PRINCIPAL AXIS INTRINSIC METHOD AND THE HIGH DIMENSIONAL TENSOR EQUATION AX－XA＝C

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Spatial Estimates for the Constrained Anisotropic Elastic Cylinder

This paper establishes spatial estimates in a prismatic (semi-infinite) cylinder occupied by an anisotropic homogeneous linear elastic material, whose elasticity tensor is strongly elliptic. The cylinder is maintained in equilibrium under zero body force, zero displacement on the lateral boundary and pointwise specified displacement over the base. The other plane end is subject to zero displacement (when the cylinder is finite, say). The limiting case of a semi-infinite cylinder is also considered and zero displacement on the remote end (at large distance) is not assumed in this case. A first approach is developed by considering two mean-square cross-sectional measures of the displacement vector whose spatial evolution with respect to the axial variable is studied by means of a technique based on a second-order differential inequality. Conditions on the elastic constants are derived that show the cross-sectional measures exhibit alternative behaviour and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay. A second approach considers cross-sectional integrals involving the displacement and its gradient and furnishes information upon the spatial evolution, without restricting the range of strongly elliptic elastic constants. Such models are principally based upon a first-order differential inequality as well as on one of second order. The general results are explicitly presented for transversely isotropic materials and graphically illustrated for a cortical bone.     

THE LINEAR BI-SPATIAL TENSOR EQUATIONφijAiXBj= C

A linear bi-spatial tensor equation which contains many often encountered equations as particular cases is thoroughly studied. Explicit solutions are obtained. No conditions on eigenvalues of coefficient tensors are imposed.     

Shape and structure recovery of an LCP droplet under a large step strain: observation and stress calculation

Shape recovery of a droplet of liquid crystalline polymer (LCP) hydroxypropylcellulose in a matrix of poly(dimethyl siloxane) subjected to a step shear strain has been studied via optical microscopy. Just after application of a large strain, the LCP droplet shape is flat ellipsoid, and then the droplet takes cylindrical shape and band texture perpendicular to the flow direction appears. The band texture fades away before emergence of poly-domain structure. In the final process with the shape of spheroid, poly-domain structure recovers very slowly. Except for the final process, the shape change is identical with that of isotropic droplet at strains smaller than 3, when the LCP viscosity in Region II is taken as an equivalent viscosity for normalization. For a 20:80 blend, the excess relaxation modulus is calculated based on the Doi-Ohta theory, taking account of the distribution of droplet size and compared with experimental modulus data.     

Partial Uniqueness and Obstruction to Uniqueness in Inverse Problems for Anisotropic Elastic Media

We consider the inverse problem of identifying the density and elastic moduli for three-dimensional anisotropic elastic bodies, given displacement and traction measurements made at their surface. These surface measurements are modelled by the dynamic Dirichlet-to-Neumann map on a finite time interval. For linear or nonlinear anisotropic hyperelastic bodies we show that the displacement-to-traction surface measurements do not change when the density and elasticity tensor in the interior are transformed tensorially by a change of coordinates fixing the surface of the body to first order. Our main tool, a new approach in inverse problems for elastic media, is the representation of the equations of motion in a covariant form (following Marsden and Hughes, 1983) that preserves the underlying physics.In the case of classical linear elastodynamics we then investigate how the type of anisotropy changes under coordinate transformations. That is, we analyze the orbits of general linear, anisotropic elasticity tensors under the action by pull-back of diffeomorphisms that fix the surface of the elastic body to first order, and derive a pointwise characterization of parts of the orbits under this action. For example, we show that the orbit of isotropic elastic media, at any point in the body, consists of some transversely isotropic and some orthotropic elastic media. We then derive the first uniqueness result in the inverse problem for anisotropic media using surface displacement-traction data: uniqueness of three elastic moduli for tensors in the orbit of isotropic elasticity tensors. ^†Partially supported by an MSRI Postdoctoral Fellowship. Research at MSRI is supported in part by NSF grant DMS-9850361. This work was conducted while the first author was a Gibbs Instructor at Yale University. ^‡Partially supported by an MSRI Postdoctoral Fellowship, and by NSF grant DMS-9801664 (9996350).     

Toward consistent formulation of Reynolds stress and scalar flux closures

Langevin stochastic differential equations provide a consistent basis for Reynolds stress, scalar transport and p.d.f. models. However, the stochastic equations must be capable of representing existing closures, like the General Linear Model, or the Rotta and Monin return to isotropy formulations. A consistent approach to derive both Reynolds stress and scalar flux transport equations, starting from a stochastic differential equation for velocity fluctuations, is presented here. A set of algebraic relations for the dispersion tensor is derived for homogeneous shear flow and for the log-layer.     

Measurement of concrete internal strain using a three-dimensional transducer system

A new three-dimensional transducer system is introduced in this paper, which enables the measurement of six internal components of strain tensor in concrete members. Laboratory tests and strain analyses were carried out to evaluate this tranducer system. The results show that the system is effective in measuring internal strains of concrete members. It was then embedded in a reinforced concrete bridge deck in Macomb County, Michigan. Field tests were conducted to measure the strains in the deck under truck wheel load. The results were used to understand the deck's strain and stress behavior under truck wheel load.     

基于微面有效应力矢量的各向异性屈服准则

基于微面模型，定义损伤变量为微面上有效承载面积的减少. 将Kachanov的一维有效 应力概念推广到三维，提出微面有效应力矢量的概念. 根据微面的有效应力矢量，将无损材 料的宏观应力张量及不变量与微面应力矢量的积分关系拓展到有损材料，得到了有损材料的 宏观有效应力张量及其不变量与宏观名义应力张量、微面面积损伤组构张量之间的关系. 将 无损材料的以应力张量不变量表示的Drucker-Prager准则推广到有损材料，建立了含缺陷 材料的各向异性屈服准则. 对有损材料，宏观有效应力张量与Murakami的有效应力张量具 有相同的形式，各向异性强度准则与Liu等提出的扩展Hill准则有相同的形式，当不考虑 静水应力对屈服的影响时，它与Hill准则具有相同的形式.     

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”     

Diffusion in isotropic and anisotropic porous systems: Three-dimensional calculations

Effective diffusion coefficients were calculated numerically for three-dimensional unit cells representative of different unconsolidated porous media. These numerical results were compared with the experimental results of Kim for packed beds of glass spheres, mica particles, and an artificial porous medium composed of mylar disks. These three-dimensional numerical results confirm that the porosity is the essential parameter for the determination of the effective diffusion coefficient in the case of unconsolidated isotropic systems. In the case of anisotropic systems, better agreement is obtained between numerical predictions and actual data when the unit cell is three-dimensional rather than twodimensional. This emphasizes the fact that three-dimensional unit cells feature more realistic geometrical properties which are needed to accurately describe anisotropic systems.