Optimal Plastic Design of a Cross Section under Torsion with Small Bending∗

A doubly connected cross section of a bar under torsion with bending is optimized in two variants; either the internal contour is given and the external one is subject to optimization, or only the inside area is given and both contours are optimized. Minimal cross-sectional area is the design objective. The material is assumed to be perfectly plastic, incompressible, and subject to the Huber-Mises-Hencky yield condition. A specially adopted perturbation method is used and only the range of small bending is analyzed. The resulting optimal bars are regarded as thick-walled, hence no wall stability constraints are applied.     

An Improved Computational Approach for Multilevel Optimum Design∗

Recently there has been a growing interest in methods that decompose complex optimization problems and employ a multilevel or hierarchical approach. One of the most irksome problems with the hierarchical approach is the discontinuous behavior of derivatives that are transferred from the lower levels of the hierarchy to the upper levels. This paper proposes a hierarchical algorithm that is free of such difficulties. A penalty function method is employed, in combination with Newton's method with approximate second derivatives, to perform the optimization. The penalty function formulation is shown to be natural for multilevel problems. A simple three-bar truss and a simple frame problem are used for demonstration     

Optimum Design of Multistory Multispan Frames for Prescribed Elastic Compliance∗

This paper presents an analytical method for minimum cost design of regular rectangular building frames for constrained elastic compliance. The method, consisting of a semi-inverse method and a design region extension procedure, is illustrated by two classes of exact solutions. It is shown that the relative story displacements are almost uniformly distributed in an optimally designed frame with almost uniform story heights and that the solutions enable one to calculate all maximum member-end stresses from member-end curvatures. It is suggested that the proposed design formulas may be utilized for design problems, subject to relative story displacement and stress constraints. Since these solutions are shown to be dependent upon the nature of the prescribed minimum stiffnesses, a method of finding a practically reasonable set of minimum stiffnesses is also presented.     

Calculation of Linear Stability of a Stratified Gas–Liquid Flow in an Inclined Plane Channel

Linear stability of liquid and gas counterflows in an inclined channel is considered. The full Navier–Stokes equations for both phases are linearized, and the dynamics of periodic disturbances is determined by means of solving a spectral problem in wide ranges of Reynolds numbers for the liquid and vapor velocity. Two unstable modes are found in the examined ranges: surface mode (corresponding to the Kapitsa waves at small velocities of the gas) and shear mode in the gas phase. The wave length and the phase velocity of neutral disturbances of both modes are calculated as functions of the Reynolds number for the liquid. It is shown that these dependences for the surface mode are significantly affected by the gas velocity.     

A Perturbation Argument for a Monge–Ampère Type Equation Arising in Optimal Transportation

We prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge–Ampère equation in order to obtain, first, interior C^1,1 estimates for the potential and, second, interior Hölder estimates for second derivatives. In particular, we take a close look at the geometry of optimal transportation when the cost function is close to quadratic in order to understand how the equation degenerates near the boundary.     

A New Optimal Growth Constant for the Hele–Shaw Instability

We study the immiscible displacement of the oil from a homogeneous porous medium by using a less viscous fluid (water). We use the Hele–Shaw model, then a sharp interface exists between the fluids. The fingering phenomenon appears, first studied by Saffman and Taylor (1958). Gorell and Homsy (1983) consider an intermediate region (I. R.) between water and oil, containing a polymer mixture. The unknown viscosity in I. R. is a parameter which can improve the stability of the I. R.–oil interface. A numerical optimal viscosity profile in I. R. is given. Carasso and Paa (1998) obtain an explicit formula for an optimal viscosity profile in I. R. An upper estimation of the growth constant is given. In this paper, a very slow viscosity profile in I. R. is defined and an optimal formula for the growth constant is obtained, less than the previous estimation of Carasso and Paa. Moreover, this formula is similar with the Saffman–Taylor result, only the water viscosity is replaced by the limit value of viscosity in I. R. on the interface with the oil. We explain the apparent contradiction between the previous results of Gorell and Homsy (1983) and Paa and Polisevski (1992).     

A Characterization of Benford’s Law in Discrete-Time Linear Systems

A necessary and sufficient condition (“nonresonance”) is established for every solution of an autonomous linear difference equation, or more generally for every sequence \((x^\top A^n y)\) with \(x,y\in \mathbb {R}^d\) and \(A\in \mathbb {R}^{d\times d}\), to be either trivial or else conform to a strong form of Benford’s Law (logarithmic distribution of significands). This condition contains all pertinent results in the literature as special cases. Its number-theoretical implications are discussed in the context of specific examples, and so are its possible extensions and modifications.     

A Theory of General Solutions of Plane Problems in Two-Dimensional Octagonal Quasicrystals

A theory of general solutions of plane problems is developed for the coupled equations in plane elasticity of two-dimensional octagonal quasicrystals. In virtue of the operator method, the general solutions of the antiplane and inplane problems are given constructively with two displacement functions. The introduced displacement functions have to satisfy higher order partial differential equations, and therefore it is difficult to obtain rigorous analytic solutions directly and is not applicable in most cases. In this case, a decomposition and superposition procedure is employed to replace the higher order displacement functions with some lower order displacement functions, and accordingly the general solutions are further simplified in terms of these functions. In consideration of different cases of characteristic roots, the general solution of the antiplane problem involves two cases, and the general solution of the inplane problem takes three cases, but all are in simple forms that are convenient to be applied. Furthermore, it is noted that the general solutions obtained here are complete in x ₃-convex domains.      

Plastic Stress Distribution in a Rotating Disk with Rigid Inclusion Under a Radial Temperature Gradient∗

Abstract

This study investigates the plastic stress distribution in a rotating disk with rigid inclusion, under an axially symmetric steady-state temperature gradient. The analysis is based on Tresca's yield condition, its associated flow rule, and linear strain hardening material behavior.

     

Linear Stability for Transition Front Solutions in Multidimensional Cahn–Hilliard Systems

We consider linear stability for planar transition front solutions \(\bar{u} (x_1)\) arising in multidimensional (i.e., \(x\in \mathbb {R}^n\)) Cahn–Hilliard systems. In previous work the author has established that the linear operator obtained from linearization about the transition front has (after Fourier transform in the transverse variable \(\tilde{x} = (x_2, x_3, \dots , x_n) \mapsto \xi \)) a leading eigenvalue that moves into the stable (Re \({\uplambda }< 0\)) half-plane at rate \(|\xi |^3\). This constitutes precisely the type of borderline case that has been effectively analyzed by the pointwise semigroup methods of Zumbrun and Howard, and we follow that approach here. In particular, the approach can be viewed as a three-step process including: (1) characterization of the spectrum of the linearized operator; (2) derivation of linear semigroup estimates, typically encoded into estimates on an appropriate Green’s function; and (3) implementation of an iterative process to accommodate nonlinearities. In this paper we address Step (2) in the case of multidimensional Cahn–Hilliard systems.     

Persistence of Exponential Trichotomy for Linear Operators: A Lyapunov–Perron Approach

In this article we revisit the perturbation of exponential trichotomy of linear difference equation in Banach space by using a Perron–Lyapunov fixed point formulation for the perturbed evolution operator. This approach allows us to directly re-construct the perturbed semiflow without using shift spectrum arguments. These arguments are presented to the case of linear autonomous discrete time dynamical system. This result is then coupled to Howland semigroup procedure to obtain the persistence of exponential trichotomy for non-autonomous difference equations.     

A Novel “Subset Splitting” Procedure for Digital Image Correlation on Discontinuous Displacement Fields

Digital Image Correlation (DIC) is an easy to use yet powerful approach to measure displacement and strain fields. While the method is robust and accurate for a variety of applications, standard DIC returns large error and poor correlation quality near displacement discontinuities such as cracks or shear bands. This occurs because the subsets used for correlation can only capture continuous deformations from the reference to the deformed image. As a result the regions around discontinuities are typically removed from the area of interest, before or after analysis. Here, a novel approach is proposed which enables the subset to split in two sections when a discontinuity is detected. This method enables the measurement of “displacement jumps”, and also of displacements and strains right by the discontinuity (for example a crack profile or residual strains in the wake). The method is validated on digitally created images based on mode I and mode II asymptotic displacement fields, for both sub-pixel and super-pixel crack opening displacements. Finally, an actual fracture experiment on a high density polyethylene (HDPE) specimen demonstrates the robustness of the method on actual images. Compared to other methods capable of handling discontinuities, this novel “subset-splitting” procedure offers the advantage of being a direct extension of the now popular standard DIC, and can therefore be implemented as an “upgrade” to that method.     

Uniform Convergence for Approximate Traveling Waves in Linear Reaction–Diffusion–Hyperbolic Systems

In this paper we study linear reaction–hyperbolic systems of the form , (i = 1, 2, ..., n) for x > 0, t > 0 coupled to a diffusion equation for p ₀ = p ₀(x, y, θ, t) with “near-equilibrium” initial and boundary data. This problem arises in a model of transport of neurofilaments in axons. The matrix (k _ij ) is assumed to have a unique null vector with positive components summed to 1 and the v _j are arbitrary velocities such that . We prove that as the solution converges to a traveling wave with velocity v and a spreading front, and that the convergence rate in the uniform norm is , for any small positive α.     

Plane-Strain Problems for a Class of Gradient Elasticity Models��A Stress Function Approach

The plane strain problem is analyzed in detail for a class of isotropic, compressible, linearly elastic materials with a strain energy density function that depends on both the strain tensor ?? and its spatial gradient ???. The appropriate Airy stress-functions and double-stress-functions are identified and the corresponding boundary value problem is formulated. The problem of an annulus loaded by an internal and an external pressure is solved.     

Graphical Illustrations for the Nur-Byerlee-Carroll Proof of the Formula for the Biot Effective Stress Coefficient in?Poroelasticity

This work presents a graphically illustrated version of the Nur-Byerlee-Carroll proof of the formula for the Biot effective stress coefficient in poroelasticity. The original elegant proof was provided by Nur and Byerlee (J. Geophys. Res. 76:6414, 1971) for isotropic materials and extended by Carroll (J. Geophys. Res. 84:7510?C7512, 1979) to anisotropic materials. Although the application of this result is in poroelasticity or in the analysis of composite materials, the proof is an analytical thought experiment in linear elasticity, and should be appreciated as such.     

A Coordinate Reduction Technique for Dynamic Analysis of Spatial Substructures with Large Angular Rotations∗

A substructuring technique is presented for transient dynamic analysis of systems composed of interconnected rigid and elastic bodies that undergo large angular displacements. Displacement of elastic bodies is represented by superposition of local linear elastic deformation on large displacement of body reference coordinate systems. Elastic bodies are thus represented by combined sets of reference and local elastic generalized coordinates. Modal analysis and substructuring of individual elastic components allow for elimination of insignificant modes. Equations of motion and constraint are formulated in terms of mixed sets of modal and reference generalized coordinates. Planar and spatial linkages with flexible elements are presented to illustrate use of the method developed.     

A New Solution Method for Some Boundary Value Problems of Elastic Beams and Plates∗

Abstract

ABSTRACT This paper presents a new solution method for several types of buckling and bending problems of beams and plates. The method is based on the use of a non-orthogonal series expansion, consisting of some specially chosen trigonometric functions for the elastic curve y of a beam or the deflection surface w of a plate. The calculations are performed using the Euler and Bernoulli polynomials, under realistic approximations of limiting values of the boundary conditions. In this method, it is not necessary to use the solution of the differential equation of the problem, Results obtained using the method are shown to be consistent with known solutions.     

Propagation of Linear Waves in Gas‐Saturated Porous Media with Allowance for Interphase Heat Transfer

The effect of gas–skeleton heattransfer processes on propagation of fast and slow waves in a porous medium is examined. Frequency intervals are identified, in which attenuation of waves in a gassaturated porous medium is mainly controlled by the heattransfer processes.     

Fracture Analysis for a Frictional Fastener Hole with Edge Cracks in an Anisotropic Plate∗

Abstract

Stress intensity factors are evaluated for a singly or doubly cracked fastener hole with frictional traction in an anisotropic plate, using a special kernel boundary integral equation (BIE) approach. The integration kernel (Green's function) used in this BIE approach has already taken the presence of the crack (or cracks) into account, thus.avoiding the need for element discretization to model the stress singularity at the crack tip. The Green's function employed is that of an infinite anisotropic plate containing an elliptical hole or crack, and subjected to an arbitrarily positioned point force. Several types of normal and shear traction conditions at the pinhole interface are considered. Numerical results are obtained for various geometrical and loading conditions and are compared with known solutions, where available, for their isotropic counterparts.