Giant Flexoelectricity in Bent Semiconductor Thinfilm.

We elucidate the flexoelectricity of semiconductors in the high strain gradient regime, the underlying mechanism of which is less understood. By using the generalized Bloch theorem, we uncover a strong flexoelectric-like effect in bent thinfilms of Si and Ge due to a high-strain-gradient-induced band gap closure. We show that an unusual type-II band alignment is formed between the compressed and elongated sides of the bent film. Therefore, upon the band gap closure, electrons transfer from the compressed side to the elongated side to reach the thermodynamic equilibrium, leading to a pronounced change of polarization along the film thickness dimension. The obtained transverse flexoelectric coefficients are unexpectedly high with a quadratic dependence on the film thickness. This new mechanism is extendable to other semiconductor materials with moderate energy gaps. Our findings have important implications for the future applications of flexoelectricity in semiconductor materials.

Directional Cell Migration Guided by a Strain Gradient.

Strain gradients widely exist in development and physiological activities. The directional movement of cells is essential for proper cell localization, and directional cell migration in responses to gradients of chemicals, rigidity, density, and topography of extracellular matrices have been well-established. However; it is unclear whether strain gradients imposed on cells are sufficient to drive directional cell migration. In this work, a programmable uniaxial cell stretch device is developed that creates controllable strain gradients without changing substrate stiffness or ligand distributions. It is demonstrated that over 60% of the single rat embryonic fibroblasts migrate toward the lower strain side in static and the 0.1 Hz cyclic stretch conditions at ≈4% per mm strain gradients. It is confirmed that such responses are distinct from durotaxis or haptotaxis. Focal adhesion analysis confirms higher rates of contact area and protrusion formation on the lower strain side of the cell. A 2D extended motor-clutch model is developed to demonstrate that the strain-introduced traction force determines integrin fibronectin pairs' catch-release dynamics, which drives such directional migration. Together, these results establish strain gradient as a novel cue to regulate directional cell migration and may provide new insights in development and tissue repairs.

Hard interfaces with microstructure: the cases of strain gradient elasticity and micropolar elasticity.

As the size of a layered structure scales down, the adhesive layer thickness correspondingly decreases from macro- to micro-scale. The influence of the material microstructure of the adhesive becomes more pronounced, and possible size effect phenomena can appear. This paper describes the mechanical behaviour of composites made of two solids, bonded together by a thin layer, in the framework of strain gradient and micropolar elasticity. The adhesive layer is assumed to have the same stiffness properties as the adherents. By means of the asymptotic methods, the contact laws are derived at order 0 and order 1. These conditions represent a formal generalization of the hard elastic interface conditions. A simple benchmark equilibrium problem (a three-layer composite micro-bar subjected to an axial load) is developed to numerically assess the asymptotic model. Size effects and non-local phenomena, owing to high strain concentrations at the edges, are highlighted. The example proves the efficiency of the proposed approach in designing micro-scale-layered devices.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Directly Observing the Evolution of Flexoelectricity at the Tip of Nanocracks.

As an electromechanical coupling between strain gradients and polarization, flexoelectricity is largely enhanced at the nanoscale. However, directly observing the evolution of flexoelectric fields at the nanoscale usually suffers from the difficulty of producing strain gradients and probing electrical responses simultaneously. Here, we introduce nanocracks in SrTiO3, Ba0.67Sr0.33TiO3, and TiO2 samples and apply continuously varying mechanical loading to them, and as a result, huge strain gradients appear at the crack tip and result in a significant flexoelectric effect. Then, using atomic force microscopy, we successfully measure the evolution of flexoelectricity around the crack tips. For the case of SrTiO3, the maximum induced electric field reaches 11 kV/m due to the tensile load increasing. The proposed method provides a reliable way to identify the significance of the flexoelectric effect. It may also open a new avenue for the study of flexoelectricity involving multiple physics phenomena including flexoelectronics, the flexo-photovoltaic effect, and others.

Interpretation of Second Law of Thermodynamics in Extended Procedures for the Exploitation of the Entropy Inequality: Korteweg Fluids and Strain-Gradient Elasticity as Examples.

In continuum physics the dissipation principle, first proposed by Coleman and Noll in 1963, regards second law of thermodynamics as a unilateral differential constraint on the constitutive equations. In 1996, Muschik and Ehrentraut provided a rigorous proof of such an approach under the assumption that, at an arbitrary instant, t0, in an arbitrary point, P0, of a continuous system, the entropy production is zero if, and only if, P0 is in thermodynamic equilibrium. In 2022, Cimmelli and Rogolino incorporated such an assumption in a more general formulation of the second law of thermodynamics. In this paper, we prove that the same conclusions hold if both the fundamental balance laws and their gradients are substituted into the entropy inequality. Such a methodology is applied to analyze the strain-gradient elasticity.

Beam theory predicts muscle deformation and vertebral curvature during feeding in rainbow trout (Oncorhynchus mykiss).

Muscle shortening underpins most skeletal motion and ultimately animal performance. Most animal muscle generates its greatest mechanical output over a small, homogeneous range of shortening magnitudes and speeds. However, homogeneous muscle shortening is difficult to achieve for swimming fish because the whole body deforms like a bending beam: as the vertebral column flexes laterally, longitudinal muscle strain increases along a medio-lateral gradient. Similar dorsoventral strain gradients have been identified as the vertebral column flexes dorsally during feeding in at least one body location in one fish. If fish bodies also deform like beams during dorsoventral feeding motions, this would suggest the dorsal body (epaxial) muscles must homogenize both dorsoventral and mediolateral strain gradients. We tested this hypothesis by measuring curvature of the anterior vertebral column with XROMM and muscle shortening in 14 epaxial subregions with fluoromicrometry during feeding in rainbow trout (Oncorhynchus mykiss). We compared measured strain with the predicted strain based on beam theory's curvature-strain relationship. Trout flexed the vertebrae dorsally and laterally during feeding strikes, yet when flexion in both planes was included, the strain predicted by beam theory was strongly and significantly correlated with measured strain (P<0.01, R2=0.60). Beam theory accurately predicted strain (slope=1.15, compared with ideal slope=1) across most muscle subregions, confirming that epaxial muscles experience dorsoventral and mediolateral gradients in longitudinal strain. Establishing this deformation-curvature relationship is a crucial step to understanding how these muscles overcome orthogonal strain gradients to produce powerful feeding and swimming behaviours.

Small-scale effects on the radial vibration of an elastic nanosphere based on nonlocal strain gradient theory.

Nonlocal strain gradient theory is widely used when dealing with micro- and nano-structures. In such framework, small-scale effects cannot be ignored. In this paper a model of radial vibration of an isotropic elastic nanosphere is theoretically investigated. The frequency equation is obtained from a nonlocal elastic constitutive law, based on a mix between local and nonlocal strain. This model is composed of both the classical gradient model and the Eringen's nonlocal elasticity model. To check the validity and accuracy of this theoretical approach, a comparison is made with the literature in certain specific cases, which shows a good agreement. Numerical examples are finally conducted to show the impact of small-scale effects in the radial vibration, which need to be included in the nonlocal strain gradient theory of nanospheres. It reveals that the vibration behavior greatly depends on the nanosphere size and nonlocal and strain gradient parameters. Particularly, when the nanospheres radius is smaller than a critical radius, the small-scale effects play a key role. Thus, the obtained frequency equation for radial vibration is very useful to interpret the experimental measurements of vibrational characteristics of nanospheres.

Mechanical Stress Modulation of Resistance in MoS2 Junctions.

Strain engineering is a powerful strategy to control the physical properties of material-enabling devices with enhanced functionality and improved performance. Here, we investigate a modulation of the transport behavior of the two-dimensional MoS2 junctions under the mechanical stress induced by a tip of an atomic force microscope (AFM). We show that the junction resistance can be reversibly tuned by up to 4 orders of magnitude by altering a tip-induced force. Analysis of the stress-induced evolution of the I-V characteristics indicates a combined effect of the tip-induced strain and strain gradient on the energy barrier height and profile. In addition, we show that the tip-generated flexoelectric effect leads to significant enhancement of the photovoltaic effect in the MoS2 junctions. A combination of the optical and mechanical stimuli facilitates reversible photomechanical tuning of resistance of the narrow-band 2D semiconductors and development of devices with an enhanced photovoltaic response.

Ultrahigh Temperature Lead-Free Film Capacitors via Strain and Dielectric Constant Double Gradient Design.

With the development of miniaturization, lightweight and integration of electronic devices, the demand for high-temperature dielectric capacitors is becoming urgent. Nevertheless, the breakdown strength and polarization are deteriorated at high temperatures due to the thermal energy assisting the electron transport and impeding the dipole alignment. Here, a structure of capacitor with double gradients of dielectric constant gradient and strain gradient is designed to achieve high breakdown strength, high working temperature, and high energy storage density simultaneously. It is found that the designed structure of BaHf0.17 Ti0.83 O3 /1mol% SiO2 doped BaZr0.35 Ti0.65 O3 /0.85BaTiO3 -0.15Bi(Mg0.5 Zr0.5 )O3 exhibits excellent energy storage performance. The energy storage density of 127.3 J cm-3 with an energy storage efficiency of 79.6% is realized in the up-sequence multilayer with period N = 2 at room temperature. Moreover, when the working temperature varies from -100 to 200 °C, the energy storage density of the N = 4 capacitor keeps stably at 84.62 J cm-3 with an energy storage efficiency 78.42% at 6.86 MV cm-1 . All these properties promise great potential applications of the designed multilayer capacitors with the double gradients in harsh environments, and the design principle can be applicable to other systems to boost working temperature.

Decay of waves in strain gradient porous elasticity with Moore-Gibson-Thompson dissipation.

We study a one-dimensional problem arising in strain gradient porous-elasticity. Three different Moore-Gibson-Thompson dissipation mechanisms are considered: viscosity and hyperviscosity on the displacements, and weak viscoporosity. The existence and uniqueness of solutions are proved. The energy decay is also shown, being polynomial for the two first situations, unless a particular choice of the constitutive parameters is made in the hyperviscosity case. Finally, for the weak viscoporosity, only the slow decay can be expected. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

Measurement of Gradient Strain Fields with Fiber-Optic Sensors.

The results of measuring gradient strain fields by embedded or mounted point fiber-optic sensors based on Bragg gratings and distributed fiber-optic sensors based on Rayleigh scattering are discussed. Along with the experiment, the results of numerical modeling of strain measurement errors associated with the assumption of uniaxial stress state in the area of the embedded Bragg grating and measurement errors by distributed fiber-optic sensors associated with gage length are presented. Experimental results are presented for 3D printed samples and samples made of polymer composite material. The geometry of the samples was chosen based on the results of numerical simulations, and provides different variants of non-uniform strain distribution under uniaxial tension, including the variant in which the derivative of the strain distribution function changes its sign. A good agreement of numerical results and experimental data obtained by distributed and point fiber-optic sensors in areas where the derivative of the strain distribution function keeps a sign and an increase in the error of strain measurement results by distributed fiber-optic sensors in areas where this derivative changes sign are demonstrated.

Strain Gradient Elasticity in SrTiO3 Membranes: Bending versus Stretching.

Young's modulus determines the mechanical loads required to elastically stretch a material and also the loads required to bend it, given that bending stretches one surface while compressing the opposite one. Flexoelectric materials have the additional property of becoming electrically polarized when bent. The associated energy cost can additionally contribute to elasticity via strain gradients, particularly at small length scales where they are geometrically enhanced. Here, we present nanomechanical measurements of freely suspended SrTiO3 crystalline membrane drumheads. We observe an unexpected nonmonotonic thickness dependence of Young's modulus upon small deflections. Furthermore, the modulus inferred from a predominantly bending deformation is three times larger than that of a predominantly stretching deformation for membranes thinner than 20 nm. In this regime we extract a strain gradient elastic coupling of ∼2.2 µN, which could be used in new operational regimes of nanoelectro-mechanics.

Flexoelectric Thin-Film Photodetectors.

The flexoelectric effect, which manifests itself as a strain-gradient-induced electrical polarization, has triggered great interest due to its ubiquitous existence in crystalline materials without the limitation of lattice symmetry. Here, we propose a flexoelectric photodetector based on a thin-film heterostructure. This prototypical device is demonstrated by epitaxial LaFeO3 thin films grown on LaAlO3 substrates. A giant strain gradient of the order of 106/m is achieved in LaFeO3 thin films, giving rise to an obvious flexoelectric polarization and generating a significant photovoltaic effect in the LaFeO3-based heterostructures with nanosecond response under light illumination. This work not only demonstrates a novel self-powered photodetector different from the traditional interface-type structures, such as the p-n and Schottky junctions but also opens an avenue to design practical flexoelectric devices for nanoelectronics applications.

Unraveling submicron-scale mechanical heterogeneity by three-dimensional X-ray microdiffraction.

Shear banding is a ubiquitous phenomenon of severe plastic deformation, and damage accumulation in shear bands often results in the catastrophic failure of a material. Despite extensive studies, the microscopic mechanisms of strain localization and deformation damage in shear bands remain elusive due to their spatial-temporal complexities embedded in bulk materials. Here we conducted synchrotron-based X-ray microdiffraction (µXRD) experiments to map out the 3D lattice strain field with a submicron resolution around fatigue shear bands in a stainless steel. Both in situ and postmortem µXRD results revealed large lattice strain gradients at intersections of the primary and secondary shear bands. Such strain gradients resulted in severe mechanical heterogeneities across the fatigue shear bands, leading to reduced fatigue limits in the high-cycle regime. The ability to spatially quantify the localized strain gradients with submicron resolution through µXRD opens opportunities for understanding the microscopic mechanisms of damage and failure in bulk materials.

Direct Observation of Huge Flexoelectric Polarization around Crack Tips.

Flexoelectricity is especially relevant for nanoscale structures, and it is expected to be largest at the tip of cracks. We demonstrate the presence of a huge flexoelectric polarization at crack tips in SrTiO3 by direct observation with scanning transmission electron microscopy. We observe an averaged polarization of 62 ± 16 µC cm-2 in the three unit cells adjacent to the crack tip, which is one of the largest flexoelectric polarizations ever reported. The polarization is screened by an electron density of 0.7 ± 0.1 e-/uc localized within one unit cell. These findings reveal the relevance of flexoelectricity for the science of crack formation and propagation.

Controllable Ferromagnetism in Super-tetragonal PbTiO3 through Strain Engineering.

The coupling strain in nanoscale systems can achieve control of the physical properties in functional materials, such as ferromagnets, ferroelectrics, and superconductors. Here, we directly demonstrate the atomic-scale structure of super-tetragonal PbTiO3 nanocomposite epitaxial thin films, including the extraordinary coupling of strain transition and the existence of the oxygen vacancies. Large strain gradients, both longitudinal and transverse (∼3 × 107 m-1), have been observed. The original non-magnetic ferroelectric composites notably evoke ferromagnetic properties, derived from the combination of Ti3+ and oxygen vacancies. The saturation ferromagnetic moment can be controlled by the strain of both the interphase and substrate, optimized to a high value of ∼55 emu/cc in 10-nm thick nanocomposite epitaxial thin films on the LaAlO3 substrate. Strain engineering provides a route to explore multiferroic systems in conventional non-magnetic ferroelectric oxides and to create functional data storage devices from both ferroelectrics and ferromagnetics.

Continuum thermomechanics of nonlinear micromorphic, strain and stress gradient media.

A comprehensive constitutive theory for the thermo-mechanical behaviour of generalized continua is established within the framework of continuum thermodynamics of irreversible processes. It represents an extension of the class of generalized standard materials to higher order and higher grade continuum theories. It reconciles most existing frameworks and proposes some new extensions for micromorphic and strain gradient media. The special case of strain gradient plasticity is also included as a contribution to the current debate on the consideration of energetic and dissipative mechanisms. Finally, the stress gradient continuum theory emerges as a new research field for which an elastic-viscoplastic theory at finite deformations is provided for the first time. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Size-dependent axial instability of microtubules surrounded by cytoplasm of a living cell based on nonlocal strain gradient elasticity theory.

Microtubules including tubulin heterodimers arranging in a parallel shape of cylindrical hollow plays an important role in the mechanical stiffness of a living cell. In the present study, the nonlocal strain gradient theory of elasticity including simultaneously the both nonlocality and strain gradient size dependency is put to use within the framework of a refined orthotropic shell theory with hyperbolic distribution of shear deformation to analyze the size-dependent buckling and postbuckling characteristics of microtubules embedded in cytoplasm under axial compressive load. The non-classical governing differential equations are deduced via boundary layer theory of shell buckling incorporating the nonlinear prebuckling deformation and microtubule-cytoplasm interaction in the living cell environment. Finally, with the aid of a two-stepped perturbation solution methodology, the explicit analytical expressions for nonlocal strain gradient stability paths of axially loaded microtubules are achieved. It is illustrated that by taking the nonlocal size effect into consideration, the critical buckling load of microtubule and its maximum deflection associated with the minimum postbuckling load decreases, while the strain gradient size dependency causes to increase them.

Phase-field modelling of ductile fracture: a variational gradient-extended plasticity-damage theory and its micromorphic regularization.

This work outlines a novel variational-based theory for the phase-field modelling of ductile fracture in elastic-plastic solids undergoing large strains. The phase-field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modelling. It is linked to a formulation of gradient plasticity at finite strains. The framework includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones, and guarantees on the computational side a mesh objectivity in post-critical ranges.

Dynamic stability of the euler nanobeam subjected to inertial moving nanoparticles based on the nonlocal strain gradient theory.

This research studied the dynamic stability of the Euler-Bernoulli nanobeam considering the nonlocal strain gradient theory (NSGT) and surface effects. The nanobeam rests on the Pasternak foundation and a sequence of inertial nanoparticles passes above the nanobeam continuously at a fixed velocity. Surface effects have been utilized using the Gurtin-Murdoch theory. Final governing equations have been gathered implementing the energy method and Hamilton's principle alongside NSGT. Dynamic instability regions (DIRs) are drawn in the plane of mass-velocity coordinates of nanoparticles based on the incremental harmonic balance method (IHBM). A parametric study shows the effects of NSGT parameters and Pasternak foundation constants on the nanobeam's DIRs. In addition, the results exhibit the importance of 2T-period DIRs in comparison to T-period ones. According to the results, the Winkler spring constant is more effective than the Pasternak shear constant on the DIR movement of nanobeam. So, a 4 times increase of Winkler and Pasternak constants results in 102 % and 10 % of DIR movement towards higher velocity regions, respectively. Furthermore, the effect of increasing nonlocal and material length scale parameters on the DIR movement are in the same order regarding the magnitude but opposite considering the motion direction. Unlike nonlocal parameter, an increase in material length scale parameter shifts the DIR to the more stable region.