On the validation of theories of thin elastic structures

This writing is meant to restitute the contents of a lecture I gave in Torino, on September 19, 2013, within the framework of the 21st AIMETA Congress. For this reason, I have made an effort to use an informal and colloquial style, even when I added some material that I did not use in Torino. Just as in my Torino lecture, there are four parts. In the introductory Sect. 1, I explain my title and justify why I think it worth to discuss the validation issue first in general (Sect. 2) and then in the context of computational mechanics (Sect. 3). In the last and longer Sect. 4, I first discuss dimension-reduction methods, mechanical and analytical, then I focus on variational validation, by the method of $\Upgamma$ -convergence, of the classic theories of linearly elastic plates.     

On fracture toughness of orthotropic metallic materials under cyclic loading

AVT-6 titanium alloy rolled sheet, which is initially isotropic and subject to kinematic hardening, is used as an example to analyze the effect of kinematic hardening on crack growth resistance under uniaxial cyclic loading. Crack growth resistance is characterized by the number of cycles to failure, critical crack length, and critical stress intensity factor. The subject of study is plane specimens with an edge notch. It is shown that prestrain changes the anisotropy of the specimens, which is determined as the ratio of the crack growth resistance in the rolling direction to that in the transverse direction. The crack path under a load applied at an angle to the axes of anisotropy is studied     

Thermomechanical cyclic behavior modeling of Cu-Al-Be SMA materials and structures

This paper concerns the behavior of Cu-Al-Be polycrystalline shape memory alloys under cyclic thermomechanical loadings. Sometimes, as shown by many experimental observations, a permanent inelastic strain occurs and increases with the number of cycles. A series of cyclic thermomechanical tests has been carried out and the origin of the residual strain has been identified as residual martensite. These observations have been used to develop a 3D macroscopic model for the superelasticity and stress assisted memory effect of SMAs able to describe the evolution of permanent inelastic strain during cycles. The model has been implemented in a finite elements code and used to simulate the behavior of antagonistic actuators based on SMA springs under cyclic thermomechanical loading with a residual displacement appearance.     

On a theory concerning the dynamical behavior of structures carrying moving masses

Summary Two new methods are presented in order to determine the behavior of a structure carrying moving masses. The first method is analytic in nature and represents a modified asymptotic method in the theory of nonlinear phenomena. The second method is an exact numerical technique general enough to be used for solving exactly a set of differential equations with singular coefficients.The results show that the analytical method is in excellent agreement with the exact one obtained by means of a numerical technique. Furthermore, it is shown that the effect of the response of the structure to the moving mass has to be properly considered.

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Übersicht Es werden zwei neue Methoden zum Bestimmen des Verhaltens von Strukturen mit bewegten Massen behandelt. Die erste Methode trägt analytischen Charakter und kann als eine Modifikation der asymptotischen Methode der Theorie nichtlinearer Schwingungen betrachtet werden. Die zweite Methode basiert auf einer exakten numerischen Integration. Sie ist allgemein genug, um das beschreibende System gewöhnlicher Differentialgleichungen exakt zu lösen. Ein Vergleich der Ergebnisse zeigt, daß ausgezeichnete Übereinstimmung erreicht wird. Die Auswertungen zeigen, daß die Einwirkung des Struktur-Verhaltens auf die bewegte Massc berücksichtigt werden muß.

     

On the elastoplastic cyclic analysis of plane beam structures using a flexibility-based finite element approach

This paper presents the extension of a flexibility-based large increment method (LIM) for the case of cyclic loading. In the last few years, LIM has been successfully tested for solving a range of non-linear structural problems involving elastoplastic material models under monotonic loading. In these analyses, the force-based LIM algorithm provided robust solutions and significant computational savings compared to the displacement-based finite element approach by using fewer elements and integration points. Although in cyclic analysis a step-by-step solution procedure has to be adopted to account for the plastic history, LIM will still have many advantages over the traditional finite element method. Before going into the basic idea of this extension, a brief discussion regarding LIM governing equations is presented followed by the proposed solution procedure. Next, the formulation is specified for the treatment of the elastic perfectly plastic beam element. The local stage for the beam behavior is discussed in detail and the required improvement for the LIM methodology is described. Illustrative truss and beam examples are presented for different non-linear material models. The results are compared with those obtained from a standard displacement method and again highlight the potential benefits of the proposed flexibility-based approach.     

Dynamical analysis of multi-stage cyclic structures

This paper deals with the dynamical analysis of a multi-stage assembly of cyclic structures such as, for example, turbomachinery compressor or turbines. If such assemblies are traditionally modelled stage by stage, the inter-stage coupling effect can sometime be important. As an answer to this issue, we propose a new method which combines a cyclic modelling of each stage with a realistic inter-stage coupling. Study cases are presented to evaluate the efficiency of the method.     

An optimal control approach to the analysis of inelastic structures under cyclic loading

Optimal control theory is used to study the asymptotic behaviour of elasto-viscoplastic structures under cyclic loading. With this approach, the asymptotic state is found as the solution of a minimization problem. General properties of this method are established. A simple thermomechanical problem is studied to illustrate and validate this approach. An interest of the proposed method lies in its capacity to handle other nonlinearities than plasticity. To illustrate this point, the approach is extended to the coupled viscoplasticity/frictionless contact problem. Some numerical results are given for an elasto-viscoplastic half-plane under cyclic frictionless indentation.     

Reconcile the perfectly elastoplastic model to simulate the cyclic behavior and ratcheting

Perfectly elastoplastic constitutive model is modified through a smoothing factor introduced by Liu [Liu, C.-S., 2003. Smoothing elastoplastic stress–strain curves obtained by a critical modification of conventional models. Int. J. Solids Struct. 40, 2121–2145]. The new model allows plasticity to happen in a non-zero-measure yield volume in stress space, rather than that of conventional zero-measure yield surface, and within the yield volume the plastic modulus is varying continuously. It endows a specific strain-hardening rule of flow stress and is able to describe the phenomena of strain hardening, cyclic hardening, the Bauschinger effect, mean-stress relaxation, strain ratcheting, out-of-phase hardening, as well as erasure-of-memory. In order to suppress the over prediction of ratcheting we consider a scalar function of smoothing factor, which can simulate the saturation behavior of uniaxial/multiaxial strain ratcheting. These effects are demonstrated through numerical examples. The existence of stress equilibrium point and limiting surface is a natural result without requiring an extra design. Moreover, the non-linear constitutive equations can be converted into a linear system for augmented stress in the Minkowski space, of which the symmetry group is a proper orthochronous Lorentz group SO_o(5, 1). The augmented stress is a time-like vector, moving on hyperboloids inside the cone. When taking the Prager kinematic hardening rule into account we can simulate some cyclic behaviors of SAE 4340 and grade 60 steels within a certain accuracy through the use of only three material constants and a fixed smoothing factor. To simulate the ratcheting behaviors of SS304 stainless steel we allow the smoothing factor to be an exponential decaying function of λ.     

On a new theory of the dynamic behavior of the structures carrying moving masses

Summary This paper represents an exact solution for a problem which has been of interest in Structural Engineering for many years. The solution is exact in the engineering sense; it converges very rapidly. The solution is given by means of an expansion of the eigenfunctions in a series. The eigenfunctions satisfying the boundary and transient conditions for masses moving over a structure have been determined exactly by performing the integration in the Stieltjes sense. The method is simple, short and elegant and is based on operational calculus. It yields the results in perfect agreement with an accurate numerical solution previously given in the literature.

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Über eine neue Theorie zum dynamischen Verhalten von Strukturen unter bewegten Massen

Übersicht In dieser Arbeit wird eine exakte Lösung für das Verhalten von Strukturen unter bewegten Massen gegeben. Die Lösung wird durch eine Reihenentwicklung in Eigenfunktionen ermittelt, wobei die Eigenfunktionen mit Hilfe von Stieltjes Integralen exakt bestimmt werden. Die Reihen konvergieren schnell; die Ergebnisse stimmen ausgezeichnet mit bekannten numerischen Lösungen überein.

     

反分析法在光泡沫金属材料动态性能实验中的应用

在实验测试泡沫金属材料的动态性能时,由于其所具有的特殊性能使得传统的SHPB技术的采用遇到较大的困难.为了实验确定泡沫金属材料的初始动力坍塌强度和"平台"应力,研究其应变率效应,在现有SHPB实验装置的基础上,利用反分析法中的反卷积技术,通过计算机模拟给出了该实验装置的传递函数,完善了SHPB实验的数据处理系统,为实验研究泡沫材料的动态特性提供了一种有效的方法.     

反分析法在泡沫金属材料动态性能实验中的应用

在实验测试泡沫金属材料的动态性能时，由于其所具有的特殊性能使得传统的SHPB技术的采用遇到较大的困难。为了实验确定泡沫金属材料的初始动力坍塌强度和平台应力，研究其应变率效应，在现有SHPB实验装置的基础上,利用反分析法中的反卷积技术，通过计算机模拟给出了该实验装置的传递函数，完善了SHPB实验的数据处理系统，为实验研究泡沫材料的动态特性提供了一种有效的方法。     

A displacement bounding principle in shakedown of structures subjected to cyclic loads

Continua or structures composed of elastic perfectly-plastic material subjected to cyclic loads which vary within the shakedown limits are considered. A theorem bounding the residual deflection at any point is presented.Some numerical examples are discussed.     

金属非晶的强度和变形特性

金属非晶发展至今已有多种体系并可实现厘米量级的块体制备,其各种性能也都有了广泛的研究。本文主要介绍金属非晶的单轴拉伸、单轴压缩、微柱压缩、薄板弯曲、拉伸-扭转等物理力学特性及关于其变形的理论分析。文章涵盖了金属非晶的以下一些力学特性：金属非晶的弹性模量和其溶剂金属的相近性―金属非晶通常具有2% 左右的弹性应变极限,对应着GPa量级的高失效强度;金属非晶单轴拉伸、压缩时的宏观塑性特征及塑性变形的典型机制;金属非晶微观上的短程与中程原子团簇结构特点及其与非晶塑性的关联;金属非晶塑性屈服与静水压力的相关性,拉扭组合时呈现的螺旋断口特征,以及Mohr-Coulomb本构模型对这些屈服特征的适用性。最后,作者也介绍了金属非晶塑性变形的微观物理模型及连续介质力学本构,以及金属非晶的断裂与疲劳特性。     

On some variational acceleration techniques and related methods for local refinement

This paper shows that the well-known variational acceleration method described by Wachspress (E. Wachspress, Iterative Solution of Elliptic Systems and Applications to the Neutron Diffusion Equations of Reactor Physics, Prentice-Hall, Englewood Cliffs, NJ, 1966) and later generalized to multilevels (known as the additive correction multigrid method (B.R Huthchinson and G.D. Raithby, Numer. Heat Transf., 9 , 511–537 (1986))) is similar to the FAC method of McCormick and Thomas (S.F McCormick and J.W. Thomas, Math. Comput., 46 , 439–456 (1986)) and related multilevel methods. The performance of the method is demonstrated for some simple model problems using local refinement and suggestions for improving the performance of the method are given. © 1998 John Wiley & Sons, Ltd.     

Mechanical requirements and structures of cyclic plasticity models

The condition of continuity in the large and in the small and Masing effect as the mechanical requirements for cyclic plasticity are first formulated. The basic characteristics of several well-known and frequently used cyclic plasticity models are examined with respect to fulfillment of these requirements, and their inevitable defects are indicated. It is concluded that among these models only the extended subloading surface model fulfills these requirements and is applicable to the prediction of elastoplastic deformation for cyclic loading, although unfortunately the other models, especially the multi, the two, and the initial subloading surface models, have been frequently adopted for metals, geomaterials, concretes, and so forth.     

Modelling the dynamic behavior of multilayered composite structures

Moscow Region. Translated from Prikladnaya Mekhanika, Vol. 24, No. 3, pp. 39–45, March, 1988.     

On the long-term response of elastic-perfectly plastic solids to dynamic cyclic loads

It is shown that the long-term response of an elastic-perfectly plastic solid subjected to dynamic actions cyclically varying in time is characterized by stresses, plastic strain rates and velocities that are all periodic with the same period of the external actions, and are in perfect analogy with the quasi-static case; on the other hand, plastic strains and displacements are in general nonperiodic (except in case of alternating plasticity) and may increase indefinitely (except when elastic or plastic shakedown occurs). Besides, the work performed by the external actions in the steady cycle equals the work performed by the elastic stresses (i.e. pertaining to the elastic response of the body to the same actions) through the plastic strain rates.

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Sommario Per un solido elastico perfettamente plastico soggetto ad azioni cicliche dinamiche si mostra che la risposta a lungo termine è caratterizzata da tensioni, deformazioni plastiche incrementali e velocità tutte periodiche con lo stesso periodo delle azioni esterne, in analogia di quanto avviene nel caso quasi-statico; per contro le deformazioni plastiche e gli spostamenti sono in generale non periodici (tranne nel caso di plasticità alternata) e possono crescere indefinitamente (tranne nel caso di adattamento elastico o plastico). Inoltre il lavoro compiuto dalle azioni esterne in un ciclo stazionario risulta eguale al lavoro delle tensioni elastiche (cioè ottenute come risposta puramente elastica del solido alle stesse azioni) attraverso le deformazioni plastiche incrementali.

     

On the energy efficiency of cyclic mechanisms

We consider cyclic mechanisms with one degree of freedom driven by engines of various types such as alternating and direct current motors, internal combustion engines, etc. We pose the problem of modifying the mechanism structure by joining additional links or by varying the parameters or operation mode of the original mechanism so as to minimize the thermal losses in the driving motor. The solution is based on the minimization of the functional determining the irreversible power losses. We show that, for the engines considered, all cyclic mechanisms with one degree of freedom should satisfy a fundamental condition ensuring the minimum of losses. We consider two examples, one of which corresponds to actually existing mechanisms.     

Design methods of rhombic tensegrity structures

As a special type of novel flexible structures, tensegrity holds promise for many potential applications in such fields as materials science, biomechanics, civil and aerospace engineering. Rhombic systems are an important class of tensegrity structures, in which each bar constitutes the longest diagonal of a rhombus of four strings. In this paper, we address the design methods of rhombic structures based on the idea that many tensegrity structures can be constructed by assembling one-bar elementary cells. By analyzing the properties of rhombic cells, we first develop two novel schemes, namely, direct enumeration scheme and cell-substitution scheme. In addition, a facile and efficient method is presented to integrate several rhombic systems into a larger tensegrity structure. To illustrate the applications of these methods, some novel rhombic tensegrity structures are constructed.