On Scalar Functionals of Nonlinear tensorial Constitutive Equations in the Theory of Plasticity for Nonisothermal Rectilinear Processes of Deformation

Based on base experiments formulated, methods are proposed to specify the scalar functional in the nonlinear equations that relate generalized stresses and finite strains in the theory of plasticity. The base experiments are conducted and the functionals are specified. It is shown that the nonlinear tensorial constitutive equations can be used to describe a nonisothermal process of deformation along a rectilinear path, which is distinct from the base ones, at high temperatures that cause creep strains     

Existence of Minima¶for Nonconvex Functionals¶in Spaces of Functions¶Depending on the Distance from the Boundary

We prove that some nonconvex functionals admit a unique minimum in a functional space of functions which depend only on the distance from the boundary of the (plane) domain where they are defined. The domains considered are disks and regular polygons. We prove that the sequence of minima of the functional on the polygons converges to the unique minimum on the circumscribed disk as the number of sides tends to infinity. Our method also allows us to determine the explicit form of the minima. (Accepted January 19, 1999)     

A Sharp Upper Bound for the¶Torsional Rigidity of Rods¶by Means of Web Functions

Using web functions, we approximate the Dirichlet integral which represents the torsional rigidity of a cylindrical rod with planar convex cross-section Ω. To this end, we use a suitably defined piercing function, which enables us to obtain bounds for both the approximate and the exact value of the torsional rigidity. When Ω varies, we show that the ratio between these two values is always larger than ¾; we prove that this is a sharp lower bound and that it is not attained. Several extremal cases are also analyzed and studied by numerical methods.     

Constitutive Functions of Elastic Materials in Finite Growth and Deformation

The constitutive functions of soft biological tissues during growth are studied. A growth, treated as addition (often non-uniform) of material points, results in deformation, residual stresses, and evolution of the constitutive functions. A theory based on the concept of equivalent material points is developed with the current configuration taken as the reference. The residual stresses developed in a spherical shell undergoing spherical growths are studied.     

On the Green's Functions and Boundary Integral Formulation of Elastic Planes with Cutouts

ABSTRACT

The problem of an infinite elastic plane that contains a hole of arbitrary shape and is subjected to a concentrated unit load is considered. The Green's function (influence function) for the problem is formulated by means of two complex potential functions. This is accomplished by mapping the region that is exterior to the hole onto a unit circle. A class of closed contour hole shapes is analyzed. Green's functions for an elliptical hole and a class of triangular holes are determined. Green's functions for a class of rectangular holes are also discussed. In order to determine stress and displacement fields for the finite plane problem, Green's function is employed and an indirect boundary integral equation is formulated, with the integrand of the integral equation incorporating the effect of the hole. The contour of the hole is no longer considered a part of the boundary and only the contour of the region that is exterior to the hole is subdivided into boundary elements. Examples for elliptical and triangular holes are solved.     

On the Existence of a Smooth Bounded Semiinvariant Manifold for a Degenerate Nonlinear System of Difference Equations in the Space m

Using the Green–Samoilenko function, we construct a bounded Frechét-differentiable semiinvariant manifold for a nonlinear system of difference equations in a Banach space of bounded number sequences.     

On a Countable-Point Nonlinear Boundary-Value Problem on a Semiaxis for Ordinary Differential Equations in the Space of Bounded Numerical Sequences

We use the idea of the Samoilenko numerical-analytic method for the investigation of a nonlinear boundary-value problem with an unbounded countable set of boundary moments on the positive semiaxis in the case where the differential equation and boundary conditions are defined in the Banach space of bounded numerical sequences.     

求解饱和半空间上弹性圆板固结沉降的积分方程

本文采用解析方法分析了弹性圆板在饮和半空间上的固结沉降。考虑弹性圆板与饮和半空间的接触面上无摩擦力,且饱和半空间表面为全部透水的。运用Ｂｉｏｔ固结理论和积分方程技术,在Ｌａｐｌａｃｅ变换域上建立了弹性圆板固结沉降的对偶积分方程,并化此对偶积分方程为第二类Ｆｒｅｄｈｏｌｍ积分方程。通过对其核函数的有效数值发得到第二类Ｆｒｅｄｈｏｌｍ积分方程的解,再利用Ｌａｐａｃｅ反演技术获得弹性板在时间域中的固结沉     

Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV

Generalized Young measures as introduced by DiPerna and Majda (Commun Math Phys 108:667–689, 1987) provide a quantitative tool for studying the one-point statistics of oscillation and concentration in sequences of functions. In this work, after developing a functional-analytic framework for such measures, including a compactness theorem and results on the generation of such Young measures by L¹-bounded sequences (or even by sequences of bounded Radon measures), we turn to investigation of those Young measures that are generated by bounded sequences of W^1,1-gradients or BV-derivatives. We provide several techniques to manipulate such measures (including shifting, averaging and approximation by piecewise-homogeneous Young measures) and then establish the main new result of this work, the duality characterization of the set of (BV- or W^1,1-)gradient Young measures in terms of Jensen-type inequalities for quasiconvex functions with linear growth at infinity. This result is the natural generalization of the Kinderlehrer–Pedregal Theorem (Arch Ration Mech Anal 115:329–365, 1991; J Geom Anal 4:59–90, 1994) for classical Young measures to the W^1,1- and BV-case and contains its version for weakly converging sequences in W^1,1 as a special case. Finally, we give an application to a new lower semicontinuity theorem in BV.     

用积分变换及边界积分方法求解多层地基的静力问题

本文利用积分变换及矩阵递推方法得到了任意n层弹性体平面应变及轴对称问题的Mindlin解。再把此解作为基本解,利用Somigliana关系式,得到计算多层弹性体内部任意点位移的简便方法。利用此法很容易编制程序,且具有较高的计算精度与速度。     

Decay and Continuity of the Boltzmann Equation in Bounded Domains

Boundaries occur naturally in kinetic equations, and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in the L ^∞ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set at the boundary. Our contribution is based on a new L ² decay theory and its interplay with delicate L ^∞ decay analysis for the linearized Boltzmann equation in the presence of many repeated interactions with the boundary.     

An Algorithm for 3D Pore Space Reconstruction from a 2D Image Using Sequential Simulation and Gradual Deformation with the Probability Perturbation Sampler

The construction of a faithful 3D pore space model of a porous medium that could reproduce the macroscopic behavior of that medium is of great interest in various fields including medicine, material science, hydrology and petroleum engineering. A computationally efficient algorithm is developed that uses the probability perturbation method and sequential multiple-point statistics simulations to generate 3D stochastic and equiprobable representations of random porous media when only a 2D thin section image is available. By employing the probability perturbation method as a gradual deformation technique, the pore patterns of a single 2D image are deformed to generate a series of 2D stochastically simulated images. The 3D pore structure is then generated by simply stacking the 2D-simulated images. The quality of the 3D reconstruction is critically dependent on the rate of deformation and a simple general procedure for choosing this parameter is presented. Various criteria such as porosity, two-point auto-correlation function, multiple-point connectivity function, local percolation probability, absolute permeability obtained by lattice-Boltzmann method (LBM), formation factor and two-phase relative permeability calculations are used to validate the results. The method is tested on two random porous solids; Berea Sandstone and synthetic Silica, for which directly measured 3D micro-CT images are available. The stochastically reconstructed 3D pore space preserves the low- and high-order spatial statistics, the macroscopic flow properties and the microstructure of the 3D micro-CT images.     

沥青路面反射裂缝问题的损伤力学守恒积分

基于损伤力学理论，建立了沥青路面反射裂缝问题的损伤力学守恒积分，证明了在反射裂缝形成过程中应变能密度的守恒性。在此基础上，导出了预估反射裂缝形成寿命的简便方法与公式。本文研究成果对于沥青路面设计与旧路补强设计具有一定理论指导意义。     

Lower Bound for the Energy of Bloch Walls in Micromagnetics

We study a two-dimensional nonconvex and nonlocal energy in micromagnetics defined over S ²-valued vector fields. This energy depends on two small parameters, β and e{\varepsilon} , penalizing the divergence of the vector field and its vertical component, respectively. Our objective is to analyze the asymptotic regime b << e << 1{\beta \ll \varepsilon \ll 1} through the method of Γ-convergence. Finite energy configurations tend to become divergence-free and in-plane in the magnetic sample except in some small regions of typical width e{\varepsilon} (called Bloch walls) where the magnetization connects two directions on S ². We are interested in quantifying the limit energy of the transition layers in terms of the jump size between these directions. For one-dimensional transition layers, we show by Γ-convergence analysis that the exact line density of the energy is quadratic in the jump size. We expect the same behaviour for the two-dimensional model. In order to prove that, we investigate the concept of entropies. In the prototype case of a periodic strip, we establish a quadratic lower bound for the energy with a non-optimal constant. Then we introduce and study a special class of Lipschitz entropies and obtain lower bounds coinciding with the one-dimensional Γ-limit in some particular cases. Finally, we show that entropies are not appropriate in general for proving the expected sharp lower bound.     

Integral theory for the dynamics of nonlinear nonholonomic system in noninertial reference frames

This paper establishes the integral theory for the dynamics of nonlinearnonholonomic system in noninertial reference frame.Firstly,based on the Routhequation of the relative motion of nonlinear nonholonomic system gives the firstintegral of the system.Secondly,by using cyclic integral or energy integral reduces theorder of the equation and obtains generalized Routh equation and Whittaker equationrespectively.Thirdly,derives canonical equation and variation equation and by usingthe first integral constructs integral invariant.And then,establishes the basic integralvariants and the integral invariant of Poincare-Cartan type.Finally,we give a series ofdeductions.     

The Cubic-to-Orthorhombic Phase Transition: Rigidity and Non-Rigidity Properties in the Linear Theory of Elasticity

In this paper we investigate the cubic-to-orthorhombic phase transition in the framework of linear elasticity. Using convex integration techniques, we prove that this phase transition represents one of the simplest three-dimensional examples in which already the linearised theory of elasticity displays non-rigidity properties. As a complementary result, we demonstrate that surface energy constraints rule out such highly oscillatory behaviour. We give a full characterization of all possibly emerging patterns for generic material parameters.     

Deformation model for brittle materials and the structure of failure waves

Constitutive equations that describe the experimentally observed failure waves are proposed to model inelastic strains of brittle materials. The complete system of equations is hyperbolic, each equation of this system has divergent form. The model is based on the assumption that continual failure is the process of transition from an intact state to a “fully damaged” state described by the kinetics of the order parameter. The structure of stationary traveling compressive waves is analyzed using a simplified model. It is shown that in a certain range of amplitudes, the wave splits into an elastic precursor and a failure wave. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 164–172, May–June, 2007.     

Upper and Lower Bounds for the Kolmogorov Entropy of the Attractor for the RDE in an Unbounded Domain

The long-time behaviour of bounded solutions of a reaction-diffusion system in an unbounded domain ⁿ , for which the nonlinearity f(u, _x u) explicitly depends on _x u is studied. We prove the existence of a global attractor, fractal dimension of which is infinite, and give upper and lower bounds for the Kolmogorov entropy of the attractor and analyze the sharpness of these bounds.