On lower-bound estimates of critical loads for cylindrical shells

The paper proposes a new approach to estimate the lower bounds of critical loads for circular cylindrical shells. These bounds are compared with the ordinary lower bound of critical load under which a shell with initial deflections loses stability. The lower bound produced by the approach is higher than the ordinary bound and can be used in design __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 78–85, October 2006.     

A lower bound estimate of the critical load for compressible elastic solids

We determine lower bound estimates for the critical load for hyperelastic solids under monotonic dead load processes. By considering the Hadamard criterion of infinitesimal stability, we first determine a lower bound for the Hadamard stability functional; then, we develop a procedure for optimal lower bound estimates for the critical load. As examples, we apply our procedure to generalized Blatz-Ko solids under simple extension, simple compression and rectilinear shear, and compare our results with other proposals contained in the literature.     

数学规划加权残值法分析薄板与薄壳的几何非线性问题

使用数学规划加权残值涯分析了薄板大挠度及薄壳的非线性稳定问题。在两者应变协调方程严格满足的情况下,论证并利用平衡方程的单调性,建立数学规划问题,首先得到无布载荷四边简支方板中心挠度的最小上界及最大下界,经典Ｌｅｖｙ解位基其间。     

On the deformation and fracture of porous materials

The constitutive behavior of porous materials (including the yield loci, the void growth rate, the macro stress-strain relation and the strain to localization instability) is examined based on the lower bound approach proposed by the present authors. These results are then compared with the experimental and the finite element results as well as those predicted by Gurson's equations. Emphasis is placed on approaching the real behavior from the upper and the lower bound analysis. Calculation is also made on the influence of void nucleation on the critical strain to instability and a modified strain-controlled nucleation criterion is proposed. Finally the instability and fracture of AISI4340 steel in plane strain tension is examined and comparison is made between theoretical and experimental results.     

Critical wave lengths and instabilities in gradient-enriched continuum theories

Continuum material models can be enriched with additional gradients in order to model phenomena that are driven by processes at lower levels of observation. For a systematic comparison of various gradient-enriched continua, dispersion analysis may be used. In this contribution, we will explore the occurence of critical wave lengths and its implications for the material stability. In particular, we will present a unifying theorem that permits to assess the stability of elastic, hardening and softening gradient-enriched continua by means of a critical wave length analysis, whereby the upper or lower bound nature of the critical wave length indicates whether the model is stable or unstable.     

Explicit and implicit gradient series in damage mechanics

Gradient enhancement series are studied in the context of damage mechanics. Distinction is made between so-called explicit series and implicit series, both of which can be derived from a nonlocal damage model. The paper focuses on the difference between second-order and fourth-order truncations for either series. Dispersion analysis and numerical simulations are used to compare the various models. It is shown that for the explicit series the fourth-order term has a detrimental influence on the response, while for the implicit series the fourth-order term leads to a slightly closer approximation of the nonlocal model. The role of the critical wave length as it emerges from the dispersion analysis is shown to be decisive. When the critical wave length acts as an upper bound, a stable response is obtained and the critical wave length equals the width of the damaging zone. On the other hand, when the critical wave length acts as a lower bound, oscillations may appear of which the periodicity is set by this critical wave length.     

Limit cycles in a sextic Lyapunov system

Employing the inverse integral factor method, the first 13 quasi-Lyapunov constants for the three-order nilpotent critical point of a sextic Lyapunov system are deduced with the help of MATHEMATICS. Furthermore, sufficient and necessary center conditions are obtained, and there are 13 small amplitude limit cycles, which could be bifurcated from the three-order nilpotent critical point. Henceforth, we give a lower bound of limit cycles, which could be bifurcated from the three-order nilpotent critical point of sextic Lyapunov systems. At last, an example is given to show that there exists a sextic system, which has 13 limit cycles.     

Buckling load of non-linear systems with multiple eigenvalues

For structural systems with a coincident lowest eigenvalue λ_c, the influence of imperfections on the buckling of the systems depends to a very large extent upon the distribution of the imperfections. Moreover, the system may buckle either at a limit point or at a bifurcation point before this limit point is reached. Considering both possibilities, a lower bound to the buckling load of the system, for a given root mean square of the imperfections, is obtained. Furthermore, with reference to a set of particular, normalized co-ordinates, it was found that the absolute minimum buckling load is given by an imperfection vector parallel to the steepest of all post-buckling paths intersecting at λ_c. At this absolute minimum buckling load the critical point is a limit point. As an example, the lower bound to the buckling load of an imperfect cylindrical shell under axial compression was calculated.     

Discontinuous energy minimizers in nonlinear elastostatics: an example of J. Ball revisited

Simple direct methods of the Calculus of Variations, together with milda priori restrictions of a constitutive nature, are exploited to show that a uniform radial displacement at the boundary fails to induce a homogeneous radial expansion of a compressible elastic ball when the boundary displacement reaches a critical value (of which an explicit lower bound is offered): rather, above the critical value, energy minimizers are radial deformations with a hole at the center, and are accompanied by a stress field with radial stress vanishing, and circumferential stress unbounded, at the surface of the hole.     

Bifurcation of limit cycles in a quintic system with ten parameters

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of the computer algebra system MATHEMATICA, the first 11 quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 11 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth, we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems. The results of Jiang et al. (Int. J. Bifurcation Chaos 19:2107–2113, 2009) are improved.     

On normal modes in classical Hamiltonian systems

Normal modes of Hamittonian systems that are even and of classical type are characterized as the critical points of a normalized kinetic energy functional on level sets of the potential energy functional. With the aid of this constrained variational formulation the existence of at least one family of normal modes is proved and, for a restricted class of potentials, bifurcation of modes is investigated. Furthermore, a conjecture about a lower bound for the number of normal modes in case the potential is homogeneous, is proved.     

板材成型中的分叉和断裂及其空洞扩展效应

用平面应力有限元方法分析空洞模型以模拟一种双相钢板材在成型过程中所遇到的微空洞损伤,经试算可使模型的总体和局部的响应与已有的实验相一致,由此可提供描述该材料的损伤本构参数并研究局部剪切带和扩散型颈缩等分叉现象,临界应变值的分布形成了成型极限图中的下限曲线,当空洞模型的总体应力急剧下降或微裂纹开始出现,其相应的总体应变值提供了上限曲线。     

微孔洞生长对板材拉伸过程中变形局部化的影响

本文在作者提出的含孔洞材料下限本构方程的基础上，采用了初始缺陷带模型对微孔洞生长及分布对板材拉伸过程中变形局部影响进行了，分析着重研究了细观损演化规律对变形局部化模式及临界应变的影响，并成功预测了ＡＩＳＩ４３４０钢板材拉伸试件变形局部化换稳为及失稳方向。     

Morphological stability of epitaxial thin elastic films by van der Waals force

Summary  The morphological stability of epitaxial thin elastic films on a substrate by van der Waals force is discussed. It is found that only van der Waals force with negative Hamaker constant tends to stabilize the film, and the lower bound for the Hamaker constant is also obtained for the stability of thin film. The critical value of the undulation wavelength is found to be a function of both film thickness and external stress. The charateristic time-scale for surface mass diffusion scales to the fourth power to the wavelength of the perturbation. Received 4 December 2000; accepted for publication 31 July 2001     

Limit states for brick masonry based on homogenization approach

In this paper, a procedure is developed for assessing the strength of brick masonry based on homogenization theory. The approach invokes a lower bound analysis whereby plastically admissible stress fields are constructed in the constituents involved, subject to periodic boundary conditions and static equilibrium requirements. The critical load is obtained by solving a constrained optimization problem. The analysis employs a set of specific loading histories such as axial tension, pure shear and biaxial tension–compression at different orientations of bed joints. The performance of this approach is verified against numerical solutions based on finite element analysis. In the second part of this paper, a methodology is outlined for identification of a macroscopic failure criterion that incorporates a critical plane approach. A quantitative verification of this criterion is carried out for different loading conditions and the results are compared with the experimental data available in the literature.     

Dynamics of martensitic phase boundaries: discreteness, dissipation and inertia

We study a fully inertial model of a martensitic phase transition in a one-dimensional crystal lattice with long-range interactions. The model allows one to represent a broad range of dynamic regimes, from underdamped to overdamped. We systematically compare the discrete model with its various continuum counterparts including elastic, viscoelastic and viscosity-capillarity models. Each of these models generates a particular kinetic relation which links the driving force with the phase boundary velocity. We find that the viscoelastic model provides an upper bound for the critical driving force predicted by the discrete model, while the viscosity-capillarity model delivers a lower bound. We show that at near-sonic velocities, where inertia dominates dispersion, both discrete and continuum models behave qualitatively similarly. At small velocities, and in particular near the depinning threshold, the discreteness prevails and predictions of the continuum models cannot be trusted.      

Vortex Rings in Fast Rotating Bose–Einstein Condensates

When Bose–Einstein condensates are rotated sufficiently fast, a giant vortex phase appears, that is, the condensate becomes annular with no vortices in the bulk but a macroscopic phase circulation around the central hole. In a former paper (Correggi et al. in Commun Math Phys 303:451–308, 2011) we have studied this phenomenon by minimizing the two-dimensional Gross–Pitaevskii (GP) energy on the unit disc. In particular, we computed an upper bound to the critical speed for the transition to the giant vortex phase. In this paper we confirm that this upper bound is optimal by proving that if the rotation speed is taken slightly below the threshold, there are vortices in the condensate. We prove that they gather along a particular circle on which they are uniformly distributed. This is done by providing new upper and lower bounds to the GP energy.     

Reduced-Stiffness Method in the Theory of Shells

The fundamentals of the reduced-stiffness method, which is used in buckling analysis of reinforced and perfect and imperfect nonreinforced shells, are set out. The method is validated analytically and experimentally. The lower bound determined by this method is very close to the experimental lower bound. Some aspects of the current state and prospects for development and generalization of the method are discussed     

On bounds for the torsional stiffness of shafts of varying circular cross section

We state upper and lower bound formulas for the torsional stiffness of shafts of varying circular cross section, in accordance with the classical Michell formulation of this problem, through use of the principles of minimum potential and complementary energy. The general results are used to obtain explicit first-approximation bounds which, for the limiting case of the cylindrical shaft, reproduce the known elementary exact results. It is conjectured that the first-approximation lower bound is significantly closer to the exact result than the first-approximation upper bound.A report on work supported by the Office of Naval Research, Washington, D.C.     

Creep life predictions for structures subject to variable loading

Previous work which established upper and lower bounds on the creep life of steadily loaded structures is extended to cater for load and temperature variations in non-homogeneous structures. The investigation is limited to the range where short term plasticity and fatigue damage can be ignored. For proportional loading, the upper bound which is based on limit analysis, is similar in form to that for constant loading. In the more general case, the upper bound is less stringent and is based on the mean load and temperature distribution over the lifetime. A lower bound on life is taken as the time for the first part of the structure to fail.The bounds are applied to three simple structures. For proportional loading the upper bound predicts the lifetime with the same accuracy as for constant loading except for extreme load variations. The presence of a temperature distribution alters the accuracy of the upper bound prediction but in most cases the change is small. In contrast, the lower bound is very sensitive to the temperature gradient.The authors use these results to develop approximate techniques for estimating the creep life of components subjected to variable loads and temperature distributions. Simplified design procedures based on the upper bound are examined and suitable amendments are proposed.