LectureVIII: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(�⃗�) ofeverymaterialpoint�⃗�.• 𝑓:ℝ( → ℝ(.𝑑:dimension(mostly𝑑 = 2,3).• Relativedisplacementfield:𝑓(�⃗�) =�⃗�+𝑢(�⃗�).

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𝑥

𝑥 + Δ𝑥

Δ𝑥

𝑓(𝑥)

𝑢(𝑥)

𝑢(𝑥 + Δ𝑥)

Δ𝑝

𝑓(𝑥 + Δ𝑥)

Δ𝑓

LocalDeformation• Taylorseries:

𝑓 �⃗� + Δ�⃗� ≈ 𝑓 �⃗� + 𝐽6Δ�⃗�• 1st-orderlinear approximation.

• As𝑓 �⃗� = �⃗� + 𝑢 �⃗� ,weget:

�⃗� + Δ�⃗� + 𝑢 �⃗� + Δ�⃗� ≈ �⃗� + 𝑢 �⃗� + 𝐽6Δ�⃗� ⟹𝑢 �⃗� + Δ�⃗� ≈ 𝑢 �⃗� + 𝐽6 − 𝐼(×( Δ�⃗�

• TheJacobians:𝐽6 =;6;<

, 𝐽= =;=;<

= 𝐽6 − 𝐼(×(.

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StretchandCompression

• Howmuchanobjectlocallystretchesorcompressesineachdirection.

• Newlength:Δ𝑓 > = 𝑓 �⃗� + Δ�⃗� − 𝑓 �⃗� > ≈ 𝐽6Δ�⃗�

>

= Δ�⃗�?∗ 𝐽6?𝐽6 ∗ Δ�⃗�• 𝐽6?𝐽6 (×(

isthe(right)Cauchy-Greentensor.

• Stretch:relativechangeinlength:

Δ𝑓 >

Δ𝑝 > ≈Δ�⃗�? ∗ 𝐽6?𝐽6 ∗ Δ�⃗�

Δ�⃗�? ∗ Δ�⃗�6

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Rigid-BodyDeformation

• Transformation:𝑓 �⃗� = 𝑅�⃗� + 𝑇

• 𝑅:rotation(constant)• 𝑇:translation.

• 𝐽6 = 𝑅,andthen𝐽6?𝐽6 = 𝐼.• Nostretch!

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Strain

• Thefractionaldeformation𝜖 = ∆𝐿 𝐿⁄• Dimensionless (aratio).• Howmuchadeformationdiffers frombeingrigid:

• Negative:compression• Zero:rigid• Positive:stretch

• Inourpreviousnotation: G6GH

= ∆IJII

= 1 + 𝜖

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𝐹𝑡

𝑡 + ∆𝑡

𝐿

𝐿 + ∆𝐿

TheGreen-LagrangeStrainTensor

• Measuresthedeviationfromrigidity:

𝐄O×O =12 𝐽6?𝐽6 − 𝐼

• Indeformationfieldterms(𝐽6 = 𝐽= − 𝐼(×():

𝐄 =12 𝐽=?𝐽= + 𝐽= + 𝐽=?

• ForStrain𝜖 = ∆𝐿 𝐿⁄ in(unitlength)direction𝑛Q:𝑛Q?𝐄𝑛Q = 𝜖 +

12 𝜖

>

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InfinitesimalStrainTensor

• Forsmallshapechanges:

𝛆 =12 𝐽=?𝐽= + 𝐽= + 𝐽=? ≈

12 𝐽= + 𝐽=?

=12 𝐽6 + 𝐽6? − 𝐼

• Notrotationinvariantanymore.• Approximate.• Butlinear.

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Poisson’sratio

• Straininonedirectioncausescompressioninanother.

• Poisson’sratio: theratiooftransversal toaxialstrain:

𝜈 = −𝑑 𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑎𝑙𝑠𝑡𝑟𝑎𝑖𝑛

𝑑 𝑎𝑥𝑖𝑎𝑙𝑠𝑡𝑟𝑎𝑖𝑛

• Equals0.5inperfectlyincompressiblematerial.

• Iftheforceisappliedalong𝑥:

𝜈 = −𝑑𝜖[𝑑𝜖<

= −𝑑𝜖\𝑑𝜖<

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Poisson’sratio

• Exampleofacubeofsize𝐿.

• Averagestrainineachdirection:𝜈 ≈ ∆I^

∆I• Approximate,becausetrueforsmallelementsanddeformation.

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𝐹

Δ𝐿

𝐿

∆𝐿′

Stress

• Magnitude ofappliedforceperareaofapplication.• largevalueó forceislargeorsurfaceareaissmall

• Pressuremeasure𝜎.• Unit:Pascal:𝑃𝑎 = 𝑁/𝑚>

• Example:gravitystressonplane:σ = 𝑚𝑔/ 𝜋𝑟>

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𝑟

𝑊

𝑚

TheLinearStressTensor• Measuringstressforeach(unit)direction𝑛 inaninfinitesimalvolumeelement:

𝜎i =𝜎<< 𝜎[< 𝜎\<𝜎<[ 𝜎[[ 𝜎\[𝜎<\ 𝜎[\ 𝜎\\

𝑛 = 𝑇𝑛

• Notethat𝑇𝑛 isnotnecessarilyparallel to𝑛!• 𝑇𝑛 = 𝜎i + 𝜏

• 𝜎i:outward/inwardnormalstress.• 𝜏:shearstress.

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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

LinearElasticity

• Stress andstrain arerelatedbyHooke’slaw• Remember𝐹 = −𝑘∆𝑥?

• Reshapetensorstovectorform:• 𝜎r = 𝜎<<, 𝜎<[,⋯ , 𝜎\\ ,andsimilarlyfor𝜖.̅• Thenthestiffnesstensor𝐂v<v holds:

𝜎r = 𝐂𝜖̅

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https://people.eecs.berkeley.edu/~sequin/CS184/TOPICS/SpringMass/Spring_mass_2D.GIF

Brownetal.“ResamplingAdaptiveClothSimulationsontoFixed-TopologyMeshes”

Hyperelastic Materials

• Seektoreturntotheir“restshape”.• Haveapotentialdeformationenergy

• Recall:springenergy:𝐸x =y>𝑘 𝑥 − 𝑥z >

• Underlyingassumption:deformationenergyisnotpath-dependent!

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LinearElasticityEnergy

• OnePossibilityis:𝐸 = y> ∫ 𝜎r, 𝜖̅ 𝑑𝑉? .

• PossibilitiesdependonthetypeofStress\Straintensorstouse.

• ThisoneispopularforlinearelasticitywithFEM.

• Weget:

𝐸 =12| 𝜖�̅�𝜖�̅�𝑉?

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DynamicElasticMaterials• Foreverypoint𝑞,ThePDEisgivenby

𝜌 ∗𝑑>𝑢𝑑𝑡> = 𝛻 � 𝜎 + �⃗�

• 𝜌:thedensity ofthematerial.• 𝑎:accelerationofpoint𝑞.• 𝛻 � 𝜎 = 𝜕 𝜕𝑥⁄ , 𝜕 𝜕𝑦⁄ , 𝜕 𝜕𝑧⁄ ∗ 𝜎 isthedivergenceofthestresstensor(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:𝛻 � 𝑢 = 0.• Irrotational (novortices):𝛻×𝑢 = 0

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Turbulentwithavortex Incompressible,irrotational flow

Steadyfield

MaterialDerivative

• Thechangeinthevelocityofthefluidparcelpassingatposition𝑥 intime𝑡.

𝐷𝑢𝑑𝑡 = 𝑢� + 𝑢 � 𝛻𝑢

• 𝑢�:unsteadyacceleration.• Howmuchvelocitychangesin𝑥 overtime.

• 𝑢 � 𝛻𝑢:convective acceleration.• Howmuchvelocitychangesduetomovementalongtrajectory.

25http://www.continuummechanics.org/images/continuity/converging_nozzle.png

Viscosity

• Resistancetodeformationbyshearstress.• Expressedbycoefficient𝜈:

𝐹𝐴 = 𝜈

𝜕𝑢𝜕𝑦

• Higher𝜈:morepressurerequiredforshearing!• Viscid fluids.

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Navier-StokesEquations• Representingtheconservationofmassandmomentumforanincompressible fluid(𝛻 � 𝑢 = 0):

𝜌 𝑢� + 𝑢 � 𝛻𝑢 = 𝛻 � 𝜈𝛻𝑢 − 𝛻𝑝 + 𝑓

• 𝑝:pressurefield• 𝜈:kinematicviscosity.• 𝑓:bodyforceperdensity(usuallyjustgravityρ𝑔).

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Inertia(pervolume) Divergenceofstress

Unsteadyacceleration

Convectiveacceleration

PressuregradientViscosity Externalbodyforces