压应力状态下细观非均匀性岩石的损伤局部化和应力应变关系分析

利用摩擦弯折裂纹模型研究了受压条件下细观非均匀性岩石的损伤局部化问题和全过程应力应变关系．模型考虑了裂纹相互作用对损伤局部化和全过程应力应变关系的影响，确定了损伤局部化发生的条件，分析了产生损伤局部化的原因．研究表明全过程应力应变关系包括线弹性阶段、非线性强化阶段、应力降和应变软化阶段．通过和实验对比分析验证了模型的正确性和有效性．     

基于裂纹尖端二阶弹性解的断裂过程区尺度

基于Westergaard应力函数裂纹尖端二阶弹性解,推导了裂纹尖端微裂区的轮廓线和特征尺寸的解析表达式;采用幂函数模型描述的拉应变软化模型,确定了在最大拉应力强度理论和最大拉应变强度理论下断裂过程区（FPZ）临界值的解析表达式;将基于Westergaard应力函数一阶弹性解及二阶弹性解、Muskhelishvili应力函数和Duan-Nakagawa模型确定的FPZ临界值进行了比较．结果表明裂纹尖端微裂区和FPZ临界值随着Poisson比的减小而增加并逐渐趋近于应用最大拉应力强度理论确定的结果;二阶弹性解确定的裂纹尖端微裂区和FPZ临界值大于一阶弹性解的值;FPZ临界值随着拉应变软化指数的增加而增加;二阶弹性解确定的FPZ临界值的精度远高于一阶弹性解确定的值.     

三轴压缩下岩石/混凝土的三维细观损伤模型

考虑三轴压应力作用下的无限大体深埋椭圆形裂纹的变形场,给出了闭合椭圆形微裂纹的能量释放率.采用能量平衡方法分析了闭合椭圆形微裂纹的扩展条件,得到了代表性体积单元中具有任意空间取向的单个闭合椭圆形微裂纹及其扩展引起的附加柔度张量.分析了闭合椭圆形微裂纹的偏折扩展,得到了微裂纹偏折扩展引起的附加柔度张量.采用Taylor方法考虑微裂纹系统对代表性体积单元变形的影响,引入概率密度函数,得到了三轴压缩下岩石/混凝土的三维细观损伤模型.分析了混凝土的单轴压缩特性,结果表明该模型能够很好地描述实验现象.     

玻璃态高聚物损伤断裂的非平衡统计理论

从细观层次的微裂纹在银纹内扩展模型出发,结合断裂力学与断裂动力学描述了裂尖前缘银纹质断裂所引起的微裂纹扩展动力学过程.把银纹几何结构和力学参数引入到微裂纹演化方程中,得到了随时间演化的微裂纹尺寸统计分布函数,并给出了微裂纹尺寸矩生成函数与任意阶宏观损伤函数.若微裂纹之间不存在相互作用且微裂纹在材料内部随机取向时,推导出了玻璃态高聚物的断裂几率及可靠性的普遍解析表达式.当微裂纹数目较大时,得到了断裂几率的极限形式——Gumbel分布及平均断裂强度与方差.     

冲击压缩下陶瓷材料中的破坏波模型

从多晶陶瓷材料细观结构非均匀性及其导致的应力奇异性分析出发，建立了陶瓷材料在冲击压缩下的本构关系，以及以表征材料损伤和破坏的非弹性体积应变为传播特征的破坏波控制方程，破坏层的非弹性体积应变包括由微裂纹成核、扩展引起的膨胀体积应变和由气孔塌陷引起的压缩体积应变两部分．结合92．93％氧化铝陶瓷板碰撞试验，数值模拟了冲击压缩下陶瓷材料中破坏波的传播过程，并对跨越破坏波阵面应力历程和剪切强度的变化规律进行了分析．     

弹性——应变软化粘塑性材料反平面剪切动态扩展裂纹尖端的渐近解

本文首先给出了一种用于描述材料软化，并存在有粘塑性的材料模型．用这种模型对反平面剪切型动态扩展状态下，裂纹尖端的弹粘塑性场进行了渐近分析，给出了弹性－应变软化粘塑性材料反平面剪切动态扩展裂纹尖端的渐近解方程．分析结果表明，在裂纹尖端应变具有（1n（R／r））~1／（n＋1）的奇异性，应力具有（1n（R／r））~－n／（n＋1）的奇异性．从而本文揭示了应变软化粘塑性材料反平面剪切动态扩展裂纹尖端的渐近行为．     

Ⅰ阶梯度损伤理论

从热力学基本定律出发,将应变张量、标量损伤变量、损伤梯度作为Helmholtz自由能函数的状态变量,利用本构泛函展开法在自然状态附近作自由能函数的Taylor展开,未引入附加假设,推导出Ⅰ阶梯度损伤本构方程的一般形式．该形式在损伤为0时可退化为线弹性应力-应变本构方程,在损伤梯度为0时可退化为基于应变等效假设给出的线弹性局部损伤本构方程．一维解析解表明,随着应力增大,损伤场逐步由空间非周期解变为关于空间的类周期解,类周期解的峰值区域形成局部化带．局部化带内的损伤变量将不同于局部化带外的损伤变量,由此可以反映出介质的局部化特征．损伤局部化并不是与损伤同时发生,而是在损伤发生后逐渐显现出来,模型的局部化机制开始启动;损伤局部化的宽度同内部特征长度成正比．     

混凝土材料的粘塑性损伤本构模型

基于不可逆热力学,引入运动硬化、等向硬化和损伤内变量,构造了相应的自由能函数和流动势函数,推导出了混凝土材料的粘塑性损伤本构模型.数值模拟的结果表明,该模型能够避开屈服面和破坏准则的基本假设来描述混凝土材料的以下特性:压缩载荷作用下的体积膨胀现象;应变率敏感性;峰值后由损伤和破坏引起的应力软化和刚度退化现象.由于此模型避开了根据各种变形阶段选择与其相应的本构模型的繁琐计算,因此更便于纳入复杂工况下应力分析有限元程序中.     

不可压缩橡胶类材料裂纹尖端应力应变场

本文对平面应变情况下不可压缩橡胶类材料裂纹尖端弹性场进行了有限变形分析．裂纹尖端场被分为收缩区和扩张区．借助于新的应变能函数和变形模式，推出了尖端场各区的渐近方程，得到了尖端场的完整描述．本文对奇异性作了讨论，得到了不可压缩橡胶类材料裂纹尖端应力及应变分布曲线，揭示了裂纹尖端应力应变场的特性．     

求解层间界面反平面剪切破坏的剪切梁模型（I）——基本特性

在反平面剪切载荷及侧压力共同作用下引起的裂纹及裂纹扩展导致的层间界面失效,是岩土工程层间界面及砌体结构中界面层上典型的失效方式。运用弹性力学和断裂力学的理论原理,提出了能够反映上述层间界面断裂失效问题力学特性的剪切梁模型。中采用具有应力软化特性的“粘性裂纹”（内聚力裂纹）模型档表述层间裂纹前方损伤过程区的本构行为。对通过粘性层结合在一起的两个弹性板,在反平面剪切载荷及侧压力共同作用下的力学行为作了解析分析计算,研究了层间界面裂纹扩展规律。     

单轴作用下岩土材料的双重介质本构模型

基于弹塑性力学和损伤力学理论，将岩土材料视为孔隙，裂隙双重介质，假设孔隙介质不发生损伤，而裂隙介质随应变的增加发生损伤，建立了单轴作用下岩土类材料的双重介质本构模型隐式表达式，并利用Newton迭代法得出了材料的全程应力-应变曲线。分析结果表明，岩土材料中裂隙空间展布的多态性（均匀展布、集中展布和随机展布）是岩土材料本构关系千变万化的根本原因。由于双重介质本构模型将岩土材料的弹性主体（孔隙介质部分）和损伤主体（裂隙介质部分）分化开来，对于研究岩土或含损伤材料的破坏具有实用价值和理论意义。     

Efficient evaluation of the material response of tissues reinforced by statistically oriented fibres

For several classes of soft biological tissues, modelling complexity is in part due to the arrangement of the collagen fibres. In general, the arrangement of the fibres can be described by defining, at each point in the tissue, the structure tensor (i.e. the tensor product of the unit vector of the local fibre arrangement by itself) and a probability distribution of orientation. In this approach, assuming that the fibres do not interact with each other, the overall contribution of the collagen fibres to a given mechanical property of the tissue can be estimated by means of an averaging integral of the constitutive function describing the mechanical property at study over the set of all possible directions in space. Except for the particular case of fibre constitutive functions that are polynomial in the transversely isotropic invariants of the deformation, the averaging integral cannot be evaluated directly, in a single calculation because, in general, the integrand depends both on deformation and on fibre orientation in a non-separable way. The problem is thus, in a sense, analogous to that of solving the integral of a function of two variables, which cannot be split up into the product of two functions, each depending only on one of the variables. Although numerical schemes can be used to evaluate the integral at each deformation increment, this is computationally expensive. With the purpose of containing computational costs, this work proposes approximation methods that are based on the direct integrability of polynomial functions and that do not require the step-by-step evaluation of the averaging integrals. Three different methods are proposed: (a) a Taylor expansion of the fibre constitutive function in the transversely isotropic invariants of the deformation; (b) a Taylor expansion of the fibre constitutive function in the structure tensor; (c) for the case of a fibre constitutive function having a polynomial argument, an approximation in which the directional average of the constitutive function is replaced by the constitutive function evaluated at the directional average of the argument. Each of the proposed methods approximates the averaged constitutive function in such a way that it is multiplicatively decomposed into the product of a function of the deformation only and a function of the structure tensors only. In order to assess the accuracy of these methods, we evaluate the constitutive functions of the elastic potential and the Cauchy stress, for a biaxial test, under different conditions, i.e. different fibre distributions and different ratios of the nominal strains in the two directions. The results are then compared against those obtained for an averaging method available in the literature, as well as against the integration made at each increment of deformation.     

考虑主应力轴旋转的土体本构关系研究进展

对目前国内外考虑主应力轴旋转的试验研究及本构模型研究进行了总结分析,并对进一步研究提出了相应的建议．基于不同的加载条件,从纯主应力轴旋转和耦合主应力轴旋转两个方面,较全面的描述了主应力轴旋转情况下土体的基本变形特性,并对考虑主应力轴旋转的土体变形试验提出了进一步研究的建议．较为系统地评述了当前较有代表性的考虑主应力轴旋转的土体本构模型（边界面模型、多机构模型、运动硬化模型和广义塑性模型）,得出了广义塑性模型更适合用来描述考虑主应力轴旋转的土体变形特性的结论．总结未来考虑主应力轴旋转的土体本构关系研究的主要方向是:把握主应力轴旋转情况下土体变形的本质特性,建立推理严密、形式简单、适用方便的本构模型,并用来指导工程实践．     

Error estimation of grain distribution function recovery for dependent orientations with allowance for grain sizes

The effect of the kernel on the smoothing of orientations in a kernel method was studied, and the influence of dependent orientations and the grain sizes on the resulting distribution was analyzed. Discrete central normal distributions on the group SO(3) were smoothed by the kernel method. This problem is motivated by the development of experimental tools for studying the texture of polycrystalline materials, especially electron microscopy, which makes it possible to measure the orientations of individual grains.     

Derivatives of a finite class of orthogonal polynomials related to inverse Gamma distribution

In this work, we consider derivatives of a finite class of orthogonal polynomials with respect to weight function which is related to the probability density function of the inverse gamma distribution over the positive real line. General properties for this derivative class such as orthogonality, Rodrigues’ formula, recurrence relation, generating function and various other related properties such as self-adjoint form and normal form are indicated. The corresponding Gaussian quadrature formulae are introduced with examples. These examples are provided to support the advantages of considering the derivatives class of the finite class of orthogonal polynomials related to inverse gamma distribution. The orthogonality property related to the Fourier transform of the derivative class under discussion is also given.     

A Variational Approach to Dynamic Recrystallization Using Probability Distributions

We investigate a model of dynamic recrystallization in polycrystalline materials. A probability distribution function is introduced to characterize the state of individual grains by grain size and dislocation density. Specifying free energy and dissipation within the polycrystalline aggregate we are able to derive an evolution equation for the probability density function via a thermodynamic extremum principle. Once the distribution function is known macroscopic quantities like average strain and stress can be calculated. For distribution functions which are constant in time, describing a state of dynamic equilibrium, we obtain a partial differential equation in parameter space which we solve using a marching algorithm. Numerical results are presented and their physical interpretation is given. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fresnel Tomography: A Novel Approach to Wave-Function Reconstruction Based on the Fresnel Representation of Tomograms

We introduce a new type of tomographic probability distribution containing complete information about the density matrix (wave function) related to the Fresnel transform of the complex wave function. We elucidate the relation to the symplectic tomographic probability distribution. We present a multimode generalization of the Fresnel tomography and give examples of applications of the present approach. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 384–393, August, 2005.     

不同先验分布下的后验分布确定土力学参数

岩土工程中各土层参数的取值是根据现场及室内试验数据,采用经典统计学方法进行确定的,但这往往忽略了先验信息的作用。与经典统计学方法不同的是,Bayes法能从考虑先验分布的角度结合样本分布去推导后验分布,为岩土参数的取值提供一种新的分析方法。岩土工程勘察可视为对总体地层的随机抽样,当抽样完成时,样本分布密度函数是确定的,故Bayes法中的后验分布取决于先验分布,因此推导出两套不同的先验分布:利用先验信息确定先验分布及共轭先验分布。通过对先验及后验分布中超参数的计算,当样本总体符合N(μ,σ2)正态分布时,对所要研究的未知参数μ和σ展开分析,综合对比不同先验分布下后验分布的区间长度,给出岩土参数Bayes推断中最佳后验分布所要选择的先验分布。结果表明:共轭情况下的后验分布总是比无信息情况下的后验区间短,概率密度函数分布更集中,取值更方便。在正态总体情形下,根据未知参数μ和σ的联合后验分布求极值方法,确定样本总体中最大概率均值μ_max和方差σ_max作为工程设计采用值,为岩土参数取值方法提供了一条新的路径,有较好的工程意义。     

Higher order asymptotic behaviour of partial maxima of random sample from generalized Maxwell distribution under power normalization

In this article, the higher order asymptotic expansions of cumulative distribution function and probability density function of extremes for generalized Maxwell distribution are established under nonlinear normalization. As corollaries, the convergence rates of the distribution and density of maximum are obtained under nonlinear normalization.