Quoc Son Nguyen

On shakedown analysis in hardening plasticity

Abstract The extension of classical shakedown theorems for hardening plasticity is interesting from both theoretical and practical aspects of the theory of plasticity. This problem has been much discussed in the literature. In particular, the model of generalized standard materials gives a convenient framework to derive appropriate results for common models of plasticity with strain-hardening. This paper gives a comprehensive presentation of the subject, in particular, on general results which can be obtained in this framework. The extension of the static shakedown theorem to hardening plasticity is presented at first. It leads by min–max duality to the definition of dual static and kinematic safety coefficients in hardening plasticity. Dual static and kinematic approaches are discussed for common models of isotropic hardening of limited or unlimited kinematic hardening. The kinematic approach also suggests for these models the introduction of a relaxed kinematic coefficient following a method due to Koiter. Some models for soils such as the Cam-clay model are discussed in the same spirit for applications in geomechanics. In particular, new appropriate results concerning the variational expressions of the dual kinematic coefficients are obtained.

Instability and friction

A review on the stability analysis of solids in unilateral and frictional contact is given. The presentation is focussed on the stability of an equilibrium position of an elastic solid in frictional contact with a fixed or moving obstacle. The problem of divergence instability and the obtention of a criterion of static stability are discussed first for the case of a fixed obstacle. The possibility of flutter instability is then considered for a steady sliding equilibrium with a moving obstacle. The steady sliding solution is generically unstable by flutter and leads to a dynamic response which can be chaotic or periodic. This dynamic response leads to the generation of stick–slip–separation waves on the contact surface in a similar way as Schallamach waves in statics. Illustrating examples and principal results recently obtained in the literature are reported. Some problems of friction-induced vibration and noise emittence, such as brake squeal for example, can be interpreted in this spirit.

On shakedown theorems in hardening plasticity

Abstract The model of generalized standard materials gives a convenient framework to extend Koiters shakedown theorems into hardening plasticity. The extension of the static shakedown theorem (Melan–Koiters 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–Pragers model) and of limited kinematic hardening. This discussion also leads to an extension of Koiters kinematic shakedown theorem and to a second kinematic coefficient.

Derivation of the Inelastic Behaviour of Plates and Shells From the Three Dimensional Models and Extensions

The elastoplastic behaviour of plates and shells is usually obtained in two different ways. First, the so-called global method assumes that the yield function can be expressed in terms of resultant in-plane stresses and bending moments. The success of such a formulation is due principally to its simplicity as a natural generalization of the elastic theory of plates and shells. Besides the well-known difficulty on an adequate expression of the adopted criterion, such a model leads to a linear variation of in-plane stresses through the thickness of the structure. If this description is widely used and seems to give entire satisfaction in limit analysis, on the contrary, in the context of elastoplastic analysis, a more refined model is sometimes necessary. The second possibility to derive the inelastic behaviour is to take account of the distribution of plastic strains along the thickness. This more complicated formulation is often preferred to the first one in computation by finite-element programs. The goal of our communication is to give a mathematical justification of the second formulation by asymptotic analysis. It consists in considering the thickness of the shell as a small parameter governing equations for plates and shells are then derived exactly from the three-dimensional description by asymptotic development with respect to the thickness. In the case of elastoplasticity, this leads to a two dimensional model with respect to the displacements of the middle surface. The plastic behaviour is characterized by multiple plastic potentials in the sense of Koiter-Mandel, with hardening parameters which are residual stresses along the thickness. The in-plane stresses are no more linear through the thickness. The derivation of the bidimensional behaviour is presented in detail in the case of standard materials for both deformation theory and incremental theory of plasticity. A comparison-between global models and the ones obtained in this paper is given for particular choice of yield criteria. A section is devoted to composite materials. It is assumed that a plasticity phenomenon can appear in the epoxy layers sticking together two layers of fibers. Then using asymptotic methods we deduce a plate model which takes into account the effect of a gliding between two layers. Finally a buckling model is suggested.

Min-Max Duality and Shakedown Theorems in Hardening Plasticity

This paper gives a comprehensive presentation on shakedown theorems in perfect and in hardening plasticity. The extension of classical shakedown theorems for hardening materials is an interesting subject in plasticity. In particular, the model of generalized standard materials gives a convenient framework to derive appropriate results for common models. The starting point is an extended static shakedown theorem for hardening plasticity. It leads by min-max duality to dual static and kinematic approaches to compute the safety coefficient with respect to shakedown. These approaches are discussed for common models of isotropic and of kinematic hardening. In particular, the kinematic approach leads to new results on the expressions of the safety coefficient.

Limit Response in Cyclic Plasticity

The objective of this communication is to present a numerical investigation on the computation of the cyclic response of elastic-plastic structures under cyclic loading. First, we investigate a cyclic approach method based upon a direct research of cyclic solutions by iterative procedure ensuring the periodic response at the local scale and at every step of iteration. This method may be a priori interesting since one deals only with cyclic responses of stress and strain.

USE OF FRACTURE MECHANICS IN TUBULAR CONNECTIONS FATIGUE CRACK GROWTH

The experimental studies carried out by IRSID (Institute de Recherches Siderurgiques) within the framework of a larger European program are backed up by a French program of theoretical and numerical studies. The purpose of this program is to relate the results obtained in the tests specimen experiments to the behaviour observed in large scale tubular joints tested in fatigue fracture mechanics. The paper presents some of the results and trends obtained, with which it should be possible to make better life predictions.