# Hourglass Modes

^{1}They produce linear strain modes, which cannot be accounted for using a standard one point integration scheme.

${\Gamma}^{4}=\left(+1,-1,+1,-1,-1,+1,-1,+1\right)$

To correct this phenomenon, it is necessary to introduce anti-hourglass forces and moments. Two possible formulations are presented hereafter.

## Kosloff and Frasier Formulation

The Kosloff-Frasier hourglass formulation ^{2} uses a simplified hourglass vector. The hourglass
velocity rates are defined as:

- $\Gamma $
- Non-orthogonal hourglass mode shape vector
- $\nu $
- Node velocity vector
- $i$
- Direction index, running from 1 to 3
- $I$
- Node index, from 1 to 8
- $\alpha $
- Hourglass mode index, from 1 to 4

This vector is not perfectly orthogonal to the rigid body and deformation modes.

All hourglass formulations except the physical stabilization formulation for solid elements in Radioss use a viscous damping technique. This allows the hourglass resisting forces to be given by:

- $\rho $
- Material density
- $c$
- Sound speed
- $h$
- Dimensional scaling coefficient defined in the input
- $\Omega $
- Volume

## Flanagan-Belytschko Formulation

In the Kosloff-Frasier formulation seen in Kosloff and Frasier Formulation, the hourglass base vector ${\Gamma}_{I}^{\alpha}$ is not perfectly orthogonal to the rigid body and deformation modes that are taken into account by the one point integration scheme. The mean stress/strain formulation of a one point integration scheme only considers a fully linear velocity field, so that the physical element modes generally contribute to the hourglass energy. To avoid this, the idea in the Flanagan-Belytschko formulation is to build an hourglass velocity field which always remains orthogonal to the physical element modes. This can be written as:

The linear portion of the velocity field can be expanded to give:

Decomposition on the hourglass vectors base gives ^{1}:

- $\frac{\partial {q}_{i}^{\alpha}}{\partial t}$
- Hourglass modal velocities
- ${\Gamma}_{I}^{\alpha}$
- Hourglass vectors base

Remembering that $\frac{\partial {v}_{i}}{\partial {x}_{j}}=\frac{\partial {\Phi}_{j}}{\partial {x}_{j}}\cdot {v}_{iJ}$ and factorizing Equation 5 gives:

is the hourglass shape vector used in place of ${\Gamma}_{I}^{\alpha}$ in Equation 2.

## Physical Hourglass Formulation

You also try to decompose the internal force vector as:

In elastic case, you have:

The constant part $\left\{{\left({f}_{I}^{\mathrm{int}}\right)}^{0}\right\}={\displaystyle \underset{\Omega}{\int}{\left({\left[{B}_{I}\right]}^{0}\right)}^{t}\left[C\right]{\displaystyle \sum _{j=1}^{8}{\left[{B}_{J}\right]}^{0}\left\{{v}^{J}\right\}}d\Omega}$ is evaluated at the quadrature point just like other one-point integration formulations mentioned before, and the non-constant part (Hourglass) will be calculated as:

Taking the simplification of $\frac{\partial {x}_{i}}{\partial {\xi}_{j}}=0;(i\ne j)$ (that is the Jacobian matrix of Strain Rate, Equation 1 is diagonal), you have:

with 12 generalized hourglass stress rates ${\stackrel{.}{Q}}_{i\alpha}$ calculated by:

and

Where, $i$ , $j$ , $k$ are permuted between 1 to 3 and ${\dot{q}}_{i}^{\alpha}$ has the same definition than in Equation 6.

Extension to nonlinear materials has been done simply by replacing shear modulus $\mu $ by its effective tangent values which is evaluated at the quadrature point. For the usual elastoplastic materials, use a more sophistic procedure which is described in Advanced Elasto-plastic Hourglass Control.

## Advanced Elasto-plastic Hourglass Control

- Plastic yield criterion
- The von Mises type of criterion is written by:$$f={\sigma}_{eq}^{2}(\xi ,\eta ,\zeta )-{\sigma}_{y}^{2}=0$$
- Elastro-plastic hourglass stress calculation
- The incremental hourglass stress is computed by:
- Elastic increment
${\left({\sigma}_{i}\right)}_{n+1}^{trH}={\left({\sigma}_{i}\right)}_{n}^{H}+\left[C\right]{\left\{\dot{\epsilon}\right\}}^{H}\text{\Delta}t$

- Check the yield criterion
- If
$f\ge 0$
, the hourglass stress correction will be done by un
radial return
${\left({\sigma}_{i}\right)}_{n+1}^{H}=P\left({\left({\sigma}_{i}\right)}_{n+1}^{trH},f\right)$

- Elastic increment

^{1}Flanagan D. and Belytschko T.,

A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control, Int. Journal Num. Methods in Engineering, 17 679-706, 1981.

^{2}Kosloff D. and Frazier G.,

Treatment of hourglass pattern in low order finite element code, International Journal for Numerical and Analytical Methods in Geomechanics, 1978.