Numerical solution of plate-stability problems

International Applied Mechanics -     

Formal solution of quasi-static problems

When the quasi-static problem is defined by a set of differential equations complemented by initial and boundary conditions, the resulting quasi-static solutions may exhibit a limited reach over the time domain. On the other hand, the infinity of equilibrium paths that can be obtained in a general non-linear problem also indicates that a proper definition of the quasi-static solution must be provided. In inelasticity problems, this infinite number of equilibrium paths occur even when no dissipative bifurcations are present. In the present paper, a general solution for quasi-static problems in Solid Mechanics is defined and explored. Special attention is addressed to material non-linearities though geometric non-linearities are also covered by the definition. Earlier concepts of path and state stability are recovered in order to reduce the number of solutions to those that are physically acceptable. The important link with the original dynamic problem is accounted for by enforcing a preferential load direction. The resulting definition relies on a time-objective criterion with straightforward applicability to the most common numerical models. In the final part of the paper, simple 1D problems are used to illustrate some of the concepts introduced in the present developments.     

A numerical solution of infiltration

A fully implicit finite difference scheme is used to evaluate one-dimensional infiltration. The method makes it possible to solve a general infiltration problem; nonhomogeneous soil profile and saturated-unsaturated seepage can be treated. To solve special problems of hydrology and soil physics, several types of boundary conditions are formulated and numerically expressed. The type of boundary conditions may vary in time depending on the values of the unknown function. High accuracy of solution is emphasized. Several applications of this method are presented.     

深梁的精确解

 抛弃所有梁理论假设, 直接从平面应力理论, 导出深梁的状态方程, 求解了简支梁精确解, 并与有限元解进行比较, 结果吻合较好.     

螺栓剪力问题的误解与正解

指出了用梁的纵向层上的剪力分配给螺栓求螺栓剪力的方法是错误的，导出了正确的求解公式，该公式显示螺栓剪力实质上和梁的弯矩相关。     

动力减振器的理论解

 以连续弹性体的振动理论为依据，推导了带有集中质量的简支梁的理论解. 在此基础上又 推导了以简支梁为模型的动力减振器的理论解. 计算结果表明：如果不考虑简支梁跨中的集 中质量，其误差成倍增加. 利用推导的计算公式，所得结论与实测结果吻合得很好.     

Numerical solution of the entrainment equation

An equation describing the change in body shape as a result of intense mass entrainment from the surface under the effect of convective aerodynamic heating under the simplest assumptions (dependence of the pressure on only the local slope of the surface, the method of local similarity to compute the heat fluxes) is considered. Mathematical questions originating during its numerical solution are investigated. The instability of explicit schemes is proved in the neighborhood of the stagnation point. A stable explicit-implicit scheme, suitable for a through computation of unsmooth solutions, is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 147–154, March–April, 1978.The author is grateful to V. V. Lunev for constant attention to the research and for useful discussions.     

有界弦的行波解及其与驻波解的关系

 有界弦的振动是介绍数学物理方程求解方法和了解 连续体振动、波动运动形态的一个典型例子. 给出了它的行波解, 并揭示了行波 解与驻波解的关系, 可以帮助加深对弦的运动形态的了解.     

Analytical solution of magnetohydrodynamic sink flow

An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of −1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD flow with an analytical closed form formula. They greatly enrich the analytical solution for the celebrated Falkner-Skan equation and provide better understanding of this equation.     

Freezing characteristics of ethylene glycol solution

An experimental and analytical study has been performed to investigate the freezing characteristics of aqueous binary solution. Ethylene glycol solutions of 5, 10, 30, 40, and 50 wt% in concentration were utilized as the testing solutions and the experiments were carried out for various initial and cooling temperatures. The effects of concentration and cooling temperature on the frozen layer of the solutions formed on a horizontal plate were extensively examined. It was found that the frozen layer became mushy dendritic ice layer except for the cases of both high concentration and quite quick or slow freezing and that the analytical results were in good agreement with the experimental ones for the dendritic ice layer.     

Continuum modelling of polyelectrolytes in solution

A phenomenological model for solutions of polyelectrolytes accounting for electromechanical interactions is proposed within the framework of continuum thermodynamics. The modelling involves the conformation of macromolecules in the form of a tensorial internal variable and the electric polarization of the solution. The paper aims to demonstrate the possible competition between flow and electricity effects on rigorous phenomenological grounds. Three kinds of dissipative mechanisms are accounted for: viscosity of the solution, electric relaxation and the relaxation of the conformation of macromolecules. Anisotropic effects induced by the applied electric field and flow-induced polarization through changes in conformation are exhibited. These effects are small. By way of application, the problem of the orientation and conformation taken by macromolecules of polyelectrolytes under the combined influence of a simple shear flow and a longitudinal or orthogonal electric bias field is treated in detail and illustrated graphically. Some optical properties are mentioned.     

力学问题求解悬赏

特别说明：对科学问题悬赏的多少并不重要,悬赏只不过是为科研生活增加点趣味,为饭后增加点谈资。另外出题人不负责提供如何解答。同时欢迎追加悬赏金额。 力学作为最古老的科学学科之一,在20 世纪得到了很大的发展,这包括有限元提出、断裂力学、生物力学的创立、稳定分岔和混沌理论、边界层理论、塑性力学和位错理论、湍流统计理论、奇异摄动理论、力学的公理化体系、克服声障和热障等,力学的确可以并且解决了大量的科学和工程实际问题。但是,力学是发展的科学,有些看起来简单的问题,有时求解并非容易。 为了鼓励创新,特别提供以下二个问题向对力学有兴趣的所有学者悬赏求解。 【1】 武际可问题：在徐秉业先生主编的《身边的力学》,北京大学出版社（1997 ）中有武先生写的一篇文章："从土豆的内伤谈起－漫谈接触问题"提出的一个问题：一个均匀的弹性球,从某一高度在重力作用下,（1 ）落到均匀的弹性半空间上,它的弹跳高度是多少？（2）落到均匀的弹塑性半空间上,它的弹跳高度是多少？ 【2】 打鸡蛋问题：二个完全相同的鸡蛋,一个运动的鸡蛋碰撞一个静止的鸡蛋,请问那个鸡蛋更容易破？     

Analytical solution of magnetohydrodynamic sink flow

An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of -1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD ...     

Exact solution of the Boltzmann equation

We build up immediate connection between the nonlinear Boltzmann transport equation and the linear AKNS equation, and classify the Boltzmann equation as the Dirac equation by a new method for solving the Boltzmann equation out of keeping with the Chapman, Enskog and Grad’s way in this paper. Without the effect of other external fields, the exact solution of the Boltzmann equation can be obtained by the inverse scattering method.     

Winkler地基上各向异性薄板弯曲的精确解-广义积分变换解

外界载荷作用下复合材料薄板的弯曲行为是工程重点关注的问题之一。针对各向同性和正交各向异性的薄板弯曲问题,研究人员已给出了经典数值解。由于计算的复杂性,针对各向异性薄板弯曲问题的解答较少。本文从薄板弯曲问题的控制方程出发,建立符合该问题的辅助特征方程,并确定相应的特征值和特征函数。利用广义积分变换的思想,建立了求解非正交铺层条件下各向异性薄板弯曲问题的数值算法,给出了各向异性薄板弯曲的精确解。与其他文献结果比较发现,该方法具有较好的收敛性和准确性。