Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

General Hosford yield functions of orthorhombic materials

Although the Hosford yield function is more suitable for describing both the yielding and the plastic deformation of orthorhombic materials than the Hill quadratic yield function, the Hosford yield function suffers from the restriction that the loading has to be coaxial with the orthotropy of the materials. To relax this restriction, herein we present a new general Hosford yield function for the orthorhombic materials. The new general Hosford yield function is suitable to any stress state of the orthorhombic materials. When η = 2, the new general Hosford yield function becomes the Hill quadratic yield function. The new general Hosford yield function is more general than the general Hosford yield function of Huang and Man (Int J Plast 41:97–123, 2013), which covers only weakly-textured sheets of cubic metals. Two examples show that the new general Horsford yield function with suitable η value gives much better fits than those of the Hill quadratic yield function (η = 2).     

Melnikov function and Poincaré map

In this paper we give the relationship between Melnikov function and Poincare map, and a new proof for Melnikov’s method. The advantage of our paper is to give a more explicit solution and to make Melnikov function for the subharmonics bifurcation and Melnikoy function which the stable manifolds and unstable manifolds intersect transversely into a formula.     

On Green's function for the reduced wave equation in a spherical annular domain with Dirichlet's boundary conditions

Summary In this study Green's function for the reduced wave equation (Helmholtz equation) in a spherical annular domain with Dirichlet's boundary conditions is derived. The convergence of the series solution representing Green's function is then established. Finally it is shown that Green's function for the Dirichlet problem reduces to Green's function for the exterior of a sphere as given by Franz and Etiènne, when the outer radius is moved towards infinity, and when a special position of the coordinate system is chosen.     

Numerical investigation of the flow and heat behaviours of an impinging jet

The flow and temperature fields of a turbulent impinging jet are rather complex. In order to accurately describe the flow and heat-transfer process, two important factors that must be taken into account are the turbulence model and the wall function. Several turbulence models, including κ–? turbulence models, κ–ω turbulence models, low-Re turbulence models, the κ–κ_l–ω turbulence model, the Transition SST turbulence model, the V2F turbulence model and the RSM turbulence model, are examined and compared to experimental data. Furthermore, for the near wall region, various wall functions are presented for comparison and they include the standard wall function, the scale wall function, the non-equilibrium wall function and the enhanced wall function. The distribution features of velocity, turbulent kinetic energy and Nusselt number are determined in order to provide a reliable reference for the multiphase impinging jet in the future.     

A kind of binary scaling function projective lag synchronization of chaotic systems with stochastic perturbation

This paper investigates a kind of binary function projective lag synchronization of uncertain chaotic systems with stochastic perturbation. In comparison with those of the existing scaling function synchronization, we assume that the given scaling function can be the binary boundary function, even an n-ary boundary function. Based on the LaSalle-type invariance principle for stochastic differential equation, the adaptive control law is derived to make the state of two chaotic systems function projective lag synchronized. Some numerical is also given to show the effectiveness of the proposed method.     

Green’s function for T-stress of semi-infinite plane crack

Green’s function for the T-stress near a crack tip is addressed with an analytic function method for a semi-infinite crack lying in an elastical, isotropic, and infinite plate. The cracked plate is loaded by a single inclined concentrated force at an interior point. The complex potentials are obtained based on a superposition principle, which provide the solutions to the plane problems of elasticity. The regular parts of the potentials are extracted in an asymptotic analysis. Based on the regular parts, Gre...     

Deformation of Euler-Bernoulli Beams by Means of Modified Green's Function: Application of Fredholm Alternative Theorem

This article deals with the determination of the static displacement function of an Euler-Bernoulli beam with two guided supports. To this end, the Green's function method is employed and exact solution is obtained. The Green's function of the problem is constructed, using pertinent boundary conditions of the problem. Nevertheless, the problem does not admit a Green's function due to a mathematical contradiction. In order to eliminate the trouble, the Fredholm Alternative Theorem is utilized and the Green's function is modified. In this case, application of this theorem adds a particular term to the Green's function which gives rise to an arbitrary constant in the Green's function. Moreover, it is shown that the problem may have no solution or an infinite number of solutions. Besides, the necessary condition for having any solution is investigated. This requirement, which states a significant rule in the mechanics of solids, is the static equilibrium of vertical forces acting on the beam. Some examples are presented and results are thoroughly discussed.     

An image treatment of elastostatic transmission from an interface layer

A basic theorem for representing the Airy stress function for two perfectly bonded semi-infinite planes in terms of the corresponding Airy function for the unbounded homogeneous plane is applied in a systematic stepwise fashion to generate the corresponding Airy stress function for a three-phase composite comprising two semi-infinite planes separated by a thick layer. The loading of the three-phase composite is arbitrary, and may be in or near the interface layer. The basic theorem is first illustrated by applying it to an elastic medium which is bounded by two unloaded straight edges which intersect at an angle π/n, where n is a positive integer. This example illustrates a case of a finite system of images, while the plane-layered medium problem leads to an infinite series of images.     

A new representation for the dynamic green's tensor of an elastic half-space or layered medium

A method for inverting the transforms of the terms in generalized ray series representations for disturbances in layered media is presented. It differs from the Cagniard reduction in that the solution of algebraic equations depending upon position x and time t is not required. This step is, in effect, replaced by contour integration of relatively simple functions. The method is applicable to anisotropic layers but it simplifies when applied to isotropic layers, for which any term in the ray series is represented as a single contour integral, around a fixed contour, of the product of a function that embodies material properties and a simple explicit function of x and t. The ‘material function’ can be tabulated and used repeatedly when the integral is evaluated for a range of values of x and t, so that the procedure is computationally quite efficient. It is illustrated by a computation of Green's function for an isotropic half-space, either free or overlaid by a fluid. Wave-front singularities are obtained explicitly from the representation and are given in an appendix.     

A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials

In this paper, a generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials is derived. The evolution equation for the active yield surface with reference to the memory yield surface is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is then derived for a general yield function for pressure insensitive and sensitive materials. Detailed incremental constitutive relations for materials based on the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker–Prager yield function are derived as the special cases. The closed-form solutions for one-dimensional stress–plastic strain curves are also derived and plotted for materials under cyclic loading conditions based on the three yield functions. In addition, the closed-form solutions for one-dimensional stress–plastic strain curves for materials based on the isotropic Cazacu–Barlat yield function under cyclic loading conditions are summarized and presented. For materials based on the Mises and the Hill anisotropic yield functions, the stress–plastic strain curves show closed hysteresis loops under uniaxial cyclic loading conditions and the Masing hypothesis is applicable. For materials based on the Drucker–Prager and Cazacu–Barlat yield functions, the stress–plastic strain curves do not close and show the ratcheting effect under uniaxial cyclic loading conditions. The ratcheting effect is due to different strain ranges for a given stress range for the unloading and reloading processes. With these closed-form solutions, the important effects of the yield surface geometry on the cyclic plastic behavior due to the pressure-sensitive yielding or the unsymmetric behavior in tension and compression can be shown unambiguously. The closed form solutions for the Drucker–Prager and Cazacu–Barlat yield functions with the associated flow rule also suggest that a more general anisotropic hardening theory needs to be developed to address the ratcheting effects for a given stress range.     

Developing a hybrid flux function suitable for hypersonic flow simulation with high‐order methods

In this paper, we develop a new hybrid Euler flux function based on Roe's flux difference scheme, which is free from shock instability and still preserves the accuracy and efficiency of Roe's flux scheme. For computational cost, only 5% extra CPU time is required compared with Roe's FDS. In hypersonic flow simulation with high‐order methods, the hybrid flux function would automatically switch to the Rusanov flux function near shock waves to improve the robustness, and in smooth regions, Roe's FDS would be recovered so that the advantages of high‐order methods can be maintained. Multidimensional dissipation is introduced to eliminate the adverse effects caused by flux function switching and further enhance the robustness of shock‐capturing, especially when the shock waves are not aligned with grids. A series of tests shows that this new hybrid flux function with a high‐order weighted compact nonlinear scheme is not only robust for shock‐capturing but also accurate for hypersonic heat transfer prediction. Copyright © 2015 John Wiley & Sons, Ltd.     

Efficient retrieval of the thermo‐acoustic flame transfer function from a linearized CFD simulation of a turbulent flame

The stability of thermo‐acoustic pressure oscillations in a lean premixed methane‐fired generic gas turbine combustor is investigated. A key element in predicting the acoustically unstable operating conditions of the combustor is the flame transfer function. This function represents the dynamic relationship between a fluctuation in the combustor inlet conditions and the flame's acoustic response. A transient numerical experiment involving spectral analysis in computational fluid dynamics (CFD) is usually conducted to predict the flame transfer function. An important drawback of this spectral method application to numerical simulations is the required computational effort. A much faster and more accurate method to calculate the transfer function is derived in this paper by using a most important basic assumption: the fluctuations must be small enough for the system to behave linear. This alternative method, which is called the linear coefficient method, uses a linear representation of the unsteady equations describing the CFD problem. This linearization is applied around a steady‐state solution of the problem, where it can consequently describe the dynamics of the system. Finally, the flame transfer function can be calculated from this linear representation. The advantage of this approach is that one only needs a steady‐state solution and linearization of the unsteady equations for calculating a dynamic transfer function, i.e. no time‐consuming transient simulations are necessary anymore. Nevertheless, as a consequence of the large number of degrees of freedom in a CFD problem, an extra order reduction step needs to be performed prior to calculating the transfer function from the linear representation. Still, the linear coefficient method shows a significant gain in both speed and accuracy when calculating the transfer function from the linear representation as compared to a spectral analysis‐based calculation. Hence, this method gives a major improvement to the application of the flame transfer function as a thermo‐acoustic design tool. Copyright © 2007 John Wiley & Sons, Ltd.     

The analytical method for three-dimensional elastic problems in finite deformations

The three-dimensional elastic problems in finite deformations are not known to have been analyzed by the usual stress function and displacement function. By applying Hasegava's presentation and Adkins perturbation method, we propose a new analytical method for three-dimensional elastic problems for compressible materials and incompressible materials, using the displacement function for axisymmetrical elastic problems in finite deformations with surface force or body force. Further, this analytical method is examined by two simple examples.     

Stretched Cartesian grids for solution of the incompressible Navier–Stokes equations

Two Cartesian grid stretching functions are investigated for solving the unsteady incompressible Navier–Stokes equations using the pressure–velocity formulation. The first function is developed for the Fourier method and is a generalization of earlier work. This function concentrates more points at the centre of the computational box while allowing the box to remain finite. The second stretching function is for the second‐order central finite difference scheme, which uses a staggered grid in the computational domain. This function is derived to allow a direct discretization of the Laplacian operator in the pressure equation while preserving the consistent behaviour exhibited by the uniform grid scheme. Both functions are analysed for their effects on the matrix of the discretized pressure equation. It is shown that while the second function does not spoil the matrix diagonal dominance, the first one can. Limits to stretching of the first method are derived for the cases of mappings in one and two directions. A limit is also derived for the second function in order to prevent a strong distortion of a sine wave. The performances of the two types of stretching are examined in simulations of periodic co‐flowing jets and a time developing boundary layer. Copyright © 2000 John Wiley & Sons, Ltd.     

Stress-Induced Phase Transformations of an Isotropic Incompressible Elastic Body Subject to a Triaxial Stress State

The present paper concerns the stable multiphase isochoric deformations for an isotropic elastic body subject to a surface traction of uniform Piola stress with two equal principal forces which are opposite to the third. To model the occurrence of such deformations, we consider a strain energy density function which depends on the first principal invariant of deformation through a non-convex function and which has an added linear dependence on the second invariant. We establish existence conditions for equilibrium multiphase deformations which give restrictions on the morphology of the connecting phases as well as on the orientation of the flat interfaces between the phases. Finally, by considering a special, but representative, form for the non-convex strain energy function, we show that there exists a “critical” value of the external load which allows for the emergence of stable coexistent deformation fields.     

Molecular shape in homogeneous flows: predictions of the rouse model

An analysis of the Rouse model is used to predict the shape of flexible, free-draining polymers in homogeneous flows. The derivation is based on the Lodge-Wu/Booij-Van Wiechen solutions [1,2] for the configurational distribution function of a Rouse chain, which consists of a free-draining string of beads connected by Hookean springs. Much of the theoretical development parallels the one used by Saab et al. [3,4] to predict link orientations for the same molecular model; rather than solving for link orientation, however, we solve for bead location. The function P_v(U, t), which describes the spatial distribution of bead v about the chain center of mass in the presence of hydrodynamic forces, is a key development of this contribution. Various averages evaluated from this function can be employed to study details of a deformed molecule's configuration. The model predicts that spatial distortion in flow is a highly nonuniform function of location in the chain; hydrodynamic forces have a greater influence on the beads near the ends of the chain than on those near the center.     

On the solution for steady state and time dependent flows in certain channels with small wall curvature

An asymptotic scheme is presented for the solution of the steady state and time dependent stream functions for flows in symmetric curved walled channels. In this scheme a class of non-linear Jeffery-Hamel solutions appear at O(1), and thus provide the first approximation to the steady state stream function. This class of Jeffery-Hamel solutions are evaluated by using a simple perturbation about Poiseuille flow. The classic Orr-Sommerfeld eigenproblem appears at O(1) in the asymptotic development of the time dependent stream function, but here there is a slow streamwise dependence. This eigenvalue problem, for a complex wave number, is solved using an algorithm which automatically provides an initial guess which is then used to iterate to the correct eigenvalue. Higher order terms in the asymptotic development, for both the steady state and time dependent stream functions, are evaluated to provide a solution for the total stream function.