Hourglass stiffness:

*Material, ...
*Hourglass, <Method> = c^HG [, c^fHG]

Use this command to control the applied artificial stiffness (so called "Hourglass stiffness") constraining (spurious) hourglass modes propagating in linear interpolated and reduced integrated elements. Excitation of these modes may lead to severe mesh distortion, with no stresses resisting the deformation. In numgeo this is done by modifying the element stiffness matrix $K^e_{IiJj}$ by adding a small artificial (hourglass) stiffness $K^{HG}_{IiJj}$ to the element that acts to constrain the hourglass mode ¹²:

$K^e_{IiJj} = K^{e,1Pt}_{IiJj} + \kappa^{HG} \text{d}\Omega^e \sum^{\alpha} \gamma_{\alpha I} \gamma_{\alpha J} \delta_{ij}$

where $K^{e,1Pt}_{IiJj}$ is the conventional element stiffness matrix evaluated at one integration point (located in the centre of the element). $\kappa^{HG} > 0$ is a numerical parameter that controls the stiffness of the hourglass resistance and can be chosen arbitrarily. However, its magnitude has to be selected such that the parameter avoid hour-glassing but at the same time does not influence the solution of the problem. Since the parameter $\kappa^{HG}$ is not a true material constant, it must be normalized to provide approximately the same degree of stabilization for any geometry and material properties ¹. In numgeo two methods for the calculation of $\kappa^{HG}$ are implemented, namely the Stiffness version and the Scaling version:

Method = Stiffness: For the stiffness method, numgeo requires the user to prescribe directly the stiffness which should be applied prevent hourglass modes:

$\kappa^{HG} = c^{HG} \frac{\partial N_I}{\partial x_i} \frac{\partial N_I}{\partial x_i}$

The user should provide numgeo with a meaningful value for $c^{HG}$, which now represents a stiffness. The choice of $c^{HG}$ however is not trivial, especially if nonlinear (inelastic) materials are modelled.

Example single-phase material: *Hourglass, Stiffness = 1000.

Example two-phase material: *Hourglass, Stiffness = 1000., 10.
Method = Scaling (single-phase material): For the scaling method an incremental shear modulus $G^\Delta_{el}$ of an equivalent linear elastic material is computed based on the deviatoric parts of the current stress and strain increments $\Delta \sigma_{ij}^* = \Delta \sigma_{ij} - \Delta \sigma_{ii}/3$ and $\Delta \varepsilon_{ij}^* = \Delta \varepsilon_{ij} - \Delta \varepsilon_{ii}/3$, respectively:

$G^\Delta_{el} = \frac{1}{2} \sqrt{ \dfrac{ \frac{1}{2} \sum_{i=1}^3 \sum_{j=1}^3 \Delta \sigma_{ij}^* \Delta \sigma_{ij}^* }{ \frac{1}{2} \sum_{i=1}^3 \sum_{j=1}^3 \Delta\varepsilon_{ij}^* \Delta \varepsilon_{ij}^*}} = \frac{1}{2} \sqrt{ \dfrac{\sum_{i=1}^3 \sum_{j=1}^3 \Delta \sigma_{ij}^* \Delta \sigma_{ij}^* }{\sum_{i=1}^3 \sum_{j=1}^3 \Delta \varepsilon_{ij}^* \Delta \varepsilon_{ij}^*}}$

The hourglass stiffness $\kappa^{HG}$ is then computed based on the following equation: - $ \kappa^{HG} = c^{HG} G^\Delta_{el} \frac{\partial N_I}{\partial x_i} \frac{\partial N_I}{\partial x_i}$

$c^{HG}$ takes here the role of a dimensionless scaling factor for the hourglass stiffness. The user has to provide numgeo with a value for $c^{HG}$.

Example single-phase material: *Hourglass, Scaling = 0.1
Method = Scaling (two-phase material): For a two-phase material, an additional hourglass scaling parameter $c^{f,HG}$ has to be given by the user. Contrary to the scaling parameter associated with the solid phase $c^{HG}$, the scaling factor $c^{f,HG}$ is associated with the fluid phase. The fluid hourglass stiffness $\kappa^{f,HG}$ is computed based on the bulk modulus of the fluid phase $K^f$:

$\kappa^{f,HG} = c^{f,HG} K^f \frac{\partial N_I}{\partial x_i} \frac{\partial N_I}{\partial x_i}$

The user has to provide numgeo with a value for $c^{f,HG}$.

Example two-phase material: *Hourglass, Scaling = 0.1, 0.01

Ted Belytschko and Lee P. Bindeman. Assumed strain stabilization of the eight node hexahedral element. Computer Methods in Applied Mechanics and Engineering, 105(2):225 – 260, jun 1993. URL: http://www.sciencedirect.com/science/article/pii/004578259390124G, doi:10.1016/0045-7825(93)90124-g. ↩↩
Ted Belytschko, Wing Kam Liu, Brian Moran, and Khalil Elkhodary. Nonlinear finite elements for continua and structures. John wiley & sons, 2013. ↩