What are the governing equations solved in FEA for structural mechanics?

Question

In CFD we focus on the NS equations as the governing equations (together with other equations) and solved them using Some numerical method (usually FVM). What are the governing equations (counter parts of the NS equation) in solid mechanics?.

Answer 1

Momentum equation

The Navier-Stokes equations represent the equations for the conservation of linear momentum. In convective form they are written as $$ \rho\frac{D\mathbf{v}}{Dt} = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{g} $$ where $\mathbf{v}$ is the velocity and the stress is $\boldsymbol{\sigma} = -p\,\mathbf{I} + \boldsymbol{\tau}$.

The linear momentum equation in solids is identical $$ \rho~\frac{D\mathbf{v}}{Dt} = \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} +\rho~\mathbf{b} $$

Constitutive relation

For compressible Newtonian fluids, the relationship between stress and velocity is $$ \boldsymbol \sigma = \lambda (\nabla\cdot\mathbf{v}) \mathbf I + 2 \mu \dot{\boldsymbol \varepsilon} $$ where $\lambda$ is the bulk viscosity, $\mu$ is the dynamic viscosity, and $$ \dot{\boldsymbol \varepsilon} = \tfrac{1}{2} \left(\nabla\mathbf{v} + ( \nabla\mathbf{v})^\mathrm{T}\right) $$

For linear elastic solids, the stress-strain relation is $$ \boldsymbol \sigma = \lambda (\nabla\cdot\mathbf{u}) \mathbf I + 2 \mu \boldsymbol \varepsilon $$ where $\lambda$ and $\mu$ are Lame parameters, $\mathbf{u}$ is the displacement, and $$ \boldsymbol \varepsilon = \tfrac{1}{2} \left(\nabla\mathbf{u} + ( \nabla\mathbf{u})^\mathrm{T}\right) $$

Eulerian vs Lagrangian

Because it typically does not make sense to solve CFD equations from the point of view of individual particles, an Eulerian FVM approach is often used. In contrast, Lagrangian methods are essential for solids because we typically need to know where each point in a structure moves to under the action of loads.