Nonlinearity

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# Nonlinearity - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Nonlinearity' - kaveri

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Presentation Transcript

### 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
• Highly erroneous results can be generated if a significant nonlinearity is ignored
• Nonlinear solution processes are much different than linear
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 [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
• 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 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]
• 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
• You are fortunate if this is nonlinear elasticity …

s

s

e

e

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
• These problems affect the {R} term …