On a counterexample to a conjecture of Saint Venant

The elliptic boundary value problem % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGHsi% slcqGHuoarcaWG1bGaeyypa0dccaGae8hiaaIaaGymaiab-bcaGiab% -bcaGiab-bcaGiaabMgacaqGUbGaaeiiaiabfM6axjaabYcaaeaaae% aacaWG1bGaeyypa0JaaGimaiab-bcaGiab-bcaGiab-bcaGiaab+ga% caqGUbGaaeiiaiabgkGi2kabfM6axjaabYcaaaaa!4E11!\[\begin{gathered}- \Delta u = 1 {\text{in }}\Omega {\text{,}} \hfill \\\hfill \\u = 0 {\text{on }}\partial \Omega {\text{,}} \hfill \\\end{gathered}\]is considered. The Saint Venant's conjecture for convex plane domains , having symmetry about two orthogonal axes, is that the maximum of |u| occurs only at the points on which are nearest to the origin. G. Sweers constructed one such domain and claimed that either the conjecture fails for or for ={(x, y);u(x, y) >}, which again is convex. We give a totally different proof of this claim. Our proof brings out clearly the reason for the failure of the conjecture and also allows us to construct many more such domains.     

HAMILTONIAN SYSTEM AND THE SAINT VENANT PROBLEM IN ELASTICITY

ＨＡＭＩＬＴＯＮＩＡＮＳＹＳＴＥＭＡＮＤＴＨＥＳＡＩＮＴＶＥＮＡＮＴＰＲＯＢＬＥＭＩＮＥＬＡＳＴＩＣＩＴＹＺｈｏｎｇＷａｎｘｉｅ（钟万勰）；ＸｕＸｉｎｓｈｅｎｇ（徐新生）；ＺｈａｎｇＨｏｎｇｗｕ（张洪武）（ＲｅｃｅｉｖｅｄＪｕｎｅ５，１９９５）Ａｂｓ...     

Existence of Pulsating Roll-Waves for the Saint Venant System

The phenomenon of roll-waves occurs when shallow water flows down open inclined channels. This flow is described by the Saint Venant’s equations with a friction term due to Chezy. In the case of a flat bottom, their existence (as entropic and periodic travelling waves) follows from a classical work due to DRESSLER [6]. The aim of this paper is to prove the existence of roll-waves when the bottom is modulated by a small periodic perturbation. Following JIN and KATSOULAKIS [15], we first compute a Burgers-type equation which possesses “pulsating” roll-waves (the wave speed oscillates around an average velocity). We prove, in a mathematically rigorous fashion, the existence of these solutions.     

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.     

A Nonlinear Korn Inequality Based on the Green-Saint Venant Strain Tensor

A nonlinear Korn inequality based on the Green-Saint Venant strain tensor is proved, whenever the displacement is in the Sobolev space \(W^{1,p}\), \(p\geq 2\), under Dirichlet conditions on a part of the boundary. The inequality can be useful in proving the coercivity of a nonlinear elastic energy.     

On the Flexure of a Saint-Venant Cylinder

The flexure of a Saint-Venant cylinder is defined from a kinematic point of view. A complete solution for the field equations is provided and the gauge choices to define the center of shear and rotation are made consequently. This seems to eliminate some ambiguities present in the literature. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Shear and Torsion Factors in the Theory of Linearly Elastic Rods

Lower bounds for the factors entering the standard notions of shear and torsion stiffness for a linearly elastic rod are established in a new and simple way. The proofs are based on the following criterion to identify the stiffness parameters entering rod theory: the rod’s stored-energy density per unit length expressed in terms of force and moment resultants should equal the stored-energy density per unit length expressed in terms of stress components of a Saint-Venant cylinder subject to either flexure or torsion, according to the case. It is shown that the shear factor is always greater than one, whatever the cross section, a fact that is customarily stated without proof in textbooks of structure mechanics; and that the torsion factor is also greater than one, except when the cross section is a circle or a circular annulus, a fact that is usually proved making use of Saint-Venant’s solution in terms of displacement components.     

含铺层拼接层合板的剪切强度

对含铺层拼接的碳纤维增强树脂基复合材料层合板进行了剪切强度实验研究.从三种不同铺层拼接角层合板上切取含缺口的剪切试件,通过实验测定了其载荷-位移曲线,得到了剪切强度值.实验结果表明,三组试件的剪切强度基本相同,即拼接层角度改变几乎不会引起层合板的剪切强度发生明显变化.采用有限元软件ABAQUS6.5对实验过程进行模拟,得到拼接层角度改变将引起拼接层中0°层出现切应力集中,但沿缺口切应力的平均值几乎不变.这也说明拼接层角度的变化几乎不影响层合板的剪切强度.     

On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E{{\mathcal E}} subject to the constraint I=?2m{{\mathcal I}=\sqrt{2}\mu}, where I{{\mathcal I}} is the wave momentum and 0 < m << 1{0 < \mu \ll 1} . Since E{{\mathcal E}} and I{{\mathcal I}} are both conserved quantities a standard argument asserts the stability of the set D _μ of minimisers: solutions starting near D _μ remain close to D _μ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schr?dinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mˉ 0{\mu \downarrow 0} .     

切应力对牛骨蠕变变形的影响研究

骨尤其是湿骨,在恒定载荷作用下会发生蠕变变形。为了确定切应力是否影响骨的蠕变变形,采用对骨薄板试样分别施加集中载荷和均布载荷的方式,测量试样挠度实时的变化曲线。结果显示,在载荷恒定时,骨试样的挠度随时间不断增加,体现了典型的蠕变特性。集中荷载下骨的蠕变变形远大于均布荷载下骨的蠕变变形,湿骨的蠕变位移比干骨高近7倍。分析认为,对试样粘弹性性质的影响不仅有正应力的作用,也有切应力的作用;切应力产生的蠕变变形约为正应力所产生蠕变变形的0.85倍。     

Study on the Influence of Surface and Geometric Factors on the Results of a Nondestructive Onsite Method to Assess Air Permeability

Experimental Techniques - While onsite measurement of air permeability provides a useful approach to assess the service life of concrete structures, the combined use of onsite and laboratory...     

弹性椭圆夹杂纵向剪切问题

获得纵向剪切下弹性椭圆夹杂问题的精确解。将复变函数的分区全纯函数理论，Cauchy型积分和Riemann边值问题相结合，求得各复势函数之间的解析关系，从而得到问题的封闭形式解，并给出了界面应力的解析表达式。本文解答与已有文献结果一致。本文发展的分析方法，为求解复杂多连通域的平面弹性问题提供了一条有效途径。     

On the Global Geometric Structure of the Dynamics of the Elastic Pendulum

We approach the planar elastic pendulum as a singular perturbation of the pendulum to show that its dynamics are governed by global two-dimensional invariant manifolds of motion. One of the manifolds is nonlinear and carries purely slow periodic oscillations. The other one, on the other hand, is linear and carries purely fast radial oscillations. For sufficiently small coupling between the angular and radial degrees of freedom, both manifolds are global and orbitally stable up to energy levels exceeding that of the unstable equilibrium of the system. For fixed value of coupling, the fast invariant manifold bifurcates transversely to create unstable radial oscillations exhibiting energy transfer. Poincaré sections of iso-energetic manifolds reveal that only motions on and near a separatrix emanating from the unstable region of the fast invariant manifold exhibit energy transfer.     

Influence of Surface Scratches on the Flexure Strength of Soda-Lime Silicate and Borosilicate Glass

Borosilicate (BS) and soda-lime silicate (SLS) glasses are being considered for use as the strike-face in transparent armor systems. This effort examined the effect of surface scratches generated with 1, 5, 10, and 30 N scratching loads on the equibiaxial flexure strength of both glasses including versions of each that had been thermally-tempered. At lower scratch loads there is more lateral cracking associated with the scratches in the BS while the scratches in the SLS predominantly exhibit plastic deformation. At the highest scratch load the damage is dominated by chips and lateral cracking within and along the scratch length. The tempered versions of the SLS appear to have less lateral cracking than the tempered version of the BS glass. In all instances the flexure strength decreases significantly when a 1 N scratch is introduced on the tin surface but when the scratch load increases any additional strength loss is minimal.     

基于应变等效性假说的损伤定义的适用条件

研究了基于应变等效性假说的损伤定义方法，分析了定义中“材料损伤前后弹性模量”的物理含义。指出该方法只适用于间接描述弹（脆）性材料损伤行为，不能简单地用受荷过程中的“卸载模量”代替假说中的“受损弹性模量”来描述具有不可逆塑性变形特征的弹塑性损伤行为，否则将可能导致对损伤行为的误判。此外，本文给出了一种耦合塑性形变和损伤机制影响的弹塑性损伤定义。     

On the Interaction of Turbulent Shear Layers with Harmonic Perturbations

The problem of coherent perturbations in a turbulent shear layer is considered for the purpose of developing a mathematical model based on a triple decomposition that extracts the coherent components of random fluctuations. The governing equations for the mean and the coherent parts of flow are derived, assuming the eddy-viscosity equivalence for the random part of flow, and solved by iterations to provide a coupled solution of the problem as a whole. Calculations agree well with experimental data in the upstream part of the layer where the mean–coherent flow interaction is the most important. In this region, the interaction changes the mean flow velocity distribution in such a manner that the neutral stability curve is shifted upstream relative to its position in the undisturbed layer and the perturbation intensity decreases further downstream. Experiments show that the coherent waves suppress the turbulent Reynolds stress production downstream of this region, but the model fails to predict the layer spreading correctly probably due to an inadequate turbulence closure of the mean flow. For the case of a turbulent mixing layer, we suggest a new closure relation that takes into account this coherent-random interaction.     

The Dependency of Shear Zone Length on the Shear Strength Profiles in Paperboard

In this work, the notched shear strength test (NST) has been further improved. In order to simplify and accelerate the testing procedure, the notches with declined slopes were used. With the proposed procedure, the shear strength profile in the thickness direction of a paperboard can be measured using one sheet only. By using the test setup, the dependency of shear zone length on shear strength was investigated. Experimental results show that both the measured shear strength values as well as the shear strength profile varied significantly with different shear zone length. Longer shear zone gave lower shear strength values and flatter profiles, while a shorter shear zone gave higher strength values and more pronounced shear strength profiles that better followed the paperboard ply structure. This proposed new method was also compared with the NST, strip shear test (SST) and rigid shear test (RST) method by using the same test material.