanical spectroscopy, where the sample experiences a harmonic stimulus (either stress or strain). 

Y]sf Findlay, JS Lai, K Onaran. Creep and Relaxation of Non-Linear Viscoelastic Materials. Amsterdam, North-Holland, 1976, 71. 

Chapters 5 and 6 discuss how the mechanical characteristics of a material (solid, liquid, or viscoelastic) can be defined by comparing the mean relaxation time and the time scale of both creep and relaxation experiments, in which the transient creep compliance function and the transient relaxation modulus for viscoelastic materials can be determined. These chapters explain how the Boltzmann superposition principle can be applied to predict the evolution of either the deformation or the stress for continuous and discontinuous mechanical histories in linear viscoelasticity. Mathematical relationships between transient compliance functions and transient relaxation moduli are obtained, and interrelations between viscoelastic functions in the time and frequency domains are given. 

In some epoxy systems ( 1, ), it has been shown that, as expected, creep and stress relaxation depend on the stoichiometry and degree of cure. The time-temperature superposition principle ( 3) has been applied successfully to creep and relaxation behavior in some epoxies (4-6)as well as to other mechanical properties (5-7). More recently, Kitoh and Suzuki ( ) showed that the Williams-Landel-Ferry (WLF) equation (3 ) was applicable to networks (with equivalence of functional groups) based on nineteen-carbon aliphatic segments between crosslinks but not to tighter networks such as those based on bisphenol-A-type prepolymers cured with m-phenylene diamine. Relaxation in the latter resin followed an Arrhenius-type equation. 

The superposition principle can be used to predict the creep and relaxation behavior at any temperature if some results are already available, with the proviso that the most reliable predictions can be made for interpolated temperatures rather than long extrapolations. 

---