Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading

This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.     

Analytical solutions for tapering quadratic and cubic rat-holes in highly frictional granular solids

New exact analytical solutions are presented for both stress and velocity fields for a Coulomb–Mohr granular solid assuming non-dilatant double-shearing theory. The solutions determined apply to highly frictional materials for which the angle of internal friction φ is assumed equal to 90°. This major assumption is made primarily to facilitate exact analytical solutions, and it is discussed at length in the Introduction, both in the context of real materials which exhibit large angles of internal friction, and in the context of using the solutions derived here as the leading term in a regular perturbation solution involving powers of 1−sinφ. The analytical velocity fields so obtained are illustrated graphically by showing the direction of the principal stress as compared to the streamlines. The stress solutions are also exploited to determine the static stress distribution for a granular material contained within vertical boundaries and a horizontal base, which is assumed to have an infinitesimal central outlet through which material flows until a rat-hole of parabolic or cubic profile is obtained, and no further flow takes place. A rat-hole is a stable structure that may form in storage hoppers and stock-piles, preventing any further flow of material. Here we consider the important problems of two-dimensional parabolic rat-holes of profile y=ax², and three-dimensional cubic rat-holes of profile z=ar³, which are both physically realistic in practice. Analytical solutions are presented for both two and three-dimensional rat-holes for the case of a highly frictional granular solid, which is stored at rest between vertical walls and a horizontal rigid plane, and which has an infinitesimal central outlet. These solutions are bona fide exact solutions of the governing equations for a Coulomb–Mohr granular solid, and satisfy exactly the free surface condition along the rat-hole surface, but approximate frictional conditions along the containing boundaries. The analytical solutions presented here constitute the only known solutions for any realistic rat-hole geometry, other than the classical solution which applies to a perfectly vertical cylindrical cavity.     

A semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater

Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation exchanging pollutants in groundwater is highly nonlinear due to the complex nonlinearities of exchange isotherms. This makes difficult to derive analytical solutions for transport equations. Available analytical solutions are valid only for binary cation exchange transport in 1-D and often disregard dispersion. Here we present a semi-analytical solution for linearized multication exchange reactive transport in steady 1-, 2- or 3-D groundwater flow. Nonlinear cation exchange mass–action–law equations are first linearized by means of a first-order Taylor expansion of log-concentrations around some selected reference concentrations and then substituted into transport equations. The resulting set of coupled partial differential equations (PDEs) are decoupled by means of a matrix similarity transformation which is applied also to boundary and initial concentrations. Uncoupled PDE’s are solved by standard analytical solutions. Concentrations of the original problem are obtained by back-transforming the solution of uncoupled PDEs. The semi-analytical solution compares well with nonlinear numerical solutions computed with a reactive transport code (CORE^2D) for several 1-D test cases involving two and three cations having moderate retardation factors. Deviations of the semi-analytical solution from numerical solutions increase with increasing cation exchange capacity (CEC), but do not depend on Peclet number. The semi-analytical solution captures the fronts of ternary systems in an approximate manner and tends to oversmooth sharp fronts for large retardation factors. The semi-analytical solution performs better with reference concentrations equal to the arithmetic average of boundary and initial concentrations than it does with reference concentrations derived from the arithmetic average of log-concentrations of boundary and initial waters.     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

Three dimensional analytical solution for finite circular cylinders subjected to indirect tensile test

This paper derives a new three-dimensional (3-D) analytical solution for the indirect tensile tests standardized by ISRM (International Society for Rock Mechanics) for testing rocks, and by ASTM (American Society for Testing and Materials) for testing concretes. The present solution for solid circular cylinders of finite length can be considered as a 3-D counterpart of the classical two dimensional (2-D) solutions by Hertz in 1883 and by Hondros in 1959. The contacts between the two steel diametral loading platens and the curved surfaces of a cylindrical specimen of length H and diameter D are modeled as circular-to-circular Hertz contact and straight-to-circular Hertz contact for ISRM and ASTM standards respectively. The equilibrium equations of the linear elastic circular cylinder of finite length are first uncoupled by using displacement functions, which are then expressed in infinite series of some combinations of Bessel functions, hyperbolic functions, and trigonometric functions. The applied tractions are expanded in Fourier–Bessel series and boundary conditions are used to yield a system of simultaneous equations. For typical rock cylinders of 54 mm diameter subjected to ISRM indirect tensile tests, the contact width is in the order of 2 mm (or a contact angle of 4°) whereas for typical asphalt cylinders of 101.6 mm diameter subjected to ASTM indirect tensile tests the contact width is about 10 mm (or a contact angle of 12°). For such contact conditions, 50 terms in both Fourier and Fourier–Bessel series expansions are found sufficient in yielding converged solutions. The maximum hoop stress is always observed within the central portion on a circular section close to the flat end surfaces. The difference in the maximum hoop stress between the 2-D Hondros solution and the present 3-D solution increases with the aspect ratio H/D as well as Poisson’s ratio ν. When contact friction is neglected, the effect of loading platen stiffness on tensile stress in cylinders is found negligible. For the aspect ratio of H/D = 0.5 recommended by ISRM and ASTM, the error in tensile strength may be up to 15% for both typical rocks and asphalts, whereas for longer cylinders with H/D up to 2 the error ranges from 15% for highly compressible materials, and to 60% for nearly incompressible materials. The difference in compressive radial stress between the 2-D Hertz solution or 2-D Hondros solution and the present 3-D solution also increases with Poisson’s ratio and aspect ratio H/D. In summary, the 2-D solution, in general, underestimates the maximum tensile stress and cannot predict the location of the maximum hoop stress which typically locates close to the end surfaces of the cylinder.     

Poromechanical response of a finite elastic cylinder under axisymmetric loading

Drained or undrained cylindrical specimens under axisymmetric loading are commonly used in laboratory testing of soils and rocks. Poroelastic cylindrical elements are also encountered in applications related to bioengineering and advanced materials. This paper presents an analytical solution for an axisymmetrically-loaded solid poroelastic cylinder of finite length with permeable (drained) or impermeable (undrained) hydraulic boundary conditions. The general solutions are derived by first applying Laplace transforms with respect to the time and then solving the resulting governing equations in terms of Fourier–Bessel series, which involve trigonometric and hyperbolic functions with respect to the z-coordinate and Bessel functions with respect to the r-coordinate. Several time-dependent boundary-value problems are solved to demonstrate the application of the general solution to practical situations. Accuracy of the numerical solution is confirmed by comparing with the existing solutions for the limiting cases of a finite elastic cylinder and a poroelastic cylinder under plane strain conditions. Selected numerical results are presented for different cylinder aspect ratios, loading and hydraulic boundary conditions to demonstrate the key features of the coupled poroelastic response.     

Axisymmetric membrane in adhesive contact with rigid substrates: Analytical solutions under large deformation

The large deformation of an elastic axisymmetric membrane in adhesive contact with a rigid flat punch is studied. Detachment of membrane is analyzed using a critical energy release rate criterion. Two types of incompressible hyperelastic material models are considered: neo-Hookean and a class of materials whose elastic energy density functions are independent of the trace of the Cauchy–Green tensor (I₂-based material). We also include pre-stretch in our formulation and study the stability of detachment process. Closed form analytical solutions for the membrane stresses, deformed profiles and energy release rate are obtained in the regime of large longitudinal stretch. For the I₂-based material, we discover an interesting “pinching” instability where the contact angle suddenly increases in a displacement controlled test. The region of validity of our analytical solutions is determined by comparing them with numerical solutions of the governing equations. We found that the accuracy of our solution improves with pre-stretch; for pre-stretch ratios greater than 1.3, our analytical solution also works well in the small deformation regime.     

A reduction method to quasilinear hyperbolic systems of multicomponent field PDEs with application to wave interaction

A reduction method is worked out for determining a class of exact solutions with inherent wave features to quasilinear hyperbolic homogeneous systems of N>2 first-order autonomous PDEs. A crucial point of the present approach is that in the process the original set of field equations induces the hyperbolicity of an auxiliary 2×2 subsystem and connection between the respective characteristic velocities can be established. The integration of this auxiliary subsystem via the hodograph method and through the use of the Riemann invariants provides the searched solutions to the full governing system. These solutions also represent invariant solutions associated with groups of translation of space/time coordinates and involving arbitrary functions that can be used for studying non-linear wave interaction. Within such a theoretical framework the two-dimensional motion of an adiabatic fluid is considered. For appropriate model pressure-entropy-density laws, we determine a solution to the governing system of equations which describes in the 2+1 space two non-linear waves which were initiated as plane waves, interact strongly on colliding but emerge with unaffected profile from the interaction region. These model material laws include the classical pressure-entropy-density law which is usually adopted for a polytropic fluid.     

A Geometrical Approach to the Study¶of Unbounded Solutions¶of Quasilinear Parabolic Equations

In this article, we are interested in the existence and uniqueness of solutions for quasilinear parabolic equations set in the whole space ? ^N . We consider, in particular, cases when there is no restriction on the growth or the behavior of these solutions at infinity. Our model equation is the mean-curvature equation for graphs for which Ecker and Huisken have shown the existence of smooth solutions for any locally Lipschitz continuous initial data. We use a geometrical approach which consists in seeing the evolution of the graph of a solution as a geometric motion which is then studied by the so-called “level-set approach”. After determining the right class of quasilinear parabolic PDEs which can be taken into account by this approach, we show how the uniqueness for the original PDE is related to “fattening phenomena” in the level-set approach. Existence of solutions is proved using a local L ^∞ bound obtained by using in an essential way the level-set approach. Finally we apply these results to convex initial data and prove existence and comparison results in full generality, i.e., without restriction on their growth at infinity.     

Theoretical analysis of a class of mixed,C0 continuity formulations for general dipolar Gradient Elasticity boundary value problems

Mixed weak formulations, with two or three main (tensor) variables, are stated and theoretically analyzed for general multi-dimensional dipolar Gradient Elasticity (biharmonic) boundary value problems. The general structure of constitutive equations is considered (with and without coupling terms). The mixed formulations are based on various generalizations of the so-called Ciarlet–Raviart technique. Hence, C⁰ continuity conforming basis functions may be employed in the finite element approximations (or even, C⁻¹ basis functions for the Cauchy stress variable). All the complicated boundary conditions, especially in the multi-dimensional scenario, are naturally considered. The main variables are the displacement vector, the double stress tensor and the Cauchy stress tensor. The latter variable may be eliminated in some of the formulations, depending on the structure of the constitutive equations. The standard continuous and discrete Babuška–Brezzi inf–sup conditions for the constraint equation, as well as, solution uniqueness for both the continuous statements and discrete approximations, are established in all cases. For the purpose of completeness, two one-dimensional mixed formulations are also analyzed. The respective constitutive equations possess general structure (with coupling terms). For the 1-D formulations, all the inf–sup conditions are satisfied, for both the continuous and discrete statements (assuming proper selection of the polynomial spaces for the main variables). Hence, the general Babuška–Brezzi theory results in quasi-optimality and stability. For multi-dimensional problems, the difficulty of deducing the inf–sup condition on the kernel is examined. Certain aspects of methodologies employed to theoretically by-pass this problem, are also discussed.     

The state vector methods for space axisymmetric problems in multilayered piezoelectric media

The state vector equations for space axisymmetric problems of transversely isotropic piezoelectric media are established from the basic equations. Using the Hankel transform, the state vector equations are reduced to a system of ordinary differential equations. An analytical solution of the problems in the Hankel transform space is presented in the form of the product of initial state vector and transfer matrix. The transfer matrices are given for the three distinct eigenvalues. Applications of the solutions are discussed. An analytical solution for the transversely isotropic semi-infinite piezoelectric media subjected to concerted point loads on the surface z=0 is presented in the Hankel transform space. Using transfer matrix and the continuity conditions at the layer interfaces, the general solution formulation of N-layered transversely isotropic piezoelectric media is given. A selected set of numerical solutions is presented for a layered semi-infinite piezoelectric solid.     

Anisotropic plastic stress fields at a slowly propagating crack tip

Under the condition that any perfectly plastic stress components at a crack tip are nothing but the functions of 0 only making use of equilibrium equations. Hill anisotropic yield condition and unloading stress-strain relations, in this paper, we derive the general analytical expressions of anisotropic plastic stress fields at the slowly steady propagating tips of plane and anti-plane strain. Applying these general analytical expressions to the concrete cracks, the analytical expressions of anisotropic plastic stress fields at the-slowly steady propagating tips of Mode I and Mode III cracks are obtained. For the isotropic plastic material, the anisotropic plastic stress fields at a slowly propagating crack tip become the perfectly plastic stress fields.     

On solutions of the problem in stresses with the use of Maxwell stress functions

The spatial problems of elasticity are mainly solved in displacements [1, 2], i.e., the Lamé equations are taken as the initial equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses. In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary, special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation method.     

Perfectly plastic stress field at a rapidly propagating plane-stress crack tip

Under the condition that all the perfectly plastic stress components at a crack tip are the functions of only, making use of the Treasca yield condition, steady-state moving equations and elastic perfectly-plastic constitutive equations, we derive the generally analytical expressions of perfectly palstic stress field at a rapidly propagating plane-stress crack tip. Applying these generally analytical expressions to the concrete crack, we obtain the analytical expressions of perfectly plastic stress field at the rapidly propagating tips of models I and II plane-stress cracks.     

基于神经网络的偏微分方程求解方法研究综述

神经网络作为一种强大的信息处理工具在计算机视觉,生物医学,油气工程领域得到广泛应用,引发多领域技术变革.深度学习网络具有非常强的学习能力,不仅能发现物理规律,还能求解偏微分方程.近年来基于深度学习的偏微分方程求解已是研究新热点.遵循于传统偏微分方程解析解、偏微分方程数值解术语,本文称用神经网络进行偏微分方程求解的方法为...     

The solution of viscous incompressible jet flows using non‐staggered boundary fitted co‐ordinate methods

A new approach for the solution of the steady incompressible Navier–Stokes equations in a domain bounded in part by a free surface is presented. The procedure is based on the finite difference technique, with the non‐staggered grid fractional step method used to solve the flow equations written in terms of primitive variables. The physical domain is transformed to a rectangle by means of a numerical mapping technique. In order to design an effective free solution scheme, we distinguish between flows dominated by surface tension and those dominated by inertia and viscosity. When the surface tension effect is insignificant we used the kinematic condition to update the surface; whereas, in the opposite case, we used the normal stress condition to obtain the free surface boundary. Results obtained with the improved boundary conditions for a plane Newtonian jet are found to compare well with the available two‐dimensional numerical solutions for Reynolds numbers, up to Re=100, and Capillary numbers in the range of 0≤Ca<1000. Copyright © 2001 John Wiley & Sons, Ltd.     

Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders

The velocity field and the adequate shear stress corresponding to the longitudinal flow of a fractional second grade fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. Initially the fluid is at rest, and at time t = 0⁺, the inner cylinder suddenly begins to translate along the common axis with constant acceleration. The solutions that have been obtained are presented in terms of generalized G functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary second grade and Newtonian fluids are obtained as limiting cases of the general solutions. Finally, some characteristics of the motion, as well as the influences of the material and fractional parameters on the fluid motion and a comparison between models, are underlined by graphical illustrations.     

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress inten- sity factors （DSIFs） for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractional wave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation （ODE）. The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional dif- ferential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.     

Hydrostatic stress-dependent perfectly-plastic stress fields at a stationary plane-stress crack-tip

Under the condition that all the stress components at a crack-tip are the functions of θ only, making use of equilibrium equations and hydrostatic stress-dependent yield condition, in this paper, we derive the generally analytical expressions of the hydrostatic stress-dependent perfectly-plastic stress fields at a stationary plane-stress crack-tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of hydrostatic stress-dependent perfectly-plastic stress fields at the tips of mode Ⅰ and mode Ⅱ cracks are obtained.     

Three-dimensional solutions for general anisotropy

The Stroh formalism is extended to provide a new class of three-dimensional solutions for the generally anisotropic elastic material that have polynomial dependence on x₃, but which have quite general form in x₁,x₂. The solutions are obtained by a sequence of partial integrations with respect to x₃, starting from Stroh's two-dimensional solution. At each stage, certain special functions have to be introduced in order to satisfy the equilibrium equation. The method provides a general analytical technique for the solution of the problem of the prismatic bar with tractions or displacements prescribed on its lateral surfaces. It also provides a particularly efficient solution for three-dimensional boundary-value problems for the half-space. The method is illustrated by the example of a half-space loaded by a linearly varying line force.