Transport solutions of the Lamé equations and shock elastic waves

The Lamé system describing the dynamics of an isotropic elastic medium affected by a steady transport load moving at subsonic, transonic, or supersonic speed is considered. Its fundamental and generalized solutions in a moving frame of reference tied to the transport load are analyzed. Shock waves arising in the medium at supersonic speeds are studied. Conditions on the jump in the stress, displacement rate, and energy across the shock front are obtained using distribution theory. Numerical results concerning the dynamics of an elastic medium influenced by concentrated transport loads moving at sub-, tran- and supersonic speeds are presented.     

Bending of thick functionally graded plates resting on Winkler–Pasternak elastic foundations

The static response of simply supported functionally graded plates (FGP) subjected to a transverse uniform load (UL) or a sinusoidally distributed load (SL) and resting on an elastic foundation is examined by using a new hyperbolic displacement model. The present theory exactly satisfies the stress boundary conditions on the top and bottom surfaces of the plate. No transverse shear correction factors are needed, because a correct representation of the transverse shear strain is given. The material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. The foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second parameter is zero. The equilibrium equations of a functionally graded plate are given based on the hyperbolic shear deformation theory of plates presented. The effects of stiffness and gradient index of the foundation on the mechanical responses of the plates are discussed. It is established that the elastic foundations significantly affect the mechanical behavior of thick functionally graded plates. The numerical results presented in the paper can serve as benchmarks for future analyses of thick functionally graded plates on elastic foundations.     

On the Geometric Properties of the Poisson Kernel for the Lamé Equation

Computational Mathematics and Mathematical Physics - It is shown that the Poisson kernel for the Lamé equation in a disk can be interpreted as a bi-univalent mapping of the projection of an...     

Diagonalization of the system of static Lamé equations of isotropic linear elasticity

We find a simplest representation for the general solution to the system of the static Lamé equations of isotropic linear elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lamé equations. The system of Lamé equations diagonalizes; i.e., it is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and some formula for producing new solutions which makes it possible, given a solution, to find new solutions to the static Lamé equations by derivations. In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov-Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of the two-dimensional elliptic systems.     

Asymptotic Expansions for Eigenvalues of the Lamé System in the Presence of Small Inclusions

We derive a complete asymptotic expansion for eigenvalues of the Lamé system of the linear elasticity in domains with small inclusions in three dimensions. By an integral equation formulation of the solutions to the harmonic oscillatory linear elastic equation, we reduce this problem to the study of the characteristic values of integral operators in the complex planes. Generalized Rouché's theorem and other techniques from the theory of meromorphic operator-valued functions are combined with asymptotic analysis of integral kernels to obtain full asymptotic expansions for eigenvalues.     

Application of Maslov’s formula to finding an asymptotic solution to an elastic deformation problem

We propose a scheme of bifurcation analysis of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic base under the assumption of two-mode degeneracy; this scheme generalizes the Darinskii-Sapronov scheme developed earlier for the case of a homogeneous beam. The consideration of an inhomogeneous beam requires replacing the condition that the pair of eigenvectors of the operator from the linear part of the equation (at zero) is constant by the condition of the existence of a pair of vectors smoothly depending on the parameters whose linear hull is invariant with respect to . It is shown that such a pair is sufficient for the construction of the principal part of the key function and for analyzing the branching of the equilibrium configurations of the beam. The construction of the required pair of vectors is based on a formula for the orthogonal projection onto the root subspace of (from the theory of perturbations of self-adjoint operators in the sense of Maslov). The effect of the type of inhomogeneity of the beam on the formof its deflection is studied.     

Some qualitative effects in the exact solutions of the Lamé problem for large deformations

Qualitative effects in the solution of a number of radially symmetric and plane axisymmetric problems for bodies made of non-linearly elastic incompressible materials are analysed for large deformations. In the case of problems of the axisymmetric plane deformation of cylindrical bodies, the lack of uniqueness of the solution for a given follower load in the case of a Bartenev–Khazanovich material and the existence of a limiting load in the case of a Treloar (neo-Hookian) material have been studied in detail and the dependences of the limiting load on the ratio of the external and internal radii of a hollow cylinder in the undeformed state have been presented. A similar study has been carried out for constitutive relations of a special form that well describe the properties of rubber. For this material, the lack of uniqueness of the solution is revealed for fairly high loads. The axisymmetric problem of the plane stress state of a circular ring made of a Bartenev–Khazanovich material has been solved and a lack of uniqueness of the solution for a given follower load was discovered in the case when the dimensions of the ring are given in the undeformed state. Similar studies have been carried out for Chernykh and Treloar materials in the case of the problem of the radially symmetric deformation of a spherical shell. It was established that, in the case of a Chernykh material, the lack of uniqueness of the solution depends considerably on the constant characterizing the physical non-linearity. The limit case of the deformation of a spherical cavity in an infinitely extended body has been investigated. The effect of an unbounded increase in the boundary stresses is observed for finite external loads, that appears in the case of the problem of the plane axisymmetric deformation of a cylindrical cavity in an infinitely extended body made of a Bartenev–Khazanovich material and in the case of the problem of the radially symmetric deformation of an infinitely extended body made of a Chernykh material with a spherical cavity.     

Maximal regularity for the Lamé system in certain classes of non-smooth domains

The aim of this article is twofold. On the one hand, we study the well-posedness of the Lamé system ${-\mu\Delta-\mu'\nabla{\rm div} }The aim of this article is twofold. On the one hand, we study the well-posedness of the Lamé system -mD-m￠?div{-\mu\Delta-\mu'\nabla{\rm div} } in L ^q (Ω), where Ω is an open subset of \mathbbRⁿ{{\mathbb{R}}^n} satisfying mild regularity assumptions and the Lamé moduli μ, μ′ are such that μ > 0 and μ + μ′ > 0. On the other hand, we prove the analyticity of the semigroup generated by the Lamé operator as well as the maximal regularity property for the time-dependent Lamé system equipped with a homogeneous Dirichlet boundary condition based on off-diagonal estimates.     

An immersed boundary–lattice Boltzmann approach to study the dynamics of elastic membranes in viscous shear flows

A combined immersed boundary–lattice Boltzmann approach is used to simulate the dynamics of elastic membrane immersed in a viscous incompressible flow. The lattice Boltzmann method is utilized to solve the flow field on a regular Eulerian grid, while the immersed boundary method is employed to incorporate the fluid–membrane interaction with a Lagrangian representation of the deformable immersed boundary. The distinct feature of the method used here is to employ the combination of simple Peskin's IBM and standard LBM. In order to obtain more accurate and truthful solutions, however, a non-uniform distribution of Lagrangian points and a modified Dirac delta function are used. Two test cases are presented. In the first case, we consider a vesicle suspended in a simple shear flow commonly known as tank-treading motion. The computed results were compared with experiments, which showed reasonably good agreement. For the second test case, we consider individual healthy (soft) and sick (stiff) RBCs suspended in a shear flow. The simulation results demonstrated that elastic deformation plays an important role in overall RBC motions characterized as tank-treading and tumbling motions, in which the natural state of the elastic membrane is an essential consideration. In addition, the results confirm that the combination of the immersed boundary and lattice Boltzmann methods permits the simulation of the complex biological phenomena.     

On the analytical approach to the linear analysis of an arbitrarily curved spatial Bernoulli–Euler beam

The equilibrium and kinematic equations of an arbitrarily curved spatial Bernoulli–Euler beam are derived with respect to a parametric coordinate and compared with those of the Timoshenko beam. It is shown that the beam analogy follows from the fact that the left-hand side in all the four sets of beam equations are the covariant derivatives of unknown vector. Furthermore, an elegant primal form of the equilibrium equations is composed. No additional assumptions, besides those of the linear Bernoulli–Euler theory, are introduced, which makes the theory ideally suited for the analytical assessment of big-curvature beams. The curvature change is derived with respect to both convective and material/spatial coordinates, and some aspects of its definition are discussed. Additionally, the stiffness matrix of an arbitrarily curved spatial beam is calculated with the flexibility approach utilizing the relative coordinate system. The numerical analysis of the carefully selected set of examples proved that the present analytical formulation can deliver valid benchmark results for testing of the purely numeric methods.     

Wiener Type Regularity of a Boundary Point for the 3D Lamé System

In this paper, we study the 3D Lamé system and establish its weighted positive definiteness for a certain range of elastic constants. By modifying the general theory developed in Maz’ya (J Duke Math 115(3): 479–512, 2002), we then show, under the assumption of weighted positive definiteness, that the divergence of the classical Wiener integral for a boundary point guarantees the continuity of solutions to the Lamé system at this point.     

On the nesting of Painlevé hierarchies: A Hamiltonian approach

We consider the phenomenon whereby two different Painlevé hierarchies, related to the same hierarchy of completely integrable equations, are such that solutions of one member of one of the Painlevé hierarchies are also solutions of a higher-order member of the other Painlevé hierarchy. An explanation is given in terms of the Hamiltonian structures of the related underlying completely integrable hierarchies, and is sufficiently generally formulated so as to be applicable equally to both continuous and discrete Painlevé hierarchies. Special integrals of a further Painlevé hierarchy related by Bäcklund transformation to the other Painlevé hierarchy mentioned above can also be constructed. Examples of the application of this approach to Painlevé hierarchies related to the Korteweg–de Vries, dispersive water wave, Toda and Volterra integrable hierarchies are considered. Our results provide further evidence of the importance of the underlying structures of related completely integrable hierarchies in understanding the properties of Painlevé hierarchies.     

Semi-Smooth Newton Methods for Optimal Control of the Dynamical Lamé System with Control Constraints

Optimal control problems governed by the dynamical Lamé system with additional constraints on the controls are analyzed. Different types of control action are considered: distributed, Neumann boundary and Dirichlet boundary control. To treat the inequality control constraints semi-smooth Newton methods are applied and their convergence is analyzed. Although semi-smooth Newton methods are widely studied in the context of PDE-constrained optimization little has been done in the context of the dynamical Lamé system. The novelty of the article is the proof that in case of distributed and Neumann boundary control the Newton method converges superlinearly. In case of Dirichlet control superlinear convergence is shown for a strongly damped Lamé system. The results are an extension of [14 A. Kröner , K. Kunisch , and B. Vexler ( 2011 ). Semi-smooth Newton methods for optimal control of the wave equation with control constraints . SIAM J. Control Optim. 49 : 830 – 858 .[Crossref], [Web of Science ®] , [Google Scholar]], where optimal control problems of the classical wave equation are considered. The control problems are discretized by finite elements and numerical examples are presented.     

Point equivalence of second-order ordinary differential equations to the fifth Painlevé equation with one and two nonzero parameters

Theoretical and Mathematical Physics - We consider the problem of the equivalence of scalar second-order ordinary differential equations under invertible point transformations. To solve this...     

Asymptotics of eigenvalues of the two-dimensional Dirichlet boundary-value problem for the Lamé operator in a domain with a small hole

The Dirichlet boundary-value problem for the eigenvalues of the Lamé operator in a twodimensional bounded domain with a small hole is studied. The asymptotics of the eigenvalue of this boundary-value problem is constructed and justified up to the power of the parameter defining the diameter of the hole.     

Homogenization of the Neumann problem for the Lamé equations of linear elasticity in domains with a periodic system of channels of small length

The problem of homogenization is considered for the solutions of the Neumann problem for the Lamé system of plane elasticity in two-dimensional domains with channels that have the form of rectilinear cylinders of length ε ^q (ε is a small positive parameter, q = const > 0) and radius a _ɛ. The bases of the channels form an ε-periodic structure on the hyperplane {x ∈ ℝ²: x ₁ = 0} and their number is equal to N _ɛ= O(ɛ⁻¹) as ε → 0. Under the limit condition lim on the parameters characterizing the geometry of the domain, the weak H ¹-limit of the generalized solution of this problem is found. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 310–322, 2005.     

An explicit exact analytical approach for free vibration of circular/annular functionally graded plates bonded to piezoelectric actuator/sensor layers based on Reddy’s plate theory

In the present study, a novel exact closed-form procedure based on the third order shear deformation plate theory is developed to analyze in-plane and out-of-plane frequency responses of circular/annular functionally graded material (FGM) plates embedded in piezoelectric layers for both close/open circuit electrical boundary conditions. Introducing a new analytical method, five governing partial deferential equations of motion beside Maxwell electrostatic equation are solved via an exact closed-form method. The high accuracy and reliability of the present approach is confirmed by comparing some of the present data with their counterparts reported in the literature. Finally, the effect of material properties, power law index and boundary conditions on the free vibration of the smart FGM plate are studied and discussed in detail.     

An elementary approach to the polynomial τ-functions of the KP Hierarchy

We give an elementary construction of the solutions of the KP hierarchy associated with polynomial τ-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial τ-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 23–36, January, 1999.