Academic literature on the topic 'Nonlinear theories. Integral equations Numerical analysis. Elasticity'

This work is devoted to modeling and the analytical/numerical analysis of tribological processes occurring on the contact surface of shields of a mechanical friction clutch. Although the considered problems have been already studied earlier, however, simplified mathematical models have been used and applied. Unlike previous works, our work takes into account elasticity and wear properties of material of shields rubbing themselves. A general nonlinear differential model of wear is considered, as well as a wear model in the integral form taking into account gradual decrease of speed of wear of shields as a result of abrasive adapting to each other in the process of the exploitation. Equations modeling contact pressure on the contact surface of shields are derived and they yield the contact pressure and the wear of the shields. An analytical/numerical analysis is carried out with the qualitative and quantitative theories of differential and integral equations, including Laplace transformation. Many interesting results are obtained, illustrated and discussed. The presented results can be widened and used in other disciplines of the science, for instance, in physics of solids or biomechanics of various human joints.

AbstractA numerical approach is used herein to study the primary resonant dynamics of functionally graded (FG) cylindrical nanoscale panels taking the strain gradient effects into consideration. The basic relations of the paper are written based upon Mindlin’s strain gradient theory (SGT) and three-dimensional (3D) elasticity. Since the formulation is developed using Mindlin’s SGT, it is possible to reduce it to simpler size-dependent theories including modified forms of couple stress and strain gradient theories (MCST & MSGT). The governing equations is derived and directly discretized via the variational differential quadrature technique. Then, a numerical solution technique is employed to study the nonlinear resonance response of nanopanels with various edge conditions under a harmonic load. The impacts of length scale parameter, material and geometrical parameters on the frequency–response curves of nanopanels are investigated. In addition, comparisons are provided between the predictions of MSGT, MCST and the classical elasticity theory.

In this paper, based on the analysis of coupled action of temperature, phase transformation and stress, the phase transformation condition and equation related to the thermal physical properties are discussed, and the transient temperature distribution of a Cr12 steel cylinder workpiece during gas quenching was obtained by solving the governing equations with a nonlinear boundary. According to the theories of thermal non-elasticity, computational mechanics and phase transformation, a new constitutive equation considering effects of phase transformation with a nonlinear surface heat transfer coefficient is proposed and is solved by means of finite element method (FEM). The thermal stress field is obtained and a way of simulating the technology of the heat treatment by computer is explored.

Mathematical modeling, theoretical/numerical analysis, and experimental verification of wear processes occurring on the contact surface of friction linings of a mechanical friction clutch are studied. In contrast to many earlier papers we take into consideration wear properties and flexibility of friction materials being in friction contact. During mathematical modeling and numerical simulations we consider a general nonlinear differential model of wear (differential wear model) and a model of wear in the integral form (integral wear model). Equations governing contact pressure and wear distributions of individual friction linings, decrease of distance between clutch shields, and friction torque transmitted by the clutch are derived and compared with experimental data. Both analytical and numerical analyses are carried out with the qualitative and quantitative theories of differential and integral equations, including the Laplace transform approach to ODEs. We show that theoretical results and numerical simulations agree with the experimental data. Finally, a numerical analysis of the proposed mathematical models was carried out in a wider range of parameters of the considered system.

In this paper, the size-dependent nonlinear pull-in behavior of rectangular microplates made from functionally graded materials (FGMs) subjected to electrostatic actuation is numerically studied using a novel approach. The small scale effects are taken into account according to Mindlin’s first-order strain gradient theory (SGT). The plate model is formulated based on the first-order shear deformation theory (FSDT) using the virtual work principle. The size-dependent relations are derived in general form, which can be reduced to those based on different elasticity theories, including the modified strain gradient, modified couple stress and classical theories (MSGT, MCST and CT). The solution of the problem is arrived at by employing an efficient matrix-based method called the variational differential quadrature (VDQ). First, the quadratic form of the energy functional including the size effects is obtained. Then, it is discretized by the VDQ method using a set of matrix differential and integral operators. Finally, the achieved discretized nonlinear equations are solved by the pseudo arc-length continuation method. In the numerical results, the effects of material length scale parameters, side length-to-thickness ratio and FGM’s material gradient index on the nonlinear pull-in instability of microplates with different boundary conditions are investigated. A comparison is also made between the predictions by the MSGT, MCST and CT.

The modeling and analysis for mechanical response of nano-scale beams undergoing large displacements and rotations are presented. The beam element is modeled as a composite consisting of the bulk material and the surface material layer. Both Eringen nonlocal elasticity theory and Gurtin–Murdoch surface elasticity theory are adopted to formulate the moment–curvature relationship of the beam. In the formulation, the pre-existing residual stress within the bulk material, induced by the residual surface tension in the material layer, is also taken into account. The resulting moment-curvature relationship is then utilized together with Euler–Bernoulli beam theory and the elliptic integral technique to establish a set of exact algebraic equations governing the displacements and rotations at the ends of the beam. The linearized version of those equations is also established and used in the derivation of a closed-form solution of the buckling load of nano-beams under various end conditions. A discretization-free solution procedure based mainly upon Newton iterative scheme and a selected numerical quadrature is developed to solve a system of fully coupled nonlinear equations. It is demonstrated that the proposed technique yields highly accurate results comparable to the benchmark analytical solutions. In addition, the nonlocal and surface energy effects play a significant role on the predicted buckling load, post-buckling and bending responses of the nano-beam. In particular, the presence of those effects remarkably alters the overall stiffness of the beam and predicted solutions exhibit strong size-dependence when the characteristic length of the beam is comparable to the intrinsic length scale of the material surface.

Recently, Kazakov, Gromov and Vieira applied the discrete Hirota dynamics to study the finite size spectra of integrable two dimensional quantum field theories. The method has been tested from large values of the size L down to moderate values using the SU (2) × SU (2) principal chiral model as a theoretical laboratory. We continue the numerical analysis of the proposed nonlinear integral equations showing that the deep ultraviolet region L → 0 is numerically accessible. To this aim, we introduce a relaxed iterative algorithm for the numerical computation of the low-lying part of the spectrum in the U (1) sector. We discuss in detail the systematic errors involved in the computation. When a comparison is possible, full agreement is found with previous thermodynamical Bethe ansatz computations.

Abstract We present a comprehensive study on the post-buckling response of nonlocal structures performed by means of a frame-invariant fractional-order continuum theory to model the long-range (nonlocal) interactions. The use of fractional calculus facilitates an energy-based approach to nonlocal elasticity that plays a fundamental role in the present study. The underlying fractional framework enables mathematically, physically, and thermodynamically consistent integral-type constitutive models that, in contrast to the existing integer-order differential approaches, allow the nonlinear buckling and post-bifurcation analyses of nonlocal structures. Further, we present the first application of the Koiter's asymptotic method to investigate post-bifurcation branches of nonlocal structures. Finally, the theoretical framework is applied to study the post-buckling behavior of slender nonlocal plates. Both qualitative and quantitative analyses of the influence that long-range interactions bear on post-buckling response are undertaken. Numerical studies are carried out using a 2D fractional-order Finite Element Method (f-FEM) modified to include a combination of the Newton-Raphson and a path-following arc-length iterative methods in order to solve the system of nonlinear algebraic equations that govern the equilibrium beyond the critical points. The present framework provides a general foundation to investigate the post-buckling response of potentially any type of nonlocal structure.

The solution of the line contact problem of elastohydrodynamic lubrication in the asymptotic régime developed by Bissett ( Proc . R . Soc . Lond . A 424, 393–407 (1989)) exhibits two regions of rapid change: a transition layer between the inlet and contact zones, and a downstream exit layer. In these regions the governing Reynolds equation of lubrication theory is essentially nonlinear, although pressure and surface displacement continue to be linearly related by the singular integral equation of plane elasticity. In combination, the system in each region reduces to a nonlinear singular integrodifferential equation with Cauchy kernel for the surface displacement, to be satisfied on either an infinite interval (transition layer) or a semi-infinite interval (exit layer). A method is developed along lines used by Spence & Sharp ( Proc . R . Soc . Lond . A400, 289 (1985), Proc . R . Soc . Lond . A422, 173 (1989)) and Spence et al . ( J . Fluid Mech . 174, 135 (1987)) for approximating the solution in either case by a finite number of trigonometric terms (up to 900). Rapid convergence is achieved by judicious allowance for end point behaviours as deduced by asymptotic analysis of the governing equations. The equations contain an eigenvalue, representing the scaled exit film thickness, which also characterizes the film thickness in the contact zone. This eigenvalue is found with high accuracy in the course of solving the transition layer problem. Close agreement with certain results of Hooke & O’Donoghue ( J . mech . Engng Sci . 14 (1), 34 (1972)) is exhibited.

A new reduced-order design synthesis technology has been developed for vibration response and flutter control of cold-stream, high-bypass ratio, shroudless, aeroengine fans. To simplify the design synthesis (optimization) of the fan, a significant order reduction of the mechanical response and stiffness-shape design synthesis has been achieved. The assumed cyclic symmetric baseline fan is modeled as a cascade of tuned, shroudless, arbitrarily shaped, wide-chord laminated composite blades, each with a reduced order of degrees of freedom using a three-dimensional (3D) elasticity spectral-based energy model (McGee et al., 2013, “A Reduced-Order Meshless Energy Model for the Vibrations of Mistuned Bladed Disks—Part I: Theoretical Basis, ASME J. Turbomach., in press; Fang et al., 2013, “A Reduced-Order Meshless Energy Model for the Vibrations of Mistuned Bladed Disks—Part II: Finite Element Benchmark Comparisons, ASME J. Turbomach., in press). The uniqueness of the mechanical analysis is that the composite fan was modeled as a “meshless” continuum, consisting of nodal point data to describe the arbitrary volume. A stationary value of energy within the arbitrarily shaped composite fan annulus was achieved using an extended spectral-based Ritz procedure to obtain the dynamical equations of motion for 3D free vibration response of a rotating composite high-bypass fan. No additional kinematical constraints (as in beam, plate, or shell theories) were utilized in the 3D elasticity-based energy formulation. The convergence accuracy of the spectral-based 3D free vibration response predictions was nearly one percent upper-bounds on the exact mechanical response of the baseline composite fan, particularly in the lowest five modes studied closely in this work, as typically seen with spectral-based Ritz procedures employed in the analysis. The spectral-based 3D predictions was validated against those predicted using a general purpose finite element technology widely used by industry. In off-design operation, the frequency margins of the lower flex-torsion modes of a fan may be dangerously close to integral-order resonant and empirical stall flutter boundaries. For a given baseline composite fan, it is proposed that to reduce the likelihood of resonant response and flutter on a Campbell diagram, design analysts can efficiently unite the newly developed reduced-order 3D spectral-based energy reanalysis within a novel reduced-order spectral-based Kuhn–Tucker optimality design synthesis procedure to fairly accurately restructure the Campbell diagram of a composite high-bypass ratio fan using stiffness optimization (by means of proper choices of angle-ply orientations of the blade laminates) and mass-balancing (shape) optimization (by way of blade thickness variation tuning of the lower aerodynamic loading portion of the blades between the dovetail root section and the midradial height section of the composite fan annulus). Fan design optima is summarized that (1) achieves multiple frequency margins and satisfies multiple empirical stall flutter constraints, (2) controls the twist-flex vibratory response in the lowest (fundamental) mode, and (3) ensures the mechanical strength integrity of the optimized angle-ply lay-up under steady centrifugal tension and gas bending stresses. Baseline and optimally restructured Campbell diagrams and design sensitivity calculations are presented, comparing optimum solution accuracy and validity of the proposed reduced-order spectral-based design synthesis technology against optimum solutions generated from open-source nonlinear mathematical programming software (i.e., NASA’s general-purpose sequential unconstrained minimization technique, Newsumt-A) (Miura and Schmit, Jr., 1979, ”NEWSUMT–A, Fortran Program for Inequality Constrained Function Minimization—Users Guide,“ NASA CR-159070). Design histories of fan stiffness and mass balancing (or shape) along with nondimensional constraints (i.e., frequency margins, reduced frequencies, twist-flex vibratory response, first-ply failure principal stress limits, and dovetail-to-midblade height thickness distribution) show that a proper implementation of fan stiffness tailoring (via symmetric angle-ply orientations) and mass-balancing (thickness) optimization of the fan assembly produces a feasible Campbell diagram that satisfies all design goals. An off-design analysis of the optimized fan shows little sensitivity to twist-flex coupling response and flutter with respect to small variability or errors in optimum design construction. Industry manufacturing processes may introduce these small errors known as angle-ply laminate construction misalignments (Graham and Guentert, 1965, “Compressor Stall and Blade Vibration,” Aerodynamic Design of Axial-Flow Compressors, Chap. XI, NASA SP-36; Meher-Hornji, 1995, “Blading Vibration and Failures in Gas Turbines, Part A: Blading Dynamics and the Operating Environment,” ASME Paper 95-GT-418; Petrov et al., 2002, “A New Method for Dynamic Analysis of Mistuned Bladed Disks Based on the Exact Relationship Between Tuned and Mistuned Systems,” ASME J. Eng. Gas Turbines Power, 124(3), pp. 586–597; Wei and Pierre, 1990, “Statistical Analysis of the Forced Response of Mistuned Cyclic Assemblies,” ASME J. Eng. Gas Turbines Power, 28(5), pp. 861–868; Wisler, 1988, “Advanced Compressor and Fan Systems,” GE Aircraft Engines, Cincinnati, Ohio (also 1986 Lecture to ASME Turbomachinery Institute, Ames Iowa)).

In this paper the mooring system of a drill ship is analyzed which is designed for South China Sea in 1500 meters depth. The analyses of the mooring lines have been developed based on the theories dealing with slender structure and cables, so the elastic rod theory is used. Elastic rod theory is developed for the analysis of line dynamics. The governing equations of mooring lines and risers are treated in the global coordinate system without transforming the coordinate system. The hydrodynamic forces on the lines together with the strain and the stress of the structures caused by geometric nonlinearity are considered. The model of the rods allows for a small elongation, and permits large deflections and finite rotations. The rods are of elasticity and arbitrary configuration, with kind of loads and tension variation along its length acting on it, including the motion of rod, hydrodynamic force resulted from the external fluid and gravity. The fluid in the riser is considered for riser analysis, and the support of the sea bottom is presented for mooring line analysis. Finite element method is used to discrete mooring lines and risers, and lines dynamic analysis is executed with time integration method. A program is developed, and its validation is checked by comparison of numerical results to exact solutions for a nonlinear, static problem. Both static analysis and dynamic analysis of the whole system are done to ensure the mooring system of the drilling ship on 1500m depth can successfully applied.