Shakedown analysis of elastoplastic structures: A review of recent developments

Classical shakedown analysis rests on the assumptions of perfectly plastic, associative temperature-independent constitutive laws, negligible inertia and damping forces and negligible geometric effects.This paper provides a survey of the recent literature on the structural behaviour under variable repeated loads, with emphasis on the developments which relaxed some of the above assumptions, but preserved the character of generalization of limit analysis typical of the ‘classical’ shakedown theory and methods of analysis and design (in contrast to evolutive, step-by-step approaches of incremental plasticity).     

DYPLAS, a finite element dynamic elastic-plastic large deformation analysis program

DYPLAS is a computer program designed to compute the response history of general three-dimensional structures subjected to transient thermal and mechanical loadings. The analysis considers nonlinearities arising from material behavior as well as from large deformations that may occur in the structure. Elastic-plastic behavior with isotropic hardening is included by means of incremental relations of the Prandtl-Reuss type combined with a variant of the ‘plastic strain-total strain’ method. Reduction of the constitutive relations to the case of generalized plane stress is also presented. Spatial discretization is achieved by applying the displacement formulation of the finite element method. In order to achieve reasonable computational time, the number of computational points used in the algorithm for numerical integration over the volume of each finite element required minimization; a triangular, flat-plate bending element is cited as an example for which this minimization was achieved satisfactorily. An explicit scheme is used for computation of dynamic response. With this scheme, formation of a global stiffness matrix is not needed. Consequently, the usual limitations arising from bandwidth or problem size are completely eliminated.     

Elastic-plastic deformation of cylindrical pressure vessels under cyclic loading

A numerical procedure based on the finite element method and incremental solution approach is presented for analyzing cylindrical pressure vessels deformed in the state of generalized plane strain. The structures considered are subjected to internal pressure, thermal gradients and axial forces, which are cyclic in nature. The materials are assumed to obey the von Mises yield criterion with kinematic strain hardening which, during the process of plastic deformation, the yield surface is permitted to translate in the stress space and may change its size as a function of temperature. Based on Drucker's postulates for stable materials in the incremental theory of plasticity, a noniso-thermal flow rule is deduced in the cylindrical coordinates. By use of the virtual work principle, the finite element equations with displacement formulations are derived. The numerical solution is then carried out by a step-by-step approach for small loading increments and an iteration scheme is employed for each loading step. Furthermore, the analysis method is extended to the determination of limit load and shakedown load of a vessel. Several problem cases are presented.     

Interaction buckling-progressive deformation

The interaction between buckling and ratchetting under cyclic additional loading is discussed for elastic-plastic structures. It is shown that progressive plastic deformation can lead to the buckling of a structure as a consequence of the resulting additional displacement. This interaction is particularly strong in perfect plasticity. In the case of kinematic hardening materials, an estimate of the primary loading is given in order to prevent the risk of ratchetting. It is shown that elastic and plastic shake-down theorems still hold when the primary load is smaller than the smallest critical load of tangent modulus.     

High temperature strength and inelastic behavior of plate–fin structures for HTGR

In this paper, both high temperature strength and inelastic behavior of plate–fin structures were discussed for applying these structures to the compact heat exchangers such as recuperative and intermediate heat exchangers for high-temperature gas-cooled reactors (HTGR). Firstly tensile, creep and fatigue tests of the brazed plate–fin model of small size were carried out to obtain the rupture strength and inelastic behavior. The influence of the braze filler metal thickness on the tensile strength was experimentally studied and a possibility of predicting both the tensile and creep strength was discussed using the data of base material of plates and fins. Secondly, we demonstrated the fabrication of large-size core with a dimension of 1000 mm, and also demonstrated that the bonding ratio in this core was improved up to almost 100% by adopting the pressurized tank system in the brazing process. Finally, we proposed the stress analysis method of plate–fin structures on the basis of the equivalent-homogeneous-solid concept, and carried out the elastic–plastic analysis of recuperative heat exchanger for HTGR. Characteristics of stress–strain behavior were discussed together with a possibility of predicting the fatigue life of the structure.     

Flaw evaluation of Japanese carbon steel piping by load curve approach

New simplified flaw evaluation method, ‘load curve approach’ was developed to evaluate the fracture load of circumferentially surface-cracked pipe. This approach has the same functions with the current two-criteria approach. Fracture stress and fracture criteria are easily estimated by two load curves based on elastic–plastic fracture mechanics and plastic collapse. Fracture analysis was conducted for Japanese carbon steel piping using this approach. The approach showed the dependency of flaw geometry and pipe diameter on pipe fracture. Z-factors were calculated from this approach and compared with Z-factors by ASME Boiler & Pressure Vessel Code, Section XI (ASME-XI) and Japanese Code. It is shown that Z-factors by the load curve approach can improve the conservativeness in the estimation of pipe fracture load.     

A knowledge-based multi-dimension discrete common cause failure model

Common cause failure (CCF) is analyzed as a manifestation of the probabilistic characteristics of component failure rate/probability stemming from stochastic environment load. CCF revolved concepts, such as ‘root cause’ and ‘coupling mechanism’ are interpreted mathematically from the viewpoint of random environment load bringing about failure dependency. Opinions about ‘inherent CCF’ and ‘additional CCF’, ‘absolute CCF’ and ‘relative CCF’ are presented and discussed. An easy-to-use CCF model is developed through multi-dimension environment load-component strength interference analysis and knowledge based parameter discretization. Owing to its strict statistical foundation, such a model has the ability of estimating component failure rate/probability and common cause failure rates/probabilities consistently, dealing with low redundancy system CCF and high redundancy system CCF uniformly, and predicting high multiplicity failure rate/probability based on low multiplicity failure data satisfactorily.     

Computational techniques for inelastic stress and fracture analyses

Details of a unified creep—plasticity theory, based on the concept of an internal time, are presented, and a geometric interpretation of such a theory is given. A ‘tangent-stiffness’ formulation, for the finite-element/boundary-element calculation of weak solutions of the strain-history in the structure, is presented. An implicit algorithm of generalized mid-point radial mapping for computing the stress-history at a material point for a given strain-history is given. Appropriate crack-tip parameters that may be used to correlate the creep-crack-growth data for single—dominate-flaws in structures operating at elevated temperatures are discussed.     

Numerical implementation of a transverse-isotropic inelastic, work-hardening constitutive model

This paper documents the numerical implementation of a transverse-isotropic inelastic, work-hardening plastic constitutive model. A brief review of the model is presented first to facilitate the understanding of its numerical implementation. This model is formulated in terms of ‘pseudo’ stress invariants, so that the incremental stress-strain relationship can be readily incorporated into existing finite-difference or finite-element computer codes. The anisotropic model reduces to its isotropic counterpart without any changes in the mathematical formulation or in the numerical implementation (algorithm) of the model.The input quantities to the algorithm are the initial stress components obtained at the end of the previous strain increment, and the new strain increments; the output quantities are the new values of the stress components. In the first step of the algorithm a set of elastic trial stresses computed, which are then tested with respect to the failure criteria. If they do not violate the failure criteria, the behavior of the material is truly elastic and these stresses are the correct stresses for the given strain increments. On the other hand, if the elastic trial stresses violate the failure criteria, they are corrected through an iteration scheme using an appropriate flow rule.A typical example of the model and its behavior in uniaxial strain and triaxial compression is presented.     

The viscoplasticity theory based on overstress applied to the analysis of beams under monotonic and cyclic creep loading

The viscoplasticity theory based on total strain and overstress can reproduce rate-dependent inelastic deformation without distinction between plastic and creep strain using two material functions. A viscosity function and an equilibrium stress-strain curve characterize rate-dependency and work hardening, respectively. The theory is used to analyze the creep and cyclic creep behavior of a beam subjected to a linearly increasing moment which is subsequently held constant.The analysis shows the existence of two possible states of equilibrium for creep deformation: termination of primary creep or secondary creep. They occur when the equilibrium stress-strain curve has positive or zero slope in the plastic range.The numerical experiments illustrate that the stress distribution at the end of the moment increase depends on the moment rate. The rate effects disappear with time when stress is redistributed. For practical purposes the equilibrium solution is obtained before 10⁷ s, when the material functions representing AISI Type 304 Stainless Steel at room temperature are used. The other equilibrium solution (secondary creep) is reached after primary creep when the constant moment is above the limiting equilibrium moment which corresponds to the plastic hinge moment of plasticity theory. The stress distribution during stationary creep is shown to be the solution corresponding to the Norton law of creep theory. The numerical experiments also illustrate the influence of various viscosity functions and equilibrium stress levels. A growth law for the equilibrium stress-strain curve is postulated and reversed loading as well as repeated loading are investigated.     

A flow rule based on Huddleston’s multiaxial rupture criterion as a yield function

An elastic–plastic constitutive equation is proposed to develop a continuum damage model for multiple flawed medium. The constitutive model takes into account the effect of hydrostatic stress to describe the decrease of rigidity and flow stress in tension and has an extended form of Huddleston’s equi-rupture function. The decrease of those properties due to an evolution of damage is described by a proper selection of parameters in the yield function in accordance with a state of damage. Some case studies are shown. The model can be extended to include the kinematic and combined hardening rules or incorporated in constitutive equations for finite element calculations such as unified models.     

Strain-rate dependent plasticity in thermo-mechanical transient analysis

The thermo-mechanical transient behavior of fuel element cladding and other reactor components is generally governed by the strain-rate properties of the material. Relevant constitutive modeling requires extensive material data in the form of strain-rate response as function of true-stress, temperature, time and environmental conditions, which can then be fitted within a theoretical framework of an inelastic constitutive model. In this paper, we present a constitutive formulation that deals continuously with the entire strain-rate range and has the desirable advantage of utilizing existing material data. The derivation makes use of strain-rate sensitive stress-strain curve and strain-rate dependent yield surface. By postulating a strain-rate dependent von Mises yield function and a strain-rate dependent kinematic hardening rule, we are able to derive incremental stress-strain relations that describe the strain-rate behavior in the entire deformation range spanning high strain-rate plasticity and creep. The model is sufficiently general as to apply to any materials and loading histories for which data is available.     

On behaviour of fuel elements subject to combined cyclic thermomechanical loads

This paper presents detailed finite element formulations on the kinematic hardening rule of plasticity included in an existing thermoelastoplastic stress analysis code primarily designed to predict the thermomechanical behaviour of nuclear reactor fuel elements. The kinematic hardening rule is considered to be important for structures subject to repeated (or cyclic) loads. The start-up/operation/shut-down and various power excursions in a reactor all can be classified as cyclic loadings. In addition to the shifting of material yield surfaces as usually handled by the kinematic hardening rule, the thermal effect and temperature-dependent material properties have also been included in the present work for the first time.A case study related to an in-reactor experiment on a single fuel element indicated that significantly higher cumulative sheath residual strains after two load cycles was obtained by the present scheme than those calculated by the usual isotropic hardening rule. This observation may alert fuel modellers to select proper hardening rules in their analyses.     

Reduced and selective integration techniques in the finite element analysis of plates

Efforts to develop effective plate bending finite elements by reduced integration techniques are described. The basis for the development is a ‘thick’ plate theory in which transverse shear strains are accounted for. The variables in the theory are all kinematic, namely, displacements and independent rotations. As only C⁰ continuity is required, isoparametric elements may be employed, which result in several advantages over thin plate elements. It is shown that the avoidance of shear ‘locking’ may be facilitated by reduced integration techniques. Both uniform and selective schemes are considered. Conditions under which selective schemes are invariant are identified, and they are found to have an advantage over uniform schemes in the present situation. It is pointed out that the present elements are not subject to the difficulties encountered by thin plate theory elements, concerning boundary conditions. For example, the polygonal approximation of curved, simply-supported edges is convergent. Other topics discussed are the equivalence with mixed methods, rank deficiency, convergence criteria and useful mass ‘lumping’ schemes for dynamics. Numerical results for several thin plate problems indicate the high degree of accuracy attainable by the present elements.     

Effective seismic input through rigid foundation filtering

This paper develops a simple yet realistic approach to account for a class of important soil-structure interaction phenomenon, namely wave scattering. The foundation is considered to act as a filter, partitioning the impinging excitation into lowpass filtered translational motion together with torsional motion (for horizontally polarized shear and Love waves) and rocking motion (for compressional, vertically polarized shear, and Rayleigh waves). These effects are evaluated for arbitrarily incident, horizontally polarized shear waves. The results are expressed in terms of filtering functions for various foundation geometries and embedment conditions and compared with ‘exact’ and other approximate solutions. Numerical results for a time history are presented in the form of translational and torsional acceleration response spectra. The formulae for these filters in the frequency domain show an interesting general relationship between the ‘effective’ translational and torsional motions of the foundation: the two motions are 90° out of phase for a given free-field harmonic wave, suggesting the possibility of computing the responses of structures to the two foundation motion components independently and combining the results by SRSS method. The future research need for adaptation of filtering concept to seismic excitations is discussed.     

Effect of reverse cyclic loading on the fracture resistance curve in C(T) specimen

Fracture resistance (J–R) curves, which are used for elastic–plastic fracture mechanics analyses, are known to be dependent on the cyclic loading history. The objective of this paper is to investigate the effect of reverse cyclic loading on the J–R curves in C(T) specimens. The effect of two parameters was observed on the J–R curves during the reverse cyclic loading. One was the minimum-to-maximum load ratio (R) and the other was the incremental plastic displacement (δ_cycle/δ_i), which is related to the amount of crack growth that occurs in a cycle. Fracture resistance tests on C(T) specimens with varying the load ratio and the incremental plastic displacement were performed, and the test results showed that the J–R curves were decreased with decreasing the load ratio and decreasing the incremental plastic displacement. Direct current potential drop (DCPD) method was used for the detection of crack initiation and crack growth in typical laboratory J–R tests. The values of crack initiation J-integral (J_I) and crack initiation displacement (δ_i) were also obtained by using the DCPD method.     

On modelling of kinematic hardening for ratcheting behaviour

A kinematic hardening rule is formulated on the assumption that each component α_i of back stress has a critical state for its dynamic recovery to be fully activated. The rule has a feature that only the projection of plastic strain rate to the direction of α_i contributes to the dynamic recovery. This is in contrast with previous rules in which the accumulated plastic strain rate enters into the dynamic recovery term. Applying the present and previous rules to experiments of modified 9Cr-1Mo steel at 550°C, we discuss effects of the feature mentioned above on simulating the multiaxial as well as uniaxial ratcheting behaviour.     

A bounding technique for dynamic plastic deformations of damped structures

A bounding principle for elastic-perfectly plastic creeping and noncreeping structures subjected to mechanical and/or thermal loads varying below or above the shakedown limit is presented. This principle contains some free “perturbation functions” which, suitably chosen, enable it to specialize, so generating bounds on a variety of deformation measures (such as inelastic work dissipated within any portion of the body, inelastic strains and displacements), some of which are new results, others recover or generalize known results. The resulting bounding technique possesses a quite unified character which is useful for computational purposes. The concept of “pseudo-plastic” strain is shown to be crucial for the derivation of bounds applicable above the shakedown limit.     

Shakedown and ratchetting under tension–torsion loadings: analysis and experiments

Structural design analyses are conducted with the aim of verifying the exclusion of ratchetting. To this end it is important to make a clear distinction between the shakedown range and the ratchetting range. The performed experiment comprised a hollow tension specimen which was subjected to alternating axial forces, superimposed with constant moments. First, a series of uniaxial tests has been carried out in order to calibrate a bounded kinematic hardening rule. The load parameters have been selected on the basis of previous shakedown analyses with the PERMAS code using a kinematic hardening material model. It is shown that this shakedown analysis gives reasonable agreement between the experimental and the numerical results. A linear and a nonlinear kinematic hardening model of two-surface plasticity are compared in material shakedown analysis.