Thermodynamic Shear Tension Insights

• Thermodynamic shear tension is a conjugate variable to shear deformations, defined to capture shear work in systems ranging from Kerr–Newman black holes to continuum solids.
• It unifies different phenomena by reinterpreting rotational work as shear work and incorporating shear variables into thermodynamic differential frameworks.
• This concept influences first-law modifications, interfacial free energies, and yielding behavior, offering practical insights across black-hole, solid-state, and fluid interfaces.

Thermodynamic shear tension is not a single universally fixed quantity across physics. In the cited literature, it denotes a thermodynamic quantity conjugate to a shear-like deformation, or more broadly the shear branch of a first-law-type description in which stress, excess shear, or interfacial tension enters as a thermodynamic variable. In quasi-local black hole thermodynamics, it is the conjugate $X$ to the geometric eccentricity $Y=a/r_+$ of a Kerr–Newman horizon, so that horizon oblateness contributes a shear work term $X\,dY$ to the first law (Campos, 22 Mar 2026). In continuum thermodynamics of solids and interfaces, the same general idea appears through conjugate pairs such as $(\tau,\gamma)$ for bulk shear and $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$ for coherent interfaces (Porporato et al., 2018).

Domain	Conjugate pair	Thermodynamic role
Kerr–Newman black holes	$Y=a/r_+$, $X$	shear work $X\,dY$
Isotropic solids under plane shear	$\tau$, $\gamma$	shear work $Y=a/r_+$0
Coherent solid–solid interfaces	$Y=a/r_+$1, interface excess shear	adsorption equation term
Sheared interfaces of complex fluids	surface tension and interfacial area	shear-dependent effective tension

1. Kerr–Newman black holes: quasi-local shear tension

The most explicit use of the term appears in the quasi-local thermodynamics of Kerr–Newman black holes. There the quasi-local internal energy is taken to be

$Y=a/r_+$2

and the rotationally induced oblateness of the horizon is encoded by the dimensionless eccentricity

$Y=a/r_+$3

Its conjugate thermodynamic shear tension is

$Y=a/r_+$4

in the internal-energy representation, or equivalently

$Y=a/r_+$5

in the enthalpy representation, with $Y=a/r_+$6. In geometric variables,

$Y=a/r_+$7

The generalized first laws become

$Y=a/r_+$8

In this construction, $Y=a/r_+$9 is a geometric shape variable measuring horizon oblateness, and $X\,dY$0 is the generalized force conjugate to that deformation; $X\,dY$1 is the corresponding shear work (Campos, 22 Mar 2026).

This framework is introduced because, for rotating horizons, the geometric volume

$X\,dY$2

is no longer a simple function of the area alone. In the spherical case there is only one radius and no shear degree of freedom, but for Kerr–Newman the breakdown of the functional dependence $X\,dY$3 requires an additional geometric variable. The paper therefore interprets the rotational sector not merely as mechanical angular work, but as a combination of bulk deformation and area-preserving horizon shear (Campos, 22 Mar 2026).

A central clarification is that the pair $X\,dY$4 is not just a relabeling of $X\,dY$5. In the Kerr limit,

$X\,dY$6

so the quasi-local formalism reorganizes rotational work into pressure–volume work plus shear work adapted to the intrinsic horizon geometry. This distinction is one of the defining features of thermodynamic shear tension in the black-hole setting (Campos, 22 Mar 2026).

2. Geometric origin, Legendre structure, and scaling

The geometric content of $X\,dY$7 is explicit in the induced horizon metric,

$X\,dY$8

Its determinant is

$X\,dY$9

which is independent of $(\tau,\gamma)$0. Therefore,

$(\tau,\gamma)$1

and variations in $(\tau,\gamma)$2 at fixed $(\tau,\gamma)$3 are traceless. In the language of the paper, the $(\tau,\gamma)$4-variation is a purely shear deformation of the horizon $(\tau,\gamma)$5-geometry: it preserves area, but changes shape by flattening the poles and enlarging the equatorial radius (Campos, 22 Mar 2026).

The enthalpy representation makes the conjugacy especially transparent. The extended enthalpy is

$(\tau,\gamma)$6

and its shear conjugate is

$(\tau,\gamma)$7

In the slow-rotation limit $(\tau,\gamma)$8, this reduces to

$(\tau,\gamma)$9

which the paper describes as a linear, Hooke-like shear response at fixed $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$0 and $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$1 (Campos, 22 Mar 2026).

The same work also shows that the shear sector modifies the differential first law without entering the Euler-type Smarr relations. With the scale assignments $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$2, $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$3, $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$4, $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$5, and $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$6 dimensionless, one obtains

$$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$7

No explicit $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$8 term appears because $$(\sigma_{3i},[V\overline{F}_{i3}/\overline{F}_{33}]_{XY}/A)$$9 is scale invariant. The paper’s interpretation is that the shear pair contributes to the differential thermodynamics, but is invisible to homogeneity-based Euler scaling (Campos, 22 Mar 2026).

3. Bulk solids and coherent interfaces

In continuum thermodynamics of isotropic materials under hydrostatic pressure and plane shear, the bulk analog of thermodynamic shear tension is explicit. The internal-energy differential is written as

$Y=a/r_+$0

and, per unit volume,

$Y=a/r_+$1

With the extended Gibbs free energy

$Y=a/r_+$2

its differential is

$Y=a/r_+$3

Here $Y=a/r_+$4 and $Y=a/r_+$5 are a thermodynamic conjugate pair in exactly the same formal sense as $Y=a/r_+$6 and $Y=a/r_+$7 (Porporato et al., 2018).

From this potential, the isothermal isobaric shear compliance is

$Y=a/r_+$8

the shear thermal deformation coefficient is

$Y=a/r_+$9

and the isothermal coefficient of dilatancy is

$X$0

Among the derived identities are

$X$1

$X$2

and

$X$3

A common misconception addressed by this formulation is that adiabatic and isothermal shear compliances are always equal; the paper shows that this equality requires an isoshear-stress transformation (Porporato et al., 2018).

A related formulation uses strain volumes $X$4, so that the mechanical work term becomes $X$5, and shear stress appears as the thermodynamic conjugate to shear strain volume exactly as normal stress is conjugate to normal strain volume. In isotropic solids this leads to the constitutive law

$X$6

which makes the shear modulus an isochoric function of the specific volume (Burns, 2018).

For coherent solid–solid interfaces under nonhydrostatic stress, the same idea appears as an interfacial excess variable. The generalized adsorption equation contains the term

$X$7

and the conjugacy relation

$X$8

For $X$9, these are interface excess shears conjugate to the shear stresses $X\,dY$0. The paper emphasizes that this property is specific to coherent interfaces, because incoherent or fluid interfaces cannot support static shear stresses parallel to the interface (Frolov et al., 2013).

4. Plasticity, yielding, and configurational thermodynamics

In non-equilibrium theories of deformation, thermodynamic shear tension usually appears as shear stress treated as a generalized thermodynamic force whose evolution is constrained by configurational disorder, defect energetics, and entropy production rather than by purely kinematic constitutive assumptions.

In thermodynamic dislocation theory for anti-plane shear, the actual shear stress is

$X\,dY$1

and the microforce balance gives

$X\,dY$2

so that

$X\,dY$3

Here $X\,dY$4 is the flow stress and $X\,dY$5 is a back stress derived from the excess-dislocation free energy $X\,dY$6. The plastic rate is governed by thermally activated depinning,

$X\,dY$7

and the formulation is constructed so that $X\,dY$8. In that setting, “thermodynamic shear tension” denotes the shear stress and flow stress whose evolution is tied to internal variables $X\,dY$9 and to a configurational free-energy structure (Le et al., 2018).

In the closely related shear-banding and yielding theories for polycrystalline solids and steel, the central constitutive ingredient is the depinning law

$\tau$0

with the plastic power $\tau$1 partitioned into dislocation storage, configurational heating, and ordinary heating. The stress is therefore not treated as an independent imposed variable: it is determined by the coupled evolution of dislocation density, effective temperature, and thermal temperature, and the same framework explains yielding transitions and adiabatic shear banding (Le et al., 2017).

A different but related line of work is the shear-transformation-zone theory of athermal amorphous materials. There the effective disorder temperature is defined by

$\tau$2

and the steady plastic shear rate is

$\tau$3

Within that theory, shear stress is a generalized thermodynamic force conjugate to plastic shear strain in the configurational subsystem. The theory concludes that the yielding transition is not truly critical, because the steady-state effective temperature approaches a nonzero $\tau$4 as $\tau$5, so the density of flow defects and the correlation length remain finite at yield (Langer, 2015).

For sheared hard-sphere materials, the corresponding state variable is not an effective temperature but the compactivity

$\tau$6

and the STZ bias takes the form

$\tau$7

This recasts shear stress as a thermodynamic driving force normalized by the product $\tau$8, which plays the role of a configurational thermal scale (Lieou et al., 2012).

5. Interfacial shear tension and shear-dependent surface tension

At interfaces in complex fluids, the phrase acquires a second meaning: a shear-dependent interfacial free energy per unit area. In a sheared glass-forming fluid, the standard pressure-drop route to surface tension yields an effective quantity that mixes bulk and interface contributions. The paper therefore separates the interfacial region from the bulk by using the pressure anisotropy $\tau$9, and defines a genuine shear-rate-dependent surface tension from the interfacial contribution alone. The resulting $\gamma$0 decreases quadratically with shear rate at low to intermediate $\gamma$1, whereas the naive whole-slab integral is strongly system-size dependent because it includes bulk normal-stress contributions (Heitmeier et al., 11 Mar 2025).

A distinct mechanism appears for surfactant-covered droplets under shear. In a phase-field model with order parameter $\gamma$2, surfactant concentration $\gamma$3, and polarization $\gamma$4, the interfacial free energy contains the term

$\gamma$5

At equilibrium, surfactant polarization aligns with the interface normal and lowers the interfacial free energy. Under tangential shear, the polarization tilts away from the normal, reducing that lowering and thereby increasing the effective surface tension. The thermodynamic definition is

$\gamma$6

and the perturbative result is

$\gamma$7

In this model, shear modifies surface tension through reorientation rather than through bulk normal-stress contamination (Hardy et al., 28 Mar 2026).

These two results show that “thermodynamic shear tension” at interfaces is not sign-definite across systems. In one case, genuine interfacial tension decreases with shear rate once bulk contamination is removed; in the other, shear increases the effective surface tension by tilting anisotropic surfactants away from the interface normal. A plausible implication is that the sign and magnitude of a shear-dependent interfacial tension are controlled by the microscopic channel through which shear couples to the interfacial free-energy density rather than by shear alone (Heitmeier et al., 11 Mar 2025).

6. Steady-state thermodynamics and conceptual limits

In Onsager-style transport thermodynamics, shear strain can be treated as an extensive “distance” and shear stress as the conjugate intensive “field.” For equilibrium shear elasticity, the fixed-strain free energy $\gamma$8 gives

$\gamma$9

while the fixed-stress ensemble uses the Legendre-dual potential $Y=a/r_+$00. In this formulation, shear stress is explicitly an intensive thermodynamic variable conjugate to strain, and strain fluctuations at fixed stress determine the reciprocal shear modulus (Palmer et al., 2016).

Steady-state thermodynamics of an ideal gas in Couette flow provides a more restrictive case. The stationary shear stress is

$Y=a/r_+$01

and the total energy balance separates volumetric work, excess heat, and excess wall work,

$Y=a/r_+$02

The resulting steady-state fundamental relation is written in terms of $Y=a/r_+$03, $Y=a/r_+$04, and a nonequilibrium entropy $Y=a/r_+$05, not in terms of an explicit shear state variable. The closest analogue of thermodynamic shear tension in that construction is therefore the mechanical shear stress together with the excess wall-work term, rather than a dedicated conjugate pair embedded in the state space (Makuch et al., 2023).

Experimental work on mesoscale liquids under oscillatory shear pushes this boundary further. In glycerol and PPG-4000, low-frequency shear produces synchronized thermal bands with amplitudes up to $Y=a/r_+$06, and the temperature variation is approximately linear in the strain amplitude above a threshold. The paper interprets this as a thermo-mechanical coupling reminiscent of thermoelasticity, with reversible conversion between shear deformation and spatially structured temperature variations rather than ordinary viscous heating. This suggests, but does not by itself establish, a thermodynamic shear variable for liquids at the mesoscale (Kume et al., 2020).

Taken together, these formulations show that thermodynamic shear tension is best understood as a family of conjugate constructions rather than a single invariant object. In black-hole thermodynamics it is a new state variable $Y=a/r_+$07 conjugate to geometric eccentricity; in stressed solids it is the ordinary shear stress $Y=a/r_+$08 conjugate to $Y=a/r_+$09; at coherent interfaces it is encoded in excess shear conjugate to $Y=a/r_+$10; and in driven complex fluids it can appear as a shear-modified interfacial free energy. The common element is the admission of shear into a thermodynamic differential structure, but the precise conjugate variable depends on the geometry, the state space, and the admissible reversible work channels (Campos, 22 Mar 2026).