Three-Dimensional Analytical Solution for an Axisymmetric Biharmonic Problem

Sburlati, Roberta

doi:10.1007/s10659-009-9195-3

Cited by 24 publications

(16 citation statements)

References 15 publications

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“…Their highly oscillatory behaviour about the mantle rapidly disappears moving towards the centre (Sburlati 2009). The radial stress at the centre is the only curve made nondimensional by dividing it by the Young's modulus gradation (3.3); it is interesting that such a normalisation leads to an almost perfectly linear curve.…”

Section: Numerical Results -Final Solutionmentioning

confidence: 99%

Three-dimensional elastic solutions for functionally graded circular plates

Sburlati¹,

Bardella²

2011

European Journal of Mechanics - A/Solids

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Three-dimensional elastic solutions are obtained for a functionally graded thick circular plate subject to axisymmetric conditions. We consider a isotropic material where the Young modulus depends exponentially on the position along the thickness, while the Poisson ratio is constant. The solution method utilises a Plevako's representation form which reduces the problem to the construction of a potential function satisfying a linear fourth-order partial differential equation. We write this potential function in terms of Bessel functions and we pointwise assign mixed boundary conditions. The analytic solution is obtained in a general form and explicitly presented by assuming transversal load on the upper face and zero displacements on the mantle; this is done by superposing the solutions of problems with suitably imposed radial displacement. We validate the solution by means of a finite element approach; in this way, we highlight the effects of the material inhomogeneity and the limits of the employed numerical method near the mantle, where the solution shows a large sensitivity to the boundary conditions.

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Section: Numerical Results -Final Solutionmentioning

confidence: 99%

Three-dimensional elastic solutions for functionally graded circular plates

Sburlati¹,

Bardella²

2011

European Journal of Mechanics - A/Solids

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“…Despite significant advances in the development of the methods aimed at the numerical analysis of the problems of mechanics and a wide choice of certified highly efficient software environments intended for the solution of these problems with the help of the finite-element method [12], the interest in analytic methods still remains fairly high. This can be confirmed, in particular, by the appearance of new scientific publications (see, e.g., [10,11,[13][14][15]) and explained by several factors. One of these factors is connected with the fact that the analytic solution of the direct problem enables one to significantly simplify the process of solution of the corresponding inverse problems.…”

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confidence: 81%

Variational Method of Homogeneous Solutions in Axisymmetric Elasticity Problems for a Semiinfinite Cylinder

Чекурін

Постолакі

2014

J Math Sci

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We develop a variational method of homogeneous solutions for the solution of axisymmetric elasticity problems for a semiinfinite cylinder with free lateral surface. We consider four types of boundary conditions imposed on the end of cylinder, namely, the conditions in stresses, in displacements, and two types of mixed conditions. The solution of the problems with the help of this method is reduced to the solution of infinite systems of linear algebraic equations. We perform the numerical analysis of convergence of the obtained solutions. We also consider an example of application of the proposed approach to the determination of stress concentration near the joint of the end of the cylinder with a perfectly rigid lateral surface.The problems of determination of the stress-strain state of cylindrical bodies appear in various scientific disciplines and, in particular, in the applied theories of the strength of solids [9]. Despite significant advances in the development of the methods aimed at the numerical analysis of the problems of mechanics and a wide choice of certified highly efficient software environments intended for the solution of these problems with the help of the finite-element method [12], the interest in analytic methods still remains fairly high. This can be confirmed, in particular, by the appearance of new scientific publications (see, e.g., [10,11,[13][14][15]) and explained by several factors. One of these factors is connected with the fact that the analytic solution of the direct problem enables one to significantly simplify the process of solution of the corresponding inverse problems.We especially mention the following two well-known approaches to the analytic solution of axisymmetric elasticity problems for a circular cylinder: the method of superposition [2, 10] and the method of homogeneous solutions [3,5].The first method is based on the representation of solutions in the form of a sum of two components: one of which guarantees the validity of boundary conditions on the end face and the other component satisfies the conditions imposed on the lateral surface. Each of these components is represented as an expansion in a complete systems of function. Thus, the systems of hyperbolic-cylindrical and trigonometric-cylindrical functions are used for this purpose in the monograph [2]. The coefficients of these expansions are determined by subordinating the solution to the boundary conditions imposed on the end face of the cylinder and on its lateral surface. As a result, an infinite system of algebraic equations is obtained, which enables one to express the coefficients of expansions via the Fourier and Fourier-Bessel coefficients of the functions appearing in the boundary conditions.The second method is based on the representation of required solutions in the form of expansions in complete systems of functions (i.e., in the so-called homogeneous solutions). The role of systems of homogeneous solutions is played by the eigenfunctions of homogeneous axisymmetric problems of the theory of elasticity...

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“…We observe that this solution technique carries out specific data on the boundary conditions of the lateral surface of the plate, as shown in details in [11]. Indeed, the boundary condition on the lateral surface (r=b) requires only transversal displacement zero for all z (see (2)) that corresponds to the usual condition in clamped or simply supported plate in structural theories (for z=0); this condition is satisfied by considering (11) and (9). The condition on zero rotation, for clamped plate, or zero moment, for simply supported plate, are not automatically verified by expansions (11).…”

Section: Solution Methodsmentioning

confidence: 99%

Three-dimensional Analyses of Functionally Graded Multi-layered Systems

Sburlati

2014

Procedia Engineering

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Functionally Graded Materials (FGMs) are composites with continuously varying volume fractions of their constituent homogeneous phases. When the characteristic length scale of such variations is much larger than the size of the phases, the model of the inhomogeneous solids is able to describe the macroscopic behaviour of these materials by means of homogenization techniques. By considering materials in which this assumption is possible, some elastic solutions for graded inhomogeneous materials have been obtained in the literature in order to tailor the material parameters for specific requirements. The main purpose of the paper is devoted to analyse the effects of FGM elements in sandwich structures, where the mismatch between the properties of the core and the face-sheets can produce interface damage especially when the structures are subjected to extreme conditions. Two different cases concerning the bending of a thick circular sandwich plate are investigated in the framework of the linear elasticity theory: (a) sandwich plates with homogeneous face sheets and FGM-core and (b) sandwich plates with FGM face sheets and homogeneous core. The related elastic solutions are obtained in the framework of the elasticity theory and explicit solutions are discussed by considering only the first contribution of the series in order; in such a way, we have a solution in closed-form and no convergence effects on the solutions. Three different aspects are studied in detail in terms of the geometric and material parameter: the semi-inverse solution method adopted with the consequence on boundary conditions, the non-linear behaviour of the radial displacement and, the shear and circumferential stress behaviour in the thickness of the plate. A comparative study between sandwich with FGM elements and conventional multi-layered systems is performed in order to highlight the effects of the inhomogeneity.

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Three-Dimensional Analytical Solution for an Axisymmetric Biharmonic Problem

Cited by 24 publications

References 15 publications

Three-dimensional elastic solutions for functionally graded circular plates

Three-dimensional elastic solutions for functionally graded circular plates

Variational Method of Homogeneous Solutions in Axisymmetric Elasticity Problems for a Semiinfinite Cylinder

Three-dimensional Analyses of Functionally Graded Multi-layered Systems

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