Chaotic vibrations of spherical and conical axially symmetric shells

Summary Chaotic vibrations of deterministic, geometrically nonlinear, elastic, spherical and conical axially summetric shells, subject to sign-changing transversal load using the variational principle, are analysed. The paper is motivated by an observation that variational equations of the hybrid type are suitableto solve many dynamical problems of the shells theory. It is assumed that the shell material is isotropic, and the Hook's principle holds. Intertial forces in directions tangent to mean shell surface and rotation inertia of a normal shell cross section are neglected. A transition form PDEs to ODEs (the Cauchy problem) is realized through the Ritz procedure. Next, the Cauchy problem is solved using the fourth-order Runge-Kutta method. Qualitative and quantitative analysis is carried out in the frame of both nonlinear dynamics and quantitative theory of differential equations. New scenarios from harmonic to chaotic dynamics are detected. Various vibration forms development versus control parameters (rise of arc; amplitude and frequency of the exciting force and number of vibrational modes accounted) are illustrated and discussed.     

弹性椭圆夹杂纵向剪切问题

获得纵向剪切下弹性椭圆夹杂问题的精确解。将复变函数的分区全纯函数理论，Cauchy型积分和Riemann边值问题相结合，求得各复势函数之间的解析关系，从而得到问题的封闭形式解，并给出了界面应力的解析表达式。本文解答与已有文献结果一致。本文发展的分析方法，为求解复杂多连通域的平面弹性问题提供了一条有效途径。     

Initial value problem for the dynamic system of anisotropic elasticity

The main object of this paper is the Cauchy problem for the dynamic system of anisotropic elasticity. Existence and uniqueness theorems of weak and smooth solutions of this problem are established by the reduction of the original elasticity system into a symmetric hyperbolic system of the first order. The numerical method of the Cauchy problem solving for anisotropic elastic system with polynomial data is obtained and its correctness is established. The simulations of the numerical solutions are presented.     

On a class of non-linear elastic materials

This article deals with a family of non-linear hyperelastic materials depending on a parameter varying from 0 to 1; is a masonry-like material and is linear elastic. Some properties of the function delivering the Cauchy stress corresponding to the infinitesimal strain E, are proved; in particular, it is shown that is strongly monotone for >0 and monotone for =0. Moreover, denoting by [u(·;), E(·;), T(·;)] the solution to the equilibrium problem for solids made of a material the convergence of [u(·;), E(·;), T(·;)] for going to 0 and 1, is investigated.     

Asymptotic Research of Nonlinear Wave Processes in Saturated Porous Media

The problem of nonlinear wave dynamics of a fluid-saturated porous medium is investigated. The mathematical model proposed is based on the classical Frenkel--Biot--Nikolaevskiy theory concerning elastic wave propagation and includes mass, momentum, energy conservation laws, as well as rheological and thermodynamic relations. The model describes nonlinear, dispersive, and dissipative medium. To solve the system of differential equations, an asymptotic modified two-scales method is developed and a Cauchy problem for initial equations system is transformed to a Cauchy problem for nonlinear generalized Korteweg--de Vries--Burgers equation for modulated quick wave amplitudes and an inhomogeneous set of equations for slow background motion. Stationary solutions of the derived evolutionary equation that have been constructed numerically reflect different regimes of elastic wave attenuation: diffusive, oscillating, and soliton-like.     

Averaged rotations at finite plane strain

The averaged rotations and other mechanical parameters at finite plane strains of an elastic material, which are characterized by a linear relation between the Cauchy stresses and the Almansi strains, are studied. The form of the elastic potential is determined. The displacement problem is reduced to a boundary-value problem for complex potentials, which is solved in terms of Cauchy-type integrals for the specified boundarys displacements. The results obtained are compared with the linear solution. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 187–196, May–June, 2000.     

Weight function theory and variational formulations for three-dimensional plane elastic cracks advancing

The weight function theory for three-dimensional elastic crack analysis received great attention after the work of Rice (1985, 1989). Several applications have been considered since then, particularly in the context of configurational stability, crack path prediction, stress intensity factor expansions, perturbation approaches. In all cases, a specific hypothesis has been made on the variation of crack shape, in order to formulate the problem in terms of Cauchy principal value. In the present note, such hypothesis is further investigated and consequences discussed. A variational statement given in Salvadori and Fantoni (2013a) is thus rephrased in terms of weight functions. Its discrete formulation shows the potential to accurate approximation of crack front propagation.     

Wavelet method applied to specific adhesion of elastic solids via molecular bonds

We propose a wavelet method to analyze the stochastic-elastic problem of specific adhesion between two elastic solids via ligand-receptor bond clusters, which is governed by a nonlinear integro-differential equation with a singular Cauchy kernel to describe the mean-field coupling between deformation of elastic materials and stochastic behavior of the molecular bonds. To solve this problem, Galerkin method based on a wavelet approximation scheme is adopted, and special treatment which transforms the singular Cauchy kernel into a smooth one has been proposed to avoid the cumbersome calculation of singular integrals. Numerical results demonstrate that the method is fully capable of solving the specific adhesion problems with complex nonlinear and singular equations. Based on the proposed method, investigations are performed to reveal the relation between steady-state pulling force and mean surface separation under different stress concentration indexes, which is crucial for assembling the overall constitutive relations for multicellular tumor spheroids and polymer-matrix microcomposites.     

Estimating the error of the beam approximation in the plane stability problem for a rectangular plate with a central crack

The plane stability problem for a rectangular, linearly elastic, isotropic plate with a central crack is solved. The dependence of the critical load on the crack length is studied using exact (the three-dimensional linearized theory of stability of elastic bodies) and approximate (beam approximation) approaches. The results produced by the beam approach are evaluated.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 117–126, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

Complementary energy,neutral equilibrium and buckling

Summary A general analytical approach of the non-linear problem of a linear elastic and isotropic straight bar of variable stiffness is indicated. The linear equivalence method, introduced by one of the authors, is applied to two fundamental cases for the isostatic straight bar, i.e. the cantilever bar (a Cauchy type problem) and the simply supported bar (a bilocal problem). Some numerical examples concerning moderate deformations and rotations are presented.

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Sommario Si propone un approccio analitico generate per la soluzione del problema non lineare di un'asta rettilinea di rigidezza non uniforme di materiale elastico-lineare isotropo.Il metodo dell'equivalenza lineare introdotto da uno degli autori è applicato a due casi fondamentali della trave isostatica rettilinea, cioè la mensola (problema alla Cauchy) e la trave appoggiata (problema bilocale). Vengono presentati alcuni esempi numerici concernenti deformazioni e rotazioni moderatamente grandi.

     

Acoustoelasticity model of inhomogeneously deformed bodies

We consider a mathematical model of dynamics of small elastic perturbations in an inhomogeneously deformed rigid body, where for the determining parameters of a local state we take the tensor characteristics of a given actual (strained) configuration (the Cauchy stress tensor and the Hencky or Almansi or Figner strain measure). An iteration algorithm is developed to solve the Cauchy problem stated in the framework of this model for a system of hyperbolic equations with variable coefficients that describes the propagation of elastic pulses in an inhomogeneous deformed continuum. In the case of two-dimensional stress fields, we obtain acoustoelasticity integral relations between the probing pulse parameters and the initial strain (stress) distribution in the direction of pulse propagation in the strained body. We also consider an example of application of the obtained integral relations in the inverse acoustic tomography problem for residual strains in a strip.     

Genesis of the multiscale approach for materials with microstructure

This paper presents an overview of the origin of multiscale approaches in mechanics. While the pioneer molecular models of linear elastic bodies by Navier, Cauchy and Poisson were contradicted by experiments, the phenomenological energetic approach by Green still seems suitable for simple materials only. Voigt’s molecular model, here reinterpreted in the light of contemporary mechanics, reconciled the two approaches providing a conceptual guideline for developing constitutive models based on a direct link between continuum and discrete solid mechanics. Such a theoretical background proves to be especially suitable for new complex materials. An example referred to masonry-like materials is given.     

Steady filtration problems with seawater intrusion: macro‐hybrid penalized finite element approximations

Macro‐hybrid penalized finite element approximations are studied for steady filtration problems with seawater intrusion. On the basis of nonoverlapping domain decompositions with vertical interfaces, sections of coastal aquifers are decomposed into subsystems with simpler geometries and small scales, interconnected via transmission conditions of pressure and flux continuity. Corresponding local penalized formulations are derived from the global penalized variational formulation of the two‐free boundary flow problem, with continuity transmission conditions modelled variationally in a dual sense. Then, macro‐hybrid finite element approximations are derived for the system, defined on independent subdomain grids. Parallel relaxation penalty‐duality algorithms are proposed from fixed‐point problem characterizations. Numerical experiments exemplify the macro‐hybrid penalized theory, showing a good agreement with previous primal conforming penalized finite element approximations (Comput. Methods Appl. Mech. Engng. 2000; 190 :609–624). Copyright © 2005 John Wiley & Sons, Ltd.     

Global Solutions for Incompressible Viscoelastic Fluids

We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near- equilibrium initial data. The results hold in both two- and three-dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals, and mixture problems.     

The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity

In this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials.     

Contact Problem for a Rectangular Punch on the Porous Elastic Half-Space

The paper is concerned with a contact problem about rigid rectangular punch forced into a half-space made of a linear elastic isotropic material with voids. We use a Cowin–Nunziato model for the half-space, and reduce the problem to a double Fredholm integral equation of the first kind. Then we apply two different approaches, to solve this equation. The first one is based on a direct collocation numerical technique. The second method is asymptotic, and we use a small parameter that is the relative width of the punch. Finally, compliance of the punch is determined, and results of the two different methods are compared with each other, as well as with a Sivashinsky–Panek–Kalker solution. Mathematics Subject Classifications (2000) 74M15.     

On the Representation of the Stored Energy for Elastic Materials with Full Transverse Response-Symmetry

We solve the representation problem for the stored energy of both transversely-isotropic and transversely-hemitropic elastic materials. Our method is based on giving the problem a form allowing application of a modified version of the classical representation theorem by Cauchy for scalar-valued mappings over the Nth power of a vector space. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture analysis on a piezoelectric sensor with a viscoelastic interface

Fracture analysis is performed on a layered piezoelectric sensor possessing a Kelvin-type viscoelastic interface. An electrically permeable anti-plane crack is situated in the piezoelectric layer and perpendicular to the interface. The crack problem is solved by the methods of integral transform and Cauchy singular integral equation. The variations of the dynamic stress intensity factor (DSIF) vs. physical and geometrical parameters are investigated. At the beginning of creep and relaxation, larger viscosity coefficient always induces smaller DSIF. With time elapsing, the effect of viscosity coefficient becomes weaker and weaker. When time approaches infinity, the viscous effect disappears, and the DSIF converges to a value corresponding to the case of an elastic interface. The effect of the viscoelastic interface on the fracture behavior of the piezoelectric layer also depends on the substrate thickness. To some extent, thicker substrate may intensify the effect of the interface.     

Investigation of instability in vertical liquid films as an initial value problem

The instability of plane-parallel vertical viscous layer downflow is investigated. We solve not the classical eigenvalue problem for the Orr-Sommerfeld equation but a Cauchy problem with respect to time and a boundary-value problem with respect to the spatial variable for a linearized system of equations. The problem is solved by means of a Laplace transformation in time and a Fourier transformation in the spatial variable. Subsequently, using the residue theorem and the method of steepest descent makes it possible to predict asymptotically the perturbation behavior as time t → ∞. The system is convectively unstable and a localized perturbation spreads out at the velocities of the trailing and leading fronts. The packet behavior is investigated over a wide range of the flow parameters.