Creep Linear viscoelasticity, prediction

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

The mechanical response of polypropylene foam was studied over a wide range of strain rates and the linear and non-linear viscoelastic behaviour was analysed. The material was tested in creep and dynamic mechanical experiments and a correlation between strain rate effects and viscoelastic properties of the foam was obtained using viscoelasticity theory and separating strain and time effects. A scheme for the prediction of the stress-strain curve at any strain rate was developed in which a strain rate-dependent scaling factor was introduced. An energy absorption diagram was constructed. 14 refs. 

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. 

First, we need a rule to predict the effect of time-varying loads on a viscoelastic model. When a combination of loads is applied to an elastic material, the stress (and strain) components caused by each load in turn can be added. This addition concept is extended to linear viscoelastic materials. The Boltzmann superposition principle states that if a creep stress ai is... 

When plastics are unloaded, the creep strain is recoverable. This contrasts with metals, where creep strains are permanent. The Voigt linear viscoelastic model predicts that creep strains are 100% recoverable. The fractional recovered strain is defined as 1 — e/cmax, where e is the strain during recovery and Cmax is the strain at the end of the creep period. It exceeds 0.8 when the recovery time is equal to the creep time. Figure 7.9 shows that recovery is quicker for low Cmax and short creep times, i.e. when the creep approaches linear viscoelastic behaviour. 

The theory of linear viscoelasticity is phenomenological there is no attempt to discover the time and frequencty response of the solid in an altogether a priori fashion. The aim is to predict behaviour under certain circumstances, having observed it under others for example, to correlate creep, stress relaxation, and (fynamic properties so that if one of these has been determined then all the others are known. This is closety related to electrical network theory, both in aim and, as will soon be apparent, in method. 

Transient Response Creep. The creep behavior of the polsrmeric fluid in the nonlinear viscoelastic regime has some different features from what were found with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation... 

Thus, for linearly viscoelastic behavior, by measuring the creep strains it is possible to draw time-modified modulus curves. Having established these curves, it is then possible to use these data to predict the behavior of the plastic under other conditions. Such time-modified modulus curves for several common thermoplastics were shown in Figure 3-58. It is important to remember that such curves are valid only for a specific temperature and for strains that do not exceed the limits of the validity of the data. 

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. 

The breadth of the scope of nonlinear phenomena can be grasped in part by considering the various time-dependent probes of linear viscoelasticity cited in Table 3.3.2 sinusoidal oscillation, creep, constrained recoil, stress relaxation after step strain, stress relaxation after steady shearing, and stress growth after start-up of steady shearing. In the linear regime— that is, at small strains or small strain rates—the experimental results of any one of these probes (in simple shear, for example) can be used to predict results for any of the other probes, not only for simple shearing defor-... 

It is clear that any theoretical explanations of the above phenomena should be able to account for the dependence of the stress and strain upon time. Ideally it should be possible to predict, for example, the stress relaxation behaviour from knowledge of the creep curve. In practice, with real polymers this is somewhat difficult to do but the situation is often simplified by assuming that the polymer behaves as a linear viscoelastic material. It can be assumed that the deformation of the polymer is divided into an elastic component and a viscous component and that the deformation of the polymer can be described by a combination of Hooke s... 

This method can also be apphed to the extrapolation of time- and temperature-dependent creep behavior. Experimental creep curves first need to be obtained at a series of different temperatures over a specific time period, and the values of comphance plotted on a logarithmic time scale. After one creep curve at a chosen temperature is defined as reference, creep curves at other temperatures are then shifted one by one along the log time scale until they superimpose to a single curve in the ideal case. Curves above the reference temperature are shifted to the right, and those below are shifted to the left. This procedure can be apphed to predict longterm creep compliance on the basis of short-term tests at different temperature levels in the range of linear viscoelasticity. 

For amorphous polymers (PC, PIB, NR) the limit increases around Tg from 1 to about 50% of strain [112]. There are some special approaches to a calculation of creep and recovery based on non-linear theories of viscoelasticity [112—114]. At present it is not possible, however, to predict with sufficient generality the onset of accelerated creep and thus of delayed yielding from non-linear viscoelastic theories. 

Creep data are invaluable for predicting die long-term functional behavior of a material or product. However, the current body of data (seldom, if ever, reported on product data sheets) cannot be compared for a series of material candidates. Polymers must be tested exactly the same way ( test mode, initial stress level, time, and temperature) in order to have a valid comparison, without relying on mathematical adjustments. The concept of the use of creep modulus has been widely adopted. However, one must still adhere to the paradigms of linear viscoelasticity to allow valid comparisons of different polymeric materials. 

The results fiom the flexural creep data give a realistic prediction of the actual part stabihty in service. There is an increase in the creep compliance of the material with a decrease in the fiber volume fraction, an increase in temperature and stress, consistent with the tensile creep and static flexural tests. Creep compliances are shghtly higher perpendicular to flow direction. Results finm DMA and actual flexmal creep tests for reinforced nylon 6/6 are consistent in the linear viscoelastic regime. 

A comprehensive analytical model for predicting long term durability of resins and of fibre reinforced plastics (FRP) taking into account viscoelastic/viscoplastic creep, hygrothermal effects and the effects of physical and chemical aging on polymer response has been presented. An analytical tool consisting of a specialized test-bed finite element code, NOVA-3D, was used for the solution of complex stress analysis problems, including interactions between non-linear material constitutive behavior and environmental effects. 

The viscoelastic behaviour of rPET polymer concrete is linear under stress levels up to 30% as a result, the two-point method of stress levels higher than 30% will probably lead to considerable error in the predicted creep deformation values. 

Upon removal of the load (or stress) at time t, a sample corresponding to the Maxwell model will retract by a value equal to its elastic contribution (so = an,olE), but will be permanently strained by a value e(t) = (an,o/il)F In a creep experiment, such a sample behaves at the onset like an elastic solid and then like a viscous liquid thus it exhibits the characteristics of a viscoelastic liquid. However, this Maxwell model also predicts a linear deformation as a function of time when subjected to a constant stress which is not realistic, because no such example could be found in the field of polymers (see Figure 12.10). 

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). 

When a mechanical part is made from a polymer, and when it is to be used as a loadcarrying component, obviously it is not necessarily always going to be subject to a constant stress as in the creep test. It generally has to be designed to withstand some history of stress variation. How will the polymer respond to the stress history Can its response be predicted Fortunately, for hnear viscoelastic behavior, predicting the response is possible, because of the principle of superposition of solutions to linear differential equations. The student, of course, remembers that if y,(Ji ) and y2(x) are both solutions of an ordinary differential equation for y x), then the sum y (x) + yj (x) is also a solution. This is the basis of the Boltzmann Superposition Principle for linear polymer behavior. 

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