Plastic collapse of a circular plate under cyclic loads

The shakedown kinematic approach is used to analyze the collapse modes for a circular plate subjected to cyclic loads distributed axisymmetrically. Depending upon the loading scheme and parameters, the plate may fail by instantaneous, incremental, or alternating plasticity collapse. For quasistatic processes the nonshakedown load may be smaller than the instantaneous plastic limit load. In the case of periodic dynamic loading, which lies outside the framework of plastic limit analysis, the frequency parameter, besides the load amplitudes, can significantly influence the nonshakedown behavior of the plate.     

Shakedown kinematic theorem for elastic–perfectly plastic bodies

The classical shakedown kinematic theorem due to Koiter for elastic–perfectly plastic bodies is re-examined and divided into separated shakedown and nonshakedown theorems. While the shakedown theorem is based on the set of Koiter's plastic strain rate cycles, the non-shakedown one involves a broader set of admissible plastic strain rate cycles, the end-cycle accumulated strains of which are deviatoric parts of compatible strain fields. For certain broad classes of practical problems the two statements are unified to yield the unique theorem in Koiter's sense.     

Failure of a circular reinforced concrete plate against dynamic fluctuating loads

Summary The kinematic shakedown method is used to evaluate the dynamic cyclic collapse loads for a clamped reinforced concrete plate. The method, based on reduced kinematic formulae recently derived by the author, requires determination of the fictitious elastic fields within the plates, in particular – their envelope corresponding to the external loads, and construction of potential incremental collapse mechanisms. The load amplitude-frequency diagrams constructed should help to choose the necessary reinforcements for the plate under given loading conditions. Received 14 January 1998; accepted for publication 25 June 1998     

Shakedown analysis of shape memory alloy structures

Phase transformational shakedown of a structure refers to a status that plastic strains cease developing after a finite number of loading cycles, and subsequently the structure undergoes only elastic deformation and alternating phase transformations with limited magnitudes. Due to the intrinsic complexity in the constitutive relations of shape memory alloys (SMA), there is as yet a lack of effective methods for modeling the mechanical responses of SMA structures, especially when they develop both phase transformation and plastic deformation. This paper is devoted to present an algorithm for analyzing shakedown of SMA structures subjected to cyclic or varying loads within specified domains. Based on the phase transformation and plastic yield criteria of von Mises-type and their associated flow rules, a simplified three-dimensional phenomenological constitutive model is first formulated accounting for different regimes of elastic–plastic deformation and phase transformation. Different responses possible for SMA bodies exposed to varying loads are discussed. The classical Melan shakedown theorem is extended to determine a lower bound of loads for transformational shakedown of SMA bodies without necessity of a step-by-step analysis along the loading history. Finally, a simple example is given to illustrate the application of the present theory as well as some basic features of shakedown of SMA structures. It is interesting to find that phase transformation may either increase or decrease the load-bearing capacity of a structure, depending upon its constitutive relations, geometries and the loading mode.     

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.     

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.     

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.     

Weak Forms of Shakedown for Elastic–Plastic Structures Exhibiting Ductile Damage

A special weak-form shakedown is studied for elastic–plastic internal-variable material models with nonlinear hardening, damageable elastic moduli and damageable yield surface, in the hypothesis of ductile damage, (i.e. damage induced by plastic strains), but the precise evolutive law of damage being left unspecified. Sufficient weak-form shakedown theorems are presented, one static and another kinematic, each assessing whether eventually plastic deformations cease together with their consequences, including ductile damage. A two-sided delimitation is provided, within which the weak-form shakedown safety factor can be located. An upper bound to the post-transient damage for a particular isotropic damage model is also proposed. A simple numerical application is presented. Sommario. Si studia una forma speciale debole di adattamento strutturale per materiali elasto-plastici con variabili interne e con incrudimento non lineare, nonché moduli elastici e superficie di plasticità soggetti a danno duttile (cioè indotto dalle deformazioni plastiche) con una non precisata legge evolutiva. Si presentano due teoremi sufficienti per l'adattamento in forma debole, ciascuno essendo capace di predire se la deformazione plastica infine cessa unitamente alle sue conseguenze, compreso il danno duttile. Si definisce una delimitazione bilaterale entro la quale ricade il fattore di sicurezza associato a questa forma debole di adattamento. Si propone altresì una delimitazione superiore per il danno accumulato nella fase transitoria nel caso di un tipo particolare di danno isotropo. Si presenta una semplice applicazione numerica.     

Structural shakedown for elastic–plastic materials with hardening saturation surface

An elastic–plastic material model with internal variables and thermodynamic potential, not admitting hardening states out of a saturation surface, is assumed as a basis to formulate a statical Melan-type shakedown theorem. Grounding on the optimality conditions relative to the shakedown load multiplier problem for a structure subjected to cyclic loads, the impending inadaptation collapse mechanism at the shakedown limit state is analyzed and discussed. It is shown that the adopted model is able to catch ratchetting collapse mode at a structural level. Numerical results for a simple structure are finally reported.     

Elastic–plastic FE analysis of a notched shaft under multiaxial nonproportional synchronous cyclic loading

An elastic–plastic finite element analysis is presented for a notched shaft subjected to multiaxial nonproportional synchronous cyclic tension/torsion loading. The elastic–plastic material property is described by the von Mises yield criterion and the kinematic hardening rule of Prager/Ziegler. The finite element program system ABAQUS is used to solve the boundary value problem. Special emphasis is given to explore the effects of the stress amplitude, the mean-stress, and the mutual interactions on the local stress–strain responses at the notch root.     

On the yield functions in Lagrangian stress space of elastic–plastic materials with internal constraints

In the present paper it is shown that the elastic range in the second Piola–Kirchhoff stress space can be chosen in a hyperplane which is through the origin of Lagrangian stress space and perpendicular to the normal of the constraint manifold at the plastic configuration, if the determinate stress response of the elastic–plastic material with simple internal constraints with some condition is correctly chosen, otherwise, it is in general in a hypersurface and the normal flow rule by Il yushin’s postulate will have an indeterminate part. The choice of determinate stress response is probable because of its indeterminacy. Therefore the yield function should be a function of the second Piola–Kirchhoff stress lying in the hyperplane so that it is more simple and the back stress as the geometric center of the elastic range in general is inside the elastic range. Finally some examples are concerned. The project supported by the National Natural Science Foundation of China (10272055).     

Theoretical and computational aspects in the shakedown analysis of finite elastoplasticity

A fully nonlinear shakedown analysis is considered for structures undergoing large elastic-plastic strains. The underlying kinematics of finite elastoplasticity are based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. It is Shown that the notion of a fictitious, self-equilibrated residual stress field of Melan's linear shakesdown theorem has to be replaced by the notion of real, self-equilibrated residual state. Path-dependent and path-independent shakedown theorems are presented that can be realized in an incremental step-by-step procedure using Finite Element codes. The numerical implementation is considered for highly nonlinear truss structures undergoing large cyclic deformations with ideal-plastic, isotropic and kinematic hardening material behavior. Path-dependency of the residual states in the case of non-adaptation and path-independency in the case of shakedown are shown, and the shakedown domain is determined taking into account also the stability boundaries of the structure.     

On shakedown theory for elastic-plastic materials and extensions

The idea that an elastic-plastic structure under given loading history may shake down to some purely elastic state (and hence to a safe state) after a finite amount of initial plastic deformation, can apply to many sophisticated material models with possible allowable changes of additional material characteristics, as has been done in the literature. Despite some claims to the contrary, it is shown; however, that the shakedown theorems in a Melan-Koiter path-independent sense have been extended successfully only for certain elastic-plastic hardening materials of practical significance. Shakedown of kinematic hardening material is determined by the ultimate and initial yield stresses, not the generally plastic deformation history-dependent hardening curve between. The initial yield stress is no longer the convenient one (corresponding to the plastic deformation at the level of 0.2%) as in usual elastic-plastic analysis but to be related to the shakedown safety requirement of the structure and should be as small as the fatigue limit for arbitrary high-cycle loading. Though the ultimate yield strength is well defined in the standard monotonic loading experiment, it also should be reduced to the so-called “high-cycle ratchetting” stress for the path-independent shakedown analysis. A reduced simple form of the shakedown kinematic theorem without time integrals is conjectured for general practical uses. Application of the theorem is illustrated by examples for a hollow cylinder, sphere, and a clamped disk, under variable (including quasiperiodic dynamic) pressure.     

A model for the through-thickness elastic–plastic behaviour of paper

An elastic–plastic material model for the out-of-plane mechanical behaviour of paper is presented. This model enables simulation the elastic–plastic behaviour under high compressive loads in the through-thickness direction (ZD). Paper does not exhibit a sharp transition from elastic to elastic–plastic behaviour. This makes it advantageous to define critical stress states based on failure stresses rather than yield stresses. Moreover, the failure stress in out-of-plane shear is strongly affected by previous plastic through-thickness compression. To cover these two features, a model based on the idea of a bounding surface that grows in size with plastic compression is proposed. Here, both the bounding and the yield surfaces are suggested as parabolas in stress space. While the bounding surface is open for compressive loads, the yield surface is bordered by the maximum applied through-thickness compression.     

Thermal stresses and shakedown in wheel/rail contact

Summary  Sliding friction between railway wheels and rails results in considerable contact temperatures and gives rise to severe thermal stresses at the surfaces of the wheels and rails. An approximate analytical solution is presented for a line contact model. The increased bulk temperature of the wheel after a long period of constant operating conditions is also taken into account. The thermal stresses have to be superimposed on the mechanical contact stresses. They reduce the elastic limit of the wheel and rail, and yielding begins at lower mechanical loads. When residual stresses build up during the initial cycles of plastic deformation, the structure can carry higher loads with a purely elastic response in subsequent load cycles. This phenomenon is referred to as shakedown. Due to the distribution of temperature, the rail surface is generally subjected to higher stresses than the wheel surface. This can cause structural changes in the rail material and hence rail damage. Received 7 May 2002; accepted for publication 3 September 2002     

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.     

Approximate elastic–plastic analysis of a notched bar under tensile and torsional fatigue

An approximate model is presented for estimating elastic–plastic stresses and strains in a notched bar subjected to synchronous non-proportional tensile and torsional cyclic loading. To begin with, it is applied to a multiaxial synchronous proportional loading system. A detailed FE analysis is performed for an axisymmetric bar of circular cross-section with a circumferential notch. The accuracy of the proposed approximate model is first established. The elastic–plastic material property is described by the von Mises yield criterion and a linear kinematic hardening rule. The comparison of the FE results with those estimated by the proposed approximate model shows good agreement for all loading cases investigated.     

Shakedown theorems for some classes of nonassociative hardening elastic-plastic material models

Criteria for quasistatic (elastic) shakedown are established by a static approach and discussed in the light of illustrative examples. Reference is made to elastoplastic solids either with nonassociative plastic flow rules or nonlinear hardening (governed by internal variables), or both. Nonassociativity, that is, lack of normality, is known to occur in the superposed spaces of stresses and strains for the constitutive models of materials with internal friction (like most geomaterials); it is shown also to manifest in the augmented space (of both measurable and internal variables) for realistically modelling a hardening behaviour of metals subjected to cyclic stressing processes. Lower and upper bounds on the shakedown limit derived for these materials from sufficient and necessary shakedown criteria, respectively, are distinct due to the nonassociativity. They suggest recourse to the notion of the reduced yield domain different from the conventional yield domain defined in the augmented space of static variables.     

ANALYSIS OF SHAKEDOWN OF FG BREE PLATE SUBJECTED TO COUPLED THERMAL-MECHANICAL LOADINGS

The static and kinematic shakedown of a functionally graded （FG） Bree plate is analyzed. The plate is subjected to coupled constant mechanical load and cyclically varying temperature. The material is assumed linearly elastic and nonlinear isotropic hardening with elastic modulus,yield strength and the thermal expansion coeffcient varying exponentially through the thickness of the plate. The boundaries between the shakedown area and the areas of elasticity,incremental collapse and reversed plasticity are determined,respectively. The shakedown of the counterpart made of homogeneous material with average material properties is also analyzed. The comparison between the results obtained in the two cases exhibits distinct qualitative and quantitative difference,indicating the importance of shakedown analysis for FG structures. Since FG structures are usually used in the cases where severe coupled cyclic thermal and mechanical loadings are applied,the approach developed and the results obtained are significant for the analysis and design of such kind of structures.     

Elastic–plastic behavior of a semicircular frame being pressed against a rigid plane

As a simplified structural model, a semicircular frame is used to study the crashworthiness behavior of an aircraft fuselage. The quasi-static large elastic-plastic deformation of a semicircular frame in the process of its being pressed against a rigid ground is analyzed. First, based on the linear elastic assumption, the quasi-static large deformation contact process of the frame can be divided into three phases, i.e., point contact, line contact and post-buckling. By means of a shooting method, the relations between the displacement and contact force as well as the distribution of bending moment in the three phases are obtained. Then, by assuming an elastic, perfectly-plastic moment-curvature relationship for the semi-circular frame, the contact process is analyzed in detail to reveal the plastic collapse mechanism, the traveling of plastic hinge and the force-displacement relationship. In order to verify the analysis, a preliminary experiment was conducted, in which two types of half rings with clamped ends were pressed by a rigid plate. In addition, a numerical simulation is also conducted by employing ABAQUS to analyze both rectangular cross-sectional beam and I-beam. Finally, the theoretical predictions are compared with the experimental results and numerical solutions, showing that the elastic-plastic analysis can predict the contact process very well.