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.     

Wave propagation in an initially stressed transversely isotropic thermoelastic solid half-space

The governing equations of thermoelasticity of transversely isotropic solid with initial stresses are formulated at uniform temperature. These equations are solved analytically in two-dimensions to show the existence of three plane quasi waves, namely, Quasi-Longitudinal (QL), Thermal (T-mode) and Quasi-Transverse (QT) waves. Reflection from a thermally insulated stress free surface of an initial stressed transversely isotropic thermoelastic solid half-space is studied. A particular model is chosen for the numerical computations of the propagation speeds, attenuation coefficients and reflection coefficients. Effects of initial stress parameter and thermal disturbances are observed on speeds of propagation, attenuation coefficients and reflection coefficients.     

Steady-state thermoelastic stresses in an infinite transversely isotropic medium containing an external circular crack

The temperature and the normal components of stress and displacement around an external circular crack in an infinite transversely isotropic body have been calculated in the present paper. The stress intensity factor has been found and a comparison of the results with those for the isotropic case has been presented graphically.     

Static plane-strain deformation of transversely isotropic magneto-electro-elastic and layered cylinders to general surface loads

Anisotropic and layered cylinders are important composite structures; however, their system of governing equations is usually solved numerically due to the complicated geometry and material anisotropy involved. In this paper, we analytically solve the plane-strain equations for general static deformation of a cylindrically anisotropic, layered magneto-electro-elastic (MEE) cylinder. We assume that the layers are perfectly bonded at the interfaces. We solve the equations through separation of variables and eigenfunction expansion. Results for each mode shape (2π periodic) are solved independently. Because the eigenspace of the mode shapes is the set of all continuous functions on the interval, any continuous loading can be applied and the corresponding solution can be found analytically through superposition of the mode-shape results. To check our formulation, we consider a cylinder with two isotropic-elastic layers under simple radial loading and reproduce the known, exact results. Then, we compare our formulation to an FEA solution for a layered piezo-electric (PE) cylinder. Finally, we apply a radial stress to three comparable MEE cylinders (one uniform MEE cylinder and two layered cylinders made of alternating piezo-electric (PE) and piezo-magnetic (PM) materials). Deformation and stress amplitudes are plotted for the first six mode shapes of each cylinder as benchmarks for further reference.     

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.

---

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.

     

Steady state stresses induced in a transversely isotropic elastic solid by a moving dislocation

An unbounded, transversely isotropic, elastic solid, is subjected to a dislocation moving at constant speed. By means of an appropriate coordinate transformation, the transient version of this problem is used to obtain the steady state solution. The solution for the plane stress field is explicit and valid for dislocation speeds which are sub-, tran-, or super-sonic with respect to the material wave speeds. The previously discovered transonic speed at which the Mach head wave was annihilated for the transient problem, is found to be present in the steady state problem also.     

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.     

A finite deformation possible in transversely isotropic materials

Zusammenfassung Der Aufsatz beweist die Existenz einer Drei-Parameter-Familie von inhomogenen Deformationen mit konstanten Deformationsinvarianten. Diese Deformationen lassen sich in jedem anfänglich homogenen transversalisotropen, inkompressiblen elastischen Material durch Oberflächenkräfte aufrechterhalten.     

Stress concentrations around openings in transversely isotropic shells

A method of investigating stress concentrations around openings in shells made of a transversely isotropic material (oriented glass-reinforced plastic [5]) is described. The use of a Timoshenko-type theory makes it possible to take into account the effect of tangential shearing stresses and the anisotropy of the elastic properties of the shell in a direction normal to the middle surface. A complex form of the resolvents of this theory, constructed in [6], is employed.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. L'vov Franko State University. Translated from Mekhanika Polimerov, No. 6, pp. 1076–1081, November–December, 1970.     

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.

---

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.

     

A transversely isotropic medium containing a penny-shaped crack subjected to a non-uniform axisymmetric loading via an anchored smooth rigid disk

A smooth rigid circular anchor disk encapsulated by a penny-shaped crack is embedded in and unbounded transversely isotropic medium. The lamellar rigid disk exerts a nonuniform axisymmetric loading to the upper face of the crack. With the aid of an appropriate stress function and Hankel transform, the governing equations are converted to a set of triple integral equations which in turn are reduced to a Fredholm integral equation of the second kind. For some transversely isotropic materials the normalized stiffness of the system falls well outside of the envelope pertinent to isotropic media. It is shown that mode I stress intensity factor is independent of the material properties and solely depends on the ratio of the radius of the rigid disk to that of the crack; moreover, for the cases where this ratio is less than about 0.9 a simple explicit approximate expression for the mode I stress intensity factor is derived. In contrast, the normalized mode II stress intensity factor is independent of the mentioned geometrical parameters but depends on the elastic properties of the material; depending on the material properties, the normalized mode II stress intensity factor can vary between 0 to ∞ for transversely isotropic materials and between 0 to π/4 for isotropic materials.     

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.     

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.

---

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.

     

Stability of a transversely isotropic cylindrical shell in axial compression

The problem of the stability of a transversely isotropic cylindrical shell subjected to axial compression is considered with the aid of three-dimensional linearized equations for small subcritical strains. Transcendental equations are obtained for the critical loads corresponding to axisymmetric and nonaxisymmetric deformation. It is shown that in the case of the axisymmetric buckling mode the Kirchhoff-Love hypothesis is asymptotically exact irrespective of the properties of the material. The dependence of the critical load on the properties of the shell material is investigated numerically. Graphs of the variation of the critical load with relative shear modulus are presented.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1064–1068, November–December, 1969.     

Three-dimensional dynamic Green’s functions in transversely isotropic tri-materials

An analytical derivation of the elastodynamic fundamental solutions for a transversely isotropic tri-material full-space is presented by means of a complete representation using two displacement potentials. The complete set of three-dimensional point-load, patch-load, and ring-load Green’s functions for stresses and displacements are given, for the first time, in the complex-plane line-integral representations. The formulation includes a complete set of transformed stress-potential and displacement-potential relations in the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For the numerical computation of the integrals, a robust and effective methodology is laid out. Selected numerical results for point-load and patch-load Green’s functions are presented to portray the dependence of the response on layering, the frequency of excitation, and type of loading.