Present Analytical Solutions Research Articles

Analysis of pullout load capacity of suction caissons in clay by a three-dimensional displacement approach

ABSTRACTA study was made to present analytical solutions of pullout load capacity for a suction caisson subjected to inclined tension in clay. The inclined tension on the skirt of the suction caisson is transformed into an equivalent system comprised of the vertical, horizontal, and moment load applied on the center of the lid. The vertical and horizontal stiffness coefficients along the skirt of the suction caisson in clay are presented by three-dimensional elastic solutions considering the nonhomogeneous and nonlinear property of clay. The vertical, horizontal, and rocking stiffness coefficient of the suction caisson on the base are presented considering the solutions of a hollow rigid cylindrical punch acting on the surface of clay. The envelopes of the horizontal and vertical ultimate load capacity for clay are presented. The yield, pullout, and failure for clay are taken into consideration. The effects of load inclination, loading depth, and aspect ratio on the pullout load capacity are shown. Behavior of the suction caisson in clay up to failure is investigated using the relationship between tensile load and displacement and that between depth, vertical, and horizontal pressure.

An analytical treatment toward solution of fractional Schrödinger equation

We have applied the new approach of homotopic perturbation method (NHPM) for Schrodinger equation featuring time-fractional derivative. A combination of NHPM, Laplace transform and multiple fractional power series form has been used the first time to present analytical solution. Then, in order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All numerical calculations in this manuscript have been carried out with Mathematica.

Studying the effect of different parameters on stress resultant and moments distribution around non‐circular cutouts in unsymmetric laminates

AbstractIn this study, using complex potential approach, the stress analysis of unsymmetric laminates with non‐circular cutouts is examined. In this study, using an appropriate mapping function the solution of perforated unsymmetric laminate with a circular cutout is extended to calculate stress distribution around non‐circular cutouts and the effect of some parameters such as lay‐up, cutout orientation and bluntness of cutouts on the stress resultants and moment resultants around them is studied. The results obtained are presented for two different lay‐ups [0/90]T and [45/–45]T. Finite element numerical solution is used to validate results obtained from the present analytical solution. The analytical solution is in conformity with the finite element solution. It is interesting to note that the unexpected results indicate that the stress resultants and moment resultants around circular cutout aren't less than other cutouts. Furthermore, if appropriate values are selected for bluntness and cutout orientation, the stress resultants and moments will be reduced in significant value.

3D thermo-mechanical bending solution of functionally graded graphene reinforced circular and annular plates

• 3D thermo-elastic bending solutions are obtained for functionally graded circular and annular plates reinforced with GPLs. • Thinner GPLs are preferred to achieve better enhancement in bending stiffness hence reduced bending deflection. • The parabolic GPL distribution offers the best reinforcing effect, followed by uniform then linear distribution patterns. • For plates reinforced by uniformly distributed GPLs, the temperature field and radial stress distribution are not affected by GPL's total content. Within the framework of three-dimensional elasticity theory, this paper investigates the axisymmetric bending of novel functionally graded polymer nanocomposite circular and annular plates reinforced with graphene nanoplatelets (GPLs) whose weight fraction varies continuously and smoothly along the thickness direction. The generalized Mian and Spencer method is utilized to obtain the analytical solutions of nanocomposite circular and annular plates under a combined action of a uniformly distributed transverse load and a through-thickness steady temperature field. Three different distribution patterns of GPLs within the polymer matrix are considered. The present analytical solutions are validated through comparisons against those available in open literature for the reduced cases. A parametric study is conducted to examine the effects of GPL weight fraction, distribution pattern, plate thickness to radius ratio, and boundary conditions on the stress and deformation fields of the plate. The results show that GPL nanofillers with a low content can have a significant reinforcing effect on the bending behavior of the thermo-mechanically loaded plate.

The impact of capillary backpressure on spontaneous counter-current imbibition in porous media

We investigate the impact of capillary backpressure on spontaneous counter-current imbibition. For such displacements in strongly water-wet systems, the non-wetting phase is forced out through the inlet boundary as the wetting phase imbibes into the rock, creating a finite capillary backpressure. Under the assumption that capillary backpressure depends on the water saturation applied at the inlet boundary of the porous medium, its impact is determined using the continuum modelling approach by varying the imposed inlet saturation in the analytical solution.We present analytical solutions for the one-dimensional incompressible horizontal displacement of a non-wetting phase by a wetting phase in a porous medium. There exists an inlet saturation value above which any change in capillary backpressure has a negligible impact on the solutions. Above this threshold value, imbibition rates and front positions are largely invariant. A method for identifying this inlet saturation is proposed using an analytical procedure and we explore how varying multiphase flow properties affects the analytical solutions and this threshold saturation. We show the value of this analytical approach through the analysis of previously published experimental data.

Three-dimensional exact magneto-electro-elastic field in an infinite transversely isotropic space with an elliptical crack under uniform loads: Shear mode

This paper studies an elliptical planar crack embedded in an infinite transversely isotropic medium in the framework of magneto-electro-elasticity. The crack is assumed to be subjected to uniformly distributed shear loads, which are anti-symmetrical with respect to the crack plane. The boundary integral equation is established via the general solution in conjunction with the generalized method of potential theory. Exact and complete field variables are obtained in terms of elliptical functions. Important parameters in fracture mechanics, e.g., the crack slip displacement, the shear stress at the crack front, and the corresponding stress intensity factor, are explicitly derived. The corresponding solutions, in the context of piezoelectricity, piezomagnetism and pure elasticity are also presented, as a byproduct of the present work. The present closed-form analytical solutions may serve as benchmarks for future simplified and numerical studies.

One spatial dimensional finite volume three-body interaction for a short-range potential

In this work, we use McGuire's model to describe scattering of three spinless identical particles in one spatial dimension, we first present analytic solutions of Faddeev's equation for scattering of three spinless particles in free space. The three particles interaction in finite volume is derived subsequently, and the quantization conditions by matching wave functions in free space and finite volume are presented in terms of two-body scattering phase shifts. The quantization conditions obtained in this work for short range interaction are L\"uscher's formula like and consistent with Yang's results in \cite{Yang:1967bm}.

Equilibrium and Sudden Events in Chemical Evolution

Abstract We present new analytic solutions for one-zone (fully mixed) chemical evolution models that incorporate a realistic delay time distribution for Type Ia supernovae (SNe Ia) and can therefore track the separate evolution of α-elements produced by core collapse supernovae (CCSNe) and iron peak elements synthesized in both CCSNe and SNe Ia. Our solutions allow constant, exponential, or linear–exponential ( ) star formation histories, or combinations thereof. In generic cases, α and iron abundances evolve to an equilibrium at which element production is balanced by metal consumption and gas dilution, instead of continuing to increase over time. The equilibrium absolute abundances depend principally on supernova yields and the outflow mass loading parameter η, while the equilibrium abundance ratio depends mainly on yields and secondarily on star formation history. A stellar population can be metal-poor either because it has not yet evolved to equilibrium or because high outflow efficiency makes the equilibrium abundance itself low. Systems with ongoing gas accretion develop metallicity distribution functions (MDFs) that are sharply peaked, while “gas starved” systems with rapidly declining star formation, such as the conventional “closed box” model, have broadly peaked MDFs. A burst of star formation that consumes a significant fraction of a system’s available gas and retains its metals can temporarily boost by 0.1–0.3 dex, a possible origin for rare, α-enhanced stars with intermediate age and/or high metallicity. Other sudden transitions in system properties can produce surprising behavior, including backward evolution of a stellar population from high to low metallicity.

Theory of linear sweep voltammetry with diffuse charge: Unsupported electrolytes, thin films, and leaky membranes.

Linear sweep and cyclic voltammetry techniques are important tools for electrochemists and have a variety of applications in engineering. Voltammetry has classically been treated with the Randles-Sevcik equation, which assumes an electroneutral supported electrolyte. In this paper, we provide a comprehensive mathematical theory of voltammetry in electrochemical cells with unsupported electrolytes and for other situations where diffuse charge effects play a role, and present analytical and simulated solutions of the time-dependent Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions for a 1:1 electrolyte and a simple reaction. Using these solutions, we construct theoretical and simulated current-voltage curves for liquid and solid thin films, membranes with fixed background charge, and cells with blocking electrodes. The full range of dimensionless parameters is considered, including the dimensionless Debye screening length (scaled to the electrode separation), Damkohler number (ratio of characteristic diffusion and reaction times), and dimensionless sweep rate (scaled to the thermal voltage per diffusion time). The analysis focuses on the coupling of Faradaic reactions and diffuse charge dynamics, although capacitive charging of the electrical double layers is also studied, for early time transients at reactive electrodes and for nonreactive blocking electrodes. Our work highlights cases where diffuse charge effects are important in the context of voltammetry, and illustrates which regimes can be approximated using simple analytical expressions and which require more careful consideration.

Set-up and production planning in hybrid manufacturing–remanufacturing systems with large returns

Hybrid systems that use both raw materials and returned products in the production process are considered. The system contains one facility, and undergoes set-ups each time it switches between two production modes. In particular, we address systems engaged mainly in remanufacturing and having a large percentage of return. This situation is encountered in companies with mature remanufacturing channels. The targeted application area is comprised of hybrid systems that uses leasing as a business model with manufacturing serving to attenuate return uncertainty. To evaluate the system performance, we take into account manufacturing and remanufacturing costs, holding costs in serviceable and return inventories, backlog and set-up costs. Our analysis of hybrid systems with high return levels reveals features that are peculiar to such systems and that differentiate them from systems with lower return rates. We first present analytical solutions for optimal production and set-up schedule, and determine the possible cycle shapes for reliable systems. Optimal policies contain intervals of manufacturing and remanufacturing at maximal rate, and intervals of on-demand and on-return remanufacturing. Failure-prone systems are studied next, using the formalism of stochastic dynamic programming. Optimal policies give rise to the trajectories converging to the patterns similar to the analytically calculated cycles.

Analysis of solution for system of nonlinear fractional Burger differential equations based on multiple fractional power series

We have applied the new approach of homotopic perturbation method (NHPM) for Burger differential system equations featuring time-fractional derivative. A combination of NHPM and multiple fractional power series form has been used the first time to present analytical solution. In order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All numerical calculations in this manuscript have been carried out with Mathematica.

A Study of Analytical Solution for the Special Dissolution Rate Model of Rock Salt

By calculating the concentration distributions of rock salt solutions at the boundary layer, an ordinary differential equation for describing a special dissolution rate model of rock salt under the assumption of an instantaneous diffusion process was established to investigate the dissolution mechanism of rock salt under transient but stable conditions. The ordinary differential equation was then solved mathematically to give an analytical solution and related expressions for the dissolved radius and solution concentration. Thereafter, the analytical solution was fitted with transient dissolution test data of rock salt to provide the dissolution parameters at different flow rates, and the physical meaning of the analytical formula was also discussed. Finally, the influential factors of the analytical formula were investigated. There was approximately a linear relationship between the dissolution parameters and the flow rate. The effects of the dissolution area and initial volume of the solution on the dissolution rate equation of rock salt were computationally investigated. The results showed that the present analytical solution gives a good description of the dissolution mechanism of rock salt under some special conditions, which may provide a primary theoretical basis and an analytical way to investigate the dissolution characteristics of rock salt.

Mode II Fracture of GFRP Laminates Bonded Interfaces under 4-ENF Test

This experiment studies the mode II fracture behavior of an adhesively bonded joint composed of GFRP laminates. A new beam model is presented to calculate the mode II ERR for GFRP bonded 4-ENF specimens. In this model, the deformation of 4-ENF specimens caused by the relative deflection angle between the upper and lower layers and by the bending deformation of the upper and lower layers, respectively, is introduced; the effect of the adhesive layer deformation is presented. The closed-form analytical solutions of compliance and energy release rate based on the crack compliance method are obtained. The high accuracy of present analytical solutions are verified by finite element analysis through bonded GFRP 4-ENF specimens and compared to the rigid joint model and the CBT model. The interfacial crack propagation is numerically simulated using shear fracture toughness determined in this experiment, from which the predicted critical load results are in good agreement with the experimental results. The conclusion indicates that the compliance and ERR can accurately be predicted using the new bonded 4-ENF beam model.

Optimized design for perforated plates with quasi-square hole by grey wolf optimizer

One major concern that has occupied the mind of the designers is a structural failure as result of stress concentration in the geometrical discontinuities. Understanding the effective parameters contribute to stress concentration and proper selection of these parameters enables the designer get to a reliable design. In the analysis of perforated isotropic and orthotropic plates, the effective parameters on stress distribution around holes include load angle, curvature radius of the corner of the hole, hole orientation and fiber angle for orthotropic materials. This present paper tries to examine the possible effects of these parameters on stress analysis of infinite perforated plates with central quasi-square hole applying grey wolf optimizer (GWO) inspired by the particular leadership hierarchy and hunting behavior of grey wolves in nature, and also the present study tries to introduce general optimum parameters in order to achieve the minimum amount of stress concentration around this type of hole on isotropic and orthotropic plates. The advantages of grey wolf optimizer are stout, flexible, simple, and easy to be enforced. The used analytical solution is the expansion of Lekhnitskii\'s solution method. Lekhnitskii applied this method for the stress analysis of anisotropic plates containing circular and elliptical holes. Finite element numerical solution is employed to examine the results of present analytical solution. Results represent that by selecting the aforementioned parameters properly, fewer amounts of stress could be achieved around the hole leading to an increase in load-bearing capacity of the structure.

Influence of Surface Energy Effects on Elastic Fields of a Layered Elastic Medium under Surface Loading

This paper presents the analysis of a layered elastic half space under the action of axisymmetric surface loading and the influence of the surface energy effects. The boundary value problems for the bulk and the surface are formulated based on classical linear elasticity and a complete Gurtin-Murdoch constitutive relation. An analytical technique using Love’s representation and the Hankel integral transform is employed to derive an integral-form solution for both displacement and stress fields. An efficient numerical quadrature is then applied to accurately evaluate all involved integrals. Selected numerical results are presented to portray the influence of various parameters on elastic fields. Numerical results indicate that the surface stress displays a significant influence on both displacement and stress fields. It is also found that the layered half space becomes stiffer with the presence of surface stresses. In addition, unlike the classical elasticity solution, size-dependent behavior of elastic fields is noted. The present analytical solutions provide fundamental understanding of the influence of surface energy on layered elastic materials. It can also be used as a benchmark solution for the development of numerical techniques such as FEM and BEM, for analysis of more complex problems involving a layered medium under the influence of surface energy effects.

Predicting dynamic response of large amplitude free vibrations of cantilever tapered beams on a nonlinear elastic foundation

The purpose of this paper is to investigate the nonlinear vibration behavior of single and double tapered cantilever beams on the nonlinear foundation and present an alternative analytical method to solve the issue. Unlike other literatures, the admissible lateral displacement function satisfying the geometric boundary conditions of a single or double tapered cantilever beam is derived by using Rayleigh-Ritz method. Analytical approximate solutions in closed and explicit form are obtained by combining the Lagrange method with the Galerkin and Newton linearization method. Compared with the exact (numeric) results, the accuracy of these approximate solutions is presented. The effects of different parameters such as vibration amplitude, and foundation stiff parameters on nonlinear frequencies and displacement response of the tapered beams are easily analyzed, with the brief expressions of the present analytical approximate solutions.

Determination of optimal parameters for perforated plates with quasi-triangular cutout by PSO

This study tries to examine the effect of different parameters on stress analysis of infinite plates with central quasi-triangular cutout using particle swarm optimization (PSO) algorithm and also an attempt has been made to introduce general optimum parameters in order to achieve the minimum amount of stress concentration around this type of cutout on isotropic and orthotropic plates. Basis of the presented method is expansion of analytical method conducted by Lekhnitskii for circular and elliptical cutouts. Design variables in this study include fiber angle, load angle, curvature radius of the corner of the cutout, rotation angle of the cutout and at last material of the plate. Also, diagrams of convergence and duration time of the desired problem are compared with Simulated Annealing algorithm. Conducted comparison is indicative of appropriateness of this method in optimization of the plates. Finite element numerical solution is employed to examine the results of present analytical solution. Overlap of the results of the two methods confirms the validity of the presented solution. Results show that by selecting the aforementioned parameters properly, less amounts of stress can be achieved around the cutout leading to an increase in load-bearing capacity of the structure.

Three-dimensional fundamental thermo-elastic field in an infinite space of two-dimensional hexagonal quasi-crystal with a penny-shaped/half-infinite plane crack

In the present work, a penny-shaped/half-infinite plane crack problem is investigated in the framework of thermo-elasticity of two-dimensional quasi-crystals. In view of the symmetry with respect to the crack plane, the original problem is formulated by a mixed boundary-value problem defined in a half-space. The boundary integral equations are derived by virtue of the general solution and the method of generalized potential theory. For the crack subjected to a pair of point temperature loadings, the corresponding fundamental thermo-elastic field variables are exactly and explicitly obtained in terms of elementary functions. Furthermore, the physical quantities on the crack plane, which are important in fracture mechanics, are given. Numerical calculations are carried out to discuss the validity of the present solutions and to characterize the thermo-phonon-phason coupling effect. The present analytical solutions may serve as benchmarks for future computational fracture mechanics of QCs and as a guide for the infrared thermography techniques in non-destructive detection.

Thermoelastic analysis of functionally graded graphene reinforced rectangular plates based on 3D elasticity

Thermoelastic bending behaviour of novel functionally graded polymer nanocomposite rectangular plate reinforced with graphene nanoplatelets (GPLs) whose weight fraction varies continuously and smoothly along the thickness direction is investigated. The generalized Mian and Spencer method is utilized to obtain the analytical solutions of nanocomposite rectangular plate with two opposite edges simply supported and under a uniformly distributed transverse load and a temperature change. Three GPL distribution patterns are considered. Comparison between the present analytical solutions and those available in literature is carried out to verify the accuracy of our analytical solutions. A parametric study is conducted to examine the effects of GPL’s weight fraction, distribution pattern, geometry and size as well as the temperature change and plate boundary conditions on the stress and deformation fields of the nanocomposite plates. Numerical results show that the addition of GPLs at a very low content can have a significant reinforcing effect on the thermo-mechanical response of the plate.

Analytical Solutions for Spontaneous Imbibition: Fractional-Flow Theory and Experimental Analysis

Summary We present analytical solutions for capillary-controlled displacement in one dimension by use of fractional-flow theory. We show how to construct solutions with a spreadsheet that can be used for the analysis of experiments as well as matrix-block-scale recovery in field settings. The solutions can be understood as the capillary analog to the classical Buckley-Leverett solution (Buckley and Leverett 1942) for viscous-dominated flow, and are valid for cocurrent and countercurrent spontaneous imbibition (SI), as well as for arbitrary capillary pressure and relative permeability curves. They can be used to study the influence of wettability, predicting saturation profiles and production rates characteristic for water-wet and mixed-wet conditions. We compare our results with in-situ measurements of saturation profiles for SI in a water-wet medium. We show that the characteristic shape of the saturation profile is consistent with the expected form of the relative permeabilities. We discuss how measurements of imbibition profiles, in combination with other measurements, could be used to determine relative permeability and capillary pressure.