Determination of the Rotation Tensor in the Polar Decomposition

An explicit representation for the rotation tensor which contains the lower powers of deformation gradient is proposed and used to evaluate the angle and axis of the rotation tensor. Some related equations about the rotation tensor are established. Through the approximate analysis, the relation between the S-R decomposition and the polar decomposition is examined. This revised version was published online in July 2006 with corrections to the Cover Date.     

含主应力轴旋转的土体平面应变问题弹塑性数值模拟

在含主应力轴旋转的土体本构关系研究的基础上,通过含主应力轴旋转的土体平面应变问题的弹塑性数值模拟结果分析,总结主应力轴旋转对土体应力分布与应形影响的规律,得出需要考虑主应力轴旋转的条件及影响的相对大小。     

A direct representation of the rotation tensor in polar decomposition of deformation gradient and its applications

Through a variational approach, an explicit connection between the additive and the polar decompositions of deformation gradient has been established. An exact formula for determining the rotation tensor in polar decomposition is obtained. The formula is fundamental in continuum mechanics and can be used to separate the rotation and the pure strain in deformation, by which various approximate expressions can be easily obtained.     

The general stress strain relation of soils involving the rotation of principal stress axes

1.IntroductionThestudyofsoils'constitutiverelationismostlylimitedinsuchStressstatethattherotationofprincipalstressaxesisabletobeignored.Soonlythevalueofprincipalstressneedtobeconsideredandthedirectionofprincipal'stressisregardedtobeunchangeable.Onthiscondition,theprincipalaxesof'stressincrement,strainincrementjstressandstrainarethesame.Therelevantexperimentsandengineeringsshowthattherotationofprincipalstressaxeswillgeneratesignificantplasticdeformationandthenoncoaxalityofstressandstrain.First…     

On several new methods to polar decomposition computation

In this paper, the polar decomposition of a deformation gradient tensor is analyzed in detail. The four new methods for polar decomposition computation are given: (1) the iterated method, (2) the principal invariant's method, (3) the principal rotation axis's method, (4) the coordinate transformation's method. The iterated method makes it possible to establish the nonlinear finite element method based on polar decomposition. Furthermore, the material time derivatives of the stretch tensor and the rotation tensor are obtained by explicit and simple expressions. The authors gratefully acknowledge the support rendered by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi of China in 1998.     

Dynamic rotation and stretch tensors from a dynamic polar decomposition

The local rigid-body component of continuum deformation is typically characterized by the rotation tensor, obtained from the polar decomposition of the deformation gradient. Beyond its well-known merits, the polar rotation tensor also has a lesser known dynamical inconsistency: it does not satisfy the fundamental superposition principle of rigid-body rotations over adjacent time intervals. As a consequence, the polar rotation diverts from the observed mean material rotation of fibers in fluids, and introduces a purely kinematic memory effect into computed material rotation. Here we derive a generalized polar decomposition for linear processes that yields a unique, dynamically consistent rotation component, the dynamic rotation tensor, for the deformation gradient. The left dynamic stretch tensor is objective, and shares the principal strain values and axes with its classic polar counterpart. Unlike its classic polar counterpart, however, the dynamic stretch tensor evolves in time without spin. The dynamic rotation tensor further decomposes into a spatially constant mean rotation tensor and a dynamically consistent relative rotation tensor that is objective for planar deformations. We also obtain simple expressions for dynamic analogues of Cauchy's mean rotation angle that characterize a deforming body objectively.     

单轴旋转惯导系统旋转性误差分析及补偿

对单轴旋转惯导系统因旋转而引入的各项误差进行分析,研究其误差特性及补偿方法。针对单轴正反连续旋转方案,在假定惯性测试组件的器件误差和其他非旋转性的误差在精确标定的情况下,推导了因旋转轴安装不正交引起的涡动、轴系间隙引起的晃动、测角器件误差、旋转控制引起的换向超调误差、角位置、角速度不准确等因素而引起的误差的表现形式,定性和定量地分析了各误差对于系统精度的影响。针对对系统影响显著的旋转轴不正交误差,提出了一种基于系统自身旋转轴正反旋转的误差标定及补偿方法并进行了仿真实验。在给定条件下的仿真结果表明,该方法能够准确标定出旋转轴的不正交误差,标定精度达到角秒级。     

Simulation of fabric drape using a thin plate element with finite rotation

The draping behavior of fabric is simulated by using four node quadrilateral thin plate elements with finite rotation. The finite element formulation is based on the total Lagrangian approach. An exact representation of finite rotation is introduced. The strain energy function accounting for the material symmetry is obtained by the tensor representation theory. To avoid shear locking, the assumed strain technique for transverse shear is adopted. The conjugate gradient method with a proposed line search algorithm is employed to minimize energy and reach the final shape of fabric. The draping behavior of a rectangular piece of fabric over a rectangular table is simulated.     

一种惯性测量单元非正交安装的单轴转位方法

针对单轴旋转式捷联惯导系统中旋转轴方向惯性器件误差导致系统误差积累的问题,提出一种惯性测量单元非正交安装的单轴转位方法,该方法不但可消除旋转轴垂直方向惯性器件误差对导航精度的影响,而且可减小旋转轴方向惯性器件误差引起的导航误差。基于单轴旋转调制原理,推导了非正交安装方法和正交安装方法的陀螺常值漂移和加速度计零偏在单轴旋转下引起的姿态误差,并对其进行分析,结果表明,在陀螺仪和加速度计常值漂移及零偏相同的情况下,非正交安装方法与正交安装方法相比,安装斜角为10°时72 h的定位误差降低约50%。     

主应力轴旋转对软土塑性变形影响分析

在姜洪伟所提出的软土各向异性弹塑性本构关系基础上，考虑类似于砂土的Ｒ－旋转试验条件，计算了软土由于主应力轴旋转所产生的塑性变形，对不同中主应力比情况下的变形规律进行分析，解释忽略主应力轴旋转影响将使设计偏于不安全的原因。     

含主应力轴旋转的土体本构关系研究进展

主应力轴旋转对岩土工程的影响日益受到人们的重视．本文从纯主应力轴旋转、纯应力洛德角变化、多种因素变化、排水与不排水等方面较全面地描述了含主应力轴旋转情况下土体的基本变形特性．对当前较有影响的含主应力轴旋转的土体本构模型（基于一般应力空间的土体本构模型、运动硬化模型、边界面模型、土体弹塑性应力应变关系的完全应力增量表述等等）进行了较为系统的评述．提出了合主应力轴旋转的实验研究中存在的核心问题与建立合主应力轴旋转的土体本构关系的根本途径．      

The finite rotation parameter of a deformable body and gyromagnetic effect

The derivation of Synge for the finite rotation formula in terms of Euler parameter is put into tensor form.Hence a further clarification of the geometric meaning of the orthogonal transformation obtained in the author’s paper[7] is made. The tensor properties of rotation axis vector are also discussed. By means of the method of co-moving coordinate system established in[8],[9], we explain the gyromagnetic effect,and derive a simple formula for calculating body couple which is induced by magnetization.     

On the gradients of the principal invariants of a second-order tensor

In this paper the gradients of the principal invariants of an arbitrary second-order tensor are derived in a very concise way.     

Optical phenomena in photoelastic models by the rotation of principal axes

The basic equations of three-dimensional photoelasticity are derived in a form which is simpler than that of equations known previously. Using the matrix representation of the solution of these equations, it is also shown that when rotation of principal axes is present there always exist two perpendicular directions of polarizer by which the light emerging from the model is linearly polarized. These polarization directions of the incident and emergent light are named primary and secondary characteristic directions, respectively. The experimental determination of characteristic directions, as well as of the phase retardation, gives three equations on every light path to determine the stress components in a three-dimensional model. A general algorithm of the method of characteristic directions is presented, and its application by determination of stress in shells by normal and tangential incidence is described. A further extension of the method to the general axisymmetric problem has been suggested.     

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

ＰＲＩＮＣＩＰＡＬＡＸＩＳＩＮＴＲＩＮＳＩＣＭＥＴＨＯＤＡＮＤＴＨＥＨＩＧＨＤＩＭＥＮＳＩＯＮＡＬＴＥＮＳＯＲＥＱＵＡＴＩＯＮＡＸ－ＸＡ＝ＣＬｉａｎｇＨａｏｙｕｎ（梁浩云）（ＷｕｙｉＵｎｉｖｅｒｓｉｔｙ，Ｊｉａｎｇｍｅｎ５２９０２０，Ｐ．Ｒ．Ｃｈｉｎ...     

Dynamic damage constitutive relation of mesoscopic heterogenous brittle rock under rotation of principal stress axes

A micro-mechanics-based model is developed to investigate microcrack damage mechanism of four stages of brittle rock under rotation of the principal stress axes. They consist of linear elastic, non-linear hardening, rapid stress drop and strain softening. The frictional sliding crack model is applied to analyze microcracks nucleation, propagation and coalescence. The strain energy density factor approach is applied to determine the critical condition of microcrack nucleation, propagation and coalescence. The inelastic strain increments are formulated within the framework of thermodynamics with internal variables. Rotation of principal stress axes affect the dynamic damage constitutive relationship and the failure strength of brittle rock.     

Modélisation de la turbulence en déclin avec rotation

The study presented in the present paper deals with the modeling of the rotation effects on a homogeneous turbulence evolving without a mean velocity gradient. The developed model results from a modification of the second order model of Haworth and Pope. This modification is expressed as a function of the rotation rate, of the component of the Reynolds tensor and of the characteristic length scale in the direction of the rotation axis. The obtained results have evidenced the great capacity of the present model to describe the rotation effects on the time evolution of the homogeneous decaying turbulence.     

Effect of rotation on the vibroconvective stability of a fluid layer with internal heat release

The equilibrium stability of a horizontal fluid layer with homogeneous internal heat release is investigated theoretically for the case in which the layer simultaneously undergoes high-frequency circular vibration in a horizontal plane and rotates about a vertical axis. The rotation frequency is assumed to be small as compared with the vibration frequency. It is found that the rotation has a stabilizing effect on the vibrational-gravitational convection. At the high-frequency limit the dependence of the critical values of the controlling parameters (gravitational and vibrational Rayleigh numbers) and the wave number on the rotation frequency is obtained.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 53–61. Original Russian Text Copyright © 2005 by Ivanova, Kozlov, and Kolesnikov.