Constitutive laws? Where from?

Hi someone (who is reading this post)! Writing something after a long time now. Had almost forgotten about the blog while settling into a new country. There are several things I'd like to discuss about that, but may be in a later post. This one is primarily to brush up something I am planning to present for a term paper in one of my courses.


Caveat 1: This post is neither about just the Constitution nor about just Laws, but about Constitutive laws!


This is something I've been thinking about as a result of my multifaceted background. As an aerospace engineer, we learnt ideas that were really restricted to how aeroplanes and satellites work and in engineering we focus so much on the applications that the subtlety of where the equations come from or the fundamentals are not discussed at all, that is to say that we wander far away from what is truly Physics and are mostly lost in the applications. As I say this, there'll be a thousand fingers pointing towards me saying "why 'lost'?" Isn't technology what is taking us further? To this, I say, true, but let's discuss more over a cup of tea/coffee!


I had learnt Fluid Mechanics before, but during my masters (still in engineering) I learned a bit of Kinetic Theory for Gases which actually held out the picture of what fluids or for that matter any continuum is really like! When we learnt solid mechanics or fluid mechanics, it was the macrosopic behaviour of these substances we focused on. But where does the macroscopic behaviour originate from? And is it consistent with the microscopic nature of matter? That is to say, does the laws of physics stay the same at molecular and humanly observable scales? Well turns out it does! But what do we know about the microscopic nature of substances? They are made of molecules and atoms! How does the way in which molecules interact with one another in a substance dictate its macroscopic behaviour ? Now, we shall take a quick detour and talk about something fairly related. Constitutive laws! I have talked a little bit about these in a previous post about Granular Materials. But, now that I am talking to people in Physics (while doing my PhD that is), I feel like the ambit of constitutive laws pervades more disciplines than just solids, liquids and gases. Wikipedia is such a buzz-kill, it already talks about this realization in great detail here, but don't look into it right away!


Nonetheless, first off, what is a Constitutive law?


It is a law that describes what 'constitutes' a medium or substance in a macroscopic sense and dictates how changing the same may alter the dynamics associated with it.


Let us talk about Newton's second law of motion. The force applied on an object is equal to the rate of change of momentum of the object. For speeds much lower than than the speed of light, we have,

F=ma,

which basically says that the force is directly proportional to the acceleration, and constant of proportionality is equal to the mass of the object on which the force is acting. The equation, thus, relates something that is an external 'cause' (a kinetic property) which is the force to what may be called the 'effect' (a kinematic property) which is the acceleration, via a constant which depends on how much matter is contained in the object. You could, for the sake of an argument, say that, the mass of an object does depend on the amount and arrangement of molecules in the object and in a way is a fundamental property of the substance constituting the object. If the number of molecules or the element or some microscopic property changed keeping the others constant, then the kinematics of the object would be different for the same kinetics.


Therefore, could we say that the mass of an object is a macrosopic property and is in turn dictated by the microscopic properties of the substance it is made of? Well, I think yes! If you disagree, I'd be happy to have a conversation. But remember that this argument is just a build-up for the story that is coming up ahead, so ... please be a little lenient. Thanks!


Great! So, can we call Newton's second law a Constitutive law in some sense? Well, may be!

It does indicate how changing the fundamental constitution of the object (medium) affects its dynamics.


Now, we talk about solid and fluid mechanics which are the most prevalent contexts where the term 'Constitutive law' appears! We know about Hooke's law, which says,

Stress = Young's Modulus x Strain

i.e. the stress applied on an object is directly proportional to the strain (for linearly elastic materials). We can again see the cause-effect relationship here, with the Young's modulus (YM) being the constant of proportionality and also the parameter that defines the material property. Basically, YM explains why a wooden ruler would not bend as much as a plastic one! Nice! So, the next question is, does YM depend on how the molecules in the material are arranged? The answer is, of-course yes! And that is not new.


Similarly for Newtonian fluids, we have,

Shear stress = Viscosity x Strain rate

where, the viscosity of the fluid is the material property.


For solids and fluids, Constitutive laws relate internal kinetic property (the stress) to a kinematic property (the strain/strain-rate). But one needs the governing equations, more specifically, the mass, momentum and energy balances to solve for the complete internal state of the system.


You'll find Constitutive laws in other fields like electromagnetism too. The parameter being constants like permittivity, permeability, electrical conductivity etc. which are dielectric/medium depended. May be now is the right time to look at Wikipedia's take on Constitutive laws. It is pretty comprehensive.

So, enough about what Constitutive laws are, but where do they come from?


We know that conservation laws / governing equations come from what is called Noether's theorem. Well, not many of us know that (at least, I did not, until recently, although the mathematical underpinnings are still beyond my reach). But, the interested reader can easily find resources! However, when it comes to Constitutive laws, one of the most important phrases you'll find on the Wikipedia page is that, "Some constitutive equations are simply phenomenological; others are derived from first principles."


For example, if you talk about the Young's modulus of a substance, we generally find it by performing experiments on them. Even the viscosity of a fluid which is its constitutive parameter is obtained experimentally. This is a phenomenological method.


The other method is to derive a constitutive law from first principles, which crudely translates to using math to model the medium/material behaviour starting from fundamental assumptions. Here, "a first principle (ab initio in Latin) is a basic proposition or assumption that cannot be deduced from any other proposition or assumption", says Wikipedia.


The method of using first principles was made historically significant by the Austrian genius Ludwig Boltzmann who laid the foundation of the kinetic theory of gases i.e. described how macroscopic properties of gases like pressure, temperature and specific volume are related to one another, not by performing experiments, but by assuming their atomic and molecular (microscopic) nature. LB proposed a probabilistic collisional model for gases thus obtaining the famous Ideal Gas law. It is the same theory that has now been extended to derive analytical expressions for the viscosity of fluids or the constitutive laws for granular materials. Later, significant work was done to establish other material parameters from first principles.


Let us now summarize:

In order to understand the macroscopic behaviour of a system, one may use conservation laws in conjunction with what are called constitutive laws, the latter of which may be derived from first principles (which are generally based on their microscopic properties) or obtained via experiments and observation.


So what?

Caveat 2: I am no physicist, and hence, the following is to be taken merely as an opinion.


My idea of Einstein's work on General Relativity is extremely simplistic and I have not done a course on it. However, having worked with some basic differential geometry at some point of time, I was familiarized with Christoffel symbols and other nuances of curvilinear coordinate systems which is extensively used in GR. Elon Musk says and it is probably very evident that we learn by analogy. We like finding similarities and differences between two different systems and establishing our understanding based on them. I did something similar.


That the distribution of matter bends spacetime is probably known to every science enthusiast today. For someone who has learnt solid mechanics, the word 'bend' rings a bell. When I put on the solid mechanics filter on and look at Einsteins field equations (EFE), it is easy to see that at the most basic level a constant relates the distribution of matter (energy-momentum tensor) to the deformation of spacetime (Einstein tensor), which is similar to the stress strain relationship stated earlier in the post in some way. This constant, comprising some other constants like the speed of light and the universal gravitational constant has been obtained phenomenologically through observations, just like the Young's modulus can be obtained from experiments.


Finally, the EFE must be combined with the geodesic equations which describe how freely falling (moving) particles move through curved spacetime, to solve for the evolution of matter distribution and spacetime together.


This coupling is relatable to the coupling of governing conservation laws (mass, momentum and energy balances) and constitutive relations in continuum (solid, fluid and others, crudely) mechanics. Thus, one could say that the EFE is the constitutive law here. Interesting? Yes? No? Okay?


This gives rise to two interesting questions:

  1. Is there a first principle formulation for the constant of proportionality in the EFE?

  2. Since these are universal constants, what may a different value of these constants imply? Just like a different value of Young's modulus or the Dielectric constant implies different materials.

Finally, this is not a research paper and I have to conclude this post, so I will not try to answer these questions right now; also, I feel it is a nice point to let the story end! What say?