# Origin of the concepts of Stress and Strain

## Background & My research

So, I recently studied about the concepts of Stress and Strain in my high school physics classes and wanted to know about the history behind the origin and emergence of these concepts.

Firstly, about the concept of stress, I found out that it was coined by Augustin-Louis Cauchy in 1822 when he announced his "Stress Principle" which states the following:

Upon any smooth closed orientable surface $$dV$$, be it an imagined surface within the body or the bounding surface of the body itself, there exists an integrable field of traction vectors $$t_{dv}$$, equipollent to the action exerted by the matter exterior to $$dV$$ (contact forces) and contiguous to it on that interior to $$dV$$ (body forces). [1]

Next to move towards the "normal" representation of stress as we know it I found that we firstly need to represent the total force (and total torque) acting on the body which can be represented by the integration of the traction vectors with respect to mass and surface area (and their rotational transformations for the total torque). Now, I found that the stress we talk about only concerns with the boundary surface forces(contact forces) and hence the mass forces(body forces) vanish as the surfaces tend to zero while considering the "normal" idea of stress. Also, the couple stresses (stresses due to couple forces causing rotation) are not considered in this "normal" sense of stress (which is the classical sense of continuum mechanics).

I also found out that the traction vector $$t_{dv}$$ is dependent on the normal vector $$n$$ to the point we find the traction vector on (which is known as Cauchy's Postulate essentially stating that $$t_{dv}$$ = $$t_{n_i}$$ for any normal $$n$$ and plane $$i$$). Finally, after some rearranging of terms we get that the resulting traction vector $$t_{dv}$$ (for some normal vector $$n$$, plane $$i$$ and some area $$dS$$ containing an arbitrary point $$X$$) is equal to $$dF/dS$$ (where $$dF$$ is the surface force on that point $$X$$) and is called the surface traction, also known as stress vector. Also, the state of stress at a particular point $$X$$ is then depicted by the stresses at all planes passing through that point with some normal vectors $$n_i$$ (this relation is known as Cauchy's Fundamental Theorem) and according to this theorem the state of the stress at any of those infinite planes can be represented merely by knowing the stress vectors on three mutually perpendicular planes of the form: $$t_{n_i}$$ = $$n_i \cdot T$$ where $$T$$ is the stress tensor (i.e. the linear map between the stress at an arbitrary plane $$i$$ and it's normal vector $$n$$) which is obtained by the tensor product of the traction vectors at three mutually perpendicular planes and their associated normal vectors ($$t_x,t_y,t_z \text{ and } n_x,n_y,n_z$$ which is essentially a 3$$\times$$3 matrix).

[1] Cited from Truesdell: The Creation and Unfolding of the Concept of Stress, chap. IV in Essays in the History of Mechanics.

## Question:

Now, I want to know what motivated Cauchy to develop his postulate on the dependence of the stress (or traction) vector and the normal vector to the surface (which would also explain the dependence of stress on the orientation of the surface i.e. the plane considered). Additionally, I'm curious about the experiments, if any, that Cauchy used to verify his formulations and theorems.(I read that his work was motivated by Euler's previous work on Hydrodynamics around the 1750s and his idea of breaking the whole body into two imaginary parts (one interior and one exterior) which was Euler's Cut Principle).

Also, is the 'proof' for Cauchy's theorem and Cauchy's postulate in the form of deriving the expression for the "linear dependence of traction on the normal vector", along with the "existence of a stress tensor" for inter-planar transformations (of the infinite planes) by the use of 'Cauchy's Tetrahedron,' the original proof provided by Cauchy himself, or just a modern depiction of the proof? If it is the original, are there any sources showing why he used the tetrahedron specifically?

I could not find any information on the concept of Strain whatsoever and how it originated or who coined the term (although I think that it was coined by Cauchy as well). Hence, any information on the concept of strain would be appreciated.

• Thank you for viewing my question, I apologize if the text may not be readable due to incorrect formatting so any edits will be appreciated. Also, I think the information in background I provided is correct but if there is any incorrect information there please mention it as well. Commented Dec 29, 2023 at 11:01