Excess Solid Stress: Theory and Applications

• Excess solid stress is the excess force localized at interfaces, defined by subtracting bulk contributions and governing in-plane tensions.
• It manifests in soft solid–liquid and coherent solid–solid interfaces via Shuttleworth-type relations and other excess quantifications.
• Experimental and imaging studies, including in soft gels and glioma, validate its role in elastic deformation, adhesion, and strain-induced biomarker behavior.

Excess solid stress denotes a stress-like excess quantity localized at a solid interface or defect after subtraction of bulk contributions, and it governs the in-plane forces transmitted by surfaces, solid–liquid interfaces, coherent solid–solid interfaces, and atomic steps. In soft matter, it directly controls capillarity-induced elastic deformation and contact-line force balance; in interfacial thermodynamics it appears through Shuttleworth-type relations linking interfacial free energy to strain; and in a distinct recent biomedical usage it denotes an imaging-derived stress proxy in glioma defined from strain and stiffness contrast rather than an interfacial excess in the Gibbs sense (Weijs et al., 2013, Xu et al., 2017, Frolov et al., 2013, Jaitner et al., 15 Sep 2025).

1. Definitions and thermodynamic basis

For soft solid–liquid interfaces, the Gibbs excess framework defines surface stress as the integral of the interfacial excess traction relative to the bulk value. For a planar interface with normal along $z$,

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$

where $\sigma_{ij}(z)$ is the local stress and $\sigma_{ij}^{\text{bulk}}$ is the stress far from the interface. In this form, $\Upsilon_{ij}$ is the integrated lateral force density that the interfacial zone exerts on the adjoining bulk, and it therefore directly enters the mechanics of soft wetting and elastocapillary deformation (Weijs et al., 2013).

A central distinction between solids and liquids is that solid surface stress need not equal surface free energy. For solids, the Shuttleworth relation states

$$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$

and for isotropic surfaces,

$\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$

The excess contribution is the strain-dependent term $\partial \gamma/\partial \varepsilon^s_{ij}$, or $d\gamma/d\varepsilon$ in the isotropic scalar form. This term vanishes for simple liquids, for which $\Upsilon_{ij}=\gamma\delta_{ij}$, but not for elastic solids because the surface can store and release energy under in-plane strain (Xu et al., 2017).

The same thermodynamic structure reappears for coherent solid–solid interfaces. Frolov and Mishin define the interface stress tensor as an excess over bulk stresses,

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$0

with the specific excess depending on the chosen reference extensive properties $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$1 and $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$2. Their generalized Shuttleworth-type equation is

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$3

In this setting, excess solid stress is explicitly a thermodynamic excess conjugate to in-plane elastic strain at the interface rather than a simple local stress concentration (Frolov et al., 2013).

2. Soft solid–liquid interfaces and capillarity

The microscopic model developed for amorphous soft solids combines enthalpic interactions and entropic elasticity in a density functional theory with a sharp-interface approximation. Molecular interactions are split into short-range repulsion and long-range attraction: the repulsion contributes an isotropic contact pressure in liquids or an elastic stress tensor in solids, while attraction is represented by effective potentials $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$4 and their planar integrals $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$5. These quantities recover the classical adhesion and surface energies through

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$6

Mechanical equilibrium is written microscopically as

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$7

and for amorphous soft solids the constitutive law is small-strain isotropic Hooke’s law (Weijs et al., 2013).

Within that framework, liquid–vapor and liquid–liquid interfaces satisfy $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$8, but a solid–liquid interface does not in general. Under plane strain parallel to the interface, the in-plane elastic stress is tied to the normal elastic stress through

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$9

so the local Poisson ratio $\sigma_{ij}(z)$0 of the interfacial solid layer controls the difference between stress and energy. The resulting closed-form expression for a solid–liquid interface is

$\sigma_{ij}(z)$1

with excess

$\sigma_{ij}(z)$2

The bracketed term is the work of adhesion between the solid and the liquid. Hence $\sigma_{ij}(z)$3 is controlled by interfacial compressibility and vanishes in the incompressible limit $\sigma_{ij}(z)$4 (Weijs et al., 2013).

This has immediate consequences for wetting. The elastocapillary length,

$\sigma_{ij}(z)$5

sets the scale of capillarity-induced deformation, but when $\sigma_{ij}(z)$6 it is the surface stress $\sigma_{ij}(z)$7 that enters the traction boundary condition. At a three-phase contact line on a soft substrate, the transmitted tangential elastic force per unit length becomes

$\sigma_{ij}(z)$8

For $\sigma_{ij}(z)$9, $\sigma_{ij}^{\text{bulk}}$0, recovering a liquid-like Neumann construction; for $\sigma_{ij}^{\text{bulk}}$1, a finite tangential traction appears and modifies wetting ridge geometry (Weijs et al., 2013).

The quantitative comparison reported in that work links theory to both simulation and experiment. For a lattice solid with springs yielding $\sigma_{ij}^{\text{bulk}}$2 and $\sigma_{ij}^{\text{bulk}}$3, the predicted $\sigma_{ij}^{\text{bulk}}$4 agrees with measured values $\sigma_{ij}^{\text{bulk}}$5. Experiments on soft elastomeric wires likewise showed significant tangential strain near contact lines, consistent with a compressible interfacial zone even when the bulk material behaves nearly incompressibly (Weijs et al., 2013).

3. Direct measurement and strain-dependent surface stress in soft gels

Direct measurement of strain-dependent solid surface stress in silicone gels established a linear constitutive form over accessible surface strains,

$\sigma_{ij}^{\text{bulk}}$6

with $\sigma_{ij}^{\text{bulk}}$7 mN/m and $\sigma_{ij}^{\text{bulk}}$8 mN/m, valid up to $\sigma_{ij}^{\text{bulk}}$9. Here $\Upsilon_{ij}$0 is the dilatational surface elastic modulus, and the measured ratio $\Upsilon_{ij}$1 shows that the excess term can greatly exceed the zero-strain surface stress. The surface stress increases by about $\Upsilon_{ij}$2 at $\Upsilon_{ij}$3 strain and doubles with only $\Upsilon_{ij}$4 strain (Xu et al., 2017).

The measurement protocol relied on wetting ridges under large glycerol droplets on biaxially stretched silicone gels. The macroscopic contact angle remained $\Upsilon_{ij}$5 and was unchanged by stretching, indicating $\Upsilon_{ij}$6 and negligible hysteresis, while the microscopic ridge opening angle $\Upsilon_{ij}$7 changed strongly with strain. In the symmetric case the local force balance is

$\Upsilon_{ij}$8

with $\Upsilon_{ij}$9 mN/m. This relation yields $$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$0 directly from the ridge geometry, independent of bulk elasticity. At zero applied strain, $$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$1 and $$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$2 mN/m because the local strain at the ridge is already $$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$3; at $$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$4, $$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$5 and $$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$6 mN/m (Xu et al., 2017).

These experiments also established the mechanical boundary condition for a solid free surface with surface stress,

$$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$7

which reduces to

$$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$8

for isotropic homogeneous $$\tau_{ij} = \gamma \delta_{ij} + \frac{\partial \gamma}{\partial \varepsilon^s_{ij}},$$9, and to

$\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$0

when $\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$1 varies spatially. The measured increase in $\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$2 with strain directly enlarges the elastocapillary length $\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$3, extending the capillary-dominated near field around contact lines (Xu et al., 2017).

A complementary adhesion study connected this excess stress to contact stiffness. In quasi-static pull-off experiments between untreated rigid silica spheres of radius $\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$4 $\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$5m and compliant PDMS gels with $\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$6 kPa and $\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$7, the zero-strain surface stress was $\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$8 N/m and the directly measured surface modulus on similar gels satisfied $\Upsilon = \gamma + \frac{d\gamma}{d\varepsilon}.$9. The zero-strain elastocapillary length was

$\partial \gamma/\partial \varepsilon^s_{ij}$0

Near contact, the geometry exhibited capillary-like features consistent with this scale (Jensen et al., 2017).

In that contact problem, the measured adhesive force was represented as

$\partial \gamma/\partial \varepsilon^s_{ij}$1

where the circumferential surface-stress contribution is

$\partial \gamma/\partial \varepsilon^s_{ij}$2

and under total wetting,

$\partial \gamma/\partial \varepsilon^s_{ij}$3

With fixed $\partial \gamma/\partial \varepsilon^s_{ij}$4, the capillary term increases the initial force but adds no stiffness. The missing stiffness arises from the excess, strain-dependent part of the surface stress: $\partial \gamma/\partial \varepsilon^s_{ij}$5 Using the experimentally observed exponential growth of the constant-curvature domain $\partial \gamma/\partial \varepsilon^s_{ij}$6, the study approximated $\partial \gamma/\partial \varepsilon^s_{ij}$7, leading to a nonzero $\partial \gamma/\partial \varepsilon^s_{ij}$8 that was required to match the measured initial stiffness $\partial \gamma/\partial \varepsilon^s_{ij}$9 (Jensen et al., 2017).

The two gel studies together established a common point. Macroscopic observables such as Young–Dupré contact angles report surface energies, but microscopic ridge angles, near-contact curvature, and contact stiffness are sensitive to the strain-dependent excess component of solid surface stress. Neglecting $d\gamma/d\varepsilon$0 therefore removes a mechanically important restoring force (Xu et al., 2017, Jensen et al., 2017).

4. Coherent solid–solid interfaces under nonhydrostatic stress

For coherent solid–solid interfaces, excess solid stress is formulated within a fully thermodynamic theory for multicomponent solids under nonhydrostatic mechanical stresses. The system comprises two homogeneous phases, $d\gamma/d\varepsilon$1 and $d\gamma/d\varepsilon$2, separated by a plane coherent interface with unit normal along $d\gamma/d\varepsilon$3. Coherency means that the same lattice penetrates both phases, lattice sites are conserved during interface migration, and sliding is prohibited. Because coherent interfaces can sustain shear stresses parallel to the interface, the thermodynamics contains not only interface free energy and interface stress but also interface excess shear (Frolov et al., 2013).

All interface excess quantities are defined through Cahn’s generalized excess method rather than a geometric dividing surface. For any extensive property $d\gamma/d\varepsilon$4,

$d\gamma/d\varepsilon$5

which ensures $d\gamma/d\varepsilon$6. The interface free energy is then written as an excess quantity, and the full adsorption equation contains thermal, chemical, normal-stress, shear-stress, and in-plane strain terms (Frolov et al., 2013).

In small-strain form, the adsorption equation is

$d\gamma/d\varepsilon$7

This identifies the interface stress tensor $d\gamma/d\varepsilon$8 as the quantity thermodynamically conjugate to in-plane elastic strain, while the excess shear measures $d\gamma/d\varepsilon$9 and $\Upsilon_{ij}=\gamma\delta_{ij}$0 are conjugate to $\Upsilon_{ij}=\gamma\delta_{ij}$1 and $\Upsilon_{ij}=\gamma\delta_{ij}$2 (Frolov et al., 2013).

A distinctive feature of this formalism is nonuniqueness. Because the generalized excess $\Upsilon_{ij}=\gamma\delta_{ij}$3 depends on the chosen reference pair $\Upsilon_{ij}=\gamma\delta_{ij}$4, the numerical value of $\Upsilon_{ij}=\gamma\delta_{ij}$5 also depends on that choice. Frolov and Mishin interpret this as path dependence on the coherent phase coexistence hypersurface rather than as a failure of definition. Under hydrostatic stress, the interface stress reduces to the unique form

$\Upsilon_{ij}=\gamma\delta_{ij}$6

The theory also yields Maxwell relations that couple interface stress to segregation, excess volume, excess shear, temperature, and nonhydrostatic loading (Frolov et al., 2013).

This coherent-interface formulation clarifies a common misconception inherited from liquid interfaces. For coherent solids, interface stress is not simply a scalar tension attached to area creation; it is a tensorial excess whose value depends on strain state, composition variables, and the constrained thermodynamic path used to define the excess (Frolov et al., 2013).

5. Step excess stress at faceted crystalline surfaces

At atomic steps on faceted crystalline surfaces, excess solid stress appears as the line-defect analog of surface stress. The thermodynamic theory of steps on faceted surfaces defines the step free energy per unit length $\Upsilon_{ij}=\gamma\delta_{ij}$7 through an Euler relation for a step-containing region and expresses it in terms of step excess quantities using Cahn’s determinant formalism. The resulting step adsorption equation is

$\Upsilon_{ij}=\gamma\delta_{ij}$8

where $\Upsilon_{ij}=\gamma\delta_{ij}$9 is parallel to the step and $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$00 is perpendicular to it (Freitas et al., 2017).

The corresponding step excess stress tensor is

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$01

with Shuttleworth-like relation

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$02

For the practical dividing-line choice $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$03, $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$04, the stress excess simplifies to

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$05

Thus, as for surfaces and coherent interfaces, the excess stress is distinct from the free energy itself and is intrinsically strain dependent (Freitas et al., 2017).

Atomistic calculations for $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$06 steps on Cu(111) using the Mishin EAM potential show that both excess energy and excess step stress have weak temperature dependence up to a homologous temperature of approximately $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$07, above which they increase strongly and the step stress becomes more isotropic. At $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$08 K,

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$09

so the low-temperature step stress is anisotropic, being compressive perpendicular to the step and tensile parallel to it. The step free energy decreases from $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$10 meV/ at $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$11 K to $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$12 meV/ at the melting point $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$13 K, but remains finite up to $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$14, indicating the absence of a roughening temperature for this $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$15 facet in the model (Freitas et al., 2017).

This line-defect formalism extends the meaning of excess solid stress from two-dimensional interfaces to one-dimensional defects. It also shows that the excess-stress concept is not restricted to soft matter or continuum interfaces: it remains well defined in atomistic crystalline thermodynamics through appropriate excess constructions (Freitas et al., 2017).

6. Surface-layer mechanics and generalized Young–Laplace equations for solids

A different but related reconstruction of excess solid stress arises when the surface of a solid is modeled as a finite-thickness layer with membrane and transverse shear responses. In Zaixing Huang’s formulation, the natural boundary condition derived from a bulk-plus-surface Lagrangian reduces in quasi-static conditions to

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$16

or in tensor notation,

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$17

Here $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$18 is a curvature-renormalized surface stress, with $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$19 the Tolman length and $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$20 the mean curvature (Huang, 2018).

The surface stress is decomposed into membrane and transverse shear parts,

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$21

where $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$22 is the in-plane membrane stress tensor and $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$23 is the surface transverse shear stress. Projecting the traction balance gives tangential and normal equilibrium: $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$24

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$25

These equations generalize the liquid Young–Laplace equation by including in-plane shearing and transverse shearing within the surface layer (Huang, 2018).

For the intrinsic constitutive content of the surface layer, the paper adopts a Shuttleworth–Herring-type membrane law,

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$26

and, for the purely intrinsic part without explicit in-plane strain,

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$27

The intrinsic transverse shear stress is proposed as

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$28

so it depends on the gradient of the inverse Gaussian curvature and a strain-dependent coefficient. This term vanishes on a sphere because $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$29 is constant, and then the normal balance reduces to a Tolman-corrected Laplace form; if also $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$30, the classical liquid result $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$31 is recovered (Huang, 2018).

In this framework, excess solid stress refers to the residual bulk stress induced by intrinsic surface membrane stress and transverse shear. Even without external loading, nonzero surface tractions generated by $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$32 must be balanced by internal stresses satisfying $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$33. The formulation therefore shifts emphasis from excess as a purely thermodynamic subtraction to excess as a surface-originated loading that generates residual stress fields in the interior of a solid (Huang, 2018).

7. Imaging-derived excess solid stress in glioma

In the glioma study, “excess solid stress” is defined operationally rather than as an interfacial excess. The deformation field $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$34 maps undeformed coordinates $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$35 to $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$36, with deformation gradient

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$37

and Green–Lagrange strain

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$38

Two regional strain measures are used: volumetric strain $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$39 and octahedral shear strain

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$40

The shear modulus is derived from magnetic resonance elastography as

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$41

with tissue density $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$42 kg/m$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$43, and the stiffness differential is

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$44

where $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$45 is the modulus of apparently unaffected brain and $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$46 that of peritumoral tissue (Jaitner et al., 15 Sep 2025).

Excess solid stress is then defined as the product of strain and stiffness differential: $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$47 These quantities have units of Pa. In the same study, volumetric stress and shear stress are estimated from an isotropic constitutive law,

$$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$48

with $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$49 used for volumetric stress computation (Jaitner et al., 15 Sep 2025).

The measured deformation patterns in patients were spatially heterogeneous and extended well beyond visible tumor margins. Non-tumor brain displacement was $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$50 mm in glioma patients versus $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$51 mm in healthy volunteers. Peritumoral volumetric strain was $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$52, unaffected-brain volumetric strain in patients was $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$53, and healthy-volunteer volumetric strain was $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$54. Peritumoral OSS was $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$55, compared with $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$56 in unaffected brain and $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$57 in healthy volunteers (Jaitner et al., 15 Sep 2025).

The principal clinical result concerned survival. Excess volumetric stress had mean $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$58 Pa, range $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$59 Pa to $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$60 Pa, and showed a linear correlation with survival days with slope $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$61 Pa/day, $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$62, and $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$63. Excess shear stress had mean $$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$64 Pa and weaker prognostic value ($$\Upsilon_{ij}=\int_{-\infty}^{\infty}\left[\sigma_{ij}(z)-\sigma_{ij}^{\text{bulk}}\right]\,dz,$$65). Neither deformation magnitude alone, tumor size, nor any single-region stiffness parameter correlated with survival. In this biomedical usage, excess solid stress is therefore an imaging-derived biomarker that integrates cause (strain) and consequence (stiffness degradation), rather than a surface or interface thermodynamic excess (Jaitner et al., 15 Sep 2025).

Across these literatures, excess solid stress consistently signifies a contribution that cannot be reduced to bulk elasticity alone. In soft interfaces it is the strain-dependent increment beyond surface energy; in coherent interfaces it is a generalized excess conjugate to in-plane strain; at steps it is a line excess linked to step free energy; in surface-layer mechanics it is the residual stress generated by intrinsic surface tractions; and in glioma imaging it is a regional stress proxy built from strain and stiffness contrast. The shared theme is that solids support mechanically consequential excess stresses that liquids, in their simplest description, do not.