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Investigation of free vibration of porous power and sigmoid-law sandwich functionally graded (FG) plates with different boundary conditions is presented in this paper. The FG sandwich plate includes three layers. The face layers are fabricated of functionally graded material (FGM) and middle layer (core) is isotropic (ceramic). Imperfect sigmoid FG sandwich plates with even and linear-uneven porosities and nonporous core layer are studied. Developed approach has been realized in the framework of a refined theory of the first-order shear deformation theory (FSDT) using variational methods and the R-functions theory. The analytical expressions are obtained for calculating the elastic characteristics with the assumption that the values of Poisson's ratio are the same for constituent FGM materials. For rectangular plates, the obtained results are compared with known results and a good agreement is obtained. Vibration analysis of a complex-shaped porous sandwich plate made of FGM has been performed. The effect of various parameters on the dynamic behavior of the plate, such as the type and values of porosity coefficients, power index, lay-up scheme, types of FGM, has been studied. [ABSTRACT FROM AUTHOR]

Paper is devoted to the effective stationary heat conductivity for the fibre composite materials. We are aimed on getting on analytical expression for effective thermal conductivity coefficient. Asymptotic homogenization approach, based on the multiple scale perturbation method, is used. This allows to reduce the original boundary value problem in multiply connected domain to the sequence of boundary value problems in simply connected domains. These problems include: the local problem for the periodically repeated cell and homogenized problem with effective coefficient. It is shown that for densely packed and high contrast fibre composites, the cell problem can be solved analytically. For this aim, lubrication approach (asymptotics of thin layer) has been employed. We also generalise obtained solution to the case of medium‐sized inclusions in the framework of the Padé approximants. [ABSTRACT FROM AUTHOR]

In this article, with the help of Leibniz integral rule on time scales, we prove some new dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales. These inequalities can be used as handy tools to study the qualitative and quantitative properties of solutions of the initial boundary value problem for partial delay dynamic equation. [ABSTRACT FROM AUTHOR]

Heat transfer at industrial levels has been revolutionized with the advancement of nanofluid and hybrid nanofluid. Keeping this development in view, this article aims to present the rate of heat transfer for conventional and hybrid nanofluids, incorporating the Hall Effect over a stretchable surface. The flow governing equations are obtained with the help of suitable assumptions, and the problem is attempted with the boundary value problem technique in MATLAB. The highly non-linear partial differential equations are transformed into non-dimensional forms using suitable similarity transforms. The criterion of convergence for solution or tolerance of a problem is adjusted to 10−7. Water is considered as a base fluid; copper (Cu) and silver (Ag) nanoparticles are mixed to obtain nanofluid. This novel work is incorporated for conventional and hybrid nanofluid with the effect of Hall current above the stretching/shrinking surface. Increasing the Stefan blowing parameter reduces the flow rate; it increases the heat transfer rate and nano-particle concentration of conventional and hybrid nanofluid. Both velocity components decreases by increasing the magnetic field. The Hall Effect also decreases the velocity of nanofluid. The outcomes are compared to previously published work, demonstrating that the existing study is legitimate. The heat transfer rate of the hybrid nanofluid is higher than the convential nanofluid. This study suggests more frequent use of hybrid nanofluid because of high heat transfer rates and reduced skin friction. [ABSTRACT FROM AUTHOR]

A parametric pendulum excited by a discrete wave-modulated step function of length is subjected to a mathematical analysis and numerical modeling. We observe an existence of almost periodic solutions of ordinary differential equations with linear boundary value conditions. An exemplary oscillator subject to both an almost periodic step elongation and forcing synchronizes with the forcing, tending to almost periodic motions like stable limit cycles. Conditions for that synchronization as well as trajectories of numerical solutions on time history plots and phase planes are shown to confirm correctness of the analytical derivations and dedicated numerical modeling. [ABSTRACT FROM AUTHOR]

In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized. Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge–Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement. [ABSTRACT FROM AUTHOR]

A novel numerical/analytical approach to study geometrically nonlinear vibrations of shells with variable thickness of layers is proposed. It enables investigation of shallow shells with complex forms and different boundary conditions. The proposed method combines application of the R-functions theory, variational Ritz’s method, as well as hybrid Bubnov–Galerkin method and the fourth-order Runge–Kutta method. Mainly two approaches, classical and first-order shear deformation theories of shells are used. An original scheme of discretization regarding time reduces the initial problem to the solution of a sequence of linear problems including those related to linear vibrations with a special type of elasticity, as well as problems governed by non-linear system of ordinary differential equations. The proposed method is validated by the investigation of test problems for shallow shells with rectangular planform and applied to new vibration problems for shallow shells with complex planforms and variable thickness of layers. [ABSTRACT FROM AUTHOR]

A modified three-phase composite model yielding reliable effective characteristics of composite structures has been proposed. In particular, the problem of effective heat transfer coefficient of the composite structure with periodically located inclusions of circular cross-sections located on a square net is solved. Advantages of the proposed model in comparison to the classical three-phase model are illustrated and discussed. [ABSTRACT FROM AUTHOR]

This paper focuses on the possibilities of controlling the dispersion spectra and wave characteristics of cylindrical waveguides by changing their geometry and electro-elastic properties. We consider cylinders with classical circular and hollow cross-sections, and waveguides that have sector cut of arbitrary angular measure in the cross-section. Numerical results are presented for the cylinders of all studied types with different boundary conditions. It is shown that the required wave characteristics can be obtained by a variation of the cross-section geometry of the waveguides. [ABSTRACT FROM AUTHOR]

In this paper, a new approach to find exact solutions is carried out for a generalized unsteady magnetohydrodynamic transport of a rate-type fluid near an unbounded upright plate, which is analyzed for ramped-wall temperature and velocity with constant concentration. The vertical plate is suspended in a porous medium and encounters the effects of radiation. An innovative definition of the time-fractional operator in power-law-kernel form is implemented to hypothesize the constitutive mass, energy, and momentum equations. The Laplace integral transformation technique is applied on a dimensionless form of governing partial differential equations by introducing some non-dimensional suitable parameters to establish the exact expressions in terms of special functions for ramped velocity, temperature, and constant-concentration fields. In order to validate the problem, the absence of the mass Grashof parameter led to the investigated solutions obtaining good agreement in existing literature. Additionally, several system parameters were used, such as as magnetic value M, Prandtl value P r , Maxwell parameter λ , dimensionless time τ , Schmidt number " S c ", fractional parameter α , andMass and Thermal Grashof numbers G m and G r , respectively, to examine their impacts on velocity, wall temperature, and constant concentration. Results are also discussed in detail and demonstrated graphically via Mathcad-15 software. A comprehensive comparative study between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently. [ABSTRACT FROM AUTHOR]

A boundary value problem associated with filtering governed by differential equations with time dependent coefficients and exhibiting a weak nonlinearity is solved. Namely, the boundary value problem is split into two independent boundary problems, i.e., that of Goursat for the concentration of a sorbate and that absorbed by a sorbent. Then each of the formulated problems is solved separately applying the Riemann transformation. [ABSTRACT FROM AUTHOR]

• Simplified beam equations with inertial terms and nonlinear curvatures are derived. • They can be treated as the first approximation to the Kirchhoff nonlinear equations. • Simplified equations versus boundary conditions are illustrated and discussed. • Nonlinear modes of beam vibrations are constructed for various boundary conditions. This paper is devoted to the construction of asymptotically correct simplified models of nonlinear beam equations for various boundary conditions. V.V. Bolotin mentioned that in some cases (e.g., if compressed load is near the buckling value), the so-called "nonlinear inertia" must be taken into account. The effect of nonlinear inertia on the oscillations of the clamped-free beam is investigated in many papers. Bolotin used some physical assumption and did not compare the order of nonlinear terms in original equations. Below we propose our method for deriving those, which we will named "Bolotin's equations". This approach is based on fractional analysis of original boundary value problems. [ABSTRACT FROM AUTHOR]