On the influence of the prebuckling behavior of elastic structures on their initial postbuckling behavior

Depending on whether or not bending and/or torsion are part of the prebuckling deformation state of elastic structures, the initial postbuckling behavior is either completely general or characterized by specific restrictions [1]. In this context, the term ‘completely general’ means (a) no restriction to monotonic improvement or deterioration of the initial postbuckling behavior in consequence of a monotonic change of a design parameter and (b) the impossibility of ‘zero-stiffness postbuckling’ characterized by a postbuckling path at a constant load level. The mathematical reason for restrictions of the postbuckling behavior is a constraint condition for the eigenvector ν ₁ appearing in the classical static stability condition (1) where denotes the tangent stiffness matrix [2]. This condition contains the first and the second derivative of with respect to the dimensionless load parameter λ. The theoretical findings will be verified numerically. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress

The problem of Rayleigh waves in an orthotropic elastic medium under the influence of gravity and initial stress was investigated by Abd-Alla [A. M. Abd-Alla, Propagation of Rayleigh waves in an elastic half-space of orthotropic material, Appl. Math. Comput. 99 (1999) 61-69], and the secular equation of the wave in the implicit form was derived. However, due to the uncorrect representation of the solution, the secular equation is not right. The main aim of the present paper is to reconsider this problem. We find the secular equation of the wave in explicit form. By considering some special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity of some previous studies, in which only implicit secular equations were derived.     

Dispersion relations of elastic waves in three-dimensional cubical piezoelectric phononic crystal with initial stresses and mechanically and dielectrically imperfect interfaces

A three-dimensional cubical piezoelectric phononic crystal is theoretically studied in this paper, formed by pre-stressed piezoelectric rectangular blocks and imperfect interfaces. Firstly, transfer matrices of pre-stressed piezoelectric rectangular blocks and imperfect interfaces are obtained based on the structural characteristics. The Bloch waves consist of three groups of sub-coupled elastic waves corresponding with three orthogonal periodical directions in the three-dimensional periodical structure. Furthermore, based on the transfer matrices of typical single cell and the Bloch theorem, it is established that the theoretical model of above-mentioned three-dimensional cubical piezoelectric phononic crystal to obtain the dispersion relations of Bloch waves. Finally, the influences of non-dimensional geometrical parameters (structural and modal parameters) and physical parameters (initial stress and mechanically and dielectrically imperfect interface parameters) on the dispersion relations are discussed based on the graphically numerical results. Numerical calculation results show that the existences of initial stresses and mechanically and dielectrically imperfect interfaces equivalently lead to the alterations of structural flexibilities or rigidities. The theoretical model and numerical discussions will provide a direct guidance of multi-material additive manufacturing for pre-stressed and inhomogeneous periodic structures with partial and global dispersion properties.     

The velocities of long waves in composites with interphase imperfections

We develop a model of elastic strain of a composite material with interphase imperfections. In the context of this model we construct a continuous model of first order (theory of effective models) for stratified and unidirectional fibrous compounds. We determine the velocities of propagation of long (compared with the size of the structure) waves in any direction. We exhibit the possible characteristic regularities of the influence of interphase imperfections on the value of the velocities. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 100–105.     

Mathematical modelling and the study of the influence of initial stresses on the SIF and ERR at the crack tips in a plate-strip of orthotropic material

A plate-strip fabricated from the orthotropic material and containing a crack whose edges are parallel to the face planes of the plate is considered. It is assumed that the strip is stretched (or compressed) initially along the crack edges by uniformly distributed external normal forces acting on the simply supported ends of the plate-strip. After this initial stretching (or compression) the crack edges are loaded by additional uniformly distributed normal (opening) forces. As a result of the action of these additional forces the stress concentration characterized by the stress intensity factor (SIF) of mode I or by the energy release rate (ERR) of mixed mode arises at the crack tips. In this paper, the influence of the initial stresses on the SIF or ERR is modelled mathematically by the use of the three-dimensional linearized theory of elasticity. The aim of the present investigations is to study the effect of the mechanical–orthotropic properties of the plate-strip material on this influence by the use of the finite element method (FEM) modelling of the corresponding boundary-value problem.     

Influence of initial stresses on fracture of composite materials containing interacting cracks

Considered in this study are the axially-symmetric problems of fracture of composite materials with interacting cracks, which are subjected to initial (residual) stresses acting along the cracks planes. An analytical approach within the framework of three-dimensional linearized mechanics of solids is used. Two geometric schemes of cracks location are studied: a circular crack is located parallel to the surface of a semi-infinite composite with initial stresses, and two parallel co-axial penny-shaped cracks are contained in an infinite composite material with initial stresses. The cracks are assumed to be under a normal or a radial shear load. Analysis involves reducing the problems to systems of second-kind Fredholm integral equations, where the solutions are identified with harmonic potential functions. Representations of the stress intensity factors near the cracks edges are obtained. These stress intensity factors are influenced by the initial stresses. The presence of the free boundary and the interaction between cracks has a significant effect on the stress intensity factors as well. The parameters of fracture for two types of composites (a laminar composite made of aluminum/boron/silicate glass with epoxy-maleic resin and a carbon/plastic composite with stochastic reinforcement by short ellipsoidal carbon fibers) are analyzed numerically. The dependence of the stress intensity factors on the initial stresses, physical-mechanical parameters of the composites, and the geometric parameters of the problem are investigated.     

Phase and group velocities of normal waves in an orthotropic layer

The article analyzes the distribution of the phase and group velocities of normal, transversely symmetric waves in an anisotropic layer of the orthrhombic class. The numerical investigation of a complex variance equation made it possible to establish qualitative peculiarities of the behavior of phase and group velocities for stresses different from elastoequivalent ones.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 101–105, 1987.     

Determination of the magnitude and distribution of the initial stresses in glass-reinforced plastics

Photoelastic stress analysis is used to demonstrate the effect of coupling agents on the residual microstresses, i.e., the stresses that develop at near the glass-resin interface during the curing of polyester and epoxy glass-reinforced plastics and, as a rule, remain throughout the life of the product. Coupling agents of different chemical composition applied to the surface of the glass fibers have different effects on the components of the state of stress and thus can act as a powerful regulating factor.     

Determination of the magnitude and distribution of the initial stresses in glass-reinforced plastics

The stresses that develop in the resin at and near the glass-resin interface during the curing of polyester and epoxy glass-reinforced plastics have been experimentally studied by the photoelastic method. The volume state of stress in three mutually perpendicular directions in various sections and in the danger zones is qualitatively and quantiatively described. It is established that the stresses investigated may be the cause of cracking that reduces the tightness and strength of the glass-reinforced plastic.I. M. Gubkin Moscow Institute of the Petrochemical and Gas Industry. Translated from Mekhanika Polimerov, No. 4, pp. 661–667, July–August, 1973.     

The influence of coriolis force on MHD waves

The influence of coriolis force on the propagation of MHD waves in compressible medium is investigated by using linear stability analysis. In a combined effect we find that, while compressibility gives rise to an acoustic wave, coriolis force results in a modified Alfvén wave.     

On the propagation of Lamb waves in a sandwich plate made of compressible materials with finite initial strains

The propagation of flexural Lamb waves in a prestrained sandwich plate made from compressible highly elastic materials is investigated within the scope of a piecewise homogeneous body model by utilizing TLTEWISB. The mechanical relations of layer materials are described by a harmonic-type potential, and numerical results are obtained for the first and second vibration modes. According to the results, the influence of problem parameters and of the initial stretching strain along the layers on the wave propagation speed is examined. The asymptotic values of the speed are considered in the cases of short and long wavelengths, and the influence of the initial strains on these asymptotic (limit) values are also analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 231–244, March–April, 2008.     

Examples of the influence of the geometry on the propagation of progressive waves

In this paper, we give examples of the influence of the domain of propagation on progressive waves. More precisely, we numerically investigate the propagation of reaction diffusion waves in cylinders with variable radius. We show that, when the radius rapidly expands from a very small radius to a larger one, depending on the viscosity and the nonlinearity, the travelling wave may be blocked. The aim of this paper is to give numerical illustrations and quantifications of this effect, and to propose some conjectures which could be interesting subjects for further mathematical investigations.This work is linked to the study of spreading depression (SD), a propagative mechanism in brain and various tissues which has been observed in vivo and in vitro in many species since their discovery in 1944 by Leao. As a matter of fact, their direct observation in Man is still controversial. The complex structure of gray and white matter in humans may block the propagation of SD over large distances in brain and thus explain the difficulty of observing it. Medical consequences of the current numerical studies are detailed in [M.A. Dronne, et al., Influence of brain geometry on spreading depressions: A computationnal study, Progress in Biophysics and Molecular Biology 97 (1) (2008) 54–59] and a first mathematical approach given in [M.A. Dronne, E. Grenier, H. Gilquin, Modelization of spreading depressions following Nedergaard, preprint, 2003].     

Effects of initial stress on guided waves in orthotropic functionally graded plates

Because of the limitation of the manufacturing technology, initial stress in functionally graded materials (FGM) and structures is inevitable. Based on the theory of “Mechanics of Incremental Deformations”, the guided wave propagation in FGM plates under gravity, homogeneous initial stress in the thickness direction and inhomogeneous initial stress in the wave propagation direction is investigated. The Legendre polynomial series method is used to solve the coupled wave equations with variable coefficients. The convergence of the polynomial series method is discussed through the numerical examples. The effects of the initial stress on the Lamb-like wave and on the SH wave are investigated respectively and the numerical results show they are quite distinct. The effect of the gravity on the wave propagation can be ignored. The effects of the initial stress in the thickness direction are very different from those of the initial stress in the wave propagation direction, both on the dispersion curves and on the displacement and stress distributions.     

Three-dimensional Tollmien-Schlichting waves generated by sound in the boundary layer on an elastic surface at transonic free-stream velocities

Within the framework of the triple-deck theory, the effect of surface elasticity on three-dimensional packets of Tollmien-Schlichting waves generated by acoustic disturbances induced near the boundary layer at transonic free-stream velocities is investigated. It is shown that the elasticity of the surface considerably weakens the most unstable oblique waves but does not change the characteristic horseshoe shape of wave packets with two disturbance peaks propagating at an angle to the incoming flow.     

The influence of a crack in a medium on the interference waves

The interference Kr-waves in an elastic half-space with a crack are considered. The processes of dispersion and attenuation of such waves are described and theoretical seismograms are constructed. A number of frequency and velocity characteristics are discussed, which can help to identify the waves in mining investigations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 110–115, 1989.