Higher-order Strain Research Articles

Hamiltonian System-Based Symplectic Framework for Analytical Vibration Analysis of Microplates

Abstract The classical elasticity is inadequate for the dynamic analysis of microplates due to the size effect. This study incorporates a higher-order strain gradient theory into the Hamiltonian system-based symplectic framework and derives new analytical solutions for the free vibration of microplates. The analytical solutions are obtained using rigorous mathematical techniques, including separation of variables, symplectic eigen expansion, and superposition, without relying on predetermined solution forms. Hence, they are not restricted to Lévy-type boundary conditions. Using these analytical solutions, we present comprehensive vibration results for microplates and perform detailed parametric studies to explore the impact of length scale parameters on the natural frequencies. Given the growing demand for microplates in advanced engineering applications, the obtained analytical solutions are expected to facilitate their design and performance optimization.

Hypoplastic model for solid-like and fluid-like granular flows

We present a unified constitutive model for the solid-like and fluid-like behaviour of granular materials by decomposing the stress rate tensor into frictional and collisional parts. An enhancement to the classical hypoplastic model is implemented to consider the acceleration effect in both steady and unsteady granular flows, achieved by introducing a higher-order strain rate. This model can effectively capture the rate-independent and rate-dependent behaviours observed in solid-like and fluid-like granular flows. Performance and validation of the unified model are presented through numerical simulations of diverse element tests.

Higher-order displacement, strain, and stress analyses of origami graphene auxetic metamaterial-reinforced cylindrical shell

Higher-order displacement, strain, and stress analyses of origami graphene auxetic metamaterial-reinforced cylindrical shell

Nonlinear optimization DIC method inspired by unsupervised learning for high order displacement measurement

Digital Image Correlation (DIC) is the most widely used non-contact full-field strain measurement method, but due to the influence of subset calculations in principle, measuring complex deformation with high strain gradients within the subset remains a challenge. DIC based on neural networks performs well in complex deformation measurement scenarios, but its measurement accuracy and generalization ability depend on extensive training sets. This paper proposes a nonlinear optimization DIC method inspired by unsupervised learning that requires only a single pair of images. Unlike traditional unsupervised algorithms of optical flow estimation or Particle Image Velocimetry (PIV), the shape function in traditional DIC is incorporated into the loss function. This allows the use of only the reference image and the deformed image as input dataset, and the correct displacement field can be output after parameter optimization through training. This method does not require additional training sets to measure high-order strain fields like traditional neural networks-based DIC. The proposed method has demonstrated promising results in simulation experiments for high-order strain measurement and does not require additional datasets of the same type. In the measurement of the star-shaped displacement fields, its accuracy is better than the traditional DIC based on the subset, and it maintains convergence even when the strain is large, a situation where traditional DIC fails to converge. In addition, compared with the network obtained by supervised training as a pre-trained model, the method proposed in this paper can achieve superior results. The methods of the article can theoretically be applied to the result optimization of other networks beyond those discussed in this paper.

Elastic-Gap Free Formulation in Strain Gradient Plasticity Theory

Abstract This article presents an elastic-gap free isotropic higher-order strain gradient plasticity theory that effectively captures dissipation associated to plastic strain gradients. Unlike conventional methods that divide the higher-order stress, this theory focuses on dividing the plastic strain gradient into energetic and dissipative components. The moment stress that arises from minimizing a dissipating potential demonstrates a nonlinear evolution over time, resembling the Armstrong–Frederick nonlinear kinematic hardening rule in classical plasticity. The thermodynamically consistent framework establishes additional dissipation in the dissipation inequality. The energetic moment stress saturates as the effective plastic strain increases during plastic flow. In contrast to the Gurtin-type nonincremental model, the proposed model smoothly captures the apparent strengthening at saturation without causing a stress jump. A passivated shear layer is analytically assessed to demonstrate that the proposed theory exhibits the same amount of dissipation as the existing Gurtin-type model when they show similar shear responses at saturation. It is also shown that the plastic flow remains continuous under nonproportional loading conditions using an intermediately passivated shear layer problem. Finally, the proposed theory is validated against a recent experiment involving combined bending torsion of an L-shaped beam using a 3D finite element solution. Overall, the proposed model provides an alternative approach to evaluating the size effect within the nonincremental isotropic strain gradient plasticity theory without introducing any stress jump.

The effect of the viscoelastic support and GRPL-reinforced foam material on the thermomechanical vibration response of piezomagnetic sandwich nanosensor plates

This paper investigates the vibration characteristics of a sandwich nanosensor plate composed of piezoelectric materials, specifically barium and cobalt, in the upper and lower layers, and a core material consisting of either ceramic (silicon nitride) or metal (stainless steel) foams reinforced with graphene (GPRL). The study utilized the novel sinosoidal higher-order deformation theory and nonlocal strain gradient elasticity theory. The equations of motion for nanosensor sandwich graphene were derived using Hamilton's principle, considering the thermal, electroelastic, and magnetostrictive characteristics of the piezomagnetic surface plates. These equations were then solved using the Navier method. The core element of the sandwich nanosensor plate can be represented using three distinct foam variants: a uniform foam model, as well as two symmetric foam models. The investigation focused on analyzing the dimensionless fundamental natural frequencies of the sandwich nanosensor plate. This analysis considered the influence of three distinct foam types, the volumetric graphene ratio, temperature variation, nonlocal parameters, porosity ratio, electric and magnetic potential, as well as spring and shear viscoelastic support. Furthermore, an analysis was conducted on the impact of the metal and ceramic composition of the central section of the sandwich nanosensor plate on its dimensionless fundamental natural frequencies. In this context, the use of ceramic as the central material results in a mean enhancement of 33% in the fundamental natural frequencies. In contrast, the incorporation of graphene into the core material results in an average enhancement of 27%. The thermomechanical vibration behavior of the nanosensor plate reveals that the presence of graphene-supported foam and a viscoelastic support structure in the core layer leads to an increase in thermal resistance. This increase is dependent on factors such as the ratio of graphene, porosity ratio of the foam, and parameters of the viscoelastic support. Metal foam or ceramic foam has been found to enhance thermal resistance when compared to solid metal or ceramic core materials. The analysis results showed that it is important to take into account the temperature-dependent thermal properties of barium and cobalt, which are piezo-electromagnetic materials, and the core layer materials ceramics and metal, as well as the graphene used to strengthen the core. The research is anticipated to generate valuable findings regarding the advancement and utilization of nanosensors, transducers, and nano-electromechanical systems engineered for operation in high-temperature environments.

A novel reduced-order method using mixed nonlinear kinematics for geometrically nonlinear analysis of thin-walled structures

Reduced-order methods based on the Koiter perturbation theory are applicable only to structural buckling problems, and high-order strain energy derivatives influence the computational efficiency of the methods. In this paper, a novel reduced-order method is proposed for the geometrically nonlinear analysis of thin-walled structures with large deflections. Mixed nonlinear kinematics are developed for the reduced-order method using co-rotational and updated von Kármán formulations. The co-rotational kinematics are applied to determine the internal force and tangent stiffness, whereas the third- and fourth-order strain energy derivatives are calculated using the updated von Kármán kinematics. Reduced-order models with one degree of freedom is constructed based on the perturbation theory, the solutions of which are nonlinear predictors of the geometrically nonlinear response. The use of mixed nonlinear kinematics significantly reduces the computational cost of constructing a reduced-order model. Nonlinear predictors are corrected using internal force-based residuals to ensure the accuracy of the geometrically nonlinear analysis. A geometrically nonlinear response with favorable smoothness is efficiently achieved using large step sizes in the path-following analysis. Various numerical examples, including the stiffened plate and wing structure, are provided to validate the accuracy and efficiency of the proposed method.

Nonlinear dynamic behavior of carbon nanotubes incorporating size effects

This research endeavors to investigate the geometrically nonlinear dynamic characteristics of Carbon Nanotubes (CNTs) while incorporating size effects based on the Second Strain Gradient (SSG) elasticity theory. To this end, the nonlinear governing equations and its corresponding boundary conditions are deduced in alignment with the Green–Lagrange strain tensor and Hamilton’s principle. Concurrently, the weak form is expounded through the utilization of the C3 continuum Hermite interpolation functions, which ensure the continuity and smoothness of higher-order strain and displacement fields. Subsequently, a perturbation methodology is introduced, incorporating nonlinear phenomena into the context of linear wave propagation within the framework of periodic structures theory. The wave propagation characteristics manifest a pronounced disparity between the models based on linear SSG theory and those through nonlinear SSG theory with stiffness hardening phenomenon. In contrast to the zigzag CNTs, armchair CNTs evince an elevated aptitude for wave control. The application of nonlinear SSG theory in combination with the wave finite element method is significant for comprehending the wave propagation characteristics of complex CNTs.

Higher-order anharmonicity and strain impact on the lattice thermal conductivity of monolayer InTe

In this work, we calculated the lattice thermal conductivity of monolayer InTe by means of phonon Boltzmann transport theory with first-principles calculated inter-atomic force constants. The higher-order phonon anharmonicity was found to play a strong impact on thermal transport in InTe. With the involvement of the phonon–phonon scattering process up to the fourth-order, the in-plane lattice thermal conductivity of monolayer InTe is 5.1 W m−1 K−1 at room temperature, which is 35% of that considering only third-order force constants. Furthermore, strain was found to be an effective way to manipulate the thermal transport in InTe, which reduces to one half when applying 5% in-plane tensile strain. The strain adjustment is due to the decreases in the phonon group velocity as well as the increase in the phonon scattering rates. These findings can enrich thermal transport properties of group-III monochalcogenides and benefit the material design of thermoelectrics and thermal management electronic devices.

Influences of External Magnetic Field on Thermo-Mechanical Vibration Analysis of Nanocomposite Beam using Higher-Order Strain Gradient Theory

Composite nanobeams with carbon nanotube (CNT) reinforcement are crucial components in various applications due to their unique mechanical and thermal properties. The present work investigates the frequency–aspect ratio relationship in composite nanobeams with CNT reinforcement, particularly under varying temperatures, magnetic field strengths, and Winkler elastic stiffnesses. The non-dimensional frequency variation with aspect ratio, influenced by temperature differences, reveals distinct behavior patterns. The introduction of a magnetic field alters frequency characteristics, with uniform distribution (UD) nanobeams displaying higher frequencies. The impact of Winkler elastic stiffness, temperature differences, and magnetic field strengths on frequency elucidates differing responses between UD, [Formula: see text]-type distribution (XD), and [Formula: see text]-type distribution (VD) nanobeams. Nonlocal strain gradient theory proves essential in capturing dynamic characteristics influenced by CNT distribution. These findings shed light on the intricate dynamics of composite nanobeams, providing valuable insights for the design of future nano-devices employing CNT-reinforced composite beams as fundamental elements.

A constitutive relation generalizing the Navier–Stokes theory to high-rate regimes

We propose a constitutive equation for flows of gases in high-rate regimes where the Navier–Stokes theory breaks down. The model generalizes the Navier–Stokes relation and agrees well with that model in all lower rate flows examined. Our proposed constitutive relation is calibrated with the method of objective molecular dynamics (OMD) using families of compressible and incompressible flows of Lennard-Jones argon. The constitutive relation makes use of the higher-order objective strain rates due to Rivlin and Ericksen (J. Rat. Mech. Anal., vol. 4, 1955, pp. 323–425). The constitutive relation is fully frame-indifferent, and the macroscopic flows corresponding to the OMD simulations are exact solutions for the proposed model. The model is shown to agree with atomistic results much better than the Navier–Stokes equations in the transition regime. The success of our model indicates that it is not higher gradients that become important in the high-rate regime, but rather higher rates of change of the strain rate tensor. While somewhat more complicated to implement than the Navier–Stokes relation, the proposed model is expected to be compatible with existing methods of computational fluid dynamics and may extend those methods to higher rate regimes, while preserving their ability to handle large spatial scales.

Experimental and theoretical evaluation of influence of hole size on deformation and fracture of elastic perforated plate

As an ideal benchmark problem for the researches of solid mechanics and fracture mechanics, the influence of the hole size on the deformation and the fracture behaviors of perforated plates has been studied for more than sixty years. However, the seemingly simple problem has not been thoroughly answered yet. To gain more understanding of the hole-size dependence of the strain/stress concentration, the present study measured the stress and the strain concentration factors by using tensile tests together with the digital image correlation analysis. A noticeable hole-size dependence was observed for both concentration factors. The experimental observations can be reasonably explained in terms of a new formulation of strain energy density which considers the contributions of both the conventional strain and the high-order strain. It is shown that the strain energy density gets more uniformly distributed when the hole gets smaller. Accordingly, the maximum strain decreases as well. Due to the positive contribution of the higher-order strain energy, the material around the hole hardens and results in a concentration factor lower than three. Because of the influence of the length scale parameter, the stress cannot be scaled by a reference strain and this may lead to a discrepancy between the stress and the strain concentration factors.

Strain topological metamaterials and revealing hidden topology in higher-order coordinates

Topological physics has revolutionized materials science, introducing topological phases of matter in diverse settings ranging from quantum to photonic and phononic systems. Herein, we present a family of topological systems, which we term “strain topological metamaterials”, whose topological properties are hidden and unveiled only under higher-order (strain) coordinate transformations. We firstly show that the canonical mass dimer, a model that can describe various settings such as electrical circuits and optics, among others, belongs to this family where strain coordinates reveal a topological nontriviality for the edge states at free boundaries. Subsequently, we introduce a mechanical analog of the Majorana-supporting Kitaev chain, which supports topological edge states for both fixed and free boundaries within the proposed framework. Thus, our findings not only extend the way topological edge states are identified, but also promote the fabrication of novel topological metamaterials in various fields, with more complex, tailored boundaries.

Vibration analysis of higher-order nonlocal strain gradient plate via meshfree moving Kriging interpolation method

In this paper, a higher-order nonlocal strain gradient plate model combining nonlocal elasticity theory with strain gradient theory in the form of adding signs is constructed. The governing equation for the higher-order nonlocal strain gradient plate model is derived by Hamilton’s principle. Two higher-order parameters with nonlocal stress and strain gradient are introduced here to explain the size effect and dispersive behaviour. The meshfree moving Kriging interpolation method is utilized to address the frequency of higher-order nonlocal strain gradient plate under complicated boundary conditions. Nonlocal elasticity theory, strain gradient theory in the form of add signs and minus signs on the effect of frequency are compared by degenerating the higher-order nonlocal strain gradient plate model. The other goal of this paper is to compare the present model with higher-order nonlocal strain gradient theory in the form of minus signs on the effect of frequency. It is shown that these two items can describe the size effect well. The present method can also produce lower natural frequencies corresponding to ultrahigh-order mode shapes and applied in a discrete medium.

Non-Linear Non-Classical Modeling of Microparticles Manipulation Based on Finite Element Method

A comprehensive model was presented using the finite element method to analyze the dynamics of cylindrical microparticles, considering non-linear and size effects. The governing equations of motion have been derived based on the weak form method. The stiffness and mass matrices, along with the force vectors, have been presented for non-classical non-linear models of both the Euler–Bernoulli and Timoshenko types. The manipulation dynamics of cylindrical gold microparticles was simulated using FEM. In the simulations, four models of Euler–Bernoulli and Timoshenko beams were utilized. A mesh independence study was investigated. In non-linear models, the stiffness matrix takes into account the effects of higher-order strain terms, leading to a decrease in the maximum bending of the microparticle. The examination of the effects of aspect ratio showed that decreasing it reduces the deflection. After investigating the effects of length scale parameter and aspect ratio, a gold microparticle was manipulated by 200 nm using various models, and the final position of the particle was recorded. To validate the results, the deformations of polystyrene microrods were compared with the available data. The results can be used to make accurate estimates for the positioning of cylindrical microparticles, such as nanowires, nanorods and nanotubes.

On Vibration Responses of Advanced Functionally Graded Carbon Nanotubes Reinforced Composite Nanobeams

This article presents an analytical approach to explore the free vibration behaviour of new functionally graded carbon nanotube-reinforced composite beams (FG-CNTRC) based on a two-variable higher-order shear deformation theory and nonlocal strain gradient theory. The beams resting on the Pasternak elastic foundation, including a shear layer and Winkler spring, are considered. The kinematic relations of the shaft are proposed according to novel trigonometric functions. The vibrated nanobeam’s motion equations are obtained via the classical Hamilton’s principle and solved using Navier’s steps. A comparative evaluation of results against predictions from literature demonstrates the accuracy of the proposed analytical model. Moreover, a detailed parametric analysis checks for the sensitivity of the vibration response of FG nanobeams to nonlocal length scale, strain gradient microstructure scale, material distribution, constant spring factors, and geometry. The current work presents the free vibration problem of supported (FG-CNTRC) beams reinforced by different patterns of carbon nanotube (CNT) distributions in the polymeric matrix.

Nonlinear fully coupled thermoelastic transient analysis of axial functionally graded composite panel

In this article, the two-way coupled thermoelastic behavior of axially graded composite plate and doubly-curved (cylindrical, spherical, hyperbolic, and elliptical) panels is examined under conductive-convective boundary and uniformly distributed loading conditions. Here, the material properties of the axial functionally graded panel are computed by employing the power-law-based Voigt’s scheme. The material and geometric nonlinearities are incorporated using the cubic-polynomial-based temperature-dependent constitutive model and higher-order kinematics-based Green-Lagrange strain, respectively. The temperature profile is obtained through thermally nonlinear theory associated with high temperature, which is used to extract the energy equations obtained from the first law of thermodynamics and deformation-dependent entropy relation. The weak forms of motion and heat-transfer equations are derived using Galerkin’s method and further combined into the two-way coupled field equation through 2D-finite element approximation via Lagrangian elements. The transient responses are computed via the Newmark and the Crank-Nicolson schemes, whereas the nonlinear iterations are executed through Picard’s iteration technique. The performance of the fully coupled model for an axially graded (metal/ceramic) structure is verified with analytical, numerical, and experimental results and tested through a variety of numerical illustrations under various sets of conditions.

Advances on kinematically exact rod models for thin-walled open-section members: Consistent warping function and nonlinear constitutive equation

This work proposes two advances on kinematically exact rod models for thin-walled open section members: (i) an explicit analytical expression for the cross-section warping function, including both primary and secondary warping for arbitrary section geometries, and (ii) a nonlinear elastic constitutive equation with all higher-order strain terms included, suitable for truly finite strains. By not neglecting any of the strain terms at the constitutive level, full bending, compression and torsional strain couplings are enabled. This improves the model performance in capturing buckling phenomena and describing the developed post-critical configurations. Since the stress-resultants end up with elaborate expressions, numerical rather than analytical integration over the cross-section is preferred for their computation, which explains the need for knowing a priori the warping function (and its directional derivatives) in the entire cross-section, including in the thickness direction. The advances are implemented in a 7-DOF (3 displacements, 3 rotations and one warping intensity) rod model available in an in-house finite element program. Validation is performed using existing results from the literature as well as solutions obtained with shell models from a commercial software. The proposed advances proved to fix pathological behavior of previous rod models in both torsional and lateral–torsional buckling problems, as well as numerical convergence issues in developed post-critical configurations.

A self - Consistent higher-order strain gradient plasticity analysis of micropolar dual-phase crystalline FGM made of Titanium-boride/ Titanium

A self-consistent higher order couple stress model is based on Fleck - Hutchinson Strain Gradient Plasticity to consider flexural behavior of brittle - ductile microscale structures. The dual phasic functionally graded material made of Titanium-boride/Titanium structure is modeled by defining a material flow stress which reflects a parameter of combined effective length scale, dependent on Geometrically Necessary Dislocations. The dual phasic crystalline micro-beam is built with the properties having continuously and non-linearly variation along the thickness. The perfectly plastic behavior of micropolar functionally graded material is described by incorporating a ductile –like mixture law, so-called TTO model, possessing a simple possibility of transition through defining a graded phase acting as an equilibrium mixed phase. Further, the effects of the elastic foundation and temperature rise on the plastic deformations are investigated. While an energy method is used to derive governing equations, the obtained results are verified individually in three steps.

Structural deformation and mechanical response of CrS2, CrSe2 and Janus CrSSe

In the framework of density functional theory (DFT), we investigate the structural deformation, and mechanical behavior of the Janus CrSSe, which has out-of-plane structural asymmetry, with conventional transition metal dichalcogenides (TMDs) CrS 2 and CrSe 2 . The Janus CrSSe could be a potential candidate for machinable optoelectronic and piezoelectric applications. We predict that these compounds are chemically, mechanically, and dynamically stable with the covalent bond between the TM(Cr) and chalcogen(X=S, Se) atoms. Due to the influence of tensile strain, the Cr-X bond length of each monolayers increases and the thickness decreases. Interestingly, the in-plane stiffness, shear and layer moduli, Poisson’s ratio, ultimate bi/uni-axial stress of Janus CrSSe are in between the values of CrS 2 and CrSe 2 monolayers. Similar to TMDs, the orientation-dependent in-plane stiffness and Poisson’s ratio demonstrates the isotropic behavior in Janus CrSSe. Furthermore, it can sustain a larger value of uni/bi-axial tensile strain with the critical strain equivalent to CrX 2 monolayers. By applying higher-order strain, we have also found average elastic–plastic behavior as expected. These findings demonstrate that the Janus CrSSe monolayer is a mechanically stable and ductile compound that maintains the hybrid behavior. • Due to covalent bonding between Cr and chalcogen (S/Se), CrS 2 , CrSe 2 , and the Janus CrSSe monolayers are chemically, mechanically and dynamically stable. • The in-plane stiffness, shear and layer moduli, Poisson’s ratio, and ultimate bi/uni-axial stress of Janus CrSSe are in between the values of CrS 2 and CrSe 2 monolayers. • The orientation dependent in-plane stiffness and Poisson’s ratio demonstrate the isotropic behavior in the Janus CrSSe. • The Janus CrSSe could be a potential candidate for machinable optoelectronic and piezoelectric applications.