Timoshenko Model Research Articles

Smart Bioinspired Material‐Based Actuators: Current Challenges and Prospects

This research review discusses several examples of plant movements, either depending on the direction of the triggering stimuli (tropisms) or not (nastic responses), which have served as inspiration to develop smart biomimetic actuators. In addition, it presents an overview of the multiple approaches for the development of autonomous actuators based on synthetic materials, as well as of their advantages and disadvantages, applicability, and limitations. The classification is based on structural and conformational characteristics (mono‐, bi‐, or multimaterial assemblies, their orientation, chemical structures, and geometrical configurations). Additionally, this review presents an alternative formulation and extension of the pioneering Timoshenko's model, which provides an understanding of the underlying mechanical principle of bilayer bending actuation. Finally, upscaled applications of this actuation principle are described, focusing mainly on biomimetic architecture. Attention is given to previously reported real‐life applications based on bio‐based materials and material systems. Furthermore, this review discusses the multiple challenges for synthetic materials when an upscaling perspective is intended. In this sense, key aspects such as time responsiveness and mechanical amplification, in terms of speed, displacement, and load‐bearing capability, are discussed.

A simple extension of Timoshenko beam model to describe dissipation in cementitious elements

In this paper, an extension of the Timoshenko model for plane beams is outlined, with the aim of describing, under the assumption of small displacements and strains, a class of dissipative mechanisms observed in cementitious materials. In the spirit of micromorphic continua, the modified beam model includes a novel kinematic descriptor, conceived as an average sliding relevant to a density of micro-cracks not varying along time. For the pairs of rough surfaces, in which such a distribution of micro-cracks is articulated, both an elastic deformation and a frictional dissipation are considered, similarly to what occurs for the fingers of the joints having a tooth saw profile. The system of governing differential equations, of the second order, is provided by a variational approach, endowed by standard boundary conditions. To this purpose, a generalized version of the principle of virtual work is used, in the spirit of Hamilton–Rayleigh approach, including as contributions: (i) the variation of the inner elastic energy, generated by the linear elasticity of the sound material and, in a nonlinear way, by the mutual, reversible deformation of the asperities inside the micro-cracks; (ii) the virtual work of the external actions consistent with the beam model, i.e., the distributed transversal forces and the moments per unit lengths; besides these two contributions, constituting the conservative part of the system, (iii) the dissipation due to friction specified through a smooth Rayleigh potential, entering a nonlinear dependence of viscous and Coulomb type on the sliding rate. Through a COMSOL Multiphysics implementation, 1D finite element analyses are carried out to simulate structural elements subjected to three- and four-point bending tests with alternating loading cycles. The dissipation of energy is investigated at varying the model parameters, and the predictions turn out to be in agreement with preliminary data from an experimental campaign. The present approach is expected to provide a valuable tool for the quantitative and comparative assessment of the hysteresis cycles, favoring the robust design of cementitious materials.

Analytical Solution for Pipeline Settlement Law Using Shield Tunnel Excavation Underneath Based on Vlasov–Timoshenko Model Considering Lateral Soil Action

Analytical Solution for Pipeline Settlement Law Using Shield Tunnel Excavation Underneath Based on Vlasov–Timoshenko Model Considering Lateral Soil Action

О деформировании решетчатой пластинки глаза

Several problems pertaining to deformation of the circular elastic isotropic plate of variable rigidity in the presence of elastic support on the edge were study. In the first problem Timoshenko model was considered. In the second problem the deformation of a multilayer plate with piecewise constant Lame coefficient along the thickness coordinate which are assumed to be constant within one layer was researched, the Zig-zag method is used for modeling. The basis for constructing a solution for both problems is the use of Lagrange’s variation principle of plates, numerical solution constructed using the Ritz method. During the research work the deflection and inflection points of the lamina cribrosa were compared for a healthy eye and for an eye with different stages of primary open-angle glaucoma for constant Lame coefficient as well as for their exponentially decreasing above. The problem of deflection of the lamina cribrosa when tacking into account the prelaminar layer was solved based on the Zig-zag method, the solution to this problem was also compared for a healthy eye and for an eye with various stages of primary open-angle glaucoma.

Investigation of free vibration of piezoelectric actuator reinforced with functionally graded Boron Nitride nanotube using timoshenko beam model and differential quadrature method

The vibrational characteristics and natural frequencies of Boron Nitride Nanotube (BNNT)-reinforced piezoelectric beams, based on the Timoshenko model, have been studied for the first time in this paper. Four types of functionally graded (FG) distribution patterns of BNNT reinforcements were considered for the FG-BNNT-reinforced piezoelectric beam. It is assumed that the material properties of FG-BNNT-reinforced beams are estimated using the Representative Volume Element (RVE) micromechanical model. The FG-BNNT-reinforced piezoelectric beam with different Boundary Conditions (BCs) is exposed to electrical voltage. Coupled partial differential equations of the system have been derived with the aid of Hamilton's principle and then converted into corresponding dimensionless equations using dimensionless variables. The differential quadrature method (DQM) has been used to obtain the natural frequencies and shape modes. According to the convergence analysis, twelve grid points were enough to obtain an accurate result for the analysis. To confirm the accuracy and reliability of the present study, the obtained results were validated with frequencies from the literature. Also, the effects of BNNT volume fraction and its reinforcement pattern, as well as applied voltage, on dimensionless natural frequencies have been investigated for different BCs. The results show that applying electric loadings can be utilized as well as controlling parameters to modify the resonance frequency of the composite actuator. More specifically, it is shown that applying voltages to a FG-BNNT-reinforced piezoelectric beam changes the natural frequency for each reinforcement pattern, BCs, and mode numbers. Such ability could be employed when designing reinforced piezoelectric composites for vibrational applications.

Optimal Uniform Strength Design of Frame and Lattice Structures

This paper provides a procedure to obtain the uniform strength of frame and lattice structures. Uniform strength condition is achieved by performing the shape optimization of all beam elements of the structure. The beam shape which guarantees uniform strength is analytically deduced from the one-dimensional Timoshenko model. The optimization problem presents itself as the search for the zeros of the objective-functions vector, which is a non-linear system of equations representing the kinematic-congruence and forces balance at every node of the structure. The analytical formulation of the optimization problem allows to construct the objective-functions vector without the use of external structural computation, i.e. not recurring to any Finite Element Analysis to accomplish iterations. This latter feature entails a great advantage in terms of computing time required to perform optimization. The proposed analytical formulation allows to directly insert the uniform strength condition into the objective-functions vector, transforming the optimization into an unconstrained problem. Some examples are shown in which the performance of the optimization procedure is discussed in terms of robustness and rate of computational complexity while increasing the degrees of freedom of the structure. The reliability and the quality of the optimization are verified through Finite Element Analysis.

Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay

Abstract In this study, we address a Cauchy problem within the context of the one-dimensional Timoshenko system, incorporating a distributed delay term. The heat conduction aspect is described by the Lord-Shulman theory. Our demonstration establishes that the dissipation resulting from the coupling of the Timoshenko system with Lord-Shulman’s heat conduction is sufficiently robust to stabilize the system, albeit with a gradual decay rate. To support our findings, we convert the system into a first-order form and, utilizing the energy method in Fourier space, and derive point wise estimates for the Fourier transform of the solution. These estimates, in turn, provide evidence for the slow decay of the solution.

A Graphene-Mica-Based Photo-Thermal Actuator for Small-Scale Soft Robots.

Small-scale soft robots demonstrate intricate life-like behavior and allow navigation through arduous terrains and confined spaces. However, the primary challenges in soft robotics are 1) creating actuators capable of quick, reversible 22D-to-3D shape morphing with adjustable stiffness, 2) improving actuation force and robustness for wider applications, and 3) designing holistic systems for untethered manipulation and flexible multimodality in practical scenarios. Here, mechanically compliant paper-like robots are presented with multiple functionalities. The robots are based on photothermally activated polymer bimorph actuators that incorporate graphene for the photo-thermal conversion of energy and muscovite mica, with its high Young's modulus, providing the required stiffness. Conversion of light into heat leads to thermal expansion and bending of the stress-mismatched structures. The actuators are designed on the basis of a modified Timoshenko model, and numerical simulations are employed to evaluate their actuation performance. The membranes can be utilized for light-driven programmable shape-morphing. Localized control allows the implementation of active hinges at arbitrary positions within the membrane. Integrated into small-scale soft robots in mass production, the membrane facilitates locomotion, rolling, and flipping of the robots. Further, grasping and kicking mechanisms are demonstrated, highlighting the potential of such actuators for futureapplications.

Transversely heterogeneous nonlocal Timoshenko beam theory: A reduced-order modeling via distributed-order fractional operators

Nonlocal effects are widely identified in micro/nano systems and have been studied extensively in recent decades. Nevertheless, existing studies mostly focus on the modeling of homogeneous nonlocal systems and as a result, cannot model micro/nanostructures with heterogeneously distributed nonlocal effects. This study introduces a reduced-order nonlocal Timoshenko beam model that facilitates precise modeling of micro-elastic beams with layer-wise heterogeneous nonlocal effects. First, a fully-resolved two-dimensional (2D) plane-strain fractional-order nonlocal elastic framework is developed to provide reference solutions for 2D nonlocal beam problems. Building upon this general nonlocal elasticity theory, an equivalent one-dimensional (1D) nonlocal Timoshenko model is then developed. The layer-wise heterogeneous nonlocal information is captured by using distributed-order (DO) operators, and the impacts arising from the nonlocal heterogeneity are further characterized by introducing two auxiliary parameters. Both 1D and 2D approaches are applied to simulate the mechanical responses of nonlocal beams. Direct comparisons of numerical simulations produced by either the 1D or the fully-resolved 2D model confirm that the DO Timoshenko beam formulation (together with the two auxiliary parameters) can capture not only the overall beam deflection but also the additional shear effect induced by the heterogeneous nonlocality. Computational cost assessment also indicates the proposed approach’s superior performance. The proposed DO Timoshenko model enables model-order reduction without compromising the heterogeneous nonlocal description of the material hence leading to an efficient and accurate reduced-order nonlocal modeling approach that addresses the limitations of prior nonlocal microstructural theories and extends applicability to a broader range of heterogeneous nano/microstructures.

Invariance Analysis and Closed-form Solutions for The Beam Equation in Timoshenko Model

Our research focuses on a fourth-order partial differential equation (PDE) that arises from the Timoshenko model for beams. This PDE pertains to situations where the elastic moduli remain constant and an external load, represented as F, is applied. We thoroughly analyze Lie symmetries and categorize the various types of applied forces. Initially, the principal Lie algebra is two-dimensional, but in certain noteworthy cases, it extends to three dimensions or even more. For each specific case, we derive the optimal system, which serves as a foundation for symmetry reductions, transforming the original PDE into ordinary differential equations. In certain instances, we successfully identify exact solutions using this reduction process. Additionally, we delve into the conservation laws using a direct method proposed by Anco, with a particular focus on specific classes within the equation. The findings we have presented in our study are indeed original and innovative. This study serves as compelling evidence for the robustness and efficacy of the Lie symmetry method, showcasing its ability to provide valuable insights and solutions in the realm of mathematical analysis.

Transformational deformation models of continuous thin‐walled structural elements with support elements of finite sizes: Theoretical foundations, computational, and physical experiments

AbstractA method of experimental study of forced bending vibrations of unfixed spans of continuous thin‐walled structural element fixed on rigid support elements of finite dimensions has been developed using instrumental means for registration of vibration acceleration amplitudes. The technique is realized on beams with different types of their fixing under the loading of one of the beam section with an external force that varies in time according to a harmonic law. Experimental dependences of amplitude values of vibration accelerations in specific points of certain spans of the beam on the frequency of external load have been obtained. These results indicate the passage of vibrations through the fixed sections due to the transformation of bending vibration modes of the loaded section into longitudinal transverse‐shift vibrations of the fixed sections during the transition across the boundary from the unfixed section to the fixed one (from the fixed to the unfixed section). For the theoretical study of the described phenomenon a transformational model of the flat beam deformation is constructed on the basis of the refined Timoshenko model with additional account of the deformability of the section with fixation to a rigid support element of finite length on one of its face surfaces. The corresponding equations of motion of the unfixed and fixed sections are derived. The corresponding kinematic and force conditions of their conjugation are formulated. On their basis, analytical solutions of two specific problems are found in the linear approximation. One‐dimensional finite elements for modeling the dynamic response of continuous beam of the considered class are also constructed. Numerical experiments have been carried out for a flat beam made of aluminum alloys D16AT, AMg‐6 and unidirectional fiber reinforced plastic. The presence of a significant transformation of the stress‐strain state parameters of the considered beams at the transition across the boundary from unfixed to fixed sections and a significant increase in the level of transverse tangential stresses on the fixed section of the beam in the vicinity of the interface of the unfixed section with the fixed one have been revealed.

Closed form Expressions of Shear Correction Factor for Functionally Graded Beams

Abstract In this work closed-form expressions of Shear Correction Factor (SCF) have been derived for beams with Functionally Graded Material (FGM), through Variational Asymptotic Method (VAM). An energy equivalence approach has been adopted between VAM and Timoshenko model, for estimating the SCF. A planar FGM beam has been considered and the calculation for SCF has been carried out. The formulation has been derived in a functional form that permits solutions for a large class of gradation models of FGM. In the limiting case when the material becomes homogeneous the estimated SCF matches exactly with that of the literature, thus validating the solution. A detailed discussion has been carried out on the results and conclusions have been drawn.

Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with microtemperature effect

Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with microtemperature effect

Dynamic Stability Analysis of FGM Beams Based on the Nonlinear Timoshenko Model

Dynamic Stability Analysis of FGM Beams Based on the Nonlinear Timoshenko Model

Application of generalised functions and Fourier’s integral transformation in modelling the effect of a moving load on a beam resting on an elastic foundation

The paper is dedicated to modelling the behaviour of an infinitely long beam resting on a solid elastic foundation under the concentrated moving load effect. The modelling is carried out in the context of determining the high-speed vehicle stock impact on the track structure. When developing the method of problem solving, the known elasticity theory factors, wave-propagation theory, complex variable theory, as well as integral transformations of generalised functions were used. The mobile concentrated force is represented by the Dirac delta function. The Fourier’s integral transformation is used to solve the equation of motion and determine the beam defect. Contour integration using Cauchy residue theorem is applied to calculate the defect values in the original space. The general dependence form the beam defect on the load velocity is determined. It is shown that the defect values in the beam increase with moving load increasing speed. It is shown how the moving load velocity affects the character of the beam defect time dependence. Solutions are given for Euler-Bernoulli, Rayleigh and Timoshenko models of beams on an elastic base. The considered mathematical models behaviour is compared. It is shown that the character of beam interaction with the moving load, the defect values, as well as the reaching critical velocities character do not differ significantly for the considered beam models. The character of reaching the maximum defect in the application area of the mobile load is analysed. It is shown that the maximum defect values of beams on an elastic base occur after the moving load. The presented methodology can be used to evaluate the impact of rolling stock on the track structure, including the design of high-speed lines.

Large displacements and rotations of a local linear elastic rod

AbstractThe Local Linear Timoshenko (LLT) model for the planar motion of a rod that undergoes flexure, shear and extension, was recently derived in Van Rensburg et al. (2021). In this paper we present an algorithm developed for this model. The algorithm is based on the mixed finite element method, and projections into finite dimensional subspaces are used for dealing with nonlinear forces and moments. The algorithm is used for an investigation into elastic waves propagated in the LLT rod. Interesting properties of the LLT rod include the increased propagation speed of elastic waves when compared to the linear Timoshenko beam, and the appearance of buckled states or equilibrium solutions for compressed LLT beams. It is also shown that the LLT rod is applicable to large displacements and rotations for a wide range of slender elastic objects; from beams to highly slender flexible rods.

Exact mathematical model for the behavior of a two-layer isotropic plate with partial interaction

This paper analyses the spatial variation of the deflection and slip at the interface of two-layer composite square plate simply supported under distributed static loading. The Timoshenko model is considered and the coupled equations are derived. The exact mathematical expressions for the transverse deflection, longitudinal deformations and rotation angle are obtained. The direct numerical simulation with Abaqus software is conducted. The decrease of the deflection and slip versus the coupling coefficient between the layers is plotted and one finds a good agreement between the mathematical and numerical simulation results. As one must expect, this shows that the composite square plate works like a monolithic plate when the coupling coefficient is high enough (no slip).

A 4D-Printing Inverse Design Strategy for Micromachines with Customized Shape-Morphing.

An active heterostructure with smart-response material used as "muscle" and inactive material as "skeleton" can deform over time to respond to external stimuli. 4D printing integrated with two-photon polymerization technology and smart material allows the material or characteristic distribution of active heterostructures to be defined directly at the microscale, providing a huge programmable space. However, the high degree of design freedom and the microscale pose a challenge to the construction of micromachines with customized shape morphing. Here, a reverse design strategy based on multi-material stepwise 4D printing is proposed to guide the structural design of biomimetic micromachines. Inspired by the piecewise constant curvature model of soft robot, a reverse design algorithm based on the Timoshenko model isdeveloped. The algorithm can approximate 2D features to a constant-curvature model and determine an acceptable material distribution within the explored printing range. Three Chinese "Long" (Chinese dragon heralds of good fortune) designed by the strategy candeform to the customized shape. In addition, a microcrawler printed using this method canimitate a real inchworm gait. These results demonstrate that this method can be an efficient tool for the action or shape design of bionic soft microrobots or micromachines with predetermined functions.

Modular Design of a Polymer-Bilayer-Based Mechanically Compliant Worm-Like Robot.

Soft earthworm-like robots that exhibit mechanical compliance can, in principle, navigate through uneven terrains and constricted spaces that are inaccessible to traditional legged and wheeled robots. However, unlike the biological originals thatthey mimic, most of the worm-like robots reported to date contain rigid components that limit their compliance, such as electromotors or pressure-driven actuation systems. Here, a mechanically compliant worm-like robot with a fully modular body that is based on soft polymers is reported. The robot is composed of strategically assembled, electrothermally activated polymer bilayer actuators, which are based on a semicrystalline polyurethane with an exceptionally large nonlinear thermal expansion coefficient. The segments are designed on the basis of a modified Timoshenko model, and finite element analysis simulation is used to describe their performance. Upon electrical activation of the segments with basic waveform patterns, the robot can move through repeatable peristaltic locomotion on exceptionally slippery or sticky surfaces and it can be oriented in any direction. The soft body enables the robot to wriggle through openings and tunnels that are much smaller than its cross-section.

Decay rate of the solutions to the Lord Shulman thermoelastic Timoshenko model

<abstract><p>In this work, we deal with a one-dimensional Cauchy problem in Timoshenko system with thermal effect and damping term. The heat conduction is given by the theory of Lord-Shulman. We prove that the dissipation induced by the coupling of the Timoshenko system with the heat conduction of Lord-Shulman's theory alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow.</p></abstract>