Limited Kinematic Hardening Research Articles

Elastic-Plastic Finite Element New Method for Lower Bound Shakedown Analysis

Shakedown analysis methods have been well developed to determine the shakedown limit load of structures under various repeated loading situations. However, to take the limited kinematic hardening, geometric effect, and deformation into account, further research is required. A new numerical method is proposed for the lower bound shakedown analysis that applies the extended static shakedown theorem for limited kinematic hardening. The new method determines the shakedown limit load with two kinds of elastic-plastic finite element (FE) analyses: (1) the elastic-plastic loading and unloading analysis is carried out using the hardening model with the actual material parameters and (2) the elastic-perfectly plastic model with ultimate stress as the yield stress is used for the limit analysis. An incremental FE technique for finite strain plasticity is utilized to take the geometric effect into account. Furthermore, the deformation of the structure is monitored by checking the calculated deformation parameters. The method is applied to a square plate with a small central hole subjected to cyclic tensile load. The corresponding physical shakedown experiment of the above square plate was conducted for verification. The numerical result shows a good agreement with the experimental result. The new method provides a valid method for the structural lower bound shakedown analysis taking into account limited kinematic hardening, geometric effect, and deformation.

Computational lower bound shakedown analysis with limited kinematic hardening is only safe if a specific hardening law is defined

Lower bound shakedown analysis is a well accepted and very convenient tool to determine the load bearing capacity of engineering structures subjected to thermo-mechanical loadings. In order to achieve realistic results, limited kinematic hardening needs to be taken into account. In many cases, this is achieved by using two-surface models which consider only the initial and the ultimate yield surface. In previous works, I had shown already that the solution obtained from such an approach applied to unrestricted, limited kinematic hardening can appear to be path-dependent. Nonetheless, the two-surface model is still used frequently in the literature for these scenarios. Thus, the aim of this paper is to emphasize the implications from using unrestricted hardening. Most importantly, it should be highlighted that in some cases the obtained result for the shakedown loading factor might not be a lower bound such that it cannot be used in conservative structural design.

Shakedown and Limit Analysis of Kinematic Hardening Piping Elbows Under Internal Pressure and Bending Moments

Abstract In this paper, the stress compensation method (SCM) adopting an elastic-perfectly plastic (EPP) material is further extended to account for limited kinematic hardening (KH) material model based on the extended Melan's static shakedown theorem using a two-surface model defined by two hardening parameters, namely, the initial yield strength and the ultimate yield strength. Numerical analysis of a cylindrical pipe is performed to validate the outcomes of the extended SCM. The results agree well with ones from literature. Then the extended SCM is applied to the shakedown and limit analysis of KH piping elbows subjected to internal pressure and cyclic bending moments. Various loading combinations are investigated to generate the shakedown limit and the plastic limit load interaction curves. The effects of material hardening, elbow angle, and loading conditions on the shakedown limit and the plastic limit load interaction curves are presented and analyzed. The present method is incorporated in a commercial finite element simulation software and can be considered as a general computational tool for shakedown analysis of KH engineering structures. The obtained results provide a useful information for the structural design and integrity assessment of piping elbows.

A direct approach to the evaluation of structural shakedown limit considering limited kinematic hardening and non-isothermal effect

This paper presents a novel direct method for the structural shakedown analysis considering limited kinematic hardening and non-isothermal effect. The Melan's static shakedown theorem is extended to consider limited kinematic hardening material and implemented into the Linear Matching Method (LMM) shakedown module. Instead of using a specific kinematic hardening rule and an explicit back stress field, the general nonlinear hardening laws are considered by using a two-surface hardening model. A two-stage procedure is developed in the extended LMM algorithm, which can generate the limited hardening shakedown envelope and the unlimited hardening curve efficiently and accurately. Also, the material non-isothermal effect is considered during the computation process of the shakedown limit by proposing a temperature-dependent hardening factor, in place of a constant and fictitious one. To validate the extended LMM method, a numerical test on a thin cylinder pipe with temperature-independent material properties is performed, and the results match well with ones from literature. Then, a numerical study on a typical aero-engine turbine disk is conducted to investigate the influence of temperature-dependent material properties and operating conditions. Several shakedown curves considering kinematic hardening effect are derived and adequately discussed. As a result, the extended LMM shakedown module is proven to be a robust, efficient and versatile tool for practical industrial problems.

Consistent limited kinematic hardening plasticity theory and path-independent shakedown theorems

The limited kinematic hardening plasticity theory is re-examined and presented with additional clarifying hypotheses being proposed. The general theory forms the base for construction of the path-independent shakedown theorems intended for possible broad engineering applications, and possible further specifications of particular hardening laws. Some specifications in the literature on limited kinematic hardening are critically analysed. Recent applications of the shakedown theorems in path-independent forms with two separated modes, which are only fully founded in this study, and further problems are briefly summarized with references to the respective literature.

Direct Methods of Plasticity in Aluminium Frames Accounting for Eurocode 9 Failure Criteria

Metal structures, given their broad plastic deformation capacity, have been often at the core of research for the evaluation of the post elastic behavior via the direct methods of plasticity. Limit and shakedown analysis were often exploited to determine the plastic collapse load capacity, as they provide advantages in terms of computational robustness in comparison with incremental non-linear analysis. The aforementioned approach has been widely and effectively applied for steel structures, characterized by rigid-plastic behavior. Υet in the case of aluminium structures, there are special parameters to be considered, as they exhibit a relationship of round-house type, whose trend cannot be interpreted through the classic elastic-perfectly plastic material law idealization.The present paper aims to develop a limit and shakedown analysis formulation suitable for the safety assessment of 3D aluminium frame structures against plastic collapse. Research yielded the proposed two-surface methodology in the framework of stress resultant plasticity, which is capable of estimating the plastic collapse limit state of real-life aluminium frames, incorporating Eurocode 9 codified failure criteria and limited kinematic hardening. A numerical example of a 3D frame is realized utilizing the finite element method and the 3-node Timoshenko column-beam element, in order to showcase the proposed methodology and validate the influence of hardening effect.

General analytical shakedown solution for structures with kinematic hardening materials

The effect of kinematic hardening behavior on the shakedown behaviors of structure has been investigated by performing shakedown analysis for some specific problems. The results obtained only show that the shakedown limit loads of structures with kinematic hardening model are larger than or equal to those with perfectly plastic model of the same initial yield stress. To further investigate the rules governing the different shakedown behaviors of kinematic hardening structures, the extended shakedown theorem for limited kinematic hardening is applied, the shakedown condition is then proposed, and a general analytical solution for the structural shakedown limit load is thus derived. The analytical shakedown limit loads for fully reversed cyclic loading and non-fully reversed cyclic loading are then given based on the general solution. The resulting analytical solution is applied to some specific problems: a hollow specimen subjected to tension and torsion, a flanged pipe subjected to pressure and axial force and a square plate with small central hole subjected to biaxial tension. The results obtained are compared with those in literatures, they are consistent with each other. Based on the resulting general analytical solution, rules governing the general effects of kinematic hardening behavior on the shakedown behavior of structure are clearly.

Limit states of structures in n-dimensional loading spaces with limited kinematical hardening

The load bearing capacity of engineering structures subjected to varying thermo-mechanical loading can be determined most conveniently by shakedown analysis. Admittedly, if limited kinematical hardening is considered, shakedown analysis is, as yet, restricted to either one or two independently varying loads. In consequence, the aim of this paper is to extend existing formulations for arbitrary numbers of loading, which is inevitable for most technical applications. For this, limited kinematical hardening was incorporated into Melan’s shakedown theorem by means of a two-surface model covering both alternating plasticity and ratcheting, generalized to n-dimensional loading spaces. As a result, the three-dimensional shakedown domain accounting for hardening is presented for a flanged pipe subjected to thermo-mechanical loading.

Direct evaluation of the limit states of engineering structures exhibiting limited, nonlinear kinematical hardening

The lower bound shakedown analysis is a most convenient tool to determine the load bearing capacity of engineering structures subjected to thermo-mechanical loadings. In order to achieve realistic results, limited nonlinear kinematical hardening needs to be taken into account. Although there exist different formulations incorporating limited kinematical hardening in the literature, it is still not conclusively clarified, whether or not these are applicable to generally-nonlinear hardening laws as well. Thus, the aim of this paper is to propose a method to determine the shakedown limit loads accounting for limited, generally-nonlinear kinematical hardening, and to close the discussion about the effect of the nonlinearity of the hardening law. The proposed method is based on an extension of the statical shakedown theorem by Melan using a two-surface model, which captures both incremental collapse and alternating plasticity. Furthermore, it is implemented into an interior-point algorithm, which is tailored to shakedown analysis and thus capable of handling large-scale problems. The algorithm allows for an arbitrary number of thermo-mechanical loadings. To illustrate the method’s potential, numerical results are shown for several examples from the field of power plant engineering.

Shakedown analysis of engineering structures with limited kinematical hardening

We present a numerical method for the computation of shakedown loads of engineering structures with limited kinematical hardening under thermo-mechanical loading. The method is based on Melan’s statical shakedown theorem, which results in a nonlinear convex optimization problem. This is solved by an interior-point algorithm recently developed by the authors, specially designed for lower bound shakedown analysis of large-scale problems. Limited kinematical hardening is taken into account by use of a two-surface model, such that both alternating plasticity and incremental collapse can be captured. For the yield surface as well as for the bounding surface the von Mises criterion is used. The proposed method is validated by two examples, where numerical results are compared to those of literature where available.

Special issue editorial

This special issue includes twenty-three extended, reviewed and revised papers originally presented at the Eleventh International Conference on Computational Structures Technology (CST2012) and the Eighth International Conference on Engineering Computational Technology (ECT2012) held concurrently in Dubrovnik, Croatia, 4–7 September 2012. The CST2012 Young Researcher Best Paper Prize was presented to Dr J.-W. Simon, of the RWTH Aachen University, Germany, for his paper “Limit States of Structures in n-dimensional Loading Spaces with Limited Kinematical Hardening”. The ECT2012 Young Researcher Best Paper Prize was presented to Dr L. Barbie, of CEA-Cadarache, France, for the paper “An Automatic Multilevel Refinement Technique based on Nested Local Meshes for Nonlinear Mechanics” by L. Barbie, I. Ramiere and F. Lebon. The CST2012 Prize Paper is the first paper included in this special issue; while the ECT2012 Prize paper is the second paper included in this issue. The standard of the papers considered for these prizes was very high and the conference editors wish to thank the members of the conference editorial board for their work in considering all the applicants (see Fig. 1).

Shakedown analysis for elastic-plastic bodies with limited kinematic hardening

A shakedown theory for elastic–plastic bodies with limited kinematic hardening (Prager's linear kinematic hardening rule is assumed) is developed, which indicates that the plastic modulus should not influence the shakedown behaviour of structures. A reduced kinematic inadaptation theorem without time integrals is deduced, which is separated into incremental and alternating plasticity collapse criteria. The yield limit for the alternating plasticity collapse mode is taken according to particular loading processes from low–cycle to high–cycle ones, or as the lowest one of the fatigue limit. Shakedown behaviour of a spherical shell under quasistatic as well as quasi–periodic dynamic loads illustrates the results.

On shakedown theorems in hardening plasticity

The model of generalized standard materials gives a convenient framework to extend Koiter's shakedown theorems into hardening plasticity. The extension of the static shakedown theorem (Melan–Koiter's theorem), proposed previously in [5], is considered here. It leads to the definition of safety coefficients in hardening plasticity by duality. Static and kinematic approaches are discussed for the models of isotropic hardening, of linear kinematic hardening (Ziegler–Prager's model) and of limited kinematic hardening. This discussion also leads to an extension of Koiter's kinematic shakedown theorem and to a second kinematic coefficient.