A general matrix operator for linear boundary value problems in slip-line field theory

A general matrix operator is established for solving linear boundary value problems in slip-line field theory. The operator will enable a wider class of problems to be solved than were previously possible with the matrix slip-line field technique. As an illustration, a range of solutions are generated involving the previously intractable Coulomb friction boundary condition.     

Mathematical theory of interval comparison and its application to measurement problems

We state the fundamental problem of measurement— comparison of quantities determined with accuracy up to an interval. We construct a mathematical theory of comparison for such quantities; for this theory we introduce interval relations similar to numerical relations, and we study their properties. Our results make it possible to solve measurement and control problems in interval form. Translated from Izmeritel'naya Tekhnika, No. 5, pp. 3–8, May, 1998.     

Homogenization of some variational problems connected to the theory of lubrication

An important problem in the theory of lubrication is to model and analyze the effects of surface roughness on the hydrodynamic performance. An efficient method to do this is homogenization. In this paper we prove a general homogenization result which allows us to consider unstationary variational problems, related to Reynolds type equations, where the lubricant may be Newtonian or non-Newtonian. Recently, the idea of finding upper and lower bounds on the effective behavior, obtained by homogenization, was applied for the first time in tribology. The homogenization result in this work may therefore also serve as a rigorous starting point for developing these successful results to unstationary problems.     

A natural neighbour method for linear elastic problems based on Fraeijs de Veubeke variational principle

The natural neighbour method can be considered as belonging to the meshless methods. Classically, the development of this method is based on the virtual work principle. In the present paper, we use the natural neighbour method for 2D domains starting from the Fraeijs de Veubeke variational principle and we approximate separately the displacement field, the stress field and the strain field: the assumed strains and the assumed stresses are constant over each Voronoi cell, the assumed surface reactions are constant along the edge where the displacements are imposed, while the assumed displacements are interpolated by Laplace interpolants. In the absence of body forces, it is shown that the calculation of integrals on the area of the solid domain can be avoided: only integrals on the edges of the Voronoi cells are necessary. On the other hand, displacements can be imposed in the average sense on some boundaries of the domain. Patch tests and some applications in the elastic domain are given in the paper and show the effectiveness of the method. Copyright © 2007 John Wiley & Sons, Ltd.     

A variational principle for minimizing experimental measurement errors and its application to a hybrid experimental-numerical method

In general, experimentally measured fields such as moiré-interferometry fringes contain various errors associated with the measurement and subsequent data analysis. Moreover, the measured displacement field does not necessarily satisfy the equilibrium equation in solid mechanics, especially when the measurement errors are involved. First, a variational principle minimizing the experimental measurement errors is derived to overcome such difficulties. Next, on the basis of this variational principle, a new hybrid moiré-interferometry and finite-element method is developed. Concepts of restoration energy and restoration force are also presented in the context of the present hybrid method. The present hybrid method demonstrates automatic detection and elimination of the measurement errors and smooth visualization of stress contours. The present hybrid method automatically achieves the path independence of the J integral, restoring the path-dependence caused by the measurement errors.     

A two-surface plasticity theory and its application to multiaxial loading

Summary A two-surface theory based upon experimental results is presented. The theory utilizes the concept of a yield surface and a loading surface to describe the essential features of the time-independent inelastic behavior of materials. The theory is implemented in a computer program and its predictions are compared to a variety of axial-torsional experimental data including both proportional and nonproportional loading.With 11 Figures     

A new high-order accurate continuous Galerkin method for linear elastodynamics problems

A new high-order accurate time-continuous Galerkin (TCG) method for elastodynamics is suggested. The accuracy of the new implicit TCG method is increased by a factor of two in comparison to that of the standard TCG method and is one order higher than the accuracy of the standard time-discontinuous Galerkin (TDG) method at the same number of degrees of freedom. The new method is unconditionally stable and has controllable numerical dissipation at high frequencies. An iterative predictor/multi-corrector solver that includes the factorization of the effective mass matrix of the same dimension as that of the mass matrix for the second-order methods is developed for the new TCG method. A new strategy combining numerical methods with small and large numerical dissipation is developed for elastodynamics. Simple numerical tests show a significant reduction in the computation time (by 5–25 times) for the new TCG method in comparison to that for second-order methods, and the suppression of spurious high-frequency oscillations.     

Asynchronous multi‐domain variational integrators for non‐linear problems

We develop an asynchronous time integration and coupling method with domain decomposition for linear and non‐linear problems in mechanics. To ensure stability in the time integration and in coupling between domains, we use variational integrators with local Lagrange multipliers to enforce continuity at the domain interfaces. The asynchronous integrator lets each domain step with its own time step, using a smaller time step where required by stability and accuracy constraints and a larger time step where allowed. We show that in practice the time step is limited by accuracy requirements rather than by stability requirements. Copyright © 2008 John Wiley & Sons, Ltd.     

A variable-domain variational theory using Clebsch variables for hybrid problems of 2-D transonic rotational flow

Summary By means of functional variations with variable domain the variational principles of [7], [8] are extended to inverse and hybrid flow problems.     

Finite element method for unbounded field problems and application to two-dimensional taper

Treatment of the finite element method for an unbounded field problem was proposed by McDonald and Wexler in 1972. Their method is superior to others, because it can exclude the singularities of Green's functions. This paper explains the treatment of the method in our 1979 letter which had some revisions of McDonald and Wexler's and calculated the time-harmonic field problems. Examples presented are electromagnetic fields of two-dimensional tapers which are open-ended. Electromagnetic waves propagate in the taper and radiate from the taper to free space. In this case, the exact solutions for radiation from tapers are not available because of the complicated shape, and so the finite element method is useful in solving these problems. Electromagnetic fields of tapers involving dielectric slabs are also calculated as examples of inhomogeneous problems.     

A variational approach to magnetoelastic buckling problems for systems of superconducting tori

A variational principle for the magnetoelastic stability problem of superconductors is constructed. Independently, a pair of integral equations is derived, from which the initial and the perturbed field can be computed. The integral equations are solved for in-plane buckling of a slender pair of concentric tori, and out-of-plane buckling of a slender pair of equal coaxial tori. By using the variational principle, it is shown that both cases can become unstable when the currents on the two tori are equally directed, and the pertinent buckling values are calculated. The thus obtained buckling values are compared with the results of an alternative, mathematically less rigorous, method. A good correspondence between the two methods is found (at least as long as the two tori are not too near).     

A study of optimal coordinate theory with application to several physical problems

A brief development of optimal coordinates for singular perturbation problems is presented in this paper, and several specific examples are then used to illustrate the application of the theory presented. In each of the examples discussed, sets of optimal coordinates are determined analytically in a systematic way.     

A linear finite element approach to the solution of the variational inequalities arising in contact problems of structural dynamics

The present paper deals with the theoretical and numerical treatment of dynamic unilateral problems. The governing equations are formulated as an equivalent variational inequality expressing D' Alembert's principle in its inequality form. The discretization with respect to time and space leads to a static nonlinear programming problem which is solved by an appropriate algorithm. Some properties of dynamic unilateral problems are outlined and the influence of several parameters on the solution is investigated by means of numerical examples.     

Numerical method for non‐linear wave and diffusion equations by the variational iteration method

This paper applies He's variational iteration to the wave equations in an infinite one‐dimensional medium and some non‐linear diffusion equations. A suitable choice of an initial solution can lead to the needed exact solution by a few iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

A 2-dimensional mixture theory for biaxially fiber reinforced composites with application to dynamic crack problems

A 2-dimensional mixture theory is developed for wave propagation in a laminated medium in which every layer is made of a fiber reinforced composite material with the angle of reinforcement alternating from layer to layer. The developed theory contains as special cases the 2-dimensional mixture theory for a laminated medium made of isotropic layers, as well as the equivalent modulus theories for bi-directionally and unidirectionally fiber reinforced composites. The developed mixture theory is applied to the problem of a semi-infinite crack in the composite which is under dynamic loading. The induced fracture mode of the crack is of mixed type and contains both Mode I and Mode II types of opening. A numerical method of solution is applied to the four coupled mixture equations of motion in the average displacements and results are given for the dynamic stress fields in the composite.     

A new boundary element method formulation for three dimensional problems in linear elasticity

Summary A new boundary element method (BEM) formulation for planar problems of linear elasticity has been proposed recently [6]. This formulation uses a kernel which has a weaker singularity relative to the corresponding kernel in the standard formulation. The most important advantage of the new formulation, relative to the standard one, is that it delivers stresses accurately at internal points that are extremely close to the boundary of a body. A corresponding BEM formulation for three dimensional problems of linear elasticity is presented in this paper. This formulation is derived through the use of Stokes' theorem and has kernels which are only 1/r singular (wherer is the distance between a source and a field point) for the displacement equation. The standard BEM formulation for three-dimensional elasticity problems has a kernel which is 1/r ² singular.With 2 Figures