Interaction between cracks in a piecewise-homogeneous body

Institute of Applied Problems of Mechanics and Mathematics, Academy of Sciences of the Ukraine, L'vov. Translated from Prikladnaya Mekhanika, Vol. 28, No. 12, pp. 46–56, December, 1992.     

A study of the influence of the orientation of a planar surface crack in the shape of a limaçon of pascal on the stress concentration in a half-space

We study the stress state in the vicinity of a planar surface crack whose boundary is described by the limaçon of pascal. The problem is solved by a conformal mapping of the region occupied by the crack onto part of a disk in the plane. This makes it possible to apply a numerical-analytic method for solving systems of double singular integral equations of the mathematical theory of cracks. We present the graphs of the dependence of the stress intensity factor on the angular coordinate for cracks that are part of a limaçon of Pascal.     

Reduction of three-dimensional dynamical elasticity theory problems with arbitrarily located plane slits to integral equations

By generalizing a method described earlier /1/ for reducing three-dimensional dynamical problems of elasticity theory for a body with a slit to integral equations, integral equations are obtained for an infinite body with arbitrarily located plane slits. The interaction of disc-shaped slits located in one plane is investigated when normal external forces that vary sinusoidally with time (steady vibrations) are given on their surfaces.

Problems of the reduction of dynamical three-dimensional elasticity theory problems to integral equations for an infinite body weakened by a plane slit were examined in /1, 2/. The solution of the initial problem is obtained in /1/ by applying a Laplace integral transform in time to the appropriate equations and constructing the solution in the form of Helmholtz potentials with densities characterizing the opening of the slit during deformation of the body. The problem under consideration is solved in /2/ by using the fundamental Stokes solution /3/ with subsequent construction of the solution in the form of an analogue of the elastic potential of a double layer.     

Regio- and stereoselective synthesis of truncated 3'-aminocarbanucleosides and their binding affinity at the A3 adenosine receptor

The stereoselective synthesis of truncated 3'-aminocarbanucleosides 4a-d via a stereo- and regioselective conversion of a diol 9 to bromoacetate 11a and their binding affinity towards the human A(3) adenosine receptor are described.     

Synthesis and isomerization reaction of 2-(benzoxazol-2-yl)-1,3-tropolones

An acid-catalyzed reaction of 2-methylbenzoxazole with 3,4,5,6-tetrachloro-1,2-benzoquinone leads to 2-(benzoxazol-2-yl)-5,6,7-trichloro-1,3-tropolone and 2-(benzoxazol-2-yl)-4,5,6,7-tetrachloro-1,3-tropolone. 1,3-Tropolones upon reflux in ethyl alcohol undergo the rearrangement resulted in the seven-membered ring contraction to form ethyl 2-(benzoxazol-2-yl)-5,6-dichloro-3-hydroxybenzoate and ethyl 2-(benzoxazol-2-yl)-4,5,6-trichloro-3-hydroxybenzoate. The molecular structure of the latter was established by X-ray diffraction analysis.     

Synthesis and structure of 3-arylamino-2-(quinolin-2-yl)tropones

A reaction of substituted 2-(quinolin-2-yl)-1,3-tropolones with POCl₃ leads to the new3-chloro-2-(quinolin-2-yl)tropone derivatives. New 2-(4-arylaminoquinolin-2-yl)-3-aryl-aminocyclohepta-2,4,6-trien-1-ones and 2-[4-morpholino(piperidino)quinolin-2-yl]-3-aryl-aminocyclohepta-2,4,6-trien-1-ones were obtained by nucleophilic substitution reaction of the halogen atom in 2-(4-chloroquinolin-2-yl)-3-chlorotropones and 3-chloro-2-[4-morpholino-(piperidino)quinolin-2-yl]tropones. Molecular structures of 3-chloro-2-(quinolin-2-yl)tropone and 3-arylamino-2-(quinolin-2-yl)tropone were established by X-ray diffraction analysis. Energy and structural characteristics of isomers of 3-arylamino-2-(quinolin-2-yl)tropones in the gas phase and in solution were calculated by the density functional theory method (PBE0/6-31G**).     

Non-Lipschitz points and the {\textit{SBV}} regularity of the minimum time function

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function \(T\) under controllability conditions which do not imply the Lipschitz continuity of \(T\) . We consider first the case of normal linear control systems with constant coefficients in \({\mathbb {R}}^N\) . We characterize points around which \(T\) is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call \(\mathcal {S}\) . Furthermore, we show that \(\mathcal {S}\) is countably \(\mathcal {H}^{N-1}\) -rectifiable with positive \(\mathcal {H}^{N-1}\) -measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set \(\mathcal {S}\) in a neighborhood of the origin in a similar way and prove the \(\mathcal {H}^1\) -rectifiability of \(\mathcal {S}\) and that \(\mathcal {H}^1(\mathcal {S})>0\) . In both cases, \(T\) is known to have epigraph with positive reach, hence to be a locally \(BV\) function (see Colombo et al.: SIAM J Control Optim 44:2285–2299, 2006; Colombo and Nguyen.: Math Control Relat 3: 51–82, 2013). Since the Cantor part of \(DT\) must be concentrated in \(\mathcal {S}\) , our analysis yields that \(T\) is locally \(SBV\) , i.e., the Cantor part of \(DT\) vanishes. Our results imply also that \(T\) is differentiable outside a \(\mathcal {H}^{N-1}\) -rectifiable set. With small changes, our results are valid also in the case of multiple control input.