Proposed slip line fields for indenting with rough plane and curved dies

Summary Theoretical solutions are presented for the plane strain deformation of a rigid-perfectly plastic solid for (a) indenting a plane surface with a partially rough plane rigid die; (b) crushing of a wedge by a rough plane rigid die; (c) indenting by a pair of opposed rough circular dies.     

Some New Implications of Certain Constitutive Inequalities in Plane Isotropic Elasticity

In plane isotropic elasticity a strengthened form of the Ordered–Forces inequality is shown to imply that the restriction of the strain-energy function to the class of deformation gradients which share the same average of the principal stretches is bounded from below by the strain energy corresponding to the conformal deformations in this class. For boundary conditions of place, this property (together with a certain version of the Pressure–Compression inequality) is then used (i) to show that the plane radial conformal deformations are stable with respect to all radial variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with arbitrary plane radial deformations. For the same type of boundary conditions and together with a different version of the Pressure–Compression inequality, an analogous property in plane isotropic elasticity (established in [3] under the assumption that the material satisfies a strengthened form of the Baker–Ericksen inequality and according to which the restriction of the strain-energy function to the class of deformation gradients which share the same determinant is bounded from below by the strain energy corresponding to the conformal deformations in that class) is used (i) to show that the plane radial conformal deformations are stable with respect to all variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with any plane deformation.     

Analysis of plane and cylindrical nonlinear hyperelastic waves in materials with internal structure

A comparative analysis of two types of hyperelastic waves—plane waves (with plane front) and cylindrical waves (with curved front)—is offered. The propagation of the waves is studied theoretically for quadratically nonlinear hyperelastic media and numerically for a class of unidirectional fibrous composite materials. Hyperelasticity is described using the classical Murnaghan potential and a structural model of the first order—the model of effective constants. The internal structure of materials is described by this model and is at the micro-or nanolevels in numerical analysis. Particular attention is given to the evolution of the wave profile. It is studied in three stages: (i) derivation of nonlinear wave equations, (ii) construction of solutions in the form of plane and cylindrical waves, and (iii) numerical analysis of the evolution of these waves in composites with microlevel (Thornel) or nanolevel (Z-CNT) fibers. The main similarities and differences between plane longitudinal and cylindrical waves are shown. The most unexpected result is the striking difference between the evolution patterns numerically observed for plane and cylindrical wave profiles __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 21–46, October 2006.     

Analysis of dynamic shear bands in an FCC single crystal

We study plane strain dynamic thermomechanical deformations of an fcc single crystal compressed along the crystallographic direction [010] at an average strain rate of 1000 sec⁻¹. Two cases are studied; one in which the plane of deformation is parallel tothe plane (001) of the single crystal, and another one with deformation occuring in the plane (101̄) of the single crystal. In each case, the 12 slip systems are aligned symmetrically about the two centroidal axes. We assume that the elastic and plastic deformations of the crystal are symmetrical about these two axes. The crystal material is presumed to exhibit strain hardening, strain-rate hardening, and thermal softening. A simple combined isotropic-kinematic hardening expression for the critical resolved shear stress, proposed by Weng, is modified to account for the affine thermal softening of the material. When the deformation is in the plane (001) of the single crystal, four slip systems (111)[11̄0], (111̄)[11̄0], (11̄;1̄;)[110], and (11̄1)[110] are active in the sense that significant plastic deformations occur along these slip systems. However, when the plane of deformation is parallel to the plane (101̄;) of the single crystal, slip systems (11̄;1)[110], (11̄1)[011], (111)[11̄0], and (111)[01̄1] are more active than the other eight slip systems. At an average strain of 0.0108, the maximum angle of rotation of a slip system within a shear band, about an axis perpendicular to the plane of deformation, is found to be 20.3° in the former case, and 22.9° in the latter.     

THE EIGENVALUE PROBLEM AND SAINT-VENANT DECAY RATE FOR A NONHOMOGENEOUS SEMI-INFINITE STRIP

The eigenvalue problem about a nonhomogeneous semi-infinite strip is investigated using the methodology proposed by Papkovich and Fadle for homogeneous plane problems. Two types of nonhomogeneity are considered： （i） the elastic modulus varying with the thickness coor- dinate x exponentially, （ii） it varying with the length coordinate y exponentially. The eigenvalues for the two cases are obtained numerically in plane strain and plane stress states, respectively. By considering the smallest positive eigenvalue, tile Saint-Venant Decay rates are estimated, which indicates material nonhomogeneity has a signifcant influence on the Saint-Venant end effect.     

An interpolated time-domain equivalent source method for modeling transient acoustic radiation over a mass-like plane based on the transient half-space Green’s function

Interpolated time-domain equivalent source method (ITDESM) is based on the assumption of free space, which makes it not suitable for reconstructing the transient acoustic quantities in the half space. Here, a half-space ITDESM is proposed to model the transient acoustic radiation over a mass-like plane. In this method, the free transient Green’s function existing in the conventional ITDESM is replaced by a closed-form transient half-space Green’s function for a mass-like plane. Such transient Green’s function enables one to take the reflection effect of the mass-like plane into consideration. Modeling acoustic radiation from three transient monopoles above an infinite plane with mass-like behavior is studied by numerical simulations to demonstrate the feasibility of the half-space ITDESM. The proposed method is also examined by comparing the reconstruction accuracy among a free-field model, a rigid plane model and a mass-like plane model. An experiment with an impacted steel plate lying above a table plate is conducted in the semi-anechoic room, and the results further verify the effectiveness of the proposed method.     

Three-dimensional shock intersection

The flow field in the neighborhood of the three-dimensional intersection of two shocks of different families is investigated when in the plane perpendicular to the line of intersection the flow velocity component is subsonic behind at least one of the departing shocks. In the plane case these flows are not realized. The boundary of the domain of the key parameters for which these flows are possible is determined. The characteristics of the flow field are determined when: (1) behind the departing shocks the flow is homogeneous, and (2) the velocity vectors behind the departing and arriving shocks are parallel to a single plane which contains the intersection line. The flow in Mach-type shock intersection in the neighborhood of the intersection lines (triple points in the plane) is a particular case of the problem considered. It is shown that Mach-type shock intersection is not possible when the intensity of the arriving shocks is less than for their steady-state Mach intersection in the calculation plane. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 137–143, November–December, 1998.     

The control of turbulent end-wall boundary layers using surface suction

An experimental evaluation of the effects of spatially-limited (i.e. localized) surface suction on a turbulent junction flow was performed using Particle Image Velocimetry (PIV). The results indicate that surface suction can (1) weaken both the instantaneous turbulent vortex and its associated surface interactions in the symmetry plane, (2) effectively eliminate the presence of the average turbulent necklace vortex in the symmetry plane, and (3) weaken the average downstream extensions of the vortex. It was also established that suction effectively reduces the low frequency component of the Reynolds-stress in both the symmetry plane and trailing-edge cross-stream planes, and stabilizes the behavior of the trailing vortex legs. Received: 18 May 1998/Accepted: 26 March 1999     

Integrability of the motion of a rolling disk of finite thickness on a rough plane

In this article an analytical solution of equations of motion of a rigid disk of finite thickness rolling on its edge on a perfectly rough horizontal plane under the action of gravity is given. The solution is given in terms of Gauss hypergeometric functions. The integrability results are used to construct various bifurcation diagrams of the steady motion of the disk. The bifurcations of the steady motion of a disk on a rough plane complements the author's bifurcation analysis of the steady motion of the disk on a smooth plane ( [M. Batista, Steady motion of a rigid disk of finite thickness on a horizontal plane, Int. J. Non-Linear Mech. 41 (4) (2006) 605–621]).     

On Infinitesimal Shear

The setting for this note is the theory of infinitesimal strain in the context of classical linearized elasticity. As a body is subjected to a deformation the angle between a pair of material line elements through a typical point P is changed. The decrease in angle is called the shear of this pair of elements. Here, we determine all pairs of material line elements at P which are unsheared in a deformation. It is seen, in general, that corresponding to any given material line element in a given plane through P, there is one corresponding “companion” material line element such that the given element and its conjugate are unsheared in the deformation. There are two exceptions. If the plane through P is a plane of central circular section of the strain ellipsoid, then every material line element through P in this plane has an infinity of companion elements in this plane – all pairs of material line elements in the plane(s) of central circular section of the strain ellipsoid are unsheared. If the plane through P is not a plane of central circular section of the strain ellipsoid, then there are two exceptional material line elements through P such that neither of them has a companion material line element forming an unsheared pair with it. The directions of these exceptional elements in the plane are called “limiting directions”. It is seen that it is the pair of elements along the limiting directions in a plane which suffer the maximum shear in that plane. A geometrical construction is presented for the determination of the extensional strains along the pairs of elements which are unsheared. Also, it is shown that knowing one unsheared pair in a plane and their extensions is sufficient to determine the principal extensions and the principal axes in this plane. Expressions for all unsheared pairs in a given plane are given in terms of the normals to the planes of central circular sections of the strain ellipsoid. Finally, for a given material line element, a formula is derived for the determination of all other material line elements which form an unsheared pair with the given element.     

Stress Analysis of an Orthotropic Work-Hardening Cylinder with Body Force*

ABSTRACT

An elastoplastic analysis of an axisymmetric cylinder subjected to linear body forces is presented. The effect of reinforcement and anisotropy are also included. Classical plasticity and familiar assumptions of plane stress and strain are used to arrive at closed-form solutions for the case of linear body forces. The problem is solved for the general case in which orthotropy is considered in the elastic range. For the case of plasticity, first the isotropic yield functions (von Mises and Tresca) are used and then the problem is extended to the case of Hill's yield criterion. Closed-form solutions are found for both the von Mises (plane strain) and Tresca (plane stress and strain) cases.     

Analysis of shallow spherical shell with circular base under eccentrically applied concentrated loads

In this paper,problems of a shallow sphericalshell with circular base under eccentrically appliedconcentrated loads are discussed.The solutions forsix cases of eccentrically applied concentrated loadsare given,namely:(1)Normal concentrated load,(2)Meridional tangential concentrated load,(3)Circumferential tangential concentrated load;(4)Concentrated moment in the tangential plane,(5)Concentrated moment in the meridional normalplane,(6)Concentrated moment in the circumferentialnormal plane.From the solutions of concentrated loads,thesolutions of distributed line loads in the form ofcosnθalong the circle are obtained.     

General high-order rational solutions and their dynamics in the (3+1)-dimensional Jimbo–Miwa equation

General high-order rational solutions are derived for the (3+1)-dimensional Jimbo–Miwa equation based on the Hirota bilinear form. The solutions are presented in terms of Gram determinants; the elements of determinants are connected to Schur polynomials and have simple algebraic expressions. Their dynamic behaviors are researched using three-dimensional imagery and contour plots. It is revealed that different kinds of solutions appear in (x, y) plane and (y, z) plane. When one of these internal parameters in the rational solutions is sufficiently large, in (x, y) plane Lump solutions appear with obvious geometric structures, which are deconstructed by a first-order Lump such as triangle, pentagon, and nonagon, among others; in (y, z) plane rational line soliton solutions with maximum background amplitude changing over time appear. These findings might help us comprehend the nonlinear wave propagation processes in the many nonlinear physical systems.

     

Determination of stress levels for dynamic fracture of oil shale

The dynamic tensile-stress amplitudes necessary to cause complete spall in unconfined oil-shale samples were experimentally determined in the laboratory for pulse durations typically encountered in the field (60–100 μs). Tensile stresses were generated by free-surface reflection of compressive pulses, and the stress-time history at the free surface was obtained by numerically differentiating the displacement-time history monitored by a fiber-optic proximity sensor. A computer code was written to display the stress profile, obtained experimentally, over the length of the specimen as time increased. In this manner, the amplitude of the resultant stress wave (incident plus reflected) was determined at the measured spall plane. By using many specimens, the fracture thresholds for these pulse durations were determined. Spall thresholds for competent shale of 25-gal/ton yield were found to be approximately 1800 psi (12.4 MPa) for pulses propagating normal to the bedding plane, and nearly 5,000 psi (34.5 MPa) for pulses propagating parallel to the bedding plane. For shale of 34 gal/ton yield, the spall-threshold levels were found to be 1200 psi (8.3 MPa) and 2700 psi (18.6 MPa) for pulses propagating normal and parallel, respectively, to the bedding plane.     

On the displacement potential solution of plane problems of structural mechanics with mixed boundary conditions

The present paper describes the advancement of displacement potential approach in relation to solution of plane problems of structural mechanics with mixed mode of boundary conditions. Both the conditions of the plane stress and the plane strain are considered for analyzing the displacement and stress fields of the structural problem. Using the finite difference technique based on the present displacement potential approach for the case of the plane stress and the plane strain conditions, firstly an elastic cantilever beam subjected to a pure shear at its tip is solved and these two solutions (plane stress and plane strain) are compared with Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951); secondly the above-mentioned displacement potential approach for the case of the plane stress and the plane strain conditions are applied to solve a one-end fixed square plate subjected to a combined loading at its tip. Effects of plane stress and plane strain on the elastic field of the plate are discussed in a comparative fashion. Limitations of Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951) over the displacement potential approach for the case of the plane stress and the plane strain conditions are not only discussed but also the superiority of the present displacement potential approach for the case of the plane stress and the plane strain conditions are reflected in the present research work.     

Numerical simulation of two-degree-of-freedom vortex-induced vibration of a circular cylinder close to a plane boundary

Two-degree-of-freedom vortex-induced vibrations (VIV) of a circular cylinder close to a plane boundary are investigated numerically. The Reynolds-Averaged Navier-Stokes (RANS) equations are solved using the Arbitrary Lagrangian Eulerian (ALE) scheme with a k-ω turbulence model closure. The numerical model is validated against experimental data of VIV of a cylinder in uniform flow and VIV of a cylinder close to a plane boundary at low mass ratios. The numerical results of the vibration mode, vibration amplitude and frequency agree well with the experimental data. VIV of a circular cylinder close to a plane boundary is simulated with a mass ratio of 2.6 and gap ratios of e/D=0.002 and 0.3 (gap ratio is defined as the ratio of gap between the cylinder and the bed (e) to cylinder diameter (D)). Simulations are carried out for reduced velocities ranging from 1 to 15 and Reynolds numbers ranging from 1000 to 15 000. It is found that vortex-induced vibrations occur even if the initial gap ratio is as small as e/D=0.002, although reported research indicated that vortex shedding behind a fixed circular cylinder is suppressed at small gap ratios (e/D<0.3 or 0.2). It was also found that vibration amplitudes are dependant on the bouncing back coefficient when the cylinder hits the plane boundary. Three vortex shedding modes are identified according to the numerical results: (i) single-vortex mode where the vortices are only shed from the top of the cylinder; (ii) vortex-shedding-after-bounce-back mode; (iii) vortex-shedding-before-bounce-back mode. It was found that the vortex shedding mode depends on the reduced velocity.     

Some exact solutions of the equations of a stationary monoenergetic beam of charged particles

We consider the class of invariant solutions which can describe only vortex flows (curl P 0, P is the generalized momentum) and show that they contain solutions corresponding to flows from a plane or cylindrical emitter with a linear voltage drop across it (direct heating) in the temperature-limited regime^*. The solution is obtained in analytic form for emission from a plane in a uniform magnetic field perpendicular to the flow plane. It also (for=0) defines a plane magnetron in the T-regime. The solution of the problem for a cylindrical emitter reduces to considering equations describing a cylindrical diode or magnetron in the T-regime, where the shape of the collector is given by the potential distribution curve for these cases. We can extend the results to a relativistic beam if restrictions are imposed on its relative dimensions which permit us to ignore the magnetic self-field. Brillouin type flows (including irrotational ones) are studied in which particles move without intersecting the equipotential surfaces along three-dimensional spirals on the surface of cones. An analytic solution is given for relativistic Brillouin flow in a conical diode when strict allowance is mede for the magnetic self-field.     

岩体结构面蠕变损伤机理研究

通过对伯格斯模型和西原模型的分析比较,选取西原模型研究岩体结构面的蠕变损伤特性。在τ0<τs情况下,由西原模型推出剪切模量的表达式,以剪切模量为变量定义损伤变量,得到结构面的损伤变量表达式。并以泥岩剪切试验为例,计算了相同正压力作用下的结构面剪切模量和损伤量。结果表明：当τ0<τs时,剪切模量、损伤量均随时间趋于稳定,且结构面蠕变的前两个阶段损伤量较小,而当τ0≥τs时,一段时间后,其损伤量开始突变;剪应力越大,其初始剪切模量越大,随时间降低越快,达到稳定蠕变阶段时降低量也相应越大;剪应力越大,结构面损伤量随时间增长越快,在达到稳定蠕变阶段时,损伤量也越大。     

Digital image plane holography (DIPH) for two-phase flow diagnostics in multiple planes

A technique for measuring the size and displacement of the disperse phase in two planes of a two-phase flow is presented. Digital image plane holography (DIPH) is used for the simultaneous recording and independent reconstruction of both planes. Each fluid plane is illuminated with two laser sheets propagating in opposite directions. The defocused image fields are holographically recorded at 90°, and can be reconstructed either in a defocused or in the best-focused plane. The analysis of the images in a defocused plane provides the sizes, while the cross-correlation of the focused images provides the velocity field, as in a regular particle image velocimetry (PIV) experiment. For air bubbles freely drifting in glycerine, diameters from 50 m to 400 m and displacements of up to 300 m have been measured.     

Determination of permeability in fibrous porous media using the lattice Boltzmann method with application to PEM fuel cells

The lattice Boltzmann method (LBM) is used to simulate the flow through an idealized proton exchange membrane fuel cell (PEMFC) porous transport layer (PTL) geometry generated using a Monte Carlo method. Using the calculated flow field, Darcy's law is applied and the permeability is calculated. This process is applied in both through‐ and in‐plane directions of the paper as both of these permeability values are important in computational fluid dynamics models of PEMFCs. It is shown that the LBM can be used to determine permeability in a random porous media by solving the flow in the microstructure of the material. The permeability in the through‐ and in‐plane directions is shown to be different and the anisotropic nature of the geometry creates anisotropic permeability. It is also found that fiber arrangement plays a large role in the permeability of the PTL. New correlations are presented for in‐ and though‐plane permeabilities of fibrous porous media with (0.6<ε<0.8). Copyright © 2008 John Wiley & Sons, Ltd.