A mathematical theory of materials with elastic range and the definition of back stress tensor

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｇｅｎｅｒａｌ,ｔｈｅｋｉｎｅｍａｔｉｃａｌｈａｒｄｅｎｉｎｇｂｅｈａｖｉｏｒｏｆｍａｔｅｒｉａｌｓｉｓｄｅｓｃｒｉｂｅｄｂｙａｖａｒｉａｂｌｅｃａｌｅｄｂａｃｋｓｔｒｅｓｏｒｓｈｉｆｔｔｅｎｓｏｒ．Ｉｔｓｖａｌｕｅｒｅｐ...     

Equations of motion and boundary conditions of incremental rate type for polar continua

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ[1]ｔｈｅｒｅｌａｔｉｏｎｓｏｆｖａｒｉｏｕｓｓｔｒｅｓｓｔｅｎｓｏｒｓ,ｔｈｅｅｑｕａｔｉｏｎｓｏｆｍｏｍｅｎｔｕｍａｎｄｔｈｅｃｏｒｒｅｓｐｏｎｄｉｎｇｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓｏｆｖａｒｉｏｕｓｆｏｒｍｓｆｏｒｃｌａｓｓｉｃａｌｃｏｎｔｉｎｕｕｍｍｅｃｈａｎｉｃｓａｒｅｓｙｓｔｅｍａｔｉｃａｌｌｙｄｅｒｉｖｅｄ.Ｉｎ[2]ｔｈｅｇｅｎｅｒａｌｉｚｅｄｃｏｎｔｉｎｕｕｍｆｉｅｌｄｔｈｅｏｒｉｅｓａｒｅｃｏｍｐｒｅｈｅｎｓｉｖｅｌｙｒｅｖｉｅｗｅｄａｎｄｃｌａｒ…     

Third-order nonlinear singularly perturbed boundary value problem

ＩｎｔｒｏｄｕｃｔｉｏｎＳｉｎｃｅ 1 958,Ｌｅｖｓｉｏｎａｎｄｓｅｖｅｒａｌｏｔｈｅｒａｕｔｈｏｒｓ[1]- [4 ]ｈａｖｅｍａｄｅａｓｕｂｓｔａｎｔｉａｌｓｔｕｄｙｆｏｒｅｘｉｓｔｅｎｃｅａｎｄｅｓｔｉｍａｔｅｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｔｈｉｒｄ＿ｏｒｄｅｒｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｗｉｔｈｔｗｏｐｏｉｎｔｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓｘ( 0 ) =Ａ ,ｘ′( 0 ) =Ｂ ,ｘ′( 1 ) =Ｃ ,ｏｒｓｌｉｇｈｔｌｙｅｘｔｅｎｓｉｖｅｌｙｌｉｎｅａｒｂｏ…     

Global Attractivity and Global Exponential Stability for Delayed Hopfield Neural Network Models

ＩｎｔｒｏｄｕｃｔｉｏｎＨｏｐｆｉｅｌｄｎｅｕｒａｌｎｅｔｗｏｒｋｍｏｄｅｌｉｓｏｎｅｏｆｔｈｅｍｏｓｔｐｏｐｕｌａｒｍｏｄｅｌｓｉｎｔｈｅｌｉｔｅｒｒａｔｕｒｅｏｆａｒｔｉｆｉｃｉａｌｎｅｕｒａｌｎｅｔｗｏｒｋｓ,ｗｈｉｃｈｉｓｄｅｓｃｒｉｂｅｄｂｙｔｈｅｆｏｌｌｏｗｉｎｇｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｓｅｑｕａｔｉｏｎｓ[1,2 ]:Ｃｉｄｕｉ(ｔ)ｄｔ =-ｕｉ(ｔ)Ｒｉ ∑ｎｊ=1Ｔｉｊｇｊ(ｕｊ(ｔ) ) Ｉｉ　　 (ｉ=1 ,2 ,… ,ｎ) ,( 1 )ｗｈｅｒｅｎ≥ 2ｉｓｔｈｅｎｕｍｂｅｒｏｆｎｅｕｒｏｎｓｉｎｔｈｅ…     

The Eigentensors of an Arbitrary Second-Order Tensor and their Quality Analyses

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｔｅｎｓｏｒａｎａｌｙｓｉｓｉｓａｖｅｒｙｕｓｅｆｕｌｍａｔｈｅｍａｔｉｃａｌｔｏｏｌｔｏｓｔｕｄｙｔｈｅｐｈｙｓｉｃａｌｐｒｏｂｌｅｍｓ.Ｂｅｃａｕｓｅｉｎｔｈｅｐｒａｃｔｉｃａｌｐｒｏｂｌｅｍｓｍａｎｙｐｈｙｓｉｃａｌｑｕａｎｔｉｔｉｅｓａｒｅｔｅｎｓｏｒｓｏｒｃａｎｂｅｅｘｐｒｅｓｓｅｄｉｎｔｅｒｍｓｏｆｔｈｅｔｅｎｓｏｒｓ.Ｆｏｒｅｘａｍｐｌｅ ,ｔｈｅｓｏｌｕｔｉｏｎｏｆｈｏｍｏｇｅｎｅｏｕｓｌｉｎｅａｒｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ…     

The Smooth and Nonsmooth Travelling Wave Solutions in a Nonlinear Wave Equation

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｅｘｉｓｔｅｎｃｅｏｆｐｅａｋｏｎＴＷＳｏｆａｎｏｎｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎρｔ =ｂｕｘ 12 [(ｕ2 ±ｕ2 ｘ) ρ] ｘ,　ρ =ｕ±ｕｘｘ ( 1 )ｗａｓｃｏｎｓｉｄｅｒｅｄｂｒｅｉｆｌｙｂｙＰ .Ｒｏｓｅｎａｕ (ｓｅｅ [1 ] ) .Ｅｑ.( 1 )ｉｓｆｏｕｎｄｂｙ“ｒｅｓｈｕｆｆｌｉｎｇ”Ｈａｍｉｌｔｏｎｉａｎｏｐｅｒａｔｏｒｏｆｂｉ＿ＨａｍｉｌｔｉｏｎｉａｎｓｔｒｕｃｔｕｒｅｉｎＫｄＶａｎｄｍＫｄＶｅｑｕａｔｉｏｎ (ｓｅｅ [2 ] ) .ＢｅｃａｕｓｅＥｑ.( 1 )ｈａｓｓｔｒｏｎ…     

NEW COLLECTIVELY FIXED POINT THEOREMS AND APPLICATIONS IN G-CONVEX SPACES

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ 1 991 ,Ｔａｒａｆｄａｒ[1]ｆｉｒｓｔｅｓｔａｂｌｉｓｈｅｄｔｈｅｆｏｌｌｏｗｉｎｇｃｏｌｌｅｃｔｉｖｅｌｙｆｉｘｅｄｐｏｉｎｔｔｈｅｏｒｅｍｏｎｔｈｅｐｒｏｄｕｃｔｓｐａｃｅｏｆｎｏｎｅｍｐｔｙｃｏｍｐａｃｔｃｏｎｖｅｘｓｕｂｓｅｔｓｏｆｔｏｐｏｌｏｇｉｃａｌｖｅｃｔｏｒｓｐａｃｅｓａｎｄｇａｖｅｉｔｓａｐｐｌｉｃａｔｉｏｎｓｔｏｔｈｅｅｘｉｓｔｅｎｃｅｏｆｅｑｕｉｌｉｂｒｉｕｍｐｏｉｎｔｓｆｏｒａｂｓｔｒａｃｔｅｃｏｎｏｍｉｅｓ.Ｔｈｅｏｒｅｍ 1　Ｌｅｔ Ｘｉ ｉ∈Ｉ…     

The survival analysis for gallopin' s system in a polluted environment

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｐｅｒｓｉｓｔｅｎｃｅｐｒｏｂｌｅｍｏｆａＬｏｇｉｓｔｉｃｐｏｐｕｌａｔｉｏｎｉｎａｐｏｌｌｕｔｅｄｅｎｖｉｒｏｎｍｅｎｔｗａｓｓｔｕｄｉｅｄｉｎｐａｐｅｒ [1 ] .Ｂｕｔｏｎｌｙｏｎｅｓｐｅｃｉｅｓｗａｓｕｎｄｅｒｔｈｅｃｏｎｓｉｄｅｒａｔｉｏｎｉｎｔｈｅｐａｐｅｒ.Ｉｎｆａｃｔ,ａｎｙｐｏｐｕｌａｔｉｏｎｉｎａｃｔｕａｌｂｉｏｌｏｇｉｃａｌｓｉｔｕａｔｉｏｎｓｍｕｓｔｂｅｌｉｖｉｎｇｏｎｏｔｈｅｒｐｏｐｕｌａｔｉｏｎｓ .ＡｓｉｍｐｌｅＧａｌｌｏｐｉｎ’ｓｒｅｓｏｕｒ…     

THE EXACT SOLITARY WAVE SOLUTIONS FOR THE KLEIN- GORDON-SCHRDINGER EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐｒｅｓｅｎｔｐａｐｅｒｗｅｃｏｎｓｉｄｅｒｔｈｅｃｏｕｐｌｅｄＫｌｅｉｎ＿Ｇｏｒｄｏｎ＿Ｓｃｈｒ ｄｉｎｇｅｒ (ＫＧＳ)ｅｑｕａｔｉｏｎｓａｓｆｏｌｌｏｗｓ:ｉψｔ+ 12 Δψ=-φψ ,φｔｔ-Δφ+ｍ2 φ=｜ψ｜2 .( 1 )( 2 )( 1 )ａｎｄ ( 2 )ｄｅｓｃｒｉｂｅａｃｌａｓｓｉｃａｌｍｏｄｅｌｏｆｉｎｔｅｒａｃｔｉｏｎｏｆｎｕｃｌｅｏｎｆｉｅｌｄｗｉｔｈｍｅｓｏｎｆｉｅｌｄ[1],ｗｈｅｒｅψｉｓａｃｏｍｐｌｅｘｓｃａｌａｒｎｕｃｌｅｏｎｆｉｅｌｄ ,φ＿ａｒｅａｌｍｅｓｏｎｆｉ…     

Finite element Galerkin Approach for a Computational Study of Arterial Flow

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｍｅａｓｕｒｅｍｅｎｔｏｆｓｈｅａｒｓｔｒｅｓｓｉｎａｒｔｅｒｉａｌｆｌｏｗｈａｓｂｅｅｎｂｅｓｔｏｗｅｄｂｙｎａｔｕｒｅｂｅｃａｕｓｅｏｆｔｈｅｐｏｓｓｉｂｌｅｒｅｌａｔｉｏｎｂｅｔｗｅｅｎｔｈｅｓｈｅａｒｓｔｒｅｓｓａｔｔｈｅｗａｌｌａｎｄｅｘｉｓｔｅｎｃｅｏｆａｔｈｅｒｏｓｃｌｅｒｏｓｉｓ.Ｔｈｅｆｌｏｗｐｈｅｎｏｍｅｎａｉｎｌａｒｇｅａｎｄｍｅｄｉｕｍｓｉｚｅｄａｒｔｅｒｉｅｓｉｎｆｌｕｅｎｃｅｔｈｅｄｅｖｅｌｏｐｍｅｎｔｏｆａｔｈｅｒｏｓｃｌｅｒｏｓｉｃｄ…     

A Concise Proof of the Representation Theorem for Fourth-Order Isotropic Tensors

We present a new proof of the representation theorem for fourth-order isotropic tensors that does not assume the tensor to have major or minor symmetries at the outset.     

Basic Issues Concerning Finite Strain Measures and Isotropic Stress-Deformation Relations

It is indicated that the commonly-used Rivlin–Ericksen representation formula for isotropic tensor functions exhibits some properties that might be undesirable for its reasonable and effective applications. Towards clarification and improvement, a set of three mutually orthogonal tensor generators is introduced to achieve an alternative representation formula for isotropic symmetric tensor-valued functions of a symmetric tensor. This representation formula enables us to express the unknown representative coefficients in terms of simple, explicit tensorial inner products of the argument tensor and the value tensor without involving their eigenvalues. In particular, the tensorial interpolation expressions thus obtained assume a unified form for the three different cases of coalescence of the eigenvalues of the argument tensor. Moreover, each summand in the alternative representation formula is shown to inherit the continuity and differentiability properties of the represented isotropic tensor function. These results are used to study some basic issues concerning finite strain measures and stress-deformation relations of isotropic materials, such as continuity and differentiability properties of the representation, determination of the representative coefficients in terms of experimental data for stress and deformation tensors, and computations of finite strain measures. This revised version was published online in June 2006 with corrections to the Cover Date.     

使用主值空间表示的各向同性塑性本构方程

针对各向同性材料,在内变量为标量的假定下,应用张量函数表示定理给出了其塑性应变增量的不变性表示.它的3个不可约基张量取决于应力张量、相互正交且共主轴.建立3个基张量构成的张量子空间与三维主值空间的对应关系,将共主轴的张量采用笛卡尔坐标系中的矢量描述,矢量在不同坐标系下的分量均为张量的一组不可约不变量.定义塑性应变增量对应的矢量为内变量增量,使用张量函数表示理论得到,内变量演化方程除取决于应力对应的矢量和内变量本身外,还取决于应力增量在张量子空间中的投影,该投影就是应力对应矢量的增量,因此,本构方程归结为确定主值空间中矢量之间的关系.最后表明,三维主值空间与张量子空间中的流动法则是等价的.     

Dyadic method for tensor functions

In this paper, we discuss tensor functions by dyadic representation of tensor. Two different cases of scalar invariants and two different cases of tensor invariants are calculated. It is concluded that there are six independent scale invariants for a symmetrical tensor and an antisymmetrical tensor, and there are twelve invariants for two symmetrical tensors and an antisymmetrical tensor. And we present a new list of tensor invariants for the tensor-valued isotropic function. The project supported by the Special Funds for Major State Basic Research Project “Nonlinear Science” and the National Basic Research Project “The Several Key Problems of Fluid and Aerodynamics”     

Averaging Anisotropic Elastic Constant Data

A method of averaging the data on the anisotropic elastic constants of a material is presented. The anisotropic elastic constants are represented by the elasticity tensor which is expressed as a second rank tensor in a space of six dimensions. The method consists of averaging eigenbases of different measurements of the elasticity tensor, then averaging the eigenvalues referred to the average eigenbasis. The eigenvalues and eigenvectors are obtained by using a representation of the stress-strain relations due, in principle, to Kelvin [17, 18]. The formulas for the representation of the averaged elasticity tensor are simple and concise. The applications of these formulas are illustrated using previously reported data, and are contrasted with the traditional analysis of the same data by Hearmon [9]. An interesting result that emerges from this analysis is a method dealing with variable composition anisotropic elastic materials whose elastic constants depend upon the particular composition. In the case of porous isotropic materials, for example, it is customary to regress the Young's modulus against porosity. The results of this paper suggest a structure or paradigm for extending to anisotropic materials this empirical method of regressing elastic constant data against composition or porosity.     

Derivation of the general form of elasticity tensor of the transverse isotropic material by tensor derivate

SymbolsU--FunchonofstrainenergyQ--OrthonormaltensorE--StraintensorEar--ComponentsofthestraintensorE,i,j=l,2,3n--VectorofthesymmetricaamsofthetransverseisotropicmaterialU*,E.,n*--FormsofU,EandninanothercoordinatesystemJf--MaininvariantsofstraintensorE,i=l,2,3Jf'n--InvariantsofstraintensorEconnectingwithvectorn,i=4,5Ji--TheabbreviatedformsofJf,Jf,Jf,Jf,",Jf,",i=l,2,3,4,5fi--ConstantsindependentonE,n,i=l,2,3,4,5el,e"--Thecovariantandcontravariantofthonormalbasisoftheusedcoordinatesyste…     

n the fourth order tensor valued function of the stress in return map algorithm

针对各向同性材料,基于一组相互正交的基张量,建立了一套有 效的相关运算方法. 基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则 使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴. 根据张量函数表示 定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组 基所表示. 推演结果表明：这些张量之间的运算,表现为对应系数矩阵之间的简单 关系. 其中,四阶张量求逆归结为对应的3\times3系数矩阵求逆,它对二阶张量的变换 则表现为该矩阵对3times 1列阵的变换. 最后,对这些变换关系应用于返回映 射算法的迭代格式进行了相关讨论.     

Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions

The present paper generalizes the method for solving the derivatives of sym- metric isotropic tensor-valued functions proposed by Dui and Chen(2004)to a subclass of nonsymmetric tensor functions satisfying the commutative condition.This subclass of tensor functions is more general than those investigated by the existing methods.In the case of three distinct eigenvalues,the commutativity makes it possible to introduce two scalar functions,which will be used to construct the general nonsymmetric tensor func- tions and their derivatives.In the cases of repeated eigenvalues,the results are acquired by taking limits.     

基于中间构形的大变形弹塑性模型

在大变形弹塑性本构理论中,一个基本的问题是弹性变形和塑性变形的分解.通常采用两种分解方式,一是将变形率(或应变率)加法分解为弹性和塑性两部分,其中,弹性变形率与Kirchhoff应力的客观率通过弹性张量联系起来构成所谓的次弹性模型,而塑性变形率与Kirchhoff应力使用流动法则建立联系;另一种是基于中间构形将变形梯度进行乘法分解,它假定通过虚拟的卸载过程得到一个无应力的中间构形,建立所谓超弹性-塑性模型.研究了基于变形梯度乘法分解并且基于中间构形的大变形弹塑性模型所具有的若干性质,包括:在不同的构形上,塑性旋率的存在性、背应力的对称性、塑性变形率与屈服面的正交性以及它们之间的关系.首先,使用张量函数表示理论,建立了各向同性函数的若干特殊性质,并导出了张量的张量值函数在中间构形到当前构形之间进行前推后拉的简单关系式.然后,基于这些特殊性质和关系式,从热力学定律出发,建立模型在不同构形上的数学表达,包括客观率表示的率形式和连续切向刚度等,从而获得模型所具有的若干性质.最后,将模型与4种其他模型进行了比较分析.      

Isotropic polynomial invariants of Hall tensor

The Hall tensor emerges from the study of the Hall effect, an important magnetic effect observed in electric conductors and semiconductors. The Hall tensor is third-order and three-dimensional, whose first two indices are skew-symmetric. This paper investigates the isotropic polynomial invariants of the Hall tensor by connecting it with a second-order tensor via the third-order Levi-Civita tensor. A minimal isotropic integrity basis with 10 invariants for the Hall tensor is proposed. Furthermore, it is proved that this minimal integrity basis is also an irreducible isotropic function basis of the Hall tensor.