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

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?.

• equations for stress, strain, load etc – Solar Mike Feb 21 at 8:34
• Static or dynamic equilibrium, geometric compatibility, and material constitutive equations. – alephzero Feb 21 at 9:13

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.

• @mart: The question was about the governing equations in solid mechanics used in structural FEA. My answer was designed to explain that there are no fundamental differences in how solids and fluids are described (in the usual situations). Of course, things are different for fracture, contact, cavitation, etc. – Biswajit Banerjee Feb 22 at 19:39
• ok sorry, I didn't read close enough. I'll remove my comment (Though I'm not sure I'm the only one to misunderstand you). – mart Feb 22 at 21:35
• If you start with the solid equations as that is what the OP is asking and refer to each fluid equivalent as necessary - that may help... – Solar Mike Feb 23 at 6:43
• @BiswajitBanerjee Yes the comparison with the equation governing fluid flow is helpful. Thank you. When studying usually we start off with the governing equation and later turn to the numerical methods to solve them. While studying FEM we are directly taken to the numerical method..and left wondering what the governing equations are. Usually it is derived from minimum potential energy theorem or virtual work theorem for individual cases etc..but noting is clearly stated about the GE. – GRANZER Feb 25 at 7:17
• A variational principle is as valid a starting point as a set of partial differential equations (PDE). You can derive the PDE from the variational principle. People start with variational principles, if they can, because these principles are more easily understood from a physical point of view and the correct function spaces can be chosen easily. If you start from a PDE, there are many ways of creating a weak form not all of which lead to an easily solved system of equations, e.g., the stress is retained in the weak form instead of using a purely displacement based PDE. – Biswajit Banerjee Feb 25 at 21:26