Dilatational Strain-Gradient Elasticity

• Dilatational strain-gradient elasticity is a theory that isolates the volumetric component of strain, penalizing its spatial gradients to capture key size-dependent effects.
• It employs both scalar and tensorial formulations, distinguishing pure dilatational stiffness from distortion-gradient coupling through advanced isotropic moduli.
• The framework extends to electromechanical systems and wave propagation, influencing flexoelectric behavior, nonlocal homogenization, and defect regularization.

Dilatational strain-gradient elasticity is the sector of strain-gradient elasticity in which the energetic penalty is attached specifically to spatial variation of dilatation, that is, the spherical or volumetric part of strain, rather than to the full strain-gradient tensor indiscriminately. In reduced descriptions this appears as a longitudinal, axial, or bending-induced gradient mechanism; in tensorial formulations it is represented by gradients of the trace of strain or, in nonlinear form, by gradients of local volume change. Across the literature, the concept appears in several closely related guises: as a distinct isotropic mode in two-dimensional representation theory, as a nonlinear isotropic three-dimensional stored-energy term involving $\mathrm{Grad}(\det \boldsymbol F)$, as an electromechanically renormalized bending stiffness in flexoelectric insulators, and as a higher-order regularization mechanism in beams, shells, surfaces, waves, and point-defect mechanics (Auffray, 2013, Balitactac et al., 7 Aug 2025, Harbola et al., 2020, Stengel, 2016).

1. Kinematic definition and field content

The general small-strain strain-gradient setting augments classical elasticity by admitting not only strain but also its gradient as constitutive variables. In a basic scalar form used for bending of thin SrTiO$_3$ membranes, the Gibbs free-energy density is written as

$$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$

so that ordinary elastic storage and strain-gradient storage coexist, with the second term becoming important when strain gradients are geometrically enhanced, as in bending where gradients scale as $1/t$ with thickness $t$ (Harbola et al., 2020).

In a fully tensorial small-strain formulation, the kinematic variables are the infinitesimal strain tensor $\varepsilon_{ij}$ and the strain-gradient tensor $\eta_{ijk}=\varepsilon_{ij,k}$, with Cauchy stress $\sigma_{ij}$ and hyperstress $\tau_{ijk}$ as work-conjugate quantities. For centrosymmetric media the fifth-order coupling tensor vanishes, and the constitutive law reduces to an uncoupled form between ordinary elasticity and strain-gradient elasticity (Auffray, 2013). In thermodynamic formulations with internal variables, the state space is enlarged further to include the strain gradient $g_{ij,k}$, a tensorial internal variable $_3$0, and its gradient $_3$1, with higher-order stress $_3$2 conjugate to the strain gradient (Ván, 2020).

The specifically dilatational content becomes explicit when the spherical part of strain is isolated. In the two-dimensional harmonic decomposition of $_3$3, the vector

$_3$4

represents the gradient of the spherical part of strain, while the remaining sector $_3$5 describes distortion-gradient effects (Auffray, 2013). In nonlinear isotropic three-dimensional theory, a more selective volumetric measure is used: $_3$6 so that only gradients of local volume change are penalized, rather than the full $_3$7 (Balitactac et al., 7 Aug 2025).

Reduced theories preserve the same distinction in lower dimensions. In a geometrically exact beam, the axial stretch measure

$_3$8

is supplemented by its derivative $_3$9, and the corresponding higher-order axial force is $$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$0. In that setting, $$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$1 is the beam analogue most closely associated with dilatational strain-gradient elasticity, while curvature-gradient effects remain a distinct flexural mechanism (Epstein et al., 2022).

2. Isotropic constitutive structure and dilatational moduli

A central result of isotropic two-dimensional strain-gradient elasticity is that the theory is not governed by a single higher-gradient modulus. Instead, the isotropic sixth-order tensor is represented by four independent moduli, and the physically meaningful set depends on the decomposition of $$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$2 that is chosen (Auffray, 2013).

For the decomposition tailored to distortion-gradient and dilatation-gradient mechanisms, the isotropic operator takes the block form

$$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$3

with

$$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$4

$$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$5

Here $$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$6 is the pure dilatation-gradient stiffness, and $$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$7 is the coupling modulus between the dilatation-gradient and the vector part of the distortion-gradient (Auffray, 2013).

This structure implies that isotropic dilatational strain-gradient elasticity generally remains coupled to non-volumetric mechanisms. A pure dilatation-gradient material exists only under the restrictive condition

$$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$8

which the paper associates with the solid analogue of a Cahn–Hilliard fluid (Auffray, 2013). A related consequence is that spectral decomposition alone does not isolate the dilatational mode: the repeated vector irreducible representations permit coupling unless a basis adapted to $$\phi = \frac{1}{2}E\varepsilon^2 + \frac{1}{2}K(\nabla \varepsilon)^2,$$9 is used.

In three-dimensional isotropic formulations, the distinction between full and dilatational strain-gradient elasticity is equally sharp. The Toupin–Mindlin quadratic form contains all isotropic invariants of $1/t$0, including $1/t$1, $1/t$2, and full gradient contractions. By contrast, the dilatational model retains only the gradient of dilatation, equivalently only the gradient of volume change (Balitactac et al., 7 Aug 2025).

Setting	Dilatational quantity	Representative constitutive content
2D isotropic SGE	$1/t$3	modulus $1/t$4 and coupling $1/t$5 (Auffray, 2013)
3D nonlinear isotropic model	$1/t$6	$1/t$7 (Balitactac et al., 7 Aug 2025)
Geometrically exact beam	$1/t$8	$1/t$9 (Epstein et al., 2022)
Thin-plate bending scalar model	$t$0 across thickness	$t$1 (Harbola et al., 2020)

The reduced scalar and one-dimensional forms therefore should not be read as complete tensor classifications. They are longitudinal or volumetric analogues of dilatational strain-gradient elasticity, whereas the full tensorial theories show that isotropy still permits multiple gradient mechanisms and their coupling (Auffray, 2013, Epstein et al., 2022).

3. Electromechanical and thermodynamic interpretations

In insulators, dilatational strain-gradient elasticity cannot generally be treated as a purely mechanical higher-order correction. A first-principles formulation of flexoelectricity and strain-gradient elasticity shows that the two are intertwined at the same order in gradients. The continuum Lagrangian contains ordinary elasticity, a quadratic strain-gradient term

$t$2

flexoelectric polarization

$t$3

and the Maxwell energy of the induced longitudinal field

$t$4

A central conclusion is that the SGE and flexoelectric contributions are separately reference-dependent, whereas their sum is gauge invariant; for longitudinal or dilatational deformations, only the combined electromechanical stiffness is unambiguously defined (Stengel, 2016).

The experimental case of freely suspended SrTiO$t$5 membrane drumheads makes this point concrete. There the effective modulus inferred from predominantly bending deformation follows

$t$6

because bending generates large strain gradients across the thickness, whereas stretching-dominated response does not receive the same enhancement. The authors interpret the resulting stiffening primarily as a flexoelectric-origin strain-gradient elasticity: bending creates a strain gradient, the gradient induces polarization $t$7 under zero applied field, the polarization carries electrostatic energy $t$8, and this energy appears mechanically as an effective SGE term with

$t$9

Experimentally, the linear bending modulus is three times larger than the stretching-dominated modulus for membranes thinner than 20 nm, and from the modulus mismatch at $\varepsilon_{ij}$0–14 nm the extracted coupling is $\varepsilon_{ij}$1 (Harbola et al., 2020).

A common misconception is therefore that a measured higher-order bending stiffness directly identifies an intrinsic lattice SGE constant. In the SrTiO$\varepsilon_{ij}$2 case, the principal interpretation is electromechanical, not purely mechanical, and surface piezoelectricity is mentioned as another mechanism that could contribute to an SGE-like term (Harbola et al., 2020). A related misconception is that a standalone longitudinal or dilatational SGE coefficient in an insulator is absolute; the gauge argument shows that this separation is incomplete unless the electrical reference is fixed (Stengel, 2016).

Thermodynamic internal-variable theory reaches a complementary conclusion from a different direction. In a weakly nonlocal Gibbs relation,

$\varepsilon_{ij}$3

higher-order stresses are thermodynamically conjugate to strain gradients, and gradient terms contribute to the stress even without dissipation. Under isotropy, second-order tensors are split into deviatoric symmetric, trace or spherical, and antisymmetric parts; the trace $\varepsilon_{ij}$4 is the volumetric strain. Restricting to purely volumetric deformations suppresses the deviatoric and antisymmetric sectors and yields a dilatational strain-gradient elasticity limit in which hydrostatic response depends on gradients of $\varepsilon_{ij}$5 and on the divergence of higher-order stresses (Ván, 2020).

4. Surfaces, beams, shells, and bending-dominated manifestations

Dilatational strain-gradient elasticity often becomes operationally visible through reduced-dimensional structures because thin geometries amplify strain gradients. In shell reduction from three-dimensional isotropic strain-gradient elasticity, the intrinsic length is assumed to scale with thickness, $\varepsilon_{ij}$6, and this forces the first consistent shell correction to appear at cubic order in thickness. For the nonlinear dilatational model, the reduced shell energy is

$\varepsilon_{ij}$7

The additional shell-scale penalty acts on the surface gradient of midsurface dilatation and on the trace of curvature mismatch, and the theory reduces to Koiter’s classical shell energy as $\varepsilon_{ij}$8 (Balitactac et al., 7 Aug 2025).

In fully nonlinear beam theory, the same logic appears through exact beam kinematics. The internal virtual work is

$\varepsilon_{ij}$9

with constitutive relations

$\eta_{ijk}=\varepsilon_{ij,k}$0

Here $\eta_{ijk}=\varepsilon_{ij,k}$1 is the axial stretch-gradient term and hence the beam-level dilatational mechanism, while $\eta_{ijk}=\varepsilon_{ij,k}$2 supplies an independent bending-gradient contribution. The gradient terms raise the differential order, introduce additional admissible end data, and generate stiffening and boundary-layer effects in large-deformation problems (Epstein et al., 2022).

Surface-gradient elasticity provides another reduced setting in which dilatational effects are expressed geometrically rather than through bulk volumetric strain. In a three-dimensional solid with a boundary surface $\eta_{ijk}=\varepsilon_{ij,k}$3, the total energy

$\eta_{ijk}=\varepsilon_{ij,k}$4

depends on surface stretching $\eta_{ijk}=\varepsilon_{ij,k}$5, relative normal curvature $\eta_{ijk}=\varepsilon_{ij,k}$6, and the covariant derivative $\eta_{ijk}=\varepsilon_{ij,k}$7. The new third-order tensor

$\eta_{ijk}=\varepsilon_{ij,k}$8

measures the rate of stretching of convected geodesics, or geodesic distortion. In the paper’s own interpretation, this is why the model is called, in effect, a dilatational strain-gradient elasticity: it penalizes localized stretching variation along geodesics rather than curvature alone (Rodriguez, 2023).

That added dependence on $\eta_{ijk}=\varepsilon_{ij,k}$9 is decisive in fracture. For anti-plane shear, Steigmann–Ogden surface elasticity loses its curvature regularization and singular crack-tip fields persist. With strain-gradient surface elasticity, the boundary law acquires the fourth-order term $\sigma_{ij}$0, the crack-opening problem becomes an integro-differential equation, and the solution is proved to be unique and classical, with finite stresses and strains at the crack tips (Rodriguez, 2023). This suggests that surface-level dilatational-gradient resistance can regularize loading modes that are inaccessible to curvature-only surface theories.

The SrTiO$\sigma_{ij}$1 membrane measurements supply the experimental analogue of these reductions. The non-monotonic thickness dependence of the bending-dominated modulus, the minimum near $\sigma_{ij}$2 nm, the sharp rise below that thickness, and the factor-of-three separation between bending and stretching moduli below about 20 nm all show that bending geometries expose higher-gradient effects that are effectively hidden in uniform stretching (Harbola et al., 2020).

5. Nonlocal, lattice, and homogenization viewpoints

One route to dilatational strain-gradient elasticity starts from nonlocal energy rather than nonlocal stress. For one-dimensional longitudinal waves, the potential and kinetic energy densities are averaged over finite horizons,

$\sigma_{ij}$3

and Taylor expansion generates local gradient terms. Retaining linear strain and linear velocity variations yields

$\sigma_{ij}$4

which is the first strain-gradient model with the identifications

$\sigma_{ij}$5

In this formulation, the microelastic and microinertia lengths are geometric moments of the nonlocal horizons, rather than abstract fitting parameters (Gortsas et al., 2022).

Discrete microstructural models supply a second derivation. A one-dimensional lattice with nearest-neighbor and next-nearest-neighbor interactions leads, in the continuum limit, to

$\sigma_{ij}$6

or equivalently to constitutive laws of the form

$\sigma_{ij}$7

Because the model is scalar and one-dimensional, it is best interpreted as a longitudinal or dilatational analogue rather than a full volumetric–deviatoric decomposition. Its significance is that the sign of the gradient term is not fixed a priori but is controlled by the competition between discrete couplings, especially next-nearest-neighbor interaction (Tarasov, 2015, Tarasov, 2015).

The homogenization problem shows a further complication. For a one-dimensional periodic heterogeneous bar whose microscale behavior is itself governed by strain-gradient elasticity, the overall effective behavior is not closed within the same strain-gradient class. The paper states explicitly that “the effective behavior of a heterogeneous strain gradient elastic medium is not described by a strain gradient elastic theory.” Instead, the effective constitutive law is kernel-based nonlocal elasticity,

$\sigma_{ij}$8

and over restricted scale ranges it may be approximated locally by a fractional strain-gradient model with order

$\sigma_{ij}$9

Accordingly, the average strain under dead traction obeys

$\tau_{ijk}$0

interpolating between classical elasticity at $\tau_{ijk}$1 and ordinary strain-gradient elasticity at $\tau_{ijk}$2 (Singh et al., 5 Jul 2025).

This non-closure under averaging is important for dilatational theories because it limits a common expectation: a microscopic volumetric-gradient mechanism need not survive coarse-graining as a local volumetric-gradient continuum with renormalized coefficients. A plausible implication is that dilatational strain-gradient elasticity is best viewed as one scale-dependent member of a broader hierarchy that includes local higher gradients, kernel nonlocality, and fractional approximations (Singh et al., 5 Jul 2025, Gortsas et al., 2022).

6. Waves, defects, and current operational domains

Wave propagation provides one of the clearest operational signatures of dilatational strain-gradient elasticity. In a centrosymmetric Mindlin-type continuum, the displacement equation

$\tau_{ijk}$3

leads to a generalized acoustic tensor and dispersive plane-wave propagation. For a square $\tau_{ijk}$4-symmetric lattice reduced to one dimension, the longitudinal and transverse branches decouple as

$\tau_{ijk}$5

and the corresponding phase velocities are

$\tau_{ijk}$6

Here the $\tau_{ijk}$7-wave branch is the dilatational response, with characteristic lengths

$\tau_{ijk}$8

The identified strain-gradient model remains accurate down to approximately $\tau_{ijk}$9, whereas the classical Cauchy homogenized model is accurate only down to about $g_{ij,k}$0 in the benchmark studied (Giuseppe et al., 2017).

Point-defect theory reveals another domain where dilatational gradient effects are structurally central. A dilatation centre is modeled as an isotropic eigendistortion

$g_{ij,k}$1

within gradient elasticity of bi-Helmholtz type. The strain-energy density

$g_{ij,k}$2

introduces two internal lengths and the bi-Helmholtz operator

$g_{ij,k}$3

Its Green function regularizes the classical defect core, so that displacement, first gradient, second gradient, eigendistortion, and stress become finite at the origin. The interaction energy between two dilatation centres becomes a finite bi-Yukawa potential, and the self-energy is finite as well (Lazar, 2019).

The current operational picture is therefore mixed. In nanoscale bending of high-permittivity insulators, dilatational-type strain-gradient effects can dominate the apparent modulus through electromechanical coupling (Harbola et al., 2020). In shells and beams they appear as higher-order penalties on midsurface dilatation and axial-stretch variation (Balitactac et al., 7 Aug 2025, Epstein et al., 2022). In fracture, surface dilatational-gradient resistance regularizes crack-tip fields that remain singular in curvature-only theories (Rodriguez, 2023). In waves, they extend the validity of continuum homogenization into shorter wavelengths (Giuseppe et al., 2017). Yet the same literature also shows two persistent limitations: in insulators the partition between mechanical and flexoelectric contributions is gauge-dependent (Stengel, 2016), and in heterogeneous media averaging may lead outside the strain-gradient class altogether (Singh et al., 5 Jul 2025).

Within that landscape, dilatational strain-gradient elasticity is best understood not as a single universal constitutive correction but as a family of higher-order volumetric or longitudinal mechanisms whose concrete realization depends on geometry, dimensional reduction, electromechanical environment, and scale of observation.