General Analytical Solution Research Articles

Mechanical Modeling and General Analytical Solution for the Dynamic Buckling of Plane Structures Using a Beam-Column Element with Varying End Restraints

The stability of structures is an important aspect that the designer must pay particular attention to in order to ensure safety against collapse. This investigation is concerned with analytical and numerical analyses of the dynamic buckling of plane structures. A rigorous mechanical model is proposed, consisting of a beam-column element with nodal ends possessing two rotational springs of rigidities acting in parallel with the bending stiffness of the beam-column. The model is first analyzed with respect to the dynamic behavior by investigating the influence of the variation in the stiffness of the nodal springs on the fundamental frequency of the proposed mechanical model. Compression axial loading is applied to the beam-column in order to study the nonlinear dynamic behavior by introducing buckling. This novel approach is used to highlight the interaction between the fundamental frequency and the critical buckling load. Simple examples are treated using the approach and the results are compared with those obtained from a global analysis. The results revealed that it is possible to reproduce the stability analysis of a global structure by simply analyzing a target element, taking into account all elements adjacent to it with less than 1% error on the results.

Bond strength evaluation methods of RC members strengthened with FRP composites

This article examines the bond behaviour of reinforced concrete members and fibre-reinforced polymer reinforcements in the failure stage. Sudden and brittle debonding like failure modes are the most common for such elements. Therefore, a comprehensive review of the existing FRP and concrete bond strength models is provided. In view of the existing methods and the theoretical and experimental research carried out by the authors in recent years, a general analytical solution is proposed for the calculation of the load carrying capacity of FRP strengthened RC members in bending. The proposed method is based on the principles of the built-up bars theory and consists of analytical intermediate crack induced debonding solution, taking into account the concrete cracking nature. The main idea is to evaluate the reduction of overall member stiffness due to the slip between concrete and FRP by reducing the FRP stress with an analytically derived factor. Such a reduction allows to predict the load carrying capacity of strengthened members sufficiently precisely. The numerical results are compared with the experimental ones of beams in bending. The experimental database consists of a great variety of RC beams strengthened with externally bonded and near surface mounted CFRP, GFRP sheets, laminates, strips, and rods. Experimental results were collected from various scientific publications.

Role of anisotropic invariants in numerically modeling soft biological tissues as transversely isotropic hyperelastic materials: A comparative study

Fiber-reinforced biological soft tissues are often modeled as anisotropic hyperelastic materials. Four strain invariants can be used to define a strain energy function for soft tissues reinforced with a single fiber family: two isotropic invariants I1,I2 and two anisotropic invariants I4,I5. Invariant I5 is often omitted in the strain energy functions to simplify the problem mathematically. In this study, the implications of using only I4 or the use of both anisotropic invariants I4,I5 in the numerical modeling of soft tissues were analyzed. A simple modification to the Holzapfel–Gasser–Ogden (HGO) model is proposed by adding a term that contains invariant I5. Material parameters were calculated by fitting the models with experimental data of uniaxial traction in the tibialis anterior tendon tissue of rats. General analytical solutions for the simple load scenarios were obtained. Such solutions are taken as a reference point to measure the precision of the numerical results obtained in the finite element simulations. The main differences between the models were observed in the shear behavior. The proposed model predicts three different shear responses (two responses with fiber reinforcement and one isotropic response), while the HGO model predicts two equal isotropic responses and only one with fiber reinforcement. Two sets of simple shear experimental data on a fiber-reinforced elastomer material were used to verify the shear stress prediction of the models. The experimental data show that the three shear behaviors are different; thus, the HGO model offers a limited description of the shear behavior. Furthermore, analytical solutions and experimental data suggest that invariant I5 is related to the elastic energy of the fibers when the material is subjected to shear in a direction parallel to the fibers. The models for non-homogeneous deformations were also compared. For this comparison, an irregular geometry was implemented in ABAQUS, and the traction and shear conditions were simulated by changing the fiber inclination. The most critical difference was found when the shear was parallel to the direction of the fiber and the minor difference when the fiber was at 45° to the load force.

The phase shift analysis of the colliding dissipative KdV solitons

In this work, the head-on collisions of the non-stationary dissipative soliton in ultracold neutral plasmas (UNPs) are investigated. The extended Poincare-Lighthill-Kuo (PLK) approach is adopted for reducing the fluid equations of the UNPs to two-counterpropagating damped Korteweg-de Vries (dKdV) equations. The dKdV equation is not an integrable Hamiltonian system, i.e., does not have an exact solution. Thus, one of the main goal of this paper is to find a new general approximate analytical solution to the dKdV equation for investigating the mechanism of the propagation and interaction of the non-stationary dissipative solitons. The residual error is estimated for checking the accuracy of the new obtained solution. The approximate analytical soliton solutions are adopted for deriving the temporal phase shifts after the collision. The impact of physical parameters on the nonstationary dissipative soliton profile and the temporal phase shifts is discussed. The obtained results will contribute to understand the mechanism of propagation and interaction of many nonlinear phenomena in different nonlinear mediums such as ocean, sea, optical fiber, plasma physics, etc.

A modified pulse‐decay approach to simultaneously measure permeability and porosity of tight rocks

AbstractPermeability and porosity are the two most important parameters of rocks for the evaluation and exploitation of oil/gas reservoirs. In this study, a modified pulse‐decay method has been developed to measure both permeability and porosity simultaneously. In the present method, the gas pressure in one chamber is changed (increased or decreased) instantaneously and then maintained constant, while the pressure response changing with time in the other one is monitored. A mathematical model of this procedure has been formulated, and a general analytical solution, including the early‐time and late‐time evolutions, has been obtained. The late‐time solution is presented for postprocessing of experimental data, which leads to the simultaneous measurement of permeability and porosity values of tight rocks. Our measurements agree well with those from the classical pulse‐decay and gas expansion methods. Compared with measuring the permeability and porosity separately, the proposed method can reduce the total test time and ensure the permeability and porosity are measured under the same effective stress condition.

On the dissipative extended Kawahara solitons and cnoidal waves in a collisional plasma: Novel analytical and numerical solutions

Two novel analytical solutions to the damped Gardner Kawahara equation and its related equations are reported. Using a suitable ansatz and with the help of the exact solutions of the undamped Gardner Kawahara equation, two general high-accurate approximate analytical solutions are derived. Moreover, the Crank–Nicolson implicit finite difference method is introduced for analyzing the evolution equation numerically. The comparison between the obtained solutions is examined. All the obtained solutions are able to investigate many types of the dissipative traveling wave solutions such as the dissipative solitary and cnoidal waves. Also, the obtained solutions help many researchers understand the mechanisms underlying a variety of nonlinear phenomena that can propagate in optical fiber, physics of plasmas, fluid mechanics, water tank, oceans, and seas. The obtained solutions could be applied for investigating the characteristics of the dissipative higher-order solitary and cnoidal waves in electronegative plasmas. Numerical results depending on the physical plasma parameters are presented.

Some results and applications on conformable fractional Kamal transform

In this paper, we derive conformable fractional Kamal transform from the classical Kamal transform and we present their properties. We discuss the conformable fractional kamal transform of some functions and relationship with conformable fractional Laplace transform. Moreover we solve the general analytical solution of a general conformable Bernoulli’s fractional differential equation by this new transform and Adomain polynomial method. Also we use this method to find the solution of linear and nonlinear conformable fractional differential equations.

General Analytical Solutions for Dynamic Load-Carrying Capacity of Brittle Materials Under an Arbitrary Incident Stress Wave

In this study, two general solutions for the dynamic tensile load-carrying capacity of brittle materials subject to an arbitrary incident stress wave in the form of Fourier integrals are derived. In order to verify the general solutions, we reduce them to three particular solutions, respectively, i.e., trapezoidal pulse, quadratic pulse and cubic pulse. It is found that all of them can well capture the experimental trends in the previous studies. In between, the first two particular solutions can exactly accord with the two previous analytical solutions under the same boundary conditions, respectively. Therefore, for arbitrary tensile boundary pulses, the dynamic tensile strength of the brittle materials can be calculated from static parameters and characteristics of external loading. This study improves the credibility of the previous works that working on deriving the dynamic load-carrying capacity through analytical methods, and thus, consolidate their theoretical foundation. Meanwhile, on basis of the general solution, the relation between the dynamic tensile load-carrying capacity and strain rate can be further deduced by combining static parameters, hence, it could be inappropriate to claim the strain rate effect on tensile strength as an intrinsic material property anymore. In addition, we found that the process to obtain an explicit analytical solution becomes increasingly difficult with the increased complexity of waveform for an incident pulse, or even no explicit solution can be obtained.

A General Analytical Solution for Axisymmetric Consolidation of Unsaturated Soil with Impeded Drainage Boundaries

The consolidation of soil is one of the most common phenomena in geotechnical engineering. Previous studies for the axisymmetric consolidation of unsaturated soil have usually idealized the boundary conditions as fully drained and absolutely undrained, but the boundaries of unsaturated soil are actually impeded drainage in most practical situations. In this study, we present a general analytical solution for predicting the axisymmetric consolidation behavior of unsaturated soil that incorporates impeded drainage boundary conditions in both the radial and vertical directions simultaneously. The impeded drainage boundary is modeled using the third kind boundary, and it can also realize fully drained and absolutely undrained ones by changing the drainage parameter. A general analytical solution is developed to predict the excess pore-air and pore-water pressures as well as the average degree of consolidation in an unsaturated soil stratum using the common methods of eigenfunction expansion and Laplace transform. The newly developed solution is expressed in the product of the terms of time, depth, and radius, which are derived using Laplace transform, usual Fourier, and Fourier-Bessel series, respectively. The eigenfunctions and eigenvalues are evaluated from the impeded drainage boundaries in both radial and depth dimensions. Then, the correctness of the proposed analytical solution is verified against the existing analytical solution for the case of traditional boundaries and against the finite difference solution for the case of general impeded drainage boundaries, and excellent agreements are obtained. Finally, the axisymmetric consolidation behavior of unsaturated soil involving impeded drainage boundaries is demonstrated and analyzed, and the effects of the drainage parameters are discussed. The results indicate that the larger drainage parameter generally expedites the dissipations of the excess pore pressures and further promotes the soil settling process. As the drainage parameter increases, its influence gradually diminishes and even can be neglected when it is larger than 100. The general analytical solution and findings of this study can help for better understanding the axisymmetric consolidation behavior of the unsaturated soil stratum in the ground improvement project with vertical drains as well as the gas-oil gravity drainage mechanism in the naturally fractured reservoirs.

Analytical study of the natural bending oscillations of a concave beam with parabolic change in thickness

The synthesis of factorization and symmetry methods produced a general analytical solution to the fourth-order differential equation with variable coefficients. The form and structure of the variable coefficients correspond, in this case, to the problem of the oscillations of a concave beam of variable thickness. The solution to this equation makes it possible to study in detail the oscillations of such and similar, for example convex, beams at the different fixation of their ends' sections. A practical confirmation has been obtained that the beam whose thickness changes in line with the concave parabola law H=a2x2+1, where a is the concave factor, demonstrates an increase in the natural frequencies of its free oscillations with an increase in its rigidity. As an example, the object's maximum deflection dependence on the beam rigidity factor has been established. The nature of this dependence confirmed the obvious conclusion that the deflections had decreased while the rigidity had increased. The evidence from the calculation results can be a testament to the correctness of the reported procedure of problem-solving. The considered problem and the analytical solution to it could serve as a practical guide to the optimal design of beam structures. In this case, it is very important to take into consideration the place and nature of the distribution of cyclical extreme operating stresses. The resulting ratios to solve the problem make it possible to simulate the required normal stresses in both the fixation and central zones when the rigidity parameter is changed. Designers could predict such a parabolic profile of the beam, which would ensure the required reduction of maximum stresses in the place of fixing the beam. The considered example of solving the problem of the natural oscillations of the beam with rigid fixation of the ends illustrates the effectiveness of the factoring and symmetry methods used. The developed solution algorithm could be extended to study the natural bending oscillations of the beam at other fixing techniques, not excluding a variant of a completely free beam

General analytical solution for vibrations of pipes with arbitrary discontinuities and generalized boundary condition on Pasternak foundation

Discontinuities of the shear force, bending moment, slope, and deflection can be caused by span supports, valves, and cracks etc. For arbitrary discontinuities appearing in the oil-conveying pipe resting on Pasternak foundation, the general analytical solution for vibrations of this structure is presented in this paper. To deal with discontinuity terms, generalized functions are employed to express the model displacement of the entire pipe as a single function. This function is then solved by Laplace transform. Continuity conditions at discontinuity points are also given. In addition, the generalized boundary condition (BC), which can be degenerated to many classical BCs, is considered. Numerical section testifies the feasibility and the validity of present solutions, and discusses the effects of some physical parameters on natural frequencies. It is shown that the closer the cracks are to the nearest nodes of mode shapes, the less effects they have on the corresponding natural frequencies; locations of support springs change effects of cracks on natural frequencies by affecting positions of structural nodes; foundation parameters have different effects on natural frequencies. The present method can be applied in maintaining safety and integrity of structures of pipes.

On the forced convective flow inside thermal collectors enhanced by porous media: from macro to micro-channels

PurposeAiming at finding the velocity distribution profile and other flow characteristic parameters such as the Poiseuille (Po) number, this study aims to focus on the three-dimensional forced convective flow inside rectangular ducts filled with porous media commonly used in air-based solar thermal collectors to enhance the thermal performance. The most general model for the fluid flow (i.e. the non-linear Darcy–Brinkman–Forchheimer partial differential equation subjected to slip and no-slip boundary conditions) is considered.Design/methodology/approachThe general governing equations are solved analytically based on the perturbation technique and the results are validated against numerical simulation study based on a finite-difference solution over a non-uniform but structured grid.FindingsThe analytical velocity distribution profile based on exponential functions for the above-mentioned general case is obtained, and accordingly, expressions for the Po are introduced. It is found that the velocity distribution tends to be uniform by increasing the aspect ratio of the duct. Moreover, a criterion for considering/neglecting the nonlinear drag term in the momentum equation (i.e. the Forchheimer term) is proposed. According to the sensitivity analysis, results show that the nonlinear drag term effects on the Nusselt number are important only in porous media with high Darcy numbers.Originality/valueA general analytic solution for three-dimensional forced convection flows through rectangular ducts filled with porous media for the general model of Darcy–Brinkman–Forchheimer and the general boundary condition including both no-slip and slip-flow regimes is obtained. An analytic expression to calculate Po number is obtained which can be practical for engineering estimations and a basis for validation of numerical simulations. A criterion for considering/neglecting the nonlinear drag term in the momentum equation is also introduced.

General solutions to mathcal {N}-field cosmology with exponential potentials

We construct the general analytical solution for the mathcal {N}-field product-exponential potential in an expanding FLRW background. We demonstrate the relevance of this analytical solution in more general contexts for the derivation of estimates for the transitional time between an arbitrary initial state and the slow-roll solutions. In certain cases, these estimates can also be used to demonstrate the non-linear convergence towards the slow-roll solutions. In addition, we extend the solution to include stiff matter as well.

Trajectories of Charged Particles Undergoing Brownian Motion in a Time-Dependent Magnetic Field II: Effect of External Parameters

The present article aims to study the effect of external parameters, such as the action of a constant force and the action of an external harmonic potential, on the trajectory of a charged particle in Brownian motion in a time-dependent magnetic field. The approach presented here builds on methods of statistical physics and produces an analytical and numerical general solution for the stochastic differential equations of motion.

Interaction among lump, periodic, and kink solutions with dynamical analysis to the conformable time-fractional Phi-four equation

This study’s prime goal is to explore new and general analytic solutions to the conformable time-fractional Phi-four equation by utilizing the advanced exp ( − φ ( ξ ) ) -expansion approach with the conformable derivative principle’s help. This approach is essentially a modified particular case of the generalized form of exp ( − φ ( ξ ) ) -expansion method. Concerning hyperbolic and trigonometric function solutions, an enormous class of modern precise moving wave solutions is revealed. Some interaction solutions obtained, such as lump-periodic and lump-periodic-kink structure and periodic wave, are seen by showing their three-dimensional (3D) and two-dimensional (2D) combination plots with the aid of computational software Maple. We have portrayed the figures of the estimated solutions to understand the physical phenomena.

An analytical approach to the cooling of a flat plate

The present work deals with the cooling of a flat plate having an initial non-homogeneous temperature distribution, which undergoes a sudden temperature drop at its surfaces. Hence, starting from an arbitrary steady-state temperature distribution, the two surfaces of the slab are suddenly cooled down to the value θ0, resulting in a strong perturbation of the initial temperature distribution. A general analytical solution is provided by adopting the variable separation technique to the Fourier equation in transient conditions, thus obtaining a general solution in terms of a very fast converging series, which includes a definite integral of the initial steady-state distribution. Common engineering applications are investigated and the related results, obtained thanks to an extremely low computational effort, are presented and discussed.

On the general analytic solution for unsteady turbulent incompressible flow between parallel plates

This research reports the mean-velocity field associated with a general plane-parallel flow, regardless of its being laminar, turbulent, steady-state, unsteady, or transient. The Reynolds-averaged Navier–Stokes equation is posed, with proper initial and boundary conditions, and the solution is developed in an easy-to-follow fashion. The time-dependent general analytic solution is obtained, which is the sum of three components: (i) the transient decay of the initial mean velocity, (ii) the unsteady mean velocity directly created by the mean pressure gradient, and (iii) the unsteady mean velocity originated from the flow's Reynolds shear-stress gradient. Each one of these components has a different evolution in time, resulting in an asynchronism among them that yields deformed mean-velocity profiles. The formalism is applied to study the transient flow created when an initial steady-state flow is accelerated or decelerated up to a final steady state. A rapidly accelerated version of this problem was already studied in a DNS by He and Seddighi [“Turbulence in transient channel flow,” J. Fluid Mech. 715, 60–102 (2013)]. We show that DNS results concerning mean velocity are qualitatively reproduced by our approach, including surprising features like the flattening of transient mean-velocity profiles, the middle mean-velocity overshoot, global laminarization, and the enhancement of the viscous sublayer related to the notion of hyperlaminarity already introduced by the authors. The decelerated case presents sharpening of mean-velocity profiles, middle mean-velocity undershoots, global turbulentization, and the destruction of the viscous sublayer, and poses predictions that await experimental confirmation. This research continues the exposition of the theory of underlying laminar flow initiated in previous papers.

СРАВНЕНИЕ РЕШЕНИЙ ЗАДАЧ ИЗГИБА ТРЕХСЛОЙНЫХ ПЛАСТИН НА ОСНОВАНИЯХ ВИНКЛЕРА И ПАСТЕРНАКА

Solutions of problems on axisymmetric bending of an elastic three-layer circular plate on the Winkler and Pasternak foundations are given. The bearing layers are taken as isotropic, for which Kirchhoff’s hypotheses are fulfilled. In a sufficiently thick lightweight, incompressible in thickness aggregate, the Timoshenko model is valid. The cylindrical coordinate system, in which the statements and solutions of boundary value problems are carried out, is connected with the median plane of the filler. On the plate contour, it is assumed that there is a rigid diaphragm that prevents the relative shear of the layers. The system of differential equations of equilibrium is obtained by the variational method. Three types of boundary conditions are formulated. One- and two-parameter Winkler and Pasternak models are used to describe the reaction of an elastic foundation. The solution to the boundary value problem is reduced to finding three desired functions, plate deflection, shear, and radial displacement in the filler. The general analytical solution to the boundary value problem is written out in the case of the Pasternak model in Bessel functions. At the Winkler foundation, the known solution is given in Kelvin functions. A numerical comparison of the displacements and stresses obtained by both models with a uniformly distributed load and rigid sealing of the plate contour is carried out.

Toward pricing financial derivatives with an IBM quantum computer

Pricing interest-rate financial derivatives is a major problem in finance, in which it is crucial to accurately reproduce the time-evolution of interest rates. Several stochastic dynamics have been proposed in the literature to model either the instantaneous interest rate or the instantaneous forward rate. A successful approach to model the latter is the celebrated Heath-Jarrow-Morton framework, in which its dynamics is entirely specified by volatility factors. On its multifactor version, this model considers several noisy components to capture at best the dynamics of several time-maturing forward rates. However, as no general analytical solution is available, there is a trade-off between the number of noisy factors considered and the computational time to perform a numerical simulation. Here, we employ the quantum principal component analysis to reduce the number of noisy factors required to accurately simulate the time evolution of several time-maturing forward rates. The principal components are experimentally estimated with the $5$-qubit IBMQX2 quantum computer for $2\times 2$ and $3\times 3$ cross-correlation matrices, which are based on historical data for two and three time-maturing forward rates. This manuscript is a first step towards the design of a general quantum algorithm to fully simulate on quantum computers the Heath-Jarrow-Morton model for pricing interest-rate financial derivatives. It shows indeed that practical applications of quantum computers in finance will be achievable in the near future.

Closed-form solutions for the coupled deflection of anisotropic Euler–Bernoulli composite beams with arbitrary boundary conditions

The fully anisotropic response of composite beams is an important consideration in diverse applications including aeroelastic responses of helicopter rotor and wind turbine blades. Our goal is to present exact analytical solutions for the first time for coupled deflection of Euler–Bernoulli composite beams. Towards this goal, two approaches are proposed: (1) obtaining the exact analytical solutions directly from the governing equations of Euler–Bernoulli composite beams and (2) extraction of the solutions from Timoshenko composite beam solutions. For the direct solution approach, based on Euler–Bernoulli theory, new variationally-consistent field equations are obtained, in which four degrees of freedom, i.e. in-plane bending, out-of-plane bending, twist and axial elongation are fully coupled. By expressing the coupled system of differential equations in a compact matrix form, a novel expression for the eccentricity of neutral axes from the midplane, as well as the shift in shear centre from the centre of beam, is obtained. This eccentricity matrix serves to decouple the bending in the two principal directions from in-plane and twist deformations. Then, the general closed-form analytical solutions for the decoupled system are derived simply using direct integration. Additionally, the analogous closed-form analytical solutions are retrieved from the previously obtained Timoshenko composite beam solution and it is proven that they are identical to those obtained from the current direct approach for conditions where Euler–Bernoulli beam theory apply. To study the effects of anisotropy, numerical results are obtained for a number of examples with different composite stacking sequences showing various coupled behaviours. The results are compared against the Chebyshev collocation method as well as against less comprehensive analytical solutions available in the literature, noting that excellent agreement is observed, where expected. The present exact solutions can serve as benchmark problems for assessing the accuracy and convergence of various analytical and numerical methods.