Elastic Bar Research Articles

Enlarged Curvature, Torsion and Torque in Helical Conformations and the Stability and Growth of α-Peptide under the Isochoric and Isobaric Conditions: Variatonal Optimization

The torsional deformation behavior of an elastic bar with a circular cross-section was investigated by applying invariant dyadic analysis, where the small finite displacement functions advocated by Saint-Venant (1855) were fully employed. It was found that the previously overlooked circumferential shear force field generated by pure torsion on the side walls of a bar produces an unusual torque term induced by the skew-symmetric part of the deformation tensor and exhibits quadratic length dependence along the z-axis of the bar. The adaptation of this torque term for a helical conformation of α-peptides creates moments acting on the circular cross-sections and is directed along the surface normal of circular cross-sections, which coincides with the tangent vector of the helix. The projection of this torque along the z-axis of the helix varies quadratically with the azimuthal angle. The radial component of the unusual torque, which also lies along the principal normal vector of the helix, starts to perform a precession motion by tracking a spiral orbit around the z-axis, whereas its apex angle decreases asymptotically with the azimuthal angle and finally reaches a finite value depending on the height of the helix along the z-axis. The ordinary torque terms, which are also deduced from the self- and anti-self-conjugate parts of the deformation tensor, have magnitudes half that of the full torque term reported in the literature. The present results were applied to the helical conformation of α-peptides designated by {3.611} to show that the mechanical stability of strained open-ended helical conformations can be successfully achieved by spontaneous readjustments of the surface and bulk Helmholtz free energies under isothermal isochoric conditions. It has been demonstrated that the main contribution to the mechanical stability of α-peptide 3.611 cannot come alone from the electrostatic dipole-dipole interaction potential of the anti-align excess dipole pairs but also from the surface Helmholtz free energy, which is characterized by a binding free energy of -15.5 eV/molecule (-32.56 Kcal/mole) for an alpha-peptide composed of 11 amino acid residues with a critical arc length of approximately 10 nm, assuming that the shear modulus is G = 1GPa and the surface Helmholtz specific free energy density is fs = 800 erg/cm2. This result was in excellent agreement with the experimental observations of the AH-1 conformation of (Glu)n Cys at pH 8. The present theory indicates that only two excess permanent anti-align dipole pairs for one α-Helical peptide molecule is requirement to stabilize the whole secondary structure of the protein that is exposed to heavy torsional deformation during the folding processes which amounts to 7.75 eV/molecule stored electrostatic energy compared to the interfacial Helmholtz free energy of -23.25 eV/molecule, which is exposed to hydrophobic environments.

Study of the Influence of the Relative Rigidity of an Elastic Hybrid System Without Damping, on Natural Self-Pulsation, Amplitude and Dynamic Factor Due to Forced Vibration

Abstract The paper aims to study the dynamic behavior of a hybrid elastic system without damping consisting of a beam in a cantilever supported on helical compression springs on which an electric motor is mounted. With the help of the MATHCAD professional program, the influence of the relative stiffness of the elastic bar subjected to bending and the springs subjected to compression was simulated, on the natural pulsation, the amplitude of the vibrations and the dynamic factor of the forced vibrations for two assembly variants of the electric motor

Peridynamic computations of wave propagation and reflection at material interfaces

Peridynamics describes the material in a non-local form and is very suited for the simulation of dynamic fracture. However, one significant effect regarding dynamic fracture is the correct handling of elastic deformation, like the pressure and tension waves inside a body, due to dynamic boundary conditions like an impact or impulse. Many peridynamic material formulations have been developed with differences in this regard. This study investigates the elastic wave propagation characteristics of bond-based, ordinary state-based, continuum kinematics-inspired peridynamics and a local continuum consistent correspondence formulation. Multiple parameters of a longitudinal pressure wave inside an elastic bar are studied. While all formulations demonstrate adequate wave propagation handling, all except the correspondence formulation are sensitive to incomplete horizons. The local continuum consistent formulation does not suffer from the surface effect and models the wave propagation with perfect accuracy.

Prediction of flexural strength in FRP bar reinforced concrete beams through a machine learning approach

PurposeThe purpose of this study is to assess the incorporation of fiber reinforced polymer (FRP) bars in concrete as a reinforcement enhances the corrosion resistance in a concrete structure. However, FRP bars are not practically used due to a lack of standard codes. Various codes, including ACI-440-17 and CSA S806-12, have been established to provide guidelines for the incorporation of FRP bars in concrete as reinforcement. The application of these codes may result in over-reinforcement. Therefore, this research presents the use of a machine learning approach to predict the accurate flexural strength of the FRP beams with the use of 408 experimental results.Design/methodology/approachIn this research, the input parameters are the width of the beam, effective depth of the beam, concrete compressive strength, FRP bar elastic modulus and FRP bar tensile strength. Three machine learning algorithms, namely, gene expression programming, multi-expression programming and artificial neural networks, are developed. The accuracy of the developed models was judged by R2, root means squared and mean absolute error. Finally, the study conducts prismatic analysis by considering different parameters. including depth and percentage of bottom reinforcement.FindingsThe artificial neural networks model result is the most accurate prediction (99%), with the lowest root mean squared error (2.66) and lowest mean absolute error (1.38). In addition, the result of SHapley Additive exPlanation analysis depicts that the effective depth and percentage of bottom reinforcement are the most influential parameters of FRP bars reinforced concrete beam. Therefore, the findings recommend that special attention should be given to the effective depth and percentage of bottom reinforcement.Originality/valuePrevious studies revealed that the flexural strength of concrete beams reinforced with FRP bars is significantly influenced by factors such as beam width, effective depth, concrete compressive strength, FRP bars’ elastic modulus and FRP bar tensile strength. Therefore, a substantial database comprising 408 experimental results considered for these parameters was compiled, and a simple and reliable model was proposed. The model developed in this research was compared with traditional codes, and it can be noted that the model developed in this study is much more accurate than the traditional codes.

Dynamics and wave analysis in longitudinal motion of elastic bars or fluids

We look at the bifurcation analysis, chaotic behavior, multistability, and sensitivity analysis of the van der Waals equation, which is used in science and engineering to study the dynamics of various structures. The variant van der Waals equation covers one-dimensional longitudinal isothermal motion in elastic bars or fluids, which is the main focus of this research. The Galilean transformation transfers the given model into a planar dynamical system. The power series methodology produces exact wave solutions. In addition, after accounting for the perturbation component, sensitivity analysis for various initial value problems is used to investigate quasi-periodic, chaotic, and time series behavior. Numerical simulation results show that influencing viscosity and the coefficient of interfacial capillarity influence the dynamical factors of the examined model. We are observing what happens with viscosity and the interface coefficients to the wave solution. In some cases, for different parameter values, we present plots of both one-dimensional and two-dimensional graphical representations of individual solutions.

An iteration method to study nonlinear wave propagation for a non-Green elastic 1D bar

Abstract The problem of propagation of nonlinear waves in a 1D bar is studied, wherein the linearized strain tensor is considered as a function of the Cauchy stress tensor. Specifically, two constitutive equations for non-Green elastic solids are investigated, introducing a novel numerical iterative method capable of obtaining approximate solutions of one nonlinear constitutive equation for rock, and one constitutive equation that shows a strain-limiting behaviour. The numerical results are compared with exact solutions for the case of a linearized elastic solid.

Displacement Analyses of Diaphragm Wall in Small-Scale Deep Excavation Considering Joints between Panels

This paper proposed a new method for modelling joints, using anisotropic plate elements and elastic bar elements to address the issue that joints between panels are usually disregarded in numerical modelling. For small-scale deep excavations, which are frequently performed in the construction of various working shafts but have not been sufficiently studied, two numerical models were developed, using the No.1 Shaft of Tongtu Road Utility Tunnel in Ningbo, China, as a research object. One model considered the joints between the panels as proposed, while the other disregarded the joints as conventional. In comparison to the conventional method, the proposed method was validated due to yielding wall displacements that closely matched the results of the field monitoring, with a notable reduction in the error observed in the calculated displacements for the short side of the excavation. Furthermore, 34 numerical models were developed in order to investigate the influence of excavation length, depth, and diaphragm wall thickness on the relative differences between the calculated displacements obtained by the two models. The results of this study can provide references for the development of finite element models for designing small-scale deep excavation.

Buckling analysis of a helical extension spring under combined loading

A new formula is introduced to analytically determine the critical torsional load or angle of rotation that confers instability in a tension spring under combined tension and torsion load using the theoretical framework of Greenhill's problem applied to an elastic bar as a starting point. The formula's effectiveness is evaluated with experimental torsion tests and finite element method simulations in Abaqus. The results demonstrate the formula's accuracy in predicting instability within specific stretched operating ranges, and achieving a relative error in correlation below 10 % for most operating points. The study underscores the significance of preload and pre-stretching in enhancing the spring's resistance to buckling.

Waves in elastic bars under axial constant velocity loading

Elastic bars serve as primary mechanical members, spanning from atomic to macroscopic scales. The development of theoretical frameworks for elastic bars is crucial for guiding a wide array of industrial and everyday applications. In this study, we comprehensively investigate wave propagation in elastic bars subjected to tension and compression under constant velocity loading conditions. First, axial waves were studied by analytically solving the axial wave equation with damping. Subsequently, we employed this theoretical framework in atomistic simulations to measure loading-rate effects and thus determine the conditions for distinguishing quasi-static and dynamic loading. Given the coupled compressive axial waves, deflection waves were analyzed to predict dynamic instability. Our findings revealed that a continuous increase in impact velocity may not necessarily result in a larger impact force. This study offers fresh insights into the mechanical theory on the dynamic behaviour of bars across diverse scales (atomic to macroscopic). These insights are invaluable for determining suitable loadings in atomistic simulations, and provide guidelines for engineering impact applications.

Stress wave attenuation in chocolate

The stress wave attenuation in five types of chocolate (extra dark, dark, milk, white, and ruby) was studied. The main acoustics properties of these chocolates were described in terms of longitudinal and transversal wave velocities. The attenuation of the stress waves was studied using the experimental technique known as the Split Hopkinson Pressure Bar (SHPB). The SHPB set-up consists of a specimen and linear elastic incident and transmission bar. Experimental data have been evaluated both in the time as well as frequency domain. Attenuation of the stress wave was described namely using the frequency dependence of the transmission coefficient of the stress wave. This damping increases with the plastic strain of the tested chocolate specimens. From a technological point of view, the fat content of the chocolate has the highest influence on the reduction of the longitudinal wave propagation velocity. The lowest attenuation was observed for the dark chocolate.

Dynamics of generalized time-fractional viscous-capillarity compressible fluid model

This analysis examines the time-fractional mixed hyperbolic-elliptic p-system of conservation laws by applying the new extended direct algebraic method. The p-system with generalized cubic van der Waals flux, and potential applications in the field of compressible isothermal viscosity-capillarity fluids, is investigated. In particular, this issue describes the longitudinal isothermal motion in elastic bars or fluids. A diverse periodic, kink, and singular soliton structures are extracted. The 3D dynamical behaviors and corresponding contour profiles of some obtained solitons are displayed. The fractional effects in the sense of Beta, M-truncated, and modified Riemann–Liouville, are discussed and illustrated. The method shows the straightforward, reliability, and efficiency for solving complex physical phenomena that is modeled by nonlinear partial differential equations.

On the splay deformation mode of a polar linearly elastic bar stretched by its own weight

This communication considers the fundamental linear elasticity problem of a prismatic bar, or plate, stretched by its own weight and examines the impact of its classical solution in the regime of isotropic and anisotropic polar material elasticity. Accordingly, the existing non-polar elasticity solution of such a self-stretched isotropic bar is initially extended to embrace appropriate classes of non-polar material anisotropy. This extension verifies that, in non-polar transverse isotropy and special orthotropy, the attained solution is exclusively dominated by splay-type features of deformation. Attention then focuses on the influence that the observed fiber-splay deformation mode, as well as its fiber-bending deformation counterpart, exert on the formulation and potential solution of corresponding boundary value problems met in polar linear elasticity. It is seen that, regardless of the isotropic or anisotropic material symmetries considered, the outlined process may lead to solution of relevant boundary value problems that are slightly different to their non-polar elasticity counterparts. This conclusion reinforces the role that a polar material version of the theorem of minimum potential energy, and relevant energy minimization approaches, can play in the search for full solution of boundary value problems met polar material elasticity.

On the well posedness of a mathematical model for a singular nonlinear fractional pseudo-hyperbolic system with nonlocal boundary conditions and frictional damping terms

<abstract><p>This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system with frictional damping terms. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary conditions. Upon some a priori estimates and density arguments, we establish the existence and uniqueness of the strongly generalized solution for the associated linear fractional system in some Sobolev fractional spaces. On the basis of the obtained results for the linear fractional system, we apply an iterative process in order to establish the well-posedness of the nonlinear fractional system. This mathematical model of pseudo-hyperbolic systems arises mainly in the theory of longitudinal and lateral vibrations of elastic bars (beams), and in some special case it is propounded in unsteady helical flows between two infinite coaxial circular cylinders for some specific boundary conditions.</p></abstract>

Curvature Analysis in a Bi-layered Non-linear Elastic Bar under Uniform Heating

The present theoretical paper is concerned with analysis of the curvature in a bi-layered bar of rectangular cross-section. The bar is subjected to uniform heating. The two layers of the bar have different thickness and material properties. Besides, the layers exhibit non-linear elastic mechanical behaviour that is treated by applying a power-law stress-strain constitutive relationship. The bi-layered bar curvature is studied analytically. For this purpose, the mechanical response of the bar to uniform heating is investigated. The strains in the two layers due to heating are analyzed. An equation is compiled by using the fact that the prolongations of the bar layers are equal. Two equations are worked out by considering the equilibrium of the layers. The analysis of the curvature presented in this paper is checked-up by comparing with a known solution for the curvature in a bi-layered linear-elastic bar under uniform heating. The uniform heating induced curvature in a bi-layered bar whose layers are continuously inhomogeneous in longitudinal direction is also studied.

Evaluation of ultimate load - bearing capacity of piles based on static compression test and pile dynamic analysis (PDA)

Determining the load bearing capacity of piles from field tests is a reliable method that allows for the verification of load carrying capacity according to theoretical calculations, thereby the ultimate load-bearing capacity of the piles can be evaluated, serving as the basis for selecting the design load-bearing capacity that is suitable for the site conditions. Among the various pile load testing methods, static compression testing is a traditional method known for its high reliability and early adoption. However, when piles are constructed at significant depths or large diameter drilled piles have higher load bearing capacities, static compression testing encounters difficulties due to the large experimental load and extends testing times. In contrast, the pile dynamic analysis method (PDA) introduced by the Case Institute of Technology in the United States in the 1960s determines the load bearing capacity of piles based on stress wave propagation theory in elastic bars. It offers several advantages, including shorter testing times and no need for static loading, making it suitable for challenging terrain conditions. Nowadays, this method is widely used and integrated into the design guidelines and testing standards of many countries. In this article, base on the results of axial static compression tests and PDA method on reinforced concrete piles and drilled piles at several real project to compare and evaluate the load bearing values of the piles and assess the reliability of the testing methods.

Detecting delamination via nonlinear wave scattering in a bonded elastic bar

In this paper we examine the effect of delamination on wave scattering, with the aim of creating a control measure for layered waveguides of various bonding types. Previous works have considered specific widths of solitary waves for the simulations, without analysing the effect of changing the soliton parameters. We consider two multi-layered structures: one containing delamination ‘sandwiched’ by perfect bonding and one containing delamination but ‘sandwiched’ by soft bonding. These structures are modelled by coupled Boussinesq-type equations. Matched asymptotic multiple-scale expansions lead to coupled Ostrovsky equations in soft bonded regions and Korteweg-de Vries equations in the perfectly bonded and delaminated region. We use the Inverse Scattering Transform to predict the behaviour in the delaminated regions. In both cases, numerical analysis shows that we can predict the delamination length by changes in the wave structure, and that these changes depend upon the Full Width at Half Magnitude (FWHM) of the incident soliton. In the case of perfect bonding, we derive a theoretical prediction for the change and confirm this numerically. For the soft bonding case, we numerically identify a similar relationship using the change in amplitude. Therefore we only need to compute one curve to determine the behaviour for any incident solitary wave, creating a framework for designing measurement campaigns for rigorously testing the integrity of layered structures.

Emergent elasticity relations for networks of bars with sticky magnetic ends

Magneto-elastic materials have unique mechanical properties arising from the coupling between the magnetic and elastic components, which can be used in many engineering applications, such as waveguide systems, impact mitigation, and soft robotics. Traditional designs of magneto-elastic materials embed magnets or magnetic particles in a monolithic body. Under extreme conditions, the elastic matrices are prone to permanent damage and loss of functionality. An alternative framework was previously proposed by the authors in which 2D magneto-elastic networks were created through vibration-driven self-assembly from a dilute system of elastic bars with sticky magnetic ends. While these networks were demonstrated to fail gracefully under extreme loading and re-assemble under random excitation, how their emergent elasticity depends on magnet and elastic member characteristics remains to be fully understood. In this paper, to calculate the 2D bulk modulus of a given magneto-elastic network, particle dynamics simulations are first performed. An analytical framework is then developed to predict the bulk modulus of derived networks with varying design parameters, such as bar length and magnetic strength, based on the Cauchy–Born approximation. Through dimensional analysis, we demonstrate that 2D bulk modulus is linearly proportional to a composite variable that combines magnet strength, elastic member length, and wall thickness of the magnet holder, as evidenced by the collapse of 120 systems onto a single line. The numerical and analytical investigation of the bulk modulus of this architected magneto-elastic material demonstrates the broad tunability of its emergent elasticity, thereby enabling a myriad of promising applications of these systems.

Study on Stability of Elastic Compression Bending Bar in Viscoelastic Medium

In the southeastern coastal regions of China, thick layers of marine soft soil are widely distributed, exhibiting characteristics such as high compressibility, high porosity, low strength, high sensitivity, and easy thixotropy, and these viscoelastic behaviors of foundation soil have significant implications for elastic compression bending bar, as evidenced by issues such as post-construction settlement of roadbeds and long-term operation deviation of bridge pile foundations. In this study, a mechanical model of an elastic bar embedded in an elastic and viscoelastic medium, fixed at the base and free at the top, is established based on the Winkler foundation assumption. The deflection function of a bar subjected to both axial force and locally distributed horizontal load is derived using the Rayleigh-Ritz method. Utilizing the elastic-viscoelastic correspondence principle, the viscoelastic medium surrounding the bar is modeled as an elastic medium in which the ground reaction coefficient varies within phase space formulation. This study provides a robust theoretical foundation for soft soil foundation engineering projects and fills a significant gap in the literature by offering a comprehensive framework for understanding displacement in elastic bars within viscoelastic media. Drawing upon the derivation of the deformation function for elastic rods within a viscoelastic medium, the findings of this research hold significant applicability across a range of domains. These include, but are not limited to, the expansion of roadways in regions characterized by coastal soft soil, as well as the monitoring of deformation and lifespan in bridge pile foundations.

Slow vibration via acoustoelastic coupling: Quasi-bound states in the continuum

Elastic bound states in the continuum (BICs) have recently garnered significant attention due to their remarkably high Q-factor, leading to the decoupling of the confined mode from spectrally coexisting radiational channels. In this presentation, we introduce a novel state capable of generating a slow vibration phenomenon, exhibiting multiphysics analogous to the concept of slow light observed in electromagnetically induced transparency (EIT). This state arises from the interaction of acoustoelastic coupling within a composite structure, featuring two acoustic cavities enclosed in an elastic bar. The proposed design facilitates quasi-BICs with exceptional spatial efficiency localized within a specific area, enabling tuning of the Purcell factor by approximately six orders of magnitude. The findings of such quasi-BICs may expand the BIC family and opens new avenues for applications in diverse fields, including lasing, sensing, screening, and energy storage platforms, where ultrahigh-Q-factor modes coexist with radiative channels.

Compressive Split Hopkinson Pressure Bar Apparatus: Part I - Design and Development

Optimum utilization of composite materials envisages its use under static as well as dynamic loading conditions. Hence, there is a need to assess the behavior of composite materials at high strain rates. Split Hopkinson Pressure Bar (SHPB) apparatus is widely used to assess high strain rate behavior of composites. Even though SHPB apparatus is widely used, its design methodology and development technique are not readily available. Design and devel- opment details of a compressive SHPB apparatus for characterizing mechanical properties of different materials at high strain rate loading are presented. Brief theoretical background necessary for the design of SHPB apparatus is presented based on one-dimensional wave propagation theory in elastic bars. Next, brief working principle of the apparatus and schematic arrangement are presented. Different components of the compressive SHPB appa- ratus are identified. Design, development and commissioning of the apparatus are discussed. Further, different instruments used and the calibration technique used are presented.