Gradient Elasticity Research Articles

The spectral methods for the propagation of thermoelastic waves in multilayer cylinders based on nonlocal strain gradient elasticity

The spectral methods for the propagation of thermoelastic waves in multilayer cylinders based on nonlocal strain gradient elasticity

A microsphere-homogenized strain gradient elasticity model for polymers

A microsphere-homogenized strain gradient elasticity model for polymers

First impressions count: Therapists’ impression on patients’ motivation and helping alliance predicts psychotherapy dropout

Objective With meta-analytically estimated rates of about 25%, dropout in psychotherapies is a major concern for individuals, clinicians, and the healthcare system at large. To be able to counteract dropout in psychotherapy, accurate insights about its predictors are needed. Method We compared logistic regression models with two machine learning algorithms (elastic net regressions and gradient boosting machines) in the prediction of therapy dropout in two large inpatient samples (N = 1,691 and N = 12,473) using baseline and initial process variables reported by patients and therapists. Results Predictive accuracies of the two machine learning algorithms were similar and higher than for logistic regressions: Therapy dropout could be predicted with an AUC of .73 and .83 for Sample 1 and 2, respectively. The initial evaluation of patients’ motivation and the therapeutic alliance rated by the respective therapist were the most important predictors of dropout. Conclusions Therapy dropout in naturalistic inpatient settings can be predicted to a considerable degree by using baseline indicators and therapists’ first impressions. Feature selection via regularization leads to higher predictive performances whereas non-linear or interaction effects are dispensable. The most promising point of intervention to reduce therapy dropouts seems to be patients’ motivation and the therapeutic alliance.

A size‐dependent model of strain gradient elastic laminated micro‐beams with a weak adhesive layer

AbstractA size‐dependent model for a laminated micro‐beam with a soft adhesive is developed in the framework of strain gradient elasticity theory. The layered beam is constituted by two Euler–Bernoulli strain gradient elastic isotropic beams, joined through a strain gradient spring‐type contact law at the adhesive level. The governing bending and extensional equations and boundary conditions are obtained by using the variational principle. The differential system shows a coupling between the flexural and axial behaviors of the upper and lower beams, due to the presence of interface terms related to shear and peeling stresses. Two benchmark problems have been presented through their closed‐form solutions, namely a simply‐supported laminated beam subjected to a uniform distributed load and a mode 2‐type loading configuration of a layered axially deformable beam. Size effects and non‐local phenomena, due to high strain concentrations, are highlighted. Though simple in their features, the examples prove the efficiency of the proposed approach in designing micro‐scale layered beams.

Predicting criminal offence in adolescents who exhibit antisocial behaviour: a machine learning study using data from a large randomised controlled trial of multisystemic therapy.

Accurate prediction of short-term offending in young people exhibiting antisocial behaviour could support targeted interventions. Here we develop a set of machine learning (ML) models that predict offending status with good accuracy; furthermore, we show interpretable ML analyses can complement models to inform clinical decision-making. This study included 679 individuals aged 11-17 years who displayed moderate-to-severe antisocial behaviour, from a controlled trial of Multisystemic-therapy in England. The outcome was any criminal offence in the 18 months after study baseline. Four types of ML algorithms were trained: logistic regression, elastic net regression, random forest, and gradient boosting machine (GBM). Prediction models were developed (1) using predictors readily available to clinicians (e.g. sociodemographics, previous convictions), and (2) with additional information (e.g. parenting). Model agnostic feature importance values were calculated and the most important predictors identified. Nested cross-validation with 100 iterations of random data splits and 10-fold cross-validation within each iteration was employed, and the average predictive performance was reported. Among the ML models using readily available predictors, the GBM is the strongest model (AUC 0.85, 95% CI 0.85-0.86); the other models have average AUCs of 0.82. This performance was better than using only the total number of previous offences as the predictor (0.67, 0.66-0.68), and the model simply assuming past offending status as the prediction (0.81, 0.80-0.81). Additional predictors slightly increased the performance of logistic regression and random forest models but decreased the performance of elastic net regression and gradient boosting machine-based models. The potential utility of ML approaches for accurately predicting criminal offences in high-risk youth is demonstrated. Interpretable ML-based predictive models could be utilised in youth services or research to help develop and deliver effective interventions.

Dual cross-linked polyurethane-alginate biomimetic hydrogel for elastic gradient simulation in osteochondral structures: Microenvironment modulation and tissue regeneration

The distinctive composition and functions of osteochondral structures result in constrained regeneration. Insufficient healing processes may precipitate the emergence of tissue growth disorders or excessive subchondral bone formation, which can culminate in the deterioration and failure of osteochondral tissue repair. To overcome these limitations, materials designed for osteochondral repair must provide region-specific modulation of the microenvironment and mechanical compatibility. To address these challenges, we propose a method to create continuous hydrogels with distinct structural and functional properties by a precise cross-linking method. We have developed an innovative polyurethane enriched with dimethylglyoxime, facilitating the coordinated loading and precise release of Zn2+. This strategy enables the meticulous control of alginate cross-linking, resulting in an elastic gradient hydrogel that closely resembles the osteochondral interface. The SeSe within the hydrogel effectively modulates the inflammatory microenvironment and fosters the M2 polarization of macrophages. The hydrogel's lower layer is designed to rapidly release Zn2+, thereby enhancing bone regeneration. The upper layer is intended to prevent bone overgrowth and stimulate chondrogenic differentiation. This dual-layer strategy allows targeted stimuli to each region, promoting the seamless integration of neoosteochondral tissue. Our study demonstrates the potential of this stratified hydrogel in achieving uniform and smooth osteochondral tissue regeneration.

An analytical method for thermal behavior analysis of long-span suspension bridges: Based on elastic catenary cable and gradient optimization

The thermal loads of suspension bridges will change the unstrained length of the main cable and hanger and lead to significate variation of the cable shape. Since the main cable is the main load-bearing structure of the suspension bridge, the change of cable shape will lead to significant bridge deformation. Hence, the thermal behavior analysis of suspension bridge is crucial. This paper introduces an analytical method to calculate the temperature-induced deformation and internal force changes of suspension bridges. Comparing to the prevalent Finite Element Method (FEM), this method has the advantage of clear calculation process and physical significance. First, the deformation of the cable-hanger structure is estimated based on the elastic catenary theory. Based on the differential relationship between curvature and force, the deformation of the pylon and beam is calculated. Finally, optimization equations are established by closing the coordinates of key nodes. The Levenberg-Marquardt (LM) method is employed for solving. In a numerical example bridge with a main span of 1700 m, the maximum error of deformation calculation based on this method is only 5.8 mm. This proves the effectiveness and accuracy of this method. Additionally, this paper analyzes the influence of various temperature loads and the mechanical property of bridge members on the thermal behavior. These works will provide valuable insights to engineers in bridge design.

3D coupled vibrations of asymmetric nano-particle mass sensors: Two-phase integral nonlocal strain gradient model and MD simulation

Since it is possible for the nano-particle to have any asymmetric shape and may attach to every position of the resonator-based mass sensors, any type of mass eccentricity and subsequently coupled vibrations are possible. So, in the present work, the 3D coupled axial-torsional-flexural vibrations of the mass nano-sensors are investigated considering both in-plane and out-of-plane bending as well as axial and torsional vibrations. In addition, the integral form of the two-phase local/nonlocal strain gradient (LNSG), as the consistent type of the common nonlocal strain gradient (NSG) elasticity theory, is employed for the first time to consider the size effects in the mass nano-sensors. A quasi-3D coupled FE model is constructed with no shear-locking within the complicated environment of the integral LNSG. The impacts of the LNSG elasticity and diverse types of possible coupling in changing the vibrational behavior of the mass nano-sensor are scrutinized and their crucial role in modeling of the nano-sensors is indicated. Furthermore, molecular dynamic (MD) simulations are implemented to identify and analyze the coupled vibrations of a mass nano-sensor consisting of a fixed-free carbon nanotube. Comparison between the MD results with those of the present coupled FE model reveals that considering the coupling effects reduces the modeling error, especially in the cases in which the coupling influences intensify. The present work can help to provide more accurate models of mechanical resonance-based nanosensors to detect nanoparticles with different shapes, larger sizes, and high mass ratios, which can lead to achieving sensors with higher sensitivity.

Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity

In this paper, we investigate the axisymmetric Hertzian contact problem at micro-/nanoscale. The deformation of material bulk is described by a simplified theory of strain gradient elasticity, and the influence of surface tension is integrated based on the surface elasticity theory. Using the Mindlin's potential function method and double Fourier integral transform, the normal surface displacement induced by a concentrated force is derived in a closed form. Following this, the contact between a rigid sphere and an elastic half-space is formulated in terms of singular integral equation, which is numerically solved by applying the Gauss-Chebyshev method. The results indicate that the distribution of contact pressure is distinctly different from that in classical elasticity theory. The indented substrate tends to perform stiffer due to the effects of surface tension and strain gradient elasticity. When the contact radius is comparable with the material length parameter, the indentation force (depth) can be ten (three) times of that given by classical Hertz theory.

Mixed finite element implementation of plane crack problems within strain gradient elasticity

AbstractAn approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method is used, where stresses, strains, and displacements are considered as independent variables. This significantly simplifies the pre‐requirement for approximating functions and allows one to use the simplest triangular finite elements with a linear approximation of displacements. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses in nodes are treated as local unknowns. This discrete problem is solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. The modeling plane crack problem (tension of a square plate with a central crack) is solved. The obtained solution agrees with the results available in the literature. Good convergence is achieved by refining the mesh for all scale parameters.

Variational consistent one-point integration with Taylor's expansion-based stabilization in the second-order meshfree Galerkin method for strain gradient elasticity

A generalized variational principle with five independent variables is proposed for strain gradient elasticity, including displacement, strain, strain gradient, stress, and double stress. Based on the principle, a one-point integration scheme is designed for the second order meshfree Galerkin method through nodal smoothed derivatives and their high order derivatives by Taylor's expansion. Since the proposed integration scheme meets the orthogonality conditions, it is variational consistent. The weak form expanded with Taylor's polynomials can be well evaluated by nodal smoothed derivatives and their high order derivatives on one quadrature point. Numerical one- and two-dimensional case studies show that the proposed integration scheme performs better than the standard Gaussian integration method in terms of accuracy, convergence, efficiency, and stability.

The influence of material gradation parallel to two unequal mode-III cracks in functionally graded materials via strain gradient elasticity theory

In this article, the strain gradient elasticity (SGE) theory is used to investigate the mechanical behaviour of a functionally graded material (FGM) weakened by two unequal collinear mode-III cracks. The SGE theory uses two material characteristic lengths, ℓ and ℓ′, to account for volumetric and surface strain-gradient factors, respectively. The directions of both cracks coincide with the material gradation, which is oriented parallel to the x-axis. The numerical outcomes of the problem are obtained by constructing the system of equations using the hyper-singular integral-differential equation technique. The influence of the material gradation parameter on various fracture parameters, including the stress intensity factor (SIF), strain and crack surface displacement (CSD), is numerically shown. In addition, different loads are considered while analysing CSD profiles in connection to the gradation parameter β. Furthermore, the effect of inter-crack distance on CSD profiles is comprehensively explored, showing information between crack geometry and material gradation.

A size-dependent nonlinear isogeometric approach of bidirectional functionally graded porous plates

Recent progress in materials science and fabrication technologies, such as additive manufacturing (AM) or 3D printing, has facilitated the design and production of intricate multi-directional functionally graded (FG) materials which are playing a pivotal role in interdisciplinary engineering applications. Developing mathematical modeling for such nanostructures, however, remains challenging. The primary objective of this research, therefore, is to present a mathematical model for studying the static bending characteristics considering the geometric nonlinearity of bidirectional FG nanoplates with internal pores. The mathematical formulations are established by utilizing the first-order shear deformation theory and the von-Kármán assumption within the framework of an isogeometric approach based on NURBS basis functions. To account for the influence of both strain gradient and nonlocal elasticity, we adopt the nonlocal strain gradient theory including two independent material length scale parameters to predict the geometric nonlinearity performance of structures at small-scale levels. The mechanical properties of multi-directional FG material models are assumed to be continuously graded in two directions based on power law, while internal pores can be dispersed into two typical distribution patterns. Several numerical investigations are performed to evaluate the substantial impact of certain parameters on the nonlinear deflections of bidirectional FG nanoplates with pores under various conditions. Compared to existing studies, the key contribution is to present a robust numerical model aimed at elucidating both the stiffness-softening and stiffness-hardening mechanisms of multi-directional FG nanostructures under static bending conditions with geometric nonlinearity. These findings enhance our insights into nonlinear bending performance and contribute valuable design guidelines for future small-scale engineering structures. These innovative designs have become popular and are now crucial in state-of-the-art engineering applications, such as aerospace, nuclear systems, or thermal resistance devices.

Wave solutions in nonlocal integral beams

Wave propagation in slender beams is addressed in the framework of nonlocal continuum mechanics. The elastodynamic problem is formulated exploiting consistent methodologies of pure integral, mixture and nonlocal strain gradient elasticity. Relevant wave solutions are analytically provided, with peculiar attention to reflection and near field phenomena occurring in presence of boundaries. Notably, the solution field is got as superimposition of incident, reflected, primary near field and secondary near field waves. The latter contribution represents a further effect due to the size dependent mechanical behaviour. Limit responses for vanishing nonlocal parameter are analytically evaluated, consistently showing a zero amplitude of the secondary near field wave. Parametric analyses are carried out to show how length scale parameter, amplitude of incident wave and geometric and elastic properties of the beam affect the amplitudes of reflected, primary near field and secondary near field waves. The results obtained exploiting different nonlocal integral elasticity approaches are compared and discussed.

Flexoelectric Enhancement of Strain Gradient Elasticity Across a Ferroelectric-to-Paraelectric Phase Transition.

We study the temperature dependent elastic properties of Ba0.8Sr0.2TiO3 freestanding membranes across the ferroelectric-to-paraelectric phase transition using an atomic force microscope. The bending rigidity of thin membranes can be stiffer compared to stretching due to strain gradient elasticity (SGE). We measure the Young's modulus of freestanding Ba0.8Sr0.2TiO3 drumheads in bending and stretching dominated deformation regimes on a variable temperature platform, finding a peak in the difference between the two Young's moduli obtained at the phase transition. This demonstrates a dependence of SGE on the dielectric properties of a material and alludes to a flexoelectric origin of an effective SGE.

A method to represent the strain hardening effect in the hyper-elastic model within a fully Eulerian framework

We propose a method to treat the strain hardening effect in the hyperelastic model. Coupled with the level set interface capturing method and a real ghost fluid method, it is formulated in a fully Eulerian framework. The HLLD approximate Riemann solver is used to obtain the interface state. We use the elastic inverse deformation gradient tensor to track the material deformation state. Once the solid starts to yield, along the normal direction in stress space, an iterative algorithm is used to relax the elastic inverse deformation gradient tensor to target value where the equivalent stress meets the requirement of consistency condition in plasticity. This approach is intuitive and can avoid the use of the physical delay time, which is difficult to calibrate from experiments. We use the Wilkins flying plate impact, cylindrical shell collapse, Taylor impact and the conical-nosed projectile penetration test cases to validate the method. The results show good agreement with the theoretical analysis and experiments. This approach can also apply without further extension when thermal softening effect or damage models are considered.

EFFECT OF ELASTIC GRADIENTS ON THE FRACTURE RESISTANCE OF TRI-LAYER RESTORATIVE SYSTEMS

To assess the impact of elastic gradients formed among restorative material, cement, and substrate on the fracture resistance of tri-layer restorative systems. Four CAD/CAM materials were utilized, two glass-ceramics (IPS e.max CAD, Vita Suprinity) and two resin-ceramic hybrids (Vita Enamic, Lava Ultimate). Their fracture resistance was examined by biaxial flexure (n=8) and Hertzian indentation (n=10) tests. Statistical analysis was conducted using ANOVA and Tukey tests (p=5%). Finite element analysis (FEA) was employed to simulate the Hertzian indentation test and elucidate the stress-fields formed on the intaglio surface below the loading area. The biaxial flexural strength (MPa) of glass-ceramics exceeded the hybrid materials (e.max 417a, Suprinity 230b, Enamic 138c, and Lava Ultimate 183bc). Conversely, the load-bearing capacity (N) of the materials bonded to dentin analog demonstrated the opposite trend, with the hybrid materials achieving superior results (e.max 830C, Suprinity 660D, Enamic 1822B, and Lava Ultimate 2593A). The stress-fields observed by FEA were coherent with the experimental results for Hertzian flexural stresses (MPa): e.max 501A, Suprinity 342C, Enamic 406B, whereas no tensile stress was observed at the intaglio surface of Lava Ultimate. Detailed analysis of the fracture resistance of the tri-layer restorative systems showed that the elastic gradients play a more significant role than the flexural strength of the restorative materials. The coherence of the elastic moduli between the restorative material and supporting structures results in reduced tensile stress concentration at the intaglio surface beneath the loading area and enhances the ability to withstand load.

Investigation of Thermal Impacts and Different Gradient Elasticity Theories on Wave Propagation through the Polymer Matrix Incorporated Carbon Nanotube Walls

Investigation of Thermal Impacts and Different Gradient Elasticity Theories on Wave Propagation through the Polymer Matrix Incorporated Carbon Nanotube Walls

A stretchable and self-powered strain sensor with elastomeric electret

Stretchable deformation sensors play an important role in the perception and autonomy of soft robots. While various sensors have been reported, the mechanical sensor with self-powered capability and stretchability to suit extreme environment is urgently required. In this study, a stretchable, self-powered, sensitivity adjustable, and long-term stability strain sensor based on the stretchable electret is researched for deformation monitoring. This stretchable and self-powered sensor is composed of the dielectric elastomer with designed elastic modulus gradient and the stretchable electret with elastomeric and Nano-particles. The electro-mechanical capability of this designed sensor is researched and optimized by multi-objective optimization approach. The sensor exhibits self-powered capability, stretchability (tensile strain from 0% to 20%), tunable sensitivity (optimized from 5.7 to 5.9 pC mm−1), and stability, where electromechanical performance remains stable after 8000 stretching cycles. The feasibility of its autonomous sensing which promotes a wide range of potential application in soft robotics is demonstrated. This elastic modulus gradient-induced and net charge design significantly broadened the potential applications with large deformation and low-attached stiffness.

In-Plane Vibrations of Elastic Lattice Plates and Their Continuous Approximations

This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding continuous plate. The considered lattice converges to a 2D linear isotropic elastic continuum at the asymptotic limit for a sufficiently small lattice spacing. This continuum has a free Poisson’s ratio, which must be lower than that foreseen by the rare-constant theory, to preserve the definite positiveness of the associated discrete energy. Exact solutions for the in-plane eigenfrequencies and modes are analytically derived for the discrete plate. The stiffness characterising the lattice interactions at the boundary is corrected to preserve the symmetry properties of the discrete displacement field. Two classes of constraints are considered, i.e., sliding supports at the nodes, one normal and the other parallel to the boundary. For both boundary conditions, a single equation for the eigenfrequency spectrum is derived, with two families of eigenmodes. Such behaviour of the lattice plate is like that of the continuous plate, the eigenfrequency spectrum of which has been given by Rayleigh. The convergence of the spectrum of the lattice plate towards the spectrum of the continuous plate from below is confirmed. Two continuous size-dependent plate models, considering the strain gradient elasticity and non-local elasticity, respectively, are built from the lattice difference equations and are shown to approximate the plane lattice accurately.