Creep, Stress Relaxation and Non-linear Viscoelasticity

In Chapter 4 we introduced linear viscoelasticity. In this scheme, observed creep or stress relaxation behaviour can be viewed as the defining characteristic of the material. The definition of the creep compliance function J t), which is given as the ratio of creep strain e t) to the constant stress o, may be recalled as 

The most significant aspect of Jis that it is a function of time only. This implies an exact proportionality between the magnitudes of strain and stress at any given time, and it is in this sense that the material is linear a doubling of stress always produces a doubling of strain. A single experiment is sufficient to define the material s creep behaviour in all circumstances. Similar comments apply to the stress relaxation modulus G t defined in terms of the stress o t) at constant strain e  

It can be shown that if a material is linear in the sense of Equation (10.1) then it is also linear in the sense of Equation (10.2), and vice versa J and G are mathematically related [1]. Thus, a linear material is one for which the creep compliance function or the stress relaxation modulus is a function of time only. When this is not the case, the material is non-linear. For example, the following simple forms are characteristic of non-linear viscoelastic materials  

An Introduction to the Mechanical Properties of Solid Polymers I. M. Ward and J. Sweeney 2004 John Wiley Sons, Ltd ISBN 0471 49625 1 (HB) 0471 49626 X (PB) 

Once modifications to functions of this kind have been made, the Boltzmann superposition principle can no longer be assumed to apply, and there is no simple replacement for it. This marks a significant change in the level of difficulty when moving from linear to non-linear theory. In the linear case, the material behaviour is defined fully by single-step creep and stress relaxation, and the result of any other stress or strain history then can be calculated using the Boltzmann integral. In the non-linear case we have lost the Boltzmann equation, and it is not even clear what measurements are needed for a full definition of the material. 

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Creep, stress relaxation and deformation under constant strain rate can be described assuming a viscoelastic response. Application of a constant strain can give rise to yield in thermoplastic materials. At yield the viscoelastic behaviour is non-linear, though the transition to non-linear is likely to occur prior to yield. 

Non-linear, time-dependent characteristics of viscoelastic materials such as polyethylene have been mathematically modelled and the model compared with experimental results. Mechanical properties such as creep and stress relaxation are non-linear because they include time-dependent and irreversible components. The time-dependent component is non-linear when relaxation time is longer than the timeframe of the experiment. This becomes increasingly so at high stress. Low stress will act on faster responding deformation modes and as stress increases slower modes will respond. The slower modes will be non-linear relative to the timescale of the experiment. Some slower modes such as relative translation of molecules are irreversible. Stress relaxation is complementary to creep in that strain is applied creating a stress that may relax according to the relative times of the experiment and molecular processes. 

According to the change of strain rate versus stress the response of the material can be categorized as linear, non-linear, or plastic. When linear response take place the material is categorized as a Newtonian. When the material is considered as Newtonian, the stress is linearly proportional to the strain rate. Then the material exhibits a non-linear response to the strain rate, it is categorized as Non Newtonian material. There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. This kind of materials are known as thixotropic deformation is observed when the stress is independent of the strain rate [2,3], In some cases viscoelastic materials behave as rubbers. In fact, in the case of many polymers specially those with crosslinking, rubber elasticity is observed. In these systems hysteresis, stress relaxation and creep take place. 

Inspection of this equation shows that it models reasonably well, on a very superficial level, a stress-strain curve of the type shown in Fig. 1(b), curve (4). In other words it raises the question as to whether the deviations from linear stress-strain relationships observed in constant strain-rate tests might not be merely resulting from the intrinsic time-dependence of the linear viscoelasticity, which can be more clearly studied in creep or stress-relaxation and not due to some new process starting at high stresses. It does not take long to show that at the strain-levels of 3-5% experienced at yield, the response of most polymers is highly non-linear (r(t)/ is a function of strain-rate S as well as t, and so eqn. (14) cannot adequately describe the behaviour. However, it is also clear that at... 

Under static loading conditions where either the stress or strain is keeping constant polymer materials (especially thermoplastics) show non-linear viscoelastic deformation behaviour to appear as retardation (creep) or relaxation. Long-term investigations to analyse creep or relaxation can be accomplished at flexural, indentation, or uniaxial tensile or compression loading as a function of time and loading level as well as environmental conditions such as temperature, media etc. (see [13Gre], p. 171 - 183). 

A phenomenological model has been proposed for the non-linear viscoelastic behaviour of thermorheologjcally complex polymer glasses prior to and including yield. The approach was based upon stress additivity. A linear viscoelastic material will exhibit stress-strain additivity. The molecular processes modelled were resolved into two parallel processes, each with a characteristic relaxation time spectrum. The model described the yield behaviour and creep experiments at increasing stress. " ... 

The non linear viscoelasticity of various particles filled rubber is addressed in range of studies. It is found that the carbon black filled-elastomer exhibit quasi-static and dynamic response of nonlinearity. Hartmann reported a state of stress which is the superposition of a time independent, long-term, response (hyperelastic) and a time dependent, short-term, response in carbon black filled-rubber when loaded with time-dependent external forces. The short term stresses were larger than the long term hyperelastic ones. The authors had done a comparative study for the non linear viscoelastic models undergoing relaxation, creep and hysteresis tests [20-22]. For reproducible and accurate viscoelastic parameters an experimental procedure is developed using an ad hoc nonlinear optimization algorithm. 

Viscoelasticity is a phenomenon observed in most of the polymers since they possess elastic and viscous characteristics when deformed. The properties such as creep, stress relaxation, mechanical damping, vibration absorption and hysteresis are included in viscoelasticity. If a material shows linear variation of strain upon the application of stress on it, its behavior is said to be linear viscoelastic. Elastomers and soft biological tissues undergo large deformations and exhibit time dependent stress strain behavior and are nonlinear viscoelastic materials. The non-linear viscoelastic properties of solid polymers are often based on creep and stress-... 

Two principal approaches have been used to model the yield behaviour of polymers. The first approach addresses the temperature and strain-rate dependence of the yield stress in terms of the Eyring equation for thermally activated processes [39]. This approach has been applied to many amorphous and crystalline polymers (see Section 12.5.1) and links have been established with molecular relaxation processes determined by dynamic mechanical and dielectric measurements and with non-linear viscoelastic behaviour determined by creep and stress relaxation. The Eyring approach assumes that the yield process is velocity controlled, i.e. the yield process relates to existing thermally activated processes that are accelerated by the application of the yield stress to the point where the rate of plastic deformation reaches the applied macroscopic strain rate. This approach has... 

In Chapter 5, we introduced linear viscoelasticity. In this scheme, the observed creep or stress relaxation behaviour can be viewed as the defining characteristic of the material. The creep compliance function - the ratio of creep strain e t) to the constant stress a - is a function of time only and is denoted as J t). Similarly and necessarily, the stress relaxation modulus, the ratio of stress to the constant strain, is the function G(r). Any system in which these two conditions do not apply is non-linear. Then, the many useful and elegant properties associated with the linear theory, notably the Boltzmann superposition principle, no longer apply and theories to predict stress or strain are approximations that must be supported by experiment. 

In order to predict the creep behavior and possibly the ensuing failure a number of approaches have been proposed. These are based respectively on the theory of viscoelasticity — including the concept of free volume — or on empirical representations of e(t) or of the creep modulus E(t) = ao/e(t). The framework of the linear theory of viscoelasticity permits the calculation of viscoelastic moduli from relaxation time spectra and their inter conversion. The reduction of stresses and time periods according to the time-temperature superposition principle frequently allows establishment of master-curves and thus the extrapolation to large values of t (cf. Chapter 2). The strain levels presently utilized in load bearing polymers, however, are generally in the non-linear range of viscoelasticity. This restricts the use of otherwise known relaxation time spectra or viscoelastic moduli in the derivation of e (t) or E (t). 

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