**Wave Propagation in 3D – Continuum Wave propagation velocity in 3D-continuum:**

**
**

comparison to rod :

Critical time step :

comparison of critical time steps

materials (ν = 0.5): α –> 0

Wave propagation velocity in 2D-continuum:

(twodimensional stress state)

comparison to rod :

- Solid elements : c
*3D-continuum*
- Shell elements : c
*2D-continuum*
- Beams & trusses : c
*rod*

__Remarks:__

- The wave propagation velocity of the rod c
*rod* has the smallest value in comparison to the 2D – and 3D-continuum.
- The wave propagation velocity for membrane deformations determines the critical time step for shell and beam elements.

**Time Step Control for Beam and Truss Elements**

For the Hughes-Liu beam and truss elements, the time step size is given by:

where L is the length of the element and c is the sound speed:

For the Belytschko beam the time step size given by the longitudinal sound speed is used, unless the bending-related time step size given by [Belytschko and Tsay 1982] governs

is smaller, where I and A are the maximum value of the moment of inertia and area of the cross section, respectively.

**Characteristic length lc for Time Step**

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warped elements :

several alternatives can be selected via `*CONTROL_TIMESTEP`

variable ISDO (Control Card 9, Columns 21-30), e.g.:

where β = 0 for quadrilateral and β = 1 for triangular shell elements.

**Time Step Control for Solid Shell Elements**

A critical time step size, Δ t*e* is computed for solid shell elements from

where V*e* is the element volume, A*emax* ist the area of the largest side, and c is the plane stress sound speed

**Critical Time Step for Spring Elements**

Problem : There is no wave propagation velocity c to calculate critical time step size.

Motivation : Consider free vibration of spring with nodal mass *m*1 and *m*2

Recall critical time step of rod :