# Contact models for soil mechanics

Leitung: | P. Wriggers |

Team: | C. Weißenfels |

Jahr: | 2012 |

**Goal of the project**

The installation of foundations influences strongly the load bearing capacity of the soil. The large discrepancy between experimental and numerical results, using Coulomb friction law for modeling the soil structure interaction, points out that new strategies to solve this kind of problems are necessary. Experimental observations show that for rough surfaces of the structure the friction angle at the contact zone corresponds to the friction angle of the soil. This leads to the conclusion that the contact zone lies completely within the soil. A way to improve the friction laws for soil structure interactions is to project the soil models onto the contact surface which is the motivation of this work.

**Contact formulation**

The Mortar solution technique is at the actual state of research the most robust contact discretization scheme. Based on this concept a mixed formulation is developed using an augmented Lagrangian description for the averaged normal penetration and a penalty formulation for the stick-slip motion written in a Hellinger-Reissner form

where the bar over the quantity indicates its averaged form. This leads to an exact enforcement of the non penetration condition and nonlinear friction laws based on an elasto-plastic concept can be integrated easily. Due to the the description in Hellinger-Reissner form the algorithm converges faster as in the standard penalty approach.

*Fig. 1: Ironing test: Vertical displacements (left) and tangential force distribution for different solution methods*

Comparing the mixed version with standard solution methods, like the penalty regularization, the augmented Lagrangian method or the pure Lagrange multiplier method which is implemented for the fist time into the Mortar framework within this project, the augmented Lagrangian method based on averaged gap shows the best numerical behavior and the mixed version can be improved if also the averaged gap is used instead of averaged normal penetration and averaged tangential movement.

*Fig. 2: Ironing test: Time step size and CPU time for different solution methods*

**Coupling contact and continuum**

The first coupling concept is based on the dependency of the coefficient of friction on the stress of the underlying soil model

Using the solid shell concept, a natural connection between the continuum strains and the averaged contact kinematics can be established, if the height of the zone goes to zero

Due to the analogue between Mohr Coulomb yield criterion and Coulomb friction law, the friction angle and hence the coefficient of friction of the actual soil stress state can be computed.

*Fig. 3: Initial mesh (left) and final position (right) for the direct shear test*

For all the numerical tests, the Ehlers soil model is used together with the material data of GEBA fine sand [1].

*Fig. 4: Direct shear test (version 1): Tangential force for Ehlers soil model with and without hardening (left) and friction angle for different pressures (right)*

In the second concept, the yield criterion is coupled to the friction law assuming the Lode angle to be zero and introducing a new dilatancy stress

with which it is able to cover dilatancy or contractancy effects within the contact zone based on the dilatancy angle.

*Fig. 5: Direct shear test (version 2): Tangential force for Ehlers soil model with and without hardening (left) and friction angle for different pressures (right)*

Comparing the results of the two concepts, the second method can be viewed as the asymptotic version of the fist concept for a limiting height.

**3D contact element**

Neglecting the limit of the height in the standard contact description, a three-dimensional formulation of the interface kinematics in the Mortar framework occurs

Using again the Hellinger-Reissner formulation for a faster algorithm, the weak form of the contact equation can be written in Voigt notation

With the developed concept of a three-dimensional description of the contact algorithm, it is now possible to integrate the continuum plasticity model directly into the contact algorithm which allows arbitrarily large deformations.

*Fig. 6: Initial mesh (left) and final position (right) of the direct shear test*

*Fig. 7: Tangential force at different pressures and at different heights of the interface zone*

**References**

[1] W. Ehlers, O. Avci, B. Markert: Computation of slope movements initiated by rain-induced shear bands in small-scale tests and in situ*Vadose Zone Journal*, 2011 (**10**),512-525.