Biharmonic Functions Research Articles

Three-Dimensional Analytical Solution for an Axisymmetric Biharmonic Problem

In this paper, we propose a method for the solution of the axisymmetric boundary value problem for a finite elastic cylinder with assigned stress and/or displacements acting on the ends and side. The technique utilizes the Love representation, which allows for reduction of the solution of the elastic problem to the search for a biharmonic function on a cylindrical domain. In the solution method suggested here, we write the Love function with a Bessel expansion and analyze in detail the conditions under which it is possible to differentiate the expansion term by term. We show that this is possible only for a restricted class of elastic solutions. In the general case, we introduce two new auxiliary functions of the z-coordinate. In this way, we obtain the general form of the axisymmetric biharmonic function, which is discussed in relation to certain specific boundary conditions applied on the side and ends of the cylinder. We obtain an exact explicit solution of practical interest for a cylinder with free ends and assigned displacements applied to the side.

Boundedness of the Hessian of a Biharmonic Function in a Convex Domain

We consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions.

Isolated singularities, growth of spherical means and Riesz decomposition for superbiharmonic functions

We consider Riesz decomposition theorem for superbiharmonic functions in the punctured ball. In fact, we show that under certain growth condition on surface integrals, superbiharmonic functions are represented as a sum of potentials and biharmonic functions.

Image Treatment of a 2D Vapor-Liquid Compound Droplet in a Linearized Steady Viscous Flow

The classical method of images is used to construct closed form exact solutions for the two-dimensional (2D) perturbed flow fields in the presence of a 2D vapor-liquid compound droplet in the limit of low-Reynolds number. The geometry of the multiphase droplet is composed of two overlapping infinitely long cylinders Ca and Cb of radii a and b, respectively, intersecting at a vertex angle π∕2. The composite inclusion has the shape resembling a 2D snowman type of object with a vapor cylinder Ca partly protruded into the cylinder Cb filled with another fluid whose viscosity is different from that of the host fluid. The mathematical problem with this inclusion in the Stokes flow environment is formulated in terms of Stokes stream function with mixed boundary conditions at the boundary of the hybrid droplet. General expressions for the perturbed stream functions in the two phases are obtained in a straightforward fashion using Kelvin’s inversion together with shift and reflection properties of biharmonic functions. Application of our method to other related problems in creeping flow and possible further generalizations are also discussed. The general results are then exploited to derive singularity solutions for the hybrid droplet embedded in (i) a centered shear flow, (ii) a quadratic potential flow, and (iii) an extensional flow past the 2D vapor-liquid compound droplet. The image singularities in each case depend on the two radii of the cylinders, the center-to-center distance, and the viscosity ratio. The exact solutions are utilized to plot the flow streamlines and they show some interesting patterns. While the flow fields exterior to the droplet exhibit symmetrical topological structures, the interior flow fields show existence of free eddies—enclosed in a figure-eight separatrix—and stagnation points (hyperbolic points). The flow characteristics are influenced by the viscosity and radii ratios. Furthermore, the asymptotic analysis leads to a rather surprising conclusion that there is a (subdominant) uniform flow far away from the droplet in all cases. The existence of an origin, the natural center of the drop of the composite geometry, which neutralizes the uniform flow for a particular choice of the physical parameters, is illustrated. This reveals the sensitivity of the geometry in 2D Stokes flow. The present results may be of some interest in models involving a combination of stick and slip boundaries. Moreover, the method discussed here can be useful both as a teaching tool and as a building block for further calculations.

Laguerre minimal surfaces, isotropic geometry and linear elasticity

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy \(\int (H^2-K)/K d\!A\). They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of \(\int (H^2-K)d\!A\), which are known to be an entity of Mobius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.

Operator method for solving the plane elasticity problem for a strip with periodic cuts

In the present paper, we use the conformal mapping z/c = ζ−2a sin ζ (a, c−const, ζ = u + iv) of the strip {|v| ≤ v0, |u| < ∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F = Cψ0 + Sψ*0 + x(Cψ1 − Sψ2) + y(Cψ2 + Sψ1), where C(v) and S(v) are the operator functions described in [1] and ψ0(u), …, ψ2(u) are the desired functions. The boundary conditions for the function F posed for v = ±v0 are equivalent to two operator equations for ψ1(u) and ψ2(u) and to two ordinary differential equations of first order for ψ0(u) and ψ*0(u) [2]. By finding the functions ψj(u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a = 1/4, v0 = 1) loaded on the boundary v = v0 by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X∞ and to the transverse and pure bending of the strip due to the transverse force Y∞ and the constant moment M∞, respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x = 0 and study the stress concentration effect near the cut bottom.

Axis-Symmetrical Solutions for n-Plies Functionally Graded Material Cylinders under Strain No-Decaying Conditions

The proposal of the present work is to furnish a general approach to construct exact elastic solutions for FGM cylinders, made of a central core and n arbitrary cylindrical hollow homogeneous and isotropic phases. The hypothesis of axis-symmetrical boundary conditions is here assumed in order to analyze a class of elastic problems which present no-decaying of selected mechanical quantities and in particular of the axial strains ϵ 33 in the radial direction, being x 3 the axis of the laminated cylinder. To construct a robust mathematical procedure for obtaining exact elastic solutions for axis-symmetrical n-plies-Functionally Graded Material Cylinders (n-FGMCs), a theorem is first given for qualifying the space of the solutions and then their mathematical form is identified, when the object exhibits no-decaying of the axial strain. By starting from the classical Boussinesq-Somigliana-Galerkin vector and specializing it to torsionless composite cylinders characterized by no-decaying of the axial strain, a special form of the bi-harmonic Love's function χ (i) (r, x3) is finally obtained. It is then demonstrated that the differential boundary value problem (BVP) always can be translated in an equivalent linear algebraic one, first solving an in cascade one-dimensional Euler-like differential system (field equations) and then writing the boundary conditions by means an algebraic system ruled by a (6n + 4)-order square matrix P. At the end, constructive and existence theorems are formulated and proved, showing examples in comparison with literature data.

Reconstruction of polyharmonic functions from samples

We study planar complex-valued functions that satisfy a certain Wirtinger differential equation of order k . Our considerations include entire functions ( k = 1 ), harmonic functions ( k = 2 ), biharmonic functions ( k = 4 ), and polyharmonic functions ( k even) in general. Under the assumption of restricted exponential growth and square integrability along the real axis, we establish a sampling theorem that extends the classical sampling theorem of Whittaker–Kotel’nikov–Shannon and reduces to the latter when k = 1 . Intermediate steps, which may be of independent interest, are representation theorems, uniqueness theorems, and the construction of fundamental functions for interpolation. We also consider supplements, variants, generalizations, and an algorithm.

The decomposed form of magnetoelastic beams with free faces

The decomposed form of an isotropic elastic plate is extended to a bending magnetoelastic beam with free faces, and the corresponding decomposition theorem is presented. It is shown that the stress state of the magnetoelastic beam without transverse surface loadings can be decomposed into three parts: the interior state, the Papkovich-Fadle state (in shortened form the P-F state) and the magnetic state. Based on the three new lemmata, a rigorous mathematical proof of the decomposition theorem is given concisely and directly, which is independent of the Papkovich-Fadle eigenfunction expansion of biharmonic functions. In the proof course, some basic mathematical methods are used only, so the proof is more convenient for being understood. More importantly, these three states correspond to the three equations derived in the refined theory of magnetoelastic beams: the fourth-order equation, the transcendental equation and the magnetic equation, respectively. Therefore, this work virtually proves the consistency between the decomposition theorem and the refined theory of bending magnetoelastic beams.

Thermal Stresses in an Elastic Circular Cylinder Containing an Oblate Spheroidal Cavity or a Penny-shaped Crack Subjected to a Prescribed Temperature Distribution

This paper contains an exact solution for the state of stress in an elastic circular cylinder with an oblate spheroidal cavity or an internal penny-shaped crack, the surface of which is subjected to a prescribed temperature distribution. The solution is deduced. with the aid of thermoelastic displacement potentials and Dougall's displacement potentials. Two sets of harmonic functions and two sets of biharmonic functions are given by simple expressions referring to cylindrical and oblate spheroidal coordinates. The boundary conditions on the surfaces of the cylinder and the cavity are satisfied by using the relations between the cylindrical and oblate spheroidal harmonics and biharmonics. The solution for an internal penny-shaped crack is also given by regarding as a limited case of an oblate spheroidal cavity. Numerical results for stress distributions, stress concentration factors and stress intensity factors are shown graphically.

On estimates of biharmonic functions on Lipschitz and convex domains

Using Maz ’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in ℝn. Forn ≥ 8, combinedwitharesultin[18], these estimates lead to the solvability of the Lp Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow us to show that the Lp Dirichlet problem is uniquely solvable for any 2 − e < p < ∞ and n ≥ 4.

On some properties of solutions of the biharmonic equation

In the present paper, the properties of the linear complex operator L ( f ) = zf z - z ¯ f z ¯ , which is defined on the class of complex-valued C 1 functions in the plane, are investigated. It is shown that harmonicity and biharmonicity are invariant under the linear operator L. Results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered. The operator L can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way.

Analytic solutions for a uniformly loaded circular plate with clamped edges

A bi-harmonic potential function was constructed in this study. Love solution was employed to obtain analytical solutions of uniformly loaded plates with two different types of clamped edges. The treatment of clamped boundary conditions was the same as the adopted by Timoshenko and Goodier (1970). The analytical solution for the first type of clamped boundary condition is identical with that obtained by Luoet al. (2004), and the solutions for both types were compared with the FEM results and the calculations of thin plate theory.

A representation formula for radially weighted biharmonic functions in the unit disc

Let $w\colon \mathbb{D}\to(0,\infty)$ be a radial continuous weight function. We consider the weighted biharmonic equation \Delta w^{-1}\Delta u=0\quad\text{in } \mathbb{D} with Dirichlet boundary conditions $u = f_0$ and $\partial_n u = f_1$ on $\mathbb{T} = \partial\mathbb{D}$. Under some extra conditions on the weight function $w$, we establish existence and uniqueness of a distributional solution $u$ of this biharmonic Dirichlet problem. Furthermore, we give a representation formula for the solution $u$. The key to our analysis is a series representation of Almansi type.

Théorème de Liouville et propriété de la moyenne biharmonique restreinte

We prove, under some conditions, that a bounded Lebesgue measurable function satisfying the restricted mean value for the biharmonic functions in R n , n ≠ 2 , or in an open set of R 2 with polar complement, is constant. To cite this article: M. El Kadiri, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Existence of boundary values for biharmonic functions

Criteria for the existence of an -limit and a weak -limit of a biharmonic function at the regular analytic boundary of a bounded two-dimensional domain are established.

On axisymmetric creeping flows involving a hybrid droplet

The Stokes flow involving a hybrid droplet submerged in an immiscible liquid is considered. The droplet has a two-sphere geometry with the two spherical surfaces intersecting orthogonally. The types of hybrid droplet considered here include (i) vapor–solid; (ii) vapor–vapor; and (iii) vapor–liquid, with the first two being limiting cases of (iii). It is assumed that the surface tension is sufficiently large so that the interfaces have uniform curvature. In a constructive and simple manner, steady state solutions of Stokes equations involving such configurations are constructed. The theory leading to the general expressions for the flow fields under the above-mentioned limitations exploit the inverse transformation, and the reflection and translation properties of the axisymmetric biharmonic function associated with the Stokes flow. These are cast in the form of a theorem followed by a simple proof. The theorem is then applied to construct closed form singularity solutions for several axisymmetric flow fields disturbed by a hybrid droplet. The different primary flow fields considered here include paraboloidal flow, flow due to a, single stokeslet and a pair of stokelets, and flow due to a potential-dipole. The salient features of the image singularities are discussed in each case. In all cases, the drag force is found to vary significantly with respect to the two radii associated with the two-sphere geometry of the droplet, and the viscosity ratio of the two liquids in the continuous and dispersed phases. In the case of singularity driven flows, the drag is influenced by an additional parameter namely, the location of the singularity. The flow streamlines in some cases show interesting flow patterns. In the case of paraboloidal flows, either a single eddy or a pair of eddies is observed depending on the ratio of the two radii. The sizes and shapes of these eddies vary monotonically with viscosity ratio. For flows due to a single and a pair of stokeslets with the same strength, no eddy is noticed. However, a toroidal eddy appears in the case of a pair of stokelets with opposite strengths. The locations of the stokeslets and the viscosity ratio influence the eddy structure significantly.

Stokes flow due to fundamental singularities before a plane boundary

A representation for the velocity and pressure fields in three-dimensional Stokes flow was presented in terms of a biharmonic function A and a harmonic function B. This representation was used to establish a general theorem for the calculation of Stokes flow due to fundamental singularities in a region bounded by a stationary no-slip plane boundary. Collins's theorem for axisymmetric Stokes flow before a rigid plane follows as a special case of the theorem. A few illustrative examples are given to show its usefulness.

A monotonicity estimate of the biharmonic Green function

In this note we present a refined form of the monotonicity property for the biharmonic Green function and the related so called harmonic compensator. Specialized to the harmonic compensator the estimate obtained is shown to be sharp.

Developing Software for Elastic Analysis of Pavement Structure Responses to Vertical and Horizontal Surface Loadings

In this research, a pavement is modeled as a multilayered elastic system composed of a subgrade that is assumed to be a semi-infinite medium. On top of the sublayer are layers of finite thickness infinitely extended horizontally. Furthermore, it is assumed that vertical loading, such as wheel load, and horizontal loading, such as brake load, are simultaneously applied at the surface of a road or runaway pavement. This load is assumed to be circular and uniformly distributed over the entire area. Expressing the displacement function of Navier's equation in a cylindrical coordinate system through harmonic and biharmonic functions, the analytical solution method is obtained by using Michell and Boussinesq functions as well as Hankel transformation and Hankel inverse transformation. In this research, a program called GAMES, which can find pavement responses to multiple vertical and shear loadings, was developed. Theoretical analysis uses the principle of superposition, and the solution is given by summing up the solutions from vertical and horizontal loads. The accuracy of this program is confirmed by comparing results for a number of worked examples with results obtained using the BISAR program. It is also possible to allow an interface slip modeled in a form of shear spring. The current version of this program allows for analysis of 10,000 measurement points with a maximum of 100 loads and 100 pavement layers.