Koiter's Theorem Research Articles

Static and kinematic shakedown theorems in diffusion-induced plasticity

1. Brassart L., Zhao K., Suo Z., 2013, Cyclic plasticity and shakedown in high-capacity electrodes of lithium-ion batteries, International Journal of Solids and Structures, 50, 1120-1129. CrossRef Google Scholar

Upper bound shakedown analysis with the nodal natural element method

In this paper, a novel numerical solution procedure is developed for the upper bound shakedown analysis of elastic-perfectly plastic structures. The nodal natural element method (nodal-NEM) combines the advantages of the NEM and the stabilized conforming nodal integration scheme, and is used to discretize the established mathematical programming formulation of upper bound shakedown analysis based on Koiter's theorem. In this formulation, the displacement field is approximated by using the Sibson interpolation and the difficulty caused by the time integration is solved by Konig's technique. Meanwhile, the nonlinear and non-differentiable characteristic of objective function is overcome by distinguishing non-plastic areas from plastic areas and modifying associated constraint conditions and goal function at each iteration step. Finally, the objective function subjected to several equality constraints is linearized and the upper bound shakedown load multiplier is obtained. This direct iterative process can ensure the shakedown load to monotonically converge to the upper bound of true solution. Several typical numerical examples confirm the efficiency and accuracy of the proposed method.

On shakedown and ratchetting of conforming frictional systems

Conforming contact is found on many important areas: materials with cracks, inclusions, layered structures, pinned geometries, etc. Coulomb friction is non-associative i.e. does not satisfy Drucker’s postulates for stability in plasticity theory, and this does not permit to use the classical Melan and Koiter theorems for cyclic loading which however remain obvious necessary conditions. On the other hand, the Coulomb friction cone has a peculiarity, its linear self-similarity, and this permits few general results, quite different but also powerful. For example, we show that for conforming contact, the Bree interaction diagram is open and sectors-shaped, and therefore there is no lower and upper bounds as in plasticity, since there is linear dependence on a load parameter \(\lambda\), and not a limit on \(\lambda\). As there is dependence on initial conditions in general, the Bree diagram doesn’t have the same profound meaning as in plasticity, since multiple steady states are possible. However, there is a complete scaling of the possible states (elastic shakedown, cyclic slip, ratchetting) with load factor \(\lambda\), and hence a worst-scenario can be defined. However, in order to develop residual stresses (which generally will take many cycles of steady cyclic loading), the load cannot go to zero, at least for friction coefficient below the critical value for "wedging". More in general, there needs to be a "permanent stick region" in the steady state.Almost conforming contacts, and in particular, in the case of initial interference, loose the independence on \(\lambda\) at low \(\lambda\) only, but "gain" greater possibilities to develop residual stresses. However, their study can be greatly simplified by developing a master curve approach to study the dependence on \(\lambda\). The theorems can also be considered the extension to cyclic loadings of the Dundurs results for conforming contacts under monotonic loading, in which case the contact area doesn’t vary in time and hence the problem is linear (from which the denomination "receding contacts"). However, here the problems remain fully non-linear, and the contact are not in general receding, nor there is a unique steady state, not even in the case of proportional loading. In the latter case it is nevertheless possible to derive some special results.

Shakedown analysis of a slope with cyclic groundwater level

AbstractGroundwater levels vary with precipitation and other environmental impacts, rendering wetting and drying cycles in parts of a slope. It may be quite disadvantageous, as groundwater has been implicated as a crucial controlling factor in slope stability. The wetting–drying cycles can be considered as cyclic loads for the slope. In such cases, a shakedown analysis is appropriate for the long‐term safety evaluation. This paper presents an upper bound shakedown analysis method based on strong version of Koiter's theorem to evaluate slope stability. An element integration method is employed to provide limit and shakedown safety factors for the slope. For plane slip surfaces or logarithmic spiral slip surfaces, the validity of the method is verified by two documented examples. Copyright © 2009 John Wiley & Sons, Ltd.

Slip-shakedown analysis and the assumption of small coupling in frictional contact

In the frictional contact of solids under cyclic loads, the shakedown behaviour of the relative displacement is of interest in the same spirit as the plastic deformation in plasticity. Cumulative slips may lead to the failure due to large relative displacements of the components of an assembly while cyclic slips are often undesired because of wear and fretting÷fatigue problems. Under Coulomb friction, it is well known that Melan and Koiter theorems are generally not available, except in certain particular cases. In this discussion, the particular case of small coupling between the contact pressures and the slip-displacements is considered. This assumption means that the tangent displacements have small or no influence on the contact pressures which can be then computed from the elastic response as in the uncoupling case. The pressure is thus a given time-dependent function and the Coulomb criterion is reduced to a Mises-like standard law of friction. It is shown here that Melan and Koiter theorems can be applied again as in standard plasticity. The dependence of the yield limit on the loading amplitude is however not classical and the extension of the static and kinematic approaches is discussed to obtain the critical shakedown load or the limit load. The validity of the assumption of small coupling is also explored by numerical simulation in a practical example.

Shakedown theorems in Contact Mechanics

In this Note, the phenomenon of cumulative slip of solids under cyclic loads is considered. The topic concerns with the relative displacements of two solids maintained in contact by friction. This problem of the daily-life mechanics can be compared to the shakedown of elastic plastic solids under cyclic loads. A transcription of the plastic shakedown theorems is given here for the problem of frictional contact. The statement of Melan theorem is first given for the cumulative-slip problem under some restrictive assumptions. It suggests again the introduction of a safety coefficient with respect to slips. The safety coefficient can be computed from two static and kinematic approaches in min–max duality and leads again to Koiter theorem. As in plasticity, these results are available only for associated laws and do not hold for Coulomb law of friction, except in some very particular situations. These results can be understood mathematically as a particular case of shakedown theorems in plasticity, when the plastic strain is localized on a surface. For Coulomb friction, numerical simulations by direct step-by-step calculations show that the asymptotic behaviour of the response with or without slip-shakedown could be obtained very quickly after some cycles. To cite this article: N. Antoni, Q.-S. Nguyen, C. R. Mecanique 336 (2008).

Application of shakedown maps under variable loads

A method able to compute the residual stresses caused by two-dimensional rolling and sliding contact is proposed and applied to the analysis of elastic shakedown. Firstly, the two-dimensional shakedown map is obtained for both full slip and partial slip conditions. The results are then compared with previous works and show good agreement. The method proposed here is not based on the application of Melan or Koiter theorems, but rather by reproducing the ‘real’ stress–strain history of the material subjected to rolling contact loads. As a result, the model allows different load conditions to be explored without any great difference with regard to computational implementation. For this reason the analysis is extended to variable load conditions. In particular, two sets of conditions are considered. One of them is p 0/ k = 3.75 and μ = 0, while the second one is p 0/ k = 3.75 and μ = 0.15. These conditions are imposed alternatively on the material. Surprisingly, the material does not shake down to the elastic state, although both conditions fall, if taken independently one from another, below the shakedown limit. The reason for this behaviour and its important implications on rolling contact fatigue damage are then discussed.

Adaptive F.E. method for the shakedown and limit analysis of pressure vessels

Upper bound estimates of limit and shakedown loads for pressure vessels are calculated by using the technique described in this paper. These have been achieved by applying Koiter's theorem and by discretizing the shell into finite elements. The flow law associated with an hexagonal prism yield surface, relates the plastic strain increments and curvatures to plastic multipliers. A suitable matrix also relates such a plastic strain field to a displacement field through a classical relation. A novel method enforces a consistent relationship between nodal displacements and nodal plastic multipliers by minimizing the residual between the two independent descriptions of the plastic increments, measured with respect to the energy norm. The discretized problem is then reduced to a minimization problem and solved by linear programming. An a posteriori error indicator in the energy norm is derived with and adaptive mesh refinement scheme.

A kinematic elastic nonshakedown theorem for implicit standard materials

Criteria for a priori recognition of the type of steady-state response induced by cyclic loads and prediction whether a structure will shakedown elastically or not, without the necessity of performing a step-by-step full analysis, have considerable importance. Melan and Koiter theorems provide criteria that guarantee whether elastic shakedown occurs or not under cyclic loads in case of perfect plasticity. However, there remain some aspects of the shakedown theory which deserve further study. One of these, concerned with more realistic nonassociative elastic–plastic constitutive material models, allowing for nonlinear kinematic and isotropic hardening suitable to describe the cyclic plastic behaviour of metallic materials, has strong motivation. Koiter's elastic nonshakedown theorem is reconsidered here, with the objective of extending it to the de Saxce's implicit standard material class, which contains a wide class of nonassociative elastic–plastic material behaviours. Shakedown analysis is formulated by a kinematic approach based on the plastic accumulation mechanism concept due to Polizzotto. A sufficient condition for elastic nonshakedown and a distinct necessary condition are established. Then, an upper bound to the shakedown multiplier is evaluated.

On shakedown theorems in hardening plasticity

The model of generalized standard materials gives a convenient framework to extend Koiter's shakedown theorems into hardening plasticity. The extension of the static shakedown theorem (Melan–Koiter's theorem), proposed previously in [5], is considered here. It leads to the definition of safety coefficients in hardening plasticity by duality. Static and kinematic approaches are discussed for the models of isotropic hardening, of linear kinematic hardening (Ziegler–Prager's model) and of limited kinematic hardening. This discussion also leads to an extension of Koiter's kinematic shakedown theorem and to a second kinematic coefficient.

Shakedown and steady-state responses of elastic-plastic solids in large displacements

Elastic-perfectly plastic solids (or structures) subjected to loads quasi-statically varying within a specified domain are addressed in the framework of large displacements and the additive strain decomposition rule. On the ground of Drucker’s principle of stability in the large, an appropriate stability requisite (called D-stability) is formulated as the positive definiteness property of a specific functional, sum of the second variation of the Helmholtz free energy with an additional term depending on higher-order geometry change effects. For a D-stable structure for which the additive strain decomposition rule is applicable, Melan’s and Koiter’s theorems of classical shake-down theory are reconsidered and reformulated for large displacements. The extended Melan and Koiter theorems so established save the essential features of the classical ones, but exhibit a greater formal complexity with consequent difficulties for engineering applications. For structures subjected to periodic loads, it is shown that—as long as the structure does not incur loss of D-stability—a long-term steady-state response (or steady cycle) occurs, which exhibits the same periodicity characteristics as in case of small displacements; that is, the (second Piola—Kirchhoff) stresses and the plastic (Green-Lagrange) strain rates become periodic as the load. A few illustrative numerical results are presented.

Numerical shakedown analysis of axisymmetric sandwich shells: An upper bound formulation

AbstractShakedown analysis of axisymmetric elastic–perfectly plastic sandwich shells is performed here using a new upper bound formulation based on a special form of Koiter's theorem concerning piecewise linearized yield surfaces. Starting from finite element techniques and the Tresca sandwich yield condition, shakedown analysis is reduced to a linear programming problem which is solved by a powerful simplex algorithm. Numerical results are given for a number of examples and a comparison is made with a previously computed lower bound formulation.

Shakedown analysis of elastic-plastic structures

Abstract Behavior of elastic-plastic structures under repetitive and fluctuating loads is considered in this paper. Plastic deformation either stabilizes after a finite number of cycles or continues during the cycling. In the first case the structure is said to have shaken down to the boundary at the loading program. If plastic deformation does not stabilize an elastic-plastic structure becomes unserviceable due to either alternating plasticity when yielding occurs repeatedly in the opposite senses or accumulation of plastic strains and progressive increase of permanent displacements. This paper attempts to survey the shakedown theory, including the accommodation of a structure to the prescribed loading range as well as inadaptation and unserviceability. The notion of shakedown, incremental collapse, ratchetting and alternating plastic deformation are first illustrated with examples. The fundamental theorems on shakedown and inadaptation are presented next, attention being directed to generalizations of the classical Melan and Koiter theorems. Applications of the theorems are given. Available methods for determining the shakedown range on generating self-equilibrated stress fields are discussed. Special techniques applicable to problems involving both dead and fluctuating loads are invoked. Recent studies accounting for inertia forces, geometrical changes, material hardening, variation of material properties with temperature, displacement assessment, etc. are referred to.

Using the Koiter theorem for adaptability problems of non-uniformly heated elastic-plastic bodies

Using the Koiter theorem for adaptability problems of non-uniformly heated elastic-plastic bodies