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.     

The effect of large rotation on post-buckling of annular plates

On the basis of the axisymmetrical finite deformation theory on Rissner's plates, the post-buckling behaviour of annular plates and the effect of large rotation are qualitatively and numerically discussed by the method of the bifurcation theory.The project is supported by the National Natural Science Foundation of China     

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.     

变形梯度张量极分解中转动张量的直接表示及其应用

本文通过变分途径建立了变形梯度张量的极分解和加法分解之间的联系.采用工程界通常采用的变形梯度张量的加法分解形式,得到了三维空间中极分解的转动张量和伸长张量的直接表示,即实现了转动和变形的分离.由这些直接表示,可以得到各种有用的近似表示.     

On Extended Polar Decomposition

In this study, we considered the extended polar decomposition using a more general approach than the one provided by Boulanger and Hayes [Int. J. Non-Linear Mech. 36 (2001) 399–420]. We showed that the procedure of the decomposition could be simplified by considering its rotation tensor. Our method is illustrated by examples.     

极分解的一类近似计算与比较

基于级数展开给出了极分解中右伸长张量 的级数表示,通过对级数的项的选取得到右伸长张量的不同近似表达式。针对不同级数展开表示,得到表达式最小误差的级数展开形式。进而结合一些简单实例,验证误差了近似公式的有效性。最后与黄模佳等关于计算右伸长张量 和转动张量 的近似表达式进行了比较,本文的级数展开方式得到的右伸长张量 和转动张量 的近似表达式不但简洁,而且计算精度更高、适用范围更广。     

The explicit representation to the principal rotation angle and the principal rotation axis

By using Cayley-Hamilton theorem, two kinds of explicit representation for the rotation tensor are proposed. One contains the lower powers of deformation gradient, by which the formula of the principal rotation angle and the explicit representation of principal axis are obtained; the other, a high efficient method to obtain the rotation tensor, does not contain the complicated coefficients and uses few variables. Some properties about the principal rotation angle and the principal rotation axis are obtained.     

The invariant representation of spins with applications in the theory of finite deformations

Based on the general solution given to a kind of linear tensor equations, the spin of a symmetric tensor is derived in an invariant form. The result is applied to find the spins of the left and the right stretch tensors and the relation among different rotation rate tensors has been discussed. According to work conjugacy, the relations between Cauchy stress and the stresses conjugate to Hill's generalized strains are obtained. Particularly, the logarithmic strain, its time rate and the conjugate stress have been discussed in detail. These results are important in modeling the constitutive relations for finite deformations in continuum mechanics. The project is supported by the National Natural Science Foundation of China and the Chinese Academy of Sciences (No. 87-52).     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅳ)——SURFACE COUSERVATION LAWS

The purpose is to reestablish rather complete surface conservation laws for micropolar thermomechanical continua from the translation and the rotation invariances of the general balance law. The generalized energy-momentum and energy-moment of momentum tensors are presented. The concrete forms of surface conservation laws for micropolar thermomechanical continua are derived. The existing related results are naturally derived as special cases from the results proposed in this paper. The incomplete degrees of the existing surface conservation laws are clearly seen from the process of the deduction. The surface conservation laws for nonlocal micropolar thermomechanical continua may be easily obtained via localization. Contributed by DAI Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931 ~)     

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.     

近场动力学框架下的键基对应模型

传统键基近场动力学模型存在泊松比限制的问题,为了解决这一问题发展了态基近场动力学模型。其中非常规态的近场动力学模型通过定义非局部的变形梯度将近场力和传统应力关联起来,方便使用传统本构,但是态基近场动力学计算效率低于键基近场动力学。结合态基模型和键基模型的优势,提出键基对应模型,定义了基于键的变形梯度,参考连续介质力学中变形梯度的极分解过程,将键的变形分为转动部分和伸长部分。从而进一步定义了应变,通过物理方程求应力,进而计算键传递的近场力。键基对应模型解决了键基近场动力学的泊松比限制问题,也不需要进行近场动力学微观材料常数的计算。数值算例和理论推导证明了键变形梯度定义以及近场力计算方式的正确性。     

Numerical simulation for deformation of nano-grained metals

Electro-deposition technique is capable of producing nano-grained bulk copper specimens that exhibit superplastic extensibility at room temperature. Metals of such small grain sizes deform by grains sliding, with little distortion occurring in the grain cores. Accommodation mechanisms such as grain boundary diffusion, sliding and grain rotation control the kinetics of the process. Actual deformation minimizes the plastic dissipation and stored strain energy for representative steps of grain neighbor switching. Numerical simulations based on these principles are discussed in this paper. The project supported by the National Natural Science Foundation of China (19972031)     

Tensor Universal Serendipity Elements and unsteady Taylor-Galerkin finite element method

A new kind of Universal Serendipity Element (USE)— the Tensor Universal Serendipity Element (TUSE) is constructed by using both tensor force finite elements and the basic idea of USE. The formulation of shape functions and their derivatives for TUSE is presented. TUSE can be used to study steady and unsteady transonic flow fields when combined with Taylor-Galerkin Finite Element Methods, the NND scheme in FDM, and four-stage Runge-Kutta methods. As numerical examples the transonic flow in cascades and one kind of complex unsteady transonic axisymmetric flow in engineering are studied. It is shown that the algorithm presented in this paper is efficient and robust. The project supported by the National Natural Science Foundation of China     

有限转动张量的保角参量及在多柔体系统中的应用

采用保角转动参数描述了多体系统中的大转动张量．该方法消除了传统的欧拉参数描述所必需的约束方程，并且适于大变形部件的建模需要．利用以上结果建立了含大变形梁状部件的多体系统的力学模型．     

Special properties of Eshelby tensor for a regular polygonal inclusion

When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor: Eshelby tensor at the center and the averaged Eshelby tensor over the inclusion domain are equal to that of a circular inclusion and independent of the orientation of the inclusion. Then Kawashita and Nozaki justified the properties mathematically. In the present paper, some other properties of a regular polygonal inclusion are discovered. We find that for an N-fold regular polygonal inclusion except for a square, the arithmetic mean of Eshelby tensors at N rotational symmetrical points in the inclusion is also equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Furthermore, in two corollaries, we point out that Eshelby tensor at the center, the averaged Eshelby tensor over the inclusion domain, and the line integral average of Eshelby tensors along any concentric circle of the inclusion are all identical with the arithmetic mean.The project supported by the National Natural Science Foundation of China (10172003 and 10372003) The English text was polished by Keren Wang.     

Distribution of active earth pressure of retaining wall with wall movement of rotation about top

Based on the Coulomb‘s theory that the earth pressure against the back of a retaining wall is due to the thrust exerted by the sliding wedge of soil from the back of the wall to a plane which passes through the bottom edge of the wall and has an inclination equal to the angle of O, the theoretical answers to the unit earth pressure, the resultant earth pressure and the point of application of the resultant earth pressure on a retaining wall were obtained for the wall movement mode of rotation about top. The comparisons were made among the formula presented here, the formula for the wall movement mode of translation, the Coulomb‘ s formula and some experimental observations. It is demonstrated that the magnitudes of the resultant earth pressures for the wall movement mode of rotation about top is equal to that determined by the formula for the wall movement mode of translation and the Coulomb‘s theory. But the distribution of the earth pressure and the points of application of the resultant earth pressures have significant difference.     

Stress-strain field near the notch tip of a rubber sheet

Analized in this paper is the elastostatic field near a notch tip in a rubber-like thin sheet. The asymptotic equations for the notch tip field are derived based on the constitutive relation given by Ref. [1]. Near field solutions are obtained in regions that decreases and increases in size as the notch tip is approached. Their singular character depends on the constitutive parameters as well as the angle of notch that is evaluated numerically. The project supported by the National Natural Science Foundation of China     

Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures

An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions are properly specified. Two dimensional and three dimensional microstructures are optimized for maximum stiffness designs by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained for a variety of design cases. The project supported by the National Natural Science Foundation of China (10372083, 90405016), 973 Program (2006CB601205) and the Aeronautical Science Foundation (04B53080). The English text was polished by Keren Wang.     

Micro-experimental study of large plastic deformation for high-purity polycrystalline aluminum

This paper aims at revealing various micro- deformation characteristics, such as crystalline slip and grain boundary slide, which are recorded under scanning electronic microscope for high-purity aluminum tensile specimen at room temperature. These experimental data provide us direct evidence for shear localization near the grain boundary network via multi- directional grain boundary slide. The nonuniform deformation induced in the grain interiors would have decisive effect on the plastic flow and failure of polycrystalline materials. Sponsored by the National Natural Science Foundation of China and the Fok Ying- Tung Education Foundation.