Lecture 7 - Soft-Body Physics - Universiteit Utrecht
Transcript of Lecture 7 - Soft-Body Physics - Universiteit Utrecht
LectureVII:Soft-BodyPhysics
SoftBodies
• Realisticobjectsarenotpurelyrigid.• Goodapproximationfor“hard”ones.• …approximationbreakswhenobjectsbreak,ordeform.
• Generalization:soft(deformable)bodies• Deformedbyforce:carbody,punchedorshotat.• Pronetostress:pieceofcloth,flag,papersheet.• Notsolid:snow,mud,lava,liquid.
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Grinspun etal. “DiscreteShells”
Elasticity
• Forcesmaycauseobjectdeformation.
• Elasticity:thetendencyofabodytoreturntoitsoriginalshapeaftertheforcescausingthedeformationcease.
• Rubbersarehighlyelastic.• Metalrodsaremuchless.
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ContinuumMechanics
• Adeformableobjectisdefinedbyrestshape andmaterialparameters.
• Deformationmap:𝑓(�⃗�) ofeverypoint�⃗� =𝑥, 𝑦, 𝑧 .
• 𝑓:ℝ- → ℝ- .𝑑:dimension(mostly𝑑 = 2,3).• Relativedisplacementfield:𝑓(�⃗�) = �⃗� + 𝑢(�⃗�)
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𝑝
𝑝 + Δ𝑝
Δ𝑝
𝑓(𝑝)
𝑢(𝑝)
𝑢(𝑝 + Δ𝑝)
Δ𝑝
𝑓(𝑝 + Δ𝑝)
Δ𝑓
LocalDeformation• Taylorseries:
𝑓 𝑝 + Δ𝑝 ≈ 𝑓 𝑝 + 𝐽8Δ𝑝
• 1st-orderlinear approximation.
• As𝑓 𝑝 = 𝑝 + 𝑢 𝑝 ,weget:
𝑝 + Δ𝑝 + 𝑢 𝑝 + Δ𝑝 ≈ 𝑝 + 𝑢 𝑝 + 𝐽8Δ𝑝 ⟹𝑢 𝑝 + Δ𝑝 ≈ 𝑢 𝑝 + 𝐽8 − 𝐼-×- Δ𝑝
• TheJacobians:𝐽8 ==8=> , 𝐽? =
=?=> = 𝐽8 − 𝐼-×-.
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StretchandCompression
• Howmuchanobjectlocallystretchesorcompressesineachdirection.
• Newlength:Δ𝑓 @ = 𝑓 𝑝+ Δ𝑝 − 𝑓 𝑝 @ ≈ 𝐽8Δ𝑝
@
= Δ𝑝A∗ 𝐽8A𝐽8 ∗ Δ𝑝
• 𝐽8A𝐽8 -×-
isthe(right)Cauchy-Greentensor.
• Stretch:relativechangeinlength:
Δ𝑓 @
Δ𝑝 @ ≈Δ𝑝A ∗ 𝐽8
A𝐽8 ∗ Δ𝑝Δ𝑝A ∗ Δ𝑝
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Rigid-BodyDeformation
• Transformation:𝑓 𝑝 = 𝑅𝑝 + 𝑇
• 𝑅:rotation(constant)• 𝑇:translation.
• 𝐽8 = 𝑅,andthen𝐽8A𝐽8 = 𝐼.
• Nostretch!
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Strain
• Thefractionaldeformation𝜖 = ∆𝐿 𝐿⁄• Dimensionless (aratio).• Howmuchadeformationdiffers frombeingrigid:
• Negative:compression• Zero:rigid• Positive:stretch
• Inourpreviousnotation: I8I>
= ∆JKJJ
= 1 + 𝜖
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𝐹𝑡
𝑡 + ∆𝑡
𝐿
𝐿 + ∆𝐿
TheLagrangian StrainTensor
• Measuresthedeviationfromrigidity:
𝐸 =12𝐽8A𝐽8 − 𝐼
• Indeformationfieldterms(𝐽8 = 𝐽? − 𝐼-×-):
𝐸 =12𝐽?A𝐽? + 𝐽? + 𝐽?
A
• Strain𝜖 = ∆𝐿 𝐿⁄ in(unitlength)direction�⃗�:𝜖 = 𝛼A𝐸𝛼
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Stress
• Magnitude ofappliedforceperareaofapplication.• largevalueó forceislargeorsurfaceareaissmall
• Pressuremeasure𝜎.• Unit:Pascal:𝑃𝑎 = 𝑁/𝑚@
• Example:gravitystresson plane:σ = 𝑚𝑔/ 𝜋𝑟@
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𝑟
𝑊
𝑚
TheLinearStressTensor• Measuringstressforeach(unit)direction𝑛 inaninfinitesimalvolumeelement:
𝜎] =𝜎^^ 𝜎_^ 𝜎`^𝜎^_ 𝜎__ 𝜎`_𝜎^` 𝜎_` 𝜎``
𝑛 = 𝑇𝑛
• Notethat𝑇𝑛 isnotnecessarilyparallel to𝑛!• 𝑇𝑛 = 𝜎] + 𝜏
• 𝜎]:outward/inwardnormalstress.• 𝜏:shearstress.
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Poisson’sratio
• Straininonedirectioncausescompressioninanother.
• Poisson’sratio: theratiooftransverse toaxialstrain:
𝜈 = −𝑑𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑒𝑠𝑡𝑟𝑎𝑖𝑛
𝑑𝑎𝑥𝑖𝑎𝑙𝑠𝑡𝑟𝑎𝑖𝑛
• Equals0.5inperfectlyincompressiblematerial.
• Iftheforceisappliedalong𝑥:
𝜈 = −𝑑𝜖_𝑑𝜖^
= −𝑑𝜖`𝑑𝜖^
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Poisson’sratio
• Exampleofacubeofsize𝐿.
• Averagestrainineachdirection:𝜈 ≈ ∆Ji
∆J• Approximate,becausetrueforsmallelementsanddeformation.
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𝐹
Δ𝐿
𝐿
∆𝐿′
LinearElasticity
• Stress andstrain arerelatedbyHooke’slaw• Remember𝐹 = −𝑘𝑥?
• Reshapetensorstovectorform:• 𝜎l = 𝜎^^, 𝜎^_, ⋯ , 𝜎`` ,andsimilarlyfor𝜖.̅• Thenthestiffnesstensor𝐶p^p holds:
𝜎l = 𝐶𝜖̅
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Brownetal. “ResamplingAdaptiveClothSimulations ontoFixed-TopologyMeshes”
BodyMaterial
• Theamountofstresstoproduceastrainisapropertyofthematerial.
• Isotropicmaterials:sameinalldirections.• Modulus:aratioofstress tostrain.
• Usuallyinalineardirection,alongaplanarregionorthroughoutavolumeregion.
• Young’smodulus,Shearmodulus,Bulkmodulus• Describingthematerialreactiontostress.
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Young’sModulus
• Definedastheratiooflinearstresstolinearstrain:
𝑌 =𝑙𝑖𝑛𝑒𝑎𝑟𝑠𝑡𝑟𝑒𝑠𝑠𝑙𝑖𝑛𝑒𝑎𝑟𝑠𝑡𝑟𝑎𝑖𝑛
=𝐹/𝐴∆𝐿/𝐿
• Example:
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𝐿 + ∆𝐿
𝐹𝐴
Shearmodulus
• Theratioofplanarstresstoplanarstrain:
𝑆 =𝑝𝑙𝑎𝑛𝑎𝑟𝑠𝑡𝑟𝑒𝑠𝑠𝑝𝑙𝑎𝑛𝑎𝑟𝑠𝑡𝑟𝑎𝑖𝑛
=𝐹/𝐴∆𝐿/𝐿
• Example:
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𝐿
𝐴 𝐹
∆𝐿
Bulkmodulus
• Theratioofvolumestresstovolumestrain(inverseofcompressibility):
𝐵 =𝑣𝑜𝑙𝑢𝑚𝑒𝑠𝑡𝑟𝑒𝑠𝑠𝑣𝑜𝑙𝑢𝑚𝑒𝑠𝑡𝑟𝑎𝑖𝑛
=∆𝑃∆𝑉/𝑉
• Example
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𝐴
𝑃 = 𝐹/𝐴
𝑉𝐹 𝐴
𝑃 + ∆𝑃
𝑉 +∆𝑉
𝐹 + ∆𝐹
DynamicElasticMaterials
• Foreverypoint𝑞,ThePDEisgivenby𝜌 ∗ �⃗� = 𝛻 z 𝜎 + �⃗�
• 𝜌:thedensity ofthematerial.• 𝑎:accelerationofpoint𝑞.• 𝛻 z 𝜎 = 𝜕 𝜕𝑥⁄ , 𝜕 𝜕𝑦⁄ , 𝜕 𝜕𝑧⁄ ∗ 𝜎 isthedivergence ofthestresstensor(modelinginternalforces).
• 𝐹:externalbodyforces(perdensity)• GeneralizedNewton’s2nd law!
• Remember𝐹 = 𝑚𝑎?• Similar,inelasticitylanguage.
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FluidMotion
• Describesobjectwithnofixedtopology• Airflow• Viscuous fluids• Smoke,etc.
• Keydescriptor:flowvelocity𝑢 = 𝑢(𝑥, 𝑡)
• Describingthevelocityofa“fluidparcel”passingatposition𝑥 intime𝑡.
• Eulerian description• Howcome?
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FlowVelocity
• Vectorfielddescribingmotion
• Steadyfield:-?-|= 0
• Incompressible:𝛻 z 𝑢 = 0.• Irrotational (novortices):𝛻×𝑢 = 0
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Turbulentwithavortex Incompressible,irrotational flow
Steadyfield
MaterialDerivative
• Thechangeinthevelocityofthefluidparcelpassingatposition𝑥 intime𝑡.
𝐷𝑢𝑑𝑡
= 𝑢| + 𝑢 z 𝛻𝑢
• 𝑢|:unsteadyacceleration.• Howmuchvelocitychangesin𝑥 overtime.
• 𝑢 z 𝛻𝑢:convective acceleration.• Howmuchvelocitychangesduetomovementalongtrajectory.
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Viscosity
• Resistancetodeformationbyshearstress.• Expressedbycoefficient𝜈:
𝐹𝐴= 𝜈
𝜕𝑢𝜕𝑦
• Higher𝜈:morepressurerequiredforshearing!• Viscid fluids.
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Navier-StokesEquations• Representingtheconservationofmassandmomentumforanincompressible fluid(𝛻 z 𝑢 = 0):
𝜌 𝑢| + 𝑢 z 𝛻𝑢 = 𝛻 z 𝜈𝛻𝑢 − 𝛻𝑝 + 𝑓
• 𝑝:pressurefield• 𝜈:kinematicviscosity.• 𝑓:bodyforceperdensity(usuallyjustgravityρ𝑔).
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Inertia(pervolume) Divergenceofstress
Unsteadyacceleration
Convectiveacceleration
PressuregradientViscosity Externalbodyforces