Global Constraints for Stress Constrained Materials: The Problem of Saint Venant

Based on the global constraint principle of Antman and Marlow, a new solution of Saint Venant's problem is proposed. The solutions for the six fundamental cases of loading in terms of stress are obtained with relative ease and converge to the classical Saint Venant's solution as the length of the beam is increased. It is also shown that the assumptions of a special technical rod theory are coherent with the requirements of the global constraint theory for the Saint Venant cylinder.     

Second-order effects and Saint Venant’s principle in the torsion problem of a nonlinear elastic rod

The torsion problem of a circular nonlinear elastic rod loaded by end moments is considered. The solution constructed by the method of successive approximations taking into account second-order effects is compared with the solution obtained by a semi-inverse method. It is shown that the dead-loading assumption breaks the symmetry of the Cauchy stress tensor in a certain region. A refined formulation of Saint Venant’s principle is proposed for the problem of determining integral strain characteristics. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 129–136, November–December, 2006.     

Solution of the problem of optimal cut in an elastic beam

The Kirchhoff model of an elastic beam with a transverse cut is considered. The nonpenetration condition proposed by A. M. Khludnev is formulated at the edges of the cut. The equilibrium model of a beam with a restriction on the cut is written in the form of a variational inequality. An analytical solution is obtained with the use of the projection operator. The problem of choosing optimal cuts is formulated for the criterion of minimum opening. Conditions for determining the extremum shapes of the beam are obtained and an example of the solution of the problem is given. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 149–157, September–October, 1999.     

The contact problem for a class of orthotropic elastic solids

The contact problem between two orthotropic solids is examined. The problem is solved by using Lodge's method, which permits the transformation of the boundary-value problem of an anisotropic solid to a form identical with the corresponding problem of an isotropic medium. The proposed solution is then compared with known results of certain cases and it is observed that it producesHertz's solution when used for an isotropic case,Lodge's solution when applied to contact between an orthotropic solid and a rigid plane and, finally,Love's solution if the solid is transversely isotropic with the axis of material symmetry perpendicular to the rigid plane of contact.     

Solution of the Rayleigh problem for a powerlaw non-Newtonian conducting fluid via group method

ＩｎｔｒｏｄｕｃｔｉｏｎＦｌｕｉｄｓｔｈａｔｏｂｅｙＮｅｗｔｏｎ’ｓｌａｗｏｆｖｉｓｃｏｓｉｔｙａｒｅｃａｌｌｅｄＮｅｗｔｏｎｉａｎｆｌｕｉｄｓ.Ｎｅｗｔｏｎ’ｓｌａｗｏｆｖｉｓｃｏｓｉｔｙｉｓτ=μｄｕ/ｄｔ,ｗｈｅｒｅτｉｓｔｈｅｓｈｅａｒｓｔｒｅｓｓａｎｄμｉｓｔｈｅｖｉｓｃｏｓｉｔｙ .ＮｏｔａｌｌｆｌｕｉｄｓｆｏｌｌｏｗｔｈｅＮｅｗｔｏｎｉａｎｓｔｒｅｓｓ＿ｓｔｒａｉｎｒｅｌａｔｉｏｎ .Ｓｏｍｅｆｌｕｉｄｓ ,ｓｕｃｈａｓｋｅｔｃｈｕｐ ,ａｒｅｓｈｅａｒｔｈｉｎｎｉｎｇ ,ｔｈａｔｉｓ,ｔｈｅｃ…     

Solution of a Riemann problem for elasticity

This paper describes a numerical algorithm for the Riemann solution for nonlinear elasticity. We assume that the material is hyperelastic, which means that the stress-strain relations are given by the specific internal energy. Our results become more explicit under further assumptions: that the material is isotropic and that the Riemann problem is uniaxial. We assume that any umbilical points lie outside the region of physical relevance. Our main conclusion is that the Riemann solution can be obtained by the iterative solution of functional equations (Godunov iterations) each defined in one- or two-dimensional spaces.Supported in part by AFOSR-88-0025.     

正交各向异性材料热弹性问题的三维无网格法计算模型及应用

建立了正交各向异性材料热弹性问题的三维无网格伽辽金(Element Free Galerkin, EFG)法计算模型。利用该计算模型对三维复合材料汽轮机叶轮和轴承座进行了热弹性分析,对比了材料方向角及热导率因子、热膨胀系数因子和拉压弹性模量因子不同组合情况下轴承座的最大热变形总位移和当量应力值,讨论了材料方向角及上述正交各向异性因子对热变形和当量应力的影响规律,并与各向同性材料进行了对比。结果表明:三维EFG模型的热变形总位移和当量应力相对误差范数分别比有限元法小0.1215%和0.1359%;材料方向角同时影响热变形的大小和方向,但对当量应力方向影响不大;正交各向异性材料因子主要影响热变形和当量应力的大小。在考虑热-机械载荷作用下的三维复合材料零件结构设计中,当以刚度或强度为主要需求时,材料方向角、热导率因子、热膨胀系数因子、拉压弹性模量因子分别在(45°~60°,8:1:4~10:1:5,(1/6):(1/5):1~(1/5):(1/4):1,(7/5):1:(9/5)~(3/2):1:2)或(0°~10°,(1/10):1:(1/5)~(1/8):1:(1/4),(1/5):1:(1/6)~(1/4):1:(1/5),1:(1/5):(1/10)~1:(1/4):(1/8))范围内取值能有效降低轴承座等结构的热变形和当量应力。     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULA RPLATE（I）

ＴＨＥＰＲＯＢＬＥＭＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＵＮＳＹＭＭＥＴＲＩＣＡＬＢＥＮＤＩＮＧＦＯＲＣＹＬＩＮＤＲＩＣＡＬＬＹＯＲＴＨＯＴＲＯＰＩＣＣＩＲＣＵＬＡＲＰＬＡＴＥ（Ｉ）ＱｉｎＳｈｅｎｇ－Ｉｉ（秦圣立）ＨｕａｎｇＪｉａ－ｙｉｎ（黄家寅）（Ｑｕｆ...