A kinematic elastic nonshakedown theorem for implicit standard materials

Summary  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 Saxcé'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. Received 15 February 2001; accepted for publication 18 October 2001     

Bounding solutions for creeping structures subjected to load variations above the shakedown limit

For load variations above the shakedown limit, cyclic plasticity solutions are defined for yield criteria of perfect-plasticity and of kinematic strain-hardening. The cyclic plasticity solutions are used to provide upper bounds on the work, displacements and creep energy dissipations which occur in the cyclic stationary state of a creeping structure.     

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.     

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.     

A method for the evaluation of design limits for structural materials in a cyclic state of creep

The purpose of the present paper is to demonstrate how the minimum theorems proposed in an accompanying paper (Ponter and Boulbibane, 2002) can be utilised in the prediction of the deformation and life assessment of structures subjected to cyclic mechanical and thermal loadings. The developed method, which is based upon bounding theorems and an associate programming method, the Linear Matching method, takes into account the changes in residual stress field occurring within a cycle. Although the solution provided a bound on the inelastic work, it also appears that generally the displacements predicted by this solution are smaller than those that would be predicted by the rapid cycle solution. By way of illustration a simple non-linear viscous model is adopted and a number of solutions are presented involving a Bree plate problem subjected to cyclic histories of load and temperature. An elastic follow-up factor is identified as a key design parameter for high temperature dwell periods.     

A comparison between analytical calculations of the shakedown load by the bipotential approach and step-by-step computations for elastoplastic materials with nonlinear kinematic hardening

The class of generalized standard materials is not relevant to model the nonassociative constitutive equations. The bipotential approach, based on a possible generalization of Fenchel’s inequality, allows the recovery of the flow rule normality in a weak form of an implicit relation. This defines the class of implicit standard materials. For such behaviours, this leads to a weak extension of the classical bound theorems of the shakedown analysis. In the present paper, we recall the relevant features of this theory. Considering an elastoplastic material with nonlinear kinematic hardening rule, we apply it to the problem of a sample in plane strain conditions under constant traction and alternating torsion in order to determine analytically the interaction curve bounding the shakedown domain. The aim of the paper is to prove the exactness of the solution for this example by comparing it to step-by-step computations of the elastoplastic response of the body under repeated cyclic loads of increasing level. A reliable criterion to stop the computations is proposed. The analytical and numerical solutions are compared and found to be closed one of each other. Moreover, the method allows uncovering an additional ‘2 cycle shakedown curve’ that could be useful for the shakedown design of structure.     

Shakedown theory for elastic plastic kinematic hardening bodies

Shakedown static and kinematic theorems for elastic–plastic (generally nonlinear) kinematic hardening solids are derived in classical (path-independence) spirit with new constructions. The generally plastic-deformation-history-dependent hardening curve is assumed to be limited by the initial yield stress and ultimate yield strength, and to obey a positive hysteresis postulate for closed plastic cycles, but else can be arbitrary and unspecified. The theorems reveal that the shakedown of structures is not affected by the particular form of the hardening curve, but just by the initial and ultimate yield stresses. While the ultimate yield strength is clearly defined macroscopically and attached to the incremental collapse mode with unbounded plastic deformations, the initial yield stress, which is responsible for the bounded cyclic plasticity collapse mode, should not be taken as the convenient one at a fixed amount of plastic deformation (0.2%), but is suggested to be taken as low as the fatigue limit to preserve the classical load-history-independence spirit of the shakedown theorems. Otherwise, for our pragmatic application purpose, it may be given empirical values between the low fatigue limit and high ultimate yield stresses according to particular loading processes considered, which may range anywhere between the high-cycle and low-cycle ones. The theorems appear as simple as those of Melan and Koiter for perfect plasticity but applied to the much larger class of more realistic kinematic hardening materials.     

Multiaxial creep and cyclic plasticity in nickel-base superalloy C263

Physically-based constitutive equations for uniaxial creep deformation in nickel alloy C263 [Acta Mater. 50 (2002) 2917] have been generalised for multiaxial stress states using conventional von Mises type assumptions. A range of biaxial creep tests have been carried out on nickel alloy C263 in order to investigate the stress state sensitivity of creep damage evolution. The sensitivity has been quantified in C263 and embodied within the creep constitutive equations for this material. The equations have been implemented into finite element code. The resulting computed creep behaviour for a range of stress state compares well with experimental results. Creep tests have been carried out on double notched bar specimens over a range of nominal stress. The effect of the notches is to introduce multiaxial stress states local to the notches which influences creep damage evolution. Finite element models of the double notch bar specimens have been developed and used to test the ability of the model to predict correctly, or otherwise, the creep rupture lifetimes of components in which multiaxial stress states exist. Reasonable comparisons with experimental results are achieved. The γ^′ solvus temperature of C263 is about 925 °C, so that thermo-mechanical fatigue (TMF) loading in which the temperature exceeds the solvus leads to the dissolution of the γ^′ precipitate, and a resulting solution treated material. The cyclic plasticity and creep behaviour of the solution treated material is quite different to that of the material with standard heat treatment. A time-independent cyclic plasticity model with kinematic and isotropic hardening has been developed for solution treated and standard heat treated nickel-base superalloy C263. It has been combined with the physically-based creep model to provide constitutive equations for TMF in C263 over the temperature range 20–950 °C, capable of predicting deformation and life in creep cavitation-dominated TMF failure.     

On some general variational principles for creep with applications to thin shells

A constitutive relation for a viscous material subject to small strains but finite rotations is postulated and associated variational theorems are formulated. These are similar to the principle of minimum of potential energy and the Hellinger-Reissner theorem of an elastic solid. The derivation of strain-displacement relations for thin shells subject to small strains but moderately large rotations are given. On this basis a mixed variational principle for thin viscous shells is developed. For the problem of creep collapse of long cylindrical shells under external pressure it is demonstrated that the mixed variational principle may be advantageous compared to other variational theorems. A comparison with the creep collapse theory of Hoff et al. is given.     

Variational methods for the steady state response of elastic–plastic solids subjected to cyclic loads

Solids (or structures) of elastic–plastic internal variable material models and subjected to cyclic loads are considered. A minimum net resistant power theorem, direct consequence of the classical maximum intrinsic dissipation theorem of plasticity theory, is envisioned which describes the material behavior by determining the plastic flow mechanism (if any) corresponding to a given stress/hardening state. A maximum principle is provided which characterizes the optimal initial stress/hardening state of a cyclically loaded structure as the one such that the plastic strain and kinematic internal variable increments produced over a cycle are kinematically admissible. A steady cycle minimum principle, integrated form of the aforementioned minimum net resistant power theorem, is provided, which characterizes the structure’s steady state response (steady cycle) and proves to be an extension to the present context of known principles of perfect plasticity. The optimality equations of this minimum principle are studied and two particular cases are considered: (i) loads not exceeding the shakedown limit (so recovering known results of shakedown theory) and (ii) specimen under uniform cyclic stress (or strain). Criteria to assess the structure’s ratchet limit loads are given. These, together with some insensitivity features of the structure’s alternating plasticity state, provide the basis to the ratchet limit load analysis problem, for which solution procedures are discussed.     

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.     

Extensions of the Prager-Shield theory of optimal plastic design

The Prager-Shield associated displacement field method for optimal plastic design is extended to multi-component specific cost functions and multiple load conditions, and a lower bound theorem based on kinematic requirements only is introduced. Since any statically admissible stress field results in an upper bound, the proposed theorem provides a simple method for establishing bounds on the optimal cost. By a simple substitution of parameters into the general equations presented, the optimality criteria can be obtained for particular design problems. Examples of optimal fibre-reinforced plates are given.     

Strain gradient plasticity,strengthening effects and plastic limit analysis

Within the framework of isotropic strain gradient plasticity, a rate-independent constitutive model exhibiting size dependent hardening is formulated and discussed with particular concern to its strengthening behavior. The latter is modelled as a (fictitious) isotropic hardening featured by a potential which is a positively degree-one homogeneous function of the effective plastic strain and its gradient. This potential leads to a strengthening law in which the strengthening stress, i.e. the increase of the plastically undeformed material initial yield stress, is related to the effective plastic strain through a second order PDE and related higher order boundary conditions. The plasticity flow laws, with the role there played by the strengthening stress, are addressed and shown to admit a maximum dissipation principle. For an idealized elastic perfectly plastic material with strengthening effects, the plastic collapse load problem of a micro/nano scale structure is addressed and its basic features under the light of classical plastic limit analysis are pointed out. It is found that the conceptual framework of classical limit analysis, including the notion of rigid-plastic behavior, remains valid. The lower bound and upper bound theorems of classical limit analysis are extended to strengthening materials. A static-type maximum principle and a kinematic-type minimum principle, consequences of the lower and upper bound theorems, respectively, are each independently shown to solve the collapse load problem. These principles coincide with their respective classical counterparts in the case of simple material. Comparisons with existing theories are provided. An application of this nonclassical plastic limit analysis to a simple shear model is also presented, in which the plastic collapse load is shown to increase with the decreasing sample size (Hall–Petch size effects).     

Creep life predictions for structures subject to variable loading

Previous work which established upper and lower bounds on the creep life of steadily loaded structures is extended to cater for load and temperature variations in non-homogeneous structures. The investigation is limited to the range where short term plasticity and fatigue damage can be ignored. For proportional loading, the upper bound which is based on limit analysis, is similar in form to that for constant loading. In the more general case, the upper bound is less stringent and is based on the mean load and temperature distribution over the lifetime. A lower bound on life is taken as the time for the first part of the structure to fail.The bounds are applied to three simple structures. For proportional loading the upper bound predicts the lifetime with the same accuracy as for constant loading except for extreme load variations. The presence of a temperature distribution alters the accuracy of the upper bound prediction but in most cases the change is small. In contrast, the lower bound is very sensitive to the temperature gradient.The authors use these results to develop approximate techniques for estimating the creep life of components subjected to variable loads and temperature distributions. Simplified design procedures based on the upper bound are examined and suitable amendments are proposed.     

Reduction of Kinematic: Inequality and Optimization of Perfectly Plastic Frameworks

Abstract

In the kinematic theory of structures consisting of perfectly plastic elements, an inequality between the plastic dissipation work and the load work is used. This inequality, which we will term “the kinematic inequality,” must hold for all kinematically admissible mechanisms. These mechanisms are generated by certain parameters which usually remain in the kinematic inequality and which thereby preclude the general application of the kinematic approach. In this paper we overcome this difficulty in the case of frames and provide various applications of the method. By using new theorems we eliminate the parameters and reduce the kinematic inequality to a finite system of inequalities which depend only on frame geometry and on loads. Based on these theorems, a procedure is offered for deriving a system of independent inequalities for general multistory multibay frames. New theorems are then obtained regarding the existence and the rotation of certain plastic hinges in collapse mechanisms. The overall theory is illustrated by a specific example. Finally, the formulations obtained following our method are used to minimize the mass of a fixed-base rectangular portal frame for any length, height, and system of loads.     

Stability against localization of softening into ellipsoids and bands: Parameter study

Strain-localization instabilities due to strain-softening which result from distributed damage such as cracking in heterogeneous brittle materials are analysed. Attention is restricted to the stability problem of equilibrium states. This problem is not equivalent to bifurcation of the equilibrium path, which may occur before stability of equilibrium is lost. The continuum is local but is enhanced by the localization limiter used in the crack band model, consisting of a lower bound on the minimum dimension of the strain-localization region, which is regarded as a material property. Presented are derivations of the critical state conditions for localization of initially uniform strain into ellipsoidal domains within an infinite continuum and into a planar band within a layer of finite thickness. These derivations are simpler than the previous Ba ant's derivations of the general stability conditions for these localizations. A numerical parameter study of the critical states is made for a broad range of material properties as well as various initial stress states and relative sizes of the strain-softening region. The material is described by Drucker-Pragcr plasticity with strain-softening that is caused by yield limit degradation. The Hatter the ellipsoidal domain, or the larger the size of the body (layer thickness), the smaller is found to be the strain-softening slope magnitude at which the critical state is reached. A softening Drucker-Prager material is found to be stable against planar-band localizations in infinite continuum for a certain range of softening material parameters.     

Limit analysis of masonry vaults by means of curved shell finite elements and homogenization

The study of masonry vaults should take into account the essentials of the material “masonry” – i.e. heterogeneity, almost no resistance to tension combined with a good compressive strength and a high friction coefficient, as well as the overall importance of the geometry for achieving the equilibrium.In this paper, a new six-noded triangular curved element, specifically developed for the kinematic limit analysis of masonry shells, is presented. Plastic dissipation is allowed only at the interfaces (generalized cylindrical hinges) between adjoining elements for combined membrane actions, bending moment, torsion and out-of-plane shear, as it is required for the analysis of thick (Reissner–Mindlin) shells. An upper bound of the collapse load is so obtained, since, looking at the dual formulation, the admissibility of the stress state is imposed only at the element boundaries. Masonry strength domain at each interface between contiguous triangular elements is evaluated resorting to a suitable upper bound FE homogenization procedure. The model is assessed through several numerical simulations on a number of masonry shells experimentally tested until collapse. In particular, the dependence of the collapse multiplier on the mesh and on the material parameters (sensitivity analysis) is thoroughly discussed.     

On the equivalence of slip-line fields and work principles for rigid–plastic body in plane strain

Extremum/work principles for a rigid–plastic body have been discussed in classical theory of plasticity to be of immense significance. Unfortunately, till now, these extremum theorems have been used only as a crude method of obtaining the limit load of a rigid–plastic body, using successive approximations by upper and lower bound estimates. On the other hand slip-line fields (SLF) have been extensively used not only for evaluation of limit load but also for obtaining sufficiently accurate estimates of stresses in the plastic region as well as in the vicinity of crack tip. Till now, these two methods of plastic analyses, that is, the work principles and SLF have remained more or less independent apart from the fact that both are upper bounds as they use kinematically admissible velocity fields. Recently, a new load bounding technique, modified upper bound (MUB) Approach, was proposed by Khan and Ghosh [Khan, I.A., Ghosh, A.K., 2007. A modified upper bound approach to limit analysis for plane strain deeply cracked specimens. International Journal of Solids and Structures 44 (10), 3114–3135]. In this article, a rigorous mathematical basis of this load bounding technique is presented and it is demonstrated that the method is actually a new form of the general extremum/work principles. The equivalence of this new form of work principle, that is, MUB with the classical SLF analysis, for a rigid–plastic material in plane strain, has been discussed in detail. Since plastic deformation fields depend on specimen geometry and type of loading specific cases have been considered. Both cracked and uncracked configurations have been analysed to establish this equivalence in general. Various simplifications resulting from the use of this new load bounding technique over SLF method has been demonstrated. Several standard problems of plane strain analysed by SLF method and validated by experiments in past have been considered in this article. As a novel application of the proposed method, single-edge-cracked plate under combined bending and tensile load has been analysed. For this specimen SLF solutions are available only for bending with small tensile load (defined in Section 3.2.4) while classical upper bound solutions are valid for bending with large tensile load. In this work a completely analytical formulation for yield locus for the entire range of tensile and bending load has been obtained. Apart from accurate evaluation of limit load, detailed evaluation of crack tip stresses and hence constraint near the crack tip has been performed using this new form of work principles.     

Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach

The class of generalised standard materials is not relevant to model the non-associative constitutive equations. The possible generalisation of Fenchel's inequality proposed by de Saxcé allows the recovery of flow rule normality for non-associative behaviours. The normality rule is written in the weak form of an implicit relation. This leads to the introduction of the class of implicit standard materials. This formulation is applied to constitutive equations involving non-linear kinematic hardening, indispensable to describe accurately and realistically the cyclic plasticity of metallic materials. For these plastic flow rules shakedown bound theorems can be extended; an analytical example of the shakedown of a thin-walled tube under constant traction and alternate cyclic torsion is considered and the obtained solution is proved to be exact.