A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations

This study develops a small-deformation theory of strain-gradient plasticity for isotropic materials in the absence of plastic rotation. The theory is based on a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows:

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the microstresses to depend on , the gradient of the plastic strain-rate, and

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the free energy ψ to depend on the Burgers tensor .

The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for the plastic strain. The microstresses are strictly dissipative when ψ is independent of the Burgers tensor, but when ψ depends on G the gradient microstress is partially energetic, and this, in turn, leads to a back stress and (hence) to Bauschinger-effects in the flow rule. It is further shown that dependencies of the microstresses on lead to strengthening and weakening effects in the flow rule.Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and, as an aid to numerical solutions, a weak (virtual power) formulation of the nonlocal flow rule is derived.     

A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin

This study develops a gradient theory of small-deformation viscoplasticity based on: a system of microforces consistent with its peculiar balance; a mechanical version of the second law that includes, via the microforces, work performed during viscoplastic flow; a constitutive theory that accounts for the Burgers vector through a free energy dependent on , with H^p the plastic part of the elastic-plastic decomposition of the displacement gradient. The microforce balance and the constitutive equations, restricted by the second law, are shown to be together equivalent to a nonlocal flow rule in the form of a coupled pair of second-order partial differential equations. The first of these is an equation for the plastic strain-rate in which the stress T plays a basic role; the second, which is independent of T, is an equation for the plastic spin. A consequence of this second equation is that the plastic spin vanishes identically when the free energy is independent of, but not generally otherwise. A formal discussion based on experience with other gradient theories suggests that sufficiently far from boundaries solutions should not differ appreciably from classical solutions, but close to microscopically hard boundaries, boundary layers characterized by a large Burgers vector and large plastic spin should form.Because of the nonlocal nature of the flow rule, the classical macroscopic boundary conditions need be supplemented by nonstandard boundary conditions associated with viscoplastic flow. As an aid to solution, a variational formulation of the flow rule is derived.Finally, we sketch a generalization of the theory that allows for isotropic hardening resulting from dissipative constitutive dependences on .     

Identifying preferred solutions to Multi-Objective Binary Optimisation problems,with an application to the Multi-Objective Knapsack Problem

In this paper we present a new framework for identifying preferred solutions to multi-objective binary optimisation problems. We develop the necessary theory which leads to new formulations that integrate the decision space with the space of criterion weights. The advantage of this is that it allows for incorporating preferences directly within a unique binary optimisation problem which identifies efficient solutions and associated weights simultaneously. We discuss how preferences can be incorporated within the formulations and also describe how to accommodate the selection of weights when the identification of a unique solution is required. Our results can be used for designing interactive procedures for the solution of multi-objective binary optimisation problems. We describe one such procedure for the multi-objective multi-dimensional binary knapsack formulation of the portfolio selection problem.     

The cubic Fermat equation and complex multiplication on the Deuring normal form

The arithmetic on elliptic curves in Deuring normal form is shown to be related to solutions of the Fermat equation 27X ³+27Y ³=X ³ Y ³. This arithmetic is used to give conditions for the existence of multipliers μ on supersingular elliptic curves in characteristic p for which μ ²=?3p. Together with an explicit factorization of a certain class equation, these conditions imply that the number of irreducible binomial quadratic factors (mod p) of the Legendre polynomial P _(p?e)/3(x) of degree (p?e)/3 is a simple linear function of the class number of the quadratic field \(\mathbb{Q}(\sqrt{-3p})\).     

Periodic points of algebraic functions and Deuring’s class number formula

The Ramanujan Journal - In this paper, transformation formulas for the function $$\begin{aligned} A_{1}\left( z,s:\chi \right) =\sum \limits _{n=1}^{\infty }\sum \limits _{m=1} ^{\infty }\chi...     

Influence of entrance channel properties on heavy-ion reaction dynamics

In the fusion of heavy nuclei, there is a distribution of fusion barrier energies resulting from coupling between intrinsic motion and internal degrees of freedom. Precise experimental measurements of excitation functions have allowed the extraction of the distributions by taking the second derivative using a point-difference method. In the case of statically deformed nuclei, experimental data shows that the different fusion barrier energies correspond to different physical configurations of the colliding nuclei, the latter affecting the subsequent dynamical trajectories over the potential energy surface, influencing the ultimate reaction products, as for example in quasi-fission. The fusion barrier distribution is also valuable in understanding the fusion of weakly bound nuclei, enabling a reliable prediction of the expected fusion cross-sections, and thus the determination of fusion suppression factors at above-barrier energies. Received: 1 May 2001 / Accepted: 4 December 2001     

Assessing solution quality in stochastic programs

Determining whether a solution is of high quality (optimal or near optimal) is fundamental in optimization theory and algorithms. In this paper, we develop Monte Carlo sampling-based procedures for assessing solution quality in stochastic programs. Quality is defined via the optimality gap and our procedures' output is a confidence interval on this gap. We review a multiple-replications procedure that requires solution of, say, 30 optimization problems and then, we present a result that justifies a computationally simplified single-replication procedure that only requires solving one optimization problem. Even though the single replication procedure is computationally significantly less demanding, the resulting confidence interval might have low coverage probability for small sample sizes for some problems. We provide variants of this procedure that require two replications instead of one and that perform better empirically. We present computational results for a newsvendor problem and for two-stage stochastic linear programs from the literature. We also discuss when the procedures perform well and when they fail, and we propose using ɛ-optimal solutions to strengthen the performance of our procedures.     

Non-linear age-dependent population dynamics

Archive for Rational Mechanics and Analysis -