Kinematic Shakedown Theorem Research Articles

Computation of Limit Loads for Bending Plates

The purpose of this paper is to present a method for calculating the upper bound limit loads of plate bending using a conforming Hsieh-Clough-Tocher (HCT) element. These limit loads can be obtained from Koiter’s kinematic shakedown theorem for the case of one load vertex instead of using the kinematic limit theorem. When combining this theorem with the approximated displacement field, the limit analysis turns into an optimization problem and can be effectively solved by Second-Order Cone Programming (SOCP). Several benchmark plate problems such as square, rectangular, and L-shape plates are investigated to illustrate the effectiveness of the proposed solution. The results of the proposed method show good agreement with the results of previous studies. The maximum error is only 2.91% for the fully clamped rectangular plate problem.

Strength and fatigue analysis of periodic perforated materials using the computational homogenization and ES-FEM approaches

This paper presents a computational homogenization shakedown analysis of periodic perforated materials with von Mises matrices. The plastic behaviors of the perforated materials under cyclic macroscopic loads are studied by means of kinematic shakedown theorem and computational homogenization method. The kinematic micro-fields are approximated by the edge-based smoothed finite element method. The resulting large-scale optimization problem is efficiently solved by using conic solver, enabling a large number of points on the macroscopic strength and fatigue surface to be calculated rapidly. The effects of the hole's shape and size on the overall strength and fatigue domains are also investigated.

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

Development Of A Pavement Rutting Model Using Shakedown Theory

Abstract The rutting of flexible pavements during their exploitation is considered to be one of the main problems in UK as well as worldwide. It is a serious mode of distress alongside fatigue in bituminous pavements that may lead to premature failure, as indicated by permanent deformation or rut depth along the wheel load path, and results in early and costly rehabilitation. This kind of pavement distress makes a negative impact to the serviceability characteristics of the flexible pavement, to the residual life of pavement structure and also to the safety and ride quality for traffic. Two design methods have been used to control rutting: one to limit the vertical compressive strain on the top of subgrade and the other to limit rutting to a tolerable amount usually around “12 mm”. Although experimental data and practical experience have been introduced into these design methods through empirical parameters, there is not a simple relationship between the elastic strain and the long-term plastic behaviour of pavement materials. This paper describes a method based on the kinematic shakedown theorem for constructing a mathematical model to predict the long-term behaviour of pavement structures under the action of repeated and cyclic loadings imposed by moving traffic. This method seeks the mechanism from within a class of mechanisms that minimises the shakedown limit load for pavement structures consisting of layers of Mohr-Coulomb materials. The model differs from extant models, in that the cyclic nature of the loading on a pavement is recognised from the outset, and the current method which is based upon foundation analysis, is replaced by a procedure employing shakedown theory that features the capabilities and applications of the developed technique for assessing rutting in flexible pavements. The basic concepts are outlined together with the most recent calculations of the critical design shakedown load. The influence of the design parameters such as, the strength, stiffness and depth of the granular base-course material as well as the consequences of traffic loading (number of equivalent standard axel loads – ESAL’s) are discussed.

Design of Unbound Pavement Layer Materials Based Upon the Shakedown Concept

This study aims to introduce an alternative design method for unbound pavement layer materials based upon shakedown theory. The kinematic shakedown theorem that describes the ultimate response of an elastic/plastic structure to cyclic loads is used to predict whether stable behaviour in the unbound granular materials layer occurs or excessive rutting will develop. This method seeks the mechanism, from within a class of mechanisms that minimises the shakedown limit load for pavements consisting of layers of Mohr-Coulomb material obeying the associated flow rule. The basic concepts are outlined together with the most recent calculations of the critical design shakedown load. The influence of the of design parameters such as, the strength, stiffness and depth of the base-course material as well as the consequences of load distribution are discussed.

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.

Kinematic shakedown analysis under a general yield condition with non-associated plastic flow

A nonlinear, purely kinematic approach with the finite element implementation is developed to perform shakedown analysis for materials obeying a general yield condition with non-associated plastic flow. The adopted material model can be used for both isotropic materials (e.g. von Mises's, Mohr–Coulomb and Drucker–Prager criteria) and anisotropic materials (e.g. Hill's and Tsai-Wu criteria) with both associated and non-associated plastic flow. Nonlinear yield criterion is directly introduced into the kinematic shakedown theorem without linearization and instead a nonlinear, purely kinematic formulation is obtained. By means of mathematical programming techniques, the finite element model of shakedown analysis is formulated as a nonlinear programming problem subject to only a small number of equality constraints. The objective function corresponds to plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a structure can then be obtained by solving the minimum optimization problem. A direct, iterative algorithm is proposed to solve the resulting nonlinear programming problem, where a penalty factor based on the calculation of the plastic dissipation power is used to overcome the numerical difficulty caused by the non-differentiability of the objective function in elastic areas. The calculation is entirely based on a purely kinematical velocity field without calculation of stresses. Meanwhile, only a small number of equality constraints are introduced into the nonlinear programming problem. So the computational effort is very modest. Numerical applications prove that the developed algorithm has a very good numerical stability and computational efficiency. The proposed approach can capture different plastic behaviours of materials and therefore has a very wide applicability.

Shakedown Fatigue Limits for Materials With Minute Porosity

The intention of this study is to predict the fatigue-safe long life behavior of elastoplastic porous materials subjected to zero-tension fluctuating load. It is assumed that the materials contain a dilute amount of voids (less than 5%) and obey Gurson’s model of plastic yielding. The question to be answered is what would be the highest allowable stress amplitude that a porous material can endure (the “endurance limit”) when undergoing an infinite number of loading/unloading cycles. To reach the answer we employ the two shakedown theorems: (a) Melan’s static shakedown theorem (“elastic shakedown”) for establishing the lower bound to fatigue limit and (b) Koiter’s kinematic shakedown theorem (“plastic shakedown”) for establishing its upper bound. The two bounds are formulated rigorously but solved with some numerical assistance, mainly due to the nonlinear pressure dependency of the material behavior and the complex description of the plastic flow near stress-free voids. Both bounds (“dual bounds”) are adjusted to capture Gurson-like porous materials with noninteractive voids. General residual stresses (either real or virtual) are presented in the analysis. They are assumed to be time-independent as generated, say, by permanent temperature gradient between void surfaces and remote material boundaries. Such a situation is common, for instance, in ordinary porous sleeves (used in space industry and alike). A few experiments agree satisfactorily with the shakedown bounding concept.

Shakedown theorems for elastic–plastic solids in the framework of gradient plasticity

Static and kinematic shakedown theorems are given for a class of generalized standard materials endowed with a hardening saturation surface in the framework of strain gradient plasticity. The so-called residual-based gradient plasticity theory is employed. The hardening law admits a hardening potential, which is a C 1-continuous function of a set of kinematic internal variables and of their spatial gradients, and is required to satisfy a global sign restriction (but not to be necessarily convex). The totally produced, the accumulated and the freely moving dislocations per unit volume, distinguished as statistically stored and geometrically necessary ones, are in this way accounted for. Like for a generalized standard material, the shakedown safety factor is found to depend on the (generally size dependent) yield and saturation limits, but not on the particular differential-type hardening law of the material.

A non‐linear programming approach to kinematic shakedown analysis of composite materials

AbstractUsing a Representative volume element (RVE) to represent the microstructure of periodic composite materials, this paper develops a non‐linear numerical technique to calculate the macroscopic shakedown domains of composites subjected to cyclic loads. The shakedown analysis is performed using homogenization theory and the displacement‐based finite element method. With the aid of homogenization theory, the classical kinematic shakedown theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. By means of non‐linear mathematical programming techniques, a finite element formulation of kinematic shakedown analysis is then developed leading to a non‐linear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a composite is then obtained. An effective, direct iterative algorithm is proposed to solve the non‐linear programming problem. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples. This can serve as a useful numerical tool for developing engineering design methods involving composite materials. Copyright © 2005 John Wiley & Sons, Ltd.

Shakedown analysis of unidirectional fiber reinforced metal matrix composites

In this paper the determination of the shakedown domain in the space of macroscopic stresses of ductile unidirectional composite materials is obtained solving a shakedown analysis problems defined over a Representative Volume. The analyses are based on the kinematic shakedown theorem and the assumption of plane strain condition. The mathematical programming problem generated is solved by an iterative procedure. Numerical tests are performed on Al/Al 2O 3 metal matrix composite and compared to the relevant macroscopic plastic collapse domains.

Lower and upper shakedown bounds for fatigue limit in two phase materials

The goal of this study is to find the ‘safe’ long term behavior of elasto-plastic structural materials subjected to fluctuating load (shortly ‘fatigue limit’). The materials whose fatigue limits are checked are: (a) materials reinforced with unidirectional stiff fibers; (b) materials with dilute amount of inclusions; (c) materials with minute porosity. To reach this goal we employ two different shakedown theorems: Melan’s static shakedown theorem (1936) as the lower bound and Koiter’s kinematic shakedown theorem (1960) as the upper bound. The solutions to the lower and upper bounds for the prescribed stress amplitude, (Δσ th) l,b. and (Δσ th tu.b. , are expressed in a rigorous form with three parametric entities: (i) the volume fraction of the second phase in the base matrix phase, f, (ii) the quality of the ‘bond’ between the two phases ‘m’, (iii) the magnitude of the residual stress, ‘p’, pre-existed in the material. The deviation between the two bounds represents the ‘uncertainty’ in our knowledge of the actual safe/unsafe state, where the materials fail to withstand the alternating load. It is shown that in the considered three types of materials, at certain amount of residual stresses, the safe load amplitudes based on shakedown analysis are indeed higher than their corresponding elastic limits (at least by 5%, 10% and 25% respectively). The apparent advantage of using shakedown bounds to predict the safe/unsafe loading amplitude is that no prior information on the actual complex failure mechanisms is required and no empiricism is needed. However, empirical data which were found in the open literature are ‘falling’ satisfactorily between the computed dual bounds.

An eigen-mode method in kinematic shakedown analysis

Abstract The kinematic shakedown theorem is formulated for some deformation processes as a kinematic extremum problem based on a polyhedral load domain and a deformation mode domain. An eigen-mode method is used to construct the deformation mode domain. Every kinematically admissible strain field within some time interval can be derived from the deformation domain and used in the proposed shakedown formulation to determine the safety load factor. Several simple shakedown problems are examined by using this proposed methodology. These numerical results are discussed and compared with available analytical and other numerical results.

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.

Kinematical shakedown analysis with temperature-dependent yield stress

This paper describes a direct shakedown analysis of structures subjected to variable thermal and mechanical loading. The classical kinematical shakedown theorem is modified to be implemented with any displacement-based finite elements. The plastic incompressibility condition is imposed by the penalty function method. The shakedown limit is found via a non-linear mathematical programming procedure. Two numerical shakedown methods are developed and implemented to provide alternative numerical means. The temperature-dependent material model is included in theoretical and numerical calculation in a simple way. Its effect on shakedown limit is investigated. The numerical examination for some pressure vessel structures subjected to thermal and mechanical loading shows a satisfying precision and efficiency of the methods presented. Copyright © 2001 John Wiley & Sons, Ltd.

Consistent shakedown theorems for materials with temperature dependent yield functions

The (elastic) shakedown problem for structures subjected to loads and temperature variations is addressed in the hypothesis of elastic–plastic rate-independent associative material models with temperature-dependent yield functions. Assuming the yield functions convex in the stress/temperature space, a thermodynamically consistent small-deformation thermo-plasticity theory is provided, in which the set of state and evolutive variables includes the temperature and the plastic entropy rate. Within the latter theory the known static (Prager's) and kinematic (König's) shakedown theorems — which hold for yield functions convex in the stress space — are restated in an appropriate consistent format. In contrast with the above known theorems, the restated theorems provide dual lower and upper bound statements for the shakedown limit loads; additionally, the latter theorems can be expressed in terms of only dominant thermo-mechanical loads (generally the vertices of a polyhedral load domain in which the loadings are allowed to range). The shakedown limit load evaluation problem is discussed together with the related shakedown limit state of the structure. A few numerical applications are presented.

A mechanics model of erosion of ductile materials by backward extrusion

Examination of eroded surfaces of ductile materials under a scanning electron microscope typically shows thin slivers or platelets; these subsequently break off to provide wear debris. Cousens and Hutchings [A.K. Cousens, I.M. Hutchings, Wear 88 (1983) 335–348] have studied this phenomenon in careful experiments, in which commercially pure aluminium and aluminium alloys were eroded by using glass beads and iron shot in an air blast erosion tester. It was observed that a thin surface layer hardened both due to embedding of the fragments of erodent and due to work hardening. The repeated impacts by erodents, then, caused the softer material to extrude out backwards, from discontinuities in the hardened layer. It is proposed that this `backward extrusion' is due to `plastic ratchetting' of the material below the hardened layer, which is progressively compressed and extruded out. The proposed mechanism has been modelled by using kinematical shakedown theorem of the theory of plasticity and the impact pressure which needs to be exceeded for the process to continue has been evaluated. By expressing the impact pressure in terms of impact velocity and material properties of the shot and the eroding surface a non-dimensional velocity is defined and related to the shakedown behaviour of the eroding system. It is shown that for the backward extrusion process to continue the non-dimensional impact velocity must be above a critical value.

Shakedown of creeping structures

The classical approach to shakedown in elastic-plastic structures is extended here to include unlimited time dependent creep in a new way: by way of illustration a simple time-hardening model is used. Shakedown analysis for creep is typically investigated in two steps: an elastic-plastic shakedown analysis based on the classical theorems followed by a correction for creep, for example based on isochronous stress-strain curves for large times with limited creep. This paper provides a different treatment of the problem of estimating creep strains where shakedown could occur by extending the classical shakedown approach to include unlimited creep. New static and kinematic shakedown theorems are established and it is demonstrated that a necessary and sufficient condition for shakedown with mechanical and thermal cyclic loading is the existence of a time-dependent safe residual stress field. Two kinematic theorems are introduced: the fast connected with a kinematically admissible cycle of inelastic strain rate and the second with kinematically inadmissiable plastic strain rate. In addition all of the results are expressed in terms of generalized variables for structural applications. A few simple examples are examined to verify these results.

The dual shakedown conditions for dilute fibrous composites

The intention of this study is to find the safe long term behavior of elasto-plastic material, reinforced dilutely with unidirectional stiff fibers. The composite is subjected to fluctuating load which acts primarily transverse to the fibers. The aim is to predict what would be the highest allowable stress amplitude, commonly called the endurance limit, that the composite can endure when undergoing an infinite number of cycles of loading. For generality, an additional time-independent background stress may exist in the composite in the form of a real (or virtual) residual stress, generated by, say, thermal mismatch or a similar effect. To reach this goal, we employ the two shakedown theorems relating the allowable amplitude of the fluctuating load to the lifetime of the composite, (a) Melan's static shakedown theorem is used for establishing the lower bound to the endurance limit and (b) Koiter's kinematic shakedown theorem is used for establishing its upper bound. Both theorems are adjusted to capture an isotropic elasto-plastic material with an embedded single fiber (i.e. a dilute composite). The fiber/matrix bonding quality plays a vital role in the outcome. It is defined by a dimensionless scalar (the shear factor m) which ranges from a no-slip condition ( m = 1) to a free-to-rotate condition ( m = 0). For cases of non-dilute composites, a rough assessment of the error is provided. The dual bounds are expressed here in an approximate form with two parametric entities: the volume fraction of the fibers (a known value) and the quality of the fiber/matrix bonding (an assumed value), leaving the magnitude of the residual stress as an independent variable. The solution for the bounds becomes rigorous as the volume fraction of the fibers approaches zero (representing a single fiber in an infinite matrix).

A general approximate technique for the finite element shakedown and limit analysis of axisymmetrical shells. Part 1: Theory and fundamental relations

This paper describes the theory and the fundamental relations for the development of a displacement formulation for the finite element shakedown and limit analysis of axi-symmetrical shells. The material is assumed to be elastic–perfectly plastic. The technique is developed based upon an upper bound approach using a reformulated kinematic shakedown theorem for a shell with piecewise linear yield conditions. The solution of the problem is obtained by discretizing the shell into finite elements. A consistent relationship between the kinematically admissible velocity fields and the pure plastic strain rate fields during collapse needs to be enforced. Such requirement is satisfied by using the theory of conjugate approximations to minimize the residual of the two independent descriptions of the plastic strain increments. The discretized problem is then reduced to a minimization problem and solved by linear programming. The class of displacement fields chosen assumes plastic hinge lines forming at nodal points and only meridional and circumferential plastic strains occurring within the elements with no change in curvature. Examples of the application of the method are given in the accompanying paper. © 1997 John Wiley & Sons, Ltd.