Asymptotic analysis of elastic curved rods

We consider a sequence of curved rods which consist of isotropic material and which are clamped on the lower base or on both bases. We study the asymptotic behaviour of the stress tensor and displacement under the assumptions of linearized elasticity when the cross‐sectional diameter of the rods tends to zero and the body force is given in the particular form. The analysis covers the case of a non‐smooth limit line of centroids. We show how the body force and the choice of the approximating curved rods can affect the strong convergence and the limit form of the stress tensor for the curved rods clamped on both bases. Copyright © 2006 John Wiley & Sons, Ltd.     

Derivation of the model of elastic curved rods from three-dimensional micropolar elasticity

In this paper we derive a model of curved elastic rods from the threedimensional linearized micropolar elasticity. Derivation is based on the asymptotic expansion method with respect to the thickness of the rod. The method is used without any a priori assumption on the scaling of the unknowns. The leading term, displacement and microrotation, is identified as the unique solution of a certain one-dimensional problem. Appropriate convergence results are proved.      

Approximation of the vibration modes of a Timoshenko curved rod of arbitrary geometry

Rodolfo Rodríguez ^{The aim of this paper is to analyse a mixed finite-element methodfor computing the vibration modes of a Timoshenko curved rodwith arbitrary geometry. Optimal order error estimates are provedfor displacements, rotations and shear stresses of the vibrationmodes, as well as a double order of convergence for the vibrationfrequencies. These estimates are essentially independent ofthe thickness of the rod, which leads to the conclusion thatthe method is locking-free. Numerical tests are reported inorder to assess the performance of the method.}     

Global regularity of the elastic fields of a power‐law model on Lipschitz domains

In this paper, we study the global regularity of the displacement and stress fields of a nonlinear elastic model of power‐law type. It is assumed that the underlying domains are Lipschitz domains which satisfy an additional geometric condition near those points, where the type of the boundary conditions changes. The proof of the global regularity result relies on a difference quotient technique. Finally, a global regularity result for the stress fields of the elastic, perfect plastic Hencky model is derived. This model appears as a limit model of the power‐law model. Copyright © 2006 John Wiley & Sons, Ltd.     

Existence of local‐in‐time classical solutions of a model of flow in a bounded elastic tube

This paper studies the local‐in‐time existence of classical solutions to a hyperbolic system with differential boundary conditions modelling a flow in an elastic tube. The well‐known Lax transformations used for obtaining a priori estimates for conservation laws are difficult to apply here because of the inhomogeneity of the partial differential equations (PDE). Rather, our method relies on a suitable splitting of the original system into the PDE part and the ODE part, the characteristics for the PDE part, appropriate modulus of continuity estimates and a compactness argument. Copyright © 2016 John Wiley & Sons, Ltd.     

Short time existence for the elastic flow of clamped curves

We show well‐posedness of the elastic flow of open curves with clamped boundary conditions. To show short time existence we prove that the linearised problem has the property of maximal ‐regularity and use the contraction principle to obtain the solution. Moreover, we show analyticity of the solution and its analytic dependency on the initial curve. With the developed methods we also prove long time existence of the flow if the initial curve is close to an elastica.     

Scattering of a SH-wave by an elastic fiber of nonclassical cross section with an interface crack

The problem of interaction of a plane time-harmonic SH-wave with an elastic fiber of quasi-square or quasi-triangular cross section, when an interface crack is present between an infinite elastic matrix and the fiber, is considered. The modified null-field method taking into account the asymptotic behavior of the solution at crack tips is exploited for obtaining numerical results. The effects of fiber shape, fiber/matrix material combination, debonding (crack size), and direction of wave incidence on the scattering amplitude in the far zone are analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 245–254, March–April, 2008.     

Observability of elastic vibrations of a beam

The observability problem for beam vibrations described by a fourth-order partial differential equation with various boundary conditions is considered. Dynamic observability problems are solved in terms of boundary conditions and observations of the beam state at certain fixed instants of time.     

An approach to solving the one-dimensional problem on compression of a viscoelastic layer dispersedly reinforced with elastic inclusions

A simplified approach to investigating the combined effect of a dispersed filler with different elastic characteristics, physical nonlinearity, and hereditary properties, and of temperature and moisture on the compression of a rectilinear layer is proposed. Various combinations of filler concentration, temperature, and moisture are considered. The investigation is performed in two variants. In the first one, it is assumed that the prevailing characteristics are the instantaneous ones; in the second variant, the deformation is mainly determined by the long-term material characteristics. The instantaneous deformation diagram is assumed nonlinear. For both the variants, analytical relations for the degree of compression as a function of initial parameters (in particular, of initial concentration of the rigid inclusions and their relative rigidity) are obtained. The time-dependent inclusion concentration diagrams are constructed for different levels of load, temperature, and moisture. Graphically, effects of the rigid inclusions and temperature on deformation of the layer are revealed. The combinations of filler concentration and temperature are determined at which their effects on compression of the layer annihilate each other. Analytical relations for the effect of moisture on compression of the dispersedly re in forced layer are obtained, and the corresponding calculated diagrams are presented. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 3, pp. 441-456, May-June, 2009. An erratum to this article can be found at     

On the stationary interaction of a Navier-Stokes fluid with an elastic tube wall

We study the stationary problem of a viscous, incompressible Navier-Stokes fluid flowing through a flexible tube with thickness. The behavior of the elastic walls of the tube is described by the equations of nonlinear elasticity for a St.Venant-Kirchhoff material. For smooth enough applied exterior forces we prove the existence of a solution to the coupled problem.     

On a dynamical hierarchical model for prismatic shells in the theory of elastic mixtures

The present paper is devoted to the design of a hierarchy of two‐dimensional models for dynamical problems within the theory of multicomponent linearly elastic mixtures in the case of prismatic shells with thickness which may vanish on some part of its boundary. The hierarchical model is obtained by a semidiscretization of the three‐dimensional problem in the transverse direction. In suitable weighted Sobolev spaces we investigate the well‐posedness of the two‐dimensional problems, prove pointwise convergence of the sequence of approximate solutions restored from the solutions of the reduced problems to the exact solution of the original problem and estimate the rate of convergence. Copyright © 2005 John Wiley & Sons, Ltd.     

A general mathematical model for the collision between a free-fall hammer of a pile-driver and an elastic pile

In this paper we consider a general mathematical model for the collision between the free-fall hammer of a pile-driver and an elastic pile whose ends are furnished with a bearing. When the free-fall hammer collides with the pile, the displacement of a cross-sectional area of the pile is the weak solution of an initial-boundary value problem involving a linear wave equation with memory boundary conditions. We generalize this problem into a nonlinear one with more general boundary conditions. Then we obtain the unique solvability and the regularity of the weak solution of this nonlinear problem. The unique solvability is shortly discussed in regard to the Galerkin method. The regularity result is obtained by a combination of a fixed-point technique and an energy method, and the convenience of this procedure is also pointed out.     

The effect of a surface energy term on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing among functions u:?^d?Ω→?^d, u_∣?Ω=0, and measurable subsets E of Ω. Here f_h⁺, f^? denote quadratic potentials defined on Ω¯×{symmetric d×d matrices}, h is the minimum energy of f_h⁺ and ε(u) is the symmetric gradient of the displacement field u. An equilibrium state û, Ê of J(u,E) is called one‐phase if E=?? or E=Ω, two‐phase otherwise. For two‐phase states, σ∣?E∩Ω∣ measures the effect of the separating surface, and we investigate the way in which the distribution of phases is affected by the choice of the parameters h??, σ>0. Additional results concern the smoothness of two‐phase equilibrium states and the behaviour of inf J(u,E) in the limit σ↓0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non‐zero boundary values. Copyright © 2002 John Wiley & Sons, Ltd.     

A mathematical study of the linear theory for orthotropic elastic simple shells

The theory of simple shells is a surface‐related Cosserat model for thin elastic shells. In this direct approach, each material point is connected with a triad of rigidly rotating directors. This paper presents a study of the governing equations for orthotropic elastic simple shells in the framework of the linearized theory. We establish the uniqueness of classical solutions, without any restrictive assumption on the strain energy function. The continuous dependence of solutions on the body loads and initial data is proved. Also, the existence of weak solutions to the equations of simple shells is proved by means of an inequality of Korn's type established for such directed surfaces. Copyright © 2009 John Wiley & Sons, Ltd.     

Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model

In [3], L. Berselli showed that the regularity criterion ? u ∈ (0, T; L ^q (Ω)), for some q ∈ (3/2, + ∞], implies regularity for the weak solutions of the Navier–Stokes equations, being u the velocity field. In this work, we prove that such hypothesis on the velocity gradient is also sufficient to obtain regularity for a nematic Liquid Crystal model (a coupled system of velocity u and orientation crystals vector d ) when periodic boundary conditions for d are considered (without regularity hypothesis on d ). For Neumann and Dirichlet cases, the same result holds only for q ∈ [2, 3], whereas for q ∈ (3/2, 2) ∪ (3, + ∞] additional regularity hypothesis for d (either on ? d or Δ d ) must be imposed. On the other hand, when the Serrin's criterion u ∈ (0, T; L ^p (Ω)) with some p ∈ (3, + ∞] ([16]) for u is imposed, we can obtain regularity of the system only in the problem of periodic boundary conditions for d . When Neumann and Dirichlet cases for d are considered, additional regularity for d must be imposed for each p ∈ (3, + ∞] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of a matrix finite-difference scheme for solving the problem of elastic vibrations

The stability of a matrix finite-difference scheme is analyzed. The scheme is based on central differences and is designed for the differential equation governing the vibrations of an elastic system.     

A stochastic model for a general load‐sharing system under overload condition

A load‐sharing parallel system functions if at least one unit in the system is functioning and the surviving units share the load. In most of research on load‐sharing system, the performance of the system has been studied only for the case when the lifetimes of components in the system follow exponential distributions. In this paper a load‐sharing parallel system is considered when the lifetimes of the units in the system are any continuous random variables. The reliability function of the system is derived and the problem of load allocation is also considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Scattering of antiplane shear wave by a propagating crack at the interface of two dissimilar elastic media

An analysis of the scattering of horizontally polarized shear wave by a semi-infinite crack running with uniform velocity along the interface of two dissimilar semi-infinite elastic media has been carried out. The mixed boundary value problem has been solved completely by the Wiener-Hopf technique. The effect of different values of the material parameter, the angle of incidence of incident wave and the crack propagation velocity on the stress intensity factor have been illustrated graphically.     

Existence and non‐existence of steady states to a cross‐diffusion system arising in a Leslie predator–prey model

This paper is concerned with a cross‐diffusion system arising in a Leslie predator–prey population model in a bounded domain with no flux boundary condition. We investigate sufficient condition for the existence and the non‐existence of non‐constant positive solution. We obtain that if natural diffusion coefficient of predator is large enough and cross‐diffusion coefficients are fixed, then under some conditions there exists non‐constant positive solution. Furthermore, we show that if natural diffusion coefficients of predator and prey are both large enough, and cross‐diffusion coefficients are small enough, then there exists no non‐constant positive solution. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of solution for a nonlinear model of thermo‐visco‐plasticity

We study a thermodynamically consistent model describing phenomena in a visco‐plastic metal subjected to temperature changes. We complete the model with the mixed boundary condition on displacement and stress and Neumann‐type condition for temperature. The main result is an existence of solutions.