# Viscous fluid equivalent of hyperelasticity

A hyperelastic solid is defined as one for which the stress tensor $$\sigma$$ can be written as the derivative of some stored energy function $$W$$ w.r.t. the strain $$\varepsilon$$:

$$\sigma = \frac{\partial W}{\partial\varepsilon}.$$

Hyperelastic solids are "nice" because the stored energy function can be identified with the Helmholtz free energy, and the Clausius-Duhem relation is satisfied automatically.

For many viscous incompressible fluids, it's possible to write the deviatoric stress tensor as the derivative of some function $$P$$ with respect to the strain rate tensor $$\dot\varepsilon$$:

$$\tau = \frac{dP}{d\dot\varepsilon}.$$

For example, for a Newtonian viscous fluid, $$P = \mu\dot\varepsilon : \dot\varepsilon$$. The case I'm interested in is that of a power-law fluid where $$\tau = K|\dot\varepsilon|^{m - 1}\dot\varepsilon$$ for some positive $$m$$, in which case $$P = \frac{K}{m + 1}|\dot\varepsilon|^{m + 1}$$. The particular form doesn't matter so much as the fact that $$P$$ must be convex.

This type of constitutive relation for fluids is nice for the same reasons that the hyperelastic relation for solids are -- you're automatically assured that the Clausius-Duhem inequality is satisfied, and there's a variational principle for the equilibrium state with a given external forcing. Is there a name for viscous fluids with this type of constitutive relation? If so, is there a term for the "potential" function $$P$$?

I might be misunderstanding your question, but it seems to me that the two cases are quite different. The first relation, what you call a "hyperelastic solid" is essentially the definition of the second Piola Kirchoff stress tensor (PK-II). In that case, $$\epsilon$$ is the full nonlinear strain tensor and $$W$$ is the elastic free energy, with a specific functional form of the energy in terms of the strain tensor realizing a specific constitutive equation. The standard Cauchy stress tensor, whose components give the force acting per unit area (of the deformed elastic body), is related to the PK-II stress tensor via multiplication by the deformation gradient. These are all conventional definitions in continuum mechanics irrespective of the constitutive relation.

The second case corresponding to the viscous fluid is in my opinion qualitatively different. Stress and strain are thermodynamically conjugate variables, while stress and strain-rate are not. The two ($$\tau$$ or $$\sigma$$ and $$\dot{\epsilon}$$) have opposing signs under time-reversal and the viscous constitutive law is a dissipative relation, involving a transport coefficient (viscosity) instead of a thermodynamic parameter (elastic modulus). Nominally a scalar function (what you call P) could be postulated in the simplest of cases and would correspond to a Rayleigh dissipation function. Extensions of the dissipation function to more complicated cases is quite non-trivial and many times ad hoc. Using entropy production rate instead offers a better alternative to the Rayleigh dissipation function and is thermodynamically motivated. But in either case, the form of the constitutive law remains phenomenological and works best in the linear response regime. Going beyond that to nonlinear and strong forcing is far less clear, and any constitutive equation you write down is essentially arbitrary.

• I didn't phrase my question very clearly -- thanks for giving it such a thoughtful response all the same! The two are qualitatively different, but both lend themselves to variational principles that superficially look quite similar. I've posted my own answer with what I found. – korrok Jan 12 '19 at 1:00

I've done some more reading and found that this idea is discussed in several sources, but many of them use different names for the same thing, and many of the names have really bad SEO. For example, [1] was one of the first papers to discuss what they refer to as a "flow potential" $$\Omega(\sigma)$$, i.e. a function of the stress tensor $$\sigma$$ such that

$$\dot\varepsilon = \frac{\partial\Omega}{\partial\sigma}.$$

However if you google the term "flow potential" you get a ton of hits about "potential flow" from elementary fluid mechanics textbooks. In the book [2] the same function $$\Omega$$ is referred to as the "creep potential". The books [3] and [4] refer to it as a "dissipation potential" which was the most common term I found in the literature.

Some sources ([1] and [3]) postulate that a dissipation potential has to exist and be convex, which implies the Clausius-Duhem inequality and a nice stability property of the steady state. Apparently there are some interesting cases of non-convexity resulting in phase transitions [5]. As far as I understand, the physical significance of the dissipation potential is that locally the rate of dissipation is maximized, i.e. the system evolves as a gradient flow.

Of the various sources I found, [3] was the most readable.

[1] JR Rice (1970), On the Structure of Stress-Strain Relations for Time-Dependent Plastic Deformation in Metals

[2] Betten (2002), Creep Mechanics

[3] Lemaitre and Chaboche (1990), Mechanics of Solid Materials

[4] Silhavy (1997), The Mechanics and Thermodynamics of Continuous Media

[5] Janecka and Pavelka (2018), Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics