Anisotropic contact in Zebulon

An orthotropic Coulomb friction model has been developed in which the friction depends on the direction of the shear displacment with respect to the principal axis of the friction. In sliding the shear stress magnitude is equal to the friction times the normal stress.  The direction is opposite the relative shear displacement vector.  If  and  are the coordinates of the unit shear displacement vector in the local coordinate system defined by the principal directions, then the friction is described by an ellipse.

 where   and  are the major and minor friction coefficients. 

The frictional principal axis is  assumed to be correleated to one of the surfaces involved in contact (but not both).  The principal axis always lie within the plane of the selected surface even though the surface may become deformed.  In addition, the principal axis may, due to shear, become nonorthogonal.  In this case, it is assumed that the friction law remains orthotropic with respect to a orthogonal axis whose bisect is the same as the bisect of the true principal axis which may no longer be perpendicular.

The model uses a penalized form of the “penalized Coulumb law.  If the shear stress magnitude is less than frictional stress  then a "sticking" condition exists and the change in the stress is linearly related to the shear displacment

 The stiffness  is computed automatically using the flexibility matrix and the local nodal area distribution and the penalty  is selected by the user.  Actually form of this equation used in the code is .  So that as the penalty approaches infinity a condition of zero shear displacement is enforced. In the example to follow, the penalty is rather large (=100) so that very little shear displacement is allowed in sticking.

Otherwise a “slipping” condition exists in which the magnitude of the stress is equal to the frictional stress  and lies in the opposite direction of the shear displacement.

To test this anisotropic friction model a 3D model was constructed with a deformable block sliding between two rigid planes.  One plane is initially pushed into the block to produce a uniform normal stress with the orther plane held fixed.  Afterwords the lower plane translates laterally while the upper plane rotates about an axis that goes through the center of the block and is normal to the plane.  It is assumed that sticking contact occurs between the top plate and the top of the block.  Therefore, the top surface of the block undergoes a pure rotation.  The friction between the bottom surface and the bottom plane is assumed to be orthotropic with the principal axis tied to the block (the slave surface) Initially the axis with the least friction is aligned to the direction  of the translation.  The rotation is rather slow compared to the translation.  In this case the rotational velocity for a corner point on the top surface relative to the translation velocity is .  The friction for the major and minor axis is 0.2 and 0.1 respectively.  The total frictional force acting on the bottom surface has been recorded as a function of the rotation angle (360° = 1 rotation).  In Figure 1 the force opposite the direction of the translation is shown in Figure 1.  As expected the block rotates the friction force begins to increase until the principal friction axis is parallel to the direction of the translation.  As a comparison, a similar test is conducted but with the isotropic friction set at .15.  Here as expected the friction force is constant.

Note that the block bottom surface is the “slave” surface while the rigid lower surface is the master.  The direction of the principal axis is continually updated as the surface rotates.  Because the combination of rotation and translation shear places a moment on the block, the normal stresses on the block face are not uniform.  This in turn causes a slight net force perpendicular to the translation.  If sign of the roation is changed, the sign of this perpendicular force changes.  Figure 2 shows the perpendicular force for orthotropic and isotropic friction.

Input file

****calcul
 ***mesh updated_lagrangian
 ***resolution newton
 **init_d_dof sequence
  **sequence  
   *time      0.0001  10.0
   *increment 10 100
   *algorithm p1p2p3
   *ratio     0.1
 ***bc
  **impose_nodal_dof
    y-top     U2 0.0
    bottom    U1 0.0
    bottom    U2 1.0  pushy
    bottom    U3 1.0  time
  **rotation node y-top 145 (0. 1. 0.) -18.0 time
 ***table
  **name pushy
   *time  0.0  0.0001  20.0
   *value 0.0  0.05  0.05

 ***contact
  **zone ortho_coulomb         % each zone can have different friction law 
   *impactor top/bottom        % this is the slave boundary set 
   *target   bottom/top        % this is the master boundary set
   *behavior_coeff             % 
     friction 0.2 0.1          % friction behavior specific 
     penalty_slip 100.0 100.   % coefficients and controls 
     direction 1.0 0.0 0.0     % 
     slave                     % ortho law on either master or slave 
   *warning_distance 0.2

  **solve_method direct
  **init_d_stress sequence
  **conv 1.e-3 100 1.e-3
 ***material
  **elset top     *file rotate_block.inp  1
  **elset bottom  *file rotate_block.inp  2
 ***output
  **value_at_integration
  **contour
****return