Influence of hardening rule on the elasto-plastic behaviour of a simple structure under cyclic loading

The modes of cyclic elasto-plastic deformation of a two-bar structure with unequal areas and lengths under the simultaneous action of sustained mechanical load and cyclic thermal history are investigated analytically using three types of elasto-plastic material models: perfectly plastic, linear kinematic hardening and linear isotropic hardening. This simple structure is shown to exhibit much of the behaviour of interest in design of structural components subjected to repeated thermal loads: viz, elastic shakedown, reversed plasticity and ratcheting. The cyclic plastic behaviour of the structure is developed in closed form and the effects of strain hardening, hardening rule and geometrical parameters of the two-bar assembly on the deformation modes are critically examined.     

On thin rings and short tubes subjected to centrally opposed concentrated loads

The deformation behaviour of mild steel rings and short tubes of various diameters, thicknesses and lengths loaded centrally by opposed conically-headed cylindrical punches is examined. The test equipment and procedure are briefly described and typical histories of deformation, punch load—displacement diagrams and the increase in the horizontal ring or tube radius for increasing punch travel are presented. A comparison is made between the latter and those predicted by use of a simple expression based on the assumption of rigid-plastic material deformation. Good correlation is obtained with rings of different diameter to thickness ratio.     

A model of one-surface cyclic plasticity and its application to springback prediction

This paper presents an elasto-plastic constitutive model based on one-surface plasticity, which can capture the Bauschinger effect, transient behavior, permanent softening, and yield anisotropy. The combined isotropic-kinematic hardening law was used to model the hardening behavior, and the non-quadratic anisotropic yield function, Yld2000-2d, was chosen to describe the anisotropy. This model is closely related to the anisotropic non-linear kinematic hardening model of Chun et al. [2002. Modeling the Bauschinger effect for sheet metals, part I: theory. International Journal of Plasticity 18, 571-95.]. Different with the model, the current model captures in particular permanent softening with a constant stress offset as well as the Bauschinger effect and transient behavior under strain path reversal. Inverse identification was carried out to fit the material parameters of hardening model by using uni-axial tension/compression data. Springback predicted by the resulting material model was compared with experiments and with material models that do not account for permanent softening. The results show that the resulting material model has a good capability to predict springback.     

Modelling transverse cracking damage in thin, filament-wound tubes subjected to lateral indentation followed by internal pressure

Transverse cracking is a typical mode of damage in laminated composites. A continuum damage model has been established for the constitutive relationship which describes initiation and evolution of such damage. The constitutive model has been incorporated into an FE structural analysis using a commercial code, ABAQUS, via one of its user-defined subroutines, UGENS. The developed user subroutine can be applied to simulate transverse cracking damage processes in general laminated composites. As an example, the response of a thin ±55 filament-wound tube subjected to loading and unloading by lateral indentation has been analysed. The predicted load displacement curves and damage growth and stress and strain distributions in each lamina are presented. One of the emphases in this paper is on sequential loading. Subsequent to complete unloading, the tube is subjected to a different loading condition, internal pressure, and simulation of the deformation and damage process is continued. The results have been discussed and compared with experimental data wherever available.     

Evaluation of cyclic plasticity models of multi-surface and non-linear hardening by an energy-based fatigue criterion

This study examines the performance of four constitutive models according to capacity in predicting metal fatigue life under proportional and non-proportional loading conditions. These cyclic plasticity models are the multi-surface models of Mroz and Garud, and the non-linear kinematic hardening models of Armstrong-Frederick and Chaboche. The range of abilities of these models is studied in detail. Furthermore, the plastic strain energy under multiaxial fatigue condition is calculated in the cyclic plasticity models by the stress-strain hysteresis loops. Using the results of these models, the fatigue lives that have set in the energy-based fatigue model are predicted and evaluated with the reported experimental data of 1% Cr-Mo-V steel in the literature. Consequently, the optimum model in the loading condition for this metal is chosen based on life factor.     

The prediction of creep rupture of pressurized tubes using a model for void growth and coalescence

A phenomenological formulation based on a theory of plasticity for voided solids, and power-law creep, to estimate the rupture times and strains for pressurized tubes is presented. Realistic values of the damage parameters involved are selected by satisfying the creep rupture uniaxial data for the same materials. Void growth and coalescence and hence the loss of load-carrying capacity of the tube is taken as a failure criterion. Computed values of rupture times for thin- and thick-walled tubes are shown to be mostly in close agreement with the experimental data found in the literature. Only fair agreement among predicted and experimental fracture strains is observed.     

Influence of hardening on roller life

Test results are presented for P6M5 steel rollers with a thin wear-resistant coating obtained by plasma and chemicothermal treatment.     

Stress distribution in a linear hardening annular disk of variable thickness subjected to external pressure

The subject of investigation is a linear hardening annular disk of variable thickness subjected to external pressure. The variation of thickness is considered for two different cases: (1) h = h_o (1 − tr)^k and (2) h = h_o (1 − tr^k), where h₀, t and k are real constants. The analysis is based on Tresca's yield condition and its associated flow rule. The governing differential equations are presented in terms of radial displacement. Exact elastic—plastic solutions are obtained in terms of hypergeometric functions for the radial displacement and stress distributions. Numerical results showing the influence of the thickness variation on the stress distribution for different values of the hardening parameter are presented graphically. It can be seen from the present analysis that the thickness variation influences the radial stress and displacement distributions significantly. The present investigation may provide useful information for structural designers and engineers working in these scopes.     

Non-axisymmetric buckling behaviour of elastic-plastic circular tubes subjected to a nosing operation

Theoretical investigations have been carried out to predict the non-axisymmetric buckling seen in the throat of circular elastic-plastic tubes subjected to a nosing operation along a frictionless conically shaped die. The buckling points and associated modes are determined by Hill's bifurcation theory in conjunction with a non-axisymmetric buckling mode. The results show that increasing the die semi-angle and the tube thickness or decreasing the material work-hardening rate are always beneficial to improve the buckling limit. The associated critical buckling mode number generally decreases as the tube thickness increases and does not depend drastically on the die semiangle nor on the hardening rate. The effect of the average r value is substantial and the buckling limit is improved when the r value reduces. Part of the present results predict well the results observed in existing experimental work.     

Deformation and energy absorption of aluminum foam-filled tubes subjected to oblique loading

Research to quantify the energy absorption of empty and foam-filled tubes under oblique loading with different loading angles and geometry parameters was carried out. Tests on circular tubes made of aluminum alloy AA6063 under quasi-static axial or oblique loading were performed. The collapse behavior of empty, foam-filled single and double tubes was investigated at loading angles of 0°, 5°, 10° and 15° with respect to the longitudinal direction of the tube. The tubes were fixed at both ends and oblique load was realized by applying a load at the upper end of a pair of specimens. When the foam-filled tubular structures subjected to oblique quasi-static loading, some new deformation modes, such as spiral folding mode, irregular extensional folding mode and irregular axi-symmetric or diamond deformation mode, were identified and ascribed to the bending of tubes and shearing of foam filler, as well as the interaction between the tubes and the foam. The energy absorption characteristics of empty and foam-filled single and double tube structures with respect to the load angle and wall thickness are determined. It is found that the energy-absorbing effectiveness factors of the circular tube structures with aluminum foam core are significant higher than those of the empty tubes and the energy absorption capacity of the foam-filled double tubes is better than that of the empty and foam-filled single tubes.     

Ratcheting behavior of cylindrical pipes based on the Chaboche kinematic hardening rule

In this study, cyclic loading behavior of thick cylindrical pipes are described. Effects of internal pressure level and axial strain amplitude on the ratcheting rate under different types of loading histories are investigated. The kinematic hardening theory based on the Chaboche model is used to predict the plastic behavior of the structures. An iterative method is developed to analyze the structural behavior under cyclic loading conditions based on the Chaboche kinematic hardening model.     

Estimating the creep life of structures subjected to arbitrary cyclic loading

For general cyclic loading it is shown that the creep life of a structure based on initial failure may be bounded from above by solving an extremum problem. This result is developed to obtain a design method based on shakedown analysis in a manner similar to earlier work on steady loading which produced a design method based on limit analysis. The proposed design method is shown to provide conservative life predictions when compared to theoretical solutions for some simple structures.     

Contribution of ring resistance in the behaviour of steel tubes subjected to a lateral impact

The work described in this paper forms part of a closed analytical solution for the response of tubes subjected to lateral quasi-static or dynamic impact loads. The results can be used in the analytical study of the response where the tubes are usually modeled as a series of beams resting on a deformable foundation (of rings). An attempt has been made to evaluate the characteristics of this deformable foundation or in the other words the contribution of the rings to the tube resistance. Both analytical and numerical approaches have been used. It has been found that by assuming a rigid-perfectly plastic behaviour for the ring resistance, accurate results for long tubes (L/D>15) are obtained. For tubes of moderate length, using rigid-perfectly plastic behaviour underestimates the ring resistance. A rigid-plastic expression with quadratic strengthening has been proposed for tubes of moderate length. This expression has been found to provide results which are much closer to the ring resistance for tubes with 5L/D15. In very short cylinders (L/D<5), it has been found that the ring resistance is dominated by the stiffness of the beams. The validity of these conclusions has been investigated by undertaking a parametric study. In this parametric study different material and geometrical properties have been examined.     

Numerical improvement of thin tubes hydroforming with respect to ductile damage

Fully coupled constitutive equations, formulated in the framework of the thermodynamics of irreversible processes with state variables, accounting for isotropic hardening as well as the isotropic ductile damage are used to simulate numerically, by the finite element analysis, 3D metal hydroforming processes with damage occurence. An implicit integration scheme for local time integration of the constitutive equations and a dynamic explicit resolution scheme to solve the associated dynamic equilibrium problem are used. The effects of friction coefficient, material ductility and hydro bulging condition, on the hydroformability of various thin tubes are discussed.     

Estimates of the creep rupture lives of structures subjected to cyclic loading

Constitutive relationships are discussed for materials which undergo creep rupture due to cyclic loading. Relationships are proposed which describe the strain rates in terms of the current stress and a single state variable. An approximate method is derived which enables a lower bound on the rupture life to be obtained for kinematically determinate structures subjected to cyclic load and isothermal conditions. The bound on the rupture life is expressed in terms of the energy dissipation rates within the structure corresponding to stationary-state creep. The effect of multi-axial stress creep-rupture upon the structural performance is examined and bounds are derived for materials which obey maximum tension and octahedral shear stress criteria. For both multi-axial stress rupture laws and structures subjected to cyclic and reverse load conditions formulae are derived which express the lower bound rupture life in terms of the behaviour of a steady-load uni-axial creep rupture test. Results of experiments which have been carried out on copper and aluminium beams are presented for cyclic and reverse load conditions. For both rupture laws the experimental rupture times are closely predicted by the representative rupture stresses and uni-axial data.     

Onset and growth of wrinkles in thin square plates subjected to diagonal tension

A corner type constitutive equation is developed using the general quadratic anisotropic yield function. The wrinkling point of square plates subjected to diagonal tension was obtained by the bifurcation and Mindlin type plate theories, in conjunction with the finite element approximation. The growth of wrinkles was traced by a three-dimensional analysis in terms of an isoparametric shell element.

The numerical investigation was performed for the thin plates characterized by Hill's anisotropic yield function. The effects of such material characteristics as yield stress, hardening rate, orthotropy and normal anisotropy on the wrinkling limit and wrinkle height have been clarified.     

Advanced applications of BEM to three-dimensional problems of monotonic and cyclic plasticity

An advanced formulation of the boundary-element method (BEM) has been developed for three-dimensional inelastic analysis under both monotonic and cyclic loading. The analysis uses isoparametric shape functions to model complex geometries and rapid functional variations accurately. The numerical integration of the kernels are carried out by devising suitable automatic sub-segmentation which controls the error in the integration. The formulation has been applied to a number of three-dimensional elastoplastic problems involving monotonic and cyclic loading to demonstrate that it can be used for realistic engineering plasticity problems.     

Inelastic deformations in a solid circular cylinder subjected to cyclic torsion and heating

An analysis is given of the behaviour of solid circular elasto-plastic bars subjected to torsion and a transient temperature distribution. Use is made of a ‘method of successive elastic solutions’ for which a numerical procedure is developed. A particular study is made of the transient and residual stress distributions, the variation with time of twist per unit length and the growth and disappearance of plastic zones for the situation where the transient temperature distribution is brought about by an instantaneous rise in surface temperature. The influence upon behaviour of strain-hardening in the material and of the temperature dependence of yield stress are noted in particular. The effects of cyclic variations of torque and surface temperature are examined with a view to determining a sequence that is most detrimental to the bar in terms of its incremental collapse.An account is given of experiments with mild steel bars subjected to torsion and thermal cycles of rapid surface heating and slow cooling. A reasonable agreement between theory and experiment is noted during a first thermal cycle. Repeated thermal cycles exhibit the phenomenon of incremental collapse.     

Stress distributions in energy generating two-layer tubes subjected to free and radially constrained boundary conditions

Analytical solutions are obtained for thermally induced axisymmetric elastic and elastic–plastic deformations in heat generating composite tubes having a free inner and a radially constrained outer boundaries. Tresca's yield condition and its associated flow rule are used to determine the elastic–plastic response of the assembly. Depending on the physical properties of the materials used, eight different plastic regions with different mathematical forms of the yield condition may occur in the assembly. The closed form solutions for these plastic regions are obtained by assuming linearly hardening material behavior. Various numerical examples are handled using thermal and mechanical properties of real engineering materials.