FAST ALGORITHMS FOR DETERMINING POSITIVE DEFINITENESS OF LARGE SCALE MATRICES IN SOLID STRUCTURE ANALYSIS

对于结构稳定性分析中超大规模矩阵正定性判定,必须采用并行计算方法,传统方法如计算特征值、主子式行列式及LDLT等直接方法难以实现.本文提出了一些可适用于并行的迭代判定算法.借鉴力学系统中能量下降的思想,发展了一种判定矩阵正定性的新思路,即将矩阵的正定性判定问题转化为一个优化问题,并基于优化算法来判定矩阵的正定性.提出了基于最速下降法和共轭梯度法来进行矩阵正定性判定的算法.然后考虑到力学系统刚度矩阵的稀疏性和结构刚失稳状态的弱非正定性,提出可以先截超平面后解方程求驻值点的方法来判定弱非正定矩阵的正定性.为了保证对强非正定矩阵判定的准确性,本文提出可以高效混杂使用截平面法和共轭梯度法.数值实验结果表明,本文提出的算法具有准确性和高效性.对于强非正定矩阵而言,共轭梯度算法更加高效;而对于弱非正定矩阵,则是截平面法和混杂算法更加高效.这些算法都容易在机群上实现并行计算,能够快速判定大规模矩阵的正定性.     

Existence of Shear Horizontal Surface Waves in a Magneto-Electro-Elastic Material

The existence of shear horizontal surface waves in a magneto-electro-elastic （MEE） half-space with hexagonal （6mm） symmetry is investigated. The surface of the MEE half-space is mechanically free, but subjected to four types of electromagnetic boundary conditions. These boundary conditions are electrically open/magnetically closed, electrically open/magnetically open, electrically closed/magnetically open and electrically closed/magnetically dosed. It is shown that except for the electrically open/magnetically closed condition, the three other sets of electromagnetic boundary conditions allow the propagation of shear horizontal surface waves.     

Effect of yield surface evolution on bending induced cross sectional deformation of thin-walled sections

Bend–stretch forming is commonly used to shape extruded tubular aluminum parts for automotive and other applications. The tubes are pre-stretched, pressurized and bent over rigid curved dies. Tension prevents buckling of the compressed side and helps reduce springback. An unwanted byproduct of the process is distortion of the cross section. Small amounts of pressure applied during forming can reduce this distortion. The problem was studied through a combination of experiment and analysis. In numerical models of the process the inelastic behavior of the aluminum alloy was modeled through isotropically hardening plasticity. With the limitations imposed by this model, use of a J₂-type yield surface resulted in uniform underprediction of cross sectional deformation. The predictions matched the measurements when a non-quadratic yield function appropriately “calibrated,” was used instead. The change in yield function shape alters the instantaneous normals to the yield surface which, in turn, affect the calculated strain increments. This paper demonstrates how suitably calibrated nonlinear kinematic hardening models can have the same corrective effect. The calibration involves selection of a suitable kinematic hardening rule. Changes in the hardening direction alter the instantaneous normals and therefore alter the plastic strain increments resulting in approximately the same net effect as the switch from the von Mises to the non-quadratic yield function.     

Snapping instability in prismatic tensegrities under torsion

Tensegrities are a class of lightweight and reticulated structures consisting of stressed strings and bars. It is shown that each prismatic tensegrity can have two self-equilibrated and stable states, leading to a snapping instability behavior under an applied torque. The predicted mechanism is experimentally validated, and can be used in areas such as advanced sensors and actuators, energy storage/adsorption equipments, and folding/unfolding devices.     

Energy corrugation in atomic-scale friction on graphite revisited by molecular dynamics simulations

Although atomic stick–slip friction has been extensively studied since its first demonstration on graphite,the physical understanding of this dissipation-dominated phenomenon is still very limited. In this work, we perform molecular dynamics(MD) simulations to study the frictional behavior of a diamond tip sliding over a graphite surface. In contrast to the common wisdom, our MD results suggest that the energy barrier associated lateral sliding(known as energy corrugation) comes not only from interaction between the tip and the top layer of graphite but also from interactions among the deformed atomic layers of graphite. Due to the competition of these two subentries, friction on graphite can be tuned by controlling the relative adhesion of different interfaces.For relatively low tip-graphite adhesion, friction behaves normally and increases with increasing normal load. However,for relatively high tip-graphite adhesion, friction increases unusually with decreasing normal load leading to an effectively negative coefficient of friction, which is consistent with the recent experimental observations on chemically modified graphite. Our results provide a new insight into the physical origins of energy corrugation in atomic scale friction.     

Full-field Measurement of Topography and Curvature by Coherent Gradient Sensing Method at High Temperature

Coherent gradient sensing (CGS) method, a real time, full-field optical technique, is insensitive to vibrations and able to provide slope and curvature maps and surface topographies, to investigate non-uniform deformations. In this paper, we analyze the thermal effects on the optical path in CGS due to air convection, and the influence of grating thickness and refractive index on the measurement accuracy. A modified governing equation is derived considering the grating thickness, which is demonstrated by testing a standard sample. Finally, we apply CGS method to measure the full-field deformation of a specimen at high temperature.     

Two-dimensional Hertzian contact problem with surface tension

In the present paper, we consider a two-dimensional contact problem of a rigid cylinder indenting on an elastic half space with surface tension. Based on the solution of a point force acting on a substrate with surface tension, we derive the singular integral equation of this problem. By using the Guass–Chebyshev quadrature formula, the integral equation is solved numerically to illuminate the influence of surface tension on the contact response. It is found that when the contact width is comparable with the ratio of surface tension to elastic modulus, surface tension significantly alters the pressure distribution in the contact region and the contact width. Compared to that of the classical Hertzian contact, the existence of surface tension decreases the displacements on the half plane and yields a continuous slope of normal stress and displacements across the contact fringe. In addition, it predicts the increase of hardness as the radius of indent cylinder decreasing. The obtained results are useful for the measurement of mechanical properties of materials based on the indentation technique.     

Postbuckling analysis and its application to stretchable electronics

A versatile strategy for fabricating stretchable electronics involves controlled buckling of bridge structures in circuits that are configured into open, mesh layouts (i.e. islands connected by bridges) and bonded to elastomeric substrates. Quantitative analytical mechanics treatments of the responses of these bridges can be challenging, due to the range and diversity of possible motions. Koiter (1945) pointed out that the postbuckling analysis needs to account for all terms up to the 4th power of displacements in the potential energy. Existing postbuckling analyses, however, are accurate only to the 2nd power of displacements in the potential energy since they assume a linear displacement–curvature relation. Here, a systematic method is established for accurate postbuckling analysis of beams. This framework enables straightforward study of the complex buckling modes under arbitrary loading, such as lateral buckling of the island-bridge, mesh structure subject to shear (or twist) or diagonal stretching observed in experiments. Simple, analytical expressions are obtained for the critical load at the onset of buckling, and for the maximum bending, torsion (shear) and principal strains in the structure during postbuckling.     

Multiple-cell elements and regular multifractals

Based on fractal super fibers and binary fractal fibers, the following objectives are approached in this paper： First, the concept of multiple-cell elements is induced and abstracted. Second, through multiple-cell elements, the constructability of regular multifractals with strict self-similarities is confirmed, and the universality of the con- struction mode for regular multifractals is proved. Third, through the construction mode and multiple-cell elements, regular multifractals are demonstrated to be equivalent to generalized regular single fractals with multilayer fine structures. On the basis of such equivalence, the dimension formula of the regular single fractal is extended to that of the regular multifractal, and the geometry of regular single fractals is extended to that of regular multifractals. Fourth, through regular multifractals, a few golden fractals are constructed.