Stephen L. Passman

Stress relaxation, creep, failure and hysteresis in a linear elastic material with voids

I study several specific boundary-value problems in a theory recently proposed to model linear elastic materials with voids. I show that, in addition to the known fact that the model exhibits stress relaxation, it also exhibits creep, hysteresis, and a phenomenon which can be interpreted as failure. In order to maintain plausible physical behavior, I suggest an a priori inequality not contained in the original theory.

A Thermomechanical Theory for a Porous Anisotropic Elastic Solid with Inclusions

We construct a mixture theory which describes a porous elastic anisotropic solid with inclusions. Thermal effects are taken into account. The theory is in accord with classical thermodynamics. Fully nonlinear isotropic and anisotropic materials are considered, and field equations are also given for a nontrivial special case which, though nonlinear, is controlled by a few material functions. When properly specialized, the theory reduces to the P-α model, a model widely used to describe porous solids.

Continuum Balance Equations for Multicomponent Fluids

The mixtures described in Chapter 5 are based on the concept that a mixture may be represented by “a sequence of bodies B k , all of which ...occupy regions of space ... simultaneously” [89, p. 81]. Examples of such materials are air (a mixture of nitrogen, oxygen, and other materials in small amounts), and whisky (a mixture of water, alcohol, and other materials in small amounts). However, it was commonly recognized at an early stage that so strong an assumption of intermiscibility was not appropriate to all physical situations. For example, soils, porous rock, suspensions of coal particles in water, packed powders, granular propellants, etc., consist of identifiable solid particles surrounded by one or more continuous media, or an identifiable porous matrix through which one or more of the continua are dispersed. Motions of the individual components are possible and, as long as there are no chemical reactions, each constituent retains its integrity1. We call such materials multicomponent mixtures. They are more complicated than classical mixtures in the sense that they have geometrical structure, but less complicated in the sense that the constituents are not intimately intermixed. A theory describing them should reflect these facts.

An exact solution for shearing flow of multicomponent mixtures

When a nondilute multicomponent fluid is placed in a viscometric testing device, the flow that is produced is not a viscometric flow. Rather, relatively thin layers of fluid with relatively few suspended particles accumulate near the boundaries. These layers exhibit high shear rates. The suspended particles accumulate far away from the boundaries, giving a high local viscosity and a low shear rate. We consider a simple but properly invariant theory of multicomponent fluids. We investigate steady flow between parallel plates with one plate stationary and the other plate moving parallel to it at constant speed. We give exact solutions to the field equations.

Stress in dilute suspensions

Generally, two types of theory are used to describe the field equations for suspensions. The so-called “postulated” equations are based on the kinetic theory of mixtures, which logically ought to give reasonable equations for solutions. The basis for the use of such theory for suspensions is tenuous, though it at least gives a logical path for mathematical arguments. It has the disadvantage that it leads to a system of equations which is underdetermined, in a sense that can be made precise. On the other hand, the so-called “averaging” theory starts with a determined system, but the very process of averaging renders the resulting system underdetermined. I suggest yet a third type of theory. Here, the kinetic theory of gases is used to motivate continuum equations for the suspended particles. This entails an interpretation of the stress in the particles that is different from the usual one. Classical theory is used to describe the motion of the suspending medium. The result is a determined system for a dilute suspension. Extension of the theory to more concentrated systems is discussed.

Classical Theory of Solutions

The major focus of this book is multicomponent materials. The mathematical theory appropriate to such mixtures of materials is still in a stage of development. A simpler type of material is the classical mixture, or solution. In such a material, the components are not physically distinct, that is, the mixing of the materials is at a molecular level. A kinetic theory for such materials was given by Maxwell [62], and was transposed into a form appropriate to continuum theory by Truesdell [91].1

A theory of localization of damage in creep

Abstract We construct a three-dimensional theory of a material in which damage may occur in creep. In the one-dimensional case the differential equations are simple enough so that they may be studied exactly, but complicated enough to exhibit a variety of interesting phenomena. Solutions are found to be homogeneous damage fields, coexistent homogeneous damage fields separated by singular surfaces, an homogeneous undamaged body with a smooth transition into a damaged region in the interior, which may be interpreted as a shear band, and periodic solutions with transitions between damaged and undamaged regions.

Postulational and Averaging Approaches

The approach put forward in Chapter 6, which could be called the “postulational” approach, is appealing in that references to the microstructure do not appear in the formulation. This is analogous to the similar approach to continuum mechanics, where the molecular nature of matter plays no role in the formulation. The existence of the microstructure is often simply ignored, or argued away by saying that the “limit” is stopped at some scale larger than the microstructure, but smaller than any macroscopic scale of interest. Another view of both the continuum approach to both ordinary continua and to multicomponent continua is that they are models of reality, and consequently are simplified to the point of omitting some phenomena; namely, they do not govern the evolution of microstructure. These models require taking limits as volumes shrink to zero, so that at some point, the volume is so small that it cannot contain many “corpuscles.” This feature does not render the model invalid, but instead suggests that constitutive equations are required to replace the microstructure detail lost in the model.

Equations of Motion for Dilute Flow

Some general properties of constitutive equations were discussed in Section 14.3. It is clear that any properly formulated theory of multicomponent fluids will be complex, and even in the simplest case will involve a large number of constants. Thus, the accustomed process for, e.g., finding the viscosity of a linearly viscous fluid by simple experimentation, must be superceded by a much more complex process. The current state of the art involves exploration of numerical solutions, extrapolation, exploration of analogies, and often not a small amount of guessing. For instance, additional guidance in formulating realistic constitutive equations can be obtained by assuming that the quantities computed from the “exact” solutions for single large particles in a flow can be taken as forms for constitutive equations. For example, the dynamics of the irrotational flow of an inviscid fluid about a sphere (see Section 15.3) can be used to compute the average force on a sphere and the average interfacial pressure. Constitutive equations arrived at in this way should reduce to the appropriate limit for dilute flows. As is evident from the state of our abilities to calculate the flow around assemblages of bodies of arbitrary shape, we cannot hope to obtain solutions that can be averaged to give appropriate constitutive equations for general motions. The calculated forms from “exact” solutions provide a guide to the appropriate forms for empirical testing. In order to attempt to obtain a system of equations that will predict the flow of a dispersed multicomponent material, it is helpful to find experiments that isolate appropriate phenomena and give (relatively) direct measurements of unknown coefficients involved therein. For best results, the physical experiment should be simple, and should correspond to a simple solution of the equations that depends mostly on the particular constitutive form being evaluated. Examples of such flows are sometimes called “viscometric,” or “separate effects” experiments. We give some of the solutions in Part V. It is clear that simple exact solutions are desirable.

Physical Reality, Corpuscular Models, Continuum Models

One view of physical reality is that the matter commonly perceived as filling space in fact consists principally of empty space, with an occasional bit of matter. Such a bit may be called an elementary particle. The distance between the bits of matter and the structure in which they are arranged, as well as their particular arrangement, dictates whether the gross material is perceived as a solid, as a liquid, or as a gas. A bit of matter may itself have a very rich structure, in which case the bit is no longer an elementary particle, but rather is in itself composed of elementary particles. This kind of structure is worthy of study. The scale of such structures is such that they are not accessible for purposes of describing the behavior of materials normally encountered in everyday experience. Neither is such structure interesting for most such purposes. The exceptions are, for example, that a solid is perceived as amorphous or as having a particular crystalline structure based on the symmetry of the arrangement of the matter in it. This affects its gross symmetry, that is, the mechanics of the macroscopic body reflects the arrangement of the bits of matter. The difficulty of scaling such theories up to the size of a body that interests us here makes it desirable to search for an alternative to description at this level of detail. Moreover, theories of such small structures are often revised, because they are not yet completely understood, and both deeper theoretical understanding and more sophisticated experimental work continue to appear. On the other hand, our experience of the gross behavior of the materials of everyday life, such as steel or water, is quite constant. Thus it is extremely rare for us to revise its mathematical description. The result is, for the purposes of modeling phenomena on the scales ordinarily perceived, it is appropriate to devise models that are independent of modern theories of atomic and subatomic physics. Two types of such models are corpuscular models and continuum models.