Length Scale Parameter Research Articles

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.

A novel isogeometric model for dynamic buckling analysis of doubly curved two-directional functionally graded porous shallow microshells in thermal environments via variable length-scale parameters

In this study, the thermo-mechanical dynamic buckling analysis of two-directional functionally graded porous (2D-FGP) shallow microshells resting on elastic foundations with arbitrary boundary conditions is performed using the isogeometric approach. The isogeometric approach takes advantage of non-uniform rational B-spline basic functions to exactly represent the structure geometry models and the attainment of higher order approximation conditions. The characteristics of the material vary in both the thickness and axial x orientation. Especially, a new point in this material length-scale parameter of microshell is analyzed as a function of spatial coordinates and as a function of the material gradient parameters. Using Hamilton’s principle, the modified couple stress theory, and Kirchhoff-Love’s shell theory, the governing equations of a 2D-FGP microshell lying on a Pasternak medium are obtained. A computer program that employs an isogeometric analysis method to analyze the static and dynamic buckling of a 2D-FGP doubly curved shallow microshell with various boundary conditions. The numerical results for thin cylindrical, spherical, and hyperbolic paraboloidal shallow microshells with various planforms, such as rectangular and circular, are presented. Several special cases are used to demonstrate the accuracy and efficacy of the developed model by comparing the numerical results obtained by the proposed formulations with other published data. In addition, the effects of certain parameters, such as the length scale parameters, the power-law indexes, the thickness-to-sides ratio, and the radius ratio, on the dynamic thermo-mechanical buckling response of the 2D-FGP shallow microshells with double curvature are explored in depth.

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.

The phase-field fracture model enriched by interpolation cover functions for brittle fracture problems

In this study, interpolation cover functions were introduced into the phase-field fracture model (PFM) to improve its convergence and efficiency. The traditional PFM has proven to be successful in simulating brittle fracture problems. However, one disadvantage is that it requires a relatively dense mesh. The PFM model enriched by interpolation cover functions (EPFM) was used for numerical fracture analysis under a relatively coarse mesh. The rank deficiency problem caused by the introduction of interpolation cover functions was resolved using the rank deficiency repair technique. Higher gradients were captured by introducing interpolation cover functions without changing the mesh structure. The numerical examples showed the improvement in convergence. The high efficiency of the proposed EPFM was demonstrated through a series of numerical examples. The numerical results of the EPFM and PFM under different length-scale parameters and meshes were also compared. The results showed that the proposed EPFM is an excellent alternative for evaluating engineering fracture problems.

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.

Higher-order asymptotic crack-tip fields in simplified strain gradient elasticity

Explicit representation for the higher-order in-plane crack-tip fields is derived using Papkovich–Neuber stress functions within the framework of simplified strain gradient elasticity (SGE). In the considered theory, strain energy density depends on both the infinitesimal strain tensor and its spatial gradient. The constitutive equations of simplified SGE contain an additional length scale parameter that can be correlated with the characteristic size of the material microstructure. Presented asymptotic solution has separable form and reduces to classical Williams’ series when the gradient effects are negligible. The leading terms in the derived solution coincide with the previously known asymptotic solutions for crack problems in SGE. The higher-order terms have coupled amplitude factors and modified definitions for the angular distribution as compared to the classical solution. Derived asymptotic fields are compared to the full-field numerical solution for the Mode I crack problem to quantify the amplitude factors and the zones of dominance for up to eight terms. The presented solution has practical applications in the development of advanced numerical methods, particularly concerning enriched finite elements for crack problems in SGE.

Gurson-Cohesive modeling (GCM) for 3D ductile fracture simulation

A Gurson-Cohesive Model (GCM) is proposed to investigate 3D ductile fracture accounting for the void growth, coalescence, and complete failure phenomena. The proposed model idealizes the ductile fracture process as continuum damage evolution, cohesive crack initiation, nonlinear softening along the crack surface, and complete failure. The Gurson model is employed to describe continuum damage with void growth, while the cohesive zone model is utilized to introduce discontinuous cracks and represent nonlinear softening behavior. The transition from the continuum damage to discontinuous crack is taken into account systematically using a porosity-based crack initiation criterion considering stress triaxiality. The computational results successfully reproduce the experimental results of fracture tests with 15–5 PH steel and structural carbon steel 20 within the unified modeling framework. Furthermore, strong and stable convergence of both global and local responses (e.g., load-displacement curve, area reduction, crack tip location) are demonstrated under the mesh refinement without the aid of any ad hoc characteristic length scale parameter in the simulations.

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.

3D Size-Dependent Dynamic Instability Analysis of FG Cylindrical Microshells Subjected to Combinations of Periodic Axial Compression and External Pressure Using a Hermitian C2 Finite Layer Method Based on the Consistent Couple Stress Theory.

This work develops a three-dimensional (3D) weak formulation, based on the consistent couple stress theory (CCST), for analyzing the size-dependent dynamic instability behavior of simply-supported, functionally graded (FG) cylindrical microshells that are subjected to combinations of periodic axial compression and external pressure. In our formulation, the microshells are artificially divided into nl layers. The displacement components of each individual layer are selected as the primary variables, which are expanded as a double Fourier series in the in-plane domain and are interpolated with Hermitian C2 polynomials in the thickness direction. Incorporating the layer-wise displacement models into our weak formulation, we develop a Hermitian C2 finite layer method (FLM) for addressing the current issue. The accuracy and the convergence rate of our Hermitian C2 FLM are validated by comparing the solutions it produces with the accurate two-dimensional solutions of critical loads and critical pressures of FG cylindrical macroshells and single-walled carbon nanotubes, which were reported in the literature. The numerical results show the effects of the material length-scale parameter, the inhomogeneity index, the radius-to-thickness and length-to-radius ratios, the load magnitude ratio, and the static and dynamic load factors on the first principal and first secondary instability regions of parametric resonance of simply-supported FG cylindrical microshells are significant.

Study of wave motion on the emergence of veering, locking, and coupling in periodic composite panels.

This research proposes the effect of micropolar-Cosserat (MC) parameters (length-scale parameters and Cosserat shear modulus) on the dispersion characteristics of propagating wave modes in periodic composite panels (PCPs). These inbuilt parameters are due to the assumption of the length-scale boundary conditions that allow for capturing the micro-rotational (MR) wave mode along with the flexural ones. A significant contribution of this study is the transformation of the two-dimensional (2-D) periodic composite problem into a series of one-dimensional (1-D) ones using the MC continuum theory. The analysis employs the transfer matrix method in the framework of the state-space approach to investigate periodic systems in the eigenvalue domain. Additionally, Bloch-Floquet's periodic boundary conditions (PBCs) are applied to the unit cell to ensure the periodicity of the system. The main innovation lies in observing veering, locking, and coupling phenomena, which occur due to alterations in lamina orientation and MC parameters. Moreover, the presence of inbuilt parameters renders the dispersion characteristics highly sensitive to even minor coefficient variations, with a mere 1% change significantly impacting eigenmode fluctuations. The sudden bandgap (BG) disappearing nature could be used to identify the accurate value of the coefficient for designing and analyzing PCPs.

Elastic size effect of single crystal copper beams under combined loading of torsion and bending

Among various strain gradient theories, the modified couple stress theory, which introduces a single length scale parameter as an additional material property, has garnered significant interest owing to its simplified portrayal of the material behavior. In this study, we investigated whether a single length scale parameter is sufficient to predict the mechanical behavior under two different type of strain gradients: torsion and bending. L-shaped beams made of single crystal copper with thicknesses ranging from 2.4μm to 9.1μm were fabricated, and loads were applied using an indenter. The contributions of bending and torsion were controlled by adjusting the loading position. Through these experiments, we demonstrated the existence of elastic size effect of single crystalline materials under strain gradients. Specifically, size effect was observed in both bending and torsion, with a larger effect observed in cases closer to pure bending. Moreover, we report that the modified couple stress theory and the modified strain gradient theory are not applicable for simulating size effect under combined loading. This discovery highlights the necessity for the development of a new theory capable of adequately simulating size effect under the intricate loading scenarios encountered in practical applications.

A three-dimensional micropolar beam model with application to the finite deformation analysis of hard-magnetic soft beams

The main purpose of this contribution is to develop a three-dimensional (3D) nonlinear beam model based on the micropolar continuum theory. To do so, a kinematic model based on the deformation of three directors and accounting for the micro-rotation tensor of the micropolar theory is introduced. One of the main characteristics of the present beam model is that 3D constitutive equations without any modification can be directly used in the formulation. Furthermore, it is known that a body couple field is induced in hard-magnetic soft materials (HMSMs) when subjected to external magnetic fluxes. Therefore, the stress tensor in HMSMs is asymmetric, in general. Since the asymmetry of stress is one of the main features of the micropolar theory, the present formulation can be used for analyzing the deformation of beams made of HMSMs. Accordingly, the virtual external work of the present model is formulated so that it accounts for the contribution from uniform or constant-gradient external magnetic fluxes on the beam. Moreover, a Total Lagrangian (TL) nonlinear finite element (FE) formulation to provide numerical solutions of the related problems is developed. Several numerical examples are solved to investigate the capability of the developed formulation. It is shown that the present formulation can model the size-dependent behavior of beam-like structures if the material length-scale parameter of the micropolar constitutive model is comparable to the thickness of the beam. Moreover, the proposed model can successfully predict the finite deformation of 3D beams made of HMSMs subjected to magnetic loading.

A quasi-3D hyperbolic formulation for the buckling study of metal foam microplates layered with graphene nanoplatelets-embedded nanocomposite patches with temperature fluctuations

The work analyzes the buckling behavior of size-dependent microplates with a metal foam core, covered by graphene nanoplatelets (GNPs)-embedded nanocomposite patches. Microplates rest on a bi-parameter elastic substrate, and they are immersed in a thermal environment to observe the effect of temperature fluctuations on their elastic buckling performance. All material properties for each layer of the microstructure are thickness-dependent. A novel quasi-3D shear and normal (Q-3D S-N) hyperbolic theory is here proposed to describe the kinematic relations, accounting for the transverse normal strain. At the same time, a modified couple stress theory (MCST) is employed to account for the size dependence of the mechanical behavior due to the presence of a material length-scale parameter. Using the energy method and virtual work principle, the differential equilibrium equations are derived and solved analytically, where solutions are verified against the existing literature. The study focuses on the impact of different parameters on the normalized critical buckling load (NCBL). Based on the results from the systematic investigation, it is found that the addition of GNPs to the microplate enhances its stiffness, leading to increased values of NCBL, which in turn reduce for an increased imperfection ratio.

A novel beam model suitable at the microscale

A novel beam model is proposed to address the limitations of existing microscale beam models by introducing a higher-order displacement field assumption for the modified couple stress theory. The governing equations and boundary conditions are derived using the variational principle. Validation through finite element analysis for a simply supported beam problem demonstrates the beam model successfully captures the width-dependent behavior of microbeams, which existing beam models cannot explain. Additionally, the beam model accurately represents the plane strain behavior as the beam width increases and the classical beam behavior at a small length scale parameter.

On wave propagation in nanobeams

Wave propagation in Rayleigh nanobeams resting on nonlocal media is investigated in this paper. Small-scale structure-foundation problems are formulated according to a novel consistent nonlocal approach extending the special elastostatic analysis in Barretta et al. (2022). Nonlocal effects of the nanostructure are modelled according to a stress-driven integral law. External elasticity of the nano-foundation is instead described by a displacement-driven spatial convolution. The developed methodology leads to well-posed continuum problems, thus circumventing issues and applicative difficulties of the Eringen–Wieghardt nonlocal approach. Wave propagation in Rayleigh nanobeams interacting with nano-foundations is then analysed and dispersive features are analytically detected exploiting the novel consistent strategy. Closed form expressions of size-dependent dispersion relations are established and connection with outcomes available in literature is contributed. A general and well-posed methodology is thus provided to address wave propagation nanomechanical problems. Parametric studies are finally accomplished and discussed to show effects of length scale parameters on wave dispersion characteristics of small-scale systems of current interest in Nano-Engineering.

Scale-dependent torsional frequency analysis of right triangle anisotropic advanced composite microscale rods

Despite several theoretical investigations on the torsional dynamic behavior of non-circular micro/nanorods, the torsional behavior of right triangle cross-section is still unevaluated. Therefore, Timoshenko-Gere theory in conjunction with nonlocal strain gradient theory is developed for modeling the scale-dependent torsional dynamic of right triangle microscale rods. Various anisotropic materials, i.e. hexagonal, trigonal, triclinic, and monoclinic materials are regarded as the constituent material of microscale rods. The shear stress function is developed for right triangle shape and according to the calculated shear stress function, the warping function of right triangle shape is obtained. The nonlocal governing equation of the torsional dynamic of anisotropic microrods is obtained with the help of the energy approach and solved using an analytical method by regarding two different boundary conditions. The accuracy of the proposed model is validated with recent similar investigations. Because of the lack of mechanical research on right triangle microrods, the proposed methodology was validated with equilateral triangle nanowire reported in the literature. Eventually, the various remarkable variants’ effects on change of scale-dependent torsional frequency are assessed comprehensively. It is found that hexagonal and triclinic materials have the largest and smallest values of frequency. Also, nonlocal parameter, b/h ratio play a decreasing role and length scale parameters and mode number plays an increasing role.

Simulation of crack propagation in piping components using phase field approach

Understanding the fracture behaviour in pipelines transporting steam and other resources is vital for its longevity and structural integrity. The prediction of crack initiation and propagation, involving crack branching, and the nucleation of new cracks, etc. in fracture processes are challenging tasks. The phase field method (PFM), based on variational formulation, emerged as a competitive mathematical tool to address fracture mechanics problems. PFM does not require the presence of predefined cracks, handles the crack initiation and propagation within the same system. This feature provides an added advantage to PFM and overcomes the limitations of Griffith’s approach-based linear elastic fracture mechanics (LEFM). In the present work, simulation of the crack propagation in the piping components by employing phase field methodologies has been carried out. Two parameters namely; a phase field parameter (ϕ) that varies between 0 and 1 differentiating between broken and intact material, and a length scale parameter (l_0) that approximates the sharp crack into the diffused crack with exponential distribution are introduced in the mathematical modeling. Numerical modeling is solved using the staggered algorithm in three layered scheme defining the displacement field and phase field within a 3-D finite element framework and implemented in Abaqus via user-defined element (UEL) subroutine codes. PFM simulates the crack path and predicts the load-displacement response, which is validated with benchmark problems, and extended for piping components under monotonic loading. The proposed study accurately captures the crack growth behaviour as observed in experiments on SA312 Type 304LN stainless steel pipes and hence, demonstrated the robustness of the proposed formulation.

On predictive models for the equivalent sand grain roughness for wall-bounded turbulent flows

One of the long-standing goals of rough wall fluid dynamics research is to determine the drag penalty of surfaces based solely on their topographical parameters. The most important length scale or roughness parameter that best describes a surface in relation to friction drag has not been agreed upon yet, despite the many studies that, over the years, have attempted to identify the most appropriate surface parameter. The concept of an equivalent sand-grain roughness (ks) was introduced to standardize different types of roughness in wall-bounded turbulence, serving as an input parameter for predicting the roughness function ΔU+. To anticipate ΔU+ generated by a rough surface under turbulent flow conditions, experts use roughness correlations that establish a correspondence between the topographical characteristics of the surface and ks. Therefore, a chronological compilation of roughness correlations is presented, detailing the parameter ranges and types of roughness used in their development. This study evaluates the effectiveness of predictive correlation functions and aims to formulate a universal function by exploring a comprehensive assortment of three-dimensional (3D) surface textures available in the literature. The results suggest that a correlation based on surface height skewness (ksk) and streamwise effective slope (ESx) can predict the ratio (ks/kq), where kq is the root mean square roughness height for 3D roughness in the fully rough regime. Despite the fact that the correlation is restricted to 3D surface roughness, which is a more realistic representation, the model demonstrated a high level of accuracy in predicting ks for over 120 distinct rough surfaces.

Modeling Control of Supercoiling Dynamics and Transcription Using DNA-Binding Proteins.

Nearly all natural and synthetic gene networks rely on the fundamental process of transcription to enact biological feedback, genetic programs, and living circuitry. In this work, we investigate the efficacy of controlling transcription using a new biophysical mechanism, control of localized supercoiling near a gene of interest. We postulate a basic reaction network model for describing the general phenomenon of transcription and introduce a separate set of equations to describe the dynamics of supercoiling. We show that supercoiling and transcription introduce a shared reaction flux term in the model dynamics and illustrate how the modulation of supercoiling can be used to control transcription rates. We show the supercoiling-transcription model can be written as a nonlinear state-space model, with a radial basis function nonlinearity to capture the empirical relationship between supercoiling and transcription rates. We show the system admits a single, globally exponentially stable equilibrium point. Notably, we show that mRNA steady-state levels can be controlled directly by increasing a length-scale parameter for genetic spacing. Finally, we build a mathematical model to explore the use of a DNA binding protein, to define programmable boundary conditions on supercoiling propagation, which we show can be used to control transcriptional bursting or pulsatile transcriptional response. We show there exists a stabilizing control law for mRNA tracking, using the method of control Lyapunov functions and illustrate these results with numerical simulations.

Nonlocal dual-phase-lag thermoelastic damping analysis in functionally graded sandwich microbeam resonators utilizing the modified coupled stress theory

Functionally graded (FG) sandwich structures stand as one of the most representative composite structures owing to the thermal resistance and the energy absorption property in non-unfiorm thermal environment. Additionally, accurately estimating thermoelastic damping (TED) is of great importance for the design of high-performance micro/nano-resonators. Nevertheless, the classical TED models fail on the micro/nano-scale structures due to without considering the influences of the spatial size-dependent effects related to heat transfer and elastic deformation. The nonlocal heat conduction model and modified coupled stress theory are responsible for the size-dependent effects. To address this issue, present study aims to conduct the size-dependent TED model of FG sandwich microbeam resonators for TED analysis by incorporating the nonlocal dual-phase-lag (NDPL) heat conduction model and the modified coupled stress theory (MCST). It is assumed that the FG sandwich microbeam resonators consist of a ceramic core and FG surfaces. The energy equation and the transverse motion equation are formulated, and then, the analytical solution is solved by complex frequency method. Exact and closed-form expressions for TED can be obtained through the complex frequency method. The results are validated, and the parameter effects of the nonlocal thermal parameter, the material length-scale parameter, the power-law index and the vibration modes on the TED are analyzed. The results show that the energy dissipation can be reduced by the size-dependent effects resulting in improving the quality factor of microstructures. It is expected that these results may provide a theoretical basis for predicating TED in the design of FG sandwich micro/nano-resonators with high quality factor in extreme heat transfer environment.