MLSM and Solid Mechanics - Revision history

Gkosec: /* Steady-state solution */

2018-02-24T09:54:09Z

‎Steady-state solution

Gkosec: /* Equations */

2018-02-24T09:53:46Z

‎Equations

Gkosec at 15:54, 8 December 2017

2017-12-08T15:54:03Z

Jureslak: /* Equations */

2017-07-14T00:41:29Z

‎Equations

Ipribec: /* Steady-state solution */

2017-01-26T16:30:21Z

‎Steady-state solution

Ipribec: /* Steady-state solution */

2017-01-26T16:13:50Z

‎Steady-state solution

Ipribec: /* Equations */

2017-01-26T15:47:43Z

‎Equations

Ipribec: Created page with "= 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{\sigm..."

2017-01-26T15:24:47Z

Created page with "= 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{\sigm..."

New page

= 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.

← Older revision		Revision as of 09:54, 24 February 2018
Line 1:		Line 1:
	Click here to return back to [[Solid Mechanics]]		Click here to return back to [[Solid Mechanics]]
−
−	~~= 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:~~
−
−	~~\[\begin{bmatrix}~~
−	~~(2\mu+\lambda)\partial_x u_x + \lambda \partial_y u_y & \mu(\partial_x u_y + \partial_y u_x) \\~~
−	~~\mu(\partial_x u_y + \partial_y u_x) & \lambda \partial_x u_x + (2\mu + \lambda) \partial_y u_y~~
−	~~\end{bmatrix} \cdot~~
−	~~\begin{bmatrix}~~
−	~~n_x \\~~
−	~~n_y~~
−	~~\end{bmatrix} =~~
−	~~\begin{bmatrix}~~
−	~~\bar{t}_x \\~~
−	~~\bar{t}_y~~
−	~~\end{bmatrix}\]~~

← Older revision		Revision as of 09:53, 24 February 2018
Line 1:		Line 1:
	Click here to return back to [[Solid Mechanics]]		Click here to return back to [[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 $\Gamma = \Gamma_N \cup \Gamma_D$ 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 =		= Steady-state solution =

← Older revision		Revision as of 15:54, 8 December 2017
Line 1:		Line 1:
		+	Click here to return back to [[Solid Mechanics]]
		+
	= Equations =		= Equations =

@@ Line 42: / Line 42: @@
 \[\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
+\mu(\partial_x u_y + \partial_y u_x) & \lambda \partial_x u_x + (2\mu + \lambda) \partial_y u_y
 \end{bmatrix} \cdot
 \begin{bmatrix}

@@ Line 19: / Line 19: @@
 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 at steady state: \[\mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F} = 0.\]
+# Directly solve for steady state: \[\mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F} = 0.\]

@@ Line 4: / Line 4: @@
 \[\rho \frac{\partial^2 \b{u}}{\partial t^2} = \nabla\cdot\b{\sigma} + \b{F}\]
-on the given domain $\Gamma = \Gamma_N \bigcup \Gamma_D$ with Dirichlet (essential) boundary conditions
+on the given domain $\Gamma = \Gamma_N \cup \Gamma_D$ with Dirichlet (essential) boundary conditions
 \[ \b{u} = \bar{\b{u}} \quad \text{ on } \Gamma_D,\]
 and traction (natural) boundary conditions

← Older revision		Revision as of 15:47, 26 January 2017
Line 14:		Line 14:
	\[\rho \frac{\partial^2 \b{u}}{\partial t^2} = \mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F}.\]		\[\rho \frac{\partial^2 \b{u}}{\partial t^2} = \mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F}.\]
	The solution of this equation with appropriate BC's will give us the desired displacement values across the domain.		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 at steady state: \[\mu\nabla^2\b{u} + (\lambda+\mu)\nabla(\nabla\cdot\b{u}) + \b{F} = 0.\]