Home

Federica EU

1/7

Giuseppe Mensitieri » 1.Body, state and constitutive behaviour

Fundamental laws and constitutive equations

1. Fundamental laws:

1^st, 2^nd and 3^rd laws of thermodynamics
Mass conservation
Conservation of linear momentum
Conservation of angular momentum

The relationships between macroscopic approach, microscopic approach and engineering applications.

Fundamental laws and constitutive equations

2) Constitutive equations:

Hopefully describe the macroscopic (i.e. accessible to direct measurement) behaviour of some restricted class of systems in some restricted class of phenomena, but ARE BY NO MEANS REGARDED AS UNIVERSAL TRUTH.

An example. Fluid dynamics:

$\rho \:is \:constant$
$\underline{\underline T}=-\left(P^{*}+\lambda div \underline v \right)\cdot\underline{\underline I}+2\mu \underline{\underline D}$

Where
$\underline{\underline T}$ is the ’stress’ tensor;
$\lambda$ and $\mu$ are viscosities;
$P^{*}$ is the ‘thermodynamic pressure’;
$\underline{\underline D}$ is the ‘rate of deformation’ tensor.

Once introduced into the linear momentum balance these equations give the Navier-Stokes equation. Actually thermodynamics, and specifically the 2^nd law, imposes restrictions on the allowable forms for the constitutive equations as well as for the relationships between them.

Fundamental laws and constitutive equations

Theory of constitutive equations: 2^nd law does not tell us what the constitutive equations are , but only what they cannot possibly be.

Moreover 2^nd law introduces two fundamental distinctions:
a) REVERSIBLE vs IRREVERSIBLE PHENOMENA
b) EQUILIBRIUM vs NON-EQUILIBRIUM CONDITIONS.

3) Engineering theory
Fundamental laws along with constituitve laws are used to solve engineering problems.

4)Statistical thermodynamics (microscopic theory)
Molecular-scale models are constructed and constitutive equations can be inferred from such models.

Fundamental concepts: body and state

Body and state

We introduce here some primitive concepts (a primitive concept is something of which we do not give any definition but that we describe by describing its properties).

BODY, $\mathfrak{B}$ . It is endowed with a fixed mass, $m_{t}$ , and occupies some finite region of space of volume $V_{t}$ . $V_{t}$ as well as the region of space, changes with time. A body is a closed system and $\partial \mathfrak{B}$ is the surface of the body.

Fundamental concepts: body and state

CONSTITUTIVE EQUATIONS. Constitutive equations are assumptions which may, or may not, adequately describe the behaviour of real bodies. There are several levels of assumptions:

1st level

Some quantity is a physical property of the body considered and, therefore, its value depends only on the physical condition of existence of the body considered, i.e. on its state, $\sigma _{t}$ . Such quantities are called functions of state or constitutive properties.

Hence, a mapping $\ell\left(\ \cdot \,\right)$ exists which maps the state into the value of the constitutive property, L:

$L=\ell\left(\ \sigma_{t} \,\right)$

Here and in the following, he subscript ‘t’ indicates that the quantity is referred to the body as a whole.

Fundamental concepts: body and state

2^nd level

We now make an assumption about which quantities determine the state of the body considered, i.e. to assign a mathematical structure to the state $\sigma _{t}$ .
There is a restriction: these quantities must, in principle, be measurable by measurements made only on the body itself.

3^rd level

We now assume a specific functional form for the constitutive mapping.

Example: the body is a mass of gas

$\left[1\right]\:p=f\left(\sigma_{t}\,\right)\;\rightarrow\;\left[2\right]\:p=f\left(V_{t},\,T\,\right)\;\rightarrow\;\left[3 \right]\:p=\frac{RTn_{t}}{V_{t}}$