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

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