Displacement Functions Research Articles

Fracture Toughness Characteristics of Stacked Bilayer Graphene via Far-Field Displacement Measurements.

Mechanical properties of graphene, e.g., strength, modulus, and fracture toughness are extremely sensitive to flaws. Here the fracture properties of stacked bilayer graphene sheets (SBLG) are reported, obtained via stacking two individually grown graphene sheets. The SBLG is presented here as a building block for flaw-resilient nanomaterials. The fracture properties of freestanding SBLG sheets, suspended on transmission electron microscope (TEM) grids, are characterized by stretching the TEM grid inside an scanning electron microscope (SEM) chamber and monitoring the local displacements in real-time. The fracture toughness is measured and expressed as a function of the critical displacement required to propagate existing cracks in the experiment via computational models. This approach decouples force and displacements measurements, and utilizes the known elastic modulus along with the known displacement boundary conditions at the onset of crack growth to estimate the far field force and stress. This strategy represents a breakthrough in nanoscale fracture mechanics for statistical analysis and high throughput experimens on multiple samples at a time. Results demonstrate that the SBLG is markedly tougher than as-grown single or multilayer graphene, with a mode I fracture toughness of ≈28.06±7.5 MPa . The mechanisms leading to a higher toughness of SBLG are also analyzed and discussed.

Series solution for dynamical characteristic of spatial beam system structure in offshore platform

In engineering, the beam system is often used as a support element for large shell structures to improve the support stiffness of the shell and its vibration characteristics are worth being investigated. In this paper, the dynamics of the beam system is modeled and its vibration characteristics are investigated. This paper develops a 6-degree-of-freedom beam theory to describe the displacements of the basic cells of a beam system. This facilitates the solution of the problem of coupling failure of the beam system. The strain potential energy, the boundary spring potential energy, Energy generated by external forces and the structural kinetic energy of the beam system are derived from the energy principle. The above energy expression is then used to construct a Lagrangian energy functional. The displacement function of the beam cells is constructed by improving the Fourier series. It is substituted into the energy functional. The functional is also processed using the Rayleigh Ritz method to finally obtain a dynamic analysis model of the beam system. Then convergence analysis and correctness verification are carried out to ensure the correctness of the model. Finally, the parametric analysis is carried out to investigate the effect of the geometry on its frequency characteristics.

Enrichment of three-dimensional numerical manifold method with cover-based contact theory for static and dynamic mechanical response analysis

Based on the mathematical grid of 4-node tetrahedral finite element form, the dual cover system, weight function and displacement function of three-dimensional numerical manifold method (3D-NMM) are proposed. The derivation process of element stiffness matrix, inertial force matrix, loading point matrix, etc. are systematically demonstrated in the present work. Based on the cover-based contact theory, the contact detection algorithm among discrete 3D blocks is realized, and the 3D manifold framework is programmed. The accuracy of the developed code is firstly calibrated through two continuum deformation analysis: cantilever beam bending model and axial tensile model of thin plate with circular hole. Then it is applied to three complex discontinuous instability simulation (SHPB, brick wall structure and cliff slope) to verify the effectiveness and accuracy of the proposed contact algorithm. The classic Steven low velocity impact process is also simulated, and the deformation and failure of polymer bonded composites (PBC) are predicted, which further verifies the feasibility and robustness of the developed code in dealing with dynamic impact problems.

Refined Semi-Analytical Framework to Predict the Natural Vibration Characteristics of Bistable Laminates

Bistable unsymmetrical laminates have received significant attention in morphing applications due to their ability to attain multiple shapes when subjected to thermal loads. Morphing structures in general are subjected to dynamic operating conditions. Also, the highly nonlinear snap-through transition between stable configurations possesses rich dynamic characteristics. Therefore, understanding the dynamic characteristics of bistable laminates is essential for designing morphing structures constituting bistable elements. Thus, the present study aims to explore the dynamics of bistable unsymmetrical laminates by evaluating their natural vibration characteristics associated with small-amplitude dynamic excitation around the static equilibrium configurations. A refined semi-analytical framework is proposed to analyze the natural vibration characteristics of the bistable laminate, where the potential energy is expressed only in terms of the unknown coefficients of the assumed out-of-plane displacement function. The in-plane components are separately evaluated using the in-plane equilibrium equations and compatibility conditions. In the dynamic analysis, perturbations are imposed on the static equilibrium configurations to capture the modal characteristics. A full geometrically nonlinear finite element (FE) model of the bistable laminate has been created in a commercially available FE package to compare semi-analytical solutions. To validate the proposed frameworks, an experimental strategy to capture the natural frequencies of a bistable laminate is presented in this paper. Unsymmetric laminates mounted at its center have been used for the experimental testing, where the vibrations are measured using miniature integrated electronics piezoelectric accelerometer sensors attached at the corners. The semi-analytical and FE results are validated against the experimental observations for the selected unsymmetrical cross-ply laminates. The proposed frameworks are further extended to a family of unsymmetrical variable-stiffness (VS) laminates generated using curvilinear fiber alignments. The selected VS family can generate bistable shapes without any twisting curvature similar to that of an unsymmetrical cross-ply laminate, where the designer can expand the design space with a plethora of multiple configurations. A parametric study is performed by tailoring the VS parameters to investigate the influence of curvilinear fiber alignments on the natural vibration characteristics of bistable VS laminates.

Intensity of Dynamic Stresses of Longitudinal Shear in a Periodically Layered Composite with Penny-Shaped Cracks

We consider the problem of harmonic torsional loading of an infinite elastic composite formed by alternating plane layers made of two different materials in the presence of a penny-shaped crack in one of the components of periodic structure. By using the integral representations for displacements and stresses in the frequency domain and satisfying the conditions of periodicity and perfect contact on the interfaces, we deduce a system of uncoupled boundary integral equations for the functions of tangential dynamic crack-opening displacement in a two-layered representative element of the composite. We perform the numerical analysis of the mode-III dynamic stress intensity factors in the vicinity of the analyzed crack depending on the wave number, the thicknesses of constituent layers, and the properties of their materials.

Particle Image micro-Rheology (PIR) using displacement probability density function

We present a novel approach to perform passive microrheology. A method to measure the rheological properties of fluids from the Brownian motion of suspended particles. Rheological properties are found from the particles' mean square displacements (MSDs) as a function of measurement time lag. Current state-of-the-art approaches find the MSD by tracking multiple particles' trajectories. However, particle tracking approaches face many limitations, including low accuracy and high computational cost, and they are only applicable to low particle seeding densities. Here, we present a novel method, termed particle image rheometry (PIR), for estimating the particle ensemble MSD from the temporal evolution of the probability density function of the displacement as a function of measurement time lag. First, the probability density function (PDF) of the particle displacements for each time lag is found using a generalized ensemble image cross-correlation approach that eliminates the need for particle tracking. Then, PDFs are used to calculate the MSD from which the complex viscosity of the solution is measured. We evaluate the performance of PIR using synthetic datasets and show that it can achieve an error of less than 1% in passive microrheology measurements, which corresponds to a twofold lower error than existing methods. Finally, we compare the measured complex viscosity from PIR with bulk rheometry for a polymeric solution and show agreement between the two measurements.

Stability of mixed overlapping elements in incompressible analysis

In this study, we integrated a recently presented pure displacement-based overlapping element with a displacement–pressure mixed formulation. We focused on the idea that the overlapping element that adopts low-order triangles can be effectively integrated with the displacement–pressure mixed formulation without violating the stability condition owing to its enlarged displacement function spaces. By performing numerical stability tests, we obtained the stable displacement–pressure combinations available for practical use. In addition, we demonstrated the effectiveness of the mixed overlapping elements compared with traditional mixed finite elements through numerical examples. Our results imply that the salient feature of the original overlapping element is maintained in incompressible analysis, such that the element is insensitive to mesh distortion and accurate compared to traditional mixed finite elements.

Anomalous Dynamical Scaling Determines Universal Critical Singularities.

Anomalous diffusion phenomena occur on length scales spanning from intracellular to astrophysical ranges. A specific form of decay at a large argument of the probability density function of rescaled displacement (scaling function) is derived and shown to imply universal singularities in the normalized cumulant generator. Exact calculations for continuous time random walks provide paradigmatic examples connected with singularities of second order phase transitions. In the biased case scaling is restricted to displacements in the drift direction and singularities have no equilibrium analogue.

Universal singularities of anomalous diffusion in the Richardson class.

Inhomogeneous environments are rather ubiquitous in nature, often implying anomalies resulting in deviation from Gaussianity of diffusion processes. While sub- and superdiffusion are usually due to contrasting environmental features (hindering or favoring the motion, respectively), they are both observed in systems ranging from the micro- to the cosmological scale. Here we show how a model encompassing sub- and superdiffusion in an inhomogeneous environment exhibits a critical singularity in the normalized generator of the cumulants. The singularity originates directly and exclusively from the asymptotics of the non-Gaussian scaling function of displacement, and the independence from other details confers it a universal character. Our analysis, based on the method first applied by Stella etal. [Phys. Rev. Lett. 130, 207104 (2023)10.1103/PhysRevLett.130.207104], shows that the relation connecting the scaling function asymptotics to the diffusion exponent characteristic of processes in the Richardson class implies a nonstandard extensivity in time of the cumulant generator. Numerical tests fully confirm the results.

A High-Reliability Piezoelectric Tile Transducer for Converting Bridge Vibration to Electrical Energy for Smart Transportation.

Piezoelectric energy transducers offer great potential for converting the vibrations of pedestrian footsteps or cars moving on a bridge or road into electricity. However, existing piezoelectric energy-harvesting transducers are limited by their poor durability. In this paper, to enhance this durability, a piezoelectric energy transducer with a flexible piezoelectric sensor is fabricated in a tile protype with indirect touch points and a protective spring. The electrical output of the proposed transducer is examined as a function of pressure, frequency, displacement, and load resistance. The maximum output voltage and maximum output power obtained were 6.8 V and 4.5 mW, respectively, at a pressure of 70 kPa, a displacement of 2.5 mm, and a load resistance of 15 kΩ. The designed structure limits the risk of destroying the piezoelectric sensor during operation. The harvesting tile transducer can work properly even after 1000 cycles. Furthermore, to demonstrate its practical applications, the tile was placed on the floor of an overpass and a walking tunnel. Consequently, it was observed that the electrical energy harvested from the pedestrian footsteps could power an LED light fixture. The findings suggest that the proposed tile offers promise with respect to harvesting energy produced during transportation.

Evidence of the Schwinger Mechanism from Lattice QCD

In quantum chromodynamics (QCD), gluons acquire a mass scale through the action of the Schwinger mechanism. This mass emerges as a result of the dynamical formation of massless bound-states of gluons which manifest as longitudinally coupled poles in the vertices. In this contribution, we show how the presence of these poles can be determined from lattice QCD results for the propagators and vertices. The crucial observation that allows this determination is that the Schwinger mechanism poles induce modifications, called “displacements”, to the Ward identities (WIs) relating two- and three-point functions. Importantly, the displacement functions correspond precisely to the Bethe–Salpeter amplitudes of the massless bound-states. We apply this idea to the case of the three-gluon vertex in pure Yang–Mills SU(3). Using lattice results in the corresponding WI, we find an unequivocal displacement and show that it is consistent with the prediction based on the Bethe–Salpeter equation.

Love‐like wave dispersion in a highly non‐homogeneous viscoelastic orthotropic layer under the effect of non‐local elasticity

This paper examines the effects of non‐local elasticity on the propagation of Love‐like waves in a Voigt‐type viscoelastic layer over an elastic half‐space. The model's geometry is considered orthotropic, highly non‐homogeneous, and initially stressed media in the context of non‐local continuum mechanics. The non‐homogeneity functions are regarded as binomial functions with a positive real exponent. Two different non‐local parameters have been defined by considering different internal characteristic lengths for the layer and half‐space. Modified Bessel functions of the first and second kinds are used to determine displacement functions. Asymptotic representations of derivatives of modified Bessel's functions have been derived to get the compact form of the dispersion equation. A comparison of phase velocity equations with classical Love wave equations has demonstrated the validity of our model. It is found that different modes of Love‐like waves are dispersive and highly affected by the attenuation coefficient. Additionally, non‐local parameters significantly influence the limitation of phase velocity modes of Love‐like waves. The phase and damped velocity versus non‐local parameters have been depicted with the effect of different parameters in the model. It has been found that the non‐local parameters decrease the particle displacement amplitude in elastic half‐space and rapidly eliminate waves with depth.

Visualization of hydrogen leak for electro-hydrogen coupled system based on Background Oriented Schlieren

As the world's energy demand continues to grow and the environmental pollution problem caused by fossil energy sources becomes more and more serious, the hydrogen-electric coupling system is considered a highly competitive new energy supply method in the future. A unique Background Oriented Schlieren (BOS) technique is proposed to detect hydrogen leakage in the low and high power consumption peaks. Firstly, the principle of the BOS is analyzed, and a new inverse function of gas jet concentration distribution is proposed; secondly, a BOS optical experiment system is built, and the displacements of background particles at z = 10d, 20d and 30d at three axial positions are obtained, and the formed undesirable displacement function is corrected by laser beam profile deformation technique, and the radial concentration distribution curves at three axial positions are obtained by concentration inversion. Comparing the concentration distribution curves obtained after the correction with those obtained from CFD simulation and uncorrected BOS experiments, it was found that after the correction, the concentration distribution curves of BOS were closer to the simulation results and could reflect the radial concentration distribution of hydrogen more accurately. We also find that the average expansion angle of the hydrogen jet is about 23° for different combinations of outlet pressure and tube diameter, which is very similar to the theoretical and experimental studies. The hydrogen monitoring system based on BOS technology is a simple device with low cost, high sensitivity, fast response, and high reliability, which provides a new idea and method for hydrogen leak monitoring in a hydrogen-electric coupled system.

Damage softening model and behavior of reinforced soil interface based on the Weibull distribution

A damage-softening model for a reinforced soil interface is proposed based on the trilinear model of a reinforced soil pullout interface, aiming at the softening phenomenon of the geogrid reinforced soil interface during the pullout process and considering the damage softening of the reinforced soil interface during the pullout process. The damage variable factor D is introduced, and it is assumed that D is a function of the interface displacement. The two-parameter Weibull distribution function is used to express the damage evolution law of the reinforced soil interface. Based on the basic control equation of the reinforced soil interface, the calculation model of the stress state of the geogrid at different stages under the pullout load is deduced. To verify the accuracy and applicability of the model, the prediction results of the model are compared with the test results and the ideal elastic–plastic model, hyperbolic model, elastic–exponential softening model, and the damage softening model based on a lognormal distribution function proposed in this paper. The distribution of interfacial shear stress is studied, and the influence of parameters at each stage is analyzed. The results show that in the elastic stage, with the increase of shear stiffness, the maximum shear stress at the interface increases, and the curvature of the curve also increases, while the elastic modulus is the opposite. In the softening stage, with the increase of softening length, the peak value of the curve moves to the free end. In the residual stage, the shear stress increases from the drawing end to the free end and tends to the residual stress. The research results are in accordance with the actual situation and can be applied to reinforced soil engineering.

Parametric excitation analysis for system performance of piezoelectric energy harvesters

Capitalizing on the advantages of parametric excitation and real-life low frequency vibration, the steady-state response of piezoelectric energy harvesting system composing of a cantilever beam with a tip mass and horizontal excitation at the fixed end is investigated. By applying Hamilton's principle, the nonlinear partial differential equation of a cantilever bimorph piezoelectric energy harvesting system with an additional tip mass is derived and analyzed. The cantilever is modeled as an axially non-elongated Euler Bernoulli beam with geometric and damping nonlinearities. By using the Galerkin method, the nonlinear partial differential equation is reduced to an electromechanical coupling system that governs a cantilever piezoelectric energy harvesting system with a tip mass under parametric excitation. The first-order resonance response of this harvesting system is studied by using the method of multiple scales. The analytical response expressions and first-order amplitude functions for vertical displacement, output voltage and output power are derived and analyzed. The influence of different impedance and tip mass on the piezoelectric energy harvesting system performance is summarized and concluded. For the parametric excitation system, the load resistance shows significant influence on the initial threshold of the piezoelectric energy harvesters under parametric excitation. For a short circuit, the initial threshold increases with increasing load resistance. For an open circuit, the initial threshold decreases with increasing load resistance. For an open circuit resistance or a short circuit resistance under parametric excitation, an increase in the end block mass effectively reduces the resonance frequency of the piezoelectric energy harvesters. Consequently, the energy harvesting bandwidth is increased, and the harvesting effect of the piezoelectric energy harvesters is enhanced. In this paper, a more accurate and practical theoretical model is established to predict the electromechanical coupling behavior of cantilever piezoelectric energy harvester under parametric excitation. Using this approach, it is possible to complement parametric and direct excitation such that the advantages of parametric excitation can be achieved. It is also significant to improve the energy conversion efficiency of energy harvesting system.

A grounded inerter-based oscillating TMD for suppressing harmonic and random vibrations

Inspired by the lightweight passive vibration control technology, in this work a new anti-vibration device is proposed to significatively improve the protection for stationary mechanical structures subjected to different kinds of vibratory loads. Such a device is herein termed as the grounded inerter-based two-degree-of-freedom tuned mass damper (GI-TDOF-TMD) composed by a grounded inerter and TDOF-TMD with rotational and translational motions, which controls the primary structure's dominant modal shape by using the synergistic dynamic effects of both devices. In order to fairly evaluate the GI-TDOF-TMD's vibration mitigation effectiveness with respect to other devices, the damped and undamped primary structure's nondimensional compliance and mobility transfer functions are firstly computed from a generalized mathematical model that considers four types of random and harmonic excitation inputs which are the following: force general case, base acceleration excitation, base displacement input and inertial force generated by the unbalance in rotary machinery. For harmonically excited mechanical systems, the H∞ performance measure is applied to minimize the maximum resonant peaks of the displacement and velocity dimensionless frequency response functions (FRFs). As the Extended Fixed-Points Technique (EFPT) is analogous to the H∞ criterion, quasi-optimal solutions are firstly computed to perfectly calibrate the FRF's invariant points. After performing this, the evolution of the FRF's control invariant frequencies revealed that GI-TDOF-TMD performs well at low inertance values. Therefore, the GI-TDOF-TMD's maximum control performance is obtained when the inerter's inertial force is the same as that yielded by the TMD's physical mass. In view of such a dynamic behavior, the GI-TDOF-TMD approximately provides 40% and 17% improvements when compared with respect to the classic dynamic vibration absorber (DVA) and TMD-inerter (TMDI), respectively. Then, the GI-TDOF-TMD's power dissipation capability is demonstrated by applying the H2 norm approach to the randomly excited damped mechanical structures. Additionally, it is also revealed through the stochastic energy balance that the GI-TDOF-TMD's internal power dissipation effectiveness is directly reflected on the minimization of the primary structure's kinetic energy. Moreover, for random inputs, the proposed device can broaden the effective operating bandwidth in approximately 45% and 19% when compared to the classic DVA and TMDI, respectively. Therefore, the GI-TDOF-TMD works better than the TMDI in terms of structural displacement and velocity response mitigation, which can be useful in civil engineering applications.

Response of UPVC pipes buried in sand under lateral ground movement

As a key part of global lifeline infrastructure, buried pipelines play a vital role in transporting products for the daily life of human beings. However, the ultimate soil resistance and yield displacement are only derived for rigid pipes in current design guidelines, which are difficult to guide the analysis of flexible pipes. This investigation presents large-scale laboratory tests on flexible UPVC pipes subjected to lateral ground movements. The soil failure mechanism and pipe deformation were tracked. The reaction developing on UPVC pipe was measured as a function of pipe lateral displacement, embedment depth, sand bed density and pipe wall thickness. Analysis results reveal that the failure mode of soil changed from the general shear failure to the local shear failure, in which the soil arching effect occurred in both the active and passive loading areas, as the embedment ratio increased. Current design approaches could overestimate or underestimate the ultimate soil resistance of UPVC pipe heavily under various conditions, while the measured yield displacement was 2 to 3 times the prediction from the design guidelines. Finally, a simplified design method was proposed based on the pipe yield displacement for predicting the maximum longitudinal strain of UPVC pipes under lateral loading.

Exploring the molecular mechanisms of isoliquiritin extraction using choline chloride-citric acid deep eutectic solvents

This study intends to explore the extraction mechanism of isoliquiritin by using experimental verification and molecular simulation techniques. Firstly, the Choline chloride (ChCl)-Citric acid (CA) deep eutectic solvent (DES) was prepared by the solvent method, and its structure was characterized by FT-IR, NMR, rheometer, molecular dynamics and quantum computation. The pharmacological activity and biosafety of DES were investigated by bacteriostasis and cell assay. The extraction mechanism of isoliquiritin by using synthesized ChCl-CA DES as extraction solvents were calculated by means of cohesive energy density (CED), mean square displacement (MSD) and radial distribution function (RDF). The results showed that compared with ethanol and water, DES as extraction solvent had the highest extraction yield of isoliquiritin. Through further study, it was found that the increase in the isoliquiritin extraction yield was mainly due to the strong molecular interaction between isoliquiritin and extraction solvent and the rapid diffusion of isoliquiritin.

Control Mechanisms for Mobile Devices

In this paper we consider control mechanisms for mobile devices in a stochastic environment. In particular, we consider a device in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional space subject to Brownian perturbations where a control mechanism moves the device towards its target location at a speed which is a function of its displacement. For this scenario, we construct stochastic differential equations for the mobility process and solve for the steady state probability density function of displacement. From this we are able to give general solutions to key metrics such as displacement outage (the long term probability of exceeding a given distance from the target), connectivity probability (derived from the SNR distribution in a Rayleigh channel with pathloss), the mean time at which the device first exceeds a given distance from the target, and the mean kinetic energy required by the control mechanism. We evaluate these metrics for important special cases of the control mechanism and also study the optimization problem of minimizing kinetic energy over the parameters of the control function.

THE MAGNETIC CHAIN MATHEMATICAL MODELS OF A NEW ANGULAR DISPLACEMENT CONVERTER WITH A VARIABLE ACTIVE AREA OF A MOVABLE CORE

The article develops mathematical models of the magnetic chain of new differential type angular displacement transformer converter with a variable active area of the movable core, in which the width of the movable ferromagnetic core is a function of the angular displacement of the converter. In this case, the magnetic chain of the converter, divided into three characteristic (before the mobile ferromagnetic core, ferromagnetic core and after the ferromagnetic core) sections, is considered as a magnetic chain with distributed parameters. It is established that when the magnetic resistance of the magnetic chain steel part is not taken into account, the working magnetic flux is a linear function of the angular displacement of the movable ferromagnetic core of the converter. The obtained mathematical models of the magnetic chain of the transformer converter of angular displacements in the form of analytical expressions of the magnetic flux in long ferromagnetic rods and the magnetic voltage between them for three characteristic sections allow to take into account the distribution of the parameters of the magnetic resistance of long ferromagnetic rods and the magnetic capacity of the air gap between them, as well as the magnetic capacity of the air gap between the rods and the movable ferromagnetic core.