基于S-R和分解定理的三维几何非线性无网格法

S-R(strain-rotation)和分解定理克服了经典有限变形理论的一些缺点, 使其可以为几何非线性数值分析提供可靠的理论基础. 对于大变形问题, 由于无网格法(element-free method)避免了对单元网格的依赖, 从而从根本上避免了有限单元法(finite element method, FEM)的单元畸变问题, 保证了求解精度. 因此, 将无网格法和S-R和分解定理结合起来势必能建立一套更加合理可靠的几何非线性数值计算方法. 目前基于S-R 定理的无网格数值方法研究较少并且只能用于二维平面问题的求解, 但实际上绝大多数问题都必须以三维模型来进行处理, 因此建立适用于三维情况的S-R无网格法是非常有必要的. 本文给出了适用于三维情况的S-R 无网格法: 采用由更新拖带坐标法和势能率原理推导出来的增量变分方程, 利用基于全局弱式的无网格Galerkin 法(EFG)得到了用于求解三维空间问题的离散格式. 利用MATLAB编制三维S-R 无网格法程序, 对受均布载荷的三维悬臂梁和四边简支矩形板结构的非线性弯曲问题进行了计算. 最后将所得的数值结果与已有文献进行了比较, 验证了本文的三维S-R无网格数值算法的合理性、有效性和准确性. 本文的三维S-R无网格数值算法可以作为一种可靠的三维几何非线性数值分析方法.      

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.     

再论有限变形梯度的“和分解”

本文就文献[1]提出对有限变形梯度“和分解”的疑问进行讨论。     

无粘结预应力混凝土板火灾行为的非线性分析

火灾试验研究表明,火灾下无粘结预应力混凝土受弯构件的变形非常大,其中转动也很大,尤其是单面受火的板,是典型的材料非线性、几何非线性问题。如何对其进行精确、高效地求解是值得关注的问题。而基于S-R分解原理的更新拖带坐标有限元法有诸多优点:有利于跟踪变形物体中各点的变形;保证单元的质量守恒;在有限元增量法求解时,还可以避免对坐标的修正;而且将转动作为一个独立的自由度,提高了求解效率。在升温过程中,虽然预应力钢筋的应力处处相等,但其各点的温度不同,导致各点的温度应变、蠕变、塑性应变均不同。预应力筋必然产生滑移。本文采用该方法对火灾下无粘结预应力钢筋混凝土板进行编程分析。通过实际算例验证该算法的可靠性,该方法求解效率高,精度也比较好。     

On the representation of finite rotation in nonlinear field theory of continuum mechanics

One of the important basic theoretical problems in the development of continuum mechanics is the separation of finite strain and finite rotation at a point in the displacement field. Now it is certain that S-R decomposition theorem provides a rational theoretical solution for this problem. The purpose of this paper is to clarify some misleading basic concepts of finite rotation of deformed body in current literature, and to promote further progress.The detailed abstract of this paper was published in 1985 Shanghai International Conference of Nonlinear Mechanics. In the present paper some new results are included.     

火灾下钢筋混凝土板的热弹塑性有限元分析——基于S-R分解原理(Ⅱ:算例分析)

采用基于S-R分解原理的更新拖带坐标有限元法,对火灾下钢筋混凝土板进行了数值模拟.并以标准升温为条件,探讨了不同的钢筋保护层厚度、钢筋及混凝土强度等级、荷载水平、板厚等因素,对钢筋混凝土简支板抗火性能的影响.计算结果表明,构件变形主要是由非线性温度场引起的热变形.为采取一些简单的设计思想来完善钢筋混凝土构件抗火性能提出了建议.     

Representation Theorems for Hemitropic and Tranversely Isotropic Tensor Functions

A representation theorem for transversely isotropic tensor-valued functions of a symmetric tensor variable is proved. The theorem holds in any finite dimension. The proof is based on the decomposition of a symmetric tensor of dimension N into a scalar, a vector, and a symmetric tensor of dimension N-1, and on the fact that the transverse isotropy of the original function is equivalent to the hemitropy of three functions, one scalar-valued, one vector-valued, and one tensor-valued, of the last two terms in the decomposition. Representation theorems for the three functions are obtained as generalizations of two theorems of W. Noll on isotropic functions. The proofs make use of an appropriate algebraic structure based on alternating forms. The three-dimensional case, as well as those of linear and of hyperelastic functions, are treated as special cases. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach

A nonlinear continuum theory of material bodies with continuously distributed dislocations is presented, based on a gauge theoretical approach. Firstly, we derive the canonical conservation laws that correspond to the group of translations and rotations in the material space using Noether’s theorem. These equations give us the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor is neither symmetric nor gauge-invariant. Based on the Belinfante-Rosenfeld procedure, we obtain the gauge-invariant Eshelby stress tensor which can be symmetric relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor is obtained as well. The decomposition of the gauge-invariant Eshelby stress tensor in an elastic and in a dislocation part gives rise to the derivation of the famous Peach-Koehler force.     

悬臂梁角点的应力非对称问题的实验研究

采用网格法研究弹性悬臂梁自由端角点在端部竖向均布荷载下的大变形,利用现代图像采集技术获得实验图像,借助MATLAB图像处理技术对图像数据进行处理,以应变与转动的S-R分解定理为理论基础,计算整旋角并绘制整旋角等高线分布图。等高线分布图显示越靠近角点处,整旋角的等值线越密集,即角点处的整旋角的梯度增大,从而证明体力矩的存在是引起悬臂梁自由端角点应力非对称现象的原因。     

火灾下钢筋混凝土板的热弹塑性有限元分析——基于S-R分解原理(Ⅰ:理论)

火灾试验研究表明,火灾下钢筋混凝土受弯构件的变形非常大,转动也很大,尤其是单面受火的板.基于S-R分解原理的更新拖带坐标有限元法分析这类构件,有利于跟踪变形物体中各点的变化,保证单元的质量守恒.用有限元增量法求解,还可以避免对坐标的修正,而且将转动作为一个独立的自由度,提高了求解效率,特别适合于几何非线性、材料非线性问题的求解.本文采用此方法对火灾下钢筋混凝土板进行编程分析.同时,为克服在时间步内温度路径难于确定的问题,本文给出了平面应力状态下的混凝土热弹塑性积分方案的初值表达式.通过实际编程发现,该方法求解效率高,精度也比较好.     

On the compatibility condition of displacement field for finite deformation of continuum

The vanishing of Riemann-Christoffel tensor is usually adopt-ed as the compatibility condition of finite deformation.However,we prove in this paper by the method of Cesaro that this condition is necessary but not sufficient for guarantee of a single-valued,continuous displacement field.A new general compatibility condi-tion,based on the theorem of strain-rotation decomposition(Chen[4])is derived.The displacement compatible condition reduces to Saint-Venant's condition when strain and rotation are infinitesimal.     

Internal constraints in linear elasticity

Sufficient conditions are obtained for continuous dependence of solutions of boundary value problems of linear elasticity on internal constraints. Arbitrary hyperelastic materials with arbitrary (linear) internal constraints are included. In particular the results of Bramble and Payne, Kobelkov, Mikhlin for homogeneous, isotropic, incompressible materials are obtained as a special case. In the case of boundary value problem of place, a compatibility condition is obtained between the internal constraints and the boundary data which is necessary for the existence of solutions. With a further coercivity assumption on the compliance tensor, it is shown that the compatibility condition is also sufficient for existence. An orthogonal decomposition theorem for second order tensor fields modeled after Weyl's decomposition of solenoidal and gradient fields leads to the variational formulation of the problem and existence theorems.Almost all the results here apply to materials both with or without internal constraints. For internally constrained materials however, the verification of certain hypothesis is surprisingly non-trivial as indicated by the computation in the appendix.     

壳体结构非线性分析的扁壳有限元法

用单位纵横向曲条条带法构造了扁壳单元的空间位移模式,依据拖带坐标描述去Ｓ－Ｒ分解定理,志出扁壳结构几何非线性分析的有限单元Ｕ．Ｃ．（ｕｐｄａｔｅｄ ｃｏ－ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ）列式,算例表明,该计算方法精度高,收敛性好。     

Deformation analysis of laser spectra based on S-R decomposition theorem

ＭｏｄｅｒｎｏｐｔｉｃａｌｔｈｅｏｒｙｈａｓｓｈｏｗｎｔｈａｔｔｈｅｆａｒｆｉｅｌｄｏｒＦｒａｕｈｏｆｅｒｄｉｆｆｒａｃｔｉｏｎｅｑｕｉｐｍｅｎｔｉｓｉｄｅｎｔｉｃａｌｔｏｔｈｅＦｏｕｒｉｅｒｓｐｅｃｔｒａａｎａｌｙｚｅｒ．ＩｎｔｈｅＦｏｕｒｉｅｒｓｐｅｃｔｒａｌａｎａｌｙｚｅｒｔｈｅＦｏｕｒｉｅｒｓｐｅｃｔｒａｏｒｔｈｅＦｒａｕｎｈｏｆｅｒｄｉｆｆｒａｃｔｉｏｎｐａｔｔｅｒｎｏｆａｎｉｍａｇｅｉｓｆｏｒｍｅｄｏｎｔｈｅｂａｃｋｆｏｃａｌｐｌａｎｅｗｈｅｎａｌａｓｅｒｂｅａｍｉｓｄｉｒｅｃｔｅｄ…     

A Representation Theorem for Material Tensors of?Weakly-Textured Polycrystals and?Its Applications in?Elasticity

Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill’s quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.     

Fifth-order isotropic descartes tensor

Ｉｎｔｈｉｓｌｉｔｅｒａｔｕｒｅ [1 ] ,[2 ]ｏｒｄｅｒ 0～ 4ｉｓｏｔｒｏｐｉｃｄｅｓｃａｒｔｅｓｔｅｎｓｏｒ’ｓｅｘｉｓｔｅｎｃｅａｎｄｉｔｓｇｅｎｅｒａｌｒｅｐｒｅｓｅｎｔａｔｉｏｎｆｏｒｍｕｌａａｒｅｄｉｓｃｕｓｓｅｄ .Ｗｉｔｈｒｅｇａｒｄｔｏｏｒｄｅｒｋ ≥ 5(ｄｅｓｃａｒｔｅｓｔｅｎｓｏｒｏｒａｆｆｉｎｅｏｒｔｈｏｇｏｎａｌｔｅｎｓｏｒ) ,ｔｈｅｎｕｍｂｅｒ (3 ｋ)ｏｆｉｔｓｃｏｍｐｏｎｅｎｔｓｉｓｖｅｒｙｂｉｇ ,ａｎｄｉｔｓｓｔｒｕｃｔｕｒｅｉｓｖｅｒｙｃｏｍｐｌｉｃａｔｅｄ ,ｓｏｎｏｐａ…     

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.      

Relations between cubic equation,stress tensor decomposition,and von Mises yield criterion

Inspired by Cardano's method for solving cubic scalar equations, the additive decomposition of spherical/deviatoric tensor (DSDT) is revisited from a new viewpoint. This decomposition simplifies the cubic tensor equation, decouples the spherical/deviatoric strain energy density, and lays the foundation for the von Mises yield criterion. Besides, it is verified that under the precondition of energy decoupling and the simplest form, the DSDT is the only possible form of the additive decomposition with physical meanings.