We need now to assign a better defined mathematical structure to the concept of * STATE*. In the following we will assume that the

**STATE VARIABLES:**

**External variables**: their values can be imposed at any one time, independently of each other.**Internal variables**: their values cannot be imposed at will, independently from the other state variables.

We give now two very important definitions.

Definition of * SITE*:

Definition of * EQUILIBRIUM*:

As a matter of fact, at equilibrium, a*ll the constitutive properties are only functions of the site*. For the special case of so-called * ELASTIC SYSTEMS* we have that

A * transformation* is defined as the mapping R (.) which maps time into site:

**PROCESS**:

**TRANSFORMATION:**

Processes and transformations coincide only in the case of elastic systems.

We will limit here our attention to systems in which the *only relevant deformations are changes of density*, i.e. t*he only non thermal state variable is V*. Consequently, if the work is only that related to density changes, we have:

Where *w* has the usual meaning of net rate of work per unit mass. By combining eq. (1) with the 1^{st} law, we obtain:

where *ρ = f(σ)*. We make now the constitutive assumption that:

Let’s consider two transformations, **R _{1}** and

We ask now ourselves:

If the system is * ELASTIC *σ = α and hence the answer is

But, in general, σ ≠ α and, consequently, it could happen that:

In particular, on the basis of the 2nd law and of the concept of irreversibility, we will demonstrate for the case of * VISCOUS* systems, that:

the values of the pressure being approximately the same provided both the expansion and the compression take place slowly enough. Furthermore one would expect the pressure difference to increase with increasing viscosity of the material.

Hence, in order to have the * capability of describing irreversible processes, one should take into account that the system considered be not elastic. In fact, elastic systems do not allow for any form of irreversibility and therefore the thermodynamic theory of elastic systems is essentially a reversible theory*.

We can generalize what we have discussed above to the cases in which the mechanical work is not only a pressure work. In fact, consider an infinitesimal area vector within a body, and let be the contact force acting through the area. The stress tensor is defined as the linear transformation of the area vector into the force vector:

The local form of the balance of linear momentum is:

since

here ρ is the *mechanical* pressure, *ρ** is the *thermodynamic* pressure, is the *deviatoric* part (zero trace tensor) of the stress tensor and is the *dynamical* part of the stress tensor. Here and in the following we will always use the superscript ‘*’ to indicate a value at equilibrium, which is, consequently, only a function of the site.

Multiplying scalarly eq. (3) by the velocity vector we obtain:

Since

we obtain from eq. (4):

Where *K* has the usual meaning of kinetic energy per unit mass. Moreover, the power *P* per unit mass of the forces acting on the system can be expressed as:

Since

where *V* has the usual meaning of local specific volume, we easily obtain by combining eqs. (5) and (6) that the net rate of work per unit mass has the following form:

From eq. (7) we can readily obtain the expression for the net rate of mechanical work per unit mass in the cases in which the only mechanical work is the pressure work. In fact, in such cases:

We recall, now, that the local form of mass balance in terms of material derivative is:

Moreover:

And, by combining the last two expressions we have:

i.e. the divergence of the velocity vector represents the relative rate of change of specific volume of the system. By combining eqs. (8) and (9) we finally obtain the expression for the net rate of work per unit mass for the cases in which the only mechanical work is the pressure work:

The previous derivation shows, once again, that **the body forces do not contribute nothing to the net rate of work.**

The concept of an elastic system can be easily generalized to the case where there are more than one external state variable to consider in addition to T; for instance, in an elastic solid the strain at any given point is a three-dimensional tensor. Let the site be the pair and T, where is an n-dimensional vector called the generalized volume.

Correspondingly, there will be an n-dimensional vector , called the generalized pressure, such that the net rate of work at a given point is given by:

We now go back to slide 6 where we wrote:

If we want to set up a theory able to predict that p at t* is different for the two transformations, we have to introduce new state variables, i.e.:

As a consequence σ ≠ α and the system is not an elastic system. * Systems for which the state is determined also by the rate of change of some or all the external state variables are named VISCOUS*. At equilibrium we have that:

As already mention before, * the equilibrium value of a state function depends only on the site*. We can then write:

So an order 0 approximation for p is *p = p** while and order 1 approximation is

If we consider the isothermal transformation reported in the following figure, before t_{0} and after t_{1} the system is at equilibrium, but it is not in the time interval from t0 t_{o} t_{1}. The total net work, W , done in that time interval is:

ELASTIC SYSTEMS →

SYSTEM FOR WHICH

We discuss here briefly the differences between * Viscous* and

In * VISCOUS* systems o

In fact there are materials (notably polymers) which have rather long relaxation times, so that * also the values at instant of time significantly remote from the instant of observation affect the state*. Such systems are called

Viscous systems allow irreversible processes: for the transformation we have illustrated and discussed above, the total net work done is dissipated, since the system at the end of the transformation is in the same state as at the start. Also, viscous systems allow a meaningful distinction to be made between equilibrium and non equilibrium values of the quantities of interest.

However, the type of irreversibility and of equilibrium considered in this section are not the only possible ones and, in fact, the pragmatically more important ones are those arising in systems endowed with internal state variables.

*1*. Body, state and constitutive behaviour

*3*. 1st and 2nd principles of thermodynamics. Integral and local forms. The concept of entropy

*4*. State and equilibrium. Part 1

*5*. State and equilibrium. Part 2

*6*. State and equilibrium. Part 3

*7*. State and equilibrium. Part 4

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