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 …    

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 % loadstrain energy 0.00 0.010.30 0.021.45 0.043.47 0.057.14 0.0713.39 0.1023.52 0.1339.44 0.1663.94 0.21101.00 0.26144.16 0.32214.21 0.40316.82 0.48449.93 0.59643.62 0.72921.88 0.871322.42 1.001642.80