1. Contact Stiffness Chapter Three

2. Training Manual October 15, 2001 Inventory # 001567 3-2 Contact Stiffness A. Basic Concepts Review: Recall that all ANSYS contact elements use a penalty stiffness (contact stiffness) to help enforce compatibility at the contact interface. Some finite amount of penetration, , is required mathematically to maintain equilibrium. However, physical contacting bodies do not interpenetrate (  = 0). The contact spring will deflect an amount , such that equilibrium is satisfied: F = k  where k is the contact stiffness. F 

3. Training Manual October 15, 2001 Inventory # 001567 3-3 Contact Stiffness... Basic Concepts As an analyst, you face a dilemma: –Minimum penetration gives best accuracy. Therefore, the contact stiffness should be very great. –However, too stiff a value causes convergence difficulties. The model can oscillate, with contacting surfaces bouncing off of each other. Iteration nIteration n+1 F F F contact F Iteration n+2

4. Training Manual October 15, 2001 Inventory # 001567 3-4 Contact Stiffness... Basic Concepts The contact stiffness is the most important parameter affecting both accuracy and convergence behavior. You must carefully determine an appropriate value for contact stiffness. –Balance the convergence efficiency against the required level of accuracy. –The “best” value is problem dependent, and must often be determined by trial-and-error. Estimate a trial value to use, then examine the convergence behavior and accuracy.

5. Training Manual October 15, 2001 Inventory # 001567 3-5 Contact Stiffness... Basic Concepts In addition to transmitting normal action (pressure) between surfaces, contact elements also transmit tangential action (friction). The contact elements use a tangential penalty stiffness to enforce compatibility in the tangential direction. The tangential penalty stiffness affects convergence and accuracy in exactly the same way as the normal penalty stiffness. F tangent  F tangent = k tangent 

6. Training Manual October 15, 2001 Inventory # 001567 3-6 Contact Stiffness B. Specifying a value Determining a good penalty stiffness value may require some experimentation. For surface-to-surface elements, the penalty stiffness is conveniently specified as a factor to be applied as a function of the underlying element stiffness. –As a starting estimate, try: FKN = 1.0 for bulky solids in contact. FKN = 0.01 – 0.1 for more flexible (bending-dominated) parts. –Alternatively, you can specify an absolute stiffness value, in units of (Force/Length)/Area.

7. Training Manual October 15, 2001 Inventory # 001567 3-7 Contact Stiffness... Specifying a value The node-to-node (except CONTA178) and node-to-surface contact elements require the input of an absolute value for the penalty stiffness KN. –As a starting estimate use the following: For bulk deformations: 0.1*E < KN < 1.0*E For bending: 0.01*E < KN < 0.1*E …where E is the modulus of elasticity. –Refer to the Appendix for a more detailed discussion on how to calculate an appropriate absolute stiffness value.

8. Training Manual October 15, 2001 Inventory # 001567 3-8 Contact Stiffness... Specifying a value The following procedure may be used as a guideline: 1. Use a low stiffness value to start. 2. Run the analysis to a fraction of the final load. 3. Check the penetration and number of equilibrium iterations used in each substep. As a rough, quick check, if you can visually detect penetration in a true-scale displaced plot of the entire model, the penetration is probably excessive. Increase the stiffness and restart. If many iterations are needed for convergence (or if convergence is never achieved), reduce the stiffness and restart. –Note: Penalty stiffness can be modified from one load step to another, and can be adjusted in a restart.

9. Training Manual October 15, 2001 Inventory # 001567 3-9 Contact Stiffness... Specifying a value Remember: The contact stiffness is the most important parameter affecting both accuracy and convergence behavior. –If you can fully grasp this idea, you will be able to master most contact difficulties! If you are having convergence problems, reduce the stiffness value, and rerun. You should also verify the validity of your results by varying the penalty stiffness value in a sensitivity study. –Tighten the stiffness in successive analyses, until important results items (contact pressure, max. SEQV, etc.) cease to change significantly.

10. Training Manual October 15, 2001 Inventory # 001567 3-10 Contact Stiffness Workshop Please refer to your Workshop Supplement for instructions on: W1. Contact Stiffness.

11. Training Manual October 15, 2001 Inventory # 001567 3-11 Contact Stiffness C. Appendix – Calculating a value Contact Stiffness Appendix

12. Training Manual October 15, 2001 Inventory # 001567 3-12 Contact Stiffness … Appendix – Calculating a value Frequently, you can estimate a good value for the contact stiffness as a function of the relative stiffness of the areas in contact. The stiffness of a bulky solid will generally be greater than that of a springy, bending-dominated structure. The contact stiffness will typically be correspondingly greater for bulky solid structures. –Examples of bulky contact include metal forging, wheel-on-rail, pin in a bearing block, etc. –Examples of bending-dominated contact include leaf springs, sheet metal forming, etc.

13. Training Manual October 15, 2001 Inventory # 001567 3-13 Contact Stiffness... Appendix – Calculating a value For bulky solids, the Hertz contact stiffness often provides an appropriate basis for the penalty stiffness. This stiffness can be estimated from the element size and Young’s modulus. For a uniformly-shaped 3D element, the Hertz stiffness would be approximately k Hertz  a x E, where a is the characteristic element size, and E is the Young’s modulus. For 2D elements with thickness (t), Hertz stiffness would be approximately k Hertz  t x E. –For 2D axisymmetric elements, the “thickness” is 1 radian x r, giving a Hertz stiffness of approximately k Hertz  r x E. a a a a at r a a

14. Training Manual October 15, 2001 Inventory # 001567 3-14 Contact Stiffness... Appendix – Calculating a value As a practical matter, a good first trial value for bulky contact stiffness would be k contact = f bulk x k Hertz, where f bulk is a factor usually between 0.1 and 10 for bulky solids. –Because the starting estimated value of f bulk ranges over at least two orders of magnitude, and because k contact will be adjusted by trial-and-error anyway, it is usually not justifiable to worry about the element’s size when estimating the penalty stiffness. For bulky solids, simply estimate the penalty stiffness by k = f bulk x E –where the factor f bulk is usually between 0.1 and 10, and a good starting value for f bulk is often f bulk = 1.0. –This estimate assumes an approximate “unit” element size; for very large or very small elements, you might need to adjust the starting value of f bulk accordingly.

15. Training Manual October 15, 2001 Inventory # 001567 3-15 Contact Stiffness... Appendix – Calculating a value If your contact involves two different materials, use E of the softer contacting material. Account for the reduced tangent modulus if plasticity will be active. For Mooney-Rivlin hyperelastic materials, the material law does not use a value of E. You will therefore have to estimate E for such materials. –You can simply scale a modulus value from the appropriate portion of the stress-strain curve. –Or, estimate an initial modulus by E = 6(a + b) where a and b are the first two Mooney-Rivlin constants.

16. Training Manual October 15, 2001 Inventory # 001567 3-16 Contact Stiffness... Appendix – Calculating a value For flexible components (beam-like and shell-like models) the stiffness of the system may be much lower than the Hertz contact stiffness. In this situation you might run a static analysis with a unit load applied to the expected area of contact to determine the local stiffness of the model. The contact stiffness can then be estimated from: k = f bend (P/  ) where P is the applied unit load,  is the corresponding deflection, and for flexible body contact, f bend is a factor between 1 and 100. Setting f bend = 1 is usually a good starting value. For practical simplicity, you might merely estimate the contact stiffness by: k = f bulk x E/10

17. Training Manual October 15, 2001 Inventory # 001567 3-17 Contact Stiffness... Appendix – Calculating a value The same issues about convergence and accuracy must be addressed with the tangential penalty stiffness. –Too soft a value leads to inaccurate results. –Too stiff a value causes convergence difficulties. –A “best” value should be determined by trial-and-error. As a starting estimate, try using k tangent = 0.01 k normal –This is the default value for most ANSYS contact elements. Of course, the tangent stress is limited by the value at which sliding occurs:    x  p –Friction is discussed in the next chapter.

18. Training Manual October 15, 2001 Inventory # 001567 3-18 Contact Stiffness... Appendix – Calculating a value For node-to-node and node-to-surface elements, you specify the value of penalty stiffness directly, in units of Force/Length. –For variable mesh densities, the overall surface stiffness will be greater where the mesh is denser, and will be less where the mesh is coarser. This could lead to uneven contact pressures. Because the penalty stiffness for surface-to-surface elements is specified per unit area, the overall surface stiffness varies much less with varying mesh density. (Surfaces shown separated for clarity) F Stiffer at dense mesh Softer at coarse mesh