Expression of strain tensor in orthogonal curvilinear coordinates

Based on an analysis of connotation and extension of the concept of the orthogonal curvilinear coordinates, we have deduced a platform of strain tensor expression of Cartesian coordinates, which turns out to be a function of Lame coefficient and unit vector. By using transform matrix between Cartesian coordinates and orthogonal eurvilinear coordinates, we have deduced a mathematical expression for correcting displacement vector differential in orthogonal curvilinear coordinates, and given a general expression of strain tensor in orthogonal curvilinear coordinates.

The Small Deformation Strain Tensor as a Fundamental Metric Tensor

In the general theory of relativity, the fundamental metric tensor plays a special role, which has its physical basis in the peculiar aspects of gravitation. The fundamental property of gravitational fields provides the possibility of establishing an analogy between the motion in a gravitational field and the motion in any external field considered as a noninertial system of reference. Thus, the properties of the motion in a noninertial frame are the same as those in an inertial system in the presence of a gravitational field. In other words, a noninertial frame of reference is equivalent to a certain gravitational field. This is known as the principle of equivalence. From the mathematical viewpoint, the same special role can be played by the small deformation strain tensor, which describes the geometrical properties of any region deformed because of the effect of some external agent. It can be proved that, from that tensor, all the mathematical structures needed in the general theory of relativity can be constructed.

Expression of strain and rotation tensor in geodetic coordinates

In this paper, unit moving trihedron is first constructed for a point on the surface of a revolution ellipsoid. Via translation, the origin of the trihedron coincides with that of Cartesian coordinates established at the center of the ellipsoid, and then through two coordinate rotations, the trihedron completely coincides with the Cartesian coordinates. Transformation formulae between the moving trihedron and unit Cartesian coordinate frameworks as well as transformation of point displacement between two unit coordinate frameworks are presented. Based on the above transformation formulae between two different coordinate frameworks, due to the fact that the displacement and moving trihedron of the point are both functions of the geodetic coordinates, components in the corresponding axis for differential of displacement vector and geodetic curves arc differential at the point in geodetic system can be obtained through complicated derivation. Displacement gradient matrix at the point in geodetic system is also given. Finally, expressions of strain and rotation tensor in geodetic coordinates are presented. Geometric meanings of the rotation tensor are explained in detail. The intrinsic relationship between strain tensors of sphere and ellipsoid are also discussed.

Strain Regularity and Stiffness Tensor of Reinforcers in Dense Particle Reinforced Composites

Based on the study of strain distribution in composites reinforced with high-volume-fraction, randomly distributed particles, the quantitative relation between the average strains of matrix/reinforcers and microstructure parameters of the composite is investigated. It is revealed that besides the average strains, strain fluctuation is also an important factor affecting the stiffness modulus of the composites. The relation between the strain fluctuation and microstructure parameters of the composite is also obtained to derive the stiffness tensor. The theoretical prediction is favorably compared with experimental results.

GEOMETRIC ANALYSIS OF FINITE STRAIN TENSORS IN TECTONODEFORMATION MECHANICS

According to the peculiarities of tectonodeformation this paper systematically researches the graphic method of finite strain theory in tectonomechanics, proposes a geometrical analytic method describing the strain state of a point with metric tensor, expounds and proves the structure of deformation gradient circle, and clearly illustrates the relations of various quantities in polar decomposition theory. Finally, a simple shear deformation which often occurs in tectonodeformation was calculated through various different methods in order to explain and to compare the peculiarities among various graphic methods of finite strain.

Crustal strain rates of southeastern Tibetan Plateau derived from GPS measurements and implications to lithospheric deformation of the Shan-Thai terrane

The link between the crustal deformation and mantle kinematics in the Tibetan Plateau has been well known thanks to dense GPS measurements and the relatively detailed anisotropy structure of the lithospheric mantle.However, whether the crust deforms coherently with the upper mantle in the Shan-Thai terrane(also known as the Shan-Thai block) remains unclear.In this study, we investigate the deformation patterns through strain rate tensors in the southeastern Tibetan Plateau derived from the latest GPS measurements and find that in the Shan-Thai terrane the upper crust may be coupled with the lower crust and the upper mantle.The GPS-derived strain rate tensors are in agreement with the slipping patterns and rates of major strike-slip faults in the region.The most prominent shear zone, whose shear strain rates are larger than 100×10^(–9) a^(–1), is about 1000-km-long in the west, trending northward along Sagaing fault to the Eastern Himalayan Syntaxis in the north, with maximum rate of compressive strain up to –240×10^(–9) a^(–1).A secondary shear zone along the Anninghe-Xiaojiang Fault in the east shows segmented shear zones near several conjunctions.While the strain rate along RRF is relatively low due to the low slip rate and low seismicity there, in Lijiang and Tengchong several local shear zones are present under an extensional dominated stress regime that is related to normal faulting earthquakes and volcanism, respectively.Furthermore, by comparing GPS-derived strain rate tensors with earthquake focal mechanisms, we find that 75.8%(100 out of 132) of the earthquake T-axes are consistent with the GPS-derived strain rates.Moreover, we find that the Fast Velocity Direction(FVDs) at three depths beneath the Shan-Thai terrane are consistent with extensional strain rate with gradually increasing angular differences, which are likely resulting from the basal shear forces induced by asthenospheric flow associated with the oblique subduction of the India plate beneath the Shan-Thai terrane.Therefore, in this region the upper crust deformation may be coherent with that of the lower crust and the lithospheric mantle.

RESEARCH ON THE DEFORMATIONS IN QINGHAI_TIBET PLATEAU AND ITS MARGINS BY INVERTING SEISMIC MOMENT TENSORS AND GPS VELOCITIES

We have determined approximate average rates of deformation in the Qinghai_Tibet plateau and its margins from the GPS data for last 10 years and the moment tensors from earthquakes between 1900 and 1999.We also determined the strain rate (seismic strain rate) associated with the seismic deformation using 254 M w ≥5.0 earthquakes,and estimated the shortening and extension rates for every block in the area as well.We also estimated the strain rate (geodetic strain rate)by 80 GPS sites’ velocity vectors and analyzed characteristic of kinematics by two kinds of strain rates and discussed earthquake potential in the area.As a result,the deformation rates from seismic moment tensors and from GPS velocities are basically agreed with each other.It is feasible to analyze seismic risk by comparing geodetic strain rate with seismic strain rate based on the opinion that strain energy will be released through earthquake.It is concluded that there is no strong earthquake potential (>M7) in the Qinghai_Tibet plateau and its margins,but there is earthquake potential (>M5) in middle Tibet in a few years.

Location and moment tensor inversion of small earthquakes using 3D Green＇s functions in models with rugged topography： application to the Longmenshan fault zone

With dense seismic arrays and advanced imaging methods, regional three-dimensional （3D） Earth models have become more accurate. It is now increasingly feasible and advantageous to use a 3D Earth model to better locate earthquakes and invert their source mechanisms by fitting synthetics to observed waveforms. In this study, we develop an approach to determine both the earthquake location and source mechanism from waveform information. The observed waveforms are filtered in different frequency bands and separated into windows for the individual phases. Instead of picking the arrival times, the traveltime differences are measured by cross-correlation between synthetic waveforms based on the 3D Earth model and observed waveforms. The earthquake location is determined by minimizing the cross-correlation traveltime differences. We then fix the horizontal location of the earthquake and perform a grid search in depth to determine the source mechanism at each point by fitting the synthetic and observed waveforms. This new method is verified by a synthetic test with noise added to the synthetic waveforms and a realistic station distribution. We apply this method to a series of Mw3.4-5.6 earthquakes in the Longmenshan fault （LMSF） zone, a region with rugged topography between the eastern margin of the Tibetan plateau and the western part of the Sichuan basin. The results show that our solutions result in improved waveform fits compared to the source parameters from the catalogs we used and the location can be better constrained than the amplitude-only approach. Furthermore, the source solutions with realistic topography provide a better fit to the observed waveforms than those without the topography, indicating the need to take the topography into account in regions with rugged topography.

DERIVATION OF THE GENERAL FORM OF ELASTICITY TENSOR OF THE TRANSVERSE ISOTROPIC MATERIAL BY TENSOR DERIVATE

In the paper the elasticity tensor and the relation between stress and strain of transverse isotropic material and isotropic material are deduced by tensor derivate. From the derivation the reason why there are two independent elasticity coefficients for isotropic elastic material and five for transverse isotropic elastic material can be seen more clearly.

FEA OF RUBBER WITH NOVEL STRAIN ENERGY FUNCTIONS

This paper presents the results of finite element analysis of rubber structures based on novel strain energy functions stemming from the representation theorem of tensorial function. The stress tensor is represented by Taylor expansion, using the representation theorem of tensorial function of a single tensorial argument for all terms in each order of the expansion. The scalar-valued coefficient functions of the theorem are represented by the integrity bases of the strain tensor and material constants to be determined by experiment. The computer implementation of the new constitutive laws has been verified by comparing the FE results with analytical solutions. A complicated structure of rubber bearing was analyzed. The FE results show good correlation with experimental data.

热空气作用下FM-2D橡胶材料老化本构模型研究

目的建立热空气作用下氟醚-2D(FM-2D)橡胶材料的老化本构模型,形成老化作用下橡胶材料力学响应分析方法,为准确评估橡胶密封件使用寿命提供依据。方法探究热空气作用下FM-2D橡胶材料老化机理,基于连续介质有限变形理论框架,采用热力学耗散势函数法,引入橡胶老化过程的势能函数,据此建立考虑橡胶材料老化的超弹性本构模型,基于橡胶老化试验,完成本构模型参数标定,实现老化作用下橡胶力学响应的预测。结果建立了热空气作用下橡胶材料的老化本构模型,依据老化试验数据标定模型参数,分析了热空气作用下橡胶材料本构模型的可靠性。结论建立的热空气作用下橡胶材料的老化本构模型可准确预测橡胶随老化时间演变的力学响应,有效模拟了橡胶材料的老化过程。

基于连续介质力学的泡沫混凝土应力-变形理论分析

为更好描述与表达泡沫混凝土的应力-变形关系,在连续介质力学中定义和分析笛卡儿张量,应用应力张量性质并采用与应力状态相关的方法分析泡沫混凝土的变形。理论分析表明:泡沫混凝土的平衡方程是在牛顿运动定律和柯西公式基础上推导得到;泡沫混凝土的平面应力状态满足的条件方程为σzz=σyy=σzx=0;泡沫混凝土的变形分析是通过物体内部任意两点间距离变化进行描述;泡沫混凝土考虑平面应变状态时,满足可积性条件,即协调方程。

基于线性疲劳累计损伤橡胶悬置疲劳寿命预测研究

橡胶元件疲劳寿命的有效预测是其设计开发过程中的重要环节。引入橡胶元件线性疲劳累计损伤原理,提出张量形式橡胶疲劳寿命公式,且根据橡胶材料的实际承载工况提出其失效标准。依据橡胶材料的承载变形可简化为单轴拉伸及简单切应变,设计用于承载拉伸载荷的哑铃型橡胶试柱和承载剪切载荷的环形橡胶试柱,并实测疲劳寿命数据,以最小二乘法原理拟合拉伸与剪切的疲劳寿命函数公式。以车用变速箱悬置与发动机后悬置为疲劳寿命预测研究对象,通过分析其承载位移载荷时的应变张量,利用张量形式的疲劳寿命预测公式预测两种悬置在两种典型工况下的疲劳寿命。结果发现,橡胶材料的拉伸疲劳寿命曲线与简单剪切疲劳寿命曲线的变化趋势一致、形状类似、拟合函数幂指数十分接近;张量形式的疲劳寿命预测公式可有效地预测橡胶悬置的疲劳寿命。

用应变地震观测求解震源矩张量的基本原理

震源机制解,即对地震矩张量的推断,对于地震研究具有至关重要的意义.应变地震观测是张量观测,与摆式地震仪的位移矢量观测不同,可以为地震研究提供新的数据源.本文讨论用应变地震观测求解震源矩张量的基本原理.在距离震源足够远的地方,地震波可以看成平面波,其性质决定于震源矩张量.假设平面地震波的应变张量可以由震源矩张量通过坐标变换计算得到,就可以通过观测应变地震波求解震源机制.这个假设至少对于双力偶震源机制是成立的.由此可以证明,在理想的无限介质中,只要有两个以上不同地点的应变地震波观测,就可以解出震源矩张量.这为解决震源机制问题提供了新的方法.目前的地震矩张量求解方法需要两方面的条件:或者需要很多观测点(例如体波反演),或者需要长周期地震波资料(例如面波反演).这些方法只适用于分析比较大的地震.对于小震,因为通常其震中周围不会有足够多的摆式地震仪观测点观测到其地震波,而地震波周期又短,难以利用传统方法给出可靠的震源机制解,所以只需少数观测点就能求解震源矩张量的新方法特别有意义.用应变地震观测求解震源机制,可以给出更为精确的结果.

局部加载条件下二维土质边坡稳定性的数值分析

借助 Hill 的材料应变局部化原理和土的弹塑性本构模型,应用声学张量示踪边坡失稳滑移线发生和发展,提出了局部加载条件下的确定边坡失稳滑移线的方法。在弹塑性有限元分析过程中,应用边坡材料的塑性一致性条件或滑移线两侧的应力连续条件,即应变局部化条件,检测每一个高斯点的声学张量行列式的值,当其等于或接近 0 时,标出滑移萌生的位置、方向,模拟边坡滑移线萌生和发展过程。

台湾海峡两岸地壳相对运动分析

利用1995～2003年福建GPS测站与台北和桃园核心基准站资料,计算了福建GPS各测站与台北和桃园之间的边长及其相对变化,同时利用遗传算法计算了台北和桃园测点的主应变和方向角。计算结果表明,在1995～1997年和2001～2003年间台海两岸的地壳相对运动都处于拉张状态,但两期拉张的主应变大小和方向有较大变化,这种变化可能反映了1999年台湾集集7.6级地震前后应变积累与释放的过程。

旋转椭球面上的应变与转动张量表达

以旋转椭球体面上某点为原点建立一个大地坐标单位活动坐标架.通过平移,使活动坐标架的原点与以椭球中心为原点的笛卡尔单位标架的原点相重合.然后再通过两次标架旋转,使活动坐标架与笛卡尔单位标架完全重合.本文给出了使两个单位标架相重合的转换关系式,以及该点位移在两个单位标架中的坐标转换式;在此基础上,考虑该点的位移及活动坐标架皆为该点大地坐标的函数,经复杂推导,分别给出了该点位移向量的微分在大地坐标系中的分量以及该点分别沿坐标曲线的弧微分表达式,继而导出了该点的位移梯度矩阵;最后推导出了椭球坐标系的应变张量与转动张量表达式,并对转动张量的几何含义进行了较详细的解释,且采用曲面理论对球面与椭球面的应变张量间的内在关系进行了讨论.

光流模值估计的微表情捕捉

采用力的加速度参量展开描述人脸表情的变化过程,直接反映变化速度,从而有效捕捉表情序列中由不完全肌肉运动所引起的微表情关键帧.利用Horn-Schunck(H-S)光流法对连续运动的人脸图像序列提取运动目标的运动特征;通过光学应变张量算法,结合运动特征中的光流速度估计,推导出加速度参量;利用全局阈值算法对加速度模值和速度与张量模值作分类、比较,实现微表情图像序列关键帧的提取.采用Oulu大学SMIC微表情数据库中16个实验对象的88个微表情片段作为实验样本,平均正确识别率可达80.7%,比仅利用光学张量算法的正确识别率高12.5%.实验结果表明,所提出的加速度参量对微表情提取更具有效性.

正交曲线坐标系的应变张量转换

论证任何一种正交曲线坐标系中的度量张量矩阵皆为对角矩阵,即在该坐标系中的该点的活动标架彼此正交,为直角坐标系;证明可采用直角坐标系的应变张量公式计算正交曲线坐标系的应变张量矩阵;并给出了正交曲线坐标系应变张量转换的普适方法和ITRF与WGS84之间应变张量矩阵转换的表达式。

脆性岩石破坏的演化细胞自动机(ECA)研究

演化细胞自动机模型是以细胞自动机方法的基本理论为基础,结合岩石力学的基本原理提出的。演化细胞自动机模型认为脆性岩石的应力–应变关系在细观上符合理想线弹脆性本构关系,在宏观上则认为脆性岩石材料是一种非均质的材料,并假定脆性岩石材料的非均质性符合Weibull分布。演化细胞自动机模型突破以往细胞自动机只有一类细胞的限制,采用包含节点细胞和三角形细胞在内的两类细胞的邻居模型。该方法以位移、力、应力和应变等矢量和张量作为系统的基本变量,将张量和纯量定义在三角形细胞中,而将矢量定义在节点细胞上,两类细胞共同形成细胞空间。在演化细胞自动机系统中,两类细胞互为邻居,依据模型所建立的局部作用规则进行演化,并且按照以修正的莫尔–库仑准则为依据建立的破坏演化规则进行破坏演化,分别考虑了由拉伸和剪切两种原因所引起的破坏情况,进而实现对脆性岩石材料加载及破坏过程的模拟。