<\body> > <\itemize> : <\equation*> \\=\u>|\\\=R>\\\a,>>|\T=\T\>|\\T=R>T\\a.>>|\\=>|\\\=R>\\\\a,>>>>> (matrix divergence: columns stay separate) : ,\,\]=\\(\\\)> : <\equation*> \=det>|>|>>|>|>|>>|>|>|>>>>>=[\,\,\]=|(123),>>||(123),>>||.>>>>> \\=\\>. \\)=\abc> <\eqnarray*> \>||\-\\,>>|\>||,>>|\>||>>> : <\eqnarray*> >||+\+\=([A\,\,\]+[\,A\,\]+[\,\,A\])/[\,\,\]=tr A,>>|>||\+\\+\\=([A\,A\,\]+[A\,\,A\]+[\,A\,A\])/[\,\,\]=[trA-tr A],>>|>||\\=[A\,A\,A\]/[\,\,\]=det A.>>>> : >(\\\)=(A\)\(A\)> > >=det A(A)>. det A(t)=det A*tr((\A)A)> : \> <\eqnarray*> \\>||\>>|\\)\>||(\\\)>>|\\)(\\\)>||\\(\\\)>>|\\)A>||\(A\)>>>> : Rotation around axis >> given by orthogonal matrix . |\>=>\>\(QQ)=0>,\ >Q>, >. =>\\>. <\itemize> and . : assumed regular. 0>. <\eqnarray*> (\)>||+\(\),>>|||>\\(\),>>|>||>X>\>\\>>>> : . : <\itemize> , F=U>, >. . Features: <\itemize> Is unique. is rotation of principal axes. average of all rotations. Principal axes of are >.\ (V)=\(U)>. \\>. \\\>. : /FF> SPD. : <\eqnarray*> ||(FF-Id)(:\|\\\|-\|\\\|=2\\\E\\),>>|>||(Id-FF)(Eulerian:\|\\\|-\|\\\|=2\\\E\\).>>>> : <\equation*> \(\)=(\\FF\)=\|U\\|. Has local maxima and minima when > is an eigenvector of . : <\equation*> \\a=F>\\A : Given <\equation*> >|>>|>>>>=f(R,\,Z), we have <\equation*> F=\\\\+\>\\\>+\\\\. Also expressible as from ,Z)>> to ,z)>>: <\equation*> F=*>r|\R>>|>*r|\\>>|*>r|\z>>>||1>\|\R>>||>*\|\\>>||1>\|\z>>>|*>z|\R>>|>z|\\>>|*>z|\Z>>>>>>. Caveat for mixed tensors: F>. However , , as usual. Also works for spherical basis, but more complicated. <\itemize> : \\\+\\\\>. : \\\>. : <\equation*> >|>|>>>>=(x)>>|(x)>>|>>>>,J=(R-Y)\. : <\equation*> >|>|>>>>=>>cos|l>\Z->>sin|l>\Z>>|>>sin|l>\Z+>>cos|l>\Z>>|Z>>>>>. . <\itemize> : /\t \(\,t)=0>. : focus on particle, expressions in terms of > and >Solids. : focus on point in space, expressions in terms of > and >Fluids. Lines: <\itemize> : Curve traced by a fixed particle. : Field lines of velocity in Eulerian POV. Both coincide under steady motion. : <\eqnarray*> |\>>||\|\t>+\>\\\,>>||\>>||\|\t>+(\>\\)\,>>|>>||T|\T>+(\>\T)\.>>>> : =|\>>. : >\\\>=L*F> (chain rule). requires a ``reference state'', does not. |\>=>\\=L*F\\=L\\>. Assume \=\\|\\\|>. <\eqnarray*> :>\\|>|\|\\\|>>||\L\=\\D\>>||\>>||-\(\\L\)>>>> : , >, >. >: stretching rate of a line element along the 1-direction >: (roughly) change in angle between the 1- and 2-direction. Principal axes > of are rigidly rotating about <\equation*> >=curl \ with =>\\>. : =2\>. (Letter here is also >>.) >=J*tr L=J div \>. : <\eqnarray*> >\\\\>||>\(\,t)\F\\>>||\t>>\\\\>||>|\>(\,t)\F+\(\,t)\L*F\\>>|||>|\>(\,t)+L\(\,t)\\>>>> : Similar, taking into account that >=J*F>. <\equation*> |\t>>\\\\s=>(|\>+\*tr(L)-L\)\\\s. : <\equation*> |\t>>\(\)\v=>|\>+\*tr(L)\v. Observe that >, which is zero in the incompressible case. : <\eqnarray*> >\\\\>||>curl \\\\=>>\\\>>|\>||\v>>|>|\t>>\\\\>||>|\>(\,t)+L\(\,t)\\>>|||>|\>(\,t)\\+>\v\\>>|||>\(\,t)\\>>|||>curl \\\\>>>> If =\\>, then the motion is . If circulation-preserving, then <\equation*> curl \=>|\>+>tr(L)-L>=0. Then consider the product rule on <\equation*> |\t>(J*F>)=\=0 to find : <\equation*> >=F>. Field lines of vorticity are . If the motion is circulation-preserving, these are material curves. <\itemize> : Assumption: <\eqnarray*> >||(referential).>>>> Therefore, <\eqnarray*> |\>J+\>>||>||\>J+\*J*div \>||>||\>+\*div \>||>|\+div(\\)>||>>> : <\equation*> |\t>>\\\v=>\|\>\v. : \> <\itemize> : )>> is force/unit area. : <\equation*> |\t>>\\=>\|\>=>\\\v+R>\)>\a. : \=\)>>. Derivation: <\itemize> )>=-\)>> by pillbox and balance law. Tetrahedron argument: > the general normal of the coordinate-system-boxed tetrahedron. Then other faces =a\>, where is the area of the complicated face. Volume . Apply >balance law, let 0>. Assume continuity, derive linear dependence by assuming values are locally constant. Updated balance law: <\equation*> >\\=>\\+\\\ : <\eqnarray*> |\>>||\s+\\()>>|\>||>\\+\\()>>>> : \>. (\\\=s\\\> can be directly verified.) Also called . > is . : \\\> <\itemize> : no contact torques. : <\equation*> |\t>>\\\\)>>\\\|\>=>\(\\\+\)\v+R>\\\)>\a > is body torque. Equality )> follows because >-derivatives vanish once \> is applied. Subsituting Cauchy's Theorem into the balance law gives <\eqnarray*> >\\\>\\+\\>||>\(\\\)\v+R>\\\\\a>>|>\\\>\\>||R>\\\\\a>>>> View in component form, apply Gau˙, derive \=0>>=\>. : <\eqnarray*> F>||()>>|>||()>>>> : <\eqnarray*> \\)\>||\\|\\|+(\\\)\\>>|\\)\>||\\)\.>>>> Use these identities to rewrite the |\>> as (\)> for irrotational flow. : <\itemize> : =-p*Id>>=-\p>. : |\>=0> or =0>. : \=\>. |\>=\\\\>. : >=\> or =\\>. : )> : =0>, . : Begin with deformation of spherical shell (with extent!), assume 1>. Derive ODE. : =-\\> <\itemize> Elastic or ideal flow here is circulation preserving, i.e. =-\something>. <\itemize> Have <\eqnarray*> >||>\p(\)+\>>|||>p(\)\\-\\.>>>> Define <\eqnarray*> (\)>||>>p(\)\\>>|\\(\)>||(\)\\>>>> Therefore <\equation*> \=-\(\(\)+\). For ideal fluid substitute > for >. :\ <\itemize> Flow irrotational (i.e. =-\\>): <\equation*> \\\+|2>+\(\)+\=\. Proof: Just rewrite, obtaining /2> from second term of material derivative. Flow steady:\ <\equation*> |2>+\(\)+\>=0, i.e. this quantity is constant along streamlines. Proof: Exploit \|\>=\\\(\)> Flow both irrotational and steady: <\equation*> \|2>+\(\)+\=\. : <\itemize> Assume =\+\\>, \|\1>, \\|\1>. Start with *(\\\)>, use cons. of. momentum without second order term, cons. of mass as \+\div(\)=0.> \=c\\\>, with >p>>. : assume steadiness =0>, use \|\>> in terms of >. <\eqnarray*> \(\\)>>||\|\>\1-|c>|\>>>>>> <\itemize> Supersonic nozzle 1>, 1>. : <\itemize> : <\eqnarray*> |\t>K(R)>||)+P(R)>>||\t>\\\\\v|\> K(t)>>||>tr(\D)\v|\> S(R)>+>\\\\+R>\\\\\a|\> P(R)>>>>> \; Proof: Multiply Equation of Motion by >, integrate by parts in the > term. : <\equation*> v\v>|\>>+D)|\>>=>\(\\)+\\\\|\>>. Key words for more global energy conservation: internal energy )>, heat supply per unit mass )>, heat flux through material surface. <\equation*> |\t>{K+U}=P(R)+H(R). Now, because there is a stress power loss above, there needs to be a gain here: <\equation*> |\t>U(R)=S(R)+H(R). : For the balance law <\equation*> |\t>>\\=>\s+R>f)>, we get <\equation*> [\V\+f)>]=0. > interface speed, -\\\>. ||||||||>|>>|||>>|\\>>|+\\\>>>||> per unit mass >||>>|\\>>|\\+r>>>|)>>>|> per unit area>||)>>>|\\)>>>|)>\\+h)>>>>>>> so that for example <\eqnarray*> V]>||>|V\+\)>]>||.>>>> Or for material jumps: )>]=\>. Derivation: <\itemize> Modification for moving boundary is <\equation*> -[\\]V. Then use pillbox that flattens around surface. Examples: <\itemize> Free boundary: pressure must be continuous, because otherwise there's a finite force on something massless. : <\itemize> Assume =\\>. Conservation of mass \=0>. Bernoulli's equation <\equation*> \\+|2>+>+\=const BCs: depthward, > free surface <\itemize> =0> at bottom |\t>(z-\)=0> at >>\=\\> at (!). pressure continuous at interface. Use Bernoulli's equation to rewrite as condition <\equation*> \\+g\=0 z=0. : -p=-\\>. : Large density over small density. : Wave formation. <\itemize> : A reference frame/coordinate system w.r.t. which vectors and tensors are seen. <\equation*> \>=\(t)+Q(t)\ so, for example, >=Q*F>, >=J>, >=U>, >=Q*R>. : <\eqnarray*> >(\>)>||(\)>>|>(\>)>||(\)>>|>)>||)Q>>>> Examples: , regions, normals, > Non-examples: =|\>+Q*\>, +>Q>, . : <\itemize> Must be workless, i.e. Constraint given as (C)=0\|\>=tr(\>)=0>, where F>. >=2FD*F>>F\F>. : =g(L)>. Cannot support shear stress at equilibrium. If , also cannot support shear stress when in motion. <\itemize> : >=g(L>)>. <\itemize> Choose , >=-W> to obtain that . Most general such : <\equation*> \(D)=\I+\D+\D, with >, >, > functions of the invariants of . Proof: Cayley-Hamilton. : <\equation*> \=-p*Id. : <\equation*> \=-p(\)Id : <\equation*> \=-p(\)Id+2\D : <\equation*> \\=-\p+\\\+\\ plus conservation of mass. =x/l>, |~>=\/v>, v)>, =t/l>. Then is =\/p>. : >. High: Dominated by inertial effects. Low: Dominated by viscous effects. No-slip BCs apply only for viscous fluids. : Watch for emergence of a boundary layer. : =f(F)> <\itemize> : \>, where > is the symmetry group of the material. <\equation*> \=f(F)=f(F*P) : =SO(3)>. Then choose >>=f(F)=f(V)>. : >=f(V>)>. Most general expression to satisfy this: <\equation*> \(V)=\Id+\V+\V, with >, >, > functions of the invariants. : Linearization! <\eqnarray*> ||\>>|||FF-Id\\\-(\\)>>||>|>||>|\\-(\\)>>>> Use these in <\equation*> \=ctr V*Id+cV+cV\\tr(E)Id+2\E, where >, > are the . : >the usual way to specify constitutive relations for solids Then <\equation*> \=\W|\F>|\>W|\V>R>F=\W|\V>V Invoke objectivity: Invoke isotropy: > depends only on invariants of . > 's principal axes line up with those of , i.e. > : <\equation*> \>=\>W(\,\,\)|\\>>. Incompressible: <\equation*> \>=\>W(\,\,\)|\\>>-p. Specifying in terms of >: <\equation*> \=IIIW>*Id+(W*>+\W>)B-W>B, where subscripts by ,II,III> mean partial derivatives. : <\eqnarray*> ||\\+\+\-3-2ln(J)+\(J-1),>>|\>||(\-1)+\J*(J-1),>>|\>||\-p.>>>> Solving a solids problem: <\itemize> Calculate (Kinematics) Calculate > Calculate > Apply conservation of momentum in deformed configuration. Solve for unknowns >, , using BCs. <\initial> <\collection> <\references> <\collection> > > > > > > > > <\auxiliary> <\collection> <\associate|toc> |math-font-series||1Tensor Stuff> |.>>>>|> |math-font-series||2Kinematics> |.>>>>|> |2.1Static |.>>>>|> > |2.1.1Static Examples |.>>>>|> > |2.2Dynamic |.>>>>|> > |math-font-series||3Balance Laws and Field Equations> |.>>>>|> |math-font-series||4Constitutive Laws> |.>>>>|>