Energy-minimizing deformations of elastic sheets with bending stiffness

Necessary conditions for energy-minimizing deformations are derived for a theory of sheets in which the strain energy function depends on the second derivatives of the deformation as well as its first derivatives. All of these conditions are extensions of well-known necessary conditions in classical calculus of variations. The interpretation of some of these conditions as material stability conditions is explained.     

Material instabilities in inelastic saturated porous media under dynamic loadings

Instabilities in inelastic saturated porous media are investigated here for general three-dimensional states under dynamic loadings using a perturbation approach.Under quasi-static conditions, unbounded growth of perturbations is related to the emergence of stationary discontinuities under drained or undrained conditions, while under dynamic conditions, unbounded growth is related either to the emergence of stationary discontinuities (and these are set by drained conditions) or to the appearance of the flutter phenomenon (acceleration waves).For associative behaviour unbounded growth always corresponds to localization under drained conditions and the onset of growth of perturbations occurs here only through divergence growth. It is only for non-associative flow that unbounded growth may correspond to undrained localization in quasi-static conditions and to flutter under dynamic conditions.     

弹性薄板弯曲问题的一个分区横向联结的泛函

给出了一个弹性薄板弯曲问题分区横向联结的泛函。并且证明，它的驻值条件等价于全部支配方程、外边界条件、分区公共边界处的内力平衡条件和位移连续条件。     

Entrainment Boundary Conditions for Free Shear Flows

In this paper the performance of different free stream boundary conditions has been investigated. By performing two dimensional free jet calculations, a test case for which an analytical solution is available, it has been shown that boundary conditions in which the pressure is prescribed perform significantly better than boundary conditions in which the velocity is prescribed. Furthermore, it has been shown that pressure boundary conditions allow for much smaller computational domains than velocity (free-slip) boundary conditions.     

Approximate dynamic boundary conditions for a thin piezoelectric layer

Approximate dynamic boundary conditions of different orders are derived for the case of a thin piezoelectric coating layer bonded to an elastic material. The approximate boundary conditions are derived using series expansions of the elastic displacements and the electric potential in the thickness coordinate of the layer. All the expansion functions are then eliminated with the aid of the equations of motion and boundary/interface conditions of the layer. This results in boundary conditions on the elastic material that may be truncated to different orders in the thickness of the layer to obtain approximate boundary conditions. The approximate boundary conditions may be used as a replacement for the piezoelectric layer and thus simplify the analysis significantly. Numerical examples show that the approximate boundary conditions give good results for low frequencies and/or thin piezoelectric layers.     

非齐次边界条件下轴向运动梁的非线性振动

轴向运动系统的横向非线性振动一直是国内外研究的热点课题之一.目前相关研究大都是针对齐次边界条件的.但是在工程实际中,非齐次边界条件更为常见,而针对非齐次边界条件的研究相对较少.为深入研究非齐次边界条件对轴向运动系统横向非线性振动的影响,本文以轴向变速运动黏弹性Euler梁为例,引入由黏弹性引起的非齐次边界条件,同时还引入由轴向加速度引起的径向变化张力,建立梁横向振动的积分-偏微分型运动方程,并导出了相应的非齐次边界条件.采用直接多尺度法分析了梁的次谐波参数共振.由可解性条件得到了梁的稳态响应,并根据Routh-Hurvitz判据确定了系统稳态响应的稳定性.通过数值例子讨论了黏弹性系数,轴向运动速度,轴向速度脉动幅值和非线性系数对幅频响应的影响,并详细对比分析了非齐次边界条件和齐次边界条件对幅频响应的影响.结果表明:随着黏弹性系数的增大,非齐次边界条件下的零解失稳区域和稳态响应幅值比齐次边界条件下的失稳区域和幅值大,非齐次边界条件对高阶次谐波参数共振的影响更加显著.最后,引入微分求积法来验证直接多尺度法的近似解结果.      

On the existence of solutions of boundary value problems of duffing type systems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ [1 ] ,ｕｎｄｅｒａｇｒｏｕｐｏｆｖｅｒｙｅｘｔｅｎｓｉｖｅｃｏｎｄｉｔｉｏｎｓ,ＳＨＥＮＺｕ＿ｈｅｓｔｕｄｉｅｄｔｈｅｅｘｉｓｔｅｎｃｅｏｆａｕｎｉｑｕｅ 2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｏｆｔｈｅｓｙｓｔｅｍｏｆｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｕ″(ｔ) Ｇ(ｕ(ｔ) ) =ｐ(ｔ) ,(1 )ｗｈｅｒｅＧ :Ｒｎ →Ｒｈａｓａｃｏｎｔｉｎｕｏｕｓｓｅｃｏｎｄｐａｒｔｉａｌｄｅｒｉｖａｔｉｖｅｓ,ａｎｄｐ:Ｒ→Ｒｎｉｓｃｏｎｔｉｎｕｏｕｓａｎｄ2π＿ｐｅｒ…     

Correct Solutions of Plane Elastic Problems for a Half-Plane

Continuous and integrable solutions and one-to-one relationships between boundary forces and displacements are found through the direct integration of the differential equations of the plane elastic problem for a half-plane with boundary conditions for either forces or displacements or with mixed boundary conditions. The necessary equilibrium conditions for forces and the compatibility conditions for displacements that ensure the correctness of the solutions are formulated     

Some new conditions for generalized H-matrices

By using a continuous transition method of a matrix and the estimate for spectral radius of a sub-matrix etc.,decision methods for a generalized H-matrix under positive definite matrix conditions are researched.Some new sufficient conditions for generalized H-matrices are obtained.When a block matrix degenerates a point matrix, these conditions namely become sufficient conditions of H-matrix.     

Gradient nanoscale polycrystalline elasticity: Intergrain interactions and triple-junction conditions

Using the principle of virtual power, we develop general balance equations, interface conditions, triple-junction conditions, and boundary conditions for second-grade nanocrystalline elastic materials undergoing infinitesimal deformations. We further develop thermodynamically consistent constitutive equations and provide a weak formulation of resulting boundary-value problems that automatically yields internal conditions such those that hold across interfaces and at triple junctions.     

Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

Numerical simulations of crystal defects are necessarily restricted to finite computational domains, supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment. This work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed. We formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers. Using this foundation we then present rigorous error estimates for (i) a truncation method (Dirichlet boundary conditions), (ii) periodic boundary conditions, (iii) boundary conditions from linear elasticity, and (iv) boundary conditions from nonlinear elasticity. Numerical results confirm the sharpness of the analysis.     

A new outflow boundary condition

Boundary conditions come from Nature. Therefore these conditions exist at natural boundaries. Often, owing to limitations in computing power and means, large domains are truncated and confined between artificial synthetic boundaries. Then the required boundary conditions there cannot be provided naturally and there is a need to fabricate them by intuition, experience, asymptotic behaviour and numerical experimentation. In this work several kinds of outflow boundary conditions, including essential, natural and free boundar conditions, are evaluated for two flow and heat transfer model problems. A new outflow boundary condition, called hereafter the free boundary condition, is introduced and tested. This free boundary condition is equivalent to extending the validity of the weak form of the governing equations to the synthetic outflow instead of replacing them there with unknown essential or natural boundary conditions. In the limit of zero Reynolds number the free boundary condition minimizes the energy functional among all possible choices of outflow boundary conditions. A review of results from applications of the same boundary conditions to several other flow situations is also presented and discussed.     

Projection conditions on the vorticity in viscous incompressible flows

The problem of establishing appropriate conditions for the vorticity transport equation is considered. It is shown that, in viscous incompressible flows, the boundary conditions on the velocity imply conditions of an integral type on the vorticity. These conditions determine a projection of the vorticity field on the linear manifold of the harmonic vector fields. Some computational consequences of the above result in two-dimensional calculations by means of the nonprimitive variables, stream function and vorticity, are examined. As an example of the application of the discrete analogue of the projection conditions, numerical solutions of the driven cavity problem are reported.     

Tractions, Balances, and Boundary Conditions for Nonsimple Materials with Application to Liquid Flow at Small-Length Scales

Using a nonstandard version of the principle of virtual power, we develop general balance equations and boundary conditions for second-grade materials. Our results apply to both solids and fluids as they are independent of constitutive equations. As an application of our results, we discuss flows of incompressible fluids at small-length scales. In addition to giving a generalization of the Navier–Stokes equations involving higher-order spatial derivatives, our theory provides conditions on free and fixed boundaries. The free boundary conditions involve the curvature of the free surface; among the conditions for a fixed boundary are generalized adherence and slip conditions, each of which involves a material length scale. We reconsider the classical problem of plane Poiseuille flow for generalized adherence and slip conditions.     

Stability criteria for a class of fractional order systems

This paper deals with the stability problem in LTI fractional order systems having fractional orders between 1 and 1.5. Some sufficient algebraic conditions to guarantee the BIBO stability in such systems are obtained. The obtained conditions directly depend on the coefficients of the system equations, and consequently using them is easier than the use of conditions constructed based on solving the characteristic equation of the system. Some illustrations are presented to show the applicability of the obtained conditions. For example, it is shown that these conditions may be useful in stabilization of unstable fractional order systems or in taming fractional order chaotic systems.     

Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells

We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching and self-intersecting shells. The conditions are derived by performing direct through-the-thickness integration in the global equilibrium conditions of continuum mechanics. At each regular internal and boundary point of the base surface our exact, local equilibrium equations and dynamic boundary conditions are equivalent, as expected, to the ones known in the literature. As the new equilibrium relations we derive the exact, resultant dynamic continuity conditions along the singular surface curve modelling the branching and self-intersection as well as the dynamic conditions at singular points of the surface boundary. All the results do not depend on the size of shell thicknesses, internal through-the-thickness shell structure, material properties, and are valid for an arbitrary deformation of the shell material elements.     

非均质Cosserat连续体细-宏观均匀化条件

基于平均场理论的多尺度模拟关键问题之一是给定恰当的表征元(RVE)边界条件,以使均匀化过程满足Hill-Mandel细宏观能量等价条件,也即Hill宏观均匀化条件。对于非均质Cosserat连续体,已有的研究工作只能得到合理的混合平动位移-偶应力表征元边界条件,常用的一致平动位移-转角以及周期边界条件等均不能使用,给计算均匀化算法推导和实施带来了困难,也阻碍了多尺度分析方法的进一步发展与应用。为此,本文在推导和建立一个新的Hill定理版本基础上,不仅成功地给定了多种强形式表征元边界条件,而且构造出了合理的弱形式周期边界条件,这些条件既满足细宏观能量等价也符合一阶平均场理论基本假定,可在均匀化方法中推广与应用。     

Buckling analysis of a laminated cylindrical shellunder torsion subjected to mixed boundary conditions

A new efficient method is developed in this paper for buckling analysis of a cross-plylaminated cylindrical shell under torsion subjected to mixed boundary conditions. The transverseshear is taken into account by a first-order theory with a shear correction factor of 5/6. The mixedboundary conditions include conditions in forces as well as conditions in displacements, and theseforces and displacements are selected as basic unknowns. The other displacements and forces areexpressed in terms of the basic unknowns by taking inverse of a matrix composed of operators.The equations of buckled equilibrium in terms of the basic unknowns are solved with doubletrigonometric series which satisfy the mixed boundary conditions. Comparison of the obtainednumerical results with those given in the literature based on completely clamped boundaryconditions checks with the fact that the mixed boundary conditions yield appreciably lowerbuckling load and less circumferential wave number than the completely clamped boundaryconditions. The curves in the figures show how the difference in buckling loads between the twokinds of boundary conditions varies when the length and thickness of the shell vary.     

Effective downstream boundary conditions for incompressible Navier–Stokes equations

The aim of this paper is to give open boundary conditions for the incompressible Navier–Stokes equations. From a weak formulation in velocity–pressure variables, some natural boundary conditions involving the traction or pseudotraction and inertial terms are established. Numerical experiments on the flow behind a cylinder show the efficiency of these conditions, which convey properly the vortices downstream. Comparisons with other boundary conditions for the velocity and pressure are also performed.     

波传播计算中的人工边界条件

对无限介质中波传播进行有界区域近似计算时,需要提出人工边界条件.本文分别就连续介质和离散晶格简要介绍一些典型的人工边界条件,包括精确人工边界条件和局部人工边界条件.