Exact Closed-form Solutions Research Articles

Exact closed-form dynamic stiffness matrix of damaged frames comprising Timoshenko–Ehrenfest beams

The present work proposes the extension of an exact distributional model for cracked beams to include the local influence on axial deformation on shear deformable beams. Axial, shear and flexural concentrated flexibilities are taken into account to model the damaged sections. Shear deformability is accounted for by adopting the Timoshenko–Ehrenfest model and exact closed-form solution of the mode shapes for multiple cracked beams are derived. On this ground, the final achievement of the work consists in the original formulation of the Dynamic Stiffness Matrix (DSM) with multiple cracked sections in exact closed form. The closed-form expression of the DSM allows the assemblage of the global DSM of a structure to conduct the free vibration analysis of damaged framed structures as well as for the study of frames with semi-rigid joints. This approach has the advantage to avoid any numerical procedure for the construction of the DSM as well as the introduction of further degrees of freedom (dofs) in correspondence of the damaged sections, keeping the size of the problem the same as in the intact configuration. Natural frequencies of the frames are determined by means of the Wittrick–Williams algorithm (W–W), then the mode shapes and modal forces are subsequently obtained in closed form. The proposed approach is validated with results available in literature for two portal frames characterized by the presence of cracks and semi-rigid joints, respectively. Finally, the potentiality of the approach, also in view of practical applications, is demonstrated analysing a multi-storey frame with semi-rigid connections.

Exact closed-form solution for vibration characteristics of multi-span beams on an elastic foundation subjected to axial force

This paper investigates the vibration characteristics of multi-span beams resting on an elastic foundation and subjected to axial forces. A comprehensive analytical expression of the dynamic response of multi-span beams on an elastic foundation that is developed to address various boundary conditions. The vibration equation is derived by employing Newton's second law. By Laplace transformations and the Green's function method, the solution of this governing equation can be obtained. Subsequently, a unified description is implemented for distinct types of boundary conditions using matrix representations. The correctness is verified through reference results and finite element methods (FEM). The effects of different parameters such as support stiffness, foundation elastic and shear layer stiffness, and axial force on the vibration characteristics are analyzed. This study demonstrates two findings: First, there are two thresholds for support stiffness, and the stiffness value is divided into three intervals. In the same interval, multi-span beams show the same properties. Second, for a rigidly supported multi-span beam, the critical axial force with a natural frequency of zero is just the corresponding Euler's buckling load; for elastically supported multi-span beams, the critical axial force falls between the Euler’s buckling load corresponding to single-span and multi-span beams.

A theoretical modelling of strengthening mechanism in graphene-metal nanolayered composites

Graphene-metal nanolayered composites (GMNCs) are a new generation of nano-structural composites characterized by a very high density of graphene reinforced interfaces (GRI) between metal nanolayers. Compared to traditional graphene flake reinforced composites, GMNCs have much higher strength, toughness and ductility due to the excellent ability of GRI on constraining dislocation motion and crack propagation. Despite numerous experimental and numerical studies on the mechanical behavior of GMNCs, the underlying strengthening mechanism is still not fully understood due to the absence of appropriate theoretical model. This paper proposes a continuum mechanics based theoretical model to explain the strengthening mechanism in GMNCs. In this model, the metal matrix and the GRI are simulated as homogenous elastic medium of infinite extend and inextensible thin membrane of zero thickness, respectively. Using the theoretical model, two boundary value problems namely (i) A circular prismatic dislocation loop approaching to the GRI and (ii) A mixed mode I/II penny-shaped crack near the GRI are formulated to reveal the two key strengthening mechanism: dislocation blocking and crack shielding, respectively. The two problems are solved analytically by the Generalized Kelvin's Solution (GKS) based method for 3D elasticity and Fredholm integral integration technique. Exact closed form solution for the Peach-Koehler (P-K) force on the dislocation loop is obtained. An efficient numerical scheme is developed to solve the Fredholm integral integration for the crack problem with very high accuracy. It is shown that our theoretical model can well capture and explain the strengthening mechanism observed in experiments. Moreover, the dominant role of Poisson's ratio on the strengthening efficiency is also revealed by our model. This finding implies the exciting possibility that the strength of GMNCs can be tailored by controlling the Poisson's ratio of the metal matrix. The present theoretical modeling can provide valuable insights into the mechanics-based design of GMNCs.

Long-distance high-fidelity continuous-variable quantum key distribution with non-Gaussian operations: An exact closed form solution

In this paper, we derive a closed-form expression for the output state of a CV-QKD protocol in the presence of zero-photon catalysis (ZPC) and a quantum scissor (QS). Then, based on this closed-form solution, we use a direct search algorithm to find the appropriate values of input state and QS parameters, which considerably enhance the range and the fidelity of a CV-QKD protocol. In the special case of pure loss channel, the largest range of the protocol is only 6.5% less than the fundamental limit of repeaterless quantum communication. In addition, examination of the protocol for different values of excess noise, reveals that there is a trade-off between range and fidelity, and a high value of fidelity can be obtained at the cost of a slight reduction in protocol range.

An oscillating reaction network with an exact closed form solution in the time domain

BackgroundOscillatory behavior is critical to many life sustaining processes such as cell cycles, circadian rhythms, and notch signaling. Important biological functions depend on the characteristics of these oscillations (hereafter, oscillation characteristics or OCs): frequency (e.g., event timings), amplitude (e.g., signal strength), and phase (e.g., event sequencing). Numerous oscillating reaction networks have been documented or proposed. Some investigators claim that oscillations in reaction networks require nonlinear dynamics in that at least one rate law is a nonlinear function of species concentrations. No one has shown that oscillations can be produced for a reaction network with linear dynamics. Further, no one has obtained closed form solutions for the frequency, amplitude and phase of any oscillating reaction network. Finally, no one has published an algorithm for constructing oscillating reaction networks with desired OCs.ResultsThis is a theoretical study that analyzes reaction networks in terms of their representation as systems of ordinary differential equations. Our contributions are: (a) construction of an oscillating, two species reaction network [two species harmonic oscillator (2SHO)] that has no nonlinearity; (b) obtaining closed form formulas that calculate frequency, amplitude, and phase in terms of the parameters of the 2SHO reaction network, something that has not been done for any published oscillating reaction network; and (c) development of an algorithm that parameterizes the 2SHO to achieve desired oscillation, a capability that has not been produced for any published oscillating reaction network.ConclusionsOur 2SHO demonstrates the feasibility of creating an oscillating reaction network whose dynamics are described by a system of linear differential equations. Because it is a linear system, we can derive closed form expressions for the frequency, amplitude, and phase of oscillations, something that has not been done for other published reaction networks. With these formulas, we can design 2SHO reaction networks to have desired oscillation characteristics. Finally, our sensitivity analysis suggests an approach to constructing a 2SHO for a biochemical system.

Influence of temperature dependent heat source/sink on transient MHD free convective flow in an infinite rigid impermeable vertical cylinder with chemical reaction

It is increasingly apparent that the inclusion of mass transfer aspects, together with certain thermal conditions, in the momentum and energy equations governing MHD flows leads to a numbers of real life applications. Keeping this in view, we have attempted an exact analysis of heat and mass transfer aspects in transient hydromagnetic free convective flow of an incompressible viscous fluid through a vertical pipe under an externally applied magnetic field, assuming presence of chemical reaction and heat source/sink. The governing PDEs, which simplify to a set of 3 linear ODEs in the physical set up considered here, have been solved using Laplace transform technique, with solutions for key physical variables presented in the term of Bessel and modified Bessel functions. The influence of governing non-dimensional parameters, namely, Hartmann number, Schmidt number, source/sink parameter, Prandtl number and chemical reaction parameter, has been illustrated on the developing velocity and some concentration profiles. Some important quantities of engineering interest-surface skin friction and volumetric flow rates-have been computed too and analysed. Some notable finding worth mentioning are: (a) heat source presence causes higher fluid velocity as compared to the heat sink; (b) all important surface shear stress can be suitably controlled, among others, by chemical reaction parameter and Schmidt number. The key challenge of this study has been to obtain exact closed-form solutions of the field equations, including cumbersome Laplace inverses. This study finds innovative applications in the emerging fields such as magnetic materials processing, chemical processes, solar energy systems, etc.

Unified and accurate simulation for large elastic strain responses of rubberlike soft materials under multiple modes of loading

AbstractA new and explicit form of the multi-axial elastic potential for elastic soft materials is constructed by means of two invariants of the Hencky strain. The new elasticity model with this form can bypass coupling complexities and uncertainties usually involved in parameter identification. Namely, exact closed-form solutions of decoupled nature are obtainable for stress responses under multiple benchmark modes. Unlike usual solutions with numerous coupled parameters, such new solutions are independent of one another and, as such, data sets for multiple benchmark modes can be separately matched with mutually independent single-variable functions. A comparative study is presented between a few well-known models and the new model. Results show that predictions from the former agree well with uniaxial and biaxial data, as known in the literature, but would be at variance with data for the constrained stress response in the plane-strain extension. In contrast, predictions from the new model agree accurately with all data sets. Furthermore, exact solutions for the Poynting effect of freely twisted elastic thin-walled tube are obtained from the new model.

Analysis of transversely isotropic compressible and nearly-incompressible soft material structures by high order unified finite elements

This work proposes high-order beam (1D) and plate (2D) finite element models for the large strain analysis of compressible and incompressible transversely isotropic hyperelastic media, defined within the Carrera Unified Formulation (CUF) framework. The strain energy density function adopted in fiber-reinforced hyperelastic materials modeling is presented and expressed in terms of invariants and pseudo-invariants of the right Cauchy-Green strain tensor. The explicit expression of the tangent elasticity tensor is derived through the assumption of coupled formulation of strain energy functions. Refined fully nonlinear beam and plate models are defined in a total Lagrangian formulation, deriving the governing equations of the nonlinear static analysis through the Principle of Virtual Displacements in terms of fundamental nuclei, in resulting expressions of internal and external forces vectors, and tangent stiffness matrix independent of kinematic models and approximation theories adopted. The iterative Newton-Raphson linearization scheme coupled with the arc-length constraint is adopted to obtain actual numerical solutions. Different benchmark analyses in hyperelasticity are performed to assess the capabilities of our proposed model, analyzing the three-dimensional stress field for moderate to large strain states and comparing actual numerical results with exact closed-form solutions or results available in the literature, demonstrating the capabilities and reliability of CUF models in the analysis of fiber-reinforced soft materials and structures.

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams with arbitrary boundary conditions based on the Nonlocal Elasticity Theory (NET) and the Timoshenko beam theory are presented. The NET considering the size effect of nanostructures is applied. FGM properties vary nonlinearly along the height of the beam. A crack model using two springs with stiffness depending on the crack depth is applied. The proposed solutions are provided explicitly as functions of integration constants determined from the standard boundary conditions. Frequency equations, that established in the form of the determinant of the matrix 3 × 3 order for arbitrary boundary conditions, are applied to analyse free vibration. Expressions for the mode shapes are given explicitly. The effects of geometry, material, nonlocal and crack parameters on the free vibration of the multiple cracked nanobeam are then analysed in detail.

Analytical investigation and graphical simulations for the solitary wave behavior of Chaffee–Infante equation

Chaffee–Infante equation has several useful applications in the field of electromagnetic waves, fluid dynamics, plasma physics, signal processing in optical fibers and sound waves. In this study, the (1+1)-D and (2+1)-D forms of the Chaffee–Infante equation are theoretically investigated to determine the variations in wave structure of the considered model by finding the analytical exact closed form solutions of the considered equations. The expressions for traveling wave solution are retrieved which involve rational, hyperbolic and trigonometric functions, through the (G′G,1G)-expansion method for analytical treatment of the equation for the first time. The solitons and other traveling wave structures are observed through two and three dimensional graphical simulations.

Dynamic Analysis of Slab Track with Crack Considering Rail Corrugation

To investigate the dynamic response of ballastless slab track with crack in the rail corrugation area under the load of urban rail vehicle, this study establishes a vertical coupling dynamic model of the vehicle-rail-slab track, considering crack and rail corrugation. The exact closed-form solution of the vibration mode of the Euler Bernoulli beam with crack is introduced. Furthermore, a finite element model of the slab track is developed, and the vertical dynamic model's fastening force is applied to the slab track at the rail corrugation frequency to calculate the surface compressive stress of the slab track. The results indicate that, under rail corrugation conditions of varying magnitudes, the presence of crack induces higher vertical acceleration amplitudes in the slab track at low frequencies. As the wave depth increases, the acceleration of the slab track at the crack gradually intensifies. Moreover, with an increasing wavelength, the corresponding frequency to the acceleration amplitude of the slab track at the crack gradually decreases. Notably, when the wave depth is 0.2 mm and the wavelength is 50 mm, the axle load transfer rate exhibits the greatest increase. These research findings offer valuable insights for the maintenance of urban rail transit slab track systems.

An effects of mass transpiration and inclined MHD on nanoboundary layer of an ostwald-de waele fluid due to a shrinking boundary

This paper examines the power-law nanofluid flows due to a shrinking sheet with mass transpiration under inclined MHD. The governing partial differential equations are similarly transformed into nonlinear ordinary differential equations and solved analytically. The mathematical model through analytical solutions graphically shows that the flow's dynamics depending on the Chandraselhar number, mass transpiration, nanofluid and inclined angle parameters. The main findings and methods of the present work includes the closed-form exact solutions are examined for some special cases and one of them is an algebraic solution. Specific and quantitative analytical solutions provide important understanding to the boundary layer flow for power-law nanofluid. The present study reveals that the magnetic field significantly increases the magnitude of the skin friction and mass transpiration. Cu-H2O nanofluid is used to examine the entaire calculation. Nanofluids are suspensions of nanoparticles in fluids that show significant enhancement of their properties at modest nanoparticle concentration.

New exact plasma equilibria with axial and helical symmetry

We derive new exact closed-form solutions of magnetohydrodynamics equations, with and without dynamics, which model astrophysical jets and other prolonged plasma configurations. The solutions are obtained in static and dynamic incompressible equilibrium settings, in axial and helical symmetry assumptions, and are given in terms of Whittaker, Coulomb, and Heun special functions. For each symmetry, two distinct families of physical solutions arise, corresponding to two distinct pressure profiles. One pressure profile models plasmas supported by an external pressure and is suitable for the description of plasma configurations in a medium, such as atmosphere. The second profile features higher pressure inside the plasma domain and can model plasmas residing in a vacuum. Examples of static and dynamic solutions in axially and helically symmetric settings, including solutions with boundary current sheets, are presented and discussed.

Gradient elasticity solutions of 2D nano-beams

In this study, the exact analytical solutions of a two-dimensional linear homogeneous isotropic nano-beam in gradient elasticity are studied. Four different types of two-dimensional cantilever beams and related boundary conditions are considered. The cases are a cantilever beam under a concentrated force at the end, a cantilever beam under a uniform load, a propped cantilever beam under a uniform load, and a fixed-end beam under a uniform load. The two-dimensional stress gradient fields are investigated and obtained from the analytical solutions of a linear second-order partial differential equation written in terms of the classical and the gradient Airy stress functions. Additionally, the micro-size effects in the displacement components for different loads and support conditions for the two-dimensional cantilever beams by using strain gradient elasticity theory are investigated. Furthermore, for one-dimensional Euler–Bernoulli beam model, the associated stress and strain elasticity solutions are obtained from two-dimensional analytical solutions. The graphical presentations of the exact closed-form solutions are provided and discussed.

An exact closed-form explicit solution of free transverse vibration for non-uniform multi-cracked beam

Non-uniformity and damage are the two primary subjects in studying the vibrations of the beam-type elements. An exact closed-form explicit solution for the transverse displacement of a non-uniform multi-cracked beam with any type of boundary conditions is introduced. The generalized functions and the distributional derivative concepts are adopted. Four fundamental functions are introduced. These functions make the boundary conditions' process and compute the frequency equation more convenient. By introducing the non-dimensional parameters, the non-dimensional motion equation of the damaged beam with an arbitrary count of cracks is derived, and its exact closed-form explicit solution is obtained based on the four introduced fundamental functions. The standard method of computing these functions is presented, and the closed-form of these functions is determined for eight cases like uniform and conical beams. The closed-form of the frequency equation and mode shapes of the non-uniform multi-cracked beam are derived for several boundary conditions. The influence of the count of cracks, their location and intensity, and the boundary conditions on the natural frequency and mode shape are assessed by running a numerical study. The first and second frequencies of a conical beam are computed to verify the obtained results by applying this newly presented closed-form solution and the Differential Quadrature Element Method. A good agreement is evident when the obtained results are compared.

Shielding effect of ring dislocation dipole on penny-shaped crack

Mechanical behavior of strongly crystalline materials can be significantly influenced by the internal dislocation structure and their iteration with micro crack-like defects. Thus, study on the crack-dislocation interaction problems can help to understand the failure mechanism of crystalline materials. Previous theoretical analysis have been mostly limited to plane problems, where the cracks are idealized as straight slit with infinite or zero out plane dimension and the dislocations slip along an inclined straight path starting from the crack tip. In this paper, we explore a new interaction mechanism under axisymmetric deformation state. An unconventional case of a penny-shaped crack interacts with a ring dislocation dipole is considered. Both the ring dislocation dipole and crack are modelled as axisymmetric edge dislocation loops. Solutions for the dislocation loops are obtained using the GKS based method. The mixed boundary value problem is converted into Abel integral equations, which can be solved analytically. Exact closed form solutions for the stress intensity factors (SIFs) in terms of elementary functions are obtained. Numerical studies are conducted to investigate the shielding effect of the ring dislocation dipole on the crack tip SIFs. Conditions for the dislocation to shield the crack are discussed. The present solutions can also be used as weight functions to study the axisymmetric interaction between penny shaped crack and other types of eigenstrain sources.

Exact closed-form solutions for behavior investigation of heterogeneous piezo-magneto-elastic discs under magneto-electro-hygro-thermo-mechanical loads

The purpose of this study is to propose full-analytical closed-form solutions for functionally graded magneto-electro-hygro-thermo-elastic (FG-MEHTE) discs exposed to various combinations of the magneto-electro-hygro-thermo-mechanical loads. The formulations account for the elastic, piezoelectric, dielectric, magnetic, piezomagnetic, pyroelectric, pyromagnetic, and hygrothermal properties of the disc. Different exponential through-thickness variations are adopted for distinct material properties. General hygrothermal boundary conditions are taken into consideration and the distributions of the rises in the moisture concentration and temperature are determined according to Fick and Fourier equations, respectively. Based on the constitutive equations of the axisymmetric hygro-thermo-elasticity problem, the Navier equations are derived. The series solutions that are proposed for the displacement field and the electric and magnetic potentials consist of appropriate superpositions of exponential and Bessel functions whose arguments are ascertained based on the edge conditions. The boundary conditions of the top and bottom faces of the disc are employed to determine the unknown coefficients of the mentioned series solutions. Results reveal that by increasing the volume fraction of the piezoelectric phase, the electric potential increases but the magnetic induction and the magnetic potential decrease in such a way that the resulting distributions of the axial and radial magnetic inductions become respectively, similar to those of the transverse normal and transverse shear stresses. In contrast to the reductions in the displacements and stresses, the magnitude of the transverse shear stress and electric and magnetic potentials grow with the thickness.

Thermally progressive Particle-Cu/Blood peristaltic transport with mass transfer in a Non-Uniform Wavy Channel: Closed-form exact solutions

The motivation behind writing the current article is to mathematically study the thermally driven transport of solute particles (measured in micro to nano meters) suspended in blood in an Esophagus mathematically taken as a non-uniform channel. The aim of the present study is to see the effects of solute particles with blood and nanoparticles in Esophagus to see its application in drug delivery and biomedical engineering etc. It also has many applications in microscale phenomena in physiology. Here blood is taken as a base fluid and nanofluid is prepared by the suspension of copper oxide nanoparticles in blood. Copper oxide is extremely useful in the medical domain for drug delivery and cancer treatment. To further study the impact of nanomedicine delivery we considered copper nanoparticles embedded in blood along other small sized solute particles. The mathematical formulation is performed in a rectangular coordinate system and constitutive flow equations are linearized by the implementation of approximations of low Reynolds number and large wavelength. The exact solution of reduced coupled and nonlinear set of equations for fluid particle velocity, temperature and fluid-particle composition profiles is computed through Mathematica while pressure rise is calculated by applying numerical integration. The effect of notable emerging parameters is discussed through graphs. It is observed that velocity rises with an increase in suspension and Casson fluid parameter, whereas opposite results are observed for temperature distribution. Nanoparticles volumetric fraction contribute to decelerating fluid flow and upsurge temperature. Schmidt number influences the fluid-particle composition by lessening it. PACS number: 47.10. A−; 47.35.Pq; 47.85.-g.

A non-newtonian fluid flow due to porous media with mass transfer and slip

The Casson viscous gas fluid flow with mass transpiration and radiation is demonstrated in this paper. A similarity transformation is utilized to interpret the representing construction of nonlinear PDEs by nonlinear ODEs. Then, using an exact analysis of the momentum equation, we constructed the dual solutions to the flow problem. This solution domain aids in the solution of the energy equation under the incomplete gamma function condition. The effect of radiation can also be seen in the heat equation. Physical parameters of interest, such as Brinkman number, porous media, thermal radiation number, induced slip, suction/injection and shear stress, are examined and graphically depicted. The gas slip velocity is used to simulate the slip flow model of the overall mass transfer on the moving sheet. Models of first- and second-order slip are introduced to induce the total mass transfer on the moving sheet. Additionally, the suction that causes the slip velocity rather than surface movement is investigated. The neighboring gases are forced to flow in the opposite direction of sheet movement by the mass suction-induced slip. As a result, the slip-induced suction and sheet movement cause the solution space to increase. The closed-form exact solutions are achieved for both the stretching and shrinking sheet cases. For stretched sheet instances, there is never a uniform solution. However, depending on the values of the Casson and wall mass transfer parameters, the solution in the case of the shrinking sheet may or may not exist, and if it does, it may or may not be unique or may have a dual nature. The research also demonstrates that stronger mass suction is required for a consistent flow of Casson fluid. The impacts of inverse Darcy number, induced slip parameter, Casson fluid and suction/injection on the flow and heat transfer properties of the fluid are examined under the influence of radiation, and the solution of each profile is shown in the form of figures along with the outcomes of the interface velocity and heat transfer rate at the surface.

A consistent approach to characterize random vibrations of nanobeams

While there is a numerous amount of research on the random vibrations of classical structures in the literature, merely some limited attempts exist to formulate the random vibrations of nano-sized structures. This study deals with the random vibration of elastic beams at the ultra-small scale. The mixture unified gradient theory of elasticity is invoked to appropriately take into consideration the size-effect. Within the introduced augmented elasticity framework, the nanoscopic effects associated with the strain gradient theory and the stress gradient theory are consistently integrated into the classical elasticity theory. The corresponding boundary-value problem of dynamic equilibrium is, accordingly, established. The exact closed-form solution of the space-time correlation function and the mean-square of the transverse displacements of the mixture unified gradient elastic beam with simply supported ends is obtained for the space-time white noise excitation. The spatial variation of the mean-square value of the transverse displacements of the elastic nanobeam in terms of gradient length-scale parameters is graphically illustrated and thoroughly discussed. A consistent approach to characterize random vibrations of elastic nanobeams within the context of the mixture unified gradient elasticity theory is presented providing important insight into the random vibration analysis of nanostructures.