Transient stresses in a transversely isotropic elastic solid caused by a moving dislocation

The plane stress field induced in an unbounded, transversely isotropic, elastic solid by a dislocation moving parallel to the material symmetry axis, is reduced to a residue calculation. The dislocation, which is suddenly applied, causes a jump in displacement across the expanding fault. The speed of the dislocation is subsonic with respect to the material propagation speeds. Explicit results are obtained for the shear stress along the axis containing the dislocation and are related to several hexogonal crystals.

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Zusammenfassung Das ebene Verformungsfeld, das in einem unbegrenzten transversal isotropen elastischen Körper infolge einer sich parallel zur Symmetrieachse bewegten Versetzung entsteht, läßt sich mit einer einfachen Residuenrechnung ermitteln. Die Versetzung, die plötzlich entstehen soll, verursacht eine sprunghafte Veränderung der Verschiebung über sich ausdehnenden Fehler. Die Geschwindigkeit der Versetzung ist in Bezug auf die materielle Fortpflanzungsgeschwindigkeit der Unterschallgeschwindigkeit gleich. Für die Schubspannung längs der Dislokationsachse werden explizite Resultate angegeben. Diese Resultate gelten auch für einige hexagonale Kristalle.

     

Transient stresses in a transversely isotropic elastic solid caused by a dislocation moving at tran- or super-sonic speeds

A suddenly applied dislocation, moving at tran- or super-sonic speed, induces a plane stress field in an unbounded, transversely isotropic, elastic solid. Treated in detail are the-function plane waves associated with this disturbance. It is shown that at a special dislocation speed, in the tran-sonic range, these head waves disappear. The results are applied to several hexagonal crystals.

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Zusammenfassung Eine plötzliche, sich mit transonarer- oder Überschallgeschwindigkeit ausbreitende Versetzung verursacht ein ebenes Verformungsfeld in einem unbegrenzten, transversal isotropen, elastischen Körper. Die mit dieser Störung verbundenen und mit Hilfe von Deltafunktionen beschriebenen, ebenen Wellen werden im Detail behandelt. Es wird gezeigt, dass diese Stosswellen bei einer bestimmten Geschwindigkeit im transonaren Bereich verschwinden. Die Ergebnisse werden auf einige hexagonale Kristalle angewendet.

     

Transient stresses in a transversely isotropic elastic solid caused by a moving dislocation whose path may intersect a lacuna

The-function plane waves caused by the leading edge of a dislocation moving at tran- or super-sonic speed in an unbounded, transversely isotropic, elastic solid are treated herein for situations in which a lacuna intersects the dislocation path. Generally the Mach lines emitted by the source extend backward, but if the source is located within a lacuna, forward Mach lines are also produced. Several hexagonal crystals for which this phenomenon occurs are considered.

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Zusammenfassung Der vorliegende Artikel befasst sich mit solchen, von Delta Funktionen beschriebenen, ebenen Wellen, die sich infolge einer Versetzung in einem transversal isotropen Körper mit transsonarer, oder Überschallgeschwindigkeit ausbreiten. Besonders behandelt werden Fälle, in denen sich diese Wellen auf eine störungsfreie Zone zubewegen. Normalerweise erstrecken sich die von der Störungsquelle erzeugten Machlinien nach rückwärts. Sobald sich die Störungsquelle jedoch innerhalb der störungsfreien Zone (Stillstelle) befindet, entstehen zusätzliche, nach vorwärts gerichtete Machlinien. Bestimmte hexagonale Kristalle, auf die dieses Phänomen zutrifft, werden gesondert betrachtet.

     

Steady state thermal stresses in an elastic solid bounded by two cones

Zusammenfassung Es wird für das Problem der Wärmespannungen in einem kegelförmig begrenzten elastischen Körper eine Ähnlichkeitstransformation entwickelt. Ist die Temperatur an der freien Oberfläche als Potenz des Abstandes von der Spitze vorgeschrieben, so lässt sich ein Satz von Ähnlichkeitsvariablen angeben, welcher die thermoelastischen Beziehungen auf Legendre-Gleichungen reduziert und damit der Lösung leicht zugänglich macht. Die Rechnung ist für den Sonderfall konstant gehaltener Oberflächentemperatur im einzelnen durchgeführt.

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This work has been done at the Mathematics Research Center, University of Wisconsin, under the U.S. Army Contract DA-11-022-ORD-2059.     

A strip cut in a transversely isotropic elastic solid

The three-dimensional problems of a strip cut in a transversely isotropic elastic space, when the isotropy planes are perpendicular to the plane of the cut, are investigated using the asymptotic methods developed by Aleksandrov and his coauthors. Two cases of the location of the strip cut are considered: along the first axis of a Cartesian system of coordinates (Problem A) or along the second axis (Problem B). Assuming that the normal load, applied to the sides of the cut (normal separation friction) can be represented by a Fourier series, one-dimensional integral equations of problems A and B are obtained, the symbols of the kernels of which are independent of the number of the term of the Fourier series. A closed solution of the problem is derived for a special approximation of the kernel symbol. Regular and singular asymptotic methods are also used to solve the integral equations by introducing a dimensionless geometrical parameter, representing the ratio of the period of the applied wavy normal load to the thickness of the cut strip. The normal stress intensity factor on the strip boundary is calculated using the three methods of solving the integral equations indicated.     

Plane strain displacement and stress waves induced in a transversely isotropic elastic solid by a line source

The plane strain displacement and stress field induced in an unbounded, transversely isotropic, elastic solid by a uniform line load is reduced to a simple residue calculation. Explicit results on a coordinate axis are given for three representative hexagonal crystals when the line load is normal to the axis of material symmetry. The case in which the applied load is not normal to the symmetry axis is also discussed.

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Zusammenfassung Der ebene Verformungs- und der ebene Spannungszustand, die in einem unbegrenzten transversal isotropen elastischen Körper durch eine gleichförmige Linienbelastung erzeugt werden, lassen sich auf eine einfache Residuenrechnung reduzieren. Als Anwendung werden drei repräsentative hexagonale Kristalle besprochen, wobei die Linienbelastung zunächst normal zur Symmetrieachse und dann allgemein vorausgesetzt wird.

     

Thermal stresses in a plate with transversely isotropic material

In this paper the thermal stresses in a plate with transversely isotropic material have been obtained by the method of Hankel transforms. Three cases of surface temperature over a circular region of exposure with flux of heat, paraboloidal distribution and constant temperature with surface radiation have been considered. Numerical results are presented for the case of surface radiation.     

Stress channelling in transversely isotropic elastic composites

Résumé Une théorie a déjà été proposée au sujet des matériaux à fibres de renforcement où l'on supposait que ces dernières étaient inextensibles et uniformément distribuées dans un composite considéré comme incompressible. Cependant, quelques unes des prédictions de cette théorie semblent être fondamentalement en désaccord avec la théorie classique de l'élasticité. Il est démontré ici que les résultats inattendus de cette théorie correspondent en fait à des cas limites de la théorie classique de l'élasticité pour des matériaux à isotropie transversale.     

An asymmetric wave field in a transversely isotropic elastic medium

A transversely isotropic homogeneous elastic medium excited by a point force perpendicular to the anisotropic axis is considered. The wave field in this medium is constructed and investigated. The front sets of the SV and SH waves are in contact with one another at a point. The front sets in the vicinity of this point are investigated additionally. If we consider the SH wave (or the SV wave) separately, then a false plane front set arises in this region. In considering the SH and SV waves in combination, this false front set disappears. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 163–174.     

A transversely isotropic elastic model of geomaterials

Under consideration is the choice of parameters of a transversely isotropic elastic model for describing the linear deformation of geomaterials. We also discuss some analytical and numerical methods of solving the corresponding dynamic equations.     

The asymmetrical mixed temperature problem for a transversely isotropic elastic layer

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic elastic layer (soft plate) investigated is in ideal contact with an absolutely rigid layer, deformable only by thermal expansion. The generalized plane temperature problem reduces to determining the stress-strain state of the soft anisotropic layer investigated using the equations of the mixed problem of elasticity theory. At the ends of the boundary layer of the soft plate (a thin contact layer), no conditions are imposed. On the remaining part of the ends of the soft plate, the boundary conditions correspond to a free boundary. The problem has a bounded smooth solution. Unlike the approach described earlier [1], it is proposed to seek an accurate solution in the form of ordinary Fourier series with respect to a single longitudinal coordinate. Solutions in polynomials are also used. It is shown that the existence of these solutions in polynomials enables the convergence of the Fourier series to be improved considerably.     

Fundamental solution of displacement equations for a transversely isotropic elastic medium

A fourth-order linear elliptic partial differential equation describing the displacements of a transversely isotropic linear elastic medium is considered. Its symmetries and the symmetries of an inhomogeneous equation with a delta function on the right-hand side are found. The latter symmetries are used to construct an invariant fundamental solution of the original equation in terms of elementary functions.     

Nonlinear transversely isotropic elastic solids: an alternative representation

A strain energy function which depends on five independent variablesthat have immediate physical interpretation is proposed forfinite strain deformations of transversely isotropic elasticsolids. Three of the five variables (invariants) are the principalstretch ratios and the other two are squares of the dot productbetween the preferred direction and two principal directionsof the right stretch tensor. The set of these five invariantsis a minimal integrity basis. A strain energy function, expressedin terms of these invariants, has a symmetry property similarto that of an isotropic elastic solid written in terms of principalstretches. Ground state and stress–strain relations aregiven. The formulation is applied to several types of deformations,and in these applications, a mathematical simplicity is highlighted.The proposed model is attractive if principal axes techniquesare used in solving boundary-value problems. Experimental advantageis demonstrated by showing that a simple triaxial test can varya single invariant while keeping the remaining invariants fixed.A specific form of strain energy function can be easily obtainedfrom the general form via a triaxial test. Using series expansionsand symmetry, the proposed general strain energy function isrefined to some particular forms. Since the principal stretchesare the invariants of the strain energy function, the Valanis–Landelform can be easily incorporated into the constitutive equation.The sensitivity of response functions to Cauchy stress datais discussed for both isotropic and transversely isotropic materials.Explicit expressions for the weighted Cauchy response functionsare easily obtained since the response function basis is almostmutually orthogonal.     

Initial supercritical behavior of buckled transversely isotropic elastic medium

The paper is concerned with a transversely isotropic homogeneous elastic medium subjected to uniform compression in the isotropy plane. The medium becomes unstable in the sense of Hadamard [1] at a definite level of initial strain. The critical strain is established to be uniquely determinate from the system of equations of bifurcation of equilibrium; however, there are many modes of buckling corresponding to this strain. A solution of the system of equations of bifurcation is built in the form of doubly periodic functions sinr ₁ x ₁sinr ₂ x ₂. The uncertainty of the mode of buckling consists in the fact that the wave numbers r ₁ and r ₂ remain arbitrary. In order to determine the relationship between the wave numbers we examine the initial supercritical behavior of the material. It turns out that the only possible modes are the chess-board mode (with r ₁ = r ₂) and the corrugation-type mode (when one of the wave numbers r ₁ or r ₂ vanishes). The initial supercritical equilibrium is shown as being stable.     

Stability of a transversely isotropic spherical shell coupled with an elastic foundation

The stability "in the small" of a flat spherical shell with elastic reinforcement is investigated. It is assumed that the shell is made of material (glass-reinforced plastic) with low shear resistance [7, 8], which determines the choice of calculation procedure: generalized applied shell theories of the Timoshenko and Ambartsumyan types [1, 3]. The results obtained are compared with the corresponding results of the Kirchhoff-Love theory.L'vov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 1, pp. 129–131, January–February, 1970.     

Stress field around an arbitrary thin inclusion in a transversely isotropic elastic half-space

We consider a thin flat inclusion of arbitrary shape located inside a transversely isotropic elastic half-space in the plane parallel to its boundary z = 0. An arbitrary tangential displacement is prescribed on the inclusion. The boundary of the half-space is stress-free. We need to find the complete field of stresses and displacements in this half-space. A governing integral equation is derived by the generalized method of images, introduced by the author. The case of circular inclusion is considered as an example. Two methods of solution of the governing integral equation are derived. A detailed solution is presented for the particular cases of radial expansion, torsion and lateral displacement of the inclusion. The solution is also valid for the case of isotropy. The governing integral equation for the case of isotropy is derived.     

Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders

The present study examined mixed mode cracking in a transversely isotropic infinite cylinder. The solutions to axisymmetric Volterra climb and glide dislocations in an infinite circular cylinder of the transversely isotropic material are first obtained. The solutions are represented in terms of the biharmonic stress function. Next, the problem of a transversely isotropic infinite cylinder with a set of concentric axisymmetric penny-shaped, annular, and circumferential cracks is formulated using the distributed dislocation technique. Two types of loadings are considered: (i) the lateral cylinder is loaded by two self-equilibrating distributed shear stresses; (ii) the curved surface of the cylinder is under the action of a distributed normal stress. The resulting integral equations are solved by using a numerical scheme to compute the dislocation density on the borders of the cracks. The dislocation densities are employed to determine stress intensity factors for axisymmetric interacting cracks. Finally, a good amount of examples are solved to depict the effect of crack type and location on the stress intensity factors at crack tips and interaction between cracks. Numerical solutions for practical materials are presented and the effect of transverse isotropy on stress intensity factors is discussed.     

Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.