考虑周期微结构分布特征的Mori-Tanaka方法

利用两点间应变Green函数张量概念所建立的应变场积分方程, 推导了两相复合材料中夹杂的应变集中张量。该张量较之传统Mori-Tanaka (MT)法采用的由稀疏法导出的应变集中张量, 增加了一个与夹杂体积分数和分布相关的项, 并由此发展了考虑周期微结构分布特征的MT法。传统的MT法虽然能很好地预测正六角形分布圆截面纤维增强复合材料等的有效模量, 但不能反映正方形分布时的四方对称性特征, 本文作者所发展的方法弥补了这方面的不足, 并且所预报的有效刚度和柔度仍然保持了原MT方法所具有的自洽特性。最后通过与双周期有限元计算结果的对照验证了本文方法的精度。      

Families of continuous spin tensors and applications in continuum mechanics

A new method is proposed for generating families of continuous spin tensors associated with families of corotational rates of second-order tensors using isotropic tensor functions of the same tensor arguments and different forms of continuous antisymmetric scalar spin functions of scalar arguments. Tensor functions are represented in terms of eigenprojections of a symmetric tensor S, which is one of the arguments of these functions. Each member of the generated family is represented as the sum of some basic spin tensor associated with the basic corotational tensor rate and the above-mentioned tensor function, whose structure is matched to the structure of the tensor function required to construct the twirl tensor of the triad of orthonormal eigenvectors of the tensor S (but this twirl tensor itself does not belong to the family of continuous spin tensors). The developed method is used in continuum mechanics to generate two families of continuous spin tensors associated with two families of objective corotational rates: Lagrangian and Eulerian. In these families, isotropic tensor functions are constructed using Lagrangian and Eulerian tensor arguments of the kinematic type, respectively. It is shown that if the same scalar spin function is used in deriving tensor functions of Lagrangian and Eulerian tensor arguments, then the corotational tensor rates associated with the generated spin tensors are objective (Lagrangian and Eulerian) counterparts of each other. It is shown that the spin tensors associated with the classical Eulerian corotational tensor rates (Zaremba–Jaumann, Green–Naghdi, d-rate) and their Lagrangian counterparts (including material rate) belong to the generated families of continuous spin tensors. It is also shown that both of these families of continuous spin tensors are subfamilies of the families of material spin tensors derived by Xiao et al. (J Elast 52:1–41, 1998). It is noted that the twirl tensors of the Lagrangian and Eulerian triads associated with the Gurtin–Spear corotational rates of tensors belong to the families of material spin tensors but do not belong to the families of continuous spin tensors. The final section gives expressions of continuous spin tensors from families associated with the families of Lagrangian and Eulerian corotational tensor rates which are appropriate for applications.     

张量分析中简化记法在公式推导中的应用及张量分量的计算

张量分析在计算力学中应用广泛,但其理论比较复杂,较难掌握和熟练运用。为此,给出了张量简化记法,这种记法与Fortran程序中或商业软件Matlab中的数组形似。将简化后张量的表达式应用于公式推导,可快速准确地得出结论。同时,通过引入矩阵的矢量化概念,将四阶张量的分量用一个方阵表示出来,这有利于符号运算和数值计算。以上两者结合可使得张量分析变得易于掌握,也有利于张量运算的推广。     

Metric tensor approach to surveillance analysis of phased array radar

The author shows that the metric tensor gab is sufficient to accurately derive some of the most important system surveillance characteristics of static and rotating phased array radars. The author presents a novel and generalised method using metric spaces on Riemann manifolds as a viable approach for investigating phased array radar system design characteristics that is mathematically tractable and avoids complex iterative and non-analytic procedures.     

Strongly orthotropic continuum mechanics and finite element treatment

Despite successful application to orthotropic analysis, any Lagrangian strain tensor that is symmetric can be classified as an isotropic metric, while the infinitely orthotropic case can be accurately dealt with using one‐dimensional elements, structural tensors or kinematic constraints. In this paper, we present a strongly orthotropic continuum mechanics basis that models the exact kinematic behavior of the intermediary class of materials and also show its application to multi‐axial media and treatment using the finite element method. By asserting that mechanistic strain metrics must be material property dependent and satisfy equilibrium, we are able to derive a novel orthotropic linear strain tensor that is asymmetric and thus capable of describing all levels of orthotropy, while maintaining generality to the well‐established isotropic approach. Subsequently formulated are a material principal rotation tensor, extended orthotropic compliance tensor and an extended Mohr's plot for strain relying on an additional metric denoted as aspectual strain. Using the developed finite element formulation, it is shown that identical stress results to conventional theory for an orthotropic linear problem are predicted, while offering a more informative analysis. A second numerical example demonstrates the unique capability of this approach to solve the erroneous response of strongly orthotropic materials under trellis shear as compared with a number of conventional and contemporary approaches and thus its ability to produce kinematically exact results. Copyright © 2008 John Wiley & Sons, Ltd.     

Subspace‐based techniques for 2‐d DOA estimation with uniform circular array under local scattering

Abstract

In mobile communication systems, local scatterers in the vicinity of the sources cause angular spreading of radiating signals as seen from a base station antenna array. Thus, the base station antenna array is typically situated on the roof of a high building away from potential multipath reflectors. The uniform circular array (UCA) geometry provides 360° azimuthal coverage and also provides information on source elevation angles. We consider the problem of two‐dimensional (azimuth and elevation) direction‐of‐arrival (DOA) estimation with UCA. In the multipath scenario, the base station antenna can receive many coherent signals that cause the array manifold to be different from the conventional array manifold model. Herein, parameters of the spatial signature in the presence of local scattering are presented which apply to UCA. Then, we present a fast searching technique to improve the efficiency of the MUSIC algorithm for two‐dimensional DOA estimation. The fast signal subspace‐based estimation method utilizes the ESPRIT algorithm and then adopts sequential one‐dimensional searching to save computational cost. Several simulation results are included for illustration and comparison.     

The geometric nature of plasticity laws

This paper concerns the plasticity constitutive laws in small strain. In the thermodynamic approach developed here, the key concept is that of internal variables. The differential nature of plasticity law has been pointed out for a long time. If we unite the invariance condition of these laws in a state variable transformation, this involves, ultimately, that the natural mathematic frame of plasticity theory is Differential Geometry. The system state is defined as a point of a differentiable manifold. The state variable are the local coordinates of this point in a chart. The internal stresses are the components of a covariant vector of the cotangent bundle to internal state manifold and the elastic domain is a convex part of cotangent vector space. The plastic yield criteria such as von Mises condition define a Riemannian structure over the manifold. The metric element is identified with the internal dissipation element. Constitutive laws link the covariant derivatives of the thermodynamic stress with the state variable. Hardening modulus splits up in two parts, kinematic hardening and metric hardening. This last is defined by Christoffel connection coefficients. Applied to von Mises isotropic yield condition, the metric hardening is identified with isotropic hardening. The Baltov-Sawczuk model is also analysed. The use of appropriate polar coordinates simplifies significantly the computations. Generalization to a significant category of non-differentiable yield criteria, such as Tresca condition, is considered by introducting a metric tensor family. The adaptation of Drucker's postulate to the proposed model requires the introduction of parallel transport of the internal stress covector. Generally, this transport is different over distinctive paths joining two points. This fact expresses internal state manifold curvature. The Riemann-Christoffel tensor is computed for von Mises, Baltov-Sawczuk and Tresca models.     

Application of higher-order tensor theory for formulating enhanced continuum models

Summary In this paper attention is focussed on the derivation of higher-order isotropic tensors and their application in the formulation of enhanced continuum models. A mathematical theory will be discussed which relates formal orthogonal invariant polynomial functions to isotropic tensors. Using this theory, the second-order to the sixth-order isotropic tensor will be derived. When the tensor order increases, the derivation procedure clearly reveals a repeatable character. Thereafter, an example will be given of how the higher-order isotropic tensors can be used in the formulation of an enhanced continuum model. It will be demonstrated that symmetry conditions significantly reduce the number of material parameters.     

On Euler’s stretching formula in continuum mechanics

The Lie time-derivative of the material metric tensor field along the motion is the proper mathematical definition of the physical notion of strain rate or stretching. Its expression, as symmetric part of the velocity gradient in Euclid space, is provided by a celebrated formula conceived by the genius of Leonhard Euler around the middle of the eighteenth century and since then reproduced in articles, books and treatises on continuum mechanics. We present here a formulation, in the proper geometric context of the four-dimensional space-time manifold endowed with an arbitrary linear connection and referring to a material body of arbitrary dimensionality. The expression involves the material time-derivative of the metric field and torsion-form and gradient of the velocity field, according to the connection induced on the trajectory. As an application, the expressions of the Gram matrix of the stretching in natural and in normalized (or engineering) reference systems induced by orthogonal polar coordinates are provided.     

带障板水听器基阵阵列流形的边界元计算及验证

建立一种复杂结构水听器基阵阵列流形计算模型。该模型运用声场计算的边界元方法,通过计算任意散射体表面声场分布,求出基阵阵元位置处声场响应,进而得到基阵阵列流形;基于该阵列流形,运用波束设计方法求出阵元权值,分别利用实测和理论阵列流形得到基阵实测波束以及理论波束。针对安装在半球形硬质铝壳体上的圆弧阵进行的水池实验,实验结果表明实测阵列流形和基于边界元方法的计算结果基本吻合,实测波束和理论波束相差不大,并且性能均优于按无指向性点接收器经相位补偿后相加所得到的波束。     

Large strain elastic-plastic theory and nonlinear finite element analysis based on metric transformation tensors

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate-independent finite strain analysis of solids undergoing large elastic-plastic deformations. The formulation relies on the introduction of a mixed-variant metric transformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure whose rate is shown to be additively decomposed into elastic and plastic strain rate tensors. The mixed-variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response in the elastic-plastic solid. Additionally, the plastic material behavior is assumed to be governed by a generalized J ₂ yield criterion and rate-independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on 1st and higher order Padé approximations. Estimates of the stress and strain histories are obtained via a highly stable and accurate explicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step. The development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical solution of finite strain elastic-plastic problems is presented to demonstrate the efficiency of the algorithm. Received: 17 May 1998     

Damage Constitutive Equations and its Application to Fiber Reinforced Composites under Transverse Impact

Two continuous field variables, called as continuity tensor and damage variable tensor, are used to describe the anisotropic responses of an elastic-brittle material under transverse impact load. Based on the continuum damage mechanics, anisotropic damage constitutive equations in both full and incremental forms are proposed here. The expressions of effective elastic module tensor, damage variable tensor and damage propagation force tensor are further derived, and the methods for determining the tensors are explained in detail. An example of strain and damage response of a fiber reinforced composite laminated plate under transverse impact load is employed to demonstrate the application of this theory. In the example, the damage variable coupled with geometric large deformation of laminated plate is also considered. The calculating results illustrate the influence of damage on strain field in the impacted laminated plate.     

Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor

A second rank symmetric tensor which describes the degree of orientation in orthotropic materials is presented and shown to reflect accurately patterns of experimental data. The use of this tensor to describe microstructural anisotropy is compared to currently accepted methods and is found to be more useful and accurate in experimental studies. A method for determining the anisotropy tensor in a material is given, based on measurements on any three mutually perpendicular planes, and the fundamental restriction of this method to orthotropic materials is discussed. Experimentally determined anisotropy tensors in five specimens of cancellous bone from five different human bones are given.     

On the duality principle of conservation laws in finite elastodynamics

The duality principle of conservation laws which holds in finite elastodynamics is studied using the two-point tensor method. Based on the general Noether's theorem, two basic equations of variational invariance are first derived, which correspond to the action integrals given, respectively, in Lagrangian and Eulerian representations for a finite motion of an elastic body. The dual relations between the conservation laws in both representations are given. The procedure for constructing these dual relations is to apply simultaneously the same infinitesimal transformation of either time or position coordinates as well as field variables to the dual equations of variational invariance, where the position coordinates could be taken either from the reference configuration or from the deformed configuration of the material body. Based on these dual relations it is shown that the conservation equations of material momentum and moment of material momentum possess the same structure as those of physical momentum and physical moment of momentum. Furthermore, three pairs of dual relations between stress tensors and material momentum tensors of various kinds are derived based on the duality principle by using the two-point tensor method. Finally, using the dual integral forms of conservation laws the concepts of dynamic material force and moment acting on defects are introduced and analyzed. The force and moment can be decomposed into a pure kinetic part and a pure deformation part, the latter corresponding to the path-independent integral as suggested in elastostatics.     

A differential approach to microstructure-dependent bounds for multiphase heterogeneous media

We present a new method of deriving microstructure-dependent bounds on the effective properties of general heterogeneous media. The microstructure is specified by the average Eshelby tensors. In the small contrast limit, we introduce and calculate the expansion coefficient tensors. We then show that the effective tensor satisfies a differential inequality with the initial condition given by the expansion coefficient tensors in the small contrast limit. Using the comparison theorem, we obtain rigorous bounds on the effective tensors of multiphase composites. These new bounds, taking into account the average Eshelby tensors for homogeneous problems, are much tighter than the microstructure-independent Hashin–Shtrikman bounds. Also, these bounds are applicable to non-well-ordered composites and multifunctional composites. We anticipate that this new approach will be useful for the modeling and optimal design of multiphase multifunctional composites.     

Apparent stiffness tensors for porous solids using symmetric Galerkin boundary elements

Representative volume elements (RVEs) from porous or cellular solids can often be too large for numerical or experimental determination of effective elastic constants. Volume elements which are smaller than the RVE can be useful in extracting apparent elastic stiffness tensors which provide bounds on the homogenized elastic stiffness tensor. Here, we make efficient use of boundary element analysis to compute the volume averages of stress and strain needed for such an analysis. For boundary conditions which satisfy the Hill criterion, we demonstrate the extraction of apparent elastic stiffness tensors using a symmetric Galerkin boundary element method. We apply the analysis method to two examples of a porous ceramic. Finally, we extract the eigenvalues of the fabric tensor for the example problem and provide predictions on the apparent elastic stiffnesses as a function of solid volume fraction.     

Integral Equation Method for Evaluation of Eddy-Current Impedance of a Rectangular,Near Surface Crack Inside a Cylindrical Hole

An integral equation method for solving the eddy-current nondestructive evaluation problem of a flat, rectangular, near surface crack inside of a cylindrical hole in a conducting material is presented. The method involves expanding the Green’s tensor, the incoming field, and the jump in electric potential over the crack in suitable basis functions. Here, plane waves, cylindrical waves, and basis functions related to the Chebyshev polynomials, are used. The way of discretization in this method leads to a formulation where the scattering is defined by a scattering matrix, independent of the incoming field. This presents an advantage, when conducting numerical simulations, since the scattering matrix does not have to be recalculated for every probe position. The numerical calculations are straightforward to perform and model predictions are compared with finite element results.     

Scan line void fabric anisotropy tensors of granular media

Soil fabric anisotropy tensors are related to the statistical distribution of orientation of different microstructural vector-like entities, most common being the contact normal vectors between particles, which are extremely difficult to determine for real granular materials. On the other hand, void fabric based tensors can be determined by image based quantification methods of voids (graphical approaches), which are well defined and easy to apply to both physical and numerical experiments. A promising void fabric characterization approach is based on the scan line method. Existing scan line based definitions of void fabric anisotropy tensors are shown analytically to inherit a shortcoming, since numerous small void segments in a sample have an inordinate contribution towards unwarranted isotropy. Discrete Element Method (DEM) of analysis subsequently confirms this analytical proof. The fact that such scan line void fabric tensor definitions yield acceptable results when used in conjunction with physical image-based measurements, is shown to be attributed to the natural “cut off” of smaller void segments that occurs during such measurements. This is the motivation to propose using the existing definition of void fabric tensors, with exclusion of void segments less than a “cut off” value associated with an internal length of the granular assembly. In addition, an entirely new void fabric tensor was introduced using the squared length, instead of the length of a void segment, as the weighting factor for the definition of the scan line void fabric tensor. It was found by means of DEM analysis that both alternative definitions are void of the aforementioned shortcoming and compatible with existing image quantification methods of void fabric anisotropy.