The Gel’fand Problem for the Biharmonic Operator

We study stable and finite Morse index solutions of the equation ${\Delta^2 u = {e}^{u}}$ . If the equation is posed in ${\mathbb{R}^N}$ , we classify radial stable solutions. We then construct nonradial stable solutions and we prove that, unlike the corresponding second order problem, no Liouville-type theorem holds, unless additional information is available on the asymptotics of solutions at infinity. Thanks to this analysis, we prove that stable solutions of the equation on a smoothly bounded domain (supplemented with Navier boundary conditions) are smooth if and only if ${N \leqq 12}$ . We find an upper bound for the Hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded Morse index, provided they are controlled in a suitable Morrey norm.     

Development of the Spectral Sturm–Liouville Method for the Solution of a Boundary-Value Problem for the Biharmonic Equation

We generalize the spectral Sturm–Liouville method for the solution of the biharmonic equation. The characteristic equation for the determination of eigenvalues is investigated and eigenfunctions are constructed. We determine the stress-strain state for a rectangular plate loaded by arbitrary forces on its sides. For an arbitrary external load, we obtain a relation for the stress-strain state in the form of a series in eigenfunctions. A method of integral moments for the determination of the coefficients of the series is proposed. The Saint-Venant principle is verified.     

On the Existence and Uniqueness of a Solution to the Interior Transmission Problem for Piecewise-Homogeneous Solids

The interior transmission problem (ITP), which plays a fundamental role in inverse scattering theories involving penetrable defects, is investigated within the framework of mechanical waves scattered by piecewise-homogeneous, elastic or viscoelastic obstacles in a likewise heterogeneous background solid. For generality, the obstacle is allowed to be multiply connected, having both penetrable components (inclusions) and impenetrable parts (cavities). A variational formulation is employed to establish sufficient conditions for the existence and uniqueness of a solution to the ITP, provided that the excitation frequency does not belong to (at most) countable spectrum of transmission eigenvalues. The featured sufficient conditions, expressed in terms of the mass density and elasticity parameters of the problem, represent an advancement over earlier works on the subject in that (i) they pose a precise, previously unavailable provision for the well-posedness of the ITP in situations when both the obstacle and the background solid are heterogeneous, and (ii) they are dimensionally consistent, i.e., invariant under the choice of physical units. For the case of a viscoelastic scatterer in an elastic solid it is further shown, consistent with earlier studies in acoustics, electromagnetism, and elasticity that the uniqueness of a solution to the ITP is maintained irrespective of the vibration frequency. When applied to the situation where both the scatterer and the background medium are viscoelastic, i.e., dissipative, on the other hand, the same type of analysis shows that the analogous claim of uniqueness does not hold. Physically, such anomalous behavior of the “viscoelastic-viscoelastic” case (that has eluded previous studies) has its origins in a lesser known fact that the homogeneous ITP is not mechanically insulated from its surroundings—a feature that is particularly cloaked in situations when either the background medium or the scatterer are dissipative. A set of numerical results, computed for ITP configurations that meet the sufficient conditions for the existence of a solution, is included to illustrate the problem. Consistent with the preceding analysis, the results indicate that the set of transmission values is indeed empty in the “elastic-viscoelastic” case, and countable for “elastic-elastic” and “viscoelastic-viscoelastic” configurations.     

An Exact Derivation from Three-Dimensions of the Linear Equations for the Bending of an Elastically Monoclinic Plate and Associated Three-Dimensional Solutions

The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are subject to a special set of body and surface loads that leave the analogous plate loads arbitrary.     

Exact Solution of the Wave‐Resistance Problem for a Submerged Cylinder. II. The Non‐linear Problem

. We study the problem of wave resistance for a “slender” cylinder submerged in a heavy fluid of finite depth with the cylinder moving at uniform supercritical speed in the direction orthogonal to its generators. We look for a divergence‐free, irrotational flow; the boundary of the region occupied by the fluid (consisting of the free surface, the bottom and the obstacle profile) is assumed to belong to streamlines and the Bernoulli condition is taken on the free surface. The problem is transformed, via the hodograph map, into a problem set in a strip with a cut. By using a “hard” version of the inverse function theorem and by taking account of the results obtained in Part I (which we recall here), we prove the existence of a complex velocity function satisfying all the requirements of the problem. In particular, this function is continuous up to the surface of the obstacle, and the only possible singularities appear at the end‐points where the boundary is not smooth. Moreover, two stagnation points appear near to the extremities of the submerged body. (Accepted October 14, 1998)     

Exact Electromagnetothermoelastic Solution for a Transversely Isotropic Piezoelectric Hollow Sphere Subjected to Arbitrary Thermal Shock

This paper presents analytical study for electromagnetothermoelastic transient behavior of a transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Exact solutions for the transient responses of stresses, perturbation of magnetic field vector, electric displacement and electric potential in the transversely isotropic piezoelectric hollow sphere are obtained by means of the Hankel transform, the Laplace transform and their inverse transforms. An interpolation method is used to solve the Volterra integral equation of the second kind caused by interactions among electric, magnetic, thermal and elastic fields. From the sample numerical calculations, it is seen that the present method is suitable for the transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Finally, the result can be used as a reference to solve other transient coupling problems of electromagnetothermoelasticity.     

A Frame-Independent Solution to Saint-Venant’s Flexure Problem

The paper illustrates a solution approach for the Saint-Venant flexure problem which preserves a pure objective tensor form, thus yielding, for sections of arbitrary geometry, representations of stress and displacement fields that exploit exclusively frame-independent quantities. The implications of the availability of an objective solution to the shear warpage problem are discussed and supplemented by several analytical and numerical solutions. The derivation of tensor expressions for the shear center and the shear flexibility tensor is also illustrated. Furthermore, a Cesaro-like integration procedure is provided whereby the derivation of a frame-independent representation of the displacements field for the shear loading case is systematically carried out via the use of Gibbs’ algebra. The objective framework presented in this paper is further exploited in a companion article (Serpieri, in J. Elast. (2013)) to prove the coincidence of energetic and kinematic definitions of the shear flexibility tensor and of the shear principal axes.     

Existence of a Global Smooth Solution¶for a Degenerate Goursat Problem¶of Gas Dynamics

The Goursat problem of a mixed type equation , P≥ 0, is considered. At the ends of its supports we have P=0, which means it is degenerate hyperbolic. We prove the global existence of a smooth solution to the degenerate Goursat problem up to a boundary where P=0. This problem comes from the expansion of a wedge of gas with constant velocity into vacuum, in two-dimensional pressure-gradient equations in gas dynamics, where P is the pressure and P=0 means vacuum. Accepted June 16, 2000?Published online December 6, 2000     

Numerical algorithm for natural frequencies computation of an axially moving beam model

An algorithm for numerical computation of natural frequencies of the axially moving Euler-Bernoulli beam is presented. It is tested against data found in the literature and against known analytical expressions of its limiting models—axially moving string and stationary beam—where good agreements were found. The numerical algorithm always stays within real algebra. Roots of the polynomial can be computed out of only three real numbers and the expressions for determinant evaluations are deduced in a numerically stable way.     

Fully Localised Solitary-Wave Solutions of the Three-Dimensional Gravity–Capillary Water-Wave Problem

A model equation derived by Kadomtsev & Petviashvili (Sov Phys Dokl 15:539–541, 1970) suggests that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. The theory is variational in nature. A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties. The reduced functional is related to the functional associated with the Kadomtsev–Petviashvili equation, and a nontrivial critical point is detected using the direct methods of the calculus of variations.     

Exact Solution of the Wave‐Resistance Problem for a Submerged Cylinder. I. Linearized Theory

. We consider the problem of finding a holomorphic function in a strip with a cut ${\cal A}= \{(x,y) : \, x\in\RE,\,\,0 satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for $|x|\to\infty. We consider the problem of finding a holomorphic function in a strip with a cut satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for . A special solution can be selected by fixing the value of the circulation around the cut. The problem is obtained by linearization of the equations of the wave‐resistance problem for a “slender” cylinder submerged in a heavy fluid and moving at uniform speed in the direction orthogonal to its generators. The results obtained, besides their own interest, are a crucial step for the resolution of the non‐linear problem. (Accepted October 14, 1998)     

Asymptotic Behavior of Global Classical Solutions to the Mixed Initial-Boundary Value Problem for Quasilinear Hyperbolic Systems with Small BV Data

In this paper, we investigate the asymptotic behavior of global classical solutions to the mixed initial-boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t,x)|t≥0,x≥0}. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C ¹ traveling wave solutions, provided that the C ¹ norm of the initial and boundary data is bounded and the BV norm of the initial and boundary data is sufficiently small. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space-time R ¹⁺ⁿ , are also given.     

Analysis of Constitutive Assumptions for the Strain Energy of a Generalized Elastic Membrane in a Nonlinear Contact Problem

For very thin shell-like structures it is common to ignore bending effects and model the structure using simple membrane theory. However, since the thickness of the membrane is not modeled explicitly in simple membrane theory it is not possible to use the three-dimensional strain energy function directly. Approximations must be introduced like the assumptions of: no thickness changes, generalized plane stress or incompressibility. In contrast, the theory of a Cosserat generalized membrane uses the three-dimensional strain energy function directly, it includes both thickness changes and shear deformation and it allows contact conditions to be formulated on the interface of the membrane with another body instead of on the middle surface of the membrane. A specific nonlinear contact problem is used to study these effects and comparison is made with solutions of a hierarchy of theories which include different levels of deformation through the thickness of the membrane and different formulations of the contact conditions. The results indicate that within the context of a simple membrane the assumption of generalized plane stress is best for this problem and that a generalized contact condition extends the range validity of the simple membrane solution to thicker membranes.     

Analytical analysis approach to nonlinear dynamic characteristics of hydraulically damped rubber mount for vehicle engine

This paper presents an analytical analysis approach to study of dynamic characteristics of hydraulically damped rubber mount (HDM). Analysis of experimental results verifies the effectiveness of the analytical approach. Frequency- and amplitude-dependent dynamic characteristic of HDM and superposition performance of dynamic characteristics of rubber mount, inertia track subsystem and HDM are investigated to clarify working mechanism of HDM. Parametric effect analysis is carried out on the basis of qualitative analysis and dynamic characteristic simulation, which provides useful guideline to HDM design.     

A Concise Derivation of Membrane Theory from Three-Dimensional Nonlinear Elasticity

Membrane theory may be regarded as a special case of the Cosserat theory of elastic surfaces, or, alternatively, derived from three-dimensional elasticity theory via asymptotic or variational methods. Here we obtain membrane theory directly from the local equations and boundary conditions of the three-dimensional theory.     

Numerical Simulation for Evolutionary History of Three-Dimensional Basin

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Approximate Semi-analytical Solution for Injection–Falloff–Production Well Test: An Analytical Tool for the In Situ Estimation of Relative Permeability Curves

An injection–falloff–production test (IFPT) was originally proposed in Chen et al. (in: SPE conference paper, 2006. doi: 10.2118/103271-MS, SPE Reserv Eval Eng 11(1):95–107, 2008) as a well test for the in situ estimation of two-phase relative permeability curves to be used for simulating multiphase flows in porous media. Hence, we develop an approximate semi-analytical solution for the two-phase saturation distribution in an oil–water system during the flowback period of an IFPT according to the mathematical theory of waves. In fact, we show that the weak solution we construct for the saturation equation for the flowback period satisfies the Oleinik entropy condition and hence is unique. In addition, we allow the governing relative permeabilities during the flowback period to be different from the relative permeabilities during injection. Using the saturation solution with the steady-state pressure theory of Thompson and Reynolds, we obtain a solution for the wellbore pressure during the flowback period. By comparing results from our solution with those from a commercial numerical simulator, we show that our approximate semi-analytical solution yields accurate saturation profiles and bottom hole pressures history. The use of very small time steps and a highly refined radial grid is necessary to generate a good solution from a reservoir simulator. The approximate analytical pressure solution developed is used as a forward model to match pressure and water flow rate data from an IFPT in order to estimate reservoir rock absolute permeability and skin factor in conjunction with in situ imbibition and drainage water–oil relative permeabilities.     

An Iteration Method for Integral Equations Arising from Axisymmetric Loading Problems

Let the concentrated forces and the centers of pressure with unknown density functions x(ξ)and y(ξ)respectively be distributed along the axis z outside the solid,then one can reduce anaxismmetric loading problem of solids of revolution to two simultaneous Fredholm integral equations.An iteration method for solving such equations is duscussed.A lemma equivalent to E.Rakotch’scontractive mapping theorem and a theorem concerning the convergent proof of the iteration methodare presented.