BEHAVIOR OF REAL MATERIALS

All metals in practical use exhibit a yield point at which elastic behavior ceases and irreversible plastic flow begins. If deformation occurs at a relatively low temperature, dislocations will accumulate and each successive dislocation will require a higher stress to make it move across a crystal. This is called strain hardening. The temperature beyond which strain hardening ceases is best defined in terms of a nondimensional homologous temperature (T ). 

The atoms in a crystal are in constant vibrational motion, the amplitude of which increases as temperature rises above an absolute zero temperature (-273 °C) where all motion ceases. At the melting point of a material, the vibrational motion is sufficient that atoms no longer retain a fixed relationship to their neighbors. The homologous temperature is the ratio of the absolute temperature of a material to absolute temperature at the melting point. The homologous temperature at absolute zero is zero while that at the melting point is one. 

Metals are made harder and stronger by structural defects that interfere with the movement of dislocations across a crystal. These defects may consist of grain boundaries, impurities, or precipitates that are insoluble in the base metal. The greater the number of interfering defects, the greater will be hardening or strengthening effects. 

A material that contains no dislocations, or defects capable of generating dislocations under stress, will behave in a perfectly brittle manner. It will remain elastic all the way to the point of fiacture even though the strain at fracture may be two orders of magnitude or more greater than the point where irreversible flow normally occurs. 

Materials fracture in either a brittle or a ductile fashion. In brittle fracture, the design criterion is the maximum normal strain and fracture will occur on a plane normal to that of maximum normal strain. Ductile fracture involves shear, the design criterion is then maximum shear stress, and fracture occurs on a plane of maximum shear strain. 

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The radiative behavior of real materials generally falls short of blackbody behavior, depending on the material. Figure 8.5 shows the spectral radiancy of a real body is always less than that of a blackbody, and the deviation is inconsistent with wavelength.3 The spectral emissivity is defined as the ratio... 

The diagrams discussed are highly artificial and would not justify any conclusion about the behavior of real materials if it were not for the fact that a material has a characteristic length, as discussed in the next section. 

To account for the fact that neither ideal solids nor ideal liquids exist in the real word, the rheological behavior of real materials can be approximated by a combination of the individual model elements, either in Unear two-element models of MaxweU and Voigt-Kelvin types, or in linear three-element models. In many real cases, nonlinear models have to be invoked (see Section 2.4.1.3). 

Calculations of this type have proliferated since the early 1980s, providing a wealth of useful information on the behavior of real materials. We will touch upon some topics where such calculations have proven particularly useful in chapters 9-11 of this book. It is impossible to provide a comprehensive review of such applications here, which are being expanded and refined at a very rapid pace by many practitioners worldwide. The contributions of M.L. Cohen, who pioneered this type of application and produced a large number of scientific descendants responsible for extensions of this field in many directions, deserve special mention. For a glimpse of the range of possible applications, the reader may consult the review article of Chelikowsky and Cohen [65]. 

Torvik, P.J, Bagley, R.L. (1984) On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294-298 Tricomi, F.G. (1957) Integral Equations (Interscience, New York)... 

While the exponential stress relaxation predicted by the viscoelastic analog of the Mawell element, ie., a single exponential, is qualitatively similar to the relaxation of polymeric liquids, it does not describe the detailed response of real materials. If, however, it is generalized by assembling a number of Maxwell elements in parallel, it is possible to fit the behavior of real materials to a level of accuracy limited only by the precision and time-range of the experimental data. This leads to the generalized, or multi-mode. Maxwell model for linear viscoelastic behavior, which is represented mathematically by a sum of exponentials as shown by Eq. 4.16. 

Mathematical relationships have been developed to relate models composed at varying degrees of these two aspects of behavior to real materials. 

Equation (8-8) is called Kirchhoffs identity. At this point we note that the emissivities and absorptivities which have been discussed are the total properties of the particular material i.e., they represent the integrated behavior of the material over all wavelengths. Real substances emit less radiation than ideal black surfaces as measured by the emissivity of the material. In reality, the emissivity of a material varies with temperature and the wavelength of the radiation. 

Neither simple mechanical model approximates the behavior of real polymeric materials very well. The Kelvin element does not display stress relaxation under constant strain conditions and the Maxwell model does not exhibit full recovery of strain when the stress is removed. A combination of the two mechanical models can be used, however, to represent both the creep and stress relaxation behaviors... 

A careful observation of the behavior of real substances reveals that neither of these idealizations is quite accurate. Thus for solids the stress decreases rather rapidly at short times and then more gradually at long times, approaching a hmiting value CToo- If cToo >0 . the material is likely to be considered a solid, and otherwise a liquid. The evolution of the relaxation stress for real solids and liquids in relaxation experiments is schematically illustrated in Figures 5.3a and 5.3b, respectively. It should be pointed out that the determination of is a subjective matter that depends on the nature of the material and the nature of the observation. The relaxation time, that is, the time necessary for the stress to approach completion, may be so short that it escapes observation. In this situation the experimenter... 

The model isotherm for each pore size class was calculated by methods described previously [9], modified to account for cylindrical pore geometry. These calculations model the fluid behavior in the presence of a uniform wall potential. Since the silica surface of real materials is energetically heterogeneous, one must choose an effective wall potential for each pore size that will duplicate the critical pore condensation pressure, p, observed for that size. This relationship is shown in Figure 2. The Lennard-Jones fluid-fluid interaction parameters and Cn/kg were equal to 0.35746 nm and 93.7465 K, respectively. 

The behavior of real component powders may be estimated using a yield criterion (1), which requires a knowledge of the magnitudes of the directional response to stress of the particle and knowledge of the particle size distribution and particle orientation. Much of this information is not readily available for molecular organic solids and is the subject increasing attention of academic materials scientists. Even knowing the crystal structure, the dynamic response to mechanical stresses is not currently predictable without exhaustive effort (3). However, yield behavior may be measured and correlated to the... 

Admittedly, the existence of the postulated transient mesophase would still require structural confirmation. Even so, the taking of a sharp drop In viscosity as Indicator of mesophase formation has well established precedents In the liquid crystal field. Such Is e.g. the well documented effect In Kevlar referred to above (12) which In many respects has similarities to the presently discussed PE, except that Kevlar Is "mesogenlc" and can exist as stable liquid crystal under ambient conditions, while the mesophase In the flexible PE Is "virtual". The latter "virtual" phase only becomes "real" transiently, which suffices to dramatically affect the entire flow behavior of the material, the effect through which It Is being detected. 

When the nonlinearity of the log viscosity vs. reciprocal temperature data was first observed, tests were made to insure that the curvature was real and not an artifact of the experimental apparatus. Hysteresis curves and constant temperature for extended time tests showed that the nonlinearity was not caused gf zovolatilization alkali or fluoride constituents or from thermal deviations in the furnace setup. It was found that the observed curvature of the data was not an artifact and represented the true physical behavior of the materials. The application of the Kirchoff-Rankine equation... 

This chapter is the first in a series that will make the case that many of the important features of real materials are dictated in large measure by the presence of defects. Whether one s interest is the electronic and optical behavior of semiconductors or the creep resistance of alloys at high temperatures, it is largely the nature of the defects that populate the material that will determine both its subsequent temporal evolution and response to external stimuli of all sorts (e.g. stresses, electric fields, etc.). Eor the most part, we will not undertake an analysis of the widespread electronic implications of such defects. Rather, our primary charter will be to investigate the ways in which point, tine and wall defects impact the thermomechanical properties of materials. 

Defect Interactions and the Complexity of Real Material Behavior... 

In this section we recall the reasons that we undertook the modeling efforts advocated in this book in the first place. Our end was to see to what extent one might explain the observed thermomechanical properties of real materials. Just what are these observations and what do they teach us We now take stock of the range of observed properties of relevance to the thermomechanical behavior of materials and the extent to which they have been understood both phenomenologically and mechanistically. 

As experimental techniques for measuring displacement and strain become increasingly accurate at micro- and nano- length scales, experiments must be performed to verify the accuracy of predictions of models of material behavior at these scales. In particular, the use of diffraction techniques for measurements of lattice distortion [20] combined with surface measurement techniques such as micro-Moire [21], speckle interferometry [22] and displacement mapping [23] promise to provide essential information on the local deformation behavior of metals and alloys in the vicinity of grain boundaries, voids and second phase particles. These techniques must be further developed and applied to the analysis of real materials to increase our knowledge of material behavior at these length scales. 

It should be emphasized that many constitutive models have been proposed especially for polymeric solutions and melts, and there is a great deal of current research that is aimed at both new models25 and numerical analysis of fluid motions by use of the existing models 26 The problem is that few have been carefully compared with the behavior of real fluids outside the highly simplistic flows of conventional rheometers, and then mainly under flow conditions in which the perturbations in material structure are weak. Thus there is currently no model that is known to provide quantitatively accurate or even qualitatively reliable descriptions of real complex fluids for a wide spectrum of flows. 

Since neither model adequately describes the behavior of real viscoelastic materials, a combination of the classic elements is often made to gain closer representation. The most common configuration is called the standard linear solid4 configuration, and it is illustrated in Figure 6.6. A more accurate representation of actual behavior can be obtained by a composite of multiple elements of the standard linear solid configuration into a multi-element model (Figure 6.7) with an array of coefficients for each element. 

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