Lecture 17: Local Balance Laws
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Overview
Steven M. Wise converts integral balance laws into local Eulerian equations using the transport theorem, the divergence theorem, and localization. The derivations yield the linear momentum equation, show that angular momentum balance requires a symmetric Cauchy stress, and produce local first- and second-law equations, including the assumptions that reduce the energy balance to the heat equation.
Key takeaways
- The local linear momentum equation is rho times the material acceleration = div S + rho b, derived by rewriting traction as S n and localizing the integral balance.
- Angular momentum balance requires the Cauchy stress tensor to be symmetric, S transpose = S, even for time-dependent motion.
- For symmetric stress S, mechanical power from deformation depends on the rate-of-strain tensor L, the symmetric part of grad v, through the contraction S:L.
- The local first law is rho times the material derivative of specific internal energy = S:L - div q + rho r, combining mechanical work, heat flux, and internal heating.
- Fourier heat flux q = -D grad theta, constant density, internal energy c_v theta, and suitable zero-source and stationary assumptions reduce the first law to a heat-diffusion equation.
- The local second law requires nonnegative entropy production, while heat transfer can raise or lower the total entropy of a chosen system.
Chapters
- The lecture moves from integral balance laws to pointwise differential equations in a spatial, or Eulerian, description.
- Wise notes that finite element methods later integrate equations into a weak form, even though modeling commonly begins with local laws.
- The localization theorem will convert identities over arbitrary subdomains into equations that hold pointwise.
- The integral momentum axiom balances the time rate of total momentum against boundary traction and body force density rho times b.
- Using traction t = S n and the divergence theorem converts the boundary force into a volume integral of div S.
- The transport result for an integral of rho times a field converts the momentum derivative into the integral of rho times the material acceleration.
- Localization gives rho times the material derivative of v equals div S plus rho b.
- The local consequence of angular momentum balance is that the Cauchy stress S must satisfy S transpose = S at every spatial point and time.
- The integral law tracks angular momentum through the moment arm x crossed with velocity and balances its rate of change against applied torques.
- The derivation reuses the transport theorem and the local linear momentum equation to eliminate acceleration and body-force terms.
- For a spatial vector field W, the material derivative is its partial time derivative plus grad W multiplied by v.
- Wise expands grad(x cross v) with the permutation symbol and the product rule, distinguishing free and summed indices.
- The term involving v crossed with itself vanishes, leaving the material derivative of x cross v in terms of x crossed with the material acceleration.
- Replacing traction by S n and material acceleration by (div S + rho b)/rho introduces stress and body-force terms into the angular momentum balance.
- The body-force moment appears on both sides and cancels.
- The remaining identity matches the equilibrium argument from Result 3.5, implying S transpose = S in the dynamic setting as well.
- Wise defines the rate-of-strain tensor L as one half of grad v plus its transpose, making L a symmetric second-order tensor.
- The kinetic-energy argument will establish that net working over an arbitrary region equals the integral of S contracted with L.
- This identity links mechanical power to deformation and is used to derive the local first law.
- Dotting local linear momentum with v and applying the product rule to div(S transpose v) isolates a divergence term and a stress-gradient contraction.
- The divergence theorem moves the stress-power flux to the boundary of the arbitrary region.
- Because S is symmetric, its contraction with grad v equals its contraction with the symmetric part L.
- The boundary term becomes traction power because S n = t.
- The transport theorem applied to kinetic energy, defined using one half rho times v dotted with itself, gives its time derivative as an integral involving v and material acceleration.
- Combining this kinetic-energy derivative with the stress-power calculation yields net working = integral of S:L over the region.
- The identity holds for arbitrary subdomains and supplies the mechanical-work term needed in the first-law derivation.
- The integral first law balances the time rate of internal energy against net heating and net working.
- Applying the transport theorem to rho times specific internal energy gives the integral of rho times its material derivative.
- The divergence theorem converts the boundary heat-flux term into a volume term involving div q.
- Localization gives rho times the material derivative of internal energy = S:L minus div q plus rho r.
- The terms in the first-law balance are combined under one volume integral over an arbitrary region omega_t.
- The localization theorem turns the integral identity into a pointwise equation for internal energy.
- The resulting equation includes mechanical heating S:L, heat conduction through minus div q, and the volumetric source rho r.
- Fourier's law sets q = -D grad theta, with theta denoting temperature and D a positive diffusion coefficient.
- Assuming constant density, specific internal energy equal to c_v theta, zero velocity, and no mechanical or internal heat sources reduces the energy balance to a diffusion equation.
- For constant D, the resulting temperature equation has the familiar form involving D times the Laplacian of theta, after accounting for density and heat capacity.
- The local entropy balance relates rho times the material derivative of specific entropy to the temperature-scaled heat source, the divergence of q/theta, and entropy production.
- The integral second law includes both natural entropy production and entropy transported by heating; heat exchange can make a system's total entropy rise or fall.
- The divergence theorem and localization convert the integral inequality into a pointwise condition with a nonnegative entropy-production term.
- With no heat input or flux, entropy can still increase through internal production.
Transcript excerpt
0:04 Welcome back to continuum mechanics. This is uh class number 17. It's our second lecture in chapter 5 talking about balance laws. All of the balance laws that we worked with were given in integral form to begin with and now we're engaging in the task of converting these into what are called local laws. These are essentially differential equations that we can work with and try to solve at some point. Uh if you use the finite element method, the interesting thing is that ultimately you integrate these equations and then solve them by in their what's called the weak form. And so you bring the the integrations back in order to solve the equations at least approximately. uh but to make the equations uh and the
0:54 the way people are most used to building and working with uh models is to uh convert all of our balance laws into what are called local laws. And so we started that process last time with a conservation of mass equation and um and a result related to that. Now we want to talk about the conservation of linear momentum and write these equations write this idea in a local version a local what's called oilarian version or a spatial version. Here's the result that we want to prove. Let fi be a motion with spatial velocity v and mass density row. Axiom 5.2 2 that is the balance of
1:43 linear momentum is equivalent to this equation. Row times the total time derivative of the velocity field V is equal to the divergence of the stress tensor field plus row * V. Let's make sure everything in sight makes sense. So the total time derivative of V is a vector field again. So row * V is a vector field. Row * B is
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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Steven M. Wise.