2.5 Restrictions on Plasticity Theories 31 users in the past, and these are discussed below. Both bear a superficial similarity to thermodynamic laws, and both lead to normality relationships, but neither embodies any thermodynamic principles. 2.5.1 Drucker's Stability Postulate Drucker (1951) proposed a “stability postulate” for plastically deforming mate- rials. Although not a thermodynamic statement, it bears a passing resemblance to the Second Law of Thermodynamics and is therefore referred to as a “quasi- thermodynamic” postulate for classifying materials. It can be stated in a variety of equivalent ways, but represents the idea that, if a material is in a given state of stress and some “external agency” applies additional stresses, then “The work done by the external agency on the displacements it produces must be positive or zero” (Drucker, 1959). If the external agency applies a stress increment GV ij that causes additional strains GH ij , then the postulate is that GV GH t 0 ij ij . The product G V G H ij ij is often called the second order work. In the one-dimensional case shown in Figure 2.6a, the postulate states that the area ABC must be positive. Strain-softening behaviour is thus excluded. In the one-dimensional case, a strain-softening material is mechanically unstable un- der stress control, and this is linked to the identification of the postulate as a “stability postulate”. Unfortunately, this has led to the interpretation that a material which does not obey the postulate will exhibit mechanically unstable behaviour. The obvious corollary is that a material which is mechanically stable must therefore obey the postulate. The identification of the postulate with me- chanical stability for the multidimensional case is, however, erroneous. The conclusion that mechanically stable materials must obey Drucker’s postulate is therefore equally erroneous. Figure 2.6. One-dimensional illustrations of (a,b) Drucker’s postulate and (c) Il’iushin’s postulate 32 2 Classical Elasticity and Plasticity If the external agency first applies then removes the stress increment GV ij , such that the additional strain remaining after this stress cycle is GH p ij , then it also follows from the postulate that GV GH t 0 p ij ij . Thus in the one-dimensional case shown in Figure 2.6b, the area ABD must be zero or positive. We do not elaborate the proof here, but it can be shown that Drucker’s postu- late leads to the requirement that the flow is associated for a conventional plas- ticity model ( i. e., that the yield surface and plastic potential are identical). Fur- thermore, it follows that the yield surface for a multidimensional model must be convex (or at least non-concave) in stress space. Strictly these results apply to uncoupled materials, in which the elastic properties do not depend on plastic strains. For coupled materials in which the elastic properties are changed by plastic straining, the results are modified, but for realistic levels of coupling the effects are rather minor. Many advantages follow from the use of associated flow. For instance, for per- fectly plastic materials with associated flow, it is possible to prove that (a) a unique collapse load exists for any problem of proportionate loading and (b) this collapse load can be bracketed by the Lower Bound Theorem and Upper Bound Theorem. In numerical analysis, associated flow guarantees that the ma- terial stiffness matrix is symmetrical, which has important benefits for the effi- ciency and stability of numerical algorithms. Furthermore, for many materials (notably metals), associated flow is an excellent approximation to the observed behaviour. For all these reasons, it is understandable therefore why many practi- tioners are reluctant to adopt models that depart from associated flow. Frictional materials (soils), however, undoubtedly exhibit behaviour which can only be described with any accuracy with non-associated flow. If a purely frictional material with a constant angle of friction were to exhibit associated flow, then it would dissipate no energy, which is clearly at variance with com- mon sense. With some reluctance, therefore, we must seek more general theo- ries, which can accommodate non-associated behaviour. 2.5.2 Il'iushin's Postulate of Plasticity The “postulate of plasticity” proposed by Il’iushin (1961) is similar to Drucker’s postulate, but significantly it uses a cycle of strain rather than a cycle of stress. It is simply stated as follows. Consider a cycle of strain which, to avoid complica- tions from thermal strains, takes place at constant temperature. It is assumed that the material is in equilibrium throughout, and that the strain (for a suffi- ciently small region under consideration) is homogeneous. The material is said to be plastic if, during the cycle, the total work done is positive, and is said to be elastic if the work done is zero. The postulate excludes the possibility that the work done might be negative. This is illustrated for the one-dimensional case in Figure 2.6c, where the postulate states that the area ABE must be non-negative. 2.5 Restrictions on Plasticity Theories 33 The postulate has certain advantages over Drucker’s statement because it uses a strain cycle. Drucker’s statement depends on consideration of a cycle of stress, which is not attainable in certain cases such as strain softening. On the other hand, almost all materials can always be subjected to a strain cycle. The excep- tions are rather unusual materials which exhibit “locking” behaviour (in the one-dimensional case this involves a response in which an increase in stress results in a decrease in strain). It may be in any case that such materials are no more than conceptual oddities, and we have never encountered them. A more significant limitation is Il’iushin’s assumption that the strain is homogeneous. It may well be that for some cases ( e. g. strain-softening behaviour), homogeneous strain is not possible, and bifurcation must occur. Il’iushin’s postulate seems even more like a thermodynamic statement (and specifically a restatement of the Second Law) than Drucker’s postulate, but again it is not. A cycle of strain is not a true cycle in the thermodynamic sense because the material is not necessarily returned to identically the same state at the end of the cycle. The specific recognition that a cycle of strain would result in a change of stress is an acknowledgment that the state of the material changes. In later chapters, it will be seen that one interpretation of this is that a cycle of strain may involve changes in the internal variables. Il’iushin’s postulate is therefore no more than a classifying postulate. Even though it holds intuitive appeal – that a deformation cycle should in- volve positive or zero work – it is possible to find materials (both real and con- ceptual) that violate the postulate. Such materials would release energy during a cycle of strain, and in so doing would change their state. Il’iushin showed that his postulate also leads to the requirement that, in a conventionally expressed plasticity theory, the plastic strain increment vector should be normal to the yield surface; in other words, the yield surface and plas- tic potential are identical. Since many materials, notably soils, violate this condi- tion, we must conclude on experimental grounds that Il’iushin’s postulate is overrestrictive, and in later chapters, we seek a broader, less restrictive frame- work. Chapter 3 Thermodynamics 3.1 Classical Thermodynamics 3.1.1 Introduction In the following, we establish the thermodynamic terminology we shall use, keeping as close as possible to conventional usage in the thermodynamics of fluids. We shall not attempt here a comprehensive introduction to thermody- namics. Numerous texts deal with this thoroughly. We shall assume therefore a certain familiarity with thermodynamic principles and provide simply a re- minder of some important points. It is important to note that our objective is not to provide a rigorously estab- lished generalization of thermodynamics as a field theory. For reasons discussed in Chapter 4, this is an area which is fraught with difficulties. Instead, our more limited objective in later chapters is to set out a formalism for plasticity theory that is consistent with accepted thermodynamic principles. We shall therefore define materials which are a subset of all those that could be described within a rigorously defined thermodynamic approach. It is for the reader to decide whether this subset is sufficiently wide that it describes materials of practical importance. First we establish some basic definitions. A thermodynamic closed system (for brevity just system) is a body of material separated from its surroundings by certain walls. The state of the system is characterized by a certain number of state variables. A complete definition of the system will also require knowledge of certain constants (for instance, the mass of material within the system), but these will be of less direct concern to us here. For instance, if the system were to consist of a certain mass of a “perfect gas,” then the chosen state variables could be the volume of the gas and the temperature T. A proper choice of state vari- ables is such that they are both necessary and sufficient to describe the current state of the system at the level of accuracy that a particular application demands. 36 3 Thermodynamics Any quantity that can be uniquely determined as a function of the state vari- ables is called a property. In the example above, for instance, the pressure p is a property of the system because p vR T for a perfect gas, where v is the spe- cific volume (volume per unit mass) and R the gas constant. Such a relation is called an equation of state. The existence of such relationships means that the roles of state variables and properties can be interchanged. For instance, pres- sure and temperature could be considered state variables, and the specific vol- ume would be determined as a property. Such an interchange of variables will be an important theme in Chapter 4 and afterwards in this book. The concepts of state variables and properties are rigorously defined only for systems that are in thermodynamic equilibrium. If this restriction were to be enforced strictly, however, classical thermodynamics could be applied only to processes that are infinitesimally slow. In practice, the concepts of classical thermodynamics can be applied successfully to rapidly changing systems, and we allow this possibility here. There is of course an important discipline of non- equilibrium thermodynamics, but there are a number of different approaches to studying it, and a discussion is beyond the scope of this book. In classical thermodynamics, the “Zeroth”, First, Second and Third “Laws” are defined. As for any “Laws” of nature, they are empirically based and are therefore unprovable: they could be falsified by counterexample but cannot be proven. However, they fit into a sufficiently logical framework that they are almost universally accepted as “true” by the scientific community. The Zeroth Law states that two bodies that are each in thermal equilibrium with a third body are also in thermal equilibrium with each other. It provides a rigorous basis for the definition of the temperature T as a property of a body that is internally in thermal equilibrium. (Throughout this book, we use T for the thermodynamic, or absolute, temperature, which is always positive.) We shall return to the First and Second Laws below. They establish the exis- tence of two further properties of a body in thermodynamic equilibrium: the internal energy and the entropy. The Third Law then requires that entropy is zero at zero temperature. Whilst important in other contexts, it does not enter our discussions here. 3.1.2 The First Law We consider a closed system isolated from its surroundings by certain walls. A process involves interaction between the system and its surroundings and can involve transfer of two types of energy: heat flow Q into the system from the surroundings and mechanical power input W , also from the surroundings. The First Law is usually stated in the following form: for a system in thermodynamic equilibrium, there is a property of the system, called internal energy U, such that QW U (3.1) 3.1 Classical Thermodynamics 37 Note very importantly, however, that Q and W are not each separately inte- grable with time to give properties Q and W, and in fact no such properties exist. The above equation is a somewhat simplified form of the First Law, in that it does not expressly account for energy input from, for instance, a gravitational field or from the flow of electrical current. Also it ignores the fact that some of the input power may, in general, cause an increase in kinetic and potential ener- gies, which are conventionally separated from internal energy. However, Equa- tion (3.1) expresses the essential principle of conservation of energy: the sum of all the sources of power input to a body is equal to the rate of increase of the energy of the body. Furthermore, the most important sources of power we must consider are mechanical power and heat supply. There is, however, one term that is sometimes mistakenly included in Equa- tion (3.1) but which we explicitly exclude, and that is a source of heat from within the body itself. In a number of texts, it will be found that an additional term of this sort is added, but the introduction of such a term, allowing energy to be magically conjured up inside the body, is nothing more nor less than a complete denial of the validity of the First Law! Why then do some authors resort to this approach? The explanation is that they find it necessary when they attempt to define the properties of a system in terms of an inadequate set of state parame- ters. An example offers the simplest explanation. Suppose that the body in ques- tion consists of a radioactive metal, surrounded by conducting walls. It will be observed that a significant amount of heat leaves the body, whilst the body itself appears to suffer no change in state. It is tempting to attribute this observation to a “heat source” somewhere within the body. However, a proper description of the state of the body requires the amounts of the different isotopes present to be known, and each of these will have a different internal energy. As one isotope decays to another, the internal energy of the body decreases, and it is this de- crease that is reflected in the outward flow of heat. If the internal composition in terms of isotopes is (mistakenly) ignored, then it becomes necessary to include the mysterious internal heat source in Equation (3.1). It is often convenient to consider a system that is unchanged by a certain process. This simply means that all state variables (and therefore all properties) of the system are the same after completion of the process as they were at the beginning. They may of course have taken some different values at some point during the process. Consider the process shown in Figure 3.1a, in which an unchanged system re- ceives an amount of work WX from the surroundings and also receives some amount of heat Q . Since the system is unchanged, 0U , and a trivial applica- tion of the first Law shows that QX , so that the system must reject to the surroundings exactly as much energy as it received in work. Many simple de- vices operate in the way shown in Figure 3.1a. For instance, a frictional brake (once it has reached a steady temperature) converts work input to heat output. The pure conversion of work to heat is called dissipation. 38 3 Thermodynamics XW XQ XW XQ Figure 3.1. Possible and impossible processes 3.1.3 The Second Law In the following, it will be necessary to consider a number of processes in which systems interact with their surroundings. The surroundings themselves will have certain properties, and in particular the temperature of the surroundings will be important. A part of the surroundings which is sufficiently large that its proper- ties can be regarded as unchanged by any interactions with the system is said to be a reservoir. The Second Law is considerably more subtle than the First and can be ex- pressed in a number of equivalent ways. It applies certain restrictions to the processes that can occur. For instance, one of the basic consequences of the Second Law is that work can be dissipated in the form of heat, but that heat cannot be changed into work without some side effects occurring too. Thus the process in Figure 3.1a (in which X is a positive quantity) is physically possible, whilst that in Figure 3.1b is not, even though it too obeys the First Law. We shall return to this example shortly. There are a number of ways by which the Law can be expressed, but the most useful is in the form of the Clausius-Duhem inequality. To express the First Law, we had to introduce the concept of a property called the internal energy U. The Second Law is also best expressed by making the hypothesis that there is a fur- ther property, called entropy S. Most people find entropy a more abstract con- cept than internal energy, and perhaps the most useful approach is initially to treat it simply as a mathematical abstraction without seeking a physical meaning. The Clausius-Duhem inequality states that, for a system that exchanges heat with n reservoirs at temperatures i T , the change of entropy is such that 1 n i i i Q S t T ¦ (3.2) where i Q is the rate of heat input from reservoir i. It can readily be seen that inequality (3.2) does not permit the process in Figure 3.1b. For the unchanged system, 0S , so that inequality (3.2) requires that 0Q d for a system exchang- ing heat with only a single reservoir (since the temperature must be positive). A consequence of the Second Law is that heat cannot spontaneously flow from a colder place to a hotter one. Consider the process shown in Figure 3.2 in which an unchanged system exchanges heat with two reservoirs at different temperatures. The First Law clearly requires that 12 0QQ . We shall assume 3.1 Classical Thermodynamics 39 that 1 Q is positive and 2 Q negative. Since 0S for the unchanged system, the Second Law requires 12 12 0 QQ t TT (3.3) Rearranging this as 11 12 0 QQ t TT , then using the fact that 1 T , T 2 and 1 Q are all positive gives 12 TtT, so that in a process of pure heat transfer, heat can only flow from a hotter place to a colder one. Now consider the process shown in Figure 3.3, in which an unchanged system (a heat engine) receives heat 1 Q from one reservoir at 1 T and rejects a fraction of this (so that 2 Q is negative and 21 QQ ) to another reservoir at 21 TT. The process produces a power output W , which (since 0 U for the unchanged system) is determined by the First Law as 12 0WQ Q ! . Now we consider the maximum possible thermodynamic efficiency of the system, which is the ratio of useful output work to the input heat flow: 12 2 11 1 1 QQ Q W QQ Q (3.4) Since 0 S for the unchanged system, from the Second Law, 12 12 0 QQ t TT (3.5) which can be rearranged as 22 11 Q Q T d T (3.6) 1 Q W 2 Q Figure 3.3. A heat engine exchanging heat with two reservoirs 1 Q 2 Q Figure 3.2. An unchanged system exchanging heat with two reservoirs 40 3 Thermodynamics (using the fact that both 1 Q and 2 T are positive). Substituting this in the expres- sion for the efficiency, we obtain: 2 11 1 W Q T d T (3.7) so that the maximum possible efficiency of such a process is 21 1T T , and it can be shown that this can be attained only by an ideal system in which there is no dissipation (in which case, the equality holds in (3.2)). We can see that 100% efficiency is approached only as 2 0To . So to achieve maximum efficiency from a heat engine, one needs to have available a reservoir at near absolute zero temperature! Countless other examples of the implications of the Second Law can be found in thermodynamics textbooks, but these examples serve our purpose here to demonstrate the role of entropy and the important qualitative difference that the Second Law introduces between mechanical work and heat supply. An important concept in thermodynamics is that of reversibility. A process is reversible if all the directions of the heat and work flows into or out of the sys- tem can be reversed simultaneously and the resulting process still obeys the Second Law. It is straightforward to show that reversibility is possible only when equality rather than inequality holds for the Clausius-Duhem relationship for the process. So for all reversible processes, 1 n i i i Q S T ¦ (3.8) In practice, almost all the materials we shall encounter in this book do not exhibit purely reversible behaviour, in that they are dissipative. We shall address this in much more detail below, but first we explore the behaviour of simple reversible materials to gain some familiarity with thermodynamic functions. 3.2 Thermodynamics of Fluids The classical thermodynamics of fluids uses the intensive quantities, pressure p and temperature T, which are properties that do not depend on the amount of fluid in the system. There are also extensive quantities, which are quantities that (for given values of the intensive quantities) take values directly proportional to the mass of the system. The volume is the most obvious example, but internal energy and entropy are also extensive. It is convenient to normalize all extensive quantities to obtain specific values of them, i. e. values per unit mass. We have already introduced the specific volume v (volume per unit mass). The specific internal energy is u and the specific entropy is s. By convention, lowercase letters are used for specific quantities. 3.2 Thermodynamics of Fluids 41 Consider again a simple material that can undergo reversible processes. In practice, this means that the process must be sufficiently slow that the sample can always be considered in a state of equilibrium, and that the sample does not dissipate energy. (There will be much further discussion of dissipation in the remainder of this book). A perfect gas, which is discussed further below, would be an example of such a material. We write the First and Second Laws for a vol- ume element of the material in the form, ,ii vu qp vv (3.9) , i i q s v §· t ¨¸ T ©¹ (3.10) where we note that if i q ist the heat flux per unit area, the rate of heat supply per unit volume is ,ii q , and the work input per unit volume is p vv . The terms in u and s are divided by v to convert them to a per-unit-volume basis. Now consider the reversal of the process. We make two observations. Firstly the sign of s and i q in the first part of inequality (3.10) will change, so that we can deduce that the equality must hold: , i i q s v §· ¨¸ T ©¹ (3.11) Secondly, we expand the right-hand side of this equation as ,, 2 , ii i i i i qq q T §· ¨¸ TT ©¹ T (3.12) and make the empirical observation that the direction of heat flow is always in the direction opposite to the temperature gradient, so that the sign of ,i T will change at the same time as i q , and the second term on the right-hand side of (3.12) does not change sign as the process is reversed. The only way (3.12) can be true for both forward and reverse processes is therefore, in the limit as the tem- perature gradients become sufficiently small, that this term becomes negligible. It follows that for a reversible process, ,ii q s v T (3.13) Now we can eliminate the divergence of the heat flux between Equations (3.9) and (3.13) to obtain for our “reversible” material, uspv T (3.14) Since the directions of temperature gradient and heat flux are opposed, the term ,ii qT T is always positive and is called the thermal dissipation. [...]... potential formulation and is more accessible to those unfamiliar with thermodynamics 4 .3. 1 The Laws of Thermodynamics In generalised thermodynamics, a central hypothesis is that the state of a material is entirely determined by the values of a certain set of independent variables: the kinematic variables (strain and internal variables) and the temperature Properties are functions of state The First Law of. .. transformation of f in which the roles of the stress and strain are interg changed (see Table 3. 2) It follows that ij ij and 2 2 g ij kl ij kl ij g (3. 33) 3. 3 Thermomechanics of Continua 2 The term ij g 2 is the isothermal compliance matrix and kl g 49 the matrix ij of coefficients of thermal expansion Note that g and f cannot be separately and arbitrarily defined because they are functionally related... this type of behaviour is that the response of the material depends not only on the current values of the state variables that we have already introduced, but also on the history of how the material arrived at that state There are broadly two approaches to dealing with this problem: “Rational mechanics” in which the response is made not simply a function of the current state but a functional of the whole...42 3 Thermodynamics 3. 2.1 Energy Functions Bearing in mind that the First Law states that u is a property of a material, it can be expressed as a function of an appropriate choice of state variables Equation (3. 14) suggests that just such a choice would be v and s, and we choose these as our independent variables, writing u u v , s It follows that u u v s v s Comparing with Equation (3. 14), we obtain... This approach is also called “thermodynamics with internal variables” (TIV) This is the approach that we shall adopt here For convenience, we shall introduce an internal variable ij which is tensorial in form and kinematic, that is to say strain-like in nature Although it is not strictly necessary, the internal variable will often be found to play the same role p as the plastic strain ij It may be... history of state Thermodynamic principles are then applied to ensure that the equations for the evolution of the dependent variables are consistent with the Laws of Thermodynamics “Generalised thermodynamics” in which the history of the state is encapsulated within certain internal variables A full description of the state then requires both the variables we have already introduced and the internal variables... as the dissipation due to heat flow and is often termed thermal dissipation The remaining part of the dissipation is termed mechanical dissipation Whilst the Second Law d dt qi ,i requires only that the total dissipation be non-negative, it is common to assume that the processes of mechanical dissipation and heat flow are independent, so that the thermal and mechanical dissipations are each required... well as on strain and temperature For convenience, we shall consider a single kinematic internal variable of tensorial form ij This is chosen because, in the following, it will be found that the internal variable is often conveniently identified with the plastic strain Generalization either to other forms of internal variable (in particular to a scalar) or to multiple internal variables is straightforward... reasonable, and that at a sufficiently low temperature, modification would be required to make the relationships consistent with the Third Law 3. 3 Thermomechanics of Continua 47 3. 3 Thermomechanics of Continua 3. 3.1 Terminology In the following, we use the terminology of classical thermodynamics as far as possible, but some minor changes are convenient We now highlight the areas where our notation departs... nondissipative material continue to apply, so that we can write u ij d 0 (3. 39) ij At the very minimum, one can then specify certain evolution equations for ij , and ensure by applying checks to them that they are always consistent with the inequality in Equation (3. 39) We find, however, that this approach is unsatisfactory for two reasons Firstly, it is mathematically inconvenient in that a set of tensorial evolution . system at the level of accuracy that a particular application demands. 36 3 Thermodynamics Any quantity that can be uniquely determined as a function of the state vari- ables is called a property internal energy expression is related to a par- ticular aspect of the physical behaviour of the material. Once a familiarity is established with the aspects of behaviour that are related to particular. which can be derived from the above approach; in particular, he examines the relationships between isothermal and adiabatic moduli. 3. 3 .3 Internal Variables and Dissipation Most materials which