COMPUTER MODELLING COMPUTER MODELLING Non-linearities Non-linearities Computer Modellin g Outline Why non-linear analysis? Non-linear response: origin and characterization Geometric non-linearit Mater ial non-linearit y: duct ileness vs. brit tlen ess Everything at once: challenges - Exam ple of non-linear mat erial model: Plasticity 2 Computer Modellin g Wh non-linear anal sis? Linear analysis can provide useful information BUT some phenomena are intrinsically non-linear. Analysts try to stay in the nea r re g m e, ow ev er som e ap p ca on s re qu re m ore sophisticated analysis (forensic engineering, flexible structures). Computer Modellin g Wh non-linear anal sis? Current computer power (supercomputers) allows for the use of sophisticated, more realistic models that were untreatable before. Non-linear analysis is much more complex: Model lin : man more m ode l arameters to be chosen Numerical comput ation: different solution methods required, often without guaranteed convergence a priori User needs advanced training and experience. Some commercial codes ready for non-linear analysis: CASTEM, ,
Material non-linearity: ductileness vs. brittleness
Everything at once: challenges
-
Example of non-linear material model: Plasticity
2
Computer Modelling
Wh non-l inear anal sis?
Linear analysis can provide useful information BUT somephenomena are intrinsically non-linear. Analysts try to stay in thenear regme, owever some app ca ons requre more
A system is NON-LINEAR when the action-response relation is notproportional (in materials or structures it does not follow Hooke’s
law).
Whatare thesources ofnon-linearit ?
• Geometric non-linearities
• Material non-linearities
• -
• Non-linear stress BCs (follower loads)
In fact, non-linearity is ubiquitous and one should justify why asystem can be treated as linear! From continuum mechanics weknow that the displacement-strain relation is non-linear in general.Experience shows that material non-linearities are common.
Computer Modelling
Non-linear res onse
Given a control parameter (displacement) and a response
parameter (force) one can plot for a system a response diagram of . , .
A non- near agram asshown does NOT indicate theorigin of the non-linearity nor
of the process.
u
In a neighborhood of the reference state, the system behaveslinearl in eneral (constant stiffness). The si n of the stiffness isrelated to the stability of the equilibrium state.
Computer Modelling
Non-linear res onse
Irreversible system (e.g. plasticity) vs. reversible (non-linear elasticity)
F F
u u
The area beneath the curve indicates the energy invested in thedeformation rocess. H steresis ields ener dissi ation.
Computer Modelling
Non-linear res onse
Some examples of non-linear response due to typical material non-linearities: brittlematerial lass rubber concrete.
The origin of these type of responses are physical processes at themicrostructural level.
In slender structures, geometric non-linearities manifest in “exotic”dia rams sna -throu h sna -back bifurcations etc. .
To follow complex equilibrium paths with unstable branches one needsto resort to arc length techniques.
Computer Modelling
Non-linear res onse
Computer Modelling
Geometric non-linearit
Second order buckling analysis allows us to estimate the bucklingload and mode. However, it is in many cases just a rough firstapproxma on o e non- near regme.
Arroyo and Arias, 2007
Computer Modelling
Ductileness vs. britt leness
Many physical phenomena exhibit localization: e.g. shear bands,fracture, shocks.
Independent of their nature, they can be modelled through smoothor sharp models (non-differentiable). The latter are in general moredifficultto treatnumericall and oftencall for re ularization.
Arias et al., 2004
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Ductileness vs. britt leness
Sometimes it is convenient to just do the opposite: turn a smooth(differentiable) problem into a sharp one. Plastic hinges are anexam le.
Arroyo et al. 1998
In addition to localization henomena, there exist other situations thatcall for regularization, e.g. contact.
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Solution methods
Increments and control of the solution
To follow an equilibrium path, the loading process is discretized inincrements. It is possible to control either the load, thedisplacements or perform a combined control.
A simple linearization in each load step produces drifting. Thus,
the system of non-linear equations is solved by iterative methodsewon- ap son .
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Solution methods
Stability
Algorithms are capable of giving complex equilibrium paths. In orderto establish their practical relevance, one has to bear in mind that
real systems seldom adopt unstable solutions. In addition, defects.
Dynamics
Explicit dynamics facilitates enormously the solution of non-linearproblems, since it does not require to solve any non-linear systemof equations. It suffices to compute the internal forces at each
Crashworthiness: contact-impact, non-linear loads, geometric andmaterial non-linearities, etc.
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Chal len es
1. Definition of the model
The modelling process is much more involved.
2. Lack of uniqueness or regularity of the solutions
,discontinuous solutions (shocks, fracture).
. e a y o e so u on
Part of the industry’s distrust of non-linear analysis stems from thefact that computations may not converge. The currently used
.
Modelling challenges raise concerns about the predictive ability of themodels. Therefore it is crucial to VALIDATE the models againstexperimental observation.
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Chal len es
The ALE method makes the simulations in large strains morerobust.
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Other non-linear roblems
We have focused so far on solids, but non-linearities are alsoimportant in:
u s
(Reactive) flux problems in porous media (e.g. canisters)
