A Simple Nonlinear Thermodynamic Theory of Arbitrary Elastic Beams

A ‘classical’ theory of beams (i.e., a theory in which the basic kinetic variables are a stress resultant and a stress couple) undergoing elastic, thermodynamic processes is developed by first deriving exact beamlike (one-dimensional) equations of motion and a beamlike Second Law (Clausius–Duhem inequality) by descent from three-dimensions. Then what may be considered as the three basic assumptions of a classical theory are introduced: an assumed form of the First Law (conservation of energy), a relaxed form of the Second Law, and a general form of the constitutive relations. Throughout, detailed specification of geometry, kinematics, or constitution is minimized. It is shown how the kinematic Kirchhoff hypothesis may be avoided by first introducing a mixed-energy density and then imposing a logically more satisfying constitutive Kirchhoff hypothesis. Mathematics Subject Classifications (2000) 74A15, 74B20, 74K10     

Unveiling the Mechanism Behind the Asymmetric Bending Compliance of Thin-Walled U-Shaped Strips:A Study Inspired by Plant Leaves

Inspired by the shape of some plant leaves,we find that the thin-walled U-shaped strips exhibit different compliances under bending with opposite orientations.The asymmetric bending compliance is attributed to the buckling of sidewalls of strips caused by the bending-induced compression.Integrating the Euler-Bernoulli beam theory with the Kirchhoff-Love thin plate theory,a theoretical model is derived for the in-depth understanding of the sidewall buckling.For pure bending,the critical moment applied to the strip for the sidewall buckling is found to be insensitive to the height,width and length of strip,which is the result of the compromise between the opposite geometric effects on the buckling behavior of sidewalls and the characteristics of cross sections.Then the critical moment can be approximated as a linear function of flexural rigidity D=Et3/12(1-v2),where t is the wall thickness of strip,E is Young's modulus,and v is Poisson's ratio.These predictions by our model agree well with the results obtained by finite element analysis.We also investigate the buckling behavior of sidewalls for bending under transverse loads,considering the loading conditions of concentrated force and distributed force.Our study unveils the mechanism behind the asymmetric bending compliance of thin-walled U-shaped strips.These results would offer convenient guidance for the promising engineering applications related to this structure,such as the design of soft robots with enhanced locomotion performance.     

Essay on the Contributors to the Elastic Stability Theory

This paper revisits the historical aspects of the contributions made into the elastic stability theory from the structural engineering point of view.Extensive quotations are brought in order to substantiate the claims made. It is hoped that the review constitutes a contribution to the identification of the main ideas pertinent to engineering mechanics.     

轴压下各向异性开口薄壁杆的非线性共振

本文推导了轴压下各向异性薄壁直杆的非线性动力学方程组.在推导中计及了杆壁中面的剪切变形,以及开口薄壁杆中剪应力沿厚度变化而造成的Saint-Venant扭矩,但略去了杆壁的弯曲.从这一方程组出发,用渐近方法计算了简谐变化轴向压力作用下薄壁杆共振时幅频曲线,且对各种影响因素,包括Saint-Venant扭矩的影响,作了分析.     

Mixed Formulations of Bending Problems for Homogeneous Elastic Plates and Beams

Mixed formulations of bending problems for homogeneous plates (beams) are proposed, whose essence is that the deformation of a plate (beam) near its fixed boundary is described by the threedimensional elasticity equations, and the remaining part by the conventional equations of plate (beam) bending. At the interface between these regions, the solutions of these equations are joined. The mixed formulation allows one to describe the threedimensional stress state in the neighborhood of the fixed boundaries of plates (beams) and take into account the complex nature of the fixing conditions. Finiteelement implementation is more efficient for the mixed formulations of plate (beam) bending problems than for the wellknown threedimensional formulations.     

Thin-walled Beams: A Derivation of Vlassov Theory via Γ-Convergence

This paper deals with the asymptotic analysis of the three-dimensional problem for a linearly elastic cantilever having an open cross-section which is the union of rectangles with sides of order ε and ε ², as ε goes to zero. Under suitable assumptions on the given loads and for homogeneous and isotropic material, we show that the three-dimensional problem Γ-converges to the classical one-dimensional Vlassov model for thin-walled beams.      

Stability of Singularly Perturbed Systems with Structural Perturbations. General Theory

Singularly perturbed systems with structural perturbations are analyzed for stability. New sufficient conditions of asymptotic stability and uniform asymptotic stability are established. The systems in question are widely used in control theory, engineering, etc     

弹性体表面分层压缩失稳问题的数学弹性力学解

采用数学弹性力学的稳定平衡方程并结合富氏积分变换的方法研究了含表面平行裂纹的弹性体在压缩载荷下的表面分层失稳问题。导出了一级显式的精确齐次奇异积分方程组,然后.通过Gauss-Chebyshev积分公式,得到一组齐次代数方程组,从而求出临界压缩载荷。并将结果与经典的材料力学梁板稳定的研究方法所得结果进行了比较,指出经典方法误差太大而不适于求解此问题。最后,利用数学弹性力学解求出的等效弹性支承常数给出一个简单精确的临界压缩载荷计算公式。     

Influence of Geometrical Parameters on the Critical Load in Three-Dimensional Stability Problems for Rectangular Plates and Beams

Two- and three-dimensional stability problems for a rectangular plate under uniaxial compression are solved by the mesh method. It is shown that the difference between the critical loads obtained in the three- and two-dimensional problems is not greater than 5% for arbitrary fixation conditions and arbitrary length and thickness ratio of the plate     

弹性支承滑动轴承－转子系统稳定性的研究

本文研究弹性支承滑动轴承－转子系统的动力稳定性问题，建立了弹性支承滑动轴承－Ｊｅｆｆｃｏｔｔ转子系统的力学模型，采用牛顿延拓法从理论上对该转子轴承系统的动力稳定性进行了分析，讨论弹性支承的刚度对系统稳定性的影响，从而提出提高系统稳定性的有关措施。最后，依据理论分析，利用Ｊｅｆｆｃｏｔｔ转于实验台，作者设计和加工了一副弹性支承器，通过刚性支承和弹性支承两个对比实验，对理论分析的结果进行了验证。实验结果表明，弹性支承能提高转子动力稳定性。     

连续矩形簿板弯曲和稳定的新法则

本文推广文献[1、2]结果,对变刚度连续矩形薄板弯曲和稳定计算提出了一个新法则,算例表明此法简明有效。     

多阶变截面杆的弹性稳定计算

本文研究单阶和多阶截面杆在集中和分布轴向荷载作用下的弹性稳定计算方法,首次给出了其整体稳定和局部稳定的解析解.     

Integration of an Equilibrium System in an Enhanced Theory of Bending of Elastic Plates

The complete integral of the system of partial differential equations governing the equilibrium bending of elastic plates with transverse shear deformation and transverse normal strain is constructed by means of complex variable methods. The process helps to elucidate the physical meaning of certain analytic constraints imposed on the asymptotic behavior of the solutions and shows that in the case of an infinite plate, any analytic solution has finite energy if and only if the bending and twisting moments, the transverse shear force, the displacements in vertical planes, and two other characteristic quantities vanish at infinity. An example is discussed to illustrate the theory.     

A Note on the Theory of Stability of Impulsive Systems with Structural Perturbations

New conditions of stability in two measures are established for nonlinear impulsive systems with structural perturbations. A list of two measures used in the nonlinear dynamics of pulse systems is presented     

非理想压杆稳定分析的样条函数法

采用样条函数构造的位移模式，出了非理想压杆稳定极限承载力及稳定平衡全过程的样条函数分析法。本方法可以考虑任意实际初始缺陷和材料弹塑性加卸载效应的影响。计算表明，本方法精度高，速度快。