# How are the classical set of equilibrium equations for linear elasticity derived?

In linear elasticity, the governing PDE is the equilibrium equations (absent of vibration considerations):

$$-\nabla \cdot \sigma = F$$

Is this equation simply derived from the sum of forces and moments?

In most linear elasticity papers, I see these governing equations. Is there an original source for where these equations came from? I'm looking for a more fundamental citation, but it seems so ubiquitously used and known that it's difficult for me to find the original source.

You take an arbitrary volume $$V$$ and use the translational and rotational equilibrium equations over it. They read

\begin{align*} \int\limits_A \mathbf{t}\mathrm{d}A + \int\limits_V \mathbf{f} \mathrm{d}V = 0\, ,\\ \int\limits_A \mathbf{r}\times\mathbf{t}\mathrm{d}A + \int\limits_V \mathbf{r}\times\mathbf{f}\mathrm{d}V = 0\, , \end{align*} where $$\mathbf{t}$$ are traction vectors, $$\mathbf{f}$$ the body forces, and $$\mathbf{r}$$ is the position vector. Then, due to the arbitrariness of the volume the integrals should equal 0 and you get the equation you present and the symmetry for the stress tensor (in classic elasticity).

According to @BiswajitBanerjee's comment, the first publications to discuss the topic were:

• Navier, C. L. M. H. (1821). Sur les lois des mouvement des fluides, en ayant egard a l’adhesion des molecules. Ann. Chimie, 19, 244-260.

• Cauchy, A. L. B. (1822). Recherches sur l'équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques.

You can find a recent discussion on

• Mase, George Thomas, Ronald M. Smelser, y George E. Mase. 2010. Continuum mechanics for engineers. 3rd ed. Boca Raton: CRC Press. (Chapter 5).

• Reddy, J. N. (2013). An introduction to continuum mechanics. Cambridge university press. (Chapter 5).

• Could you expand on what the "translational" and "rotational" equilibrium equations are? I'm not familiar with the terminology. It just the sum of momentum/forces? Nov 7, 2020 at 20:18
• It's the integral in this case, since now you have densities. Nov 7, 2020 at 20:21
• 1) Navier, 1821, Sur les lois des mouvement des fluides, en ayant egard a l'adhesion des molecules, Ann. Chimie. 2) Cauchy, 1822, Recherches sur l'équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques Nov 8, 2020 at 2:46

A different perspective on the question is this: Newton's law says that mass times acceleration equals the sum of all forces. You are interested in the steady state case, so the acceleration is zero and as a consequence, the sum of all forces is zero. This has to hold at each point of the solid if you want the body to not move.

The sum of all forces equals the external forces $$F$$ (actually, a force density, because we're looking at individual points) acting at each point of the body plus the internal forces $$\nabla \cdot \sigma$$ due to the stresses.

In other words, the equation you quote is simply a force balance.

You can find a discussion in AF Bower's book.

Applied Mechanics of Solids 1st Edition ISBN-13: 978-1439802472, ISBN-10: 1439802475

The book is available online at AF Bower's website

http://solidmechanics.org/Text/Chapter2_3/Chapter2_3.php#Section2_3_1

• Link-only answers are suboptimal since links might change in the future. Could you expand your answer adding the bibliographic information (since it is also a book) and adding a excerpt from the book. Nov 8, 2020 at 17:31