Bending Solutions Research Articles

Modeling direct and converse flexoelectricity in soft dielectric rods with application to the follower load

Dielectric rods have been employed in various electromechanical applications, including energy harvesters and sensors. This paper develops a general framework to model large deformations in dielectric rods, considering both direct and converse flexoelectric effects. Initially, we derive the governing differential equations for a three-dimensional dielectric continuum solid to model large deformations, incorporating converse flexoelectricity. Then, we derive the equilibrium equations for the flexoelectric strain-gradient special Cosserat rod. Subsequently, we establish its constitutive relations and identify the corresponding work conjugates. To solve these governing differential equations numerically, we implement a quaternion-based numerical approach and obtain flexoelectricity-based solutions corresponding to the follower load. Moreover using these constitutive relations, we have also obtained nonlinear analytical solutions for bending under the follower load that show an excellent agreement with our numerical results. Bending under the follower load is also compared with the transverse load to understand the electric field generation. Unlike, under the application of the transverse load, where the electric field increases monotonically, for the follower load, the electric field gradually switches its sign. The role of direct and converse flexoelectric coefficients has also been examined, and several interesting conclusions have been drawn. Finally, we analyze the effect of mechanical and electrical length scale parameters. The electromechanical response from the follower load can be utilized to fabricate flexoelectric sensors for nanoelectromechanical systems.

Analytical solutions for large-deflection bending of variable-thickness inhomogeneous functional graded composite circular plates with parameterized boundaries under hygro-thermo-mechanical loads

A brand-new hygro-thermo-mechanical bending model of the inhomogeneous concave and convex composite circular plates having varying thickness undergoing large deformation resting on nonlinear tri-parameter spring and shear elastic layers is proposed. Three-dimensional analytical hygrothermal field of circular plate is demonstrated with non-uniform thermal and moisture diffusion coefficients in axial variation, while thickness and elastic module of material are modeled in quadratically change and sufficiently summarized in normalization. The circled boundaries with arbitrary translational and rotational constraints are parameterized with feasible region of elastic parameters highlighted in rectangular domain by Linear Programming method. Two reduced-order integro-differential governing equations for the composite circular plates under extreme load have been derived, while analytical bending solutions are obtained by an employed homotopy-based analytical scheme with accuracy verified and convergence accelerated by truncation and iteration. Whether thickness or elastic module of composite material is variable, the final outcome on plate structural strength is the discrepancy of bending stiffness with different load capacity, with the former revealing more sensitive than the latter under the condition of same values. Once amplitude and distribution of hygrothermal load are determined, influence on large-deflection bending of circular plates exists at a limited range of nonlinearity, while non-uniform distribution of thermal expansion and moisture concentration aggravates bending effects only within this range.

Analytical solutions for strain-driven Timoshenko nanobeam bending using generalized functions

Static nanobeam bending problems have yielded conflicting results from many researchers. Methods suggested by numerous researchers have shown inconsistencies in capturing size effects and satisfying essential and natural boundary conditions for nanobeam configurations when subjected to different types of loading. This work proposes an alternative approach using Eringen’s strain-driven integral nonlocal elasticity to arrive at analytical solutions. The theory of generalized functions and the Dirac-delta identity are used to revisit Timoshenko nanobeam modeling aspects, focusing on nanobeam elastostatics. Proposed analytical solutions for fundamental and derived quantities based on the strain-driven nonlocal model have been compared with the results of existing approaches. A crucial drawback with the strain-driven models of Eringen in both integral and differential form is its inability to satisfy equilibrium equations, which is addressed extensively in this work.

Sensitivity Analysis of the Factors Affecting the Ground Heave Caused by Jet Grouting

Jet grouted piles are widely used to reduce post-construction settlement of soft clay roadbeds. Nevertheless, it is easy to cause ground heave due to the jet grouted pile. According to the analytical method and numerical method, a sensitivity analysis of the factors affecting ground heave caused by a single jet grouted pile was performed. It is found that the influence of each parameter on ground heave is in the following order: grout pump pressure > embankment load > soil type (including the cohesion, friction angle, and Young’s modulus) > pile diameter > pile length. Considering the effect of the pump pressure on the ground heave is more significant, based on the analytical method of ground heave caused by a single jet grouted pile combined with the solution of small-deflection bending of a circular thin plate, the calculation method for the suggested limit grout pressure for construction under different embankment heights was established. Suggested values of theoretical grout pump pressure were given to prevent ground heave from harming the pavement of operating highways. This study provides some theoretical basis for the subsequent research on the jet grouted pile.

Approximate Solutions for Bending of Beams and Buckling of Columns Made of Functionally Graded Materials

Approximate Solutions for Bending of Beams and Buckling of Columns Made of Functionally Graded Materials

An analytic solution for bending of multilayered structures with interlayer-slip

Layered structures are prevalent in both natural environments and engineered composite materials. The elastic bending behavior of these structures is primarily governed by properties of their abundant interfaces. While the behavior of two- and three-layered beams has been extensively studied, this research shifts the focus to the impact of elastic shearing at interfaces on the deflection of multilayered structures comprising a substantial number of layers. We present an analytical solution indicating that the bending properties of multilayered beams and plates are nonlinearly dependent on interfacial stiffness. Denoting Se as the effective bending stiffness of an n-layered beam of length L, and S0 as the bending stiffness of a perfectly bound counterpart, we arrive at SeS0=11+(n2−1)tanhαLαL where αL represents a dimensionless parameter related to geometry and material properties. The analytical solutions, validated through finite element simulations, highlight the substantial variations in stiffness across different layered structures. This solution could also be instrumental in assessing interfacial damage and delamination in lamellar composites.

Torsional buckling of a tape-spring: Review and renew

Tape-springs – straight strips with transversely curved cross-sections – undergo a flattening of the cross-section upon bending, and accompanying torsional buckling in equal-sense bending which then “recedes” at greater deformation. The exact analytical solution for tape-spring bending was solved by Mansfield; however, the resulting expression for the twisting curvature at the onset and recession of torsional buckling cannot be solved in closed-form. Approximate closed-form expressions are therefore sought: firstly, by utilising Taylor series approximations of Mansfield’s extensive functions; and secondly, by assuming a uniform transverse curvature, thereby rendering the problem algebraic in nature, and deriving new governing expressions. Further approximations assume the cross-section does not deform upon bending, and that the tape-spring remains a developable surface in the inextensible limit of zero thickness. All approximate solutions are shown to agree closely with Mansfield’s solution at small curvatures, with the uniform curvature approximation also capturing the recession of torsional buckling to a reasonable degree of accuracy. The accuracy of the uniform curvature approximation is related to a pertinent dimensionless parameter, which governs the size of a boundary layer in Mansfield’s exact solution: as this parameter tends to infinity, the uniform curvature approximation and Mansfield’s exact solution are shown to converge; as it tends to zero, all solutions are shown to converge on the inextensible limit.

3D nonlocal solution with layerwise approach and DQM for multilayered functionally graded shallow nanoshells

A nonlocal layerwise 3D bending solution for multilayered functionally graded shallow nanoshells is presented. Eringen’s nonlocal elasticity theory is used for approaching the small length scale effect. Curvilinear orthogonal coordinate system is employed for writing the constitutive and equilibrium equations. The middle-plane of displacements are assumed as a Navier solution which are only valid for simply supported spherical, cylindrical panels and rectangular plates. The thickness domain is discretized by the well-known Chebyshev–Gauss–Lobatto, and its derivatives are calculated numerically by the Differential Quadrature method (DQM). Lagrange interpolation polynomials are utilized as the basis functions for DQM. The traction conditions at the top and the bottom of the shell panel and continuity conditions of the interlaminar adjacent layers for transverse stresses are considered, so this method presents layerwise capabilities. The results are compared with higher order theories proposed in the literature, and due to the 3D nature of the presented formulation, benchmark solutions are provided.

Microstructure-based modeling of primary cilia mechanics.

A primary cilium, made of nine microtubule doublets enclosed in a cilium membrane, is a mechanosensing organelle that bends under an external mechanical load and sends an intracellular signal through transmembrane proteins activated by cilium bending. The nine microtubule doublets are the main load-bearing structural component, while the transmembrane proteins on the cilium membrane are the main sensing component. No distinction was made between these two components in all existing models, where the stress calculated from the structural component (nine microtubule doublets) was used to explain the sensing location, which may be totally misleading. For the first time, we developed a microstructure-based primary cilium model by considering these two components separately. First, we refined the analytical solution of bending an orthotropic cylindrical shell for individual microtubule, and obtained excellent agreement between finite element simulations and the theoretical predictions of a microtubule bending as a validation of the structural component in the model. Second, by integrating the cilium membrane with nine microtubule doublets and simulating the tip-anchored optical tweezer experiment on our computational model, we found that the microtubule doublets may twist significantly as the whole cilium bends. Third, besides being cilium-length-dependent, we found the mechanical properties of the cilium are also highly deformation-dependent. More important, we found that the cilium membrane near the base is not under pure in-plane tension or compression as previously thought, but has significant local bending stress. This challenges the traditional model of cilium mechanosensing, indicating that transmembrane proteins may be activated more by membrane curvature than membrane stretching. Finally, we incorporated imaging data of primary cilia into our microstructure-based cilium model, and found that comparing to the ideal model with uniform microtubule length, the imaging-informed model shows the nine microtubule doublets interact more evenly with the cilium membrane, and their contact locations can cause even higher bending curvature in the cilium membrane than near the base.

On the variable length scale parameter in functionally graded non-porous and porous microplate/nanoplate

In this article, the solution for free vibration and static bending of nonporous and porous square microplate/nanoplate is studied by practicing the modified couple stress theory (MCST). The impact of incorporating the length scale parameter of MCST is confirmed by comparing the results to that of molecular dynamics simulations. This study is aimed at investigating the influence of implementing a variable length scale parameter on the static and dynamic behavior of the small-scaled plates. Results are obtained adopting the analytical technique and compared to cases of presupposing constant numerical values of material length scale throughout each plate. The novelty of this theoretical study is the approach employed to achieve the formulation of the material length scale parameter, which varies along the thickness of both porous and nonporous plates. The results indicate that the inclusion of the variable length scale parameter increases the fundamental frequency and stiffness of plates. When comparing two cases of variable length scale parameter, one approximating the scale parameter of each constituent based on the bending parameter and Poisson’s ratio of the material, and the other adopting the numerical values estimated by prior experimental studies, reveals that the latter gives higher results and is chosen correspondingly.

Analytical solutions for the thermo-mechanical bending of FG beams using a higher order shear deformation theory (HSDT)

The aim of this paper is to analyze a thermo-mechanical behavior of functionally graded beam in thermal environment. Therefore, this research work investigates a thermal, mechanical and thermo-mechanical bending of functionally graded beam resting on simple supports and using a higher order shear deformation theory. the proposed kinematic uses only three variables, which is even less than the first order shear deformation theory (FSDT) and the conventional higher order shear deformation theories (HSDTs). The present model adopts a new displacement field, The material properties of the FG beam are varied along the thickness direction according to the power law distribution (P-FGM). The governing equations for the thermo-mechanical bending investigation are obtained through the principal of virtual work and solved via Navier-type solution method was used to obtain results. All the results are presented in non-dimensional form and validated it by developing the classical beam theory (CBT), and other higher order theories. Parametric studies are thus elaborated to see the influence of the characteristic’s geometries and mechanical in particular: thickness of the FG beam, material index and ratio (width, length).

Analytical and isogeometric solutions of flexoelectric microbeams based on a layerwise beam theory

The theory of layerwise beam, which considers both microstructure and flexoelectric effects, is used in the curvature-based flexoelectricity theory. The displacement field function of the layerwise beam model is given. The governing equations and boundary conditions are obtained using a variational formulation based on Hamilton's principle. Analytical solutions for static bending and free vibration of a simply supported layerwise beam are derived. Isogeometric analysis (IGA) with higher-order continuity considering the flexoelectric effect is presented to numerically solve the static bending and free vibration problems. The results obtained from the analytical and IGA approaches are compared through several examples in which the microstructure and flexoelectric effects on the mechanical responses of layerwise nanobeams are analyzed.

Heat Pipe Bending Effect on Cooling Effectiveness in Electrical Machines

Heat pipes (HPs) are being extensively explored in motor cooling scenarios for enhanced cooling capacity. HPs are commonly bent to adapt to the compact structure of electrical machines, whereas the bending effect on motor cooling effectiveness still requires further investigation. This article analytically and experimentally studies the effect of the bending process, including bending angle and bending radius, on HP thermal properties. Cooling density is defined and derived to analyze the tradeoff between HP thermal performance and additional volume due to HP installation. Practical guidelines for feasible HP bending solutions under different space constraints are provided to achieve a higher cooling density. Finally, the HP bending effect on motor cooling effectiveness under various cooling methods is quantitatively evaluated through a validated thermal model based on a stator-winding assembly. The bending process can degrade the HP equivalent thermal conductivity by up to 76%, thus leading to a temperature difference of 4.8K under liquid cooling conditions.

An analytical model for nanoscale flexoelectric doubly curved shells

The phenomenon of flexoelectricity, wherein the generation of electrical polarization results from strain gradients, has garnered significant attention in electromechanics. This phenomenon holds immense potential for diverse applications in nanoelectromechanical systems. Doubly curved shells present a particularly compelling structure for implementing flexoelectric devices, representing the most general scenario. To comprehensively understand the mechanical behavior of nanoscale flexoelectric doubly curved shells, we present a novel analytical model in this study. Our model incorporates a reformulated strain gradient elasticity theory, a modified expression of electric enthalpy density considering Maxwell’s self-field gradient, and a moderately thick shell configuration. To further elucidate this model, we establish complete formulations and analytical solutions of static bending and free vibration problems. This study reveals the crucial roles that various parameters such as principal curvature radii, length scale parameters, transverse structural sizes, and the Winkler–Pasternak foundation play in controlling the mechanical behavior of nanoscale flexoelectric doubly curved shells. These theoretical findings offer valuable insights into the design of nanodevices based on the direct flexoelectric effect.

Curvature-based flexoelectric nanobeams: Analytical and numerical isogeometric analyses

In this paper, new curvature-based flexoelectric Euler-Bernoulli and Timoshenko beam models are derived based on the couple stress and its flexoelectric theory. These models consider both the microstructure and flexoelectric effects. The governing equations and boundary conditions are obtained using the variational formula of Hamilton's principle. Analytical solutions for the static bending and free vibration problems of Euler-Bernoulli and Timoshenko beam models with a simply supported nanobeam are presented. To extend the applicability of the curvature-based flexoelectric beam models, an isogeometric analysis (IGA) numerical approach which satisfy the needs of C1 continuity is developed. Numerical results are provided to verify the accuracy of the analytical and IGA numerical solutions, and the influence of flexoelectric effects, microstructure effects and boundary conditions on the mechanical behavior of nanobeams is analyzed.

Peeling of Finite-Length Plates From an Elastomeric Foundation: A 1D Cylindrical Bending Solution

Abstract Quasi-static peeling of a finite-length, flexible, horizontal, one-dimensional (1D) plate (strip, thin film) from a horizontal, thin, elastomeric layer (foundation) is considered. The displaced end of the plate is subjected to an upward deflection or to a rotation. The top of the interlayer is perfectly bonded to the plate, and its lower surface is bonded to a rigid, flat substrate. A transversality (debonding) condition is derived for peeling, based on the common fracture mechanics approach. Whereas debonding from a Winkler foundation can be expressed in terms of the displacement (or equivalently the foundation stress2) at the bond termination, the sixth-order formulation required for elastomeric foundations involves a more complex debonding criterion. Transversality relationships are used to describe this limit state (here the onset of debonding) in terms of co-state variables, herein the deflection and slope at the peel front. In the analysis, bending is assumed to be paramount, linear Kirchhoff–Love (classical) plate theory is used to model the deformation, and therefore displacements are assumed to be small. The foundation is linearly elastic and incompressible. The effects of the work of adhesion, the length of the plate, and the initial nonbonded length of the plate are investigated. The results are compared to those for a Winkler foundation. By replacing the shear modulus of the interlayer by viscosity, and displacements by their time derivatives, the results are expected to apply to viscous liquid interlayers as well.

Deformation of corrugated hydrogel panel subject to chemo-mechanical coupled loading

In this paper, a novel plate theory is proposed for corrugated soft panels with a chemically coupled constitutive model of hydrogel, which is formulated via perturbation analysis on the physically based constitutive theory. Here, the elastic moduli naturally incorporate the chemical effects that could reduce to the classical linear Hookean law or second-order Murnaghan model. As a result, the resultant constitutive model is concise in form and comprehensive in physical integrity, where no further coupling is required between the deformation of solid and diffusion of fluid. Applying the symplectic elasticity method, the exact bending solution is then derived for corrugated hydrogel panels with two opposite sided clamped. The main effects of chemical potential, Flory parameter, and geometric factors on the equivalent stiffnesses and transverse deflections are investigated in detail. Some new results could be used as a reference for the designing of flexible and wearable devices.

The exact solution of annular sandwich plate undercoated with radial-FGM subjected to radial pressure or transverse load

This study aims to derive the exact stretching and bending solutions of the radially FGM annular sandwich plate subjected to either radial pressure or transverse load. The radially FGM annular sandwich plate consists of two homogeneous rings and an FGM-undercoated layer with uniform Poisson’s ratio and radially graded Young’s modulus based on the power function. Although the radially FGM annular sandwich plate is nonhomogeneous, the stretching and bending problems can be analyzed separately based on the elastic plate theory. Then the exact solution corresponding to the power-law graded FGM can be evaluated by solving the simultaneous equations of the boundary and continuous conditions. The theoretical results are validated through the comparison with the finite element solutions. When degenerated, our results coincide with those of the homogeneous annular plate in the literature, making the obtained solutions reliable. Using the radial-FGM as the undercoated layer makes the strains and stresses smoothly continuous at the interfaces of the layers. The effects of the material index on mechanical behavior are under investigation.

Analytic solutions for static bending and free vibration analysis of FG nanobeams in thermal environment

Beam, plate, and shell structures manufactured from functionally graded materials (FGMs) are becoming increasingly prevalent in practice. The manufacture of these structures is not free from flaws, one of which is that the thickness distribution of the material does not change continuously. This is the first study to apply two distinct analytical solutions to examine the static and free vibration responses of imperfect FG nanobeams in terms of material distribution and temperature influence. The calculation formulas are based on the novel shear strain theory, which, despite being simple, accounts for both bending and shear strain, rendering the calculation procedure highly convenient. Either of the two precise solutions is computed using direct integrals and can be computed for various boundary conditions, which is not possible with Navier’s solutions. The results of the verification indicate that the approaches given in the article guarantee the needed precision. This study also analyzes the effect of several geometrical, material, nonlocal, and imperfection characteristics on the static bending and free oscillation response of imperfect FG nanobeams.

Analytical Solution of Bending, Buckling, and Vibration of Functionally Graded Nanobeams based on Nonlocal Elasticity

Analytical Solution of Bending, Buckling, and Vibration of Functionally Graded Nanobeams based on Nonlocal Elasticity