Deformable Boundary Conditions Research Articles

Elastic medium and torsional spring effects on the nonlocal dynamic of functionally graded porous nanotubes

In this study, Eringen’s nonlocal elasticity theory that applies the small size effects in functionally graded porous nanotubes embedded in an elastic matrix is discussed. The material properties of functionally graded porous nanotubes are taken into account to vary over the radius direction with a rule of mixture. The free torsional vibration relation according to nonlocal elasticity theory, via Hamilton’s principle, is obtained and an eigenvalue solution is constructed for the free torsional vibration response of the presented work. The presented analytical model is validated by comparing the calculated mathematical results for homogeneous nanotubes with rigid and non-rigid boundary conditions. Special attention is given to deformable boundary conditions, porosity coefficient, material grading coefficient and also to the influence of elastic medium on the free torsional vibration frequencies. In this paper, it has been proven that the influence of length, elastic medium, elastic torsional spring rigidities, material grading and porosity coefficients on the vary in the torsional vibration frequency of the functionally graded nanotube is not small.

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.

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.

Rotary inertia effect on dynamic analysis of embedded FG porous nanobeams under deformable boundary conditions with the effect of neutral axis

Rotary inertia effect on dynamic analysis of embedded FG porous nanobeams under deformable boundary conditions with the effect of neutral axis

Numerical modeling of hydrodynamic added mass and added damping for elastic bridge pier

This paper presents a numerical model using the boundary element method for determining the hydrodynamic added mass and added damping of an elastic bridge pier with arbitrary cross-section. Combining the Euler–Bernoulli beam theory with the constant boundary element method, the modal superposition method is used to consider the deformable boundary conditions on the surface of elastic piers to couple the interaction between the elastic pier and water, and the equations for the hydrodynamic added mass and added damping of a general section pier considering the effect of pier-water coupling are derived. The accuracy of the developed model is verified by a benchmark experiment. The developed model is calculated for circular piers and compared with the added mass analytical formulation. The effects of oscillating frequency and structure geometry on the added mass and added damping are further investigated. Results demonstrate that the developed model can be used to solve the hydrodynamic added mass and added damping problems of the elastic bridge pier. Compared to the analytical formula, the developed method incorporates the consideration of added damping in the analysis of the pier-water coupling problem. Oscillating frequency and structure geometry have significant effects on added mass and added damping.

Winkler‐Pasternak foundation effect on the buckling loads of arbitrarily rigid or restrained supported nonlocal beams made of different FGM and porosity distributions

AbstractThe present research investigates lateral stability of a functionally graded nanobeam using Eringen's differential nonlocal elasticity model under rigid (clamped, pinned, free) and deformable (lateral, rotational restraints) boundary conditions. Sigmoid and power law have been employed as grading laws to study the influence of the material distribution on the snap‐buckling analysis of a nanobeam with arbitrary boundary conditions. Moreover, Fourier sine series with Stokes’ transformation are employed to investigate the effects of boundary conditions on the stability response of nanobeams embedded in a Pasternak foundation. A parametric study has been performed to investigate the effect of deformable boundaries, Pasternak foundation and small‐scale parameters on the stability response of the nanobeam and the results have been presented in a series of tables and figures. It has been observed that consideration of the small‐scale parameter, Pasternak foundation, deformable boundaries and functionally grading index (of sigmoid and power‐law) are essential while analyzing the static stability response. The obtained analytical results may be used as benchmarks in future researches of functionally graded nanobeams embedded in an elastic medium.

Simulation of the deformable lateral boundaries in biaxial test using DEM

Applying the discrete element method (DEM) in soil mechanics can provide abundant information at the particle-scale and facilitates illustration of the macro-mechanical behaviour of soils based on the inter-particle mechanisms. The triaxial test is one of the most common laboratory methods to study the macro-mechanical behaviour of particulate materials such as soil. However, many problems in geotechnical design can be assumed and simplified as a plane strain phenomenon. Therefore a biaxial test can be conducted to reproduce the macro-mechanical behaviour of soil, where the sample is enclosed by two horizontal rigid platens and a vertical latex membrane, which is a deformable continuous element and allows the enclosed specimen to deform freely while maintaining confining stress during loading. This paper presents an algorithm to represent physical and mechanical characteristics of latex membrane in the 2D DEM simulation of biaxial test using the PFC2D code. To investigate the impact of the lateral boundary conditions on the micro-macro mechanical behaviours of soil samples, two sets of DEM biaxial tests are conducted, i.e. with rigid and deformable lateral boundary conditions. The DEM modeling results indicate that the lateral boundary conditions have a significant effect on the micro-scale fabric properties, thickness and inclination of the shear band. The comparison between these two simulations also demonstrates that the lateral boundary conditions play a major role in the peak and post-peak stress-strain behaviours as well as the dilation and critical state behaviours of granular soils.

Size‐dependent Levinson beam theory for thermal vibration of a nanobeam with deformable boundary conditions

AbstractIn this study, an eigen‐value problem for deformable boundary conditions in nonlocal elasticity is described using Levinson beam theory. Firstly, the nanobeam has been modeled by placing two springs that can be deformed in the downward direction. These springs control the amount of downward displacement at the ends. In the analytical solution, the displacement points are defined by two coefficients and the interior part of the nanobeam deflection is expressed by Fourier sine series. Stokes’ transformation is preferred to enforce the boundary conditions to the desired point. After the mathematical operations, a matrix of coefficients including the general elastic spring constants has been found. The eigenvalues of this coefficient matrix give the frequencies of the Levinson nanobeam. The effect of some parameters on the free vibration frequencies is shown in a series of graphs and tables.

Torsional static and vibration analysis of a non-circular restrained micro/nanowire

This paper aims to study the static and free torsional vibration of non-circular micro/nanowires using the modified couple stress theory with several deformable and non-deformable support conditions. The geometrical properties of the micro/nanowires are considered to have triangular, elliptical and circular profiles. Equation of motion in torsional direction is derived based on torsional equilibrium and systems of linear equations are computed by adopting Fourier sine series with Stokes’ transformation. Moreover, deformable boundary conditions have been taken into account to formulate eigenvalue problem by employing the higher-order force boundary conditions. Material length scale parameter and the shape of cross-section have considerable impacts on torsional vibration frequencies of restrained microwire. The obtained frequency curves of restrained microwires are shown to be dependent on various parameters such as the shape of cross-section, deformable boundaries, material length scale parameter and also warping function.

Thermal vibration of perforated nanobeams with deformable boundary conditions via nonlocal strain gradient theory

Abstract Due to nonlocal and strain gradient effects with rigid and deformable boundary conditions, the thermal vibration behavior of perforated nanobeams resting on a Winkler elastic foundation (WEF) is examined in this paper. The Stokes transformation and Fourier series have been used to achieve this goal and to determine the thermal vibration behavior under various boundary conditions, including deformable and non-deformable ones. The perforated nanobeams’ boundary conditions are considered deformable, and the nonlocal strain gradient theory accounts for the size dependency. The problem is modeled as an eigenvalue problem. The effect of parameters such as the number of holes, elastic foundation, nonlocal and strain gradient, deformable boundaries and size on the solution is considered. The effects of various parameters, such as the length of the perforated beam, number of holes, filling ratio, thermal effect parameter, small-scale parameters and foundation parameter, on the thermal vibration behavior of the perforated nanobeam, are then illustrated using a set of numerical examples. As a result of the analysis, it was determined that the vibration frequency of the nanobeam was most affected by the changes in the dimensionless WEF parameter in the first mode and the changes in the internal length parameter when all modes were considered.

Stability analysis of arbitrary restrained nanobeam embedded in an elastic medium via nonlocal strain gradient theory

A novel stability model is analytically reformulated for the nano-sized beam resting on a one-parameter elastic foundation. The stability solution is based on the nonlocal strain gradient elasticity theory. To corporate the small size effects, two small scale parameters are introduced. The six-order ordinary differential form of the buckling equation, together with two force boundary conditions, are utilized to examine the stability equation in terms of lateral deflection. The infinite terms of linear equations are discretized with the help of the Stokes’ transformation and Fourier sine series. The present work can investigate the effects of elastic spring parameters at the ends, nonlocal properties, elastic medium properties, strain gradient parameter, and buckling behavior of the nanobeam. The predictions of the proposed analytical model with deformable boundary conditions are in agreement with those available in the scientific literature for the nanobeam on elastic foundation based on a closed form of solution. The presence of the deformable conditions, elastic foundation, nonlocal, and strain gradient properties change the buckling loads and buckling mode shapes.

Nonlinear stability analysis of embedded restrained nanobeams using the Stokes’ transformation

The geometrically nonlinear stability analysis of restrained nanobeams under an elastic medium is considered in the presented manuscript. The investigation is based on the nonlocal strain gradient elasticity theory, Euler-Bernoulli beam theory and geometrical nonlinearity which is a significant change in geometry. To the best of the authors’ knowledge, the nonlinear stability response of a nanobeam with elastic boundary conditions and on an elastic foundation has not been presented before via the theory of nonlocal strain gradient. The aim of this paper is to fill this gap in the literature by offering a method that can give a solution under general elastic boundary conditions. Using a Fourier sine series, a Fourier coefficient suitable for constructing an eigenvalue problem is computed. The constructed problem has been treated in the transformed region with the help of the Stokes’ transform, then the stability analysis is executed for arbitrary boundary conditions (rigid or restrained) based on the nonlocal strain gradient elasticity. The influences of various nonlinear parameters, deformable boundary conditions, small-scale parameters and elastic foundation parameters on the stability of the constrained nanobeam are studied. It is observed that the investigated variables produce significant changes in buckling loads.

Critical buckling loads of embedded perforated microbeams with arbitrary boundary conditions via an efficient solution method

Abstract In the present work, the small size effects on stability properties of perforated microbeams under various types of deformable boundary conditions are studied considering the Fourier sine series solution procedure and a mathematical procedure known as Stokes’ transformation for the first time. The main benefit of the present method is that, in addition to considering both the gradient elasticity and the size effects, the kinematic boundary conditions are modeled by two elastic springs as deformable boundary conditions. The deformable boundary conditions and corresponding stability equation are described using the classical principle which are then used to construct a linear system of equations. Afterward, an eigenvalue problem is adopted to obtain critical buckling loads. The correctness and accuracy of the present model are demonstrated by comparing results with those available from other works in the literature. Moreover, a numerical problem is solved and presented in detail to show the influences of the perforation properties, geometrical, and the variation of small-scale parameters and foundation parameters on the stability behavior of the microbeams. In addition, according to the best knowledge of the authors, there is no study in the literature that examines the buckling behavior of perforated microbeams on elastic foundation with the gradient elasticity theory.

Thermal buckling analysis of a saturated porous thick nanobeam with arbitrary boundary conditions

This manuscript aims to research the thermal buckling of saturated thick (Timoshenko) nanobeams under different boundary conditions using Fourier sine and cosine series for the first time. The equation of motion and related boundary conditions are derived by using the kinematic relations of shear deformation (Timoshenko beam) theory to contain the shear effect. Consequently, the rotary inertia and transverse shear strain are considered through the mathematical analysis. Fourier sine and cosine series are used to compute the critical buckling temperature of the thick saturated nanobeam with deformable boundaries. To check the validity of the presented method, the developed Fourier series method with Stokes’ transformation is applied to validate the thermal buckling of rigidly supported thick nanobeam by giving the proper values to spring parameters. In particular, the presented analytical procedure can also be degenerated to the thin Euler-Bernoulli nanobeam by assigning proper value to shear correction factor. Influence of the deformable boundary conditions, small scale parameter and the saturated parameter (porosity coefficient) on the thermal buckling temperature are discussed in detail. The theoretical models and eigen-value formulation presented herein should also be served to solve the thermal stability response of different micro sized-structures with various temperature effects and supporting conditions.

Nonlocal Free Vibration of Embedded Short-Fiber-Reinforced Nano-/Micro-Rods with Deformable Boundary Conditions.

An efficient eigenvalue algorithm is developed for the axial vibration analysis of embedded short-fiber-reinforced micro-/nano-composite rods under arbitrary boundary conditions. In the formulation, nonlocal elasticity theory is used to capture the size effect, and the deformable boundary conditions at the ends are simulated using two elastic springs in the axial direction. In addition, to determine the reinforcing effect of restrained nano-/micro-rods, a new system of linear equations with the concept of the infinite power series is presented. After performing the mathematical processes known as Fourier sine series, Stokes’ transformation and successive integration, we finally obtain a coefficient matrix in terms of infinite series for various rigid or deformable boundary conditions. Some accurate eigenvalue solutions of the free axial vibration frequencies of the short-fiber-reinforced micro-/nano-composite rods with and without being restrained by the means of elastic springs are given to show the performance of the present method. The presence of the elastic spring boundary conditions changes the axial vibration frequencies and corresponding mode shapes.

Size dependent torsional vibration of a restrained single walled carbon nanotube (SWCNT) via nonlocal strain gradient approach

A torsional vibration model of a restrained nano-sized rod in which deformable boundary effect is considered in the context of nonlocal strain gradient theory is proposed in this work. The strain gradient nonlocal theory introduced from Eringen’s nonlocal model contains additional small scale gradient parameter that represent the high-order strain gradient effects besides the Eringen’s size effect parameter depicting interactions of particles. Within the framework of Fourier infinite series and Stokes’ transform, an eigenvalue problem is employed to obtain the torsional vibration frequencies. The validity of the presented work is checked by comparing the calculated numerical results by giving proper values to elastic spring parameters and good agreement is achieved. This proposed model can be effectively applied to practical and non-idealized torsionally vibrating problems involving deformable boundary conditions. The two size parameters influencing on torsional dynamics of restrained nanorods are study in detail, meanwhile, validation and comparisons of present work with other non-classical elastic models, and the relations between rigid and deformable boundary conditions are shown in this paper.

Bayesian calibration of GPU–based DEM meso-mechanics Part I: Parallelization of RVEs

Calibration of Discrete Element Method (DEM) parameters is essential for modeling geotechnical applications. This task can, however, be extremely tedious or sometimes even impossible to undertake. This is largely due to two issues namely: (1) a large sample size of DEM simulations and number of sampling iterations are necessary to accurately infer the probability distribution of a model over a large parameter space and (2) DEM is computationally intractable compared to other numerical methods. In the scope of reducing the number of sampling iterations, automatic calibration techniques are available to extract and make use of the hidden contact mesostructure correlations through adaptive sampling. Coincidentally, to improve computational speed, significant advances toward Graphics Processor Unit (GPU) based DEM algorithms have been achieved over the past years on particle parallelism. Nevertheless, the problem remains that DEM simulations are serialized during the calibration processes. While the companion paper addresses parameter calibration, this study presents a novel algorithm to parallelize independent simulations within a sample set. The selected system is the Representative Volume Element (RVE) which is widely used in geotechnics for solving soil response in the static regime. The algorithm includes the following key features: (1) simulation level parallelism of non-interacting RVEs through highly efficient hierarchical memory groups and access patterns (2) a low latency and memory-efficient implementation of deformable periodic boundary conditions (PBC) which uses lookup tables and bitmasks (3) modified Uniform Grid and Bounding Volume Hierarchy (BVH) contact detection algorithms which partitions the RVE index into the hashing keys. The drained DEM triaxial compression is used to validate the algorithm on dry graded quartz. Three performance degrading factors for the calibration processes are considered: (1) the number of particles per RVE (2) calibration sample size and (3) sequential launch time per calibration step. This algorithm shows a factor of about 9.8 times speedup when parallelizing 100 DEM RVEs in one batch.

Torsional and longitudinal vibration analysis of a porous nanorod with arbitrary boundaries

In the present study, free torsional and axial vibrations of porous nanorods with torsional, axial elastic boundary conditions are presented via Eringen’s nonlocal elasticity theory. This theory takes into account the small size effect into the differential formulation due to dealing with nano-sized structures. Even though free axial and torsional vibrations based analysis of nanorod is a widely investigated topic, there are only few researches which exist in the scientific literature pertaining to the vibration analysis of porous nanostructures with nonrigid (deformable) boundary conditions. This study brings a contribution to the literature by presenting both porosity and small size effects with deformable boundary conditions. It is also planned to examine the effects of porosity and small size in detail. A model that allows to examine all of these effects within the framework is created within the scope of this study. To this end two torsional and axial springs are attached to the porous nanorod at both ends separately. Present model bridges the gap between deformable and non-deformable (rigid) boundary conditions. An eigen-value problem is constructed including torsional and axial coefficients by using Stoke transformation and the Fourier sine series. The novelty of this work is that it seeks to construct a general eigen value algorithm for the torsional and axial vibration frequencies of the porous nanorods subjected to the deformable and rigid boundary conditions for the first time. Moreover, the effects of some parameters such as porosity index, axial and torsional restraints on the free vibration frequencies are investigated and some conclusions are drawn.

Longitudinal vibration analysis of FG nanorod restrained with axial springs using doublet mechanics

In the current paper, the free longitudinal vibration response of axially restrained functionally graded nanorods is presented for the first time based on the doublet mechanics theory. Size dependent nanorod is considered to be made of functionally graded material consist of ceramic and metal constituents. It is assumed that the material properties of the functionally graded nanorod are assumed to vary in the radial direction. The aim of this study is that to investigate the influences of various parameters such as functionally graded index, small size parameter, length of the nanorod, mode number and spring stiffness on vibration behaviors of functionally graded nanorod restrained with axial springs at both ends. For this purpose, Fourier sine series are used to define the axial deflection of the functionally graded nanorod. Then, an eigenvalue approach is established for longitudinal vibrational frequencies thanks to Stokes’ transformation to deformable axial springs. Thus, the presented eigenvalue solution method is attributed to both rigid and deformable boundary conditions for the axial vibration of the functionally graded nanorod. With the help of the results obtained with the presented eigenvalue problem, it is observed that the parameters examined cause significant changes in the frequencies of the functionally graded nanorod.

Buckling analysis of restrained nanobeams using strain gradient elasticity

Size-dependent stability analysis of nanobeams under axial loading is investigated by strain gradient elasticity. An effective general model is proposed for different deformable boundary conditions by combining Fourier sine series and Stokes’ transformation. The theoretical model presented here bridges the gap between clamped and pinned-pinned supporting conditions, which is of great significance for the application of the strain gradient small scale effects to nanobeam. Analytical solutions are obtained and comparisons with results obtained by the researchers for rigid boundary conditions are carried out. The outcomes can be useful for the optimization and design of nano-sized beams and provide new benchmarks for nanomechanical analyses.