Q:

What's the difference between constitutive laws and governing equations?

I'm studying about the finite element method in a class but I don't come from a civil engineering background. Anyways, it hasn't been made clear to me what the difference between constitutive laws and governing equations are. To me they both relate physical quantities with one another.

A:

A constitutive law is generally an algebraic relation which tells you the coefficients of a differential equation, while the governing equations are the differential equations themselves.
For example, if I have a metal piston on top of a gas, I can write down the equation of motion for the piston
$$m \ddot X - PA = 0$$
Where P is the pressure in the gas and A is the area of the piston. Without knowing how the pressure depends on the piston position, this is not a closed equation--- it refers to an undetermined quantity, the pressure. But the ideal gas law, that the pressure $P=C/(V-AX)$ where C,V are constants, determines the pressure in terms of X, and gives
$$ m \ddot X -{ AC\over (V - AX)} =0$$
Now the equation is closed--- it tells you the future behavior of X knowing X alone. The ideal gas law is the constitutive relation in this case, while the differential equation is the governing equation.
Constitutive equations are algebraic, governing equations are differential.

A:

A constitutive law is often an approximate solution to another differential (governing) equation which has much smaller transient scale. In the example with a piston we used a static law for pressure in a dynamical situation. We supposed that sound waves and other complications in the gas due to piston's moving relax much faster than $X$ changes, so we do not solve the other differential equation with unknown piston position, but we use its solution with a given (static) piston position. Thus we reduce the number of coupled governing equations. 
Another example is a relationship between the time-dependent electric field $\vec{E}(t)$ and current $\vec{j}(t)$ via conductivity $\sigma$: $$\vec{j}(t) = \sigma \vec{E}(t)$$ 
Here the inertial properties of electrons are neglected (no retardation, no relaxation time, immediate reaction to ( or "following") the external filed). As a matter of fact, there is a governing equation for electron velocity containing an inertial term: something like $m_e \frac{d\vec{v}}{dt}$ in the left-hand side and the forces in the right-hand side. The forces are the external electric field and the internal friction proportional to velocity (electric resistance). When we neglect the right hand side, the solution is obtained from equality of forces (driving and friction); this is how one obtains the solution $\vec{j}(t) = \sigma \vec{E}(t)$ (a constitutive law) in place of a governing equation.

