Lagrangian Formulation Research Articles

Effect of volcano-like textured tools on machining of Ti-6Al-4V alloy: a numerical and experimental analysis

Higher temperature and lower thermal conductivity between the tool-chip contact face easily result in adhesion and wear on the tool surface which negatively affects the tool life and the machining quality of the workpiece in the machining Ti-6Al-4V alloy process. To address these problems, the investigation, combining the simulation and experiment, introduced the volcano-like textures of different diameters with excellent anti-adhesion performance on the tool rake face. Deform-3D software with updated Lagrangian formulation was used for numerical simulation, and the thermo-mechanical analysis was performed using the Johnson-Cook material model to predict the cutting forces, cutting temperature and tool wear. In machining experiments, volcano-like textures (VT) with different diameters were fabricated by fiber laser on the rake face of cemented carbide tools close to the main cutting edge. Experiments in machining Ti-6Al-4V alloy were carried out with the textured tools and non-textured tool under rough machining, semi-finishing and finishing conditions. Then, cutting force, cutting temperature and tool wear were investigated. The results showed that textured tools generally perform better than non-textured tool. The cutting force of VT1 was reduced by 31.2% and 50%, respectively, compared with the non-textured tool under semi-finishing and finishing conditions. With the refinement of machining, the cutting performance of the textured tools is improved, and the service life of the tool is extended. Therefore, the investigation can provide a basis for the surface texture design and optimization of the carbide tool.

Teğet Sınır Şartlarının Ortasından Tekil Yüklü Katmanlı Kompozit Silindirik Panellerin Nonlineer Davranışlarına Etkisi

In this study, the influence of straight edge tangential restraints on the nonlinear response of symmetrically laminated and balanced composite cylindrical panels subject to a pinching force is investigated. An 8-node degenerated nonlinear shell element, formulation of which is based on the Total Lagrangian Formulation, is employed for geometrically nonlinear analysis and the Arc-Length Method is used to trace the nonlinear path. First, the element is validated for geometrically non-linear analysis by solving two verification problems. Then, numerical results for different rotational boundary conditions are presented for two different stacking sequences, and thickness values. The numerical results presented show that there is no significant difference between the tangentially unrestrained and restrained clamped panels when only one edge is tangentially unrestrained. However, it is observed that the simply supported panels demonstrate a much less stiff behavior when one of the straight edges is tangentially unrestrained.

Lagrangian Formalism for the Analysis of Acoustic Properties of Two-Phase Composite Materials with Damaged Interfaces

Lagrangian Formalism for the Analysis of Acoustic Properties of Two-Phase Composite Materials with Damaged Interfaces

Global dynamical analysis of an electronic spin–orbit coupling system

By means of a canonical generalized momentum and a canonical conjugate spin variable, a complete canonical Hamiltonian formalism is designed to describe a coulomb field with electronic spin–orbit coupling in a semi-classical and non-relativistic way. After this operation, unlike the existing Lagrange formulation, the concepts of hidden momentum, hidden angular momentum and spin kinetic energy are not used in the canonical formalism. Besides, it is easy to find that there are four first integrals involving the conserved total energy and the conserved total angular momentum vector in an 8-dimensional phase space of the system. In this sense, the global dynamics is typically integrable, regular and non-chaotic, and each orbit in the phase space is a quasi-periodic 4-dimensional Kolmogorov-Arnold-Moser(KAM) torus.

Updated Lagrangian unsaturated periporomechanics for extreme large deformation in unsaturated porous media

Unsaturated periporomechanics is a strong nonlocal poromechanics based on peridynamic state and effective force state concept. In the previous periporomechnics the total Lagrangian formulation is adopted for the solid skeleton of porous media. In this article as a new contribution we formulate and implement an updated Lagrangian unsaturated periporomechanics framework for modeling extreme large deformation in unsaturated soils under drained conditions. In this new framework the so-called bond-associated sub-horizon concept is utilized to enhance the stability and accuracy at extreme large deformation of the solid skeleton. The stabilized nonlocal velocity gradient in the deformed configuration is used to update the effective force state from a critical state based visco-plastic constitutive model for unsaturated soils. The updated Lagrangian periporomechanics paradigm is numerically implemented through an explicit Newmark scheme for high-performance computing. Numerical examples are presented to demonstrate the stability of the computational updated Lagrangian periporomechanics paradigm and its efficacy and robustness in modeling extreme large deformation in porous media under drained conditions.

Existence results in large-strain magnetoelasticity

We investigate variational problems in large-strain magnetoelasticity, in both the static and the quasistatic settings. The model contemplates a mixed Eulerian–Lagrangian formulation: while deformations are defined on the reference configuration, magnetizations are defined on the deformed set in the actual space. In the static setting, we establish the existence of minimizers. In particular, we provide a compactness result for sequences of admissible states with equi-bounded energies which gives the convergence of the composition of magnetizations with deformations. In the quasistatic setting, we consider a notion of dissipation which is frame-indifferent and we show that the incremental minimization problem is solvable. Then we propose a regularization of the model in the spirit of gradient polyconvexity and we prove the existence of energetic solutions for the regularized model.

Large amplitude free vibration of sandwich beams with flexible core and FG Graphene Platelet Reinforced Composite (FG-GPLRC) face sheets based on extended higher-order sandwich panel theory

The present article deals with the nonlinear free vibration analysis of sandwich beams with flexible core integrated with face sheets which are reinforced with graphene platelets (GPLs). Each layer of the composite media may have a different amount of graphene platelets which leads to a functionally graded (FG) distribution. The elastic modulus of the reinforced composite (RC) is calculated via the Halpin–Tsai model. Following the extended higher order sandwich panel theory (EHSAPT) which captures the core thickness compressibility effect and the axial stress of the core, the complete set of governing equations is obtained using the Lagrangian formulation. To analyze the effect of the inclusion of nonlinear terms in the core, hitherto not reported in the literature, results are computed considering the von Kármán type of geometrical nonlinearity assumptions for the core and face sheets. With the aid of the Ritz-based method, Chebyshev functions are implemented as admissible functions of the displacement fields. Furthermore, the nonlinear governing equations are solved using a direct displacement control technique. As the EHSAPT can be applied with any combination of the core and face sheets and not only the soft cores that the other theories demand, the results for the sandwich beam with stiff core and consist of functionally graded carbon nanotube-reinforced composite face sheets provided for validation. Since this study is the first in the open literature to develop nonlinear vibration of the sandwich beam based on EHSAPT, Timoshenko beam theory (TBT) is also applied to investigate the problem and to extract nonlinear frequencies with various stiffness of the core, meanwhile, the results are compared with those obtained by the EHSAPT. Parametric studies are conducted in detail to examine the influences of weight fraction and distribution pattern of graphene platelets, core to face sheet thickness ratio, boundary conditions, length to thickness ratio, and amplitude of vibrations. It is found that by increasing the stiffness of the core, the inclusion of nonlinear effects in the core as well as the face sheet strain nonlinearity is more significant. Finally, the nonlinear vibration analysis of sandwich beams can change the variation trend of frequency predicted in the linear analysis, and the different trends are observed depending on the vibration amplitude.

Arbitrary Lagrangian-Eulerian Remeshing in FE Simulations of Tire Forming

ABSTRACT Tire forming simulation is challenging for Lagrangian finite element method codes due to large changes in the geometry of the tire in the course of molding. This Technical Note briefly describes the use of Arbitrary Lagrangian-Eulerian (ALE) adaptive remeshing in the context of tire molding and curing simulations. The ALE concept generalizes the pure Lagrangian formulation, where the solution within a time step is split into a mesh smoothing step, a history remapping step, and a Lagrangian step. Mesh distortion is reduced in the smoothing step by optimizing the node positions of spatial and material meshes without changing the data structure. The advantage of the ALE approach is demonstrated by a tire forming example.

Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold II

We have previously developed the BRST quantization on the hypersurface V N − 1 embedded in N -dimensional Euclidean space R N in both Hamiltonian and Lagrangian formulation. We generalize the formalism in the case of L -dimensional manifold V L embedded in R N with 1 ≤ L < N . The result is essentially the same as the previous one. We have also verified the results obtained here using a simple example of particle motion on a torus knot.

Intelligent Neural Integral Sliding-mode Controller for a space robotic manipulator mounted on a free-floating satellite

In this paper, a fractional neural integral sliding-mode controller based on the Caputo-Fabrizio derivative and Riemann–Liouville integral for a robot manipulator mounted on a free-floating satellite is proposed and developed. The mathematical model is obtained by using the Euler–Lagrange formalism and generalized through the Caputo-Fabrizio derivative. The conventional neural integral sliding-mode controller reported in the literature were also developed to make a comparative analysis between both controllers. The controllers are tuned using a swarm intelligence algorithm. External disturbances are also considered. Simulation results show the superiority and robustness of the proposed control system for trajectory tracking tasks through the root-mean-square error. Additionally, two additional reference trajectories were considered. In this sense, the numerical simulations were done by using the same orders and gains, showing the robustness of our fractional-order control strategy under different operating conditions. Further, the proposed controller uses the fractional derivative in the sense of Caputo-Fabrizio which has not been used enough in the area of robotics, so this is a start to analyse the contributions that arise when using said derivative in this robotic.

Brans–Dicke Unimodular Gravity

We propose a unimodular version of the Brans–Dicke theory designed with a constrained Lagrangian formulation. The resulting field equations are traceless. The vacuum solutions in the cosmological background reproduce the corresponding solutions of the usual Brans–Dicke theory but with a cosmological constant term. A perturbative analysis of the scalar modes is performed and stable and unstable configurations appear, in contrast with the Brans–Dicke case for which only stable configurations occur. On the other hand, tensorial modes in this theory remain the same as in the traditional Brans–Dicke theory.

Volume of a rotating black hole in 2+1 dimensions

In this article we apply the technique for maximal volume estimation of a black hole developed by Christodoulou and Rovelli [1] for Schwarzschild black hole and by Zhang et al. [3] for non rotating BTZ black hole, to the case of a rotating black hole in 2+1 dimensions. We derive the equation of the maximal hyper-surface for the rotating BTZ black hole using the Lagrangian formulation demonstrated in [1]. Further we use maximization technique illustrated earlier by Bengtsson et al. [4] for Kerr black hole to arrive at the similar result for our case. We argue that the maximum contribution to the volume of the hyper-surface comes from what we call the steady state radius, which we show depends on mass M and the AdS length scale. We demonstrate that this steady state radius can be derived using independent considerations of vanishing extrinsic curvature. We show that the volume of this segment of the maximal hyper-surface, the CR volume, depends on mass, AdS length scale and angular momentum J. We further compute the entropy of a scalar field living on the maximal hyper-surface for a near extremal black hole and show that it is proportional to the horizon entropy of the black hole.

Dynamics of pendulum-based systems under human arm rotational movements

Wearable devices have applications in health monitoring and patient assistance devices. Harvesting energy from human motion kinetic energy by the pendulum-based electromagnetic generator is a promising way of extending the battery life of these devices. In this paper, the oscillatory and rotary dynamical response of pendulum-based systems under an open-circuit condition is investigated for two different models considering the lower arm’s rotation. The arm pitching model (in-plane excitation) represents the swing of the lower arm during walking. The arm rolling rotation model (out-of-plane excitation) represents the twisting type of motion of the lower arm movement associated with a specific kind of disorder, for example, tremors experienced by Parkinson’s patients. The governing equations of motion around stable equilibria are derived using Lagrange formulation, and their nondimensional forms are introduced for the two models. A special case of these models is developed for experimental validation and system identification, where the system parameters are identified by fitting frequency response functions on the experimental data. The system’s nonlinear dynamical response shows various bifurcations, including limit point bifurcation (i.e., saddle-node bifurcation), pitchfork bifurcation, a cascade of period-doubling bifurcations under amplitude or frequency sweeps. When the system shows a bi-stable response, the rotary response usually has a higher angular velocity amplitude than the oscillatory response. Basins of attraction plots for different operating conditions are presented to show the effect of initial conditions on the system’s type of steady-state response: oscillatory or rotary.

Effect of Geo-Material on Dynamic Response of Tunnel Subjected to Surface Explosion

Prime materials involved in a problem such as underground structures are concrete, reinforcement steel, and geo-material surrounding the tunnel. Among these three materials, concrete and steel are manufactured materials and their properties can be controlled up to a certain extent. However, geo-material is a naturally occurring material whose constitutive properties vary from region to region, making it highly unpredictable. Findings from one study cannot be applied to other geotechnical problems directly, especially in the case of tunnels subjected to surface explosions. The blast wave generated has to travel through the geo-material before it interacts with the tunnel. As the shock wave propagates radially, its characteristics are likely to be altered by the geo-material. Limited study has been carried out considering this problem. In the present study, the effect of various types of geo-material on the blast response of tunnels subjected to surface explosions is investigated. Finite element analysis has been carried out using LS-DYNA®, wherein the problem has been modeled using the multi-material arbitrary Lagrangian–Eulerian (MM-ALE) method. Materials with fluid behavior such as air, explosives, and soil are modeled using ALE formulation. Other materials including tunnel lining, reinforcement steel, and rock are modeled using Lagrangian formulation. Blast loading is simulated using the Jones–Wilkins–Lee (JWL) equation of state. Geo-materials considered for the comparative study are sandy loam, saturated clayey soil, sandstone, and granite. Vertical displacement measured at the crown of the tunnel is used to determine the response of the tunnel. Sandy loam soil, being a highly compressible soil, exhibits non-linear and fluid-like behavior under high-strain loading such as explosions. Tunnels undergo extreme deformation in the case of sandy loam soil and clayey soil compared to rock cases. Further, the effect of saturation in sandy loam on tunnel stability is studied. It is observed that with the increase in saturation of soil, more blast energy is transmitted to the structure, which results in higher deformation. Lastly, the effect of the weathering of rock on the tunnel’s response is investigated in the case of sandstone and granite. It was observed that weathering in rock led to more displacement of tunnel crown when compared to intact rock.

Vehicle crashworthiness performance in frontal impact: Mathematical model using elastic pendulum

Vehicle occupant injuries due to collisions cause many fatalities every year. Safe vehicle design plays a critical role in averting serious injuries to occupants and vulnerable road users in the event of a crash. In this paper we study a full frontal vehicle crash against a rigid barrier introducing a Lumped Parameter Model (LPM) inspired by the elastic pendulum motion. The model uses polar coordinates to simplify the problem and the governing equations have been defined using Lagrangian formulation. The Simulink model has been validated against Finite Element (FE) data demonstrating good correlation with pitching angle and maximum crush of the vehicle. These parameters are crucial for designing vehicles which efficiently protect occupants.

Lagrangian formulation of the de Broglie–Proca for space with noninteger dimension

Fractals are the fundamental characteristics of the universe. In this research, the Lagrangian formulation of the de Broglie–Proca (dBP) field for space with noninteger dimension is examined. Based on the Laplace of the dBP’s equations in noninteger space, the solutions of these equations are found where the dimensionality of the space is [Formula: see text]. Also, we obtain the wave equation for the electric field and the wave equation for the magnetic field in noninteger space.

Installation of groups of stone columns in clay: 3D Coupled Eulerian Lagrangian analyses

This paper describes the results of three-dimensional (3D) finite element analyses investigating the installation effects of groups of stone columns in purely cohesive soils. Installation of stone columns is simplified to the insertion of rigid cylindrical elements with conical tips in a single homogeneous soil layer (Tresca plasticity and a quasi-incompressible elastic law). Installation of a single column is simulated as the reference case and the installations of two columns and a group of nine columns are considered to study the interaction between the installation of several columns and the influence of the installation sequence. The process is simulated using a Coupled Eulerian Lagrangian formulation. Stone column installation alters the surrounding soil and the numerical results show the increase in horizontal stresses and pore pressures. The installation effects of several columns at common spacings overlap between each other and accumulate, producing higher horizontal stresses and pore pressures in a larger area. The installation sequence is mainly visible around the last column installed, where the radial stresses are lower.

Reconstruction and stability analysis of some cosmological bouncing solutions in F(ℛ,T) theory

This paper investigates the possibility of reconstruction of the generic function in [Formula: see text] gravitational theory by considering some well-known cosmological bouncing models, namely, exponential evaluation, oscillatory, power law and matter bounce model, where [Formula: see text] and [Formula: see text] are Ricci scalar and trace of energy–momentum tensor, respectively. Due to the complexity of dynamical field equations, we propose some ansatz forms of function [Formula: see text] in perspective models and examine which type of Lagrangian is capable of reproducing bouncing solution via analytical expression. It is seen that for some cases of exponential, oscillatory and matter bounce models, it is possible to get analytical solution while in other cases, it is not possible to achieve exact (general) solutions so only complementary solutions can be discussed. However, for power-law model, all forms of generic function can be reconstructed analytically. Next we analyze the energy conditions and stability of these reconstructed cosmological bouncing models which have analytical forms. It is found that these models are stable for linear forms of Lagrangian only but the reconstructed solutions for power law are unstable for some nonlinear forms of Lagrangian. Further, we determine the observable quantities like spectral index ([Formula: see text]) and tensor-to-scalar ratio ([Formula: see text]) for the simplest reconstructed form of [Formula: see text] function. As a result, we directly confront the reconstructed linear form of Lagrangian in [Formula: see text] model with 2018 Planck observations. Furthermore, we analyze that [Formula: see text] gravity with dark energy epoch is consistent with Sne-Ia+BAO+H(z)+CMB data and show that bounce can unify with dark energy epochs in [Formula: see text] gravity.

Nonlinear (chiral) p-form electrodynamics

In our previous article Phys. Rev. Lett.127 (2021) 271601, we announced a novel ‘democratic’ Lagrangian formulation of general nonlinear electrodynamics in four dimensions that features electric and magnetic potentials on equal footing. Here, we give an expanded and more detailed account of this new formalism, and then proceed to push it significantly further by building the corresponding Lagrangian theories of higher form field interactions in arbitrary dimensions. Special attention is given to interactions of chiral 2k-forms in 4k + 2 dimensions, with further details for 2-forms in 6 dimensions and 4-forms in 10 dimensions. We comment more broadly on the structure of covariant equations of motion for chiral fields, and on the place of our Lagrangian theories in this context. The Lagrangian theories we develop are simple and explicit, and cover a much broader class of interactions than all past attempts in the literature.

Dynamic analysis of arch dams considering fluid-structure interaction

The social, economic, and environmental consequences of the failure of an arch dam make it essential to evaluate its dynamic response to mitigate the risk of a disaster. Since the slenderness and flexibility of these dams tend to increase the fluid-structure interaction during an earthquake, this work compares the dynamic response of a hypothetical arch dam in the Marañón river, in northern Perú, for both full and empty-reservoir conditions. Three formulations were used to estimate the hydrodynamic pressures: Westergaard’s added mass, Eulerian and Lagrangian. The comparisons were performed for earthquakes of distinct seismogenic sources, previously matched to a uniform hazard spectrum with a return period of 10000 years. The finite element method was used to derive the seismic demands of the dam-reservoir-foundation system in COMSOL Multiphysics software, carrying out time-history analyses assuming linear elastic behavior of the dam and foundation domains and a massless foundation approach, ignoring the effect of waves propagation in the foundation but considering its stiffness. The results show that the Lagrangian and Eulerian formulations produce similar seismic demands, while Westergaard’s added mass formulation is conservative. The full-reservoir condition generally increases the seismic demands, but the results will depend on the boundary conditions assumed for the fluid and the characteristics of the earthquake, among other factors. Earthquakes matched to the same uniform hazard spectrum do not necessarily produce equal dynamic responses.