Nonlinearity

1. Nonlinearity Structural Mechanics Displacement-based Formulations

2. Nonlinearity • Linearity: • Response of the system is directly proportional to the action that produces it • All mechanics problems are nonlinear • Many involve nonlinearity small enough to be ignored • Highly erroneous results can be generated if a significant nonlinearity is ignored • Nonlinear solution processes are much different than linear

3. Nonlinear Overview • Our FEA problems eventually become: • Solve for {D}, that’s it (secondary quantities derive from {D}) • What if [K] and/or {R} are functions of {D} (or functions of a secondary quantity like stress calculated from {D}?

4. Global Stiffness Matrix • Global stiffness matrix [K] is assembled from element stiffness matrices [k] • Element stiffness matrices [k] are functions of: • [B], which contains information about node point locations • [E], which contains information about material properties • If either of these changes in moving from the undeformed to the deformed configuration, a true solution requires iteration

5. Solution to Equations • What does it mean to find a “solution”? • Discrete interpretation: Find a deformed configuration that satisfies equilibrium, stress/strain and compatibility • Integral interpretation: Find a deformed configuration that minimizes system energy • If we base the solution on the undeformed configuration we don’t get the correct solution, we get the solution for a problem that would have “moved into” that geometry • For linear analyses we ignore this distinction

6. Geometric Nonlinearity • Geometric nonlinearity affects [k] calculations, and thus the assembled [K] • If you have any doubts, run as nonlinear and compare with the linear solution • Most modern commercial codes are adept enough to do this without difficulty • MSC.Marc actually defaults to a nonlinear solver

7. Changes in [E] • Recall the element stiffness formulation: • Clearly if [E] changes as a function of deformation, then we cannot get to the correct solution without updating material properties

8. Modulus Changes • You are fortunate if this is nonlinear elasticity … s s e e

9. Material Nonlinearity • Like geometric nonlinearity, material nonlinearity affects [k] calculations, and thus the assembled [K] • You will likely know if there is any elastic change in modulus for a problem from the materials involved (elastomers) • You can track plasticity to some extent by updating [E], but not for repeated loadings …

10. Contact • These problems affect the {R} term …

11. Contact Convergence