An analytical elastic-perfectly plastic contact model

A new formulation for elastic-perfectly plastic contact in the normal direction between two round surfaces that is solely based on material properties and contact geometries is developed. The problem is formulated as three separate domains: the elastic regime, mixed elastic–plastic behavior, and unconstrained (fully plastic) flow. Solutions for the force–displacement relationship in the elastic regime follow from Hertz’s classical solution. In the fully plastic regime, two well supported assumptions are made: that there is a uniform pressure distribution and there is a linear force–deflection relationship. The force–displacement relationship in the intermediate, mixed elastic–plastic regime is approximated by enforcing continuity between the elastic and fully plastic regimes. Transitions between the three regimes are determined based on empirical quantities: the von Mises yield criterion is used to determine the initiation of mixed elastic–plastic deformation, and Brinell’s hardness for the onset of unconstrained flow. Unloading from each of these three regimes is modeled as an elastic process with different radii of curvature based on the regime in which the maximum force occurred. Simulation results explore the relationship between the impact velocity and coefficient of restitution. Further comparisons are made between the model, experimental results found in the literature, and other existing elastic–plastic models. The new model is well supported by the experimental measurements of compliance curves for elastic–plastic materials and of coefficients of restitution from impact studies, and in elastic-perfectly plastic regimes is demonstrated to be more accurate than existing models found in the literature.     

The shakedown load boundary of an elastic-perfectly plastic structure

In the hypothesis of small displacements and combined time-variable/steady loads, the geometrical-mechanical properties of the shakedown load boundary are investigated. It is shown that, in the load space, the shakedown load boundary plays the role of yield surface, and that a certain plastic strain accumulation vector—characterizing some impending inadaptation collapse mechanism—obeys the normality rule, whereas a specific form of the maximum plastic work theorem constitutes an effective tool for the evaluation of the shakedown limit load corresponding to a specified inadaptation collapse mode. The equations governing the state of the structure at the shakedown limit are provided and the related collapse mechanism is shown to specify the shape of the steady-state response of the structure to a periodic load enveloping the load domain with an intensity slightly above the shakedown limit.

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Sommario Nell'ipotesi di piccoli spostamenti e di plasticità perfetta, si studiano le proprietà geometrico-meccaniche della frontiera del dominio dei carichi di adattamento. Si mostra come, nello spazio dei carichi, la suddetta frontiera giuoca il ruolo di superficie di plasticizzazione ed un particolare vettore di deformazione plastica accumulata—che caratterizza il meccanismo di non adattamento—segue la legge della normalità, mentre una forma specifica del teorema del massimo lavoro plastico costituisce un valido strumento per la valutazione del carico limite corrispondente ad un dato meccanismo di non adattamento. Si forniscono altresì le equazioni che governano lo stato della struttura al limite di adattamento, mostrando la capacità del relativo meccanismo di collasso a rappresentare la risposta a lungo termine della struttura a dei carichi periodici inviluppanti il dominio dei carichi con una intensità un poco al di sopra del limite di adattamento.

     

Scaling relationships in conical indentation of elastic-perfectly plastic solids

Using dimensional analysis and finite element calculations we derive several scaling relationships for conical indentation into elastic-perfectly plastic solids. These scaling relationships provide new insights into the shape of indentation curves and form the basis for understanding indentation measurements, including nano- and micro-indentation techniques. They are also helpful as a guide to numerical and finite element calculations of conical indentation problems. Finally, the scaling relationships are used to reveal the general relationships between hardness, contact area, initial unloading slope, and mechanical properties of solids.     

Predicting the coefficient of restitution of impacting elastic-perfectly plastic spheres

The current work presents a different methodology for modeling the impact between elasto-plastic spheres. Recent finite element results modeling the static deformation of an elasto-plastic sphere are used in conjunction with equations for the variation of kinetic energy to obtain predictions for the coefficient of restitution. A model is also needed to predict the residual deformation of the sphere during rebound, or unloading, of which several are available and compared in this work. The model predicts that a significant amount of energy will be dissipated in the form of plastic deformation such that as the speed at initial impact increases, the coefficient of restitution decreases. This work also derives a new equation for the initial critical speed which causes initial plastic deformation in the sphere that is different than that shown in previously derived equations and is strongly dependant on Poisson’s Ratio. For impacts occurring above this speed, the coefficient of restitution will be less than a value of one. This work also compares the predictions between several models that make significantly different predictions. The results of the current model also compare well with some existing experimental data. Empirical fits to the results are provided for use as a tool to predict the coefficient of restitution.     

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.

     

Higher order asymptotic solution of mode I crack growth in elastic-perfectly plastic media

A higher order asymptotic solution of near-tip field is studied for plane-atrain Mode-I quasi-static steady crack growth in the incompressible (v=1/2) elastic perfectly-plastic media. The results show that the near-tip stress and strain are fully continuous, and the strain possesses In (A/r) singularity at the crack tip. The expressions of the stress, strain and velocity in each region are also given. The project supported by National Natural Science Foundation of China     

THE PRECISE AND NEW ANALYSIS FOR A MODE Ⅲ GROWING CRACK IN AN ELASTIC-PERFECTLY PLASTIC SOLID

The near crack line field analysis method has been used to investigate into ModeⅢ quasistatically propagating crack in an elastic-perfectly plastic material.Thesignificance of this paper is that the usual small scale yielding theory has been brokenthrough.By obtaining the general solutions of the stresses and the displacement rate ofthe near crack line plastic region,and by matching the general solutions with theprecise elastic fields(not the usual elastic K-dominant fields)at the elastic-plasticboundary,the precise and new solutions of the stress and deformation fields,the sizeof the plastic region and the unit normal vector of the elastic-plastic boundary havebeen obtained near the crack line.The solutions of this paper are sufficiently precisenear the crack line region because the roughly qualitative assumptions of the smallscale yielding theory have not been used and no other roughly qualitative assumptionshave been taken,either.The analysis of this paper shows that the assumingly“steady-state cas     

Quadratic programming with parametric vectors in three-dimensional contact problems

The present study is concerned with 3-D contact problems by parametric quadratic programming(PQP). The stiffness matrix of a 3-D contact element is introduced by means of penalty function expression of the contact pressure and frictional force, and two constitutive relations of 3-D contact problems can be obtained, similar to elasto—plastic problems. The penalty factors can be eliminated by using a definite numerical technique, so that a parametric linear complementary problem is obtained. As an application of the theory presented, some numerical examples are illustrated.     

Quadratic programming algorithm for wall slip and free boundary pressure condition

Wall slip is often observed in a highly sheared fluid film in a solid gap. This makes a difficulty in mathematical analysis for the hydrodynamic effect because fluid velocity at the liquid–solid interfaces is not known a priori. If the gap has a convergent–divergent wedge, a free boundary pressure condition, i.e. Reynolds pressure boundary condition, is usually used in the outlet zone in numerical solution. This paper, based on finite element method and parametric quadratic programming technique, gives a numerical solution technique for a coupled boundary non‐linearity of wall slip and free boundary pressure condition. It is found that the numerical error decreases with the number of elements in a negative power law having an index larger than 2. Our method does not need an iterative process and can simultaneously gives rise to fluid film pressure distribution, wall slip velocity and surface shear stress. Wall slip always decreases the hydrodynamic pressure. Large wall slip even causes a null hydrodynamic pressure in a pure sliding solid gap. Copyright © 2005 John Wiley & Sons, Ltd.     

弹塑性结构优化的并行算法

本文给出了弹塑性结构优化设计的序列二次规划算法，分别考虑了弹塑性结构分析，灵敏度分析和二次规划的Ｌｅｍｋｅ算法的并行计算。针对弹塑性结构灵敏度的不连续性，比较了连续模型和间断模型的计算结果，结果是接近的。算例表明此算法有很好的并行性。     

EXACT SOLUTIONS OF NEAR CRACK LINE FIELDS FOR MODE I CRACK UNDER PLANE STRESS CONDITION IN AN ELASTIC－PERFECTLY PLASTIC SOLID

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Dissipative mesoscopic structures in plastic deformation

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 128–139, July–August, 1993.     

Plastic analysis of structures and duality in linear programming

Summary The possibility of formulating the static and kinematic methods of plastic analysis theory as dual linear programming problems is extended to the cases of monodimensional structures under combined stresses with piecewise linear yield curves (or surfaces).     

Strain path theory of plastic constitutive relation

Based on Illiushin's two hypotheses, a new plastic constitutive equation of integral type is proposed. The deviatoric stress induced by a strain path consists of two parts which vary according to different laws. Comparison of theoretical calculation with recent experiments is satisfactory. The present theory is not an endochronic theory.     

Limit analysis of arch-beam structures by dynamic programming

Summary We study one-dimensional structures like arch-beams in the limit state of plastic collapse, on the ground of a two-dimensional yielding surface (bending moment and normal generalized stress). The proposed method, which is able to give a numerical solution of the problem of finding the limit load, rests on the upper bound theorem of limit analysis and uses dynamic programming. We examine also some questions linked with numerical procedures. A future work devoted to applications will complete the treatment.

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Sommario Si studiano le strutture monodimensionali ad arco nello stato limite di collasso plastico, sulla base del dominio bidimensionale (momento flettente e sforzo normale). Il metodo proposto, atto a fornire una soluzione numerica del problema della ricerca del carico limite, si fonda sul teorema cinematico dell'analisi limite e sull'impiego della programmazione dinamica. Si prendono anche in esame talune questioni connesse con l'algoritmo di calcolo. Un successivo lavoro di carattere applicativo completerà la trattazione.

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This paper has been supported by National (italian) Research Council (C.N.R.).