Variable temperature history

Formulae (1.7.2) apply only to the case where temperature does not vary with time. If the temperature is time-dependent, then the first point to be observed is that the non-aging hypothesis may break down. Consider a sudden constant strain e applied at Then at a later time t. 

Let us divide the time interval [to, t] into small sub-intervals Atj and make the approximation that the temperature has a constant value 7, during each of these. We now assume that each sub-interval of time is scaled according to the temperature prevailing at that time, so that 

This result was first given by Morland and Lee (1960), who refer to as the reduced time or the pseudo-time. Note that 

The strain in these equations should be considered to include a contribution due to thermal expansion of the material. If the temperature depends on position, the viscoelastic function will also, and space homogeneity is lost. Equation (1.7.8) remains valid in fact but the dynamical equations, which will be introduced in Sect. 1.8, are altered in a significant manner. 

The dependence on temperature of the viscoelastic functions is clearly nonlinear, though the mechanical constitutive relation, discussed in Sect. 1.2, is linear. If the temperature variation is small, the theory may be completely linearized. Let 

---

There is a striking contrast between the millennial variability of polar interglacial and glacial temperature histories. Whereas climate changes were frequent and large throughout... 

When reactions are fast relative to the mixing rate, not only are the apparent reaction rates affected but the whole time and temperature history of the reaction mechanism is also affected, yielding different selec-tivities and yields, depending on the intensity of the mixing. This often leads to a scale-up/scale-down problem, where yields of the desirable products in a plant-scale reactor are not as good as those in a small-scale reactor in the laboratory or the pilot plant. If the yield drops from the pilot-scale to the plant-scale reactor when all other important variables (temperature, pressure, and composition) have been held constant, then there is a mixing problem. Fast... 

In conclusion, many different processes occur during fat crystallization, at widely varying rates, all depending on temperature (history) and on fat composition. This makes it very difficult to predict the mechanical properties of plastic fats. However, the various processes involved have been identified, and their dependencies on composition and several external variables have been established. This means that trends can often be predicted. 

From one point of view, the solution depends on obtaining a relationship between temperature and composition, so the terms of equation (1-164) can be expressed in terms of a single dependent variable and, hopefully, integrated analytically. Alternatively, one could look for a temperature-time relationship. This may not be easy to do, however, since the time-temperature or composition-temperature history of a reaction in which heat is evolved or consumed is a function of the rate itself. Obviously, one must look to another relationship in addition to that of mass conservation in order to obtain this history. 

External factors that will influence polymer mechanical properties are temperature or thermal treatment, temperature history, large differences in pressure, and environmental factors such as humidity, solar radiation, or other types of radiation. The mechanical properties of polymer are also sensitive to the methods and variables used for testing, such as strain deformation as well as the rate at which the strain is performed. Finally, the mechanical behavior of polymeric materials and the values of their mechanical properties will be sensitive to the kind of strain that is imposed by the applied force, namely, tension, compression, biaxial, or shear. 

Equation 17 can be solved numerically for any specified temperature history. However, the mesophase temperature history is a function of both time and position (see Fig. 4), and so the transient temperature distribution T(x, t) must be specified in order to obtain an analytic solution for equation 17. If the temperature distribution is static during steady burning of a thick sample as the surface x = 0 recedes at constant velocity v = (dx/df)r and the mesophase is a thin surface layer, then the rate of temperature rise of the mesophase is constant, (dT/df)a =o = -v(dT/dx)t = The assumption of a constant heating rate for the mesophase transforms the independent variable in equation 17 from time to temperature and allows for a solution in terms of the mass fraction of polymer remaining at temperature T... 

We have repeatedly used the term hydrodynamic, and we now give it a more precise definition. By a hydrodynamic process we mean one for which the local thermodynamic variables, temperature, chemical potential (or density), and velocity, are determined by the past history of their boundary values. The normal solution to the Boltzmann equation, as well as its generalization obtained in the previous Sections, then clearly corresponds to a hydrodynamic process. The significance of the term hydro-dynamic may be clarified by the consideration of some processes of non-hydrodynamic type. A process of relaxation in momentum space in a spatially uniform gas is clearly non-hydrod5mamic, since the local thermod5mamic variables are not at all pertinent to its description. Another example is provided by processes in a Knudsen gas. Here there is an essential dependence on the particular form of the boundary forces. An insensitivity to the nature of the boundary forces is implied in the definition of a hydrod5mamic process, for which it is immaterial whether a thermal reservoir is constructed of, say, copper or aluminum, and... 

During pyrolysis, the yield of gases and liquid products may vary from 25 to 70 percent by weight. The yield depends on a number of variables, such as coal type, gas atmosphere, final pyrolysis temperature, time-temperature history, and pressure. A certain operating condition may lead to increased product yield, but achievement of this process condition often is obtained at added expense. 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

It is important to note that the state determined by this analysis refers only to the pressure (or normal stress) and particle velocity. The material on either side of the point at which the shock waves collide reach the same pressure-particle velocity state, but other variables may be different from one side to the other. The material on the left-hand side experienced a different loading history than that on the right-hand side. In this example the material on the left-hand side would have a lower final temperature, because the first shock wave was smaller. Such a discontinuity of a variable, other than P or u that arises from a shock wave interaction within a material, is called a contact discontinuity. Contact discontinuities are frequently encountered in the context of inelastic behavior, which will be discussed in Chapter 5. 

Contact discontinuity A spatial discontinuity in one of the dependent variables other than normal stress (or pressure) and particle velocity. Examples such as density, specific internal energy, or temperature are possible. The contact discontinuity may arise because material on either side of it has experienced a different loading history. It does not give rise to further wave motion. 

Object in this section is to review how rheological knowledge combined with laboratory data can be used to predict stresses developed in plastics undergoing strains at different rates and at different temperatures. The procedure of using laboratory experimental data for the prediction of mechanical behavior under a prescribed use condition involves two principles that are familiar to rheologists one is Boltzmann s superposition principle which enables one to utilize basic experimental data such as a stress relaxation modulus in predicting stresses under any strain history the other is the principle of reduced variables which by a temperature-log time shift allows the time scale of such a prediction to be extended substantially beyond the limits of the time scale of the original experiment. 

---