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Nonlinearity. Structural Mechanics Displacement-based Formulations. 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

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nonlinearity

Nonlinearity

Structural Mechanics

Displacement-based Formulations

nonlinearity1
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
nonlinear overview
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}?
global stiffness matrix
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
solution to equations
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
geometric nonlinearity
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
changes in e
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
modulus changes
Modulus Changes
  • You are fortunate if this is nonlinear elasticity …

s

s

e

e

material nonlinearity
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 …
contact
Contact
  • These problems affect the {R} term …