Deformation bounds for elastic-plastic solids within and out of the creep range

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.     

Mathematical programming methods for displacement bounds in elasto-plastic dynamics

In elastic-plastic structures subjected to dynamic external actions, if unbounded plastic deformations are developed, either local failure due to plastic fatigue (alternating plasticity) or gradual divergence of the deformed configuration (incremental collapse) will occur. Therefore, the boundedness in time of total plastic strains, and hence of total plastic work (usually referred to as adaptation or shakedown) is necessary for structural safety, in the sense that it rules out the occurrence of the above critical phenomena. Necessary and sufficient conditions for shakedown have been established by several authors. However, in many instances adaptation is not sufficient to ensure safety. In fact, even if plastic deformations can be proved to be finite, they can exceed some critical limit or exhaust the material ductility. In particular, for dynamic loading histories that cease after a certain time, a structure will certainly shakedown under any load amplitude, so that a safety criterion based on this event is clearly meaningless. Typical histories of this kind are earthquake or blast loadings.When the loading history is known, it is possible, in principle, to assess safety by following the actual plasto-dynamic evolution of the system, but this is often a laborious task and in several cases it provides far more information than is actually needed. On this remark rests the interest of methods capable to provide some essential information, such as upper bounds on maximum deflections or strains, through a moderate computational effort. In recent years several alternative techniques have been developed to bound from the above various quantities of interest. With the exception of very simple situations, the best bounds that can be obtained through any one of these methods involve the solution of constrained optimization problems.In this paper a study of several deformation bounding techniques is performed. The problem is formulated and the main previous results are outlined first with reference to general continua made of hardening materials. Then a class of discrete structural models (such as some finite element discretizations) is considered and, on this basis, two categories of deformation bounding techniques are described from the previous main results. All these techniques, some of which are new, permit the optimization of the upper bound by solving one or more mathematical programming problems of special forms. Some of the bounding procedures are shown to have merely theoretical interest, since they lead to cumbersome numerical procedures or to very coarse bounds. The formulations that appear to have practical application are compared from various standpoints (type of loading history, different hardening rules, influence of second order geometric effects, quantities to be bounded) and first assessment of their practical usefulness is attempted. Generalizations to second-order geometric and thermal effects and to situations in which the time history is not completely known are envisaged.     

Development of constitutive equations for computer analysis of stainless steel components

Engineering application makes conflicting demands of constitutive equations which are difficult to satisfy simultaneously, so forcing considerable approximation. The difficulty is compounded by the frequent need, in anything but room temperature application, to be able to describe the behaviour of the structural material over a range of temperatures. This is illustrated by considering the spectrum of behaviour of Type 316 stainless steel from room temperature to operation at 600°C. It is found that a simple plasticity model describes the behaviour well at 400°C but is less adequate at 20°C in the presence of “cold creep”. There is a discussion of the way plasticity and creep can both be described, with a systematic interaction but without the restrictions of a single “unified relation” for all inelastic deformation.     

Probabilistic methods in plastic structural analysis

The redistribution of stresses in ductile structures, although beneficial from the safety viewpoint, introduces another source of uncertainty, which requires specific methods when the probabilistic approach to reliability evaluation is followed.Practicable procedures have been developed for structures that satisfy the classical assumptions of plastic limit analysis. In particular, two theorems that allow to find rigorous upper and lower bouds on the probability of full plastic collapse under given loads, are presented. Other methods for probabilistic limit analysis are also indicated, including in particular a specifically developed parametric simulation procedure.The last part of the paper is devoted to the reliability analysis of plastic structures subject to loads varying (slowly) in time.It is recalled first that probabilistic limit analysis can be easily extended to the shakedown—incremental collapse problem, provided the loads vary within a finite domain: however, the significance of such an approach for stochastically varying loads is questioned. In fact, as time increases, the probability also increases that the loads cross any given threshold. Therefore, it is more appropriate to speak of “plastic adaption” rather than “shakedown”, and to focus the attention on the probability of reaching, in any given time interval, a certain permanent deformation. Again, only approximate solutions (in the form of upper and lower bounds) can be found to this question, but this appears to be a more rational and promising approach to the problem.     

Development of guidelines for inelastic analysis in design of fast reactor components

The interim guidelines for the application of inelastic analysis to design of fast reactor components were developed. These guidelines are referred from “Elevated Temperature Structural Design Guide for Commercialized Fast Reactor (FDS)”. The basic policies of the guidelines are more rational predictions compared with elastic analysis approach and a guarantee of conservative results for design conditions. The guidelines recommend two kinds of constitutive equations to estimate strains conservatively. They also provide the methods for modeling load histories and estimating fatigue and creep damage based on the results of inelastic analysis. The guidelines were applied to typical design examples and their results were summarized as exemplars to support users.     

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).     

A dynamic model for element reliability

A dynamic model is developed for a system element reliability distribution over a generalized strength space. A differential equation is obtained describing the time-dependence of the reliability distribution function (RDF). The equation covers a wide class of power reactor system components which perform under intense stress conditions where a standard subdivision into a “burn-in” period, a “chance failures” range and a “wear-our” period is inapplicable.The hazard distribution function (HDF) over strength is introduced within the model and it is shown that a standard hazard rate is a strength-averaged failure intensity parameter with the RDF as a weighting function.It is shown that a well-known “bathtub” form of the hazard rate function corresponds to an analytical solution of the principal RDF transfer equation under some simplifying assumptions.     

Inelastic stress-strain response for notched specimen of Cr-1Mo steel at 600°C

Following a series of cooperative studies A-I and A-II (phase III) concerning the inelastic behaviour of high temperature materials under uniform state of stress, finite element analyses were carried out on circumferential notched cylinders subjected to plasticity-creep interaction conditions. Using an electric capacitance type extensometer “Strain-Pecker”, which is capable of measuring a local strain response with a gauge length of 0.5 mm under high temperature conditions, stress-strain responses for both global and local regions near the notch root were evaluated. Ten kinds of inelastic constitutive model were introduced into a finite element code, and the responses for four kinds of loading pattern were examined for two types of notch shape.     

The frequency-dependent elements in the code SASSI: A bridge between civil engineers and the soil–structure interaction specialists

After four decades of the intensive studies of the soil-structure interaction (SSI) effects in the field of the NPP seismic analysis there is a certain gap between the SSI specialists and civil engineers. The results obtained using the advanced SSI codes like SASSI are often rather far from the results obtained using general codes (though match the experimental and field data). The reasons for the discrepancies are not clear because none of the parties can recall the results of the “other party” and investigate the influence of various factors causing the difference step by step. As a result, civil engineers neither feel the SSI effects, nor control them. The author believes that the SSI specialists should do the first step forward (a) recalling “viscous” damping in the structures versus the “material” one and (b) convoluting all the SSI wave effects into the format of “soil springs and dashpots”, more or less clear for civil engineers. The tool for both tasks could be a special finite element with frequency-dependent stiffness developed by the author for the code SASSI. This element can represent both soil and structure in the SSI model and help to split various factors influencing seismic response. In the paper the theory and some practical issues concerning the new element are presented.     

Elementary catastrophe theory modelling of duffing's equation for seismic excitation of nuclear power facilities

Elementary catastrophe theory can provide conceptual insight into some aspects of a variety of problems in dynamics. It is a qualitative tool with some quantitative results. In this paper, it is applied to forced nonlinear vibrations of seismic disturbances, which may be approximated by Duffing's equation. The behavior of such a system fits naturally into ECT modelling, where changes in parameters of the system lead to “jump” type behavior. The important conclusion is that nonlinear oscillators can exhibit elementary catastrophes, but the design engineer may be able to manipulate characteristics of the system in order to avoid the “jump” behavior of the response.     

Failure strains and proposed limit strains for an reactor pressure vessel under severe accident conditions

The local failure strains of essential design elements of a reactor vessel are investigated. The size influence of the structure is of special interest. Typical severe accident conditions including elevated temperatures and dynamic loads are considered.The main part of work consists of test families with specimens under uniaxial and biaxial load. Within one test family the specimen geometry and the load conditions are similar, but the size is varied up to reactor dimensions. Special attention is given to geometries with a hole or a notch causing non-uniform stress and strain distributions typical for the reactor vessel. A key problem is to determine the local failure strain. Here suitable methods had to be developed including the so-called “vanishing gap method”, and the “forging die method”. They are based on post-test geometrical measurements of the fracture surfaces and reconstructions of the related strain fields using finite element models.The results indicate that stresses versus dimensionless deformations are approximately size independent up to failure for specimens of similar geometry under similar load conditions. Local failure strains could be determined. The values are rather high and size dependent. Statistical evaluation allow the proposal of limit strains which are also size dependent. If these limit strains are not exceeded, the structures will not fracture.     

Development of elevated temperature structural design guide for Japanese demonstration breeder reactor

Programs to develop the “elevated temperature structural design guide for the demonstration fast breeder reactor” (DDS) in Japan have been conducted since 1987. The DDS is to be developed on the basis of the “elevated temperature structural design guide for class 1 components of prototype fast breeder reactors” (ETSDG), by considering structural and material features of the demonstration fast breeder reactor (DFBR) and incorporating results of the latest R&D. This paper describes the progress of the R& D concept of the DDS, and discusses some typical results of current studies on the DDS.     

The prediction of 2D thermal detonations and resulting damage potential

The main purpose of this paper is to introduce a new concept for the processes responsible for the escalation and propagation of steam explosions. The concept recognizes that initially only a small quantity of coolant around each coarsely premixed melt mass “sees” the fragmenting debris coming off it, hence it is called the concept of “microinteractions”. We also derive the analytical basis for it, define the nature of the requisite constitutive laws and related experimental data, and demonstrate that this concept is essential for the prediction of steam explosion energetics in large-scale premixtures in 2D geometries. We also provide the first numerical illustrations of this concept, implemented in the computer code .m. Further, we provide the first numerical results of steam explosions in large water pools, i.e. ex-vessel explosions. These results reveal two important mechanisms for explosion “venting” and thus for reducing the dynamic loads on adjacent structures. We conclude that, taken together, the “microinteractions” and “venting” make realistic predictions of steam explosion loads feasible and within reach in the near future.     

Discretizing the diffusion equation on unstructured polygonal meshes in two dimensions

We derive a discretization of the two-dimensional diffusion equation for use with unstructured meshes of polygons. The scheme is presented in r–z geometry, but can easily be applied to x–y geometry. The method is “node”- or “point”-based and is constructed using a finite volume approach. The scheme is designed to have several important properties, including second-order accuracy, convergence to the exact result as the mesh is refined (regardless of the smoothness of the grid), and preservation of the homogeneous linear solution. Its principle disadvantage is that, in general, it generates an asymmetric coefficient matrix, and therefore requires more storage and the use of non-traditional, and sometimes slowly-converging, iterative linear solvers. On an unstructured triangular grid in x–y geometry, the scheme is equivalent to the linear continuous finite element method with “mass-matrix lumping”. We give computational examples that demonstrate the accuracy and convergence properties of the new scheme relative to other schemes.     

Probabilistic analysis of inelastic structures

Probabilistic methods for the analysis of linear elastic structures, although sometimes unsatisfactory for the purpose of reliability assessment, are well stated and codified. By contrast, when the beneficial contribution of the redistribution of stresses has to be investigated, numerical procedures of analysis are involved and the randomness of the structural response is generally investigated by simulation procedures as no probabilistic method is applicable.However, an irrational use of simulation techniques does not produce probabilistic results, but only useless numerical results. These procedures are criticized and the adoption of “hybrid approaches” is suggested as a rationalization.Special emphasis is devoted to the dynamic analysis of inelastic structures; the problem of establishing adequate probabilistic failure criteria is also discussed.     

On a simplified inelastic analysis of structures

In this paper two main problems are considered: the derivation of cyclic constitutive relations during inelastic regime where hardening, softening and creep can occur, and the development of the eventual periodical state in the structure during cyclic thermodynamical loadings.We give a very simple and practical framework to solve these problems in one unique manner.Its essential feature consists in the introduction of a family of internal parameters which characterize local inelastic mechanisms and the family of transformed internal parameters which are linearly linked to the previous ones through a symmetrical non-negative matrix and are indeed the opposite of the associated residual stresses. Thanks to that, the treatment of the local plastic or viscoplastic yield conditions can be easily made from only the classical simple purely elastic (or viscoelastic) analysis.This property allows important results during cyclic loadings: conditions for elastic shakedown, plastic shakedown, ratcheting and bounds for the limiting state.Several examples are given in the text.     

The use of energy absorbers to protect structures against impact loading

When a flying missible impacts a fixed structure, the interface loading is dependent on the deformation characteristics of both impacting and impacted bodies. If both are too rigid to accommodate the amount of gross deformation required to neutralize the incoming kinetic energy, or if such energy absorption has a chance to proceed in uncontrolled and unreliable ways, then there is a need to interpose a specifically designed “energy absorber” between missile and structure, from which a well-defined load time history can be derived during the course of impact.

The required characteristics of such an energy absorption material are:

• the capability to accommodate large permanent deformation without structural failure; and

• the reliable and controlled “load-deformation” (or “stress-strain”) behaviour under dynamic conditions, with an aim at an optimal square shape curve.

Consideration must also be given to environmental or other disturbing effects, like temperature, humidity, and “out of plane” loading. A short survey is presented of the wide range of energy absorbers already described in technical papers or used in a number of practical safety applications within varied engineering fields (from vehicle crash barriers to high energy pipe whipping restraints). However, with such open a literature, information is usually lacking in the specific data required for design analysis.

The following “energy absorption” materials and processes have thus been further experimentally investigated, with an a aim at pipe whipping restraint application for nuclear power plants:

1. (1) plastic extension of austenitic stainless steel rods;

2. (2) plastic compression of copper bumpers; and

3. (3) punching of lightweight concrete structures.

Dynamic “stress-strain” characteristics have been established for stainless steel bars at several temperatures under representative loading conditions. For this purpose, a test rig has been specifically designed to incorporate a number of adjustable parameters and to behave as a representative “slice” of an actual pipe whipping restraint; typical strain rates are in the 10 sec⁻¹ range. The behaviour of copper bumpers has been compared under static and dynamic conditions (using a conventional drop weight test (DWT) machine); as no significant strain rate effects were emphasized, only static tests have been further developed. The DWT rig was used again to investigate crushing or punching of cellular concrete under varying geometries and loading conditions. To remedy certain deficiencies of the regular commercial grades of cellular concrete, special lightweight mixtures have been studied to optimize material toughness and provide a wider range of specific resistance.Results of this experimental program are presented and discussed. The use of energy absorbers is then illustrated for a few typical pipe whipping restraints. The design of restraints is based on real dynamic characteristics of “energy absorption” material as produced by the test program. To derive design loads of restraints, a number of methods can be used ranging from a simplified “energy balance” graph to sophisticated plastodynamic computer analysis. Typical results are presented and discussed to compare the efficiency of these alternative methods.     

A two-dimensional model for fluid-structure interaction in curved pipes

A “channel” model was developed for the purpose of simulating the interactive fluid-structural response of curved pipes to pressure pulses. Simulation is shown to have been achieved analytically in both the axisymmetric (“breathing”) and transverse (“bending”) modes of interactive behavior.An experimental program which was aimed at the validation of the model is also described. Tests were run in both straight and curved pipe configurations. Comparisons between measurements and model calculations demonstrate the validity of the model within the range of parameters under consideration.The model was implemented into the DISCO code for nonlinear fluid-shell interaction.     

Corrosion of vessel steel during its interaction with molten corium: Part 2: Model development

An experimental examination of the cooled vessel steel corrosion during the interaction with molten corium is presented. The experiments have been conducted on “Rasplav-2” test facility and followed up with physico-chemical and metallographic analyses of melt samples and corium-specimen ingots. The results discussed in the first part of the paper have revealed specific corrosion mechanisms for air and inert atmosphere above the melt. Models have been proposed based on this information and approximate curves constructed for the estimation of the corrosion rate or corrosion depth of vessel steel in conditions simulated by the experiments.