MLSM and Solid Mechanics

Equations

The goal in computational solid mechanics is to solve the equation of motion for a solid body \[\rho \frac{\partial^2 \b{u}}{\partial t^2} = \nabla\cdot\b{\sigma} + \b{F}\]

on the given domain $\Gamma = \Gamma_N \bigcup \Gamma_D$ with Dirichlet (essential) boundary conditions \[ \b{u} = \bar{\b{u}} \quad \text{ on } \Gamma_D,\] and traction (natural) boundary conditions \[ \b{\sigma}\cdot\b{n} = \bar{\b{t}},\quad \text{ on } \Gamma_N,\] where the quantities with the bar sign $\bar{}$ indicate prescribed displacements and surface tractions, respectively. Another type of boundary condition are mixed (Robyn) type boundary conditions. In this case the displacement might be given in one direction, and the traction will be given for the other direction. In case the right-hand side value is zero, the boundary conditions are called homogeneous. Homogeneous Dirichlet boundary conditions mean the the body is fully fixed (constrained), while zero traction boundary conditions allow the surface to move freely.

Using Hooke's law and the strain-displacement relation, the equation of motion can be transformed into the Navier-Cauchy equations \[\rho \frac{\partial^2 \b{u}}{\partial t^2} = \mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F}.\] The solution of this equation with appropriate BC's will give us the desired displacement values across the domain.

Steady-state solution

To obtain the steady state solution we have two possible strategies:

1. Add a (linear) dampening term to the Navier equation, and simulate the dynamic behaviour until the motion ceases: \[\rho \frac{\partial^2 \b{u}}{\partial t^2} + \eta_C \frac{\partial \b{u}}{\partial t} = \mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F}.\]
2. Directly solve at steady state: \[\mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F} = 0.\]