Optimal Orthotropy for Minimum Elastic Energy by the Polar Method

The polar method is a minimal invariant representation in plane elasticity. A plane orthotropic elastic behaviour is expressed by five polar invariants related to the elastic symmetries. In this paper, considering the orthotropy orientation and the polar invariants as optimisation parameters, we discuss the problem of minimising the elastic energy for a given state of stress. The minimisation with respect to the orientation is solved in order to find the associated optimal elastic energy for given polar invariants. Then, this quantity is minimised with respect to the polar invariants which characterise the magnitude of the anisotropic components of the elastic stiffness tensor. Optimal uncoupled composite laminates corresponding to this optimum are presented for membrane and bending loadings.     

GIANT ANHYSTERETIC RESPONSE OF FERROELECTRIC SOLID SOLUTIONS WITH MORPHOTROPIC BOUNDARIES: THE ROLE OF POLAR ANISOTROPY

Computer modeling and simulation for the Pb(Zr 1 x Ti x )O 3 (PZT) system reveal the role of polar anisotropy on the giant anhysteretic response and structural properties of morphotropic phase boundary (MPB) ferroelectrics. It is shown that a drastic reduction of the composition- dependent polar anisotropy near the MPB flattens energy surfaces and thus facilitates reversible polarization rotation. It is further shown that the polar anisotropy favors formation of polar domains, promotes phase decomposition and results in a two-phase multidomain state, which response to applied electric field is anhysteretic when the polar domain reorientation is only caused by polarization rotation other than polar domain wall movement. This is the case for the decomposing ferroelectrics under a poling electric field with the formation of a two-phase multidomain microstructure, wherein most domain walls are pinned at the two-phase boundaries. Indication of the microstructure dependence of the anhysteretic strain response opens new avenues to improve the piezoelectric properties of these materials through the microstructure engineering.     

基于伪地球坐标系的惯导系统全球导航算法

由于极区特殊的地理、电磁条件,惯性导航因其自主性和信息完备性使之成为极区导航的首选。然而在考虑全球执行能力时,现有常用的任何一种力学编排方案都不能单独的实现全球导航。通常采用组合编排的方式,这样则不利于惯导算法的全球统一。基于此提出了基于伪地球坐标系的全球导航算法。该方案在全球导航时可以实现惯导算法的内在统一,并可保证物理平台的平稳切换,从而实现平台惯导与捷联惯导系统编排方案的统一。另外,该方案也更方便同其它局部惯导系统进行交互通信,仅不同的参数转换单元是必需的。同时,简单的切换逻辑也可以减小程序设计的复杂度和降低计算机负担。最后通过仿真证明了该算法的可行性。     

An Integrable Model of Nonstationary Rotationally Symmetrical Motion of Ideal Incompressible Liquid

We consider a partially invariant solution of the Eulerequations with respect to a six-parameter Lie group admitted by thissystem where the vertical component of velocity is a function of thevertical coordinate and time only while two other components andpressure do not depend on the polar angle in a cylindrical coordinatesystem. The analysis of the corresponding overdetermined system leads totheir special (but nontrivial) dependence of the polar radius. Afterthis, the nonlinear factor-system for invariants of the group is reducedto a system of ordinary differential equations by introduction ofLagrangian coordinates. As a result, we obtain a wide class of new exactsolutions which describes vortex motions of an ideal incompressibleliquid including motions with singularities.     

General solution of A 2-D weak singular integral equation with constraint and its applications

Hertz'ssolutionTheProjectsupportedbytheNaturalScienceFoundationofGuangdongProvinceofChina[.IntroductionTheintegralequationsubjecttoconstraintp(s,te)=o,for(s,W)-(r,o)6Q={(r,O)lF(r,o)>c.}'(l.2)hasbeenstudiedin[l].Themultipleintegrationsimilartothelefthandsi…     

Hydrodynamics of active polar systems in a (Visco)elastic background

We derive the full set of macroscopic equations necessary to describe the dynamics of systems with active polar order in a viscoelastic or elastic background. The active polar order is manifested by a second velocity, whose non-zero modulus is the polar order parameter and whose direction is the polar preferred direction. Viscoelasticity is described by a relaxing strain field allowing for a straightforward change from transient to permanent elasticity. Relative rotations of the elastic structure with respect to the polar direction are taken into account. The intricate coupling between active polar order and (transient) elasticity leads to a combined relaxation of the polar order parameter and the strains. The rather involved sound spectrum contains a specific excitation due to a reversible coupling between elasticity and polar order. Effects of chirality are also considered.     

On 2D Viscoelasticity with Small Strain

An exact two-dimensional rotation–strain model describing the motion of Hookean incompressible viscoelastic materials is constructed by the polar decomposition of the deformation tensor. The global existence of classical solutions is proved under smallness assumptions only on the size of the initial strain tensor. The proof of global existence utilizes the weak dissipative mechanism of motion, which is revealed by passing the partial dissipation to the whole system.     

A note on the stress singularity at a non-uniformly moving crack tip

The stress-singularity at a crack tip moving arbitrarily in an elastic plate under plane strain conditions is investigated. By formulating the wave-equations in a polar coordinate system attached to the crack-tip, it is found by an asymptotic analysis that the angle-dependence of the singularity is only dependent on the instantaneous cracktip velocity. This result is used to derive a relation between the dynamic stress-intensity factor and the energyrelease rate.     

微粗糙平面上翻身陀螺的稳定性

本文讨论微粗糙平面上翻身陀螺绕直立的极轴永久转动的稳定性。应用关于部分变量的稳定性定理,导出在粘性和库仑两种摩擦条件下的解析形式稳定性判据。该判据用永久转动角速度的大小而不是进动角速度的大小作为判别准则,较为合理。     

Well-posedness of two-phase local/nonlocal integral polar models for consistent axisymmetric bending of circular microplates

Previous studies have shown that Eringen's differential nonlocal model would lead to the ill-posed mathematical formulation for axisymmetric bending of circular microplates. Based on the nonlocal integral models along the radial and circumferential directions, we propose nonlocal integral polar models in this work. The proposed strainand stress-driven two-phase nonlocal integral polar models are applied to model the axisymmetric bending of circular microplates. The governing differential equations and boundary conditions (BCs) as well as constitutive constraints are deduced. It is found that the purely strain-driven nonlocal integral polar model turns to a traditional nonlocal differential polar model if the constitutive constraints are neglected. Meanwhile, the purely strain-and stress-driven nonlocal integral polar models are ill-posed, because the total number of the differential orders of the governing equations is less than that of the BCs plus constitutive constraints. Several nominal variables are introduced to simplify the mathematical expression, and the general differential quadrature method (GDQM) is applied to obtain the numerical solutions. The results from the current models (CMs) are compared with the data in the literature. It is clearly established that the consistent softening and toughening effects can be obtained for the strain-and stress-driven local/nonlocal integral polar models, respectively. The proposed two-phase local/nonlocal integral polar models (TPNIPMs) may provide an e-cient method to design and optimize the plate-like structures for microelectro-mechanical systems.     

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.     

Plane stationary flows with ionizing shock waves in a magnetic field

The formulation of stationary, plane, and self-similar problems is considered when the flow parameters depend only on the polar angle, and the magnetic field lies in the flow plane. The case in which the magnetic field is perpendicular to the flow plane has been examined in [1]. The conditions are found under which the solution depends on an arbitrary parameter and the reasons for this nonuniqueness are explained. Self-similar solutions are constructed to describe the flow around an insulating wedge and a wall.     

磁电复合材料中拓扑磁结构的力学调控

磁性斯格明子是在一些铁磁材料中存在的一种重要拓扑磁结构,由于其具有独特的磁-电-力-热多场耦合特性,在未来新型自旋电子器件中有着广泛的应用前景。然而,磁性斯格明子一般需要在外加磁场下才能稳定存在,极大地限制了其在自旋电子器件中的实际应用。本文基于实空间下磁电材料的相场模拟,发现铁电和铁磁复合薄膜中铁电斯格明子可以通过界面变形来稳定铁磁斯格明子。由于力电耦合效应,铁电层中铁电斯格明子的非均匀分布极化在界面产生周期性的非均匀界面变形。界面变形通过力磁耦合效应,使铁磁层中的磁性斯格明子在没有外加磁场的条件下能够稳定存在。本文的研究结果表明,基于磁电复合材料中的力-电-磁耦合效应,通过优化设计复合材料中不同组元的结构,可以实现拓扑磁结构的力学调控,从而为设计基于拓扑磁结构的新型自旋电子器件提供了新思路。     

Mesoscale modeling of dynamic compression of boron carbide polycrystals

An anisotropic nonlinear elastic model is advanced for crystals belonging to either of two polytypes of boron carbide ceramic. Crystals undergo transformation to an isotropic, amorphous phase upon attainment of a local state-based criterion associated with a loss of intrinsic stability. The model is implemented using the dynamic finite element method, and is demonstrated on a representative volume consisting of fifty polyhedral grains subjected to uniaxial strain at a uniform high strain rate and shock compression at axial pressures ranging from 10 to 50 GPa. Predicted stress–strain behavior is in close agreement with experimental data. For polycrystals consisting of both polytypes, amorphization initiates at stress levels slightly below the experimental Hugoniot elastic limit, and occurs more readily than observed in experiment. For polycrystals consisting only of the CBC (polar) polytype, amorphization initiates at impact pressures similar to those suggested by experiment. In either case, transformation is promoted by dynamic stress interactions and elastic coefficient mismatch among anisotropic crystals. Results support a previous conjecture that amorphization is related to shear instability and cross-linking of the CBC chain in the polar polytype.     

Topographic effects on polar low and tropical cyclone development in simple theoretical model

The polar low and tropical cyclone type vortices over topography are assumed to be the axisymmetrical and thermal-wind balanced systems, which are solved as an initial value problem of a linearized vortex equation set in cylindrical coordinates. The roles of the sensible and latent heating, friction, and topography in the structure and intensification of the polar low and tropical cyclone type vortices are analyzed. The radial velocity, vertical velocity, azimuthal velocity, and the unstable growth rate including the topography effects are obtained. It is shown that the interaction between the flow and the topography plays a significant role in the structure and intensification of the polar low and tropical cyclone system. The analysis of the topography term indicates that, in the up-slope side of the mountain, the radial inflow and the vertical ascent forced by the mountain can intensify the polar low and tropical cyclone type vortex and increase the unstable growth rate. However, in the lee side of the mountain, the radial inflow and the vertical descent forced by the mountain can weaken the polar low and tropical cyclone type vortex and decrease the unstable growth rate of the polar low and tropical cyclone system. In addition, the evolutionary process and the spatial structure of the polar low observed over the Japan Sea on 19 December 2003 are investigated with the observational data to verify this theoretical result.     

Spline sectorial elements

In this paper, the quadratic and cubic splines local interpolation on a sectorial element in polar coordinates is discussed and a class of spline sectorial elements for analyses of plane and thin, problems are presented. A reasonable treatment of the assumed displacement fields for elements with nodes at the origin, (r=0) has been made so that the elements can not only characterize the geometrical properties at the origin but also remove the singularity of strains and stresses there. Some numerical examples are given to show the efficiency of the proposed elements.     

平面问题极坐标下几何方程的一种解析法

经典弹性力学教材中都是通过单元的几何变形来推导平面问题极坐标下的几何方 程. 这里, 用解析法从直角坐标系下的几何方程推出极坐标下的几何方程.     

正多边形弹性薄板的弯曲解

提出了一种在极坐标下求解正多边形薄板弯曲的新方法。在均布荷载条件下,通过选取满足特定对称条件的试函数,得到正多边形弹性薄板弯曲问题的级数展开解法,在挠度或弯矩表达式中仅取前5项,就可获得较高精度的结果,该方法能较好地解决特定正多边形薄板的弯曲问题,既可适用于求解均布荷载下的任意正多边界薄板的弯曲问题,也可以推广到承受任意荷载的任意正多边形薄板。