Latent hardening in polycrystalline copper and comparison with high density polyethylene

When a parallelepiped specimen of polycrystalline copper is compressed in thex-direction (primary direction) while holding thez-dimension unchanged by a vice, the specimen is anisotropically hardened as follows: when thez-direction is subsequently compressed while holding thex-dimension unchanged the yield stress (0.2% offset) is higher than the final flow stress of the primary deformation. This is similar to latent hardening in single crystals. On the other hand, if the second compression is in they-direction instead of thez-direction, the yielding (0.2% offset) occurs at a stress less than the final flow stress of the primary deformation. Both effects are reported here together with the results of two successive compressions in two mutually perpendicular directions without any constraints in either compression. These results are compared with the earlier results of high density polyethylene.     

Numerical experiments on paper-fluid interaction — permeability of a three-dimensional anisotropic fibre network

The effect of paper structure on flow characteristics of various fluids is one of the most important, fundamental problems in papermaking, coating, and printing. A computer code based on a cellular automaton model, in particular the lattice-gas Boltzmann model, has been developed to simulate flow numerically in a random fibre network. As a preliminary investigation, a numerical experiment has been conducted on the three-dimensional permeability of an interpenetrable fibre network. It was found that the in-plane permeability and the z-directional (thickness direction) permeability are very sensitive to the distribution of fibre segments in the z-direction. At a constant porosity, the z-directional permeability increases and the in-plane permeability decreases with increasing z-directional fibre orientation.     

Generalized plane problem of a cracked piezoelectric layer bonded to dissimilar layers

Summary In this paper, we proposed a model to study the electro-elastic crack problem for a cracked piezoelectic layer bonded to two elastic layers of finite thickness. The crack is assumed to be through thez-direction and the crack faces perpendicular to they-direction. Fourier transforms technique is used to reduce the problem to the solution of singular integral equations. The model is general enough to account for arbitrary electrical polarized direction and material anisotropy, for any mechanical or electrical mode of loading. Numerical results are plotted to illustrate the influence of the crack face electrical boundary condition on crack tip fields for different layer thickness.     

Levitron: multi-scale analysis of stability

The Levitron is a typical system for complex Hamiltonian dynamics. Using accurate integrators numerical stability studies were done, especially with respect to variations of the magnetic moment μ for different initial positions x and z. For large μ, equivalent to stronger magnetic fields, the region of stable trajectories is splitted into two parts, whereas for small μ, only one stability region is observed. A linear ansatz is not sufficient to explain this splitting of regions. Vertical and transverse stability conditions have to be combined to understand this behaviour using a multi-scale ansatz. For different hole sizes in the magnetic base plate, the same behaviour appears. For larger holes one has to use stronger magnetic fields. Physically, the stability limits can be identified as critical gradients (forces) of the underlying potential. In x-direction, the stability boundaries are determined by a maximum x-gradient of the potential, which is allowed to act on the top. This derivative of the potential determines a force acting on the top. If this force in x-direction gets too strong, the top deviates from a stable trajectory and gets unstable.     

Axisymmetric elasticity solutions for a uniformly loaded annular plate of transversely isotropic functionally graded materials

Summary The axisymmetric problem of a functionally graded, transversely isotropic, annular plate subject to a uniform transverse load is considered. A direct displacement method is developed that the non-zero displacement components are expressed in terms of suitable combinations of power and logarithmic functions of r, the radial coordinate, with coefficients being undetermined functions of z, the axial coordinate. The governing equations as well as the corresponding boundary conditions for the undetermined functions are deduced from the equilibrium equations and the boundary conditions of the annular plate, respectively. Through a step-by-step integration scheme along with the consideration of boundary conditions at the upper and lower surfaces, the z-dependent functions are determined in explicit form, and certain integral constants are then determined completely from the remaining boundary conditions. Thus, analytical elasticity solutions for the plate with different cylindrical boundary conditions are presented. As a promising feature, the developed method is applicable when the five material constants of a transversely isotropic material vary along the thickness arbitrarily and independently. A numerical example is finally given to show the effect of the material inhomogeneity on the elastic field in the annular plate.     

MHD boundary-layer flow due to a moving extensible surface

The flow due to a moving extensible sheet that obeys a more general stretching law is considered. The sheet occupies the negative x-axis and is moving continually in the positive x-direction, in an incompressible viscous and electrically conducting fluid. The sheet somehow disappears in a sink that is located at (x, y) = (0, 0). The governing system of partial differential equations is first transformed into a system of ordinary differential equations, and the transformed equations are solved numerically using a finite-difference scheme, namely the Keller-box method. The features of the flow and heat-transfer characteristics for different values of the governing parameters are analyzed and discussed. It is found that dual solutions exist for the flow near x = 0, where the velocity profiles show a reversed flow.     

ON THE USE OF FFT ALGORITHM FOR THE CIRCUMFERENTIAL CO-ORDINATE IN BOUNDARY ELEMENT FORMULATION OF AXISYMMETRIC PROBLEMS

A new numerical method is proposed for the boundary element analysis of axisymmetric bodies. The method is based on complex Fourier series expansion of boundary quantities in circumferential direction, which reduces the boundary element equation to an integral equation in (r–z) plane involving the Fourier coefficients of boundary quantities, where r and z are the co-ordinates of the (r, θ, z) cylindrical co-ordinate system. The kernels appearing in these integral equations can be computed effectively by discrete Fourier transform formulas together with the fast Fourier transform (FFT) algorithm, and the integral equations in (r–z) plane can be solved by Gaussian quadrature, which establishes the Fourier coefficients associated with boundary quantities. The Fourier transform solution can then be inverted into (r, θ, z) space by using again discrete Fourier transform formulas together with FFT algorithm. In the study, first we present the formulation of the proposed method which is outlined above. Then, the method is assessed by using three sample problems. A good agreement is observed in the comparisons of the predictions of the method with those available in the literature. It is further found that the proposed method provides considerable saving in computer time compared to existing methods of literature. © 1997 by John Wiley & Sons, Ltd.     

A semi-analytical solution for stress analysis of moderately thick laminated cylindrical panels with various boundary conditions

Stress analysis of moderately thick laminated cylindrical panels with different loading and boundary conditions is presented. Boundary conditions include clamped, simply supported and free while uniform and sinusoidal distributed loadings are considered. Assuming effects of shear deformation and initial curvature, governing equations of the problem are derived. The governing partial differential equations (PDEs) in terms of three displacement components, two rotations and ten stress resultants include a system of 15 first order PDEs. Application of the extended Kantorovich method (EKM) to the governing equations yields to a double set of algebric-differential equations in terms of x and θ. The resulted systems are then solved iteratively with very fast convergence. It is demonstrated that the method converges rapidly independent of initial guess functions. Comparisons of the EKM predictions with other analytical and FEM analyses are in close agreement. More results for panels with particular boundary conditions are presented for future studies.     

Nonlinear oscillations of laminated plates using an accurate four-node rectangular shear flexible material finite element

The objective of the present paper is to investigate the large amplitude vibratory behaviour of unsymmetrically laminated plates. For this purpose, an efficient and accurate four-node shear flexible rectangular material finite element (MFE) with six degrees of freedom per node (three displacements (u, v, w) along thex, y andz axes, two rotations (θ _x and θ _y ) abouty andx axes and twist (θ _xy )) is developed. The element assumes bi-cubic polynomial distribution with sixteen generalized undetermined coefficients for the transverse displacement. The fields for section rotations θ _x and θ _y , and in-plane displacementsu andv are derived using moment-shear equilibrium and in-plane equilibrium equations of composite strips along thex- andy-axes. The displacement field so derived not only depends on the element coordinates but is a function of extensional, bending-extensional coupling, bending and transverse shear stiffness as well. The element stiffness and mass matrices are computed numerically by employing 3×3 Gauss-Legendre product rules. The element is found to be free ofshear locking and does not exhibit any spurious modes. In order to compute the nonlinear frequencies, linear mode shape corresponding to the fundamental frequency is assumed as the spatial distribution and nonlinear finite element equations are reduced to a single nonlinear second-order differential equation. This equation is solved by employing thedirect numerical integration method. A series of numerical examples are solved to demonstrate the efficacy of the proposed element.     

An efficient method for solving implicit and explicit stiff differential equations

When the s‐stage fully implicit Runge–Kutta (RK) method is used to solve a system of n ordinary differential equations (ODE) the resulting algebraic system has a dimension ns. Its solution by Gauss elimination is expensive and requires 2s³n³/3 operations. In this paper we present an efficient algorithm, which differs from the traditional RK method. The formal procedure for uncoupling the algebraic system into a block‐diagonal matrix with s blocks of size n is derived for any s. Its solution is s²/2 times faster than the original, nondiagonalized system, for s even, and s³/(s−1) for s odd in terms of number of multiplications, as well as s² times in terms of number of additions/multiplications. In particular, for s=3 the method has the same precision and stability properties as the well‐known RK‐based RadauIIA quadrature of Ehle, implemented by Hairer and Wanner in RADAU5 algorithm. Unlike RADAU5, however, the method is applicable with any s and not only to the explicit ODEs My′=f(x, y), where M=const., but also to the general implicit ODEs of the form f(x, y, y′)=0. The block‐diagonal form of the algebraic system allows parallel processing. The algorithm formally differs from the implicit RK methods in that the solution for y is assumed to have a form of the algebraic polynomial whose coefficients are found by enforcing y to satisfy the differential equation at the collocation points. Locations of those points are found from the derived stability function such as to guarantee either A‐ or L‐stability properties as well as a superior precision of the algorithm. If constructed such as to be L‐stable the method is a good candidate for solving differential‐algebraic equations (DAEs). Although not limited to any specific field, the application of the method is illustrated by its implementation in the multibody dynamics described by both ODEs and DAEs. Copyright © 2000 John Wiley & Sons, Ltd.     

A local study of a double critical point in Taylor-Couette flow

Summary The steady flow, governed by Navier-Stokes equations for an incompressible viscous fluid, between concentric cylinders is considered. The inner cylinder is assumed to be rotating and the outer one at rest. The flow is assumed to be axisymmetric and periodic in the axial direction. At an (m, n) critical point, the eigenfunctions of the operator, linearized around the exact solution, the Couette flow, consist ofm andn axial waves. In a neighbourhood of such a double critical point, using Liapunov-Schmidt method, bifurcation equations are obtained, in ². Expressions for the leading coefficients in the truncated system of 2 equations are derived. Using these, the coefficients are computed at a (2, 4) critical point for 2 different radii ratios and the local bifurcation diagrams obtained. Available numerical solutions of the Navier-Stokes system near this double critical point confirm that the reduced bifurcation equations reproduce the qualitative behaviour adequately.     

聚变堆包层第一壁缩比部件激光选区熔化成形研究

目的 研究激光选区熔化(SLM)成形第一壁缩比结构的组织性能。方法 以316L粉末为原材料,运用Inspire软件对不同成形姿势下第一壁缩比结构的应力与变形情况进行数值模拟,选择最佳成形姿势进行SLM成形,以控制整体变形,并对成形零件进行显微组织观察与力学性能测试。结果 实验结果表明,与立放和侧放2种成形姿势相比,平放时残余应力与变形最小,最大残余应力为29.68 MPa,最大变形量为0.29 mm。成形件微观组织呈现各向异性,x–y方向主要为粗大的胞状晶组织,z–x方向为细长的柱状晶组织。力学测试结果显示,x–y方向的抗拉强度为672.1 MPa,伸长率为48.2%,冲击韧性为100.6 J/cm²;z–x方向的抗拉强度为646.9 MPa,伸长率64.4%,冲击韧性为136.3 J/cm²。结论 组织的差异性主要是由扫描工艺与熔池内部复杂的温度场引起的,微观结构的各向异性会造成力学性能的差异,x–y方向的强度高于z–x方向的,z–x方向上的塑性韧性更高。     

Second order effects in an elastic half-space acted upon by a non-uniform shear load

Summary A closed form solution to the second order elasticity problem, when an isotropic compressible elastic half-space undergoes a deformation owing to a non-uniformly distributed shear load, is presented. The method of integral transform is employed to determine the solutions. An example is discussed in detail to illustrate the second order effects. Numerical calculations for the second order elastic material for thez-direction displacement and the stresst _rz are carried out. It is found that the second order effect is to reduce thez-direction displacement and to dereaset _rz inside the circle but to increase its value outside the circle.     

A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities

This article proposes a four-variable shear deformation refined beam theory for thermo-mechanical vibration characteristics of porous, functionally graded (FG) beams exposed to various kinds of thermal loadings by using an analytical method. Thermo-mechanical properties of functionally graded material (FGM) beams are supposed to vary through the thickness direction, and are estimated through the modified power-law rule in which the porosities with even and uneven types are approximated. The material properties of FGM beams are supposed to be temperature dependent. Porosities possibly occur inside FGMs during fabrication because of technical problems that lead to the creation of microvoids in these materials. The variation of pores along the thickness direction influences the mechanical properties. Thus, it is incumbent to predict the effect of porosities on the thermo-mechanical vibration behavior of FG beam in the present study. Four types of thermal loading, namely, uniform, linear, nonlinear, and sinusoidal temperature rises through the z-axis direction are discussed. The governing differential equations and boundary conditions of FG porous beams subjected to thermal loadings are formulated through Hamilton's principle, based on a four-variable refined theory that considers a constant transverse displacement and higher order variation of axial displacement through the depth of the beam without the need of any shear correction factors. An analytical solution procedure is used to achieve the natural frequencies of porous FG beams subjected to various temperature fields. The impact of several specific parameters such as power-law exponent, porosity volume fraction, different porosity distribution, and thermal effect on the vibration of the porous FG beams is perused and discussed in detail. It is deduced that these parameters play a notable role on the thermo-dynamic behavior of porous FG beams. Presented numerical results can serve as benchmarks for the future analyses of FG beams with porosity phases.     

A laminar walljet on a moving wall

Summary The laminar wall jet from a momentum source at the leading edge on a wall which is moving in the same direction with uniform velocity is considered. It is shown that a solution is possible starting at the leading edge and proceeding all the way downstream. For smallx (x measures distance along the wall) we find the solution by using a natural coordinate expansion in powers ofx ^1/2. For largex, the asymptotic solution is approached through eigensolutions and the two coordinate expansions are then joined by a numerical solution of the full equations.With 2 Figures     

A mixed semi analytical solution for functionally graded (FG) finite length cylinders of orthotropic materials subjected to thermal load

A simplified and accurate analytical cum numerical model is presented here to investigate the behavior of functionally graded (FG) cylinders of finite length subjected to thermal load. A diaphragm supported FG cylinder under symmetric thermal load which is considered as a two dimensional (2D) plane strain problem of thermoelasticity in (r, z) direction. The boundary conditions are satisfied exactly in axial direction (z) by taking an analytical expression in terms of Fourier series expansion. Fundamental (basic) dependent variables are chosen in the radial coordinate of the cylinder. First order simultaneous ordinary differential equations are obtained as mathematical model which are integrated through an effective numerical integration technique by first transforming the boundary value problem into a set of initial value problems. For FG cylinders, the material properties have power law dependence in the radial coordinate. Effect of non homogeneity parameters and orthotropy of the materials on the stresses and displacements of FG cylinder are studied. The numerical results obtained are also first validated with existing literature for their accuracy. Stresses and displacements in axial and radial directions in cylinders having various l/r _i and r _o/r _i ratios parameter are presented for future reference.     

Fully coupled solution of the equations for incompressible recirculating flows using a penalty-function finite-difference formulation

This paper describes two new solution algorithms for steady recirculating flows that use a penalty formulation to eliminate the pressure from the finite difference form of the governing equations. One algorithm uses successive substitution to linearize the equations, while the other employs the Newton-Raphson linearization. In both cases, the equations are solved in a fully coupled manner using a sparse matrix form of LU decomposition. The D'Yakonov iteration is used to avoid unnecessary factorizations of the coefficient matrix, significantly improving the computational efficiency. The Newton-Raphson linearization leads to faster convergence, but the execution times of the two methods are comparable. The algorithms converge rapidly and are robust to changes in grid size and Reynolds number. In a number of laminar two-dimensional flows, the new methods proved to be two to ten times faster than some conventional iterative methods.List of symbols A coefficient matrix - A _e area of east control-volume face - a coefficient in the discretization equations - â coefficients in the modified momentum equations in the penalty formulation, Eqs. (13) - b constant term in the discretization equations - F symbolic form of nonlinear system of equations, F ₍₎=A –b=0 - H characteristic length - J Jacobian matrix - n iteration number - P dimensionless pressure - p pressure - Re Reynolds number - U dimensionless u velocity - u x-direction velocity - V dimensionless v velocity - v y-direction velocity - X, Y dimensionless coordinates, X=x/H, Y=y/H - x, y physical coordinates - x x-direction width of the control volume - normalized error in the unconverged solution, Eqs. (19) - dimensionless penalty parameter, Eq. (7) - dimensional penalty parameter, Eq. (10) - viscosity - density     

Large deflection analysis of beams with variable stiffness

Summary. In this paper, the Analog Equation Method (AEM), a BEM-based method, is employed to the nonlinear analysis of a Bernoulli-Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe nonlinear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled nonlinear differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious load distributions. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.     

Hydrodynamic lubrication of squeeze-film porous bearings

Summary Closed-form analytical solutions for three different types of squeeze-film porous bearing are introduced in this paper. The effects of the permeability parameter on the pressure profile, load-carrying capacity, and time required to squeeze the fluid out of the lubricated conjunction are presented. The results show that as the permeability parameter increases, both the pressure profiles and the load-carrying capacity of the bearing decrease in the case of pure squeeze motion. Furthermore, the results show that for dimensionless permeability parameters less than 0.001, the effect of the porous layer on the hydrodynamic lubrication of squeeze-film porous bearings can be neglected.Notation c Clearance, m - e Eccentricity, m - h Film thickness, m - h _p Porous layer thickness, m - k _x Permeability of the porous layer inx-direction, m² - k _y Permeability of the porous layer iny-direction, m² - k _z Permeability of the porous layer inz-direction, m² - k ₁ Permeability ratio - p Pressure within film region, Pa - p ^* Pressure within porous layer, Pa - P Dimensionless pressure within film region - P ^* Dimensionless pressure within porous layer - r Radial coordinate - Dimensionless radial coordinate - u _a Velocity of surfacea inx-direction, m/s - u _b Velocity of surfaceb inx-direction, m/s - v _a Velocity of surfacea iny-direction, m/s - v _b Velocity of surfaceb iny-direction, m/s - w Squeeze velocity, –h/t, m/s (w _a =–w,w _b =0) - w ₀ Flow velocity into porous layer inz-direction, m/s - w _z Load-carrying capacity per unit width, N/m - W _z Dimensionless load-carrying capacity - x Coordinate, m - X Dimensionlessx-coordinate - y Coordinate, m - z Coordinate, m - Z Dimensionlessz-coordinate - Dimensionless parameter,l/h _p - Lubricant viscosity within film region, Pa s - ^* Lubricant viscosity within porous layer, Pa s - ₀ Lubricant viscosity at atmospheric pressure, Pa s - Lubricant density within film region, Kg/m³ - ^* Lubricant density within porous layer, Kg/m³ - Circumferential coordinate, rad - Dimensionless permeability parameter, - Eccentricity ratio     

一种模量双向变化的梯度复合平面梁的级数解

3D打印技术的发展使复杂梯度结构的制造更加容易,有必要对复杂梯度问题的求解开展研究;目前,关于梁结构模量沿轴向或厚度方向梯度变化问题的研究已经较多,但对模量沿2个方向同时变化的研究较少。因此,通过复数形式傅里叶分解的方法对模量以指数形式沿厚度方向和轴向同时变化梯度平面复合梁问题进行了求解。首先,采用弹性力学半逆解法得到了问题的四阶变系数偏微分控制方程的通解;然后,利用级数展开,求解了对称载荷作用下该梁的特解;最后,通过与有限元结果进行对比,说明了级数解的正确性。结果表明:当梯度双向变化时,梁结构的应力分布和变形情况更加复杂,模量较高的位置应力较大,而模量较低的位置应力较小。提出的级数解还可推广至其他相关的梯度双向变化非均匀平面和半平面问题的研究。