Notes Multi-Dimensional Plasticity I am back, but still catching up - PowerPoint PPT Presentation

Notes Multi-Dimensional Plasticity � I am back, but still catching up � Simplest model: total strain is sum of elastic and plastic parts: � = � e + � p � Assignment 2 is due today (or next time I � m in the dept following today) � Stress only depends on elastic part (so rest state includes plastic strain): � Final project proposals: � = � ( � e ) • I haven � t sorted through my email, but make � If � is too big, we yield, and transfer some sure you send me something now (even quite vague) of � e into � p so that � is acceptably small • Let � s make sure everyone has their project started this weekend or early next week cs533d-term1-2005 1 cs533d-term1-2005 2 Multi-Dimensional Yield criteria Yielding � Lots of complicated stuff happens when � First note that shear stress is the important materials yield quantity • Metals: dislocations moving around • Materials (almost) never can permanently • Polymers: molecules sliding against each other change their volume • Etc. • Plasticity should ignore volume-changing � Difficult to characterize exactly when plasticity stress (yielding) starts � So make sure that if we add kI to � it • Work hardening etc. mean it changes all the time too doesn � t change yield condition � Approximations needed • Big two: Tresca and Von Mises cs533d-term1-2005 3 cs533d-term1-2005 4

Tresca yield criterion Von Mises yield criterion � This is the simplest description: � If the stress has been diagonalized: 2 + � 2 � � 3 2 + � 3 � � 1 2 � � Y 1 • Change basis to diagonalize � ( ) ( ) ( ) � 1 � � 2 2 • Look at normal stresses (i.e. the eigenvalues of � ) 2 � 1 2 � � Y � More generally: ( ) � F 3 Tr � 3 • No yield if � max - � min � � Y 2 � Tends to be conservative (rarely predicts � This is the same thing as the Frobenius norm of the deviatoric part of stress yielding when it shouldn � t happen) • i.e. after subtracting off volume-changing part: � But, not so accurate for some stress states ( ) I F � � Y • Doesn � t depend on middle normal stress at all 2 � � 1 3 Tr � 3 � Big problem (mathematically): not smooth cs533d-term1-2005 5 cs533d-term1-2005 6 Linear elasticity shortcut Perfect plastic flow � For linear (and isotropic) elasticity, apart � Once yield condition says so, need to start changing plastic strain from the volume-changing part which we � The magnitude of the change of plastic strain cancel off, stress is just a scalar multiple of should be such that we stay on the yield surface strain • I.e. maintain f( � )=0 • (ignoring damping) (where f( � ) � 0 is, say, the von Mises condition) � So can evaluate von Mises with elastic � The direction that plastic strain changes isn � t as strain tensor too (and an appropriately straightforward � p = � � f scaled yield strain) � “Associative” plasticity: ˙ �� cs533d-term1-2005 7 cs533d-term1-2005 8

Algorithm Sand (Granular Materials) � After a time step, check von Mises criterion: � Things get a little more complicated for ( ) F � � Y > 0 is ? f ( � ) = 2 dev � sand, soil, powders, etc. 3 � Yielding actually involves friction, and thus is pressure (the trace of stress) dependent � If so, need to update plastic strain: new = � p + � � f � Flow rule can � t be associated � p �� � See Zhu and Bridson, SIGGRAPH � 05 for dev ( � ) = � p + � quick-and-dirty hacks… :-) 3 dev ( � ) F 2 • with � chosen so that f( � new )=0 (easy for linear elasticity) cs533d-term1-2005 9 cs533d-term1-2005 10 Multi-Dimensional Fracture Fluids � Smooth stress to avoid artifacts (average with neighbouring elements) � Look at largest eigenvalue of stress in each element � If larger than threshhold, introduce crack perpendicular to eigenvector � Big question: what to do with the mesh? • Simplest: just separate along closest mesh face • Or split elements up: O � Brien and Hodgins SIGGRAPH � 99 • Or model crack path with embedded geometry: Molino et al. SIGGRAPH � 04 cs533d-term1-2005 11 cs533d-term1-2005 12

Fluid mechanics Inviscid Euler model � We already figured out the equations of motion for � Inviscid=no viscosity � ˙ x = � � � + � g continuum mechanics ˙ � Great model for most situations • Numerical methods usually end up with viscosity-like error terms � Just need a constitutive model anyways… � ij = � p � ij ( ) � = � x , t , � ,˙ � � Constitutive law is very simple: • New scalar unknown: pressure p � We � ll look at the constitutive model for “Newtonian” fluids • Barotropic flows: p is just a function of density next (e.g. perfect gas law p=k( � - � 0 )+p 0 perhaps) • Remarkably good model for water, air, and many other simple • For more complex flows need heavy-duty thermodynamics: an fluids equation of state for pressure, equation for evolution of internal • Only starts to break down in extreme situations, or more complex energy (heat), … fluids (e.g. viscoelastic substances) cs533d-term1-2005 13 cs533d-term1-2005 14 Lagrangian viewpoint Eulerian viewpoint � We � ve been working with Lagrangian methods � Take a fixed grid in world space, track how so far velocity changes at a point • Identify chunks of material, � Even for the craziest of flows, our grid is always track their motion in time, nice differentiate world-space position or velocity w.r.t. � (Usually) forget about object space and where a material coordinates to get forces chunk of material originally came from • In particular, use a mesh connecting particles to • Irrelevant for extreme inelasticity approximate derivatives (with FVM or FEM) • Just keep track of velocity, density, and whatever else � Bad idea for most fluids is needed • [vortices, turbulence] • At least with a fixed mesh… cs533d-term1-2005 15 cs533d-term1-2005 16

Conservation laws Conservation of Mass � Identify any fixed volume of space � Also called the continuity equation (makes sure matter is continuous) � Integrate some conserved quantity in it (e.g. mass, momentum, energy, …) � Let � s look at the total mass of a volume (integral of density) � Integral changes in time only according to how fast it is being transferred from/to � Mass can only be transferred by moving it: flux must be � u surrounding space � � � ( ) � n = � q f q • Called the flux � t � � � � � � � = � � u � n • [divergence form] q t + � � f = 0 � t � � � ( ) = 0 � t + � � � u cs533d-term1-2005 17 cs533d-term1-2005 18 Material derivative Finding the material derivative � A lot of physics just naturally happens in the � Using object-space coordinates p and map x=X(p) to world-space, then material derivative is just Lagrangian viewpoint • E.g. the acceleration of a material point results from Dt q ( t , x ) = d D ( ) dt q t , X ( t , p ) the sum of forces on it • How do we relate that to rate of change of velocity = � q � t + � q � � x measured at a fixed point in space? � t • Can � t directly: need to get at Lagrangian stuff = q t + u � � q somehow � Notation: u is velocity (in fluids, usually use u but occasionally v or V, � The material derivative of a property q of the and components of the velocity vector are sometimes u,v,w) material (i.e. a quantity that gets carried along with the fluid) is Dq Dt cs533d-term1-2005 19 cs533d-term1-2005 20

Compressible Flow Incompressible flow � In general, density changes as fluid compresses or � So we � ll just look at incompressible flow, expands where density of a chunk of fluid never � When is this important? changes • Sound waves (and/or high speed flow where motion is getting • Note: fluid density may not be constant close to speed of sound - Mach numbers above 0.3?) • Shock waves throughout space - different fluids mixed � Often not important scientifically, almost never visually together… significant � That is, D � /Dt=0 • Though the effect of e.g. a blast wave is visible! But the shock dynamics usually can be hugely simplified for graphics cs533d-term1-2005 21 cs533d-term1-2005 22 Simplifying Conservation of momentum D � � ˙ x = � � � + � g � Incompressibility: � Short cut: in ˙ Dt = � t + u � � � = 0 use material derivative: � Conservation of mass: ( ) = 0 � t + � � � u � Du Dt = � � � + � g � t + � � � u + � � � u = 0 ( ) = � � � + � g � u t + u � � u � Subtract the two equations, divide by � : � � u = 0 � Or go by conservation law, with the flux due to transport of momentum and due to stress: � Incompressible == divergence-free velocity • Equivalent, using conservation of mass • Even if density isn � t uniform! ( ) t + � � u � u � � ( ) = � g � u cs533d-term1-2005 23 cs533d-term1-2005 24