Strain Gradient Theory Research Articles

Peridynamic correspondence model with strain gradient elasticity for microstructure dependent size effects

This study presents a peridynamic (PD) correspondence model with strain gradient elasticity (SGE) to capture size effect on strength of nano- and micro-scale structures. The classical elasticity theory lacking a length scale parameter does not account for the microstructure-dependent size effects. Strain Gradient (SG) elasticity is an extension of the classical elasticity with a length scale parameter(s) which can be linked to the microstructure of the material. Peridynamic theory introduces damage into the constitutive relations in a natural way by allowing for interactions of a material point within its horizon which is referred to as the internal length parameter. The correspondence model of the PD theory permits the incorporation of the constitutive relationship of SG theory. Therefore, the present approach combines the effects of PD and SGE length scale parameters on the stiffness and strength of the material. The resulting equations present two length parameters: the horizon of a material point in PD theory and the characteristic length in SGE theory. The PD equation of motion with SGE (PDSG) is free of spatial derivatives and allows for the imposition of classical and nonclassical boundary conditions. The PDSG is first applied to investigate the deformation response of a one-dimensional nanoscale film subjected to a quasi-static tensile load and a dynamic load through an initial constant strain. The static response is compared with the analytical solution for varying ratio of SG length parameter to PD horizon. The dynamic response is compared with a computational solution while satisfying the nonclassical boundary conditions. Subsequently, the PDSG is applied to study the deformation response of a two-dimensional nanoscale film subjected to a quasi-static axial load and a specified tangential end displacement. These two cases are considered under special displacement constraints to compare with the available analytical solutions. Subsequently, a two-dimensional nanoscale film with or without an internal crack is subjected to uniform tensile loading. Finally, crack propagation is simulated under incremental displacement loading. The results capture the analytical predictions and the expected increase in stiffness with increasing ratio of SG length parameter to PD horizon.

Strain gradient theory-based vibration analyses for functionally graded microbeams reinforced by GPL

On the basis of modified strain gradient theory (MSGT), the free vibration of graphene nanoplatelets (GPL) reinforced functionally graded (FG) microbeam on elastic foundation is studied. Firstly, material properties of the composites are calculated by Halpin-Tsai model and linear mixed model. Then, the MSGT and Reddy beam theory are adopted to establish the fundamental equations of the microbeam on the elastic foundation. And the governing differential equations related to the size effect are derived by the Hamiltonian principle and solved by the Navier method. Finally, the influences of the graded distribution, mass fraction, geometric shape of GPL and size effect on the vibration behaviors are discussed in detail. Results show that these parameters have significant influence on the natural frequencies. The methods to improve the natural frequencies of the microbeams are proposed considering the influence of internal microstructure on the overall characteristics. The research results are of great significance to design this kind of micro-nano structures.

Coupled bandgaps and wave attenuation in periodic flexoelectric curve nanobeams

Curve beams are widely used in structural engineering like civil engineering and aircraft structures. Due to their coupling effects of bending, shear, and torsion, curve beams have a significant role in wave propagation fields as complete bandgaps can be generated. With the miniaturization of devices, it becomes increasingly imperative to investigate wave characteristics of curve beams at the nanoscale, taking into account the flexoelectric effect. In this study, a theoretical model for the periodic flexoelectric curve beams is established under the strain-gradient electro-elasticity theory. Subsequently, a customized state-space-based transfer matrix method for flexoelectric curve beam is specifically proposed, referred to as FCB-TMM. According to the Floquet-Bloch theorem, the dispersion relations for the periodic flexoelectric curve beams with intriguing coupling characteristics can be obtained and verified. Additionally, the influence of the flexoelectric effect and strain gradient effect on its complete bandgaps is analyzed. The results indicate that the flexoelectric effect plays a vital role in wave propagation at small scales, causing a shift in the bandgaps towards higher frequencies. Ultimately, the synergistic effects of significant geometric and material parameters combined with the flexoelectric effect on the bandgaps are systematically discussed. It is discovered that changing flexoelectric coefficients may alter the pattern of bandgap variation with these geometric and material parameters, while the influence of flexoelectric coefficients on bandgaps varies depending on the values of these parameters. Our investigation offers an insight into understanding the wave propagation of the curve nanobeams, and thus further guides the application and development of flexoelectric wave components in MEMS/NEMS.

Nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator: Introducing machine learning approach to measure nonlinear phase velocities

In today's rapidly changing corporate environment, business executives are turning to cutting-edge technology to solve complex problems efficiently. A machine learning approach allows for a more thorough understanding and quantification of decision-making outcomes, leading to increased confidence in business decisions. In this work for the first time, machine learning approach and mathematical simulation are used to model nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator. Using mathematical modeling, the current work studies the nonlinear phase velocity in a sandwich nanoshell composed of a bi-directional functionally graded (BD-FG) core and piezoelectric patch. The modeling uses first-order shear deformation theory with the fairly thick hypothesis to describe the displacement fields of piezoelectric layers. For core layer, a higher order shear deformation theory with terms of higher order in the Taylor series expansion is presented. The nanosystem is accurately modeled using nonlocal strain gradient theory, incorporating nonlocal and length scale characteristics. The solution approach section outlines the multiple scale method for the time domain and the harmonic solution methodology for the displacement domain to address nonlinear equations. To reduce the computational expenses, a machine-learning approach is introduced to address the nonlinear vibration issue in the current study. Validating the conclusions of the current study by comparing them with machine-learning solutions and another published paper can confirm their accuracy. The results section presents the impacts of factors like area of piezoelectric patch, length scale, applied voltage, nonlocality, geometry condition and FG power index on the nonlinear phase velocity of the BD-FG nanoshell coupled with piezoelectric patch.

Analysis of planar arbitrarily curved microbeams with simplified strain gradient theory and Timoshenko–Ehrenfest beam model

As the first endeavor in the context of Mindlin’s strain gradient theory, this study contributes a systematic and rigorous derivation for governing equations and boundary conditions of planar arbitrarily curved microbeams. The Timoshenko–Ehrenfest beam model is incorporated into a simplified version of Mindlin’s strain gradient theory. Kinematic unknowns include displacement components of the beam axis in the local coordinate system and the rotation of cross-section. Since the derived governing equations and boundary conditions are extremely complex, analytical solutions are not achievable for microbeams having non-uniform curvature. To facilitate the numerical analysis, two isogeometric collocation formulations are proposed, that is, displacement-based and mixed formulations. Several tests are designed to evaluate the accuracy and reliability of the proposed isogeometric collocation formulations, especially with respect to the well-known locking pathology. It is found that the mixed formulation is more accurate and robust than the displacement-based one. Therefore, the mixed formulation is then used to numerically investigate the size-dependent behavior and stiffening effect. Furthermore, some informative tests are performed to delineate the significance of the curviness in the prediction of structural responses of planar arbitrarily curved microbeams, which appears to be still an unanswered issue.

In-plane surface waves propagating in a coated half-space based on the strain-gradient elasticity theory

By employing the strain gradient elasticity theory, we investigate the propagation of in-plane surface waves in a coated half-space with microstructures. We first investigate the general case of the present problem, that is, both the surface layer and the half-space are described by the strain-gradient elasticity theory. We formulate the boundary and continuity conditions of the general case and derive the dispersion relations of the surface waves. Then we investigate two special cases: (1) the surface layer is described by the strain-gradient elasticity theory, while the half-space by the classical elasticity theory; (2) the surface layer is described by the classical elasticity theory while the half-space by the strain-gradient elasticity theory. We examine the effects of strain-gradient characteristic lengths on the dispersion curves of surface waves in all cases. This study helps to further understand the propagation characteristics of elastic waves in materials with microstructures.

Nonlocal-Strain-Gradient-Based Anisotropic Elastic Shell Model for Vibrational Analysis of Single-Walled Carbon Nanotubes

In this study, a new anisotropic elastic shell model with a nonlocal strain gradient is developed to investigate the vibrations of simply supported single-walled carbon nanotubes (SWCNTs). The Sanders–Koiter shell theory is used to obtain strain–displacement relationships. Eringen’s nonlocal elasticity and Mindlin’s strain gradient theories are adopted to derive the constitutive equations, where the anisotropic elasticity constants are expressed via Chang’s molecular mechanics model. An analytical method is used to solve the equations of motion and to obtain the natural frequencies of SWCNTs. First, the anisotropic elastic shell model without size effects is validated through comparison with the results of molecular dynamics simulations reported in the literature. Then, the effects of the nonlocal and material parameters on the natural frequencies of SWCNTs with different geometries and wavenumbers are analyzed. From the numerical simulations, it is confirmed that the natural frequencies decrease as the nonlocal parameter increases, while they increase as the material parameter increases. As new results, the reduction in natural frequencies with increasing SWCNT radius and the increase in natural frequencies with increasing wavenumber are both amplified as the material parameter increases, while they are both attenuated as the nonlocal parameter increases.

Vibration frequency and mode localization characteristics of strain gradient variable-thickness microplates

Based on the modified strain gradient theory and Kirchhoff–Love assumptions, a free vibration model is established for variable-thickness microplates with three material length scale parameters, and the corresponding Euler–Lagrange equation is derived. Keeping the microplate volume constant, three kinds of thickness variations along the X-direction are considered: linear, parabolic, and cubic variations. Further, a C2-type four-node 36-degree-of-freedom variable-thickness differential quadrature finite element is developed to solve the resulting variable-coefficient boundary value problem by combining the Gauss–Lobatto and differential quadrature rules. Galerkin approximate solution for fully simply supported microplates is provided as a benchmark for numerical method verification. The convergence and accuracy of the model are verified through several examples. Finally, the vibration frequencies of the microplates under different thickness variation types, taper ratios, thickness ratios, material length scale parameters, and boundary conditions are examined through parametric analysis. Also, the influence of each factor on the mode localization of the microplates is quantitatively characterized by the modal assurance criterion (MAC) for the first time. The numerical results show that the thickness variation has a minor effect on the vibration frequency but a significant influence on the mode shape when the volume of the microplate is constant. The strain gradient, taper ratio, and discontinuous boundary have a remarkable effect on the distribution of higher-order mode shape contour lines and the regions where vibration localization occurs.

Static and free vibration responses of nanobeams considering flexoelectricity and surface effect

In this paper, the electro-mechanical behavior of a flexoelectric nanobeam, considering the surface effect, is studied with an induced electric potential. Based on the modified strain gradient theory and Hamilton’s variational principle, the governing differential equations of the nanobeam and the corresponding boundary conditions are obtained. The exact deflection solution of the flexoelectric nanobeam is derived from static bending. The relationship between the induced electric potential, including the surface effect, and the angle of the end of the cantilever beam is presented. Moreover, the characteristic equation of natural frequency is obtained by using a separate variable method under an open circuit with surface electrodes and an induced electric potential condition (OCI). The simulated results indicate the electro-mechanical response of the cantilever beam can be controlled by adjusting the flexoelectric coefficient, residual surface stress, and material length scale parameters of strain gradient theory. The free end of the beam with a platform phenomenon is also found by setting the appropriate parameters. The results also show that the residual surface stress and the ratio of beam thickness to material length scale parameters have a more significant effect on the effective frequency shift of the flexoelectric nanobeam. Therefore, considering the induced electric potential and surface effect is of great significance for the study of flexoelectric nanobeam sensors.

Nonlinear thermal buckling analysis of temperature-dependent porous annular and circular microplates reinforced by graphene platelets by using isogeometric analysis method

In this research, by using isogeometric analysis (IGA) method, nonlinear thermal buckling analysis of porous circular and annular microplates reinforced by graphene platelets (GPLs) under uniform temperature rise is performed by taking both the temperature dependence of the material properties and the size effects into account. The closed-cell metal foam scheme combined with the Halpin-Tsai micromechanics model and the rule of mixture is utilized to identify the effective material properties of the porous nanocomposites for three GPL dispersion patterns and three porosity distributions. With the aid of the virtual work principle, the classical and the non-classical motion equilibrium formulation based on the higher-order shear deformation theory in conjunction with the modified strain gradient (MSG) theory for the nonlinear temperature-dependent buckling behaviour are established. The higher-order NURBS basis functions capable of accomplishing easily the smoothness of arbitrary continuity order are successfully utilized to solve the nonlinear governing equations by satisfying the C2-continuity of the displacement field required by the MSG theory. The thermal buckling behaviour is assessed in detail by using various parameters such as the thickness-to-radius ratio, the porosity coefficient, the GPL weight fraction and the material length scale as well as the porosity distribution scheme and the GPL dispersion pattern. The results show the significance of incorporating the temperature dependency of the material properties when analyzing the thermal buckling behaviour and present the combination scheme of porosity and GPL distributions for enhancing the thermal buckling resistance.

Dynamic analysis of short-fiber reinforced composite nanobeams based on nonlocal strain gradient theory

This study deals with the dynamic behavior of short-fiber reinforced composite nanobeams. It is assumed that short-fibers are aligned or randomly distributed in the composite nanobeams. Nonlocal strain gradient theory is applied to composite nanobeam mechanics including Euler–Bernoulli and Timoshenko beam models. The transverse vibration of these composite nanobeams is investigated for various boundary conditions. Approximate Ritz method is used for obtaining the natural frequencies of short-fiber reinforced composite nanobeams. In addition to vibration analysis, wave propagation in short-fiber reinforced composite nanobeams is investigated and wave dispersion relations are analytically obtained for both Euler-Bernoulli and Timoshenko beam models. The vibration and wave dispersion results of short-fiber reinforced composite nanobeams are obtained for aligned and randomly distributed cases. The results obtained from this paper showed that there is no significant difference between the aligned and randomly oriented short-fiber composite nanobeams. This provides great convenience to designers where it is not possible to orient the reinforcement material in composites. The present study may be useful for the mechanical analysis and design of micro/nano-electromechanical systems (MEMS/NEMS), nanoprobes, nanosensors, nanoactuators, and atomic force microscopes.

Vibration analysis of embedded porous nanobeams under thermal effects using boosting machine learning algorithms and semi-analytical approach

This study presents a thermal vibration analysis of functionally graded porous nanobeams using boosting machine learning models and a semi-analytical approach. Nonlocal strain gradient theory is employed to explore vibration behavior, accounting for thermal and size effects. A semi-analytical approach solution utilizing Fourier series and Stokes’ transform to establish an eigenvalue problem capable of examining vibrational frequencies of porous nanobeams in both rigid and deformable boundary conditions is presented. Four boosting models including gradient boosting (GBoost), light gradient boosting (LGBoost), extreme gradient boosting (LGBoost), and adaptive boosting (AdaBoost) are employed to study the impact of seven crucial parameters on natural frequencies of a nanobeam. Sobol quasi-random space-filling method is used to generate the samples by varying input feature combinations for different porous nanobeam distributions. The model performance is assessed using statistical metrics, visualization tools, 5-fold cross-validation, and SHAP analysis for feature importance. The results highlight the effectiveness of boosting ML models in predicting natural frequencies, particularly XGBoost, which achieved an exceptional R2 value of 0.999, accompanied by the lowest MAE, MAPE, and RMSE values among the models assessed. LGBoost and AdaBoost follow XGBoost in performance, while GBoost exhibits relatively lower effectiveness, as highlighted by radar plots. The SHAP analysis revealed the significant impact of the foundation parameter on frequency prediction, with porosity coefficients notably influencing higher vibration modes.

Vibration analysis of functionally graded carbon nanotubes reinforced composite nanoplates

This work presents the analytical analysis for free linear vibration behavior of functionally graded-carbon nanotubes reinforced composite (FG-CNTRC) nanoplates in the framework of nonlocal strain gradient theory (NSGT) and the first-order shear deformation plate theory (FSDPT). The nanoplate is considered made of a mixture of an isotropic polymer matrix and reinforced carbon nanotubes (CNTs). Four different distributions of CNTs are examined including uniformly distributed and FG reinforcements (FG-O, FG-X, and FG-V). The governing equations of motion are established based on the Hamilton’s principle. The closed-form analytical solution for the natural frequency of FG-CNTRC nanoplates with simply supported all edges is carried out by using the Navier-type solution. The impact of some key parameters on the natural frequencies of FG-CNTRC nanoplates is also studied and discussed. The result shows that FG-CNTRC nanoplates reveal the softening- or hardening-stiffness effects depending on the relationship between the nonlocal parameter and the material length scale parameter. The aspect ratios of FG-CNTRC nanoplates, the volume fraction, and the distribution pattern of CNTs have also an important impact on the vibration behavior of FG-CNTRC nanoplates.

Vibration characteristics of multilayer functionally graded microplates with variable thickness reinforced by graphene platelets resting on the viscoelastic medium under thermal effects

Due to their improved mechanical properties and adaptability, microplates with tailored variable thickness profiles are becoming essential parts of advanced micro- and nanoelectromechanical systems (MEMS and NEMS). This study conducts a thorough analytical analysis of the vibration properties of thermally loaded, multilayer functionally graded graphene platelet-reinforced composite (FG-GPLRC) microplates of linearly or parabolically varying thickness resting on viscoelastic medium under different boundary conditions. The Halpin–Tsai micromechanical model and the law of mixtures are employed to calculate the effective material characteristics for various reinforcement distributions in the microplate. These distributions encompass uniformly symmetric and asymmetric arrangements. The study utilized the first-order shear deformation theory (FSDT) in conjunction with the modified strain gradient theory (MSGT) and Hamilton's principle to generate the dynamic governing equations for the structure, accounting for size-dependent effects. The resulting equations are afterwards solved using the utilization of the Galerkin technique. This enables the evaluation of the proposed solution's correctness and precision. The impact of various factors on vibration behavior is investigated through numerical analysis. These factors encompass length scale parameters, temperature fluctuations, temperature distribution profiles, boundary conditions, the distribution pattern of the GPL, taper constants in both unidirectional and bidirectional scenarios, the weight fraction of the GPL.

Nonlinear wave propagation in graphene incorporating second strain gradient theory

The exploration of graphene has attracted extensive interest owing to its significant structural and mechanical properties. In this research, we numerically investigate wave propagation in defect-free single-layer graphene, considering its geometrically nonlinear behavior through second-strain gradient elasticity. To capture the geometric nonlinearity, firstly, the nonlinear strain–displacement relations are introduced. The governing equation and associated boundary conditions are derived using Hamilton’s principle. Then, the weak form, including the element matrices, is established. The eigenvalue problem is solved for 2D wave propagation through periodic structures theory. Finally, the dynamical properties such as band structures, mode shapes, energy flow, and wave beaming effects are analyzed. The numerical results reveal that the geometric nonlinearity through the second strain gradient influences the wave propagation characteristics in graphene. The findings are significant and contribute to the understanding of graphene’s dynamic response, with implications for the engineering applications of graphene-based nanostructures.

Torsional vibration behavior of a restrained non-circular nanowire in an elastic matrix

This study introduces an approach to analyze torsional vibration in non-circular nanowires within magnetic fields, considering various boundary conditions on an elastic foundation. Analytical formulas for natural angular frequencies are obtained using nonlocal strain gradient theory. The analysis covers three non-circular cross-sections, incorporating the warping effect. Elastic springs at the wire ends simulate support conditions, restricting rotation around the wire’s axis. The torsion function around the axis is represented by a Fourier series, discretized at the spring points and linked using the Stokes’ transform alongside the boundary values. This leads to an eigenvalue problem that includes higher-order material size parameters (strain gradient, nonlocal), spring parameters, and the warping function. The study’s novelty lies in effectively solving torsional vibration for non-circular sections, addressing warping function, elastic medium, and size effects under both deformable and non-deformable boundary conditions. The presented solution is capable of determining vibration frequencies for both rigid and deformable boundary conditions. This is accomplished by specifying the torsional spring stiffness values, thereby obviating the necessity for additional recalculations. In order to verify the obtained results and compare them with the existing literature, nanowires with free and clamped boundary conditions are solved numerically by changing the spring parameters in the eigenvalue problem. In the formulation of natural angular frequency; length scale parameters, warping, magnetic field, elastic medium effects are included. In addition, since the support conditions are modeled with elastic springs, the resulting formulas are quite general and can be used to solve different types of torsional vibration problems.

Size parameter calibration of nonlocal strain gradient theory based on molecular dynamics simulation of guided wave propagation in aluminum plates

The nonlocal strain gradient theory (NSGT) describes long-range interatomic interactions and higher-order strain gradients by introducing size parameters. However, the NSGT is constrained in studying the elastic wave propagation problems due to the difficulty in obtaining the precise size parameters. In this paper, the molecular dynamics (MD) simulations of the elastic wave propagation in the aluminum nano-plate are conducted, and the size parameters in the NSGT are calibrated based on the MD simulation results. The MD simulation process is divided into three stages: relaxation, excitation, and free vibration analysis. The embedded atom method is used to describe the interactions between the metal atoms. During the MD simulations, appropriate boundary conditions and excitation methods are proposed to obtain the group velocities of the elastic bulk, Lamb and SH waves. In addition, based on the three-dimensional (3D) elasticity theory and analytical integration Legendre polynomial method, the theoretical dispersion curves of the elastic wave propagation in the NSGT model are obtained. It is found that both the nonlocal theory and the NSGT can predict the guided elastic wave group velocity, and the NSGT prediction and MD simulation can be well matched when the size parameters are restricted to a certain calibrated regional band. The vibration of atoms in the Lamb wave propagation process is more complex than that in the SH wave propagation, leading to a more substantial size effect on the Lamb wave dispersions.

Saint-Venant torsion based on strain gradient theory

In this study, the Saint-Venant torsion problem based on strain gradient theory is developed. A total form of Mindlin's strain gradient theory is used to acquire a general Saint-Venant torsion problem of micro-bars formulation. A new Finite Element formulation based on strain gradient elasticity theory is presented to solve the Saint-Venant torsion problem of micro-bars. Moreover, the problem is solved for both micro and macro scale. In addition, six different cross sections in solid and hollow condition are considered, which the Saint-Venant torsion problem is solved on them. Torsional response of micro-bars is predicted by using the modified couple stress theory and the strain gradient theory of Aifantis which both of them employ only one length scale parameter. Some micro-bars with different cross-sectional area are investigated to show the performance of current study and conclusions are drawn from the micron-scale torsional feedback for micro-bars.

Size-dependent thermal-electro-mechanical behaviors of a piezo-flexoelectric micro-beam based on nonlocal strain gradient theory and dual-phase-lagging heat model

Size-dependent thermal-electro-mechanical behaviors of a piezo-flexoelectric micro-beam based on nonlocal strain gradient theory and dual-phase-lagging heat model

Explicit Formulas for the Deformation of Chiral Porous Circular Beams in Gradient Thermoelasticity

Chirality and porosity are characteristic properties of nanostructured materials. Their effects on the mechanical behaviour of structural elements, such as shells, plates and beams, cannot be disregarded. In this paper, we study the thermoelastic deformation of a chiral porous circular beam loaded with an axial force and torque. The beam is also under the action of a constant temperature field. The analytical solution is obtained using the results established in a paper recently published by the Author within the context of the strain gradient theory proposed by Papanicopolous. In the constitutive equations, the chirality is introduced by a material constant parameter and the porosity is described by means of a scalar function. Displacements, microdilatation function, and stress and strain fields are expressed in explicit form and in terms of engineering constants. Explicit formulas of the stiffness of chiral porous circular beams are presented and the effects of right and left chirality are discussed.