Difference between revisions of "MLSM and Solid Mechanics"
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To obtain the steady state solution we have two possible strategies: | To obtain the steady state solution we have two possible strategies: | ||
# Add a (linear) dampening term to the Navier equation, and simulate the dynamic behaviour until the motion ceases: \[\rho \frac{\partial^2 \b{u}}{\partial t^2} + \eta_C \frac{\partial \b{u}}{\partial t} = \mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F}.\] | # Add a (linear) dampening term to the Navier equation, and simulate the dynamic behaviour until the motion ceases: \[\rho \frac{\partial^2 \b{u}}{\partial t^2} + \eta_C \frac{\partial \b{u}}{\partial t} = \mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F}.\] | ||
| − | # Directly solve | + | # Directly solve for steady state: \[\mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F} = 0.\] |
| + | |||
| + | Displacement boundary conditions are very simple to implement with both approaches. The desired values are simply prescribed for dynamic simulation with explicit methods, In case of implicit methods where we need to solve a system of equations the prescribed displacements are placed into the right-hand side of the system, and the value 1 is placed in the diagonal of the matrix (for a given node). | ||
| + | |||
| + | For the traction boundary condition | ||
| + | |||
| + | \[\begin{bmatrix} | ||
| + | \sigma_{xx} & \sigma_{xy} \\ | ||
| + | \sigma_{yx} & \sigma_{yy} | ||
| + | \end{bmatrix} \cdot | ||
| + | \begin{bmatrix} | ||
| + | n_x \\ | ||
| + | n_y | ||
| + | \end{bmatrix} = | ||
| + | \begin{bmatrix} | ||
| + | \bar{t}_x \\ | ||
| + | \bar{t}_y | ||
| + | \end{bmatrix}\] | ||
| + | |||
| + | we first use Hooke's law and the strain-displacement relations to express the traction boundary in terms of displacements: | ||
| + | |||
| + | \[\begin{bmatrix} | ||
| + | (2\mu+\lambda)\partial_x u_x + \lambda \partial_y u_y & \mu(\partial_x u_y + \partial_y u_x) \\ | ||
| + | \mu(\partial_x u_y + \partial_y u_x) & \lambda \partial_x u_y + (2\mu + \lambda) \partial_y u_y | ||
| + | \end{bmatrix} \cdot | ||
| + | \begin{bmatrix} | ||
| + | n_x \\ | ||
| + | n_y | ||
| + | \end{bmatrix} = | ||
| + | \begin{bmatrix} | ||
| + | \bar{t}_x \\ | ||
| + | \bar{t}_y | ||
| + | \end{bmatrix}\] | ||
Revision as of 17:13, 26 January 2017
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:
- Add a (linear) dampening term to the Navier equation, and simulate the dynamic behaviour until the motion ceases: \[\rho \frac{\partial^2 \b{u}}{\partial t^2} + \eta_C \frac{\partial \b{u}}{\partial t} = \mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F}.\]
- Directly solve for steady state: \[\mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F} = 0.\]
Displacement boundary conditions are very simple to implement with both approaches. The desired values are simply prescribed for dynamic simulation with explicit methods, In case of implicit methods where we need to solve a system of equations the prescribed displacements are placed into the right-hand side of the system, and the value 1 is placed in the diagonal of the matrix (for a given node).
For the traction boundary condition
\[\begin{bmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{bmatrix} \cdot \begin{bmatrix} n_x \\ n_y \end{bmatrix} = \begin{bmatrix} \bar{t}_x \\ \bar{t}_y \end{bmatrix}\]
we first use Hooke's law and the strain-displacement relations to express the traction boundary in terms of displacements:
\[\begin{bmatrix} (2\mu+\lambda)\partial_x u_x + \lambda \partial_y u_y & \mu(\partial_x u_y + \partial_y u_x) \\ \mu(\partial_x u_y + \partial_y u_x) & \lambda \partial_x u_y + (2\mu + \lambda) \partial_y u_y \end{bmatrix} \cdot \begin{bmatrix} n_x \\ n_y \end{bmatrix} = \begin{bmatrix} \bar{t}_x \\ \bar{t}_y \end{bmatrix}\]
