Loss of Ellipticity in Plane Deformation of a Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solid

Change of type in the governing equations of equilibrium is examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for strength of reinforcement. Plane deformations interpreted in terms of both local and global plane strain are considered. Loss of ordinary ellipticity is found to occur for sufficiently large strength of reinforcement under sufficiently severe deformation which necessarily involves contraction in the reinforcing direction. Loss of ellipticity in local plane strain is easily characterized, and its incipient breakdown is associated with the possible emergence of surfaces of weak discontinuity with orientation normals in the reinforcing direction. Loss of ellipticity in global plane strain is given a two-dimensional manifold characterization in a space involving 2 deformation parameters and the strength of reinforcing parameter. Orientation normals for the associated surfaces of weak discontinuity at incipient breakdown do not in general conform to the reinforcing direction. This revised version was published online in August 2006 with corrections to the Cover Date.     

Relaxed Energy for Transversely Isotropic Two-Phase Materials

The paper gives a simple derivation of the relaxed energy W ^qc for the quadratic double-well material with equal elastic moduli and analyzes W ^qc in the transversely isotropic case. We observe that the energy W is a sum of a degenerate quadratic quasiconvex function and a function that depends on the strain only through a scalar variable. For such a W, the relaxation reduces to a one-dimensional convexification. W ^qc depends on a constant g defined by a three-dimensional maximum problem. It is shown that in the transversely isotropic case the problem reduces to a maximization of a fraction of two quadratic polynomials over [0,1]. The maximization reveals several regimes and explicit formulas are given in the case of a transversely isotropic, positive definite displacement of the wells. This revised version was published online in June 2006 with corrections to the Cover Date.     

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

Regioselective synthesis of 2,4-disubstituted thiophenes

A highly regioselective route was established to 2-aryl-, 2-cyclohexyl-, and 2-(2-arylethyl)4-alkylthiophenes, which are potential candidates as liquid crystalline compounds of low viscosity. The key synthetic intermediates, 2-substituted-4-(chloromethyl)thiophenes 6, 14, and 20 were prepared respectively from the reactions of β, γ-epoxycarbonyl compounds 5, 13, and 19 with Lawesson's reagent in the presence of a catalytic amount of p -toluenesulfonic acid. The epoxycarbonyl compounds were obtained from the TiCl₄-mediated reactions of 2-(chloromethyl)-3-(trimethylsliyl)propene (10) with acid chlorides followed by epoxidation with m-chloroperoxybenzoic acid, or from prior epoxidation followed by oxidation with pyridinium dichromate of homoallylic alcohols 3. The homoallylic alcohols 3 were synthesized from the reactions of 2-(chloromethyl)-3-(trichlorosilyl)propene (2) with aldehydes in N, N-dimethylformamide. Copper (I) catalysed cross-coupling reactions of 2-substituted-4-(bromomethyl)thiophenes (which were prepared by transhalogenation of 2-substituted-4- (chloromethyl)thiophenes with NaBr in acetone) with Grignard reagents afforded 2,4-disubstituted thiophenes. Using this method, eleven 2,4-disubstituted thiophenes were synthesized and their potentials as liquid crystalline compound of low viscosity were examined. The synthesized 2-(4-cyanophenyl)-4-pentylthiophene was observed to have a lower melting point than the corresponding 2,5-disubstituted thiophene. This observation is consistent with the expectation from the basis of molecular linearity which can affect the viscosity and/or melting point of crystalline compounds.     

Elastic Deformation of Dispersely Failing Unidirectionally Reinforced Composites

A phenomenological model is proposed for describing the elastic deformation of a unidirectionally reinforced composite capable of accumulating scattered microdamages during its loading. The composite is considered as a homogeneous transversely isotropic solid. Its damaged state at every point is characterized by a centrally symmetric scalar function on the unit sphere (the damage function), which is used to account for variations in the elastic properties of the material during its deformation. The damage itself depends on the history of some equivalent strain, for which four simplified variants are suggested. The relation between strains and stresses is defined in a differential form. Dependences are presented for determining all unknown constants from simple mechanical experiments. As an example, an actual unidirectionally reinforced GFRP is considered, for which the main two-dimensional sections of corresponding failure surfaces are also constructed.     

Influence of the orientation of the isotropy axis on the stability of transversely isotropic semibounded laminates

The surface-buckling problem for elastic transversely isotropic laminates is solved using the three-dimensional linearized theory of stability with the assumption of small prebuckling strains. The influence of the orientation of the isotropy axis on the critical parameters of the problem and the equilibrium state of the material is studied analytically and numerically __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 47–55, October 2006.     

Magnetic properties and microstructure of HDDR isotropic Nd-Dy-Fe-Co-B bonded magnets with high coercivity

The effect of small amount of Co and Dy addition on the magnetic properties of HDDR isotropic Nd-Dy-Fe-Co-B bonded magnets was investigated. The experimental results show that the intrinsic coercivity H_cj and the reversible temperature coefficient of remanence of (Nd_0.65 Dy_0.35)_12.5-(Fe_0.9Co_0.1)₈₁B_6.5 magnet were 1.53 MA·m^-1(19.3 kOe) and -0.059%/°C (25–155°C), respectively. The high coercivity and low temperature coefficient of the magnet are due to the enhanced anisotropy field, increased Curie temperature and improved microstructure by Dy and Co addition.     

A Deformation Model of Monotropic Plastic Foams in Shear Perpendicularly to the Isotropy Plane

Based on an ellipsoidal model for the cellular structure of monotropic plastic foams, their deformational properties in shear perpendicularly to the isotropy plane are predicted. The theoretical results are obtained for the case of pure shear. The deformation energy is calculated by the method of coordinates assuming that the polymer struts of the structure can change both their length and orientation. Numerical values of shear moduli are found for monotropic and isotropic plastic foams, which agree satisfactorily with the experimental data for polyurethane and polyvinylchloride foams available in the literature.     

Three-dimensional surface current loops in broadband responsive negative refractive metamaterial with isotropy

We propose a bulk negative refractive index (NRI) metamaterial composed of periodic array of tightly coupled metallic cross-pairs printed on the six sides of a cube for applications of superlenses. The structural characteristics of the three-dimensional (3D) metamaterial consist in the high symmetry and the superposition of metallic cross-pairs, which can increase the magnetic inductive coupling between adjacent cross-pairs and realize a broadband and isotropic NRI. The proposed 3D structure is simulated using the CST Microwave Studio 2006 to verify the design validity. The simulation results show that the proposed structure can not only realize simultaneously an electric and magnetic response to an incident electromagnetic (EM) wave, but also exhibit a broadband NRI whose relative bandwidth can reach up to 56.7%. In addition, the NRI band is insensitive to the polarization and the incident angle of the incident EM wave. Therefore, the proposed metamaterial is a good candidate material as three-dimensional broadband isotropic NRI metamaterial.     

Room temperature Compton profiles of conduction electrons in α-Ga metal

Room temperature Compton profiles of momentum distribution of conduction electrons in α -Ga metal are calculated in band model. For this purpose, the conduction electron wave functions are determined in a temperature-dependent non-local model potential. The profiles calculated along the crystallographic directions, (100), (010), and (001) are found to be nearly isotropic. This conclusion is in reasonable agreement with experimental observations