Lecture 19: Frame Indifference and Material Constraints
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Overview
Steven M. Wise derives the material form of the Clausius-Duhem inequality, then shows how frame indifference constrains constitutive laws and how incompressibility produces a reactive stress associated with pressure. The lecture also defines isothermal and closed processes, connecting energetic passivity to a nonnegative time-integrated stress power and preparing for the derivation of the incompressible Navier-Stokes equations in Chapter 6.
Key takeaways
- The material Clausius-Duhem inequality follows by transforming entropy, heat supply, and boundary heat flux to the reference configuration, applying the divergence theorem, and localizing over arbitrary material subdomains.
- Frame indifference requires scalars to remain unchanged, vectors to transform by Q, and second-order tensors to transform by QSQ^T under a superposed rigid motion.
- Constitutive laws involving the rate of strain must respect its transformation L* = QLQ^T; this forces objective transformation rules for energy, heat-flux, and stress models.
- Incompressibility is expressed materially as det F = 1 and spatially as div v = 0, the constraint used in incompressible Navier-Stokes.
- A constraint-induced reactive stress is proportional to D gamma(F); for incompressibility it becomes a pressure-like multiple of the identity in Cauchy stress.
- For energetically passive bodies, closed isothermal processes imply nonnegative integrated stress power, linking the reduced second law to the mechanical equations developed next.
Chapters
- The second law tracks entropy change through heat supply, heat flux across the boundary, and nonnegative entropy production.
- The lecture aims to convert the integral entropy balance into a local material, or Lagrangian, inequality.
- The temperature, entropy, heat supply, and heat flux are expressed in material coordinates.
- Changing variables from the current domain to the reference configuration introduces the Jacobian J = det F.
- Mass conservation gives rho J = rho_0, allowing spatial mass-density factors to be replaced by reference density.
- The material heat-flux vector requires the Piola transformation J F^-1 q, rather than simply reusing the spatial heat flux.
- The divergence theorem converts the boundary heat-flux integral into a material-domain integral.
- Because the reference subdomain is arbitrary, localization makes the integrand a pointwise entropy inequality.
- For isothermal processes, the temperature time-derivative and temperature-gradient terms vanish, yielding the reduced Clausius-Duhem inequality for free energy.
- Two admissible motions are related by x* = Q(t)x + c(t), where Q is a proper rotation and c is a translation.
- The framework is Galilean: observers share synchronized clocks, unlike the relativistic setting mentioned for contrast.
- The observer transformation is used to determine which measured quantities must be invariant or rotate with the observer.
- The deformation gradient transforms as F* = QF, while the right stretch U and right Cauchy-Green tensor C = F^T F remain unchanged.
- The rotation in the polar decomposition transforms as R* = QR, and the left stretch transforms as V* = QVQ^T.
- The spatial rate-of-strain tensor transforms objectively as D* = QDQ^T.
- Taking the material gradient of x* = Qx + c eliminates the spatially uniform translation and gives F* = QF.
- The identity Q^TQ = I gives U* = U and C* = C directly from their definitions.
- Uniqueness of the polar decomposition yields R* = QR, with the corresponding relation V* = QVQ^T.
- A frame-indifferent scalar has the same value for both observers, while a vector transforms by Q.
- A second-order tensor transforms as QSQ^T, distinguishing its objectivity rule from those for scalars and vectors.
- The frame-indifference axiom applies these rules to density, temperature, entropy, internal energy, Cauchy stress, and Fourier heat flux.
- Constitutive models for internal energy, heat flux, and Cauchy stress are written as functions of density, temperature, and rate of strain.
- Frame indifference requires scalar energy laws to be unchanged under L transforming to QLQ^T.
- Heat-flux laws must rotate by Q, and stress laws must transform by Q on the left and Q^T on the right; a model depending only on temperature satisfies the scalar condition.
- A material constraint is expressed as gamma(F) = 0 for every admissible motion.
- For incompressibility, gamma(F) = det F - 1, so the local volume ratio is J = 1.
- The equivalent spatial condition is div v = 0, the familiar incompressibility equation in Navier-Stokes.
- The constraint axiom decomposes the first Piola-Kirchhoff stress into active and reactive parts.
- The active stress follows a constitutive equation, while the reactive stress does zero stress power: P_R:Fdot = 0.
- The reactive component enforces the material constraint without contributing power along permitted motions.
- For a regular constraint, the most general reactive stress is proportional to the derivative D gamma(F), with an unknown scalar multiplier.
- For J = 1, the derivative of det F is J F^-T, so the reactive Piola stress has the form of a scalar times JF^-T.
- Transforming to Cauchy stress for an incompressible material gives a reactive term proportional to the identity tensor, identifying the pressure contribution in Navier-Stokes.
- An isothermal process has temperature constant across both material position and time.
- Stirring water and slowly compressing a spring illustrate approximately isothermal mechanical processes, although friction can cause small temperature increases.
- In the isothermal formulation, the lecture retains the second-law inequality while setting aside the first-law energy balance under the stated heat-supply and heat-flux assumptions.
- A process is closed over an interval when each particle returns to its initial position at the interval endpoints; the endpoint velocity and temperature fields must also match.
- An energetically passive body has nonnegative material free-energy change over every closed process.
- These definitions support an inequality relating stress power over time to the free-energy difference.
- For a closed isothermal process in an energetically passive body, integrating the reduced Clausius-Duhem inequality gives nonnegative time-integrated first Piola-Kirchhoff stress power.
- The derivation uses positivity of temperature, entropy production, and J, followed by the fundamental theorem of calculus for the free-energy time derivative.
- The corresponding spatial-coordinate result is left as an exercise; Chapter 6 will apply these ideas to isothermal Navier-Stokes flows.
Transcript excerpt
0:04 Welcome back to continue mechanics. This is lecture 19 and the last lecture in chapter five if everything goes goes well. So chapter five is all about balance laws and converting integration integral um yeah integral bound uh balance laws into local uh balance laws whether that be uh spatial that is oilarian or material that is lrangeian. So I want to end or I want to um first start with our last balance law um that we were working through. We were working through these last time converting the
0:53 integration laws into local lrangeian forms and the last one that we uh need to do is the second law of thermodynamics. So here's the statement. Let phi be a motion of a continuum body. Then the second law of thermodynamics implies an equation of this form which is really a which is really an inequality. It's called the classius doom inequal inequality. And the reason why it's usually written as an in inequality is this term here is a production term which we don't always quantify but we do know that it's always greater than or equal to zero. So so usually this written this law is written
1:41 without this term and with this equality replaced by the inequality. That's why it's called the Clausius duhamm inequality. All right. So, it's not very meaningful, I think, to talk about what each piece or term means. It's much easier if I'd give you or remind you what the integration law looks like. Okay. Uh before I do that though, I should say all of these are written in terms of lrangeian or material fields. So, this is the entropy density in a material description. This is the heat source in a entropy description. This is the temperature or sorry in a material
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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.