On adaptation of elastic plastic anisotropic damage structures to cyclic loading

Summary.  Conditions for adaptation (shakedown) of structures of elastic plastic materials with anisotropic damage and strain hardening as subjected to cyclic loading are in question. The nature of damage is assumed ductile, so that the damage process ceases along with the cessation of plastic deformation process. The consideration is based on a material model of classical type. The demand of thermodynamical admissibility of deformation processes results in the conclusion: the quadratic form σ : ˙L : σ has to be positive where L denotes the current value of the elastic compliance tensor. This inequality makes it possible to show that the rate of plastic deformation tensor tends to zero, and the total plastic dissipation is bounded, if the classical Melan condition of shakedown holds for the initial state of the material. Received February 15, 2002; revised January 24, 2003 Published online: June 12, 2003 This research was implemented due to support from the Israeli Ministry of Science and the Hebrew University of Jerusalem.     

A plastic shakedown constitutive curve based on uniaxial ratcheting behavior of 304 stainless steel at room temperature

On the basis of the Bauschinger effect, a relationship between the elastic space defined in this study and the accumulated plastic strain is measured in uniaxial ratcheting tests of 304 stainless steel at room temperature. According to this relationship, a new model of uniaxial ratcheting is established and used to simulate uniaxial ratcheting behavior. The results of simulation agree well with the experimental results. These results demonstrate that the relationship between the elastic space and the accumulated plastic strain plays an important role in uniaxial ratcheting simulation. Furthermore, by taking into account the interaction of ratcheting and viscoplasticity, the relationships among elastic space, accumulated plastic strain and loading cases are discussed. It can be seen that, when the stress ratio R of valley stress versus peak stress is not less than zero, the accumulated plastic strain is a function of the peak stress. So, a constitutive curve is obtained to describe the stable states of plastic shakedown for 304 stainless steel material under the stress ratio R ≥ 0. It can be used to determine the accumulated plastic strain of engineering structures under cyclic loading only by an elastic–plastic analysis.     

FATIGUE DAMAGE AND SHAKEDOWN IN METAL MATRIX COMPOSITE LAMINATES

SUMMARY

Fatigue failure of metal matrix composite laminates is often preceded by a substantial loss of stiffness associated with cyclic plastic straining and subsequent low-cycle fatigue crack growth in the matrix. Experimental observations suggest that two principal crack patterns are involved; these are related here to the deformation modes predicted by the bimodal plasticity theory of fibrous composites. The relation is utilized in modelling the damage process such that matrix crack growth is regarded as a shakedown mechanism leading to a saturation damage state. For a given program of variable cyclic loading, evaluation of the saturation state is formulated as a non-linear optimization problem, where the total damage in a laminate is minimized subject to non-linear constraints derived from the ply yield criterion, hardening rule, and physically motivated bounds on the damage parameters. Effective elastic stiffness reduction and local stress redistribution predicted by the optimization procedure are compared with experimental measurements on several B/AI laminates. Stress transfer to and overloading of the fibres in certain plies appears to cause final fatigue failure of the laminate.     

A unified theory of elasticity and plasticity

It is shown that a certain set of rate theory type constitutive equations can give rise to both elastic behavior and plastic yield in infinitesmal strain both in loading and unloading. Elastic behavior corresponds to uniqueness of solutions of initial value problems of differential equations and holds when the equations satisfy a Lipschitz condition. This Lipschitz condition fails when a von Mises yield condition holds, and the corresponding nonuniqueness gives rise to plastic yield during loading. During plastic yield, the basic constitutive relations automatically turn into the Prandtl-Reuss equations. Upon unloading from yield, the stable solution of the equations dictates reentry into an elastic regime. The transition to and from plastic yield is exact and sudden, not asymptotic, but the transition is smooth. It is shown that linear behavior in the elastic regimes can be approximated arbitrarily closely. The theory can be extended to include strain hardening.     

Numerical shakedown analysis of axisymmetric sandwich shells: An upper bound formulation

Shakedown analysis of axisymmetric elastic–perfectly plastic sandwich shells is performed here using a new upper bound formulation based on a special form of Koiter's theorem concerning piecewise linearized yield surfaces. Starting from finite element techniques and the Tresca sandwich yield condition, shakedown analysis is reduced to a linear programming problem which is solved by a powerful simplex algorithm. Numerical results are given for a number of examples and a comparison is made with a previously computed lower bound formulation.     

Shakedown limit of rail surfaces including material hardening and thermal stresses

Sliding friction between railway wheels and rails results in elevated contact temperatures and gives rise to severe thermal stresses at the wheel and rail surfaces. The thermal stresses have to be superimposed on the mechanical contact stresses. Due to the distribution of stresses, the rail surface is generally subjected to higher stresses than the wheel surface. The elastic limit is reduced and yield begins at lower mechanical loads. During the first cycles of plastic deformation, the material hardens and residual stresses build up. The residual stresses provide the structure to shake down to pure elastic behaviour in subsequent load cycles up to a shakedown limit. The kind of hardening observed for rail steel has a considerable influence on the shakedown limit. The shakedown limit is dropped to lower mechanical loads due to the thermal stresses in the rail surface as well. This might cause structural changes in the rail material and rail damage.     

循环载荷下层间界面反平面剪切破坏的解析解:2.再加载及综合响应分析

本研究利用“剪切梁”模型,研究了在定常侧压力和周期性的反平面剪切载荷共同作用下,层间界面的破坏规律。本文为本研究的第二部分。对应一个完整的载荷周期,解析地给出了界面层上剪应力及位移场在再加载过程中的分布规律及演化规律。计算结果表明,由于裂纹前方损伤过程区的存在,当再加载过程不能覆盖卸载过程造成的损伤区时,会产生新的“应力锁死”现象。文末对结构整体的载荷－位移行为的分析,揭示了周期载荷作用下的两种结构响应模式：１）　接触塑性安定;２）　增量破坏。     

Evolution of mechanical properties for a dual-phase steel subjected to different loading paths

The evolution of the mechanical properties of a dual-phase (DP590) steel sheet after being prestrained by uniaxial tension, plane strain and equal biaxial stretching was investigated. Specimens were first loaded using the three prestraining modes. Then, from the prestrained specimens, a few sub-sized samples were machined along the rolling direction and the transverse direction for further uniaxial tension testing. Six loading paths were provided. Equal biaxial stretching was performed using a cruciform specimen. The evolution of work hardening performance, elastic modulus, yield stress and tensile stress under the six loading paths were discussed in detail. The results indicate that loading paths can affect the latent work hardening performances, strain hardenability, yield stress and tensile stress evolution as well as the elastic modulus decrease during plastic deformation. The uniaxial tension–uniaxial tension path results in a cross-softening phenomenon, the largest yield stress enhancement and a mild maximum tensile stress increase. The equal biaxial stretching-uniaxial tension path leads to a cross-hardening phenomenon, the least yield stress enhancement and the largest tensile strength increase maximum tensile strength. The elastic modulus of DP590 steel not only changes with the accumulated plastic strain but also varies with the loading paths. The largest decrease of the elastic modulus equal biaxial stretching–uniaxial tension can reach 12.7% beyond 8% equivalent strain, which is 5.2% greater than that in the monotonic uniaxial tension path.     

Finite element analysis of elastic–plastic materials displaying mixed hardening

The problem of non-radial loading (in the stress space) of structures of elastic–plastic metals is discussed. A concept of combined isotropic and kinematic hardening is proposed. For a material displaying such mixed hardening, constitutive equations are derived and incorporated in a finite element program for the analysis of elastic–plastic 2D-structures. When determining the numerical solution of the non-linear equations, the program enables the investigator to select an optimal combination of step by step and iteration procedures. The computed results are compared with experimental results and with results from calculations based on the overlay model, Reference 25.     

On the low cycle fatigue failure of insulated rail joints (IRJs)

Insulated rail joints (IRJs) are safety critical components in the signalling system of railway corridors which provide a break in the continuity of the rail steel to locate trains. IRJs connect the two rail ends at the discontinuity to achieve geometric and mechanical requirements of rail. The bending stiffness of an IRJ is about one third that of continuous rail. As a result, the IRJs, especially those in heavy haul tracks, exhibit early failure predominantly due to ratchetting or alternating plasticity of railhead metal in the vicinity of the endpost insulators.A three-dimensional (3D) finite element numerical simulation is carried out to examine failures of railhead material in the vicinity of the endpost of an insulated rail joint considering high frequency dynamic wheel loading. A dynamic wheel load of 182 kN is applied through a contact patch; the distribution of contact pressure is considered using a non-Hertzian formulation. A 12 m long global IRJ model and a sub-model for localised analysis are employed. The shakedown theorem is employed in this study. Nonlinear isotropic/kinematic elastic–plastic material modelling is employed in the simulation. A peak pressure load lower than the shakedown limit is considered as the input load.The equivalent plastic strain plot for this load case lower than the shakedown limit demonstrates the railhead damage captured through a localised stress analysis in the vicinity of the endpost using the sub-modelling technique. The sub-surface plastic deformation of railhead material extends down to 8 mm from the railhead top surface. The critical crack initiating stress components are at 2–4 mm sub-surface depth. As such, the railhead material fails due to alternating plasticity through low cycle fatigue. Laboratory tests were performed to verify the simulation results and found that test and simulation results correlated well.     

Fatigue crack initiation life estimation in a steel welded joint by the use of a two-scale damage model

This work deals with the fatigue behaviour of S355NL steel welded joints classically used in naval structures. The approach suggested here, in order to estimate the fatigue crack initiation life, can be split into two stages. First, stabilized stress–strain cycles are obtained in all points of the welded joint by a finite element analysis, taking constant or variable amplitude loadings into account. This calculation takes account of: base metal elastic–plastic behaviour, variable yield stress based on hardness measurements in various zones of the weld, local geometry at the weld toe and residual stresses if any. Second, if a fast elastic shakedown occurs, a two-scale damage model based on Lemaitre et al. 's work is used as a post-processor in order to estimate the fatigue crack initiation life. Material parameters for this model were identified from two Wöhler curves established for base metal. As a validation, four-point bending fatigue tests were carried out on welded specimens supplied by 'DCNS company'. Two load ratios were considered: 0.1 and 0.3. Residual stress measurements by X-ray diffraction completed this analysis. Comparisons between experimental and calculated fatigue lives are promising for the considered loadings. An exploitation of this method is planned for another welding process.     

Stress triaxiality effect on void nucleation in ductile metals

The stress triaxiality effect on the strain required for void nucleation by particle‐matrix debonding has been investigated by means of micromechanical modelling. A unit‐cell model considering an elastic spherical particle embedded in an elastic‐plastic matrix was developed to the purpose. Particle‐matrix decohesion was simulated through the progressive failure of a cohesive interface. It has been shown that the parameters of matrix‐particle cohesive interface are correlated with macroscopic material properties. Here, a simple relationship for the maximum cohesive opening at interface failure as a function of material fracture toughness and yield stress has been derived. Results seem to confirm that, increasing stress triaxiality, the strain at which void nucleation is predicted to occur decreases exponentially in a similar way as for fracture strain. This result has substantial implications in modelling of ductile damage because it indicates that if the stress triaxiality is high enough, ductile fracture can occur at plastic strain lower than that necessary to nucleate damage for moderate or low stress triaxiality regime.     

Kinematical shakedown analysis with temperature‐dependent yield stress

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

Linear Matching Method for Design Limits in Plasticity

In this paper a state-of-the-art numerical method is discussed for the evaluation of the shakedown and ratchet limits for an elastic-perfectly plastic body subjected to cyclic thermal and mechanical load history. The limit load or collapse load, i.e. the load carrying capacity, is also determined as a special case of shakedown analysis. These design limits in plasticity have been solved by characterizing the steady cyclic state using a general cyclic minimum theorem. For a prescribed class of kinematically admissible inelastic strain rate histories, the minimum of the functional for these design limits are found by a programming method, the Linear Matching Method (LMM), which converges to the least upper bound. By ensuring that both equilibrium and compatibility are satisfied at each stage, a direct algorithm has also been derived to determine the lower bound of shakedown and ratchet limit using the best residual stress calculated during the LMM procedure. Three practical examples of the LMM are provided to confirm the efficiency and effectiveness of the method: the behaviour of a complex 3D tubeplate in a typical AGR superheater header, the behaviour of a fiber reinforced metal matrix composite under loading and thermal cycling conditions, and effects of drilling holes on the ratchet limit and crack tip plastic strain range for a centre cracked plate subjected to constant tensile loading and cyclic bending moment.     

Simulation of cyclic stress/strain evolutions for multiaxial fatigue life prediction

This study deals with simulation for cyclic stress/strain evolutions and redistributions, and evaluation of fatigue parameters suitable for estimating fatigue lives under multiaxial loadings. The local cyclic elastic–plastic stress–strain responses were analyzed using the incremental plasticity procedures of ABAQUS finite element code for both smooth and notched specimens made of three materials: a medium carbon steel in the normalized condition, an alloy steel quenched and tempered and a stainless steel, respectively. Emphasis is on the studying of ‘intelligent’ material behaviors to resist fracture, such as stress redistribution and relaxation through plastic deformations, etc. For experimental verifications, a series of tests of biaxial low cycle fatigue composed of tension/compression with static and cyclic torsion were carried out on a biaxial servo-hydraulic testing machine (Instron 8800). Different multiaxial loading paths were used to verify their effects on the additional cyclic hardening. The comparisons between numerical simulations and experimental observations show that the FEM simulations allow better understanding on the evolutions of the local cyclic stress–strain and it is shown that strong interactions exist between the most stressed material element and its neighboring material elements in the plastic deformations and stress redistributions. Based on the local cyclic elastic–plastic stress–strain responses, the energy-based multiaxial fatigue damage parameters are applied to correlating the experimentally obtained lives. Improved correlations between the predicted and the experimental results are shown. It is concluded that the improvement of fatigue life prediction depends not only on the fatigue damage models, but also on the accurate evaluations of the cyclic elasto-plastic stress/strain responses.     

The plastic potential in the theory of anisotropic elastic-plastic solids

In the paper the yield condition is proposed for the most general anisotropic material. It is one of the possible generalizations of the Huber-Mises-Hencky yield condition for the case of anisotropy. The body considered is anisotropic elastically as well as plastically. It is assumed that the plastic anisotropy tensor is a definite function of the elastic anisotropy tensor. The corresponding flow function is a part of the strain energy and its value remains unchanged when all normal components of stress are increased by the same value. The theory of the eigen states for fourth order tensors is used. The plastic anisotropy tensor proposed has the same deviatoric eigen states as the elastic anisotropy tensor. The proposed yield condition reduces to that of Huber-Mises-Hencky when the anisotropy is vanishingly small. The method presented in this paper can be also applied to describe other types of plastic anisotropy tensor.     

Bounds to Shakedown Loads for a Class of Deviatoric Plasticity Models

The problem of estimating bounds to shakedown loads for problems governed by a class of deviatoric plasticity models including those of Hill, von Mises, and Tresca is addressed. Assuming that an exact elastic solution is available, an upper bound to the elastic shakedown multiplier can be obtained relatively easily using the plastic shakedown theorem. A procedure for computing this upper bound for arbitrary load domains is presented. A number of problems are then examined and it is found that the elastic shakedown factor is given as the minimum of the plastic shakedown factor and the classical limit load factor. Finally, some exact solutions to a number of two dimensional problems are given.     

Thermal stresses in a shrink fit due to an inhomogeneous temperature distribution

Summary Based on Tresca's yield condition and its associated flow rule, a semianalytical method is presented for the calculation of thermal stresses due to steady-state thermal loading in an assembled shrink fit. The calculation is evaluated assuming plane stress conditions, linear elastic-perfectly plastic materials, and linearly temperature dependent yield stresses. Depending on the temperature gradient, different combinations of pure elastic and plastic zones arise in the shaft and in the hub.     

Bounds on transient phase plastic deformations in optimal design of steel frames subjected to cyclic loads

A minimum volume multicriterion design of elastic perfectly plastic steel frames subjected to a combination of quasi-static fixed and cyclic loads, bounding the transient phase plastic deformations, is proposed. The problem is formulated according to a plastic shakedown criterion, so that incremental and instantaneous collapse are certainly prevented when the frame is subjected to very strongly amplified cyclic loads. The further condition that the structure must also behave elastically in serviceability conditions is imposed. Since the steady-state loading history is known it is possible to directly bound the steady-state plastic deformations. By applying a suitable own bounding theorem, it is also possible to indirectly bound the transient plastic deformations, whichever the unknown transient real loading history is. A numerical application confirms the fundamental role played by the transient phase plastic deformations both on the design as well as on the structural behaviour and the great performance of the proposed formulation.