Low-dimensional structures, from atomically thin sheets to one-dimensional nanowires, have emerged as a cornerstone of modern science and technology, offering unprecedented opportunities to explore fundamental physics and engineer novel devices with tailored functionalities. Moreover, advances in manufacturing are enabling the production of such small-scale structures. A wide application of these small-scale structures requires the efficacy of design and simulation. Examples include micro-beams utilized in MEMS (Micro-Electro-Mechanical Systems) as actuators [1] and sensors [2], where the beam thickness is usually of the order of microns. Further, many of these applications employ thin films for specific performance needs [3,4].

Classical continuum mechanics is based on the assumption that material behavior can be described using only the first gradient of displacement. Although effective at larger scales, this framework is inadequate for micro- and nanoscales, where size effects on elastic response become prominent. Experimental investigations using high-precision instruments, such as nanoindenters, have confirmed that classical theories cannot accurately capture the elastic response at these small scales [5,6]. Lam et al. [6] reported increased bending rigidity in experiments on epoxy micro-beams, while McFarland and Colton [7] observed similar stiffening behavior in micro-plates. At low dimensions, the structural response shows significant deviations from the predictions of classical elasticity theory. A key limitation of classical continuum theories is their inability to incorporate a material length scale that reflects the influence of microstructure. This omission makes them inadequate for problems in which the microstructure significantly affects the mechanical behavior. Both experimental and theoretical studies have shown that incorporating such length scales is essential to accurately capture the influence of microstructure on material behavior [8–12]. Further, classical continuum mechanics cannot eliminate singularities that arise from point loads, crack tips, dislocation lines, or interface discontinuities, making such problems mathematically ill-posed [13]. To address these limitations, Eringen [14,15] introduced microcontinuum field theories as an extension of classical models, tailored for materials characterized by microscale dimensions and dynamic behavior over short time scales.

In these microcontinuum theories, materials are modeled not as a collection of point-like particles, as in classical mechanics, but as assemblies of deformable entities with inherent orientation. A conceptual example often used to illustrate this framework is the crystal lattice of cesium chloride (CsCl), as shown in Fig. 1. Unlike classical elasticity, where a material point is rigid and has only translational degrees of freedom, this approach allows the material point to have three deformable directional vectors (directors). These directors add nine new degrees of freedom collectively referred to as micro-deformations to the three translational degrees of freedom (macro-deformations) of classical theories.

This approach to modeling structural response is relevant for materials and systems with internal microstructures, such as liquid crystals with dipolar molecules, polymers, solids with defects such as microcracks or dislocations, biological fluids with deformable particles (e.g., blood platelets) and composite materials [15,17–19]. Further, various types of microcontinuum theories have evolved depending on the nature of internal constraints applied to these deformable elements. The most general form, the micromorphic theory [14,20], includes all nine degrees of freedom of the microstructure. Simplified variants include the microstretch theory, which disallows shear deformations between directors, and the micropolar theory, which treats the directors as rigid. Together, these micromorphic, microstretch, and micropolar formulations are known as 3M microcontinuum theories [15]. Note that the 3M continuum theories introduce additional degrees of freedom to capture microstructural effects, resulting in a more complex extension of classical continuum mechanics. Alternatively, modeling of micro-deformations can also be explored by employing the gradients of macro-deformation variables to reduce the number of degrees of freedom [21]. In detail, Mindlin [21] addressed the effects of size by formulating higher-grade continuum models, where the mechanical behavior of solids is described using higher order gradients of existing displacement field variables. Thus, in gradient elasticity, the strain energy function involves both the strain terms and their gradients, resulting in stresses that depend on higher-order derivatives of the displacement field. For detailed historical insights, the reader may refer to Mindlin [22].

The origins of this strain gradient elasticity approach can be traced back to Cauchy and Augustin [23], who proposed an infinite series representation of constitutive relations for isotropic materials with periodic crystal structure, and Voigt [24] further analyzed the kinematic constitutive equations of discrete lattice models, incorporating molecular displacements and rotations. In the early twentieth century, the Cosserat brothers [25] advanced a generalized continuum theory, introducing enriched kinematics in which material particles possess both translational and rotational degrees of freedom. However, these developments largely went unnoticed until the 1960s, when interest in gradient continuum theories was significantly revived. Key contributions during this period include foundational works by Toupin [26], Kröner [27], Mindlin [21,28,29], and Green and Rivlin [30]. Building upon the seminal work by Mindlin [21] in connecting micro-deformations to macro-deformation gradients, Mindlin and Tiersten [28] simplified the couple stress theory by expressing it in terms of rotation gradient tensors derived from macroscopic displacement vectors. This approach can be viewed as a reduced form of the broader framework developed by Mindlin et al. [21,31], which considers all components of the higher-order displacement gradients. Variants of this general theory have been proposed depending on the specific macro-deformation tensors used to evaluate these gradients [31]. The initial work by Mindlin included only first-order strain gradients [31], but was later expanded to include second-order gradients [29]. This allowed for the introduction of the significant effects of surface elasticity at low dimensions. In the early 1970s, Germain [32,33] developed the equilibrium relations for the first-strain gradient elasticity theory, including the effect of microstructure via additional degrees of freedom, using the method of virtual power. A second resurgence in gradient continuum theories occurred in the 1980s, driven by Eringen [9]. This formulation proposes a gradient elasticity framework via reformulation of earlier work on the integral approach to nonlocal elasticity, replacing integral-type constitutive equations with differential equations, offering a more tractable framework for these theories. However, these are fundamentally different from the Strain gradient elasticity (SGE) framework proposed by Mindlin [31].

The seminal studies reported above by Mindlin et al. [21,29,31] paved the way for numerous simplified SGE frameworks. A substantial body of literature explores analytical and numerical techniques for the solution of governing differential equations (GDEs) for a simplified SGE framework. The challenge lies in these partial differential equations (PDEs) being of higher order compared to classical elasticity. The higher-order governing differential equations arising from these strain gradient formulations have been extensively solved through exact and analytical solutions [34–37], without imposing prior assumptions on the displacement field beyond the necessary continuity conditions in the structural domain. Further, given the higher-order partial differential equations for SGE, interpolation schemes with enhanced continuity are sought in numerical formulations [38,39]. To address this, various numerical methods have been developed, including mixed formulations [40], C1 continuous elements [41], and isogeometric analysis (IGA) [42].

Strain gradient theory offers significant advantages, particularly in modeling micro- and nano-scale structures where classical elasticity falls short. Firstly, the model incorporates length scale parameters to account for the deformation of the material microstructure. This enables an accurate description of the structural response at low dimensions, where the effects of microstructure are significant. Furthermore, unlike alternate higher-order theories (3M), SGE does not introduce additional degrees of freedom [31]. Finally, the constitutive framework of the SGE remains consistent with the classical elasticity even when the material length scale tends to zero, ensuring broad applicability. Its versatility extends to static, dynamic, and stability problems, making it a powerful tool in the design and analysis of advanced materials and structures.

In particular, the presence of strain gradients was theorized to present a novel multiphysics coupling that can be leveraged to develop interesting applications at low dimensions [43]. This opens exciting avenues for micro- and nano-architected smart structures [44,45]. Among these, the coupling between the electric field and the strain gradients was first studied theoretically in crystals nearly fifty years ago [46]. Much later to these theoretical advancements, Cross and colleagues conducted experimental investigations in various dielectric materials to experimentally demonstrate this behavior [47,48]. It has been observed that non-uniform strains in a general dielectric material can induce a non-zero electric field, a phenomenon known as the flexoelectric effect. These experiments often focused on deformation of structures in bending (flexure), which generates significant strain gradients; thereby drawing the name ‘flexoelectricity’. Of significant interest is the fact that a coupling involving strain gradients is observed to be universal, without restriction to limited (non-centrosymmetric for piezoelectric) crystal structures [43]. Subsequently, extensive research has been conducted to explore the flexoelectric effect in solids, leading to advances in the design of actuators, sensors, and energy harvesting devices based on this phenomenon. Analogously to the electro-mechanical coupling above considering strain gradients, its coupling with other field variables such as the magnetic [49] and thermal field [50] variables are also noted in the literature.

A detailed time history and a quantitative assessment of the literature produced in the general area of strain gradient theory are provided in Fig. 2. An exponential increase in the literature on SGE in the last 20 years indicates a growing interest in fundamental and applied research on strain gradients. In particular, research on strain gradients has been carried out predominantly in the domain of plasticity. Conventional plasticity theory lacks an intrinsic length scale in its constitutive laws and therefore cannot capture size-dependent effects. However, numerous plasticity phenomena exhibit clear size effects, where smaller structures show a stronger mechanical response [51–53]. For example, indentation hardness in metals and ceramics increases with decreasing indenter size, due to smaller reinforcing particles leading to higher rates of strengthening and work hardening in metals for the same volume fraction [54,55]. Although these effects may arise from different mechanisms, all require a length scale for proper interpretation. A common approach to capture such size effects is to make the yield stress a function of both the strain and the strain gradient [56–60]. In this article, we review the literature on strain gradients, highlighting their role on the elastic and multi-physics coupled response of low-dimensional structures, areas that have received less attention.

We begin this review article with a description of the constitutive models for SGE in Section 2. This includes a detailed derivation of the constitutive relations for strain gradient theory, including an overview of the simplified SGE models and the approaches used in the literature to estimate the corresponding material constants. This is followed by a discussion of the various analytical and numerical approaches (finite element (FE) and non-FE based) used to solve the governing differential equations for SGE in Section 3. Before proceeding to a discussion of the literature on applications of SGE and related multi-physics coupling, Section 4 discusses the physical significance of SGE. Subsequently, Sections 5 and 6 highlight the foundational and applied research on multiscale elastic and multiphysics coupling (flexo- electric, magnetic, and caloric effects) within the framework of strain gradient theories. Finally, Section 7 presents a review of alternate integral and integro-differential approaches to capture nonlocal effects beyond strain gradient elasticity. We conclude with some probable directions for future research on strain gradient elasticity.

The deformation energy density of an elastic continuum considering microstructural effects can be expressed as [15,21]: (1)𝒲(ε,η)=𝒲CT(ε)+𝒲SGE(η),where 𝒲CT(ε) is the classical (strain) deformation energy density which is a function of the first gradients of displacement (strain, ε) [61]. 𝒲SGE(η), expressed in terms of a higher-order deformation metric η, is an additional deformation energy metric that introduces size effects. The classical theories of elasticity can be recovered by ignoring the contribution of the latter term towards the complete deformation energy.

Above, we introduce the additional contribution in 𝒲SGE(η) by carefully choosing the higher-order deformation metric, η. To begin with, Eringen proposed “3M” microcontinuum theories to account for the energy associated with higher-order deformation metrics 𝒲SGE. This was achieved by introducing additional deformation variables, such as micro-rotations of the directors in a microstructure by Cosserat [25]. A schematic illustration of the deformable directors is provided in Fig. 1 with the example of a crystalline cesium chloride (CsCl) lattice structure. This was further extended by Mindlin in his seminal work [21] to develop a framework for incorporating the effects of microstructure via micro-rotations on mechanical behavior of the macro structure. Later, these studies were extended to include additional deformation variables such as micro-stretch by Eringen [14,15]. However, considering the complexity of introducing additional degrees of freedom, alternative simpler frameworks were developed. In these frameworks, η is expressed as higher-order gradients of conventional deformation measures. Examples include the couple stress theory [26,28] in which the micro-rotation variable is replaced by a rotation gradient tensor defined in terms of the second-order strain tensor.

Another example of a higher-order deformation metric is the first gradient of the second-order strain tensor proposed by Mindlin [31]. This approach incorporates the effects of the micro-structure within the constitutive relations via gradients of the classical (second-order) strain tensor. Thereby, no additional degrees of freedom are introduced within the constitutive model following the strain gradient approach to capture microstructure effects. It is worth noting that the set of higher-order metrics can be expanded further to include contributions from the second gradient of the strain tensor, enriching the total deformation energy density described in Eq. (1).

We present here the details of a constitutive model for the first strain gradient theory. More specifically, the constitutive relations for SGE considering the first derivative of strain gradients are detailed. Later, a discussion on the alternate forms for the first strain gradient theories (either as second gradients of displacements or first gradients of the strain tensor or their components) is provided. Subsequent to this, simplified forms of the SGE proposed in literature, following from the generalized form, are introduced, and details on their relevance to different studied are provided. We follow this with brief discussions of the characterization of the additional material constants introduced via SGE.

Before proceeding further, we note that an experimental observation of a non-zero polarization in centrosymmetric crystals [47,62] that cannot be explained by piezoelectric coupling. Moreover, when a strain gradient is applied across the symmetric crystal domain, the symmetry of the centrosymmetric crystal can be disrupted, resulting in net polarization upon deformation. This phenomenon is illustrated in Fig. 3. As shown, prior to deformation, the positive ion aligns with the centroid of the negative charges in the lattice, leading to zero net polarization. When a strain gradient is applied, the centroid of the negative charges shifts relative to the positive ion, and a net polarization is induced in the centrosymmetric crystal. This electro-mechanical coupling follows from the strain gradient referred to as the flexoelectric effect. This will be discussed in detail later in Section 6.

This section presents a concise overview of approaches for solving boundary value problems, including strain gradient terms within the constitutive relations (example is Eq. (26a)). We present a detailed account of the analytical and numerical methods explored in the literature for solving these equations. Before we begin, we note that Hosseini and Niiranen [136] established uniqueness of a solution for these higher-order GDEs, including SGE terms, assuming continuity and coercivity of the bilinear form (available thanks to the energy framework introduced in Eq. (1)). Although their proof is available for a 3D boundary-value problem (BVP) considering the SGE framework, these equations are similar to those developed previously in Eq. (26a)), and hence the observations can be extended. This result is significant because it provides a foundation for analytical and numerical methods, including the FE, meshfree, and IGA methods.

As discussed in Section 1, the origins of SGE go back to enriched kinematics for material particles possessing a microstructure. In particular, unlike classical elasticity theories that treat particles as rigid, the Cosserat brothers [25] introduced enriched kinematics in which material particles possess both translational and rotational degrees of freedom. These additional degrees of freedom correspond to the deformation of the microstructure of the particle; a schematic example of the same for CsCl is shown in Fig. 3. Subsequently, a continualization of the microstructure effects introduces higher-order gradients within the constitutive relations. More specifically, a variety of higher-order continuum theories have been introduced in Section 2, each incorporating different choice of higher-order gradients into the continuum framework to capture the specific effects of the microstructure. In particular, gradient elasticity formulations have initially been proposed to better approximate the effect of discrete lattice behavior in wave propagation studies [97,201]. The effect of lattice deformations over high-frequency (low-wavelength) wave propagation is studied using micro-deformations of the directors within the lattice (see Fig. 3) [21]. Among higher-order continuum theories, the strain gradient elasticity theory successfully models size-dependent behavior in small-scale structures. It introduces three material length scale parameters, each associated with dilatation, deviatoric stretch, and symmetric rotation gradients. This theory has been used to study the static and dynamic responses of microscale Bernoulli–Euler beams [144,202,203] and Timoshenko beams [204–206]. By setting two length scale parameters corresponding to the dilatation and deviatoric terms to zero, it simplifies to the modified couple stress theory by Yang et al. [65]. This highlights the specific applicability of the strain gradient elasticity theories to capture diverse deformation mechanisms of the microstructure. In all these cases, gradients are used to enhance the representation of the complex microstructure over the structural response. This approach allows gradient to be used to regularize the solution by reducing singularities or discontinuities and smoothen the heterogeneity of the material [56,57,120,207]. In other cases, gradients are used to introduce heterogeneity, enhancing the representation of the complex microstructure over the structural response.

Higher-order continuum models in the literature differ in several aspects, particularly in the manner of incorporating the strain gradient terms. Some examples of the same are discussed in Section 2.2 above. Some models introduce these gradients directly, either through the energy functional or the constitutive equations, following ad hoc approaches. Other models derive higher-order terms from discrete lattice structures using homogenization techniques. This approach preserves a connection with the microstructure of the material, allowing additional material parameters to be interpreted in terms of microscale properties. In this section, we propose to demonstrate the physical significance of the SGE following both of these approaches. More specifically, we derive the SGE terms within the deformation energy following from a 1D chain of spring mass representation for an axial bar. Thereby, we demonstrate the higher-order terms corresponding to long-range interactions, beyond immediate neighbors, within this discrete structure way the foundation for strain gradient elasticity. We conclude this derivation via a continualization of the discrete structure and illustrate the higher-order terms reducing to strain gradients. Thereafter, the higher-order gradients are also connected to the specific deformation modes in a complex microstrfollows.

An infinite one-dimensional lattice of identical particles, each with mass M, is considered, arranged in a collinear manner along the x-axis with a spatial period of le, as illustrated in Fig. 8. The particles are restricted to longitudinal motion. The position and displacement of the i-th particle (i∈R) at time t are denoted by xi(t) and ui(t), respectively. Interactions between particles are modeled using lumped springs with stiffness ki,j, where i and j represent the indices of interacting particles with i≠j. Here, the comma in the subscript indicates particle indices, not differentiation. Assuming all springs are initially unstressed at t=0, the potential energy stored in the i-th cell of the lattice corresponding to its interaction with j-th particle is given by: (29)𝒰i,j=12ki,j|ui−uj|2.

Assuming small displacement gradients (O(ε)), a Taylor series expansion about the point xi yields: (30)ui−uj=(xi−xj)Dxj1uj+12(xi−xj)2Dxj2uj+H.O.T.

The first term in the above expression caters to the interaction of point i with the immediate neighbors. The second-order terms introduced above expands the interaction to points beyond the immediate neighbors, resulting in a weakly nonlocal interactions being captured here. It is well established that cohesive forces between particles weaken with increasing interatomic distance. At the continuum level, particularly in strain gradient formulations, this behavior is typically represented through convolution terms involving power-law kernels of order one in the stress–strain constitutive relations. In lattice models, the stiffness of springs connecting distant particles serves a similar role to these kernels given as follow: (31)ki,j=k0[c1|xij|+c2|xij|2].

Here, |xij|=|xi−xj| denotes the distance between the i-th and j-th particles. It is important to note that the above equation is valid only for i≠j, as i=j would imply a particle is connected to itself, which is not physically meaningful. The coefficients c1 and c2 are defined as functions chosen to ensure both dimensional consistency and frame invariance of the formulation. Furthermore, the constant k0, which has the units of classical stiffness [MT−2], will be further interpreted when deriving the continuum limit of the lattice model. Substituting Eqs. (30) and (31) in Eq. (29). (32)𝒰i,j=k02[c1(Dxj1uj)2⏟strain+14c2(Dxj2uj)2⏟strain gradients].

By assuming a small le and applying a continualization process, the discrete variables representing the particle positions and displacements can be replaced with their continuum counterparts (xi→x). In the continuum limit, the constant k0 can be defined as: (33)k0=EAle.

Here, E and A represent the Young’s modulus and the cross-sectional area of the equivalent one-dimensional continuum, respectively. Thus, the constant k0 can be interpreted as the equivalent spring stiffness for an axial element characterizing the nearest-neighbor interaction forces in the lattice, which models the microstructure without considering size-effects. Furthermore, the constants c1 and c2 in Eq. (31) are defined as: (34)c1=le2,  c2=le4.

The above choice follows from scaling the contributions of strain and strain gradient terms towards the total deformation energy, and to maintain the dimensional consistency. Under these assumptions, the continuum limit of the discrete sum in Eq. (32) ∀ i,j can be expressed as the following strain gradient representation: (35)𝒰(x)=EA2l∗[le2(Dx1u)2+le44(Dxj2uj)2].

The deformation energy density at a point x in the continuum can be expressed as: (36)Π(x)=𝒰(x)Ale=E2[(Dx1u)2⏟strain+le24(Dxj2uj)2⏟strain~gradient].

The second term in the above integral represents the deformation energy density corresponding to strain gradient elasticity. The preceding discussion highlights that incorporating strain gradients into the continuum model enables it to be weakly nonlocal across the domain. Further, the above derivation is limited to axial deformations, but can be extended to more general deformation modes as well.

We proceed to interpret the strain gradient terms as (additional) micro-deformation variables [21,208]. This allows us to capture the deformation of the director vectors within the microstructure. In this discussion, the second-order tensor e describes the state of strain of the microstructure (similar to ε employed earlier). Furthermore, the third-order micro-displacement/deformation tensor, denoted by ϕ and having dimensions of length, is analogous to the hyperstress tensor defined in Eq. (7), and therefore physically related to the strain-gradient tensor (η). The physical/kinematic interpretation of these tensors in the context of strain gradient theory is schematically illustrated in Fig. 9. The micro-displacements ϕpij can be interpreted as the displacements of the free ends of a triplet of fibers of length ℓ emerging from a generic point x and aligned with the coordinate axes. These fibers are typically chosen along the deformable directors illustrated in Fig. 3. In particular, the micro-displacement components, defined as ϕpij=ℓ2eij,p (i,j,p=1,2,3), are similar to the hypertresses available following SSGET defined in Eq. (16). These components represent the configuration change of a triplet of orthogonal fibers of length ℓ aligned with the reference coordinate axes along the direction xp of the gradient. Therefore, for each p∈1,2,3, the component ϕpij describes the displacement of the free ends of the fibers in the (i,j) plane when i≠j, and the displacement of the fiber i along its own axis when i=j. These components collectively capture the local microstructural deformation using strain gradients.

Classical continuum theory is widely used to predict the elastic response in structural analysis and design. However, it fails to accurately predict structural behavior at the micro- and nanoscales. For instance, experimental studies on quasistatic bending of the human compact bone demonstrated a stiffening of up to a factor of two with reducing dimension, which cannot be accounted for following classical elasticity [12]. This is demonstrated to be better captured following the micropolar and couple stress (see Section 2.2) theories of elasticity. Similarly to this, Yang and Lakes repeated the tests on torsion response of the human compact bone specimen [11,12], and experimentally demonstrated the size-effects on torsional stiffness. More specifically, as the dimensions of structure decrease, the classical continuum theory tends to underestimate the structural stiffness [12,209]. To address these limitations of classical elasticity theory, higher-order elasticity theories that incorporate length scale parameters corresponding to the material microstructure have been proposed. These include Cosserat theory [25], couple stress theory [28], micromorphic theory [14], micropolar theory [210], and a family of strain gradient elasticity theories [26,29,31]. Among these, we focus here on the application of strain gradient theories for modeling the size-dependent elastic response.

Structural response: Several studies have explored gradient-based (higher-order) theories to improve the predictions of structural behavior at the micro- and nanoscales. Lam et al. [6] conducted early research in this area, observing an increase in the bending stiffness (compared to classical elasticity predictions) of epoxy microbeams as their thickness was reduced from 115 to 20 µm. Most recently, similar observations were observed from studies on the bending response of epoxy microbeams (thickness, h≈8−170µm) by Liebold and Müller [211], Fu et al. [212], and Choi et al. [213]. An illustration of the increase in bending stiffness with decreasing thickness is depicted in Fig. 10a. This observation is different from the constant (normalized) bending stiffness for different dimensions predicted by the classical continuum theory. The additional deformation modes corresponding to the strain gradients are proposed to capture the effect of microstructure on the elastic response of the low-dimensional structures. For instance, the size-dependent response of micro-cantilever beams noted earlier in [6] is captured via MSGT given in Eq. (13). Moreover, the effect of the microstructure, and thereby the additional deformation modes introduced by the higher-order gradients, is negligible for structures with dimensions atleast a couple of orders greater than the characteristic length scale corresponding to the microstructure. This is also reflected in Fig. 10, where the results of higher-order and classical theories agree with increasing thickness. Therefore, the efficacy of SGE theories is demonstrated to capture the effects of microstructure over structural behavior at low dimensions.

Recall the detailed comparison of the different higher-order strain gradient elasticity theories provided in 2.2. The simplified constitutive models differ according to the specific deformation modes they account for. For instance, the MCST proposed by [65] accounts for only the rotation gradients of the deformable directors (see Fig. 3). In contrast to this, the MSGT provided in Eq. (13) also considers the gradients of the stretch and deviatoric deformation modes of these directors within the microstructure. Finally, the GFSGET considers all the probable modes of deformation of the directors within the microstructure. This specific choice of the constitutive framework depends on the microstructure and the mechanics of its deformation. A comparison of the relative size effects realized following the GFSGET (see Eq. (11)), MSGT (Eq. (13)) and MCST over the bending response of an elastic beam is provided in Fig. 10b. Therefore, appropriate choice of the SGE theory must be chosen for modeling the structural response of low-dimensional structures composed of different material types.

Elastodynamics: The study of the propagation of elastic waves in solids is of immense theoretical and practical significance, offering valuable insights into the elastic behavior of materials and structures. Given the growing importance of micro- and nano-structures, extensive research has been conducted to analyze wave propagation in low-dimensional structures [214–218]. The effect of increasing stiffness with reducing dimensions is reflected in different forms in elastic behavior; for instance, with reference to the current context, an increase in natural frequencies with reducing microstructure dimensions [37,219]. In addition, a dispersion of elastic waves is another example of the effect of microstructure on the behavior of the material. Recall that the governing differential equation for strain gradient elasticity given in Eq. (26a). These GDEs can be further simplified using Navier’s solution. The resulting Navier’s governing equations for the first strain gradient theory of elasticity, assuming the body forces to be absent, may be expressed as [97]: (37)(λ+2μ)(1+lλ2∇2)∇(∇.u)−μ(1+lμ2∇2)∇×∇×u=ρu¨−ρld2∇2u¨,where the term ρld2 corresponds to micro-inertia, with ld2=d23 and the characteristics lengths lλ and lμ correspond to the effect of the strain gradients on the longitudinal and transverse waves, respectively. These length scales are given by: (38)lλ=a1+a2+a3+a4+a5λ+2μ,  lμ=a2+2a4+a54μ.

Furthermore, if the SGE effects are neglected (ai=0, i=1,...,5), both lλ and lμ reduce to zero, effectively simplifying the formulation to that of classical elasticity. More importantly, the phase velocity of the longitudinal and transverse waves following GFSGET are given by: (39)Vpl=λ+2μρ1+lλ2ξ21+ld2ξ2,Vpt=μρ1+lμ2ξ21+ld2ξ2where ξ is the wavenumber. The dependence of the phase velocities on the wavenumber confirms the dispersive behavior on account of microstructure. These observations are unlike the predictions of classical elasticity, where the wave propagation through the solid is predicted to be non-dispersive. However, this agrees with a prediction of dispersive wave propagation by Gazis et al. [220], as early as 1960, following an investigation of surface elastic waves in cubic crystals. Their study used both continuum mechanics and discrete particle models to analyze wave displacements, velocities, and attenuation constants for different elastic parameters.

To better illustrate the effect of SGE over wave propagation, we note that in the long-wavelength limit (wavelength>>characteristic length scale of microstructure) the phase velocities are given as: (40a)Vpl|ξ→0=λ+2μρ,Vpt|ξ→0=μρthereby reducing to the velocities evaluated from classical elasticity. Alternatively, in the short-wavelength limit, the phase velocities of the longitudinal and transverse waves reduce as: (40b)Vpl|ξ→∞=lλldλ+2μρ,Vpt|ξ→∞=lμldμρ

In the short-wavelength limit, the influence of the microstructure, captured by the SGE, is clearly significant for the elastodynamic behavior. This observation agrees with the seminal work of Mindlin [21] on elastic wave propagation, which also incorporates microstructural effects through SGE.

Building upon this, Savin et al. [37,141] explored wave dispersion in low-dimensional structures incorporating SGE. Moreover, employing the results of these studies, Savin experimentally determined the elastic constants of polycrystalline metals using ultrasonic techniques. Later, Mühlhaus et al. [201] proposed a one-dimensional Cosserat model for longitudinal wave propagation in granular media, introducing additional rotational degrees of freedom to capture grain rotations. Vardoulakis et al. [221] extended Mindlin’s theory to develop a linear gradient-elastic model with surface energy (see Section 2.3), effectively predicting Shear horizontal surface wave motions in a homogeneous half-space. Georgiadis et al. [222] further demonstrated that the couple-stress elasticity theory with microstructure enhances the accuracy of Rayleigh wave modeling, bridging classical continuum mechanics and atomistic lattice models. Incorporating SGE allowed for improved agreement with both experimental data and high-frequency predictions of discrete particle theories. Askes et al. [223] introduced four simplified versions of SGE theory, focusing on their dispersive properties, causality in accordance with Einstein’s principles, and behavior in basic boundary and initial value problems. Further, Vavva et al. [224] employed the simplified Mindlin Form-II (dipolar gradient elasticity theory) to investigate symmetric and antisymmetric wave modes in 2D stress-free gradient plates. Their study showed that variations in elastic constants highlight the influence of microstructure, which significantly affects the propagation of bulk longitudinal and shear waves by introducing both material and geometrical dispersion.

Papargyri et al. [96] highlighted that incorporating shear and rotary inertia corrections in beams and plates mimics the inclusion of micro-elastic and micro-inertia terms, contributing to wave dispersion in structural elements. Extending this perspective, Sidhardh and Ray [97] investigated the dispersion of Rayleigh–Lamb waves in micro-plates using the generalized first strain gradient elasticity theory (GFSGET). More recently, Gao et al. [142] proposed a model for elastic wave band gap analysis in periodic composite beams, accounting for surface energy, shear deformation, and rotational inertia using harmonic wave solutions—further enriching the modeling of wave propagation in gradient elastic media.

Composite homogenization: More than half a century ago, Boutin [225] studied the effects of microstructure on periodic elastic composites. In this study, higher-order terms in the form of higher-order gradients of macroscopic strain are included to introduce nonlocal effects due to the microstructure. Similar homogenization techniques have attracted renewed interest from researchers, particularly due to recent advances in manufacturing that enable the fabrication of complex microstructures. Although studies based on classical elasticity establish the influence of geometry and volume fraction of the inclusion phase on the effective properties of composites [226], several researchers [227–230] observed that the elastic properties of composites vary with the dimensions of the inclusion phase. In particular, size effects due to nonlocal interactions of low-dimensional inclusions are realized over the homogenized properties of the composite.

To address this, Eshelby’s inhomogeneity problem has been reworked within the framework of gradient elasticity theory. Recently, Sidhardh and Ray developed the modified Eshelby’s tensor for spherical, cylindrical, and elliptical inclusions considering the generalized first strain gradient elasticity theory (GFSGET in Eq. (11)) [152,153]. As expected, the Eshelby’s tensor in the SGE framework depends on the length scale of the microstructure controlling the nonlocal interactions, and varies for different simplified forms of the SGE discussed in Section 2.2. This is unlike a constant value for the Eshelby’s tensor, within the inclusion, following classical elasticity; see Fig. 11. This study and the observations therein follow from similar works by Zhang and Sharma [150] and Gao and Ma et al. [151] for Eshelby’s tensor of spherical and cylindrical inclusions within the couple stress and SSGET frameworks (discussed in Eq. (16)), respectively.

Using this approach and an extension of Eshelby’s integral representation to SGE, an exact solution was derived for the problem of a spherical inhomogeneity with interphase [231]. This solution was later applied to study the scale effects on the effective material properties at low dimensions. Furthermore, within the same extended Eshelby integral framework, the problem of coated spherical inhomogeneity with surface cohesive phenomena was analyzed [232] for predicting the effective properties of the composite material. Ameen et al. [233] applied the classical and higher order method to a two-phase microstructure and compared with a rigorous full-scale numerical solution. The accuracy of classical and higher-order homogenized solutions depends only mildly on the degree of stiffness contrast. For the particle-reinforced system considered here, the error increases with increasing stiffness contrast but eventually stabilizes in the rigid-particle limit. It is noted that, higher-order periodic homogenization can effectively refine the classical solution with sufficient accuracy at low scale ratios.

Reduced-order modeling: Modeling complex 3D structures and time-dependent problems presents high computational demands due to a large number of system of equations being involved. Among the several reduced-order models explored for improvement in the computational burden, modeling via nonlocal theories, and more specifically SGE, has attracted the research community. The nonlocal interactions presented by SGE (see Section 4) enable to capture the complex interactions within the microstructure. More specifically, the SGE allows a development of accurate reduced-order models for complex macrostructures by modeling complex interactions via nonlocal effects. For instance, modeling a beamlike lattice structure by 1D micropolar beam was carried out in [234]. Recently, Tran et al. [143] demonstrated the effectiveness of SGE theory in analyzing 2D triangular lattice structures, from the linear regime to von Kármán–type geometric nonlinearity. In addition to this, substantial computational savings were demonstrated following this approach, thanks to reduced numbers of the degrees of freedom, while still maintaining high accuracy compared to conventional 2D finite element simulations. Furthermore, Karttunen et al. [112,235] proposed a localization method to extract the response of periodic classical beam from micropolar formulations. This 1D beam model was later applied to linear bending and vibration analyses of 2D web-core sandwich panels with flexible joints, demonstrating excellent agreement with both experimental results and two-dimensional finite element beam frame simulations. Further, Nampally et al. [120] extended the formulation to a nonlinear analysis of micropolar Timoshenko beams by employing a two-scale energy method. This approach was used to derive the micropolar constitutive equations for various core topologies, including web, hexagonal, Y-frame, and corrugated cores, enabling a more comprehensive representation of their mechanical behavior. This emerging direction holds considerable promise for employing SGE in modeling complex engineering structures. For instance, an example of the lattice core structure studied using micropolar theories is illustrated in Fig. 6b.

MEMS and NEMS: MEMS technology has emerged as a significant high-tech industry, following the rise of microelectronics. However, as MEMS devices scale down in size, their surface area to volume ratio increases significantly. This leads to pronounced size effects, where mechanical behavior deviates from that of macro-scale systems, and surface effects become increasingly dominant [236–238]. Therefore, various studies have explored size-effects over the dynamic behavior of MEMS and NEMS [239].

An example of the size-effects on performance of MEMS is the effect of nonlocal elastic interactions over pull-in instability; in addition to its effect on the elastic behavior. One critical challenge in designing MEMS is avoiding pull-in instability, which can cause the microbeam to collapse and lead to device failure. For instance, Xue et al. [240] investigated the size-effects in MEMS devices using a mechanism-based strain gradient approach, and noted a significant increase in mechanical strain energy (when compared to classical theory) in the digital micromirror device. Further, Hamid [241] studied the influence of vibrational amplitude on pull-in instability and natural frequency of actuated microbeams using a second-order frequency–amplitude model. Incorporating electrostatic and fringing field effects within strain gradient elasticity, the analysis showed that this theory predicts a higher pull-in voltage compared to classical and modified couple stress theories. Kahrobaiyan et al. [242] developed a FE model for Timoshenko beams based on strain gradient theory, capable of reproducing various classical and non-classical beam formulations (strain gradient Euler–Bernoulli beam element, modified couple stress Timoshenko and Euler–Bernoulli beam elements, and also classical Timoshenko and Euler–Bernoulli beam elements). This model was used to compute the static pull-in voltage of an electrostatically actuated microswitch, showing strong agreement with experimental results and outperforming classical FEM predictions. Soroush et al. [243] investigated the impact of Van der Waals forces on the stability of NEMS, considering gradient effects in nanobeams under electrostatic loading. Their findings show that stronger intermolecular forces lead to reduced pull-in deflection and voltage of the actuators. These studies underscore the necessity of advanced mathematical and computational methods for analyzing and designing MEMS and NEMS, considering the limitations of classical theories and the importance of long-range interactions at this scale.

Carbon nano-structures: Since their discovery in the early 1990s [244], carbon nanotubes (CNTs) have attracted attention because of their exceptional electrical, thermal, and chemical properties [245]. However, experimental findings on elastic behavior of CNTs often vary significantly from predictions by classical theories, reflecting discrepancies due to differing test environments and methodologies [246–249]. This can be partly owed to the size-effects realized at these length scales. Therefore, several studies have focused on the elastic response of CNTs including size-effects via SGE constitutive theories, when subject to a myriad of load types and corresponding deformation modes (axial/bending, structural stability, static/dynamic) [250–254]. The improvement in predictions of the elastic response after including SGE is demonstrated via comparison with atomistic simulations. The work of Sun and Liew [255] provides a clear example of this agreement, showing that higher-order gradient theories accurately predict Single Walled NanoTubes (SWNTs) buckling as determined by atomistic simulations. In most studies, it is reported that the introduction of strain gradients within the potential energy given in Eq. (1) presents a greater stiffness of the structure with reducing dimensions (below µm). Further, SGE theories have been employed to explore the elastic response of 2D nanostructures like graphene. Specifically, experimental and atomistic simulation studies have revealed a pronounced size-effect in mechanical properties of graphene sheets [52,256–258]. To address the limitation to classical elasticity in capturing this effect, several studies employed SGE constitutive theories for the elastic response of graphene sheets [259–262]. For example, Lu et al. [263] proposed a nonlocal strain gradient third-order beam model to analyze the nonlinear bending response of functionally graded porous micro/nano-beams reinforced with graphene platelets. Notably, upon including SGE within the constitutive framework for graphene, some exciting observations were realized. In their study, Kundalwal et al. [264] observed that, strain gradients in non-piezoelectric graphene sheets affect both ionic positions and the asymmetric redistribution of electron density. This induces a strong polarization to obtain an apparent piezoelectric behavior from a graphene sheet containing non-centrosymmetric trapezoidal shaped pore, such as shown in Fig. 12. More details on exploitation of the strain gradients in nanostructures for multiphysics coupling will be discussed in the next section.

Ferroelectric materials are widely used in actuators, sensors, memory devices, electro-optics, and MEMS because of their multifunctional properties driven by interactions between internal polarization and external factors like temperature, pressure, and electric fields. Among these, piezoelectric effect, presenting a non-zero electrical polarization to applied mechanical pressure (or vice versa), is well established. This coupling of the electro-mechanical field variables is limited to the non-centrosymmetric crystalline (and semi-crystalline) materials. Surprisingly, experiments detected a non-zero polarization in ferroelectrics even in their centrosymmetric crystalline phase. More specifically, Ma and Cross [47,48,62] note a non-zero polarization upon bending of ferroelectric crystals in their centrosymmetric phase at very low dimensions. This is attributed to a loss in symmetry and a resultant polarization in the materials as a result of the strain gradients across the structure. The loss in centrosymmetry and the resulting net polarization across the crystals due to the presence of a strain gradient is called the phenomenon of flexo-electricity. The suffix flexo- follows from initial observations of non-zero polarization in centrosymmetric crystals when subject to bending (flexure). This observation is significant because it reveals that electro-mechanical coupling in smart materials is a universal property, extending beyond the limitations of non-centrosymmetric structures. Similar observations of multiphysics coupling involving strain gradients are realized over magneto-elastic and thermo-elastic field variables, and discussed briefly in the section below.

Electro-mechanical coupling: The flexoelectric effect, coupling strain-gradients and electrical polarization, was theorized long before any experimental evidence. It may be interpreted as an electro-mechanical phenomenon in which a material exhibits a non-zero electric polarization in response to a strain gradient (non-uniform mechanical deformation) as demonstrated in Fig. 3b. As seen in this figure, strain gradients (following from a non-uniform strain across the thickness due to a bending load) present an asymmetric redistribution of electrons. Thereby, a net polarization is realized in centrosymmetric material. Understanding these interactions is key for advancing nanoscale electro-mechanical coupling with relevance for energy harvesting and actuation technologies.

The constitutive relations for the flexoelectric solid can be derived from the following expression for the internal energy density in centrosymmetric solids [265]: (41)𝒲=12aijPiPj+12bijklPi,kPj,l⏟Electrical+12Cijklεijεkl+12gijklmnηijkηlmn⏟Mechanical; See \hyperlink{ref-1}{Eq. (1)}−12flijk(Plηijk−εijPl,k)⏟Flexoelectricwhere Pi and Pi,j are the polarization field and its first gradient. The above expression for energy can be decomposed into three distinct parts: electrical, mechanical, and electro-mechanical coupling. The terms corresponding to elastic deformation energy are already introduced in Eq. (1). Additional terms for electrical and coupled energy are introduced above to model flexoelectric solids. a and b are second-order dielectric susceptibility and fourth-order polarization gradient tensors, respectively, and f is the (effective) flexocoupling tensor. The complete definition of this coupling material constant depends on the material symmetry; analogous to the definition for g corresponding to SGE. Mindlin [43] presented the definition of this material constant for isotropic solids. The coupling term in the above expression presents a non-zero electric polarization induced for a non-uniform strain across the domain, referred to as the direct flexoelectric effect. Additionally, converse flexoelectric field presents a non-uniform strain across the domain in response to applied electric polarization. The internal energy density corresponding to the piezoelectric effect is omitted above, owing to the assumption of centrosymmetry (the third-order material coefficient tensor is identically zero). From the above expression for internal energy, the constitutive relations for mechanical stresses (σ and τ; see Eq. (7)) and electrical field can be derived [266].

In the late 1950s, Mashkevich [267] experimentally observed the presence of the flexoelectric effect in crystals. Further, Kogan [46] and Harris [268] presented additional insights into the flexoelectric effect and improved understanding of its underlying mechanisms. Following the seminal work on SGE by Mindlin [31], the coupling of strain gradients with electrical polarization was explored in [43,269,270]. In particular, in this work, Mindlin extended the classical piezoelectric theory and developed a framework for electro-mechanical coupling involving strain gradients (the term ‘flexoelectricity’ was not coined yet). This phenomenon may also be interpreted as the size effect involving multi-physics coupling realized as a consequence of the strain gradients in low-dimensional structures. Therefore, the effect of the coupling decreases with increasing dimensions of the structure/crystal, confirmed in later numerical studies [271].

In the later experiments by Indenbom [272], the flexoelectric coefficients coupling polarization and strain gradient were observed to be four orders of magnitude higher than the initial theoretical estimates. Despite the important role of the flexoelectric effect, experimental measurements of flexoelectric coefficients have been limited to a few ferroelectric materials [62,273], biological structures [274–276], and polymeric materials [277–279]. For instance, Lu et al. [280] experimentally determined the flexoelectric coefficients of macroscopic polyvinylidene fluoride (PVDF) polymers, revealing notable differences between their bulk and thin film forms. Given the several number of material constants involved, a series of experiments is necessary for the characterization of the flexoelectric coupling tensor. A schematic illustration of the experimental setup used for the direct measurement of flexoelectric constants is presented in Fig. 13 [281]. Further, Sharma et al. [282,283] theoretically estimated the flexoelectric coefficients using molecular dynamics simulations. However, significant discrepancies between theoretical predictions and experimental findings have highlighted a promising area of research in material modeling. This growing interest was further reflected in the comprehensive review by Yudin and Tagantsev [284].

Building on the initial modeling framework developed by Mindlin [43], Hu and Shen [266,285] developed a variational approach to analyze the flexoelectric behavior of structures. This was possible by incorporating extended linear theory for dielectrics including surface effects. These frameworks make it possible to conduct analytical and numerical studies on flexoelectric solids for various structural configurations. For instance, the implications of flexoelectricity at the nanoscale were explored by Majdoub et al. [286,287], using Bernoulli-Euler beam theory to model lead zirconate titanate (PZT) and Barium titanate (BT) nanowires. Their results highlighted enhanced electromechanical coupling because of the combined influence of piezoelectricity and flexoelectricity, suggesting promising applications in energy harvesting. The analysis is further extended to thick beams where shear effects become more prominent [204,288]. The shear deformation effects on the flexoelectric coupling were explored.

Following from a coupling involving the strain gradients, the flexoelectric effect is expected to demonstrate size-dependent behavior. This underscores the critical role of flexoelectricity in the electro-mechanical behavior of nanoscale piezoelectric structure. For instance, Yan and Jiang [289] investigated the impact of flexoelectric coupling on the static bending behavior of piezoelectric nanobeams within the framework of the Euler–Bernoulli beam model. Their study revealed that the flexoelectric effect significantly enhances the electroelastic response as the beam thickness decreases, these indicating a strong size dependence. Further, Liang et al. [290] investigated the combined influence of surface piezoelectricity (also gains relevance with reducing dimensions) and bulk flexoelectricity on the bending response of piezoelectric nanobeams. Their findings revealed a substantial enhancement in the effective electromechanical coupling coefficient when both surface and bulk effects were considered. The inclusion of bulk flexoelectricity contributed to a stiffening of the structure, resulting in reduced deflections compared to predictions made by conventional piezoelectric models.

Although flexoelectricity follows from a non-zero strain gradient, realized in low-dimensional structures, most studies have ignored the simultaneous effect of strain gradients over the elastic behavior. In particular, several studies on flexoelectricity proceeded with ignoring the effect of SGE (follows from ignoring the deformation energy due to strain gradients in Eq. (41) while retaining the strain gradients in flexoelectric coupled energy). Since both flexoelectricity and strain gradients can significantly alter the electromechanical behavior of these structures, a unified analysis is essential. In this direction, Sidhardh and Ray [92] explored the combined effect of GFSGET and flexoelectricity on the electro-mechanical response of flexoelectric solids as an actuator and energy harvester. The enhanced stiffness resulting from SGE produced a significant reduction in the deformations induced by the flexoelectric coupling. These observations are drawn from an exact solution developed in their study, which is devoid of assumptions such as displacement or electrical field variable distributions across the domain. Static analyses offer limited insight into piezoelectric and flexoelectric energy harvesting, typically following from dynamic behaviors such as harmonic vibrations. Therefore, based on theoretical and numerical frameworks to investigate the flexoelectric response for static loading conditions, several studies [53,291] explored the dynamic response of flexoelectric solids. Further, Liang et al. [271] developed analytical expressions for the maximum electric potential generated through the combined piezoelectric and flexoelectric effect in various dielectric materials. Wang and Wang [292] explored the distribution of electric potential in piezoelectric nanowires by incorporating combined flexoelectric and surface effects. It is observed that the surface effect significantly influences the electro-mechanical response (displacements and electric potential). Unlike the studies above on the monolithic dielectric structure that demonstrates a flexoelectric effect, Qi et al. [51] explored the flexoelectric layers attached to the surface of a substrate beam for actuation.

Leveraging the large flexoelectric coefficients in ferroelectric materials and the high strain gradients present in small-scale dielectrics, various applications have been developed. Among these, the most interesting is the universal nature of flexoelectricity being exploited to develop apparent piezoelectric solids. An example of the same is depicted in Fig. 12. In this example, under external load, a non-zero strain gradient is realized in the vicinity of an asymmetric cutout in the 2D graphene sheet [264]. This presents an electrical response to the mechanical load that follows from the flexoelectric coupling. A similar observation was noted earlier in molybdenum oxide (MoO2) by Sharma et al. [293], and it was proposed to enhance the (otherwise low) electro-mechanical coupling coefficients by introducing asymmetric cutouts in the domain. Further to this, as an electromechanical coupling phenomenon, the flexoelectric effect has potential for use in sensors, actuators, transducers, and energy harvesting devices [291,294]. To complement these theoretical insights, Fig. 14a presents an example of flexoelectricity in soft elastomers for application in a robotic appendage [295]. Ferroelectric behavior has also been observed in a specific class of soft materials, particularly biological membranes; see Fig. 14b. These membranes, composed of key biopolymers within the extracellular matrix of various soft tissues, show notable ferroelectric characteristics [296,297]. Within biological systems, membrane flexoelectricity has been postulated to contribute critically to fundamental processes, including mechanotransduction and auditory function [298,299]. Recently, Zelisko et al. [300] developed an analytical model based on statistical mechanics, supported by molecular dynamics simulations, to propose a plausible mechanism underlying biological ferroelectricity in tropoelastin. Their study predicted piezoelectric constants that surpass those of any known polymer. These studies offer compelling evidence highlighting the significance of flexoelectricity in soft lipid bilayers [297].

Magneto-elastic coupling: Just as flexoelectricity couples strain gradients with electrical polarization, a similar coupling occurs for magneto-elastic field variables, involving strain gradients. As noted previously, most existing research on strain gradients is centered on strain gradient elasticity or the flexoelectric effect. In contrast, the magneto-elastic coupling in crystals arising from an interaction between strain gradients and induced magnetic fields has received relatively little attention [301]. For instance, even in studies on magneto-elastic coupling in low-dimensional structures, the coupling of the strain gradient and magnetic field variables is not considered [302]. This interaction, known as the flexomagnetic effect, leads to the generation of a non-zero magnetic field under inhomogeneous strain. A schematic illustration for the same is depicted in Fig. 15b, where perovskite ABO3 lattice undergoes a deformation from symmetric cross-section (square) to asymmetric cross-section (rectangular and inclined parallelogram), thereby causing a spontaneous flexoeffect in nanowires. It is considered a specific case of the broader flexomagnetoelectric effect, a concept introduced by Bobylev and Pikin [49] to explore the combined magneto-electric and elastic behavior in liquid crystals. Flexo-electromagnetism fundamentally arises from the coupling between mechanical strain gradients and electromagnetic fields. A time-varying strain gradient induces a dynamic electric field. In response, the electromechanical interaction generates a magnetic field. This coupled mechanism results in the excitation of electromagnetic waves [303], as schematically illustrated in Fig. 15a.

Restricting the above coupling to magnetic and elastic field variables, a flexomagnetic effect is proposed in the literature. Anaologus to Eq. (41), the free energy density of a centrosymmetric ferromagnetic solid is given as [145,304,305]: (42)𝒲=−12aijmHiHj⏟Magnetic+12Cijklεijεkl+12gijklmnηijkηlmn⏟Elastic; See \hyperlink{ref-1}{Eq. (1)}−fijklmHiηjkl⏟Flexomagnetic.where Hi is the magnetic field vector. The above expression for energy can be decomposed into three distinct parts: magnetic, elastic, and magneto-elastic coupling. The terms corresponding to elastic deformation energy are already introduced in Eq. (1). Additional terms for magnetic and magneto-elastic energy are introduced above for modeling flexomagnetic solids. In the above expression, am is the second-order magnetic permeability tensor, and the fourth-order tensor fm, coupling strain gradient and magnetic field is, the flexomagnetic tensor. The superscripts in am and fm are introduced here to differentiate from the constants introduced previously in Eq. (41) for flexoelectricity. The complete form of the flexomagnetic material constant for different magnetic classes is provided by Eliseev et al. [305]. The constitutive relations for mechanical stresses and magnetic induction can be derived from the above expression using thermodynamic balance laws [145]. Following from the coupling energy term above, we see a non-uniform strain inducing a magnetic field across the domain, referred to as the direct flexomagnetic effect. Furthermore, a converse flexomagnetic effect presenting non-uniform strain (strain gradient) across the domain in response to an applied magnetic field.

Eliseev et al. [304–306] conducted extensive theoretical analyses using symmetry theory to determine the form of the flexocoupling tensor in various magnetic classes. Their findings suggest that all 90 magnetic classes exhibit a non-zero flexomagnetic effect, at least near the surface. This is a significant observation, as this renders the flexomagnetic effect to be global and not restricted to a limited (mostly non-centrosymmetric in case of piezomagnetism) class of materials. Furthermore, Lukashev et al. [307,308] performed first-principles studies on Mn-based antiperovskites to compute the coupling tensor coefficients. These results imply the potential for the development of piezomagnetic-like structures that exploit flexomagnetic coupling without relying on traditional piezomagnetic materials.

Pyatakov and Zvezdin [309] investigated the combined electro-magneto-elastic coupling via third-order flexomagnetoelectric coupling tensors in antiferromagnetics, demonstrating its influence on inhomogeneous magnetoelectric coupling in various multiferroic systems. Expanding on the idea of inhomogeneous magnetoelectric coupling, the flexomagnetoelectric effect is characterized by a nonlinear interaction that is linearly dependent on polarization and quadratically on magnetization. This phenomenon was explored in detail by [310,311].

This phenomena can be visualized via the mechanism demonstrated in Fig. 16 above. The strain gradient defines a polar axis within the crystal, enabling the development of electric polarization due to symmetry considerations. This coupling between mechanical bending and electrical polarization is known as the flexoelectric effect, observed in solids (Fig. 16a) and in liquid crystals (Fig. 16b) [312]. A related phenomenon, spin flexoelectricity, refers to the polarization induced by spatial variations in magnetization. A notable example is the fan-like spin cycloid structure depicted in Fig. 16c, which emerges in magnetic atom monolayers [312,313] and is attributed to interfacial electric fields, demonstrating the effects of spin flexoelectric interactions [314].

Recently, Sidhardh and Ray [145] developed a constitutive model of the flexomagnetic effect and employed it to evaluate the magnetoelastic response of a beam, providing a deeper understanding of the interplay between strain gradients and magnetic field generation in multiferroic systems. This formulation simplified a detailed analysis of the flexomagnetic effect in solids. Malikan et al. [315,316] extended this constitutive framework for a magnetoelastic response analysis of cobalt-ferrite magnetic nanostructures (CFMNs), focusing on bifurcation buckling and post-buckling phenomena. This was achieved using the Euler-Bernoulli beam theory, nonlinear Lagrangian-von Kármán strains, and the nonlocal strain gradient elasticity (NSGT) approach. Further, Borkar et al. [317] developed a framework to study the effect of macroscopic inhomogeneous strain gradients on magnetic moments in multiferroic composites. Their results show that strong magneto-mechanical coupling under inhomogeneous strain can significantly enhance magnetic susceptibility at lower magnetic fields compared to relaxed Pb[(Zr0.52Ti0.48)0.60(Fe0.67W0.33)0.40]O3]0.80−[CoFe2O4]0.20 (PZTFW-CFO) composites. The application of inhomogeneous strain also led to increased zero-field-cooled (ZFC) and field-cooled (FC) magnetizations by reducing surface spin disorder.

Thermo-elastic coupling: In recent years, the pursuit of more efficient and eco-friendly cooling technologies has gained momentum, driven by the need to reduce harmful greenhouse emissions. Among various emerging approaches, flexocaloric cooling has shown significant promise. In general, caloric effects refer to reversible temperature changes in solid materials triggered by external fields, such as magnetic, electrical, or mechanical. The effect may arise from variations in magnetic spin states, electric-dipole alignments, or structural configurations. The thermodynamic cooling mechanisms include both vapor compression and ferroic-based cycles. Significant thermal responses are generally observed in materials that exhibit spontaneous phase transitions near the target operating temperature [318]. Specifically, the flexocaloric effect involves temperature changes resulting from the application of mechanical stress [319,320].

Existing studies indicate that the flexocaloric effect induced by a strain gradient is generally realized through an asymmetric electron distribution within the material commonly observed in structures such as truncated pyramids [321] or the bending of thin films made of non-centrosymmetric materials [322]. The flexoelectric coupling significantly alters the misfit strain–temperature phase diagram by promoting out-of-plane polarization phases due to a built-in field. Considering both flexoelectric and work function-induced fields leads to a notable enhancement in electrocaloric performance. For instance, Bai et al. [323] investigated the size-dependent behavior of paraelectric Ba0.67Sr0.33TiO3 (BST) truncated pyramids using a phenomenological thermodynamic framework. Their findings revealed that the flexoelectricity-induced entropy and adiabatic temperature changes are size dependent (which is obvious from previous studies involving strain gradients) and vary with applied stress and working temperature. Specifically, as the thickness of the pyramid decreases and the external stress increases, the resulting strain gradient is enhanced. Consequently, a greater flexocaloric effect is realized, leading to greater entropy and temperature changes. Further, Patel et al. [321,324] examined the caloric effect induced by flexoelectricity in truncated pyramids of BST ceramics of varying heights. The results suggest that ferroelectric materials can effectively generate cooling even beyond their Curie temperature, which was previously thought to be a limiting factor. Qiu et al. [325] employed an extended nonlinear thermodynamic framework to model the flexocaloric effect in BaTiO3 thin films. Furthermore, BaTiO3 and BST thin films have been extensively studied for their high caloric effects, which complement the electrocaloric effect and have potential for solid-state cooling applications [326].

Recently, an alternative approach has emerged that involves the induction of strain gradients in uniform structures through the varying material composition; for instance using functionally graded materials (FGM) [327]. Sharma et al. [322] developed a numerical framework to study the flexocaloric effect in FGMs. The flexocaloric effect is realized by virtue of the flexoelectric polarization generated in the FGM due to the inhomogeneous material composition.

These findings highlight the exciting possibilities for achieving novel approaches for multi-physics coupling involving a strain gradient coupling with electrical, magnetic, and thermal field variables. In particular, we also note the rather global nature of such couplings, which extend beyond a limited choice of materials typically used for multiphysics coupling.

The fundamental idea behind nonlocal theories is that all particles within a defined region, usually referred to as the horizon of nonlocality, interact with one another through long-range cohesive forces. This interaction can be modeled using either gradient or integral relations in the constitutive equations, leading to “weak” gradient methods or “strong” integral methods, respectively. A schematic illustration of the nonlocal interactions across the domain is provided in Fig. 8. We note that, unlike the local elastic constitutive relations, the mechanical response in a nonlocal solid depends on the relevant field variables at all points within the domain of nonlocal influence. We have provided a detailed account of the gradient models for nonlocal elasticity in this review article. We conclude with a brief overview of some integral theories for nonlocal interactions proposed in the literature.

We begin by presenting some limitations of the gradient-based theories documented in the literature. To begin with, we revisit the governing equations and corresponding boundary conditions for the gradient elasticity developed in Eq. (26b). The physical motivation behind the additional boundary conditions (essential and natural) for SGE is unclear, leading to ambiguity in enforcing them and problems with their application in practical problems [10]. This is in addition to the difficulties in arriving at analytical or numerical solutions for the GDEs in Eq. (26a) on account of the higher-order derivatives. This aspect has been discussed in detail in Section 3.2. Finally, gradient elastic models have a limited range for the nonlocal horizon [328]. For instance, recall from Section 5 that microstructure effects were realized only in structures not much larger (at most one order of magnitude) than the nonlocal length scale. Owing to this limitation, these models are not suitable for capturing the nonlocal interactions over a larger geometric span.

Integral models for nonlocal elasticity, also referred to as the strong form of nonlocal elasticity, are mathematically better suited to account for the long-range interactions across the domain. In addition, these constitutive models can be divided into three different approaches: (i) the strain-driven approach, (ii) the stress-driven approach, and (iii) the displacement-driven approach. A brief overview of these models is provided below.

This review provides a comprehensive overview of strain gradient elasticity, highlighting its importance in addressing the limitations of classical elasticity at small length scales. In addition, various constitutive models for SGE are explored here, from generalized higher-order gradient theories to simplified versions tailored for specific applications, including second-order SGE introducing surface effects. The physical relevance of SGE, particularly its ability to capture size effects and (weakly) nonlocal interactions, has been discussed. Furthermore, the review has covered the analytical and numerical techniques developed to solve the higher-order governing differential equations presented by the SGE framework. We present detailed applications of SGE in multiscale elasticity and multiphysics coupling scenarios, highlighting its ability to bridge the limitations of classical continuum mechanics. Finally, a discussion of alternative integral and integro-differential theories provides a broader perspective on non-classical continuum mechanics.

Potential Areas for Future Research

Despite significant progress in the field of strain gradient elasticity, several avenues remain open for future research and development. Moreover, since most of the literature has concentrated on simplified theories of SGE, the entire potential of the gradient effects on the solids has not yet been realized. Some potential areas for future research are:

Fundamental gaps: As mentioned earlier, the physical relevance of the additional boundary conditions derived from the variational principles within the SGE framework remains unclear. This ambiguity is compounded by mathematical inconsistencies in higher-order gradient terms within the SGE constitutive laws. For instance, the consistency of the symmetric and asymmetric parts of rotational gradients in higher-order elasticity, proposed almost half a century ago, is still being debated. Consequently, a fundamental investigation into the physical and mathematical significance and consistency of different deformation mechanisms captured via higher-order continuum theories requires to be explored. Future research in SGE should also explore bridging the gap between the theoretical framework and its physical interpretation, as well as improving experimental validation to increase its practical utility.

By addressing these potential research areas, the field of strain gradient elasticity can continue to evolve and provide increasingly accurate and insightful tools for understanding and designing materials and structures at small (and also higher) length scales, paving the way for advancements in various technological domains.