Bar Of Square Cross Section Research Articles

Bending of a Cosserat Elastic Bar of Square Cross Section: Theory and Experiment

Pure bending experiments on prismatic bars of square cross section composed of reticulated polymer foam exhibit deformation behavior not captured by classical elasticity theory. Perhaps the clearest example of this is the observed sigmoidal deformation of the bars' lateral surfaces, which are predicted by classical elasticity theory to tilt but remain planar upon pure moment application. Such foams have a non-negligible length scale compared to the bars' cross-sectional dimensions, whereas classical elasticity theory contains no inherent length scale. All these facts raise the intriguing question: is there a richer, physically sensible, yet still continuum and relatively simple elasticity theory capable of modeling the observed phenomenon in these materials? This paper reports our exploration of the hypothesis that Cosserat elasticity can. We employ the principle of minimum potential energy for homogeneous isotropic Cosserat linear elastic material in which the microrotation vector is taken to be independent of the macrorotation vector (as prior experiments indicate that it should be in general to model such materials) to obtain an approximate three-dimensional solution to pure bending of a prismatic bar having a square cross section. We show that this solution, and hence Cosserat elasticity, captures the experimentally observed nonclassical deformation feature, both qualitatively and quantitatively, for reasonable values of the Cosserat moduli. A further interesting conclusion is that a single experiment—the pure bending one—suffices to reveal directly, via the observation of surface deformation, the presence of nonclassical elastic effects describable by Cosserat elasticity.

Momentum and heat transfer characteristics of a long parallelepiped submerged in power-law fluids in the laminar vortex shedding regime

The governing partial differential equations describing the flow and forced convection heat transfer characteristics of power-law fluids past a long parallelepiped at α=45° incidence have been solved numerically in the two-dimensional laminar flow regime. In particular, the major thrust of this work is to elucidate the influence of Reynolds number (Re), Prandtl number (Pr) and power-law index (n) on the detailed and macroscopic momentum and heat transfer characteristics in the laminar vortex shedding regime. The detailed features of the flow and heat transfer phenomena are visualized and analyzed in terms of the instantaneous streamlines, iso-vorticity and isotherm contours whereas the surface pressure and local Nusselt number distributions over the surface of the parallelepiped provide the next level of description of these phenomena. The gross features of the flow are captured in terms of the time-averaged drag coefficient, rms value of the lift coefficient, Strouhal number and Nusselt number. The trends seen here are qualitatively similar to that reported in the literature for a circular cylinder and a bar of square cross-section (with zero angle of incidence). Broadly, shear-thinning viscosity promotes heat transfer due to the lower effective viscosity of the fluid in the close proximity of the parallelepiped bar. As expected, shear-thickening behaviour shows the opposite effect. The ranges of the pertinent dimensionless groups spanned in this study are as follows: 0.3⩽n⩽1.8; 45⩽Re⩽120 and 0.7⩽Pr⩽80. Over this range of Reynolds number, the flow was found to be truly periodic. Finally, the numerical heat transfer results are correlated using a simple analytical form which facilitates the interpolation of the present results for the intermediate values of the parameters.

Nonlinear diffusion-induced stresses in a long bar of square cross section

This paper is to investigate the nonlinear effect of the self-induced electric field on the diffusion-induced stresses in a long bar. We first approximate the nonlinear concentration-dependent diffusivity as a series of third-degree polynomials by the least-squares curve-fitting techniques, and then calculate the distributions of concentration by the Galerkin method. Afterwards, the diffusion-induced stresses inside the bar are determined analytically by introducing the Goodier displacement potential and Airy stress function. It is found that the nonlinear self-induced electric fields can depress both the concentration gradient and the maximum diffusion-induced stresses apparently, and these effects are more significant at short times than at long times.

Three-Dimensional Numerical Analysis of Dynamic Thermo-Elastic/Viscoplastic Problem by the Method of Characteristics

This paper is concerned with a numerical technique based on the method of characteristics for three-dimensional dynamic thermo-elastic/viscoplastic problems. A constitutive model covering a wide range of strain rates and a wide range of temperatures, proposed by Tanimura, is used. As a numerical example, the three-dimensional stress wave propagation in a thermo-elastic/viscoplastic bar of square cross section subjected to both an impact loading and a thermal shock is presented. The stability and convergence of these numerical solutions are examined by checking the error in the total energy of the system.

Nonhydrostatic Stress Effects on Solid Phase Epitaxial Growth in Silicon

AbstractThe dependence of solid phase epitaxial growth in Si on uniaxial compression applied perpendicular to the amorphous-crystal interface is investigated. Long, thin pure Si bars of square cross section are ion-implanted to produce amorphous layers on the end faces. The bars are placed end-to-end and uniaxially loaded at temperature to partially regrow the amorphous layers. The resulting growth rates are measured ex situ by re-heating the samples on a hot stage and using time-resolved reflectivity to deduce interface depths. Preliminary results are that uniaxial compression is more effective than hydrostatic pressure for enhancing the growth rate, in qualitative but not quantitative agreement with previously made predictions.

Diffusion-induced stresses in a long bar of square cross section

The diffusion-induced stresses in a long bar of square cross section have been investigated. Two mass transfers consisting of constant average concentration and constant surface concentration were considered. The concentration distribution was obtained in an analytic form. The displacement and stress fields were obtained numerically using a Jacobi iterative method. The maximum stress is located at the corner for constant average concentration and at the midedge for constant surface concentration. A comparison of stresses built up in the bar of a square cross section and in a thin slab was made.

Finite-element simulation of induction heat treatment

An efficient finite-element procedure has been developed for the analysis of induction heat treatment problems involving nonisothermal phase changes. The finite-element procedure first simulates the magnetic field developed when currents flow through an induction coil by solving Maxwell’s electromagnetic field equations; at the following step, it calculates the temperature distribution in the workpiece due to eddy currents induced by the magnetic field. The final stage of the simulation involves the determination of the distributions of residual stress, hardness, and microstructure in the workpiece. The finite-element analysis includes temperature-dependent material properties, changes in permeability of the workpiece at the Curie temperature, a mixed hardening rule to describe the material constitutive model, and the incorporation of time-temperature-transformation (TTT) diagram. The procedure was applied to the simulation of the induction hardening of 1080 steel bar. Firstly, the magnetic field and temperatures developed in the workpiece during (a) the induction heating of an infinitely long 1080 steel cylinder by a single encircling coil and (b) the induction heating of a semi-infinite half-space by a single coil suspended above it were calculated using the finite-element procedures. These were validated by comparing them with analytical solutions derived for these configurations using a Green’s function method. Finally, to demonstrate the predictive capability and practical applicability of the current finite-element procedure, two examples pertaining to the induction heat treatment of an infinite 1080 steel bar of square cross section and a notched finite 1080 steel cylinder of circular cross section were analyzed to predict the magnetic field, temperature, and residual stress distributions. The current finite-element procedure could be used as a powerful design tool for linking induction heat treating parameters with the mechanical property attributes of the heat treated component.

Dynamic Behavior of Elastic/Viscoplastic Bars of Square Cross Section Subjected to Longitudinal Impact.

Numerical solutions are presented for three-dimensional waves in an elastic/viscoplastic bar of square cross section, subjected to a uniform velocity impact. The governing hyperbolic partial differential equations are solved by the method of numerical integration along bicharacteristics. The numerical results for the bar of square cross section are compared with those for that of circular cross section in order to examine the effect of the difference in the shape of the cross section on the wave propagation in the bar. It is shown that the distributions of the longitudinal stress or strain across the transverse section in the vicinity of the impact end of the square elastic bar do not become virtually uniform.

Torsion of a micropolar elastic prism of square cross-section

An analytical solution is presented for the problem of torsion of a prismatical bar of square cross-section of a micropolar (Cosserat) elastic solid. Warp of the cross-sections is found to differ from the warp in a classically elastic solid. Contrary to the classical case, a non-zero shear strain is predicted to occur at the edge of the bar. Novel experimental modalities are suggested on the basis of this analytical solution.

Applications of the white light speckle method to interior displacement measurement

The white light speckle method is used to obtain the interior displacement at different planes of a block with a central idented load, and of a prismatic bar of square cross section under torsion. The problem of decorrelation of speckles due to tilt is not encountered in this method, resulting in reasonable fringe patterns, and good agreement between the displacements obtained with this method and that predicted by the theory of elasticity is found.

A New Tire Cord Adhesion Test

Abstract A new tire cord adhesion test is reported. Although developed primarily for measuring adhesion of rubber to tire cord, the test is applicable to any cord- or wire-reinforced composite, for example, rubber hose or belting. The specimen is a rubber bar of square cross section containing two partially embedded cord ends opposite each other. Upon the application of sufficient tensile force to the cords, failure is initiated at the tip of the embedded end and proceeds along the cord to the exterior, resulting in pull-out of one of the ends. This result contrasts with the initiation of failure in a specimen of the pull-through type containing a throughgoing cord. In the latter specimen, failure initiates where the cord emerges from the rubber and runs back into the interior. Failure is adhesive when pull-out occurs. Under special conditions cohesive failure (tear) occurs and the specimen is cleaved transversely. Reproducibility is excellent and the test has high discriminating power. A theoretical equation permits calculation of the energy of adhesion.

Magnons in a square bar

We present a theoretical investigation of magnon excitations within a simple cubic, ferromagnetic crystal bar of square cross section formed by the intersection of [100] surfaces. The eigenfrequency spectrum for spin waves in the bulk, and localized at the surfaces and edges of the bar is obtained and examined in the presence of dipolar, and nearest and next nearest neighbor exchange spin-spin interactions, for values of the one-dimensional wavevector extending form the origin to the boundary of the one-dimensional Brillouin zone for the bar. The modes obtained are wavelike in the direction parallel to the bar axis. The localization of the modes is determined from an examination of their eigenvectors.

Magnetostatic modes of gyrotropic cylinder and bar

We present a theory of magnetostatic modes which are wavelike in the direction parallel to the axis of an infinitely long, ferromagnetic bar of square cross-section and are localized at the faces and edges of the bar. We write H = −▽ ϑ, where φ is a magnetic scalar potential, and obtain as the equations determining ϑ, Σ μ αβ∂ 2σ ∂X α∂X β =0 for the interior of the bar, and ▽ 2 ϑ = 0 for the exterior, where μ αβ ( ω) is the magnetic permeability tensor of the bar. We transform to cylindrical coordinates and solve the resulting second order partial differential equations by assuming a solution for ϑ of the form ϑ( r, θ, z) = e iqx f( r, θ). The use of cylindrical coordinates allows us to project out of our expressions for ϑ the parts which belong to each of the irreducible representations of C 4, the group of proper rotations which leave the bar invariant. This has the consequence that the boundary conditions need to be applied along only one side of the bar, and are then automatically satisfied along the remaining three sides as well. The boundary conditions that ϑ and the normal component of B be continuous across the bar surfaces are satisfied at a discrete set of points along one side of the bar, and two coupled eigenvalue equations are obtained which are solved simultaneously for the frequencies of the corresponding magnetostatic modes. The convergence of this method is found to be quite good. We also present the dispersion curves for the magnetostatic modes of a gyrotropic right circular cylinder which are localized at the surface of the cylinder.

Elastic-Plastic Torsion of Nonhomogeneous Bars

This paper considers the mathematical formulation for elastic-plastic torsion of uniform and nonuniform nonhomogeneous bars subjected to monotonically increasing angle of twist. Each material in a given bar is assumed to be isotropic and to behave in an elastic perfectly-plastic manner. The method of solution employs a statically admissible minimum rate principle of plasticity, finite elements, and stress function formulations of the torsion problems to reduce the solutions for stress-like quantities to a problem in quadratic programming. Typical results are presented for a nonhomogeneous nonuniform bar of square cross section and a nonhomogeneous nonuniform bar composed of two thick-walled cylinders of different material properties and different diameters.

Extremum principles for certain hyperelastic materials

A hyperelastic material is here said to be of class H m if the elastic potential is a homogeneous function of order m + 1 in the components of the Lagrangian displacement gradient. It is shown that a single solution to a boundary value problem generates an infinite family of solutions to a family of related boundary value problems. Assuming that a solution to a boundary value problem exists, it is shown that it is unique provided that the material is stable in the sense of Hill in a deleted neighbourhood of the stress-free state. A minimum theorem concerning the strain energy and the virtual work of the prescribed forces is established for the equilibrium configurations, and a maximum theorem concerning the virtual work of the prescribed surface displacements and the complementary stress energy is established for compatible stress fields. As an application, upper and lower bounds are found for the torsional stiffness of a cylindrical bar of square cross section under infinitesimal deformation.

Infrared spectra of the hydrosulfide ion in potassium halide crystals: Cheng-Ken Chi and Eugene R. Nixon, Department of Chemistry and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104

We present a theoretical investigation of magnon excitations within a simple cubic, ferromagnetic crystal bar of square cross section formed by the intersection of [100] surfaces. The eigenfrequency spectrum for spin waves in the bulk, and localized at the surfaces and edges of the bar is obtained and examined in the presence of dipolar, and nearest and next nearest neighbor exchange spin-spin interactions, for values of the one-dimensional wavevector extending form the origin to the boundary of the one-dimensional Brillouin zone for the bar. The modes obtained are wavelike in the direction parallel to the bar axis. The localization of the modes is determined from an examination of their eigenvectors.

An Approximate Theory Governing Symmetric Motions of Elastic Rods of Rectangular or Square Cross Section

Variational equations of motion are developed for symmetric motions of linear elastic bars of rectangular cross section. In the finite term approximation, sufficient terms are retained to allow a longitudinal mode, two thickness-stretch modes, and two thickness-shear modes of vibration in an infinite bar of square cross section. Modes for complex wave numbers are also investigated. Adjustment factors in the strain energy and kinetic energy potentials are used to match exact and experimental solutions. Experimental frequency versus wave number results for four modes are reduced by Fourier synthesis and compared both to the approximate theory and to the exact solution for circular cylinders. Theory is intended to predict behavior of thick rectangular bars for which the plane stress solution is not accurate.

Torsional Resonance Vibrations of Uniform Bars of Square Cross Section.

Relations by which the shear modulus may be computed from the fundamental and overtones of the torsional resonance frequencies of square bars have been established empirically.The results are analyzed in terms of a proportionality factor, R, defined by the equation G = (2l fn/n)2ρR. R is found to increase with increasing cross section to length ratio. Also, the overtones are less than integral multiples of the fundamental by an amount which increases with increasing cross section to length ratio.

Plastic deformation of silver chloride I. Internal stresses and the glide mechanism

The experiments of Obreimow & Schubnikoff (1927) on the birefringence produced by the plastic deformation of single crystals of rock salt have been extended to a polycrystallirie material. Rolled sheets of silver chloride have been recrystallized and then deformed plastically in various ways—by simple extension and by bending, for example. The sheets are transparent and very ductile and, since silver chloride is cubic in structure, the birefringence patterns observed under the microscope provide a picture of the distribution of the internal stresses uncomplicated by natural double refractions. It is suggested that results obtained with this optical method are applicable to metals. Silver chloride appears to deform by glide, and when the glide packets are observed on edge the glide plane and glide direction to the crystal structure has been studied by making observations upon these bands and upon the glide lines formed on the surfaces of bars of square cross-section consisting effectively of chains of single crystals. The orientations of the fifteen sets of glide bands examined in this way were all consistent with glide movements in a <110> direction; the glide plane, however, was not always a crystallographic plane of low indices. In the six cases in which the measurement was possible, it lay within 9° of the plane in the.<110> zone on which the maximum shear stress, resolved in the <110> direction, acted. It is concluded that silver chloride deforms by ‘pencil glide’, the mechanism postulated by Taylor & Elam in 1926 to explain the plastic behaviour of a-iron. The transmission of pencil glide across grain boundaries is discussed. The residual stresses observed by the optical method in polycrystalline sheets may be divided into three groups: (1) A system of stresses set up between the glide zones of each grain and alternating with a period equal to the spacing of the glide zones. A detailed analysis of these is given in the second paper (part II). (2) Alternating stresses produced when a system of glide zones meets a grain boundary. (3) ‘Heyn stresses’ produced by the nonuniformity of plastic deformation from grain to grain.

An attempt to detect some electro-optical effects

The following paper contains a description of some experiments made with the object of detecting possible effects due to electric and magnetic fields and moving matter on the velocity of propagation of light in glass. The results obtained were negative, but it seems worth while to publish a short account of the experiments. The optical part of the apparatus is a simple form of interferometer which proved very easy and convenient to work with. It consists of a square glass frame made up of glass bars of square cross-section cemented together with canada balsam. Three of the corners are cut off at 45°, as shown in the figure, and the fourth corner contains a half silvered surface FF. Light entering in the direction of the arrow A is divided into two beams by the silver film, which pass round the frame in opposite directions, being totally reflected at the cut-off corners. Half of each beam emerges in the direction of the arrow B, and the two beams at B are in a condition to interfere with each other.