# 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

^{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.

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

## Flanagan-Belytschko Formulation

^{1}:

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

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

## Physical Hourglass Formulation

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:

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:
(13) $$f={\sigma}_{eq}^{2}(\xi ,\eta ,\zeta )-{\sigma}_{y}^{2}=0$$ - 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.