7. Constitutive laws

Conservation needs material behavior

Conservation constrains storage and transport, but usually does not say how a material carries flux or stress. A constitutive law supplies that closure. Its parameters and assumptions describe the modeled material, not a universal identity.

Learning outcomes

After this chapter, you should be able to:

• separate a balance law from the constitutive relation that closes it;
• read Newtonian viscosity and isotropic linear elasticity in residual form;
• check the dimensions of material parameters; and
• identify when a constitutive assumption is unsuitable even though its source is well typed.

Why does heat flow down a temperature gradient?

For the stationary slab, conservation gave $c_v\partial_tT+\nabla\cdot\boldsymbol j=s_v$. Two unknown fields, $T$ and $\boldsymbol j$, remain. A simple material relation is Fourier’s law,

$$\boldsymbol j=-\kappa\nabla T,\qquad \kappa>0.$$

For isotropic material, conductivity $\kappa$ is a scalar. The minus sign makes heat flow toward lower temperature. This is a phenomenological material law, not a consequence of the divergence theorem. Its elementary slab form is introduced in OpenStax, section 1.6.

Return to the slab gradient $-200\,\mathrm{K/m}$. With illustrative $\kappa=0.1\,\mathrm{W/(m\,K)}$, the predicted flux is $+20\,\mathrm{W/m^2}$. Increasing conductivity by a factor of two doubles the flux for the same prescribed endpoint temperatures. It does not change the linear temperature profile in this source-free homogeneous example. That last qualification matters: with prescribed flux instead, changing conductivity changes the gradient.

Integrating the constant-gradient law gives $\dot Q=\kappa A(T_h-T_c)/L$. Therefore the thermal resistance assumed in the network chapter is $R=L/(\kappa A)$. The component-level relation has now been derived from the continuum law and geometry. For $L=0.2\,\mathrm m$ and $A=0.1\,\mathrm{m^2}$, it is $20\,\mathrm{K/W}$ and the 40 K temperature difference transfers $2\,\mathrm W$.

Which parts of a linear law are assumptions?

Constant conductivity assumes its variation over the temperature range is unimportant. Isotropy assumes every direction conducts alike. Locality assumes the flux at one point responds to the gradient at that point. A laminate can require a tensor conductivity; a wide temperature interval can require $\kappa(T)$. Conservation still holds when we replace the constitutive law.

The same separation appears in mechanics: conservation balances forces, kinematics describes deformation or motion, and a material law turns those kinematic measures into stress. The two linear examples below follow that pattern; real materials are not always linear.

Newtonian viscous stress

For the steady fluid component, the Cauchy stress is represented by

$$\boldsymbol{\sigma} = 2\mu\,\operatorname{sym}(\nabla\boldsymbol{u}) - p\boldsymbol{I}.$$

Velocity gradient has dimension $1/T$. Dynamic viscosity $\mu$ has dimension $M/(LT)$, so their product has pressure dimension $M/(LT^2)$. The isotropic pressure term has the same dimension and tensor shape.

The fluid balance places this stress inside a divergence. Find dynamic_viscosity in the source linked below: its declared unit is kg / (m * s). Its physical value describes a material at the conditions of interest; choosing it is a modeling decision.

Isotropic linear elasticity

For small displacement $\boldsymbol{d}$, the solid package uses

$$\boldsymbol{\sigma} =2\mu_s\,\operatorname{sym}(\nabla\boldsymbol{d}) +\lambda\,(\nabla\!\cdot\boldsymbol{d})\boldsymbol{I}.$$

Both strain measures are dimensionless. The Lamé parameters $\mu_s$ and $\lambda$ therefore have stress dimension. The solid component applies divergence to this stress and balances it against the body force. Open IsotropicBalance2d in the source below to see the expression in its full context.

The plane-strain and plane-stress components derive different Lamé parameters from Young’s modulus and Poisson’s ratio. They share an equilibrium structure while making different kinematic assumptions.

Read the standard solid components.

Test a material law in a simple motion

Consider plane shear $\boldsymbol u=(\dot\gamma y,0)$ with constant shear rate $\dot\gamma$. The gradient has one nonzero off-diagonal entry. Symmetrization places $\dot\gamma/2$ in both off-diagonal entries, so Newtonian shear stress is $\sigma_{xy}=\mu\dot\gamma$. The factor 2 in the constitutive formula is necessary: it cancels the half introduced by the symmetric part. A constant translation has zero gradient and produces no viscous shear stress.

For a solid, $\boldsymbol d=(\varepsilon x,0)$ gives constant small strain $\varepsilon$ in the $x$ direction. A rigid infinitesimal rotation $\boldsymbol d=(-\omega y,\omega x)$ has an antisymmetric gradient and zero symmetric part, so the linear law assigns no strain energy to that motion. These hand checks tell us more about the formula than a plausible displacement plot alone.

Recognize the equations inside reusable components

The direct flow model in the fields chapter wrote its stress and balances explicitly. The repository’s fluid source gives those same equations names: SteadyNewtonianBalance2d and SteadyStokes2d. Inspect how SteadyStokes2d composes the balance with its boundary interface. The solid package similarly separates IsotropicBalance2d from material parameters and boundary components.

These source components show how named laws collect equations you now understand. For a runnable packaged model, use the bundled Eqiora.Fluid.Incompressible or Eqiora.Solid.LinearElasticity examples in the continuum reference. Choosing a component still leaves the material assumptions to you. The fluid path and structural path develop the complete assemblies and physical assumptions; heat transfer continues with Fourier conduction.

Parameters do not validate the material model

Dimensional checking can reject a viscosity supplied in metres or a Young’s modulus supplied in seconds. It cannot establish that a fluid is Newtonian, that deformation is small, or that an isotropic material adequately describes the specimen. Those judgments come from the phenomenon, scale, and intended use of the model.

Deliberate failure: give viscosity the pressure dimension

If dynamic_viscosity is declared as kg / (m * s ^ 2), multiplying it by grad(velocity) produces pressure per time rather than pressure. It can no longer be subtracted from isotropic_lift(pressure). The dimensional error points to the parameter declaration; rearranging the momentum equation does not repair it.

Exercises

1. Verify the dimensions of both terms in the Newtonian stress.
2. Explain which part of the fluid example is conservation and which part is constitutive closure.
3. Show that displacement gradient is dimensionless.
4. Compare the plane-stress and plane-strain parameter expressions in the standard package. Which balance and interface structures stay unchanged?
5. Name a material behavior for which isotropic linear elasticity would be a poor assumption, even if every unit checks.

Exercise guidance

For exercise 1, $[\mu]=M/(LT)$ and $[\nabla\boldsymbol u]=1/T$, giving $M/(LT^2)$ as required. For exercise 4, distinguish suppressed out-of-plane strain from suppressed out-of-plane stress before comparing parameter expressions. They are different physical reductions, not two numerical solvers for an identical problem.

Boundary-dependent prediction. Double $\kappa$ in the slab while holding heat flux, rather than endpoint temperatures, fixed. Derive the new temperature drop. Then explain why the earlier fixed-temperature experiment had a different observable response.

References

• Samuel J. Ling, Jeff Sanny, and William Moebs, University Physics, volume 2 (OpenStax, 2016), section 1.6, Mechanisms of heat transfer, for conduction and thermal conductivity.
• Carol Livermore, Henrik Schmidt, James H. Williams, and Simona Socrate, Mechanics & Materials I, MIT OpenCourseWare 2.001 (2006), lecture notes, lectures on strain, stress, and elasticity. Use these to develop the small strain constitutive interpretation beyond the shear checks here.

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