Shakedown of a cohesive-frictional half-space subjected to rolling and sliding contact

The problem of rolling and sliding contact of a cylinder on the surface of a half-space of cohesive-frictional material is considered. Three shakedown multipliers, of which two are upper bounds and one is exact are computed using a simple numerical procedure. This latter solution differs significantly from previously published analytical solutions which, for realistic material parameters, typically overestimate the shakedown load by a factor of 1.5–2.5.     

Application of the kinematical shakedown theorem to rolling and sliding point contacts

In the plane-strain conditions of a long cylinder in rolling line contact with an elastic-perfectly-plastic half-space an exact shakedown limit has been established previously by use of both the statical (lower bound) and kinematical (upper bound) shakedown theorems. At loads above this limit incremental strain growth or “ratchetting” takes place by a mechanism in which surface layers are plastically sheared relative to the subsurface material.In this paper the kinematical shakedown theorem is used to investigate this mode of deformation for rolling and sliding point contacts, in which a Hertz pressure and frictional traction act on an elliptical area which repeatedly traverses the surface of a half-space. Although a similar mechanism of incremental collapse is possible, the behaviour is found to be different from that in two-dimensional line contact in three significant ways: (i) To develop a mechanism for incremental growth the plastic shear zone must spread to the surface at the sides of the contact so that a complete segment of material immediately beneath the loaded area is free to displace relative to the remainder of the half-space, (ii) Residual shear stresses orthogonal to the surface are developed in the subsurface layers, (iii) A range of loads is found in which a closed cycle of alternating plasticity takes place without incremental growth, a condition often referred to as “plastic shakedown”.Optimal upper bounds to both the elastic and plastic shakedown limits have been found for varying coefficients of traction and shapes of the loaded ellipse. The analysis also gives estimates of the residual orthogonal shear stresses which are induced.     

Limit analysis for a general class of yield conditions

The paper describes a generalisation of the programming method described by Ponter and Carter (1997) for the evaluation of optimal upper bounds on the limit load of a body composed of a rigid/perfectly plastic material. The method is based upon similar principles to the `Elastic Compensation' method which has been used for design calculations for some years but re-interpreted as a non-linear programming method. A sufficient condition for convergence is derived which relates properties of the yield surface to those of the linear solutions solved at each iteration. The method is demonstrated through an application to a Drucker–Prager yield condition in terms of the Von Mises effective stress and the hydrostatic pressure. Implementation is shown to be possible using the user routines in a commercial finite element code, ABAQUS. The examples chosen indicate that stable convergent solutions may be obtained. There are, however, limits to the application of the method if isotropic linear solutions are used for an isotropic yield surface. In an accompanying paper (Ponter and Engelhardt, 2000) the method is extended to shakedown and related problems.     

Bounds for shakedown of cohesive-frictional materials under moving surface loads

In this paper, shakedown of a cohesive-frictional half space subjected to moving surface loads is investigated using Melan’s static shakedown theorem. The material in the half space is modelled as a Mohr–Coulomb medium. The sliding and rolling contact between a roller and the half space is assumed to be plane strain and can be approximated by a trapezoidal as well as a Hertzian load distribution. A closed form solution to the elastic stress field for the trapezoidal contact is derived, and is then used for the shakedown analysis. It is demonstrated that, by relaxing either the equilibrium or the yield constraints (or both) on the residual stress field, the shakedown analysis leads to various bounds for the elastic shakedown limit. The differences among the various shakedown load factors are quantitatively compared, and the influence of both Hertzian and trapezoidal contacts for the half space under moving surface loads is studied. The various bounds and shakedown limits obtained in the paper serve as useful benchmarks for future numerical shakedown analysis, and also provide a valuable reference for the safe design of pavements.     

A minimum theorem for cyclic load in excess of shakedown,with application to the evaluation of a ratchet limit

Although the shakedown theorems for perfect plasticity have been known since Koiter's 1960 review paper, extensions of the theory to situations where ratchetting or reverse plasticity occurs in excess of shakedown have not appeared in the literature. In this paper a generalisation of the upper bound theorem is derived which reduces to the upper bound shakedown theorem in the limiting case when the load point approaches the shakedown boundary. The new theory is used to develop a method for identifying the ratchet limit for a class of loading histories through the sequential minimisation of two functionals. A programming method, based on the Elastic Compensation method for shakedown is then derived and convergence proven. Numerical examples of the application of the method to practical problems are discussed by us in an accompanying paper.     

理想塑性结构极限与安定分析的数值方法

极限分析和安全分析的近代发展方向是寻找通用性强，计算效率高的数值方法。本文介绍将有限单元法和数学规划法相结合的、同时适用于极限分析和安全分析的统一数值方法，包括下限格式和上限格式。     

Shakedown limits for an oscillating,elastic rolling contact with Coulomb friction

We examine experimentally and theoretically the effect of frictional shakedown of a three-dimensional elastic rolling contact. Small oscillations of the local normal forces lead to incremental sliding processes within the area of contact. Consequently, this causes a macroscopic slip motion of the two contacting bodies. If the oscillation amplitude is sufficiently small, the frictional slip ceases after the first few loading periods and a safe shakedown occurs. Otherwise the slip motion is continued and the contact fails.     

On kinematic method in shakedown theory: I. Duality of extremum problems

A kinematic method for determining the safety factor in shakedown problems is developed. An upper bound kinematic functional is defined on a set of kinematically admissible time-independent velocity fields. Every value of the functional is an upper bound for the safety factor. Using convex analysis methods, conditions are established under which the infimum of the kinematic upper bounds equals the safety factor, in particular, conditions under which it is sufficient to consider only smooth velocity fields for the safety factor calculation. The method generalizes that recently proposed for the case of spherical yield surfaces by Kamenjarzh and Weichert. The extension covers a wide class of yield surfaces and inhomogeneous bodies. A shakedown problem for a beam subjected to a concentrated load is considered as an example.     

A method for the evaluation of a ratchet limit and the amplitude of plastic strain for bodies subjected to cyclic loading

An extension of the upper bound shakedown theorem to load histories in excess of shakedown has been presented elsewhere in this issue. Here the minimisation process described therein is applied to the solutions of the ratchet limit as well as shakedown and limit load for a range of simple problems. The solutions provide an estimate of the maxima of the varying plastic strain magnitudes, which is compared with the Neuber approximate values. The position of the ratchet boundary is confirmed by comparison step-by-step analysis.     

Lower bound shakedown analysis by the symmetric Galerkin boundary element method

In this paper, the static shakedown theorem is reformulated making use of the symmetric Galerkin boundary element method (SGBEM) rather than of finite element method. Based on the classical Melan’s theorem, a numerical solution procedure is presented for shakedown analysis of structures made of elastic-perfectly plastic material. The self-equilibrium stress field is constructed by linear combination of several basis self-equilibrium stress fields with parameters to be determined. These basis self-equilibrium stress fields are expressed as elastic responses of the body to imposed permanent strains obtained through elastic–plastic incremental analysis. The lower bound of shakedown load is obtained via a non-linear mathematical programming problem solved by the Complex method. Numerical examples show that it is feasible and efficient to solve the problems of shakedown analysis by using the SGBEM.     

Upper Bound Shakedown Analysis of Plates Utilizing the C1 Natural Element Method

This paper proposes a numerical solution method for upper bound shakedown analysis of perfectly elasto-plastic thin plates by employing the C1 natural element m...     

Minimum Theorems and Iterative Solutions Methods for Creep Cyclic Loading Problems

In recent years a particular programming method, the linear matching method, has been particularly successful in the evaluation of optimal upper bounds to shakedown limits for an elastic perfectly plastic body. The method applies to any convex yield condition with an associated flow rule and sufficient conditions for convergence exist. For creep constitutive equations and for a body under cyclic loading, there exist a class of cyclic solutions, the so called 'rapid cycle' solutions for which the residual stress field remains constant throughout the cycle. In this paper an upper bound theorem for the rapid cycle solution is derived and related to the upper bound shakedown theorem. This allows the linear matching method to be extended to this class of creep problems. A sufficient condition for convergence is derived. For a flow potential expressed in terms of a Von Mises effective stress, the sufficient condition is shown to be a simple and common property of creep equations. Sommario. Recentemente, un particolare metodo di programmazione, detto del materiale elastico equivalente, si è rivelato particolarmente efficiente nella valutazione della delimitazione superiore ottimale del limite di adattamento di solidi idealmente elasto-plastici. Il metodo vale con riferimento a qualunque condizione di plasticità convessa con legge di scorrimento associata e sono disponibili condizioni sufficienti di convergenza. Nel caso di legami costitutivi viscosi, per solidi soggetti a carichi ciclici esiste una classe di soluzioni, dette di 'ciclo rapido', in cui gli sforzi residui si mantengono costanti nel ciclo. In questo lavoro si deriva un teorema di delimitazione superiore per le soluzioni di ciclo rapido, che viene relazionato al corrispondente teorema di adattamento. Ciò permette di estendere il metodo del materiale elastico equivalente a questa categoria di problemi viscosi. Una condizione sufficiente per la convergenza del metodo viene anche dimostrata. Nel caso di un potenziale espresso in termini dello sforzo equivalente di von Mises, tale condizione si rivela essere una semplice e comune proprietà del legame costitutivo.     

An Improved Upper Bound to Maximum Deflections of Elastic-Plastic Structures at Shakedown

ABSTRACT

A method is presented providing an upper bound to the maximum shakedown deflections for elastic-perfectly plastic structures. The influence of plastic zones is taken into account. Residual stresses required by the Melan theorem can be expressed in terms of stresses due to ideal plastic hinges and of stresses due to the finite extent of plastic zones. Making use of this fact a bound on the total energy dissipated in a shakedown process as well as a bound to the permanent displacement have been derived. This bound permits an estimate of the deflections at shakedown. Application of the method is illustrated by means of two examples.     

Shakedown analysis for a class of strengthening materials within the framework of gradient plasticity

The classical shakedown theory is extended to a class of perfectly plastic materials with strengthening effects (Hall–Petch effects). To this aim, a strain gradient plasticity model previously advanced by Polizzotto (2010) is used, whereby a featuring strengthening law provides the strengthening stress, i.e. the increase of the yield strength produced by plastic deformation, as a degree-zero homogeneous second-order differential form in the accumulated plastic strain with associated higher order boundary conditions. The extended static (Melan) and kinematic (Koiter) shakedown theorems are proved together with the related lower bound and upper bound theorems. The shakedown limit load problem is addressed and discussed in the present context, and its solution uniqueness shown out. A simple micro-scale structural system is considered as an illustrative example. The shakedown limit load is shown to increase with decreasing the structural size, which is a manifestation of the classical Hall–Petch effects in a context of cyclic loading.     

Multiple-zone sliding contact with friction on an anisotropic thermoelastic half-space

The paper studies a class of multiple-zone sliding contact problems. This class is general enough to include frictional and thermal effects, and anisotropic response of the indented material. In particular, a rigid die (indenter) slides with Coulomb friction and at constant speed over the surface of a deformable and conducting body in the form of a 2D half-space. The body is assumed to behave as a thermoelastic transversely isotropic material. Thermoelasticity of the Green–Lindsay type is assumed to govern. The solution method is based on integral transforms and singular integral equations. First, an exact transform solution for the auxiliary problem of multiple-zone (integer n > 1) surface tractions is obtained. Then, an asymptotic form for this auxiliary problem is extracted. This form can be inverted analytically, and the result applied to sliding contacts with multiple zones. For illustration, detailed calculations are provided for the case of two (n = 2) contact zones. The solution yields the contact zone width and location in terms of sliding speed, friction, die profile, and also the force exerted. Calculations for the hexagonal material zinc illustrate effects of speed, friction and line of action of the die force on relative contact zone size, location of maximal values for the temperature and the compressive stress, and the maximum temperature for a given maximum stress. Finally, from our general results, a single contact zone solution follows as a simple limit.     

Dynamic shakedown of elastic-plastic solids for a set of alternative loading histories

The paper deals with dynamic shakedown of an elastic-perfectly plastic solid body subjected to a loading history which is unknown but is allowed to belong to a given set of loading histories. In the hypothesis of a piecewise linear convex set, a sufficient shakedown theorem is given and a bounding principle for the plastic work produced is formulated in terms of the dynamic elastic responses to a discrete set of loading histories. The solution of a minimization problem gives the most stringent bound which also proves to possess a local character, i.e., it regards the plastic work density at any point.     

Shakedown analysis of engineering structures with limited kinematical hardening

We present a numerical method for the computation of shakedown loads of engineering structures with limited kinematical hardening under thermo-mechanical loading. The method is based on Melan’s statical shakedown theorem, which results in a nonlinear convex optimization problem. This is solved by an interior-point algorithm recently developed by the authors, specially designed for lower bound shakedown analysis of large-scale problems. Limited kinematical hardening is taken into account by use of a two-surface model, such that both alternating plasticity and incremental collapse can be captured. For the yield surface as well as for the bounding surface the von Mises criterion is used. The proposed method is validated by two examples, where numerical results are compared to those of literature where available.     

Transition from stick to slip in Hertzian contact with “Griffith” friction: The Cattaneo–Mindlin problem revisited

Classically, the transition from stick to slip is modelled with Amonton–Coulomb law, leading to the Cattaneo–Mindlin problem, which is amenable to quite general solutions using the idea of superposing normal contact pressure distributions – in particular superposing the full sliding component of shear with a corrective distribution in the stick region. However, faults model in geophysics and recent high-speed measurements of the real contact area and the strain fields in dry (nominally flat) rough interfaces at macroscopic but laboratory scale, all suggest that the transition from ‘static’ to ‘dynamic’ friction can be described, rather than by Coulomb law, by classical fracture mechanics singular solutions of shear cracks. Here, we introduce an ‘adhesive’ model for friction in a Hertzian spherical contact, maintaining the Hertzian solution for the normal pressures, but where the inception of slip is given by a Griffith condition. In the slip region, the standard Coulomb law continues to hold. This leads to a very simple solution for the Cattaneo–Mindlin problem, in which the “corrective” solution in the stick area is in fact similar to the mode II equivalent of a JKR singular solution for adhesive contact. The model departs from the standard Cattaneo–Mindlin solution, showing an increased size of the stick zone relative to the contact area, and a sudden transition to slip when the stick region reaches a critical size (the equivalent of the pull-off contact size of the JKR solution). The apparent static friction coefficient before sliding can be much higher than the sliding friction coefficient and, for a given friction fracture “energy”, the process results in size and normal load dependence of the apparent static friction coefficient. Some qualitative agreement with Fineberg's group experiments for friction exists, namely the stick–slip boundary quasi-static prediction may correspond to the arrest of their slip “precursors”, and the rapid collapse to global sliding when the precursors arrest front has reached about half the interface may correspond to the reach of the “critical” size for the stick zone.     

Deflection Analysis of Elastic-Plastic Frames at Shakedown

ABSTRACT

A method is presented providing an upper bound solution to the maximum deflections which may occur in elastic-plastic frames under loadings that vary arbitrarily within prescribed limits.

The problem is reduced to linear programming; a numerical example is given. The results show that the theory of shakedown remains valid even for very low values of the safety factor against incremental collapse.