### Chicken or Egg: Causality Dilemma

No, this is not a philosophical article referencing the vicious circle of cause and consequence. I just used a catchy title to draw your attention. So what I really wish to answer is what is more fundamental: Stress or Strain? Now, a few of you might argue as to why this question and tell that stress-strain always co-exist. To answer that, I must tell you that it is by looking back that we have a clear vision of the future. Thus differentiating the primary from the auxiliary, felicitates us in making groundbreaking discoveries. And besides, it’s always good to know how the human brain works and what is more instinctive to the homo sapiens.

Alright, so let’s get the mess sorted. Probably many of us believe that stress has to be fundamental. We apply a force on a body, it develops stresses and leads to deformations/strains. But, this approach has a basic flaw. How do you measure stresses? All the instruments we presently have all measure deflections and it is by calibrating these instrument do we get the stresses. So, we can imagine that we are measuring stress but it is the strains that we are always measuring. Also consider this view: stress is a quantity that measures the internal forces in a body. So zooming in to the molecular level, we see that the neighbouring molecules cannot exert forces on each other unless and until their mutual equilibrium is disturbed.

So, the case that I have presented until now is strongly in favour of strain. Come to think, this sounds a logical choice. Strain is essentially a normalized form of displacement, and displacement makes no sense without space. And anything to do with space is automatically related to the geometry. Ah! Now we have hit the bull’s-eye. You see, every form of math is but a variant of geometry. Geometry is the first human endeavor to understand nature and its language. The Greeks used geometry to do basic arithmetic: add, subtract, multiply, divide and even take a square root! “Euclid’s Elements” is referred as the most influential textbook ever written. It has never ran out of print(since the inception of printing) and is second only to the Bible in the number of editions published. So, now you can understand why all the fuss around geometry. It took us five and a half centuries to figure out algebra, geometry on the other hand is as natural to us as is sight.

Alright! Now you would say that I always knew that strain was more fundamental than strain, that is why it is always an independent quantity when we plot the famous stress-strain relationship. But, there is a catch in this statement. One can plot strain-stress as good as stress-strain, as long as one uses the natural strain. The problem with our “Engineering” strain is that, we engineers love to live in the world of statics and simplify every system to a static one. We see that the area of the cross-section keeps changing but ah, what the heck. The deformations are small and thus the change in area is small and is generally ignored. When Leonhard Euler first proposed the equation1, Giordano Riccati first played with the idea and later Thomas Young popularized the philosophy, they lived in the elastic range of the material. This range as was later found was surprisingly very narrow when compared to the deformation of the material at failure (The plastic strain can go as high as 8 to 10 times the yield strain and the fracture strain 15 to 30 times). So, the Engineer’s static simplification of area is short lived. But we still adhere to this assumption which gives rise to the problem of having multiple strains for a single value of stress. So, it is the definition of a function which forces the strain to be plotted as the abscissa.

Puff! That was a long discussion. But, I really hope that now you understand that why seeing is believing or why closing your eyes to the assumptions can lead to major mistakes. And of course, we all know that it is not possible for an animal to evolve into another species in a single lifetime and the fact that egg-laying species pre-date the existence of chickens; it is biologically necessary that the egg came first.

Notes:

1 Robert Hooke in 1678 published a paper suggesting that the displacement under a load is related to the applied force. In 1705, James Bernoulli proposed that the proper way to study deformation of geometry is by giving force per unit area or stress as a function of elongation per unit length, strain. Leonhard Euler, a student of James Bernoulli’s brother Johann Bernoulli, proposed the linear relationship s=E*e.

1. What about the cases where there is STRAIN without STRESS and vice versa ?
For example

Strain without Stress Case:

Temperature Change produces strains in a statically determinate structures without creating any corresponding stresses.

Stress without Strain Case:

In statically indeterminate structures there are stresses resulting from the restricted movement arising from temperature changes with no strain.

1. Well, the case is, when you resist something you will have stresses developed inside the material. Temperature changes in a statically determinate structure, which means that you are allowing the material to expand. This leads to zero stress inside the material and that is correct. And then you can never have stress strain relation.

And the other case of stress without strain, well the bar is fixed at both ends and you heat it, definitely till some point it will not show any signs of longitudinal strain. But after some time, if it is a long bar it will buckle which is a sign of strain, if it is a short bar it will try to bulge. If you cool the bar, it will develop tension but till the fracture point is reached it will not fail. If you consider the second case I would like to tell that here the longitudinal strains will not be developed but there will definitely be formation of lateral strains.

So I wish to point out that, there cannot be zero stain and some stress, but there can be zero stress with strains developed in it. That is why we are saying that strain is a fundamental quantity, not stress.

I hope you got the answer... :D

The problem with strain is that it is an artificial quantity. It is constructed to quantify deformation from the natural length. When we heat an object, its natural length is the length after its thermal expansion. So when you heat a free ended rod(statically determinate structure), it goes to its natural length. There so to speak no natural strain. When you heat a fixed fixed beam(statically indeterminate structure), the beam is not allowed to expand as it naturally wants to. So we have a sort of natural compression by the ends. So it is in some sort of strain.
Hope this helps.

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