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.     

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.

     

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.     

Pullback attractors for a non-autonomous incompressible non-Newtonian fluid

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors AV in space V (has H²-regularity, see notation in Section 2) and AH in space H (has L²-regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing AV=AH, which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.     

Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.     

Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.     

Determining nodes of the global attractor for an incompressible non-Newtonian fluid

This paper estimates the finite number of the determining nodes to the equations for an incompressible non-Newtonian fluid with space-periodic or no-slip boundary conditions. The authors prove that, whenever the second order derivatives of two different solutions within the global attractor have the same time-asymptotic behavior at finite number of points in the physical space, then the two solutions possess the same time-asymptotic behavior at almost everywhere points of the physical space.     

Stress state of an isotropic shell with an elastic inclusion

Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 76–80, 1990.     

The coupled problem of a solid oscillating in a viscous fluid under the action of an elastic force

The torsional oscillations are studied of a solid of revolution under the action of elastic torque inside a container with a viscous incompressible fluid. We prove the asymptotic stability of the static equilibrium. We use the two approaches: the direct Lyapunov and linearization methods. The global asymptotic stability is established using a one-parameter family of Lyapunov functionals. Then small oscillations are studied of the fluid-solid system. The linearized operator of the problem of a solid oscillating in a fluid can be realized as an operator matrix obtained by appending two scalar rows and two columns to the Stokes operator. This operator is therefore a two-dimensional bordering of the Stokes operator and inherits many properties of the latter; in particular, the spectrum is discrete. The eigenvalue problem for the linearized operator is reduced to solving a dispersion equation. Inspection of the equation shows that all eigenvalues lie inside the right (stable) half-plane. Basing on this, we justify the linearization. Using an abstract theorem of Yudovich, we prove the asymptotic stability in a scale of function spaces, the infinite differentiability of solutions, and the decay of all their derivatives in time.     

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.

     

Propagation of discontinuity waves of any order through an elastic solid saturated with an inviscid fluid

Summary In this paper we study the propagation of discontinuity waves of any order through an elastic solid saturated with an inviscid fluid by using the linear theory derived in [1] by G. Szefer. By means of the singular surfaces theory we obtain the possible normal speeds of propagation of the wave front and the evolution law along the normal trajectories for transverse and longitudinal propagation.

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Riassunto In questo lavoro studiamo la propagazione di onde di discontinuità di ogni ordine in un mezzo elastico poroso saturato da un fluido non viscoso nell'ambito della teoria lineare formulata da G. Szefer in [1]. Usando il metodo delle superfici singolari otteniamo le possibili velocità normali di avanzamento del fronte d'onda e la legge di evoluzione delle discontinuità lungo le corrispondenti traiettorie normali sia per propagazione trasversale, sia per propagazione longitudinale.

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This work was supported by the G. N. F. M. of C. N. R.     

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.     

Unique solvability for a class of non-Newtonian fluids for a reacting mixture with vacuum

In this paper,we consider a class of non-Newtonian fluids for a reacting mixture in one-dimensional bounded interval, provided the initial data satisfying a compatibility condition. The main ingredient is that we allow the initial density vacuum.     

Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid

The authors construct the trajectory attractor and global attractor for an autonomous two-dimensional non-Newtonian fluid.     

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.     

On some unsteady flows of a non-Newtonian fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows impulsively started from rest by the motion of a boundary or two boundaries or by sudden application of a pressure gradient. Flows considered are: unsteady flow over a plane wall, unsteady Couette flow, flow between two parallel plates suddenly set in motion with the same speed, flow due to one rigid boundary moved suddenly and one being free, unsteady Poiseuille flow and unsteady generalized Couette flow. The results obtained are compared with those of the exact solutions of the Navier–Stokes equations. It is found that the stress at time zero on the stationary boundary for the flows generated by impulsive motion of a boundary or two boundaries is finite for a fluid of second grade and infinite for a Newtonian fluid. Furthermore, it is shown that for unsteady Poiseuille flow the stress at time zero on the boundary is zero for a Newtonian fluid, but it is not zero for a fluid of second grade.