Navier and Beltrami-Michell Equations

Summary of equations for infinitesimal linear isotropic elasticity (static equilibrium only):

There are 3 equilibrium equations (6 unknowns):

(1) $\begin{equation*} \frac{\partial \sigma_{ij}}{\partial x_j}+\rho b_i=0\text{ } (\text{no sum on } i) \end{equation*}$

There are 6 constitutive equations (6 more unknowns):

(2) $\begin{equation*} \sigma_{ij}=2\mu\epsilon_{ij}+\lambda\epsilon_{kk}\delta_{ij} \end{equation*}$

There are 6 strain-displacement equations (3 more unknowns – i.e. displacement in 3 directions):

(3) $\begin{equation*} \epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) \end{equation*}$

TOTAL = 15 equations ; 15 unknowns

By this point, we have dealt with all of these equations enough to recognize that there are indeed 15 unknowns and 15 equations. However, in order to stem any possible confusion about the physical interpretation of these equations and unknowns, it will be mentioned one last time that rotational values of “ $u$ ” as well as “moment” equilibrium of stress only appear in structural analysis formulas that have been simplified. In solid mechanics, concepts like “moment equilibrium” are not considered, but would be satisfied naturally in any well-posed problem.

eq. 3 $\longrightarrow$ eq. 2 $\longrightarrow$ eq. 1
$\longrightarrow$ 3 equations, 3 unknowns (3 displacements):

(4) $\begin{equation*} (\lambda + \mu)\frac{\partial^{2}u_j}{\partial x_i \partial x_j}+\mu \frac{\partial^{2} u_i}{\partial x_j \partial x_j}+b_i=0 \quad i=1,2,3 \end{equation*}$

The three equations of eq. 4 are known as the Navier Equations

Just as the Navier Equations (eq. 4) depict a “displacement” relationship, we can alternatively develop a “stress” relationship, known as the Beltrami-Michell Equations.

Without going through the derivation:

(5) $\begin{equation*} \sigma_{ij,kk}+\frac{1}{1+\nu}\sigma_{kk,ij}-\frac{\nu}{1+\nu}\sigma_{mm,kk}\delta_{ij}=0 \end{equation*}$

The six equations of eq. 5 are known as the Beltrami-Michell Compatibility Equations. In 2D, we can derive a similar expression with the aid of the so-called Airy Stress Function. Energy methods can also be used in order to arrive at similarly concise expressions. Graduate-level courses in solid mechanics devoted to linear infinitesimal elasticity would investigate these kinds of concepts in detail.

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