Thermodynamics

The central physical theory on which we base our development of continuum mechanics is thermodynamics. In the first half of this chapter, we will review the elements of thermodynamics as it applies to a general system interacting with an environment. We will subsequently generalize and extend these notions to make them applicable for continuum mechanics.

Review of Thermodynamics

The subject matter that we will review here is variously called equilibrium thermodynamics, or, more appropriately, thermostatics. The primary concern of thermodynamics is a global description of the behavior of a system interacting with an environment, possibly exchanging energy and/or mass in the process. We will formalize these notions below. One of the hallmarks of the thermodynamic description of a phenomena is its generality - it is applicable for any macroscopic phenomena.

Trivia

There is no better way to summarize the status of thermodynamics among other theories of physics than the introduction provided by Gerard Maugin in his book The thermomechanics of nonlinear irreversible behaviors: An introduction:

"Perhaps, after all, the wise man's attitude towards thermodynamics should be to have nothing to do with it. To deal with thermodynamics is to look for trouble. This is not the citation of a famous scientist, but the result of a deep cogitation following mere observations. Why do we need to get involved in a field of knowledge which, within the last hundred years, has exhibited the largest number of schizophrenics and megalomaniacs, imbalanced scientists, paranoiacs, egocentrists, and probably insomniacs and sleepwalkers? Is there any other field of scientific activity such as this one where, apparently, each of the bricklayers has unhesitatingly revindicated an originality that [...] we may see but with confusion.

For some thermodynamicists and for their enemies as well, thermodynamics is the science of everything, so that for the second category of individuals, it is, by mere logic, the science of nothing, and thus not a science at all; and the "everything" is naturally seen as a sign of megalomania from which some have not escaped. [...]"

The excerpts given above should give you an idea of the effect thermodynamics can have on you if you dig deeper. In any case, keep in mind that every thermodynamicist worth their salt has their own point of view, often a very strong one!

Basic definitions

A thermodynamic system, or just a system, is the part of the universe that we are interested in. The rest of the universe is called the environment. The implicit understanding in this description is that the system is in contact with its environment. We also say that the system interacts with its environment. The interaction of the system with the environment is understood to be through one of two mechanisms: work and heat. Both these are special kinds of energies; the difference is that work refers a controllable form of energy, while heat refers to the part of the energy that is uncontrollable and often associated with dissipative internal mechanisms in the system.

Remark

An important assumption that we will make throughout this course is that all interactions are short-ranged This means that there are no long range forces in the system, or in the interactions between the system and its environment.

A defining feature of the thermodynamic approach is the study of a complex system as a black box, without any detailed consideration of the microscopic constitution of the system. The system is thus described by means of measurable physical quantities called thermodynamic variables, also known as state variables. The thermodynamic variables together determine the thermodynamic state, also referred to as just the state, of the system. It is important to note, however, that there is no universal recipe to determine the thermodynamic variables to describe a system; it is contextual and depends entirely on the system being investigated. It is conventional to classify thermodynamic variables as being extensive, or intensive. An extensive state variable is one whose value is doubled if we double the system size, whereas an intensive variable is one which is independent of system size. We will see examples of these shortly.

An important extensive state variable that is common to all thermodynamic systems is the internal energy, $E$. Loosely, put, this the internal energy of a system is a measure of the total energy that the system posseses. In addition to the internal energy, we will typically characterize the sytem by means of a finite number of extensive state variables, which we will denote as $Y_1, Y_2, \ldots, Y_n$.

Example

As a simple example, consider a gas enclosed in cylinder, one face of which is controlled by a piston. The state variables in this case are the internal energy, volume of the cylinder, and the number of particles in the gas (in appropriate unis).

Note

The fact that the internal energy is an extensive variable is dependent on the assumption that all the interactions are short-ranged. In the presence of long-range interactions, the extensivity of the internal energy is not a valid assumption in general.

The system is said to be in thermodynamic equilibrium with its environment if the state variables of the system do not change with time. Implicit in this description of equilibrium is the choice of a time scale with respect to which the variation of the state variables is negligible.

Note

The notion of thermodynamic equilibrium is distinct from the notion of mechanical equilibrium. The latter is characterized by a net balance of forces acting on the system. This is not true in thermodynamic equilibrium.

In addition to the internal energy, another extensive variable that is common to all thermodynamic systems is the entropy of the system, $S$. The best means to gain some intuition about entropy is from statistical mechanics. For the present purposes, it is sufficient to think of entropy loosely as a measure of disorder in the system, or, equivalently, a measure of how much the energy supplied to a system degrades to a form that is unusable.

Remark

If you are really interested in the statistical mechanical underpinnings of the notion of entropy, you are welcome to register for the course AE 731: Multiscale Modeling of Materials. If you're tired of my lecturing and would rather prefer learning it on your own, you can consult the many excellent textbooks on statistical mechanics out there; for instance, a simple and elegant discussion of the notion of entropy from the statistical mechanical perspective can be found here.

A fundamental postulate of equilibrium thermodynamics is that the entropy is a well defined function of the other extensive variables that characterize the system: $$S = \hat{S}(E,Y_1, \ldots, Y_n).$$ This equation is called the fundamental equation of the thermodynamic system. Further, we will assume that $\hat{S}$ is a monotonically increasing function of $E$. This assumption permits us to invert the fundamental relation to obtain the internal energy $E$ as a function of the entropy $S$ and the other extensive variables of the system: $$E = \hat{E}(S, Y_1, \ldots, Y_n).$$ Both these forms of the fundamental equation are useful in practice.

The temperature of the system, $T$, is defined either in terms of the function $\hat{E}$ as $$T = \frac{\partial \hat{E}(S,Y_1, \ldots, Y_n)}{\partial S},$$ or in terms of the function $\hat{S}$ as $$\frac{1}{T} = \frac{\partial \hat{S}(E, Y_1, \ldots, Y_n)}{\partial E}.$$ Both these definitions are equivalent. The temperature thus defined is also known as the absolute temperature of the system. Note that $T$ is an intensive variable since both $E$ and $S$ are extensive variables.

We will now briefly consider the partial derivatives of $\hat{E}$ with respect to the other thermodynamic variables. We define the thermodynamic driving force $X_i$, where $1 \le i \le n$, as $$X_i = \frac{\partial \hat{E}(S, Y_1, \ldots, Y_n)}{\partial Y_i}.$$ The thermodynamic driving force $X_i$ is said to be conjugate to the thermodynamic variable $Y_i$. It is helpful to think of $X_i$ as the generalized force corresponding to the generalized displacement $Y_i$.

Suppose now that the system is in a particular state, say $A$, and as a consequence of its interactions with the environment, it evolves to a different state $B$. The path followed by the system in the space of admissible thermodynamic variables in going from the state $A$ to the state $B$ is called a thermodynamic process. We will particularly be interested in reversible thermodynamic processes - these have the property that the process can be localy reversed in time, and further that the system is in thermodynamic equilibrium at every intermediate state of the process. It is convenient to think of a reversible thermodynamic process as a quasi-static process - this means that the process is carried out slow enough that the system is in equilibrium at every step of the process. A thermodynamic process that is not reversible is said to be irreversible.

First law of thermodynamics

The first law of thermodynamics is essentially a statement of conservation of energy. Recall that the energy of a system is not necessarily constant since it interacts with the environment. But the sum of the energies of the system and the environment is a constant. We can alternatively state this is as follows: the change in energy of the system during a thermodynamic process that takes it from state $A$ to state $B$ is exactly equal to the energy supplied to, or, extracted from the system as a consequence of its interaction with the environment during this thermodynamic process. Since the only means by which a system can interact with its environment is through work and/or heat, we see that $$\Delta E_{AB} = W_{AB} + Q_{AB}.$$ Here, $\Delta E_{AB}$ is the change in the internal energy of the system as it moves from state $A$ to state $B$. $W_{AB}$ is the work done on the system by the environment durign the thermodynamic process $AB$, and $Q_{AB}$ is the heat supplied to the system by the environment during the same thermodynamic process. It is important to note that $\Delta E_{AB} = E_B - E_A$. In other words, the change in internal energy is simply a difference of the initial and final internal energies, and does not depend on the particular details of the path connecting the states $A$ and $B$. The terms $W_{AB}$ and $Q_{AB}$, on the other hand, are typically path-dependent and need a detailed specification of the path from state $A$ to state $B$. The work done in moving from state $A$ to state $B$ can be written as $$W_{AB} = \sum_{i=1}^n \int_{A}^{B} X_i(S, Y_1, \ldots, Y_n) \, dY_i.$$ Here, $(S, Y_1, \ldots, Y_n)$ is an intermediate state in the thermodynamic path connecting the states $A$ and $B$.

Remark

Notice the similarity of the expression with the expression of the work done in elementary mechanics as the line integral, along a specified path, of the inner product of the instantaneous force and the corresponding infinitesimal displacement.

Second law of thermodynamics

The second law of thermodynamics is a more subtle law which states that the change in entropy $\Delta S_{AB}$ of the system as it goes from state $A$ to state $B$ is always non-negative, irrespective of whether the thermodynamic path $AB$ is reversible or irreversible: $$\Delta S_{AB} \ge 0.$$ Here, $\Delta S_{AB} = S_B - S_A$ denotes the total change in entropy of the system. If $Q_{AB}$ denotes the total heat supplied to the system during the thermodynamic process $AB$, and if $T$ denotes the temperature of the environment during the process $AB$, we can write the change in entropy $\Delta S_{AB}$ as follows: $$\Delta S_{AB} = \frac{Q_{AB}}{T} + \Delta S_i, \qquad \Delta S_i \ge 0.$$ The change in entropy entirely due to the influx of heat is given by $Q_{AB}/T$, while $\Delta S_i$ denotes internal irreversible production of entropy in the system as a consequence of the thermodynamic process. The fact that $\Delta S_i \ge 0$ can be equivalently written as $$\Delta S_{AB} \ge \frac{Q_{AB}}{T}.$$

Remark

A generalized version of this inequality, known as the Clausius-Duhem inequality, will serve as the primary statement of the second law of thermodynamics in our study of continuum thermodynamics.

In the special case when the process $AB$ is reversible, $\Delta S_i = 0$ and we can write the change in entropy as follows: $$\Delta S_{AB} = \frac{Q_{AB}}{T}.$$ Note the equality in this expression! Using this in conjunction with the first law of thermodynamics and the expression for the work done by the thermodynamic driving forces, we see that $$\Delta E_{AB} = \sum_{i=1}^n \int_{A}^{B} X_i(S, Y_1, \ldots, Y_n) \, dY_i + \frac{Q_{AB}}{T}.$$ This form of the first law is especially useful in deriving a number of equivalent thermodynamic potentials for reversible thermodynamic processes. We will define those potentials as and when we need them later on.

Continuum Thermodynamics

We will now introduce a thermodynamic approach to modeling continua, and formulate, in particular, the first and second laws of thermodynamics for continua. This will serve as the starting point for our subsequent study of both balance principles and constitutive behavior.

Note

One of the early drivers for the thermodynamic approach to continuum mechanics is the so-called rational thermodynamics school. The most vocal proponents of this school of thought were Clifford Truesdell, Bernard Coleman and Walter Noll. One of the defining traits of this school is its axiomatic structure with a fairly large share of postulated laws for a field theoretic approach to the model the mechanical and thermal response of continua. Though a lot of the mathematics we develop in this course will look similar to the approach taken in rational thermodynamics, we will not entirely adopt the Truesdellian approach to modeling continua.

There are a few important points to note in order that we can properly understand the development of a thermodynamic theory of continua as a a non-equilibrium field theory. Let us unwrap this in stages:

• To understand a field theory means, recall that in our review of thermodynamics, the state variables were defined for the whole system. As a simple example, consider the case of a gas in a cylinder with a piston on one end. The volume occupied by the gas is an example of an extensive state variable that is globally defined with respect to the system. Let us now consider an elastic body undergoing deformation as a consequence of external loads acting on it. We will see that the equivalent of the volume of the gas in this case turns out to be the elastic strain. But as should be clear from our study of kinematics, strain is defined pointwise, and is actually a field variable; it is not a global definition as is the case with state variables in conventional thermodynamics.

• What makes the situation more complicated is that the field variables exhibit both spatial and temporal variations - the strain in an elastic solid can vary over the body, and the entire strain field can evolve over time. This is a defining trait of a system that is out of equilibrium. One of the difficulties in constructing a continuum theory of such non-equilibrium processes is that even in the elementary thermodynamic case, a rigorous theory of systems out of equilibrium with the same generality and depth as equilibrium thermodynamics is not available.

• In practice, we get away with this problem in continuum mechanics by making a series of assumptions, whose eventual validity is decided solely based on the agreement of the predictions of the resulting theory of continua with experimental data. The most important of this is that it is possible define non-equilibrium field theoretic analogues of quantities like internal energy, entropy, and other state variables that characterize the system. Further, the laws of thermodynamics will be appropriately extended based on these non-equilibrium field variables.

• The modern theory of dissipative phenomena in continuum mechanics postulates additional field variables called internal variables. These are variables that cannot be controlled, but nevertheless are required to represent the non-equilibrium state of the sytem satisfactorily. We will not delve too much into the thermodynamics of internal variables in this course, primarily because we won't have enough time to cover them in the lectures.

• A significant issue with this (i.e. the so-called rational) approach to thermomechanics which postulates the existence of field variables like internal energy and entropy is that there is no longer a clear means to distinguish between thermodynamic equilibrium and the extent of deviation from thermodynamic equilibrium in this framework. Nevertheless, the development to follow can be viewed as a first look at the thermomechanics of continua that provides a working approximation that works well practice.

With the foregoing caveats in place, let us now turn to the generalization of the first and second laws of thermodynamics for continua. Suppose that a continuum body $\mathcal{B}$ occupies a region $B_t \subseteq \mathbb{R}^3$ at the current instant of time. Let us start with the total energy $E(t)$ of the body at the current instant of time $t$. This can be written as the sum of the internal energy $U(t)$ of the body, and its kinetic energy $T(t)$, as follows: $$\begin{split} E(t) &= T(t) + U(t),\\ T(t) &= \int_{B_t} \frac{1}{2}\rho(\mathsf{x},t)\mathsf{v}(\mathsf{x},t)\cdot\mathsf{v}(\mathsf{x},t)\,dv,\\ U(t) &= \int_{B_t} \rho(\mathsf{x},t)e(\mathsf{x},t)\,dv. \end{split}$$ Here, $\rho$ is the mass density that we studied earlier in the context of kinematics, and $\mathsf{v}$ is the spatial velocity. The quantity $e(\cdot,t):B_t \to \mathbb{R}$ is the internal energy per unit mass. To understand what this means, consider a small volume $\Delta v$ centered at $\mathsf{x} \in B_t$, the internal energy of the material in this volume is given by $\rho(\mathsf{x},t)e(\mathsf{x},t)\Delta v$.

Remark

A useful mental picture to understand the meaning of the internal energy $\rho(\mathsf{x},t)e(\mathsf{x},t)\Delta v$ is obtained by picturing the material inside the elementary volume $\Delta v$ as consisting of a (large!) collection of balls connected by springs, as shown in the figure below:

The internal energy of the material in the region $\Delta v$ can be understood as the sum of the potential energy stored in the springs and the kinetic energy associated with the vibration of the balls about their mean positions. The center-of-mass velocity of this spring-mass system corresponds to the spatial velocity $\mathsf{v}(\mathsf{x},t)$. Note that the internal energy consists of both the potential and kinetic components, though the latter is not the kinetic energy of the macroscopic continuum. This discussion can be made precise using averaging tools from statistial mechanics - this will be discussed in the companion course Multiscale Modeling of Materials.

The motion of the body in three dimensional space is viewed as a thermodynamic process. Since this is a continuous-in-time process, we will be working with the rate at which the energy of the body changes, $$\dot{E}(t) = \frac{d}{dt} \int_{B_t} \rho(\mathsf{x},t)\left(e(\mathsf{x},t) + \frac{1}{2}\mathsf{v}(\mathsf{x},t)\cdot\mathsf{v}(\mathsf{x},t)\right)\,dv,$$ instead of infinitesimal changes in the internal energy as in the case of conventional thermodynamics. We will now extend the first law of thermodynamics for continua as follows: $$\dot{E}(t) = P_m(t) + P_h(t),$$ where $P_m(t)$ is mechanical power, and $P_h(t)$ is the rate at which heat is addded to the body.

To compute the mechanical power, we need to introduce two distinct kinds of forces that act on a continuum body. The first, called a body force, acts on every volume of material inside the body. The second, called a surface force, acts on the external surface of the body.

Example

As an example, consider the force of gravity acting on a body in a gravitational field. It is clear that every sub-volume of the body experiences a gravitational force. We thus see that gravity is an example of a body force. As an example of a surface force, suppose that we are holding a block of material in our hands. The block is held in place by the frictional forces between the hand and the block, and acts only on the associated contact surface.

To formalize this notion, we introduce a body force density $\mathsf{b}(\cdot,t):B_t \to T\mathbb{R}^3$ so that the total body force acting on a small volume $\Delta v$ at $\mathsf{x} \in B_t$ is given by $\rho(\mathsf{x},t)\mathsf{b}(\mathsf{x},t)\Delta v$. In an analogous fashion, we introduce the surface force density at $\mathsf{x} \in \partial B_t$ as the vector $\mathsf{t}(\mathsf{x},t,\mathsf{n}(\mathsf{x},t)) \in T_{\mathsf{x}}B_t$, where $\mathsf{n}(\mathsf{x},t)$ denotes the outward normal to the surface $\partial B_t$ at $\mathsf{x} \in \partial B_t$. This means that the force acting on a small surface area $\Delta a$ at $\mathsf{x} \in B_t$ with normal $\mathsf{n}(\mathsf{x},t)$ is given by $\mathsf{t}(\mathsf{x},t,\mathsf{n}(\mathsf{x},t))\Delta a$. Using these definitions, we can write the mechanical power due to the body and surface forces as $$P_m(t) = \int_{B_t} \rho(\mathsf{x},t)\mathsf{b}(\mathsf{x},t)\cdot\mathsf{v}(\mathsf{x},t)\,dv + \int_{\partial B_t} \mathsf{t}(\mathsf{x},t,\mathsf{n}(\mathsf{x},t))\cdot\mathsf{v}(\mathsf{x},t)\,da.$$ Notice that the body force density is the body force per unit mass, while the surface force density is the surface force per unit area.

As the final ingredient to formulating the first law of thermodynamics for continua, write the rate at which heat is added to the body is written as $$P_h(t) = \int_{B_t} \rho(\mathsf{x},t)r(\mathsf{x},t)\,dv - \int_{\partial B_t} h(\mathsf{x},t,\mathsf{n}(\mathsf{x},t))\,da.$$ Here, $r(\cdot,t):B_t \to \mathbb{R}$ is the rate at which heat is produced inside the body, and $h(\mathsf{x},t,\mathsf{n}(\mathsf{x,t}))\Delta a$ is the rate at which heat flows out of $B_t$ into the environment through an infinitesimal area $\Delta a$ centered at $\mathsf{x} \in \partial B_t$.

Putting all this together, we can write the first law of thermodynamics for continua, $\dot{E}(t) = P_m(t) + P_h(t)$, as $$\begin{split} \frac{d}{dt} \int_{B_t} \rho(\mathsf{x},t)\left(e(\mathsf{x},t) + \frac{1}{2}\mathsf{v}(\mathsf{x},t)\cdot\mathsf{v}(\mathsf{x},t)\right)\,dv &= \int_{B_t} \rho(\mathsf{x},t)\mathsf{b}(\mathsf{x},t)\cdot\mathsf{v}(\mathsf{x},t)\,dv\\ &+ \int_{\partial B_t} \mathsf{t}(\mathsf{x},t,\mathsf{n}(\mathsf{x},t))\cdot\mathsf{v}(\mathsf{x},t)\,da\\ &+ \int_{B_t} \rho(\mathsf{x},t)r(\mathsf{x},t)\,dv\\ &- \int_{\partial B_t} h(\mathsf{x},t,\mathsf{n}(\mathsf{x},t))\,da. \end{split}$$ We will develop the pointwise version of the first law for continua later on.

Let us now turn our attention to the second law of thermodynamics for continua. In exact analogy to the foregoing developments, we can generalize the statement of the second law in conventional thermodynamics, $\Delta S_{AB} \ge Q_{AB}/T$, as discussed earlier, as follows: $$\frac{d}{dt}\int_{B_t} \rho(\mathsf{x},t)\eta(\mathsf{x},t)\,dv \ge \int_{B_t} \frac{\rho(\mathsf{x},t)r(\mathsf{x},t)}{\theta(\mathsf{x},t)}\,dv - \int_{\partial B_t} \frac{h(\mathsf{x},t,\mathsf{n}(\mathsf{x},t))}{\theta(\mathsf{x},t)}\,da.$$ This form of the second law is also known as the Clausius-Duhem inequality. To understand the new terms introduced here, consider a small volume $\Delta v$ centered at $\mathsf{x} \in B_t$. Then, $\eta(\mathsf{x},t)$ is the entropy per unit masss such that the total entropy of the material in the volume $\Delta v$ is given by $\rho(\mathsf{x},t)\eta(\mathsf{x},t)\Delta v$. The quantity $\theta(\mathsf{x},t)$ denotes the absolute temperature of the material contained in the volume $\Delta v$.

This completes our preliminary discussion of the first and second laws of thermodynamics adapted for continua. In the following, we will use the laws of thermodynamics as presented here to derive all the balance principles of continuum mechanics.