Boundary integral equation methods in the theory of elasticity of hemitropic materials: A brief review

The theory of elasticity of hemitropic materials has recently been the object of rigorous mathematical analysis. In particular, the potential method and the theory of pseudodifferential equations have been used in studying the solvability in various function spaces of the main boundary value and transmission problems, in smooth and in Lipschitz domains. The main features and results of this boundary integral equations approach are briefly reviewed here.     

Synthesis and structure of copper(II) trans-diaqua-bis(3-hydroxybenzoylhydrazine) nitrate dihydrate [Cu(C7H8O2N2)2(OH2)2](NO3)22H2O

Synthesis of [Cu(m-HBH)₂(OH₂)₂](NO₃)₂·2H₂O, where m-HBH = C₇H₈O₂N₂ (3-hydroxybenzoylhydrazine), is described. The structure of the compound was studied by X-ray phase analysis and IR spectroscopy; crystal data are a = 57.415(6) Å, b = 19.760(2) Å, c = 7.586(2) Å; Fdd 2, Z = 16, R(F) = 0.053. The compound consists of [Cu(m-HBH)₂(OH₂)₂]²⁺ complex cations, NO ₃ ^? anions, and two water molecules. The similarity between the IR spectra of Cu(m-HBH)₂(NO₃)₂·nH₂O and Co(m-HBH)₂(NO₃)₂·5H₂O, element analysis data, and crystal data obtained at the first stage of X-ray analysis show that the structures and compositions of these compounds are identical relative to the type of surroundings of the central atom. In contrast to the cobalt compound [Co(m-HBH)₂(OH₂)₂](NO₃)₂·3H₂O, in which the cobalt atom has a nearly regular octahedron as a coordination polyhedron, the copper(II) compound has a square bipyramid around the copper atom; c.n. is 6 = 4 + 2 (planar distances: 2.013(2) Å, 2.021(2) Å, 2.033(3) Å, 2.087(3) Å; axial distances: 2.367(3) Å, 2.374(3) Å) and lacks one crystallization water molecule.     

The Basic Mixed Problem for an Anisotropic Elastic Body

The mixed boundary value problem is considered for an anisotropic elastic body under the condition that a boundary value of the displacement vector is given on some part of the boundary and a boundary value of the generalized stress vector on the remainder. Using the potential method and the theory of singular integral equations with discontinuous coefficients, the existence of a solution of the mixed boundary value problem is proved.     

Contact problems for two anisotropic half-planes with slits

The problem of a stressed state in a nonhomogeneous infinite plane consisting of two different anisotropic half-planes and having slits of finite number on the interface line is investigated. It is assumed that the difference between the displacement and stress vector values is given on the interface line segments; on the edges of the slits we have the following data: boundary values of stress vector (problem of stress) or displacement vector values on one side of the slits, and stress vector values on the other side (mixed problem). Solutions are constructed in quadratures.     

Crystalline and molecular structure of diaminoguanidinium tetrachlorozincate

A single-crystal X-ray diffraction study of a single crystal of the composition (AgunH)₂[ZnCl₄], where AgunH = (NH₂)₂C-NH-NH₂)⁺, has been performed. Crystals are monoclinic, space group P2₁/c, Z = 4, a = 7.642(2) Å, b = 24.123(4) Å, c = 7.496(2) Å, β = 81.77(2)°, V = 1368.0(2) ų, T = 293 K. The finite value R = 0.045 was obtained for 1878 independent reflections with I > 3σ(I). The structure was built from cations of aminoguanidinium and tetrahedral complex anions [ZnCl₄]^2?.     

Structure of trans-diaqua-bis(3-hydroxybenzoylhydrazine)copper(II) sulfate monohydrate, [Cu(C7H8O2N2)2(OH2)2]SO4·H2O

Synthesis, X-ray phase analysis (XRPA), and IR spectroscopic study were performed to investigate the structure of [Cu(m-HBH)₂(OH₂)₂]SO₄·H₂O, where m-HBH = C₇H₈O₂N₂ (3-hydroxybenzoylhydrazine); a = 7.154(3) Å, b = 8.012(2) Å, c = 20.061(2) Å, β = 90.56(2)°; P2₁, Z = 2, R(F) = 0.046. The structure of the compound consists of the [Cu(m-HBH)₂(OH₂)₂]²⁺ complex cations, SO ₄ ^2? anions, and one water molecule. The copper atom is surrounded by a tetragonal bipyramid; c.n. is 6 = 4 + 2 (the distances are 2.030(5) Å, 2.054(6) Å, 2.061(5) Å, and 2.097(3) Å (equatorial); 2.459(4) Å and 2.53(4) Å (axial)). In complex compounds of 3d metals containing m-HBH, the function of the latter is probably invariable; the distribution of the anions and water molecules in the structure depends on the type of the metal and on the anion shape and geometry for metals with slightly different ionic radii.     

X-ray study of M(Aet)2·8H2O (M = Sr, Ba) and the crystal structure of [Sr(Aet)(OH2)5](Aet)·3H2O

An X-ray phase and X-ray structural analysis of M(Aet)₂-8H₂O compounds, where M(II) = Sr and Ba; (Aet)^- = (C₁₀H₁₁N₄O₂S₂) is the sulfaethiodole (2-(para-aminobenzene-sulfamido)-5-ethyl-1,3,4-thiadiazole) anion is carried out. Symmetry and unit cell parameters of Sr(Aet)₂•8H₂O and Ba(Aet)₂-8H₂O are found: P2₁/c, Z= 4, a = 10.390(2) ?, b= 14.609(3) ?, c = 21.931(5) ?, β = 92.42(2)°, R = 0.041; and P2₁/c, Z= 4, a = 9.793(2) ?, b = 15.408(4) ?, c = 22.553(6) ?, β = 94.98(2)° respectively. The considered compounds are isostructural. The structure of Sr(Aet)₂-8H₂O is fully studied; the crystals of the compound are composed of the complex centrosymmetric dimeric [(H₂O)5Sr(Aet)₂Sr(OH₂)₅]²⁺ cations, (Aet)^- anions, and water molecules; the independent part of the structural formula of the compound is [Sr(Aet)(OH₂)₅](Aet)•3H₂O. The sulfaethiodole anion is coordinated to the complexing metal atom by oxygen and nitrogen atoms with the formation of a four-membered ring.     

Crystal structure of cis-diaquabis(sulfapyridazinato) nickel(II)

We have previously found that ethazole and sulfadimethoxine (sulfanilamide representatives) do not coordinate to Mg(II), Mn(II), Co(II), Ni(II), Cu(II), and Zn(II) metals, and they are outer-sphere anions in the complex compounds of these metals. This work presents the results of the structural study of yet another sulfanilamide representative: the [Ni(A)₂(OH₂)₂] compound containing sulfapyridazine. Here A is the sulfapyridazine anion (C₁₁H₁₁N₄O₃S is 6-(p-aminobenzenesulfamido)-3-methoxypyridazine) coordinated by nitrogen atoms (N(1) and N(2)), thus functioning as a bidentate cyclic ligand. The symmetry and unit cell parameters are as follows: monoclinic, C2/c space group, a = 20.116(3) ?, b = 8.221(2) ?, c = 16.923(4) ?, β = 75.19(5)°, V = 2705.6(7) ?³, T = 293 K, and Z = 4. The final value of R = 0.048 was obtained for 1930 independent reflections with I > 2σ(I). The structure consists of complex molecules.     

Crystal and molecular structure of guanidinium dicitratoferrate(II)

Synthesis and X-ray investigation of a (GunH)₂[Fe(Cit)₂] single crystal are described [a = 10.327(2) Å, b = 10.414(2) Å, c = 11.267(2) Å; β = 117.25(3)°, V = 1077.2(4) ų, P2₁/n, Z = 2, R(F) = 0,0239, 1196 reflections with I > 2σ(I )], where GunH is the guanidine cation (C(NH₂)₃)⁺, and Cit is the citric acid anion (C₆H₆O₇)^2?. The structure is composed of guanidinium cations and centrosymmetrical sym-cis-octahedral complex anions [Fe(Cit)₂]^2?. The Cit^2? anion acts as a cyclic tridentate ligand and forms condensed fiveand six-membered metal-containing cycles.