A note on general solution of Stokes equations

Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).     

Relaxation and retardation functions of the Maxwell model with fractional derivatives

A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition. This model is used to determine the relaxation and retardation functions. The relaxation function was found in the time domain with the help of a power law series; a direct solution was used in the Laplace domain. The solution can be presented as a product of a power law term and the Mittag-Leffler function. The retardation function is determined via Laplace transformation and is solely a power law type.The investigation of the relaxation function shows that it is strongly monotonic. This explains why the model with fractional derivatives is consistent with thermodynamic principles.This type of rheological constitutive equation shows fluid behavior only in the case of a fractional derivative of the stress and a first order derivative of the strain. In all other cases the viscosity does not reach a stationary value.In a comparison with other relaxation functions like the exponential function or the Kohlrausch-Williams-Watts function, the investigated model has no terminal relaxation time. The time parameter of the fractional Maxwell model is determined by the intersection point of the short- and long-rime asymptotes of the relaxation function.     

Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions

The paper provides an exact analytical solution for the equilibrium configurations of a cantilever rod subject to inclined force and tip moment acting on its free end. The solution is given in terms of Jacobi’s elliptical functions and illustrated by several numerical examples and several graphical presentations of shapes of deformed cantilevers. Possible forms of the underlying elastica of a cantilever are discussed in detail, and various simple formulas are given for calculating the characteristic dimensions of the elastica. For the case when a cantilever is subject only to applied force, three load conditions are discussed: the follower load problem, the load determination problem, and the conservative load problem. For all cases, either a formula or an effective procedure for determining the solution is provided. In particular, a new efficient procedure is given to determine all possible equilibrium shapes in the case of the conservative load problem.     

Damage theory of materials with microstructure

A new form of damage theory of materials is proposed, that is valid for the case of nonconservative stresses. The partial entropy, strain and microstructure parameters are taken as the state variables. Without assuming the free energy to be a state function, the basic governing equations are derived. According to the balance of released and dissipated energy, the general form of damage evolution equation is obtained. Further, assuming the existence of independent damage mechanisms, the normality of damage evolution equation is proven. The generalized damage variables are discussed. Finally, some examples are given to show the applications of the theory. Projects Sponsered by the Joint Seismological Science Foundation.     

On simplified solutions of contact problems for elastic foundations

In the paper, a method of averaging displacements in a circular area lying in a linearly elastic transversally isotropic foundation is developed for the four modes of motion: vertical, horizontal, rocking and torsional. The corresponding formulae are constructed in a general form which does not depend on the kind of Green functions. For vertical and horizontal modes, uniform load distribution are applied and the simple integral mean is considered, whereas for rotational modes the load proportional to the distance from an axis of rotation is used, and angles of rotation for individual points are averaged with weight of the distance squared. Along with the case of equal radii of circles of loading and averaging, the case of different radii is studied, which allows one to consider contact problems for embedded axisymmetric foundations having the radius varying with depth. As examples the following contact problems are studied: static stiffness for a cone embedded in a homogeneous isotropic half-space in vertical motion, and dynamic stiffness for a disk on a layer resting on a homogeneous half-space for four modes of motion. Comparisons with the corresponding exact solutions are carried out.     

The integrated singular basis function method for the stick-slip and the die-swell problems

We further develop a new singular finite element method, the integrated singular basis function method (ISBFM), for the solution of Newtonian flow problems with stress singularities. The ISBFM is based on the direct subtraction of the leading local solution terms from the governing equations and boundary conditions of the original problem, followed by a double integration by parts applied to those integrals with singular contributions. The method is applied to the stick-slip and the die-swell problems and improves the accuracy of the numerical results in both cases. In the case of the die-swell problem it considerably accelerates the convergence of the free surface profile with mesh refinement. The advantages and disadvantages of the ISBFM when compared to other singular methods are also discussed.     

Trigonometric function used to formulate a multi-nodal finite tubular element

It is presented an alternative formulation to solve the problem of the deformation analysis for tubular element under pinching loads. The solution is based on a new displacement field defined from a total set of trigonometric functions. The solution is developed in a multi-nodal finite tubular ring element with a total of eight degrees of freedom per section considered. The purpose of this paper is to provide an easy alternative formulation when compared with a complex finite shell element or beam element analysis for the same application. Several case studies presented have been compared and discussed with numerical analyses results reported by other authors and the results obtained with a shell element from a Cosmos/M^® programme.     

CONSTRUCTION SPACE HARMONIC FUNCTIONS IN POLYN0MIAL FORM AND SPHERICAL FUNCTIONS BY COMPLEX-FUNCTIONAL

I-Intr0ductionSeeingthattheharmonicfunctionsarewidelyapplied,inthispaperwehaveconstructedthespaceharmonicfunctionsinpolynomialformbyapplyingthetheoryofcomplex-functiona1.Thenwehaveobtainedthesphericalfunctionstoo,includingtheLegendrepolynomialandtheassoci…     

Construction space harmonic functions in polynomial form and spherical functions by complex-functional

In this paper, applying the theory of complex-functional, not only the space harmonic functions in polynomial form, but also the spherical functions are obtained.     

An Analytical Solution for Fully Developed Flow in a Curved Porous Channel for the Particular Case of Monotonic Permeability Variation

A theoretical analysis of the Dean problem in heterogenous porous media is presented for the specific case of monotonic permeability variation in the vertical direction. The solutions are presented in terms of the curvature ratio η which is shown to affect the flow patterns. No multiple vortex solutions were noted for all values of the curvature ratio η.     

Representation Theorems for Hemitropic and Tranversely Isotropic Tensor Functions

A representation theorem for transversely isotropic tensor-valued functions of a symmetric tensor variable is proved. The theorem holds in any finite dimension. The proof is based on the decomposition of a symmetric tensor of dimension N into a scalar, a vector, and a symmetric tensor of dimension N-1, and on the fact that the transverse isotropy of the original function is equivalent to the hemitropy of three functions, one scalar-valued, one vector-valued, and one tensor-valued, of the last two terms in the decomposition. Representation theorems for the three functions are obtained as generalizations of two theorems of W. Noll on isotropic functions. The proofs make use of an appropriate algebraic structure based on alternating forms. The three-dimensional case, as well as those of linear and of hyperelastic functions, are treated as special cases. This revised version was published online in August 2006 with corrections to the Cover Date.     

Systematic errors in digital image correlation due to undermatched subset shape functions

Digital image correlation techniques are commonly used to measure specimen displacements by finding correspondences between an image of the specimen in an undeformed or reference configuration and a second image under load. To establish correspondences between the two images, numerical techniques are used to locate an initially square image subset in a reference image within an image taken under load. During this process, shape functions of varying order can be applied to the initially square subset. Zero order shape functions permit the subset to translate rigidly, while first-order shape functions represent an affine transform of the subset that permits a combination of translation, rotation, shear and normal strains. In this article, the systematic errors that arise from the use of undermatched shape function, i.e., shape functions of lower order than the actual displacement field, are analyzed. It is shown that, under certain conditions, the shape functions used can be approximated by a Savitzky-Golay low-pass filter applied to the displacement functions, permitting a convenient error analysis. Furthermore, this analysis is not limited to the displacements, but naturally extends to the higher-order terms included in the shape functions. This permits a direct analysis of the systematic strain errors associated with an undermatched shape function. Detailed numerical studies are presented for the case of a second-order displacement field and first- and second-order shape functions. Finally, the relation of this work to previously published studies is discussed.     

Bianalytic functions, biharmonic functions and elastic problems in the plane

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Radial basis function (RBF)‐based parametric models for closed and open curves within the method of regularized stokeslets

The method of regularized Stokeslets (MRS) is a numerical approach using regularized fundamental solutions to compute the flow due to an object in a viscous fluid where inertial effects can be neglected. The elastic object is represented as a Lagrangian structure, exerting point forces on the fluid. The forces on the structure are often determined by a bending or tension model, previously calculated using finite difference approximations. In this paper, we study spherical basis function (SBF), radial basis function (RBF), and Lagrange–Chebyshev parametric models to represent and calculate forces on elastic structures that can be represented by an open curve, motivated by the study of cilia and flagella. The evaluation error for static open curves for the different interpolants, as well as errors for calculating normals and second derivatives using different types of clustered parametric nodes, is given for the case of an open planar curve. We determine that SBF and RBF interpolants built on clustered nodes are competitive with Lagrange–Chebyshev interpolants for modeling twice‐differentiable open planar curves. We propose using SBF and RBF parametric models within the MRS for evaluating and updating the elastic structure. Results for open and closed elastic structures immersed in a 2D fluid are presented, showing the efficacy of the RBF–Stokeslets method. Copyright © 2015 John Wiley & Sons, Ltd.     

Enhancing control of grid distribution in algebraic grid generation

Three techniques are presented to enhance the control of grid-point distribution for a class of algebraic grid generation methods known as the two-, four- and six-boundary methods. First, multidimensional stretching functions are presented, and a technique is devised to construct them based on the desired distribution of grid points along certain boundaries. Second, a normalization procedure is proposed which allows more effective control over orthogonality of grid lines at boundaries and curvature of grid lines near boundaries. And third, interpolating functions based on tension splines are introduced to control curvature of grid lines in the interior of the spatial domain. In addition to these three techniques, consistency conditions are derived which must be satisfied by all user-specified data employed in the grid generation process to control grid-point distribution. The usefulness of the techniques developed in this study was demonstrated by using them in conjunction with the two- and four-boundary methods to generate several grid systems, including a three-dimensional grid system in the coolant passage of a radial turbine blade with serpentine channels and pin fins.     

Comparison of three spectral methods for the Benjamin-Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions

The Benjamin-Ono equation is especially challenging for numerical methods because (i) it contains the Hilbert transform, a nonlocal integral operator, and (ii) its solitary waves decay only as O(1/|x|²). We compare three different spectral methods for solving this one-space-dimensional equation. The Fourier pseudospectral method is very fast through use of the Fast Fourier Transform (FFT), but requires domain truncation: replacement of the infinite interval by a large but finite domain. Such truncation is unnecessary for a rational basis, but it is simple to evaluate the Hilbert Transform only when the usual rational Chebyshev functions TB_n(x) are replaced by their cousins, the Christov functions; the FFT still applies. Radial basis functions (RBFs) are slow for a given number of grid points N because of the absence of a summation algorithm as fast as the FFT; because RBFs are meshless, however, very flexible grid adaptation is possible.     

On 3-d analog of the second basic problem of the theory of elasticity for a cylindrical solid

We formulate and solve the 3-d analog of the second basic problem for a cylindrical elastic solid.     

On free energy-based formulations for kinematic hardening and the decomposition F = fpfe

Within the framework of linear plasticity, based on additive decomposition of the linear strain tensor, kinematical hardening can be described by means of extended potentials. The method is elegant and avoids the need for evolution equations. The extension of small strain formulations to the finite strain case, which is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, proved not straight forward. Specifically, the symmetry of the resulting back stress remained elusive. In this paper, a free energy-based formulation incorporating the effect of kinematic hardening is proposed. The formulation is able to reproduce symmetric expressions for the back stress while incorporating the multiplicative decomposition of the deformation gradient. Kinematic hardening is combined with isotropic hardening where an associative flow rule and von Mises yield criterion are applied. It is shown that the symmetry of the back stress is strongly related to its treatment as a truly spatial tensor, where contraction operations are to be conducted using the current metric. The latter depends naturally on the deformation gradient itself. Various numerical examples are presented.