Linear elastic chain with a hyper-pre-stress

To account for surface relaxation in ultra-thin films, we consider the simplest one-dimensional discrete chain with harmonic interactions of up to second nearest neighbors. We assume that the springs, describing interactions of the nearest neighbors (NN) and next to nearest neighbors (NNN) have incompatible reference lengths, which introduce a hyper-pre-stress and results in a formation of the exponential surface boundary layers. For a finite body loaded by a system of (double) forces at the boundary, we explicitly find the displacement field and compute the energies of the inhomogeneous stressed and reference configurations. We then obtain a simple expression for the hyper-pre-stress-related contribution to the surface energy and show an unusual scaling of the total energy with the film thickness. For ultra-thin films we report an anomalous stiffness increase due to the overlapping of the surface boundary layers. Implications of the micro level hyper-pre-stress in fracture mechanics and in the theory of non-Bravais lattices are also discussed.     

Interaction of plane elastic harmonic waves with a perfectly bonded elastic inclusion

The interaction of plane harmonic waves with a thin elastic inclusion in the form of a strip in an infinite body (matrix) under plane strain conditions is studied. It is assumed that the bending and shear displacements of the inclusion coincide with the displacements of its midplane. The displacements in the midplane are found from the theory of plates. The priblem-solving method represents the displacements as discontinuous solutions of the Lamé equations and finds the unknown discontinuities solving singular integral equations by the numerical collocation method. Approximate formulas for the stress intensity factors at the ends of the inclusion are derived     

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.     

Interaction of an elastic wave with a plate

Within the framework of the dynamic theory of elasticity a numerical solution is set up for an axisymmetric problem which arises in connection with the problem of measuring stresses on the boundary between a solid medium and a rigid wall. In the cylindrical r, z coordinates the medium fills the cylinder z > 0, r < R, the case R being possible when the medium occupies the half-space z > 0. The elastic medium borders on a rigid wall which in the plane z = 0 has a deformable part in the form of a circular elastic plate clamped along the edges. A plane longitudinal wave in the form of a semiinfinite step falls from infinity. The interaction of this wave with the plate is investigated, with the main attention given to the study of the effect of the problem parameters on the deflection of the plate subjected to the wave.Deceased.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 74–85, March–April, 1972.Concluding, the author thanks the participants of the Seminar on Dynamics of Solids, at the Institute of Problems of Mechanics, for the discussion of the paper.     

Stability loss of a series of fibers in an elastic matrix with highly elastic deformation

Ternopol' Branch, L'vov Polytechnic Institute. Translated from Prikladnaya Mekhanika, Vol. 27, No. 7, pp. 47–52, July, 1991.     

Vibrations of an elastic orthotropic layer with a cavity

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–97, January–February, 1991.     

Bending of elastic plates with a physically nonlinear inclusion

An infinite elastic isotropic plate with an elliptical, physically nonlinear inclusion loaded at infinity by uniformly distributed moments is considered. Surface loads are absent. The problem of the stress-strain state of the plate is solved in a closed form. It is shown that, for reasonably general stress-strain relations for the inclusion, the bending-moment field (and the corresponding curvatures) in the inclusion is homogeneous. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 152–157, November–December, 2006.     

Some static problems for elastic bodies with a crust

Summary This paper studies the diffusion of stresses in a linear isotropic elastic body coated with a thin elastic shell called the ?crust?. Two static problems are considered. The first is that of the semi-infinite half-space covered by a thin plate subject to a point load at the origin. In the second problem a solid elastic sphere enveloped by a thin spherical elastic shell with two opposite point loads at the poles is investigated. Solutions for the displacement within the thin crust are obtained in the two problems by means of the Hankel transform and the Legendre polynomial respectively.

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Sommario Questo lavoro studia la diffusione degli sforzi in un corpo elastico, lineare, isotropo rivestito da un guscio elastico sottile denominato la ?crosta?. Sono considerati due problemi elastostatici. Il primo è quello di un semispazio coperto da una piastra sottile soggetta and un carico concentrato nell'origine. Nel secondo problema si studia una sfera solida elastica avvolta da un guscio sottile elastico con due carichi concentrati opposti nei poli. Nei due problemi le soluzioni in termini di spostamenti entro la crosta si ottengono rispettivamente mediante la trasformata di Hankel e i polinomi di Legendre.

     

Squeeze flow of a Bingham-type fluid with elastic core

The squeeze flow of a Bingham-type material between finite circular disks is considered. The material is modelled assuming that the unyielded region behaves like a linear elastic core. A lubrication approximation is considered. It is shown that no paradox can arise, such as that has been pointed out for many years by various authors when the unyielded region in the fluid is supposed to be perfectly rigid. The unyielded region is shown to be always detached from the axis of symmetry. Some numerical simulations are worked out for different squeezing rates.     

Momentless elastic reinforced shells with a zero gaussian curvature

In [1], a number of statements were formulated of the problem of the attainment of a momentless stressed state in elastic reinforced shells with an arbitrary form of their middle surface. The present work is devoted to the solution of three of the statements advanced in [1], for the case where the middle surface has a zero Gaussian curvature.     

The delamination of a growing elastic sheet with adhesion

We study the onset of delamination blisters in a growing elastic sheet adhered to a flat stiff substrate. When the ends of the sheet are kept fixed, its growth causes residual stresses that lead to delamination. This instability can be viewed as a discontinuous buckling between the complete adhered solution and the buckled solution. We provide an analytic expression for the critical deformation at which the instability occurs. We show that the critical threshold scales with a single dimensionless parameter that comprises information from the geometry of the sheet, the mechanical parameters of material and the adhesive features of the substrate.