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This paper is concerned with the shakedown limit states of porous ductile materials with Drucker-Prager matrix under cyclically repeated loads. Using the hollow sphere model and Melan’s shakedown theorem based on time-independent residual stress fields, a macroscopic fatigue criterion is derived for the general conditions of cyclic loads . First, the case of the hollow sphere subjected to pure hydrostatic loading is studied and the limit states of collapse by fatigue or by development of mechanism are derived. Then, the general case involving shear effects with any arbitrary cyclic load fluctuations ranging from the pulsating load to the alternating one is considered. The key idea is in two steps: (i) the choice of appropriate trial stress and trial residual stress fields and (ii) then maximizing the size of the load domain in the spirit of the standard lower shakedown theorem. The new macroscopic shakedown criterion depends on the porosity, the friction angle , Poisson’s ratio, the two stress invariants of the effective stress tensor and the sign of the third one. Together with the limit analysis-based yield criterion corresponding to the sudden collapse by development of a mechanism at the first cycle, it defines the safety domain of porous materials subjected to cyclic load processes. Interestingly, it is found that the safe domain is little sensitive to variations of the friction angle, however, it is considerably reduced compared to the one under monotonic loads obtained by limit analysis. Finally, a comparative study between the analytical results and numerical predictions performed by micromechanics-based finite element simulations is conducted for different porosities and friction angles.

The shakedown safety domain of porous materials is investigated in this work by a direct numerical method based on Melan’s theorem. Considering the critical loading path of the load domain instead of the whole history, the statical shakedown condition is transformed to a large size optimization problem, of which the objective function is the shakedown limit factor, with the discretization of a three dimensional model composed of von Mises matrix. As in limit analysis of the hollow sphere model, we consider pure hydrostatic and deviatoric loads to produce the reference elastic stresses in the fictitious elastic body, for the purpose of capturing the effects of compressive/tensile and shear effects. By the use of a non-linear optimizer IPOPT based on the interior-point method, the proposed optimization problem is solved efficiently, providing not only the shakedown limit load factor, but also the distribution of residual stresses for the limit state. The established method is applied to axisymmetric and non-axisymmetric cyclic loadings, as well as the two independent variable loadings, in order to discover the influence of the loading condition and the solid matrix characteristics on the shakedown safety domain of porous medium (hollow sphere). The obtained results are illustrated and fully compared with reference solutions from literature and with the one from the incremental step-by-step FEM computations. The present method has been accessed and validated by the comparative study.

A large number of analytical strength criteria have been developed for porous materials having a Drucker-Prager solid matrix by using different up-scaling methods. A number of representative exisitng criteria are selected and summarized. These criteria are first evaluated and compared both analytically and numerically with new FEM solutions obtained in this work. A wide variety of new numerical results are then provided from finite element simulations to enlarge the comparisons. In particular, the strength under the pure shear condition is discussed and studied in depth, either numerically and theoretically. Based on these comparisons, a new improvement of the prediction of pure shear strength is established. With this improvement, we derive a new macroscopic strength criterion for the porous materials with a Drucker-Prager type solid matrix. The new criterion is validated against those numerical solutions for a wide range of porosity and frictional coefficient. The special case of a porous material with a von Mises type solid matrix is also studied with a good validation.

An approximate analytical criterion is established in the present study to describe the macroscopic strength behavior of porous materials with a pressure sensitive and tension-compression asymmetry solid matrix. The solid matrix is assumed to obey a Mises-Schleicher type criterion at the local scale. By using the stress variational homogenization method (SVH) with a strictly statically admissible stress field, an analytical criterion is first obtained. This criterion can retrieve the exact solution for the hydrostatic loading and provide a good prediction of the strength for the pure deviatoric loading (Σeq/σo). However, for small values of stress triaxiality in compressive domain, the analytical criterion underestimates the strength. Based on the accurate value of Σeq/σo obtained by the SVH method and taking into account some special conditions and requirements, an approximate improved strength criterion is derived. For a wide validation of the proposed criterion, new numerical results are presented based on the finite element simulations. The improved criterion is then verified through comparisons with both the upper and lower bounds given in Pastor, Kondo, and Pastor (2013) and the finite element results for a large range of porosity and tension-compression asymmetry ratio of the matrix. It is found that this new criterion improves existing ones. Finally, the criterion is applied to describe the plastic yield stresses of porous chalk and plaster.

This paper provides the analytical solution of the elastic hollow sphere subjected to axisymmetric and pure deviatoric surface tractions within the framework of the infinitesimal strains. The expressions of the stress and displacement fields are derived in closed-form in terms of spherical harmonics by using Boussinesq-Neuber-Papkovitch potentials. The obtained solution is valid for thin and thick hollow sphere. It is shown that, for the J 2 -plasticity, the hollow spherical shell undergoes incipient first plastic strains at the pole θ = π / 2 located on the internal surface boundary. In the perspective of shakedown analysis of ductile porous materials, the macroscopic stress and strain fields of the hollow sphere model are obtained from their local counterparts by the volume average operator.

International audience; The aim of work is to propose a macroscopic criterion of porous ductile materials with von Mises type matrix using the Stress Variational Homogenization (SVH) recently proposed by the authors. Starting with Hill's stress principle for rigid plastic materials, the yield condition on the microscopic stress field is relaxed by means of Lagrange multiplier. On the other hand, the trial stress field is strictly statically admissible allowing to improve a previous model in which the stress condition on the pore boundary was only satisfied in a mean sense. By solving the resulting Saddle point problem and performing a suitable approximation of the integral occurring in it, we derive a closed form formula. The main feature of the new macroscopic criterion is a more accurate value obtained for the pure shear loading case. We also derive the macroscopic flow rule as well as the porosity evolution equations. The obtained results are fully discussed and compared to existing models, available numerical data and to Finite Elements results.

The paper is devoted to the determination of shakedown limit states of porous ductile materials under cyclically repeated loads by considering the hollow sphere model with von Mises matrix. We adopt Melan's statical approach based on time-independent residual stress fields. First of all, we determine the exact solution for the pure hydrostatic loading. It turns out that the collapse occurs by fatigue. Next, suitable trial stress fields are built with additional terms to capture the shear effects. This theoretical fatigue criterion of porous materials agreed with numerical simulations and experimental results provided in literature according to which, in ductile metals, the strain to fracture under cyclic loading is considerably lower than the one reached monotonically. Finally, the built criterion is compared to numerical simulations derived by both shakedown direct method and incremental simulations.

In a recent paper (Zhang et al., 2017), we have analytically derived a macroscopic criterion to determine the shakedown domain for ductile porous media with von Mises matrix under repeated loads by the use of Melan’s theorem. The proposed criterion in Zhang et al. (2017) concerns only two limit cyclic loading cases: (i) the alternating and (ii) the pulsating loads. In this paper, we generalize this approach for all possible intermediate cyclic loading cases ranging between the tow limit cases. The key idea relies on the choice of a more appropriate trial residual stress field allowing analytical computations and the improvement of the previous model simultaneously. The new analytical criterion depends on the first and second macroscopic stresses invariants and the sign of the third one. A more accurate value for pure shear loading case is obtained. The obtained results are fully discussed and compared to the previous criterion (Zhang et al., 2017) and numerical results derived by both shakedown direct method and incremental elastic plastic simulations.

International audience; In a recent paper (Hasbroucq et al., 2010 [1]), we have analyzed the long-term behavior of a discrete mechanical system with temperature-dependent elastic properties under cyclic thermomechanical loadings. In particular, we have shown that the residual stress and strain fields are time-dependent when shakedown occurs and thus the Halphen's (2005) [2] conjecture is not a necessary shakedown condition. Also, we have shown that there is loss of convexity of elastic and shakedown domains in the Bree diagram. In order to examine if these facts are related to the finite dimensional character of the structure and to the uniaxial path load in Hasbroucq et al. (2010) [1], we consider here a continuum media, namely a thin plate undergoing a biaxial loading. Linear and quadratic dependence of Young's modulus and Poisson's coefficient with respect to the temperature are considered. Closed-form expressions of the asymptotic responses of the plate under a given thermomechanical history loads are presented for shakedown, alternating plasticity and ratcheting regimes. In addition, the present study shows that in the Bree diagram, the elastic domain is convex while the shakedown zone is not so and, contrary to Hasbroucq et al. (2010) [1], the residual (elastic) strain field is time-independent for shakedown. Therefore, the present example falls in the Halphen's shakedown conjecture.

This paper presents a short review and new results about the self-excited responses under the form of stick-slip regimes. First, the Van-der Pol oscillator with one degree of freedom is considered. Then it is shown that it is possible to build semi-analytical and numerical (by the FEM.) solutions of stick-slip-separation waves for a brake-like system. Then, we present new results concerning the mechanical model composed of a rigid half space in frictional sliding with an elastic half-space. The method of solution, based on periodic complex Radoks potentials, is novel and differs from those in literature. Besides, in contrast with many works, we shall consider the longitudinal elongation which plays a crucial rule in the solution procedure. A unique and weakly singular solution is found and satisfies all stick-slip conditions except over a narrow zone at transition points which implies a cracklike behaviour at the stick-slip borders.

International audience; In Gurson's footsteps, different authors have proposed macroscopic plastic models for porous solid with pressure-sensitive dilatant matrix obeying the normality law (associated materials). The main objective of the present paper is to extend this class of models to porous materials in the context of non-associated plasticity. This is the case of Drucker–Prager matrix for which the dilatancy angle is different from the friction one, and classical limit analysis theory cannot be applied. For such materials, the second last author has proposed a relevant modeling approach based on the concept of bipotential, a function of both dual variables, the plastic strain rate and stress tensors. On this ground, after recalling the basic elements of the Drucker–Prager model, we present the corresponding variational principles and the extended limit analysis theorems. Then, we formulate a new variational approach for the homogenization of porous materials with a non-associated matrix. This is implemented by considering the hollow sphere model with a non-associated Drucker–Prager matrix. The proposed procedure delivers a closed-form expression of the macroscopic bifunctional from which the criterion and a non-associated flow rule are readily obtained for the porous material. It is shown that these general results recover several available models as particular cases. Finally, the established results are assessed and validated by comparing their predictions to those obtained from finite element computations carried out on a cell representing the considered class of materials.

The paper analyzes the frictional sliding crack at the interface between a semi-infinite elastic body and a rigid one. It gives solutions in complex form for non-homogeneous loading at infinity and explicit solutions for polynomial loading at the interface. It is found that the singularities at the crack tips are different and that they are related to distinct kinematics at the crack tips. Firstly, we postulate that the geometry of the equilibrium crack with crack-tip positions b and a is determined by the conditions of square integrable stresses and continuous displacement at both crack tips. The crack geometry solution is not unique and is defined by any compatible pair ( b , a ) belonging to a quasi-elliptical curve. Then we prove that, for an equilibrium crack under given applied load, the “energy release rate” G tip , defined at each crack tip by the J e -integral along a semi-circular path, centered at the crack tip, with vanishing radius e , vanishes. For arbitrarily shaped paths embracing the whole crack, with end points on the unbroken zone, the J -integral is path-independent and has the significance of the rate, with respect to the crack length, of energy dissipated by friction on the crack.

This paper is concerned with a semi-analytical approach to the solution of the axisymmetric indentation problem for a multilayered elastic half-space. The stress and displacement fields for each layer and the substrate are derived in closed form by using the Papkovich-Neuber potentials and the Hankel transform. The bonded or sliding interface conditions between the sub-layers are handled by the use of the appropriate transfer matrix, and then the mixed boundary value problem is reduced to a Fredholm integral equation. Symbolic and numerical computations of the solution are implemented in the symbolic software Mathematica in the form of a fast and efficient numerical algorithm, allowing rapid determination of the load-displacement curves and composite elastic properties for an arbitrary rigid indenter shape. A series of results for different indenters (flat, conical, spherical and blunted conical punch shapes) and different multilayered composites is presented and discussed. The complete set of symbolic and numerical computations are provided as supplementary resources with the paper. © 2013 Elsevier Ltd. All rights reserved.

International audience; The paper is devoted to the determination of plastic limit state of a hollow sphere with a Drucker-Prager matrix and subjected to hydrostatic loading. There are two possible plastic regimes corresponding respectively to the tensile and compressive stresses. For the associated case (dilation angle equal to the friction angle), the collapse is complete (the whole sphere is plastified) with a unique regime. For the non-associated cases, we consider weaker solutions (partial collapse and regime change). Nevertheless, we show the collapse is complete and exhibits a single regime. Consequently, the collapse stress field and the limit load do not depend on the value of the dilation angle. This theoretical result is confirmed by numerical simulations.

In this paper, the construction of analytic solutions of stick-slip waves propagating along the interface between an elastic half-space and a moving rigid one is investigated. The contact between the solids is governed by unilateral constraints and Coulomb friction law and the deformable body is loaded by remote uniform stresses $$\tau^{*}_{yy}0$$ . The method of solution is based on the analytic continuation of Radok’s complex potentials within the framework of steady elastodynamical problems. The governing equations combined with the boundary conditions are reduced to a Riemann–Hilbert problem with discontinuous coefficient. This approach for the stick-slip study is novel and differs from those in literature, namely the series method and the Weertman’s dislocation formulation. We present the closed-form solution of the Riemann–Hilbert problem and show that the principal unknowns are of number two: the wave celerity and the ratio of the slip length by the stick one. The considered loading introduces an additional velocity V* related to the longitudinal elongation $$\varepsilon^{*}_{xx}$$ due to the normal stress $$\tau^{*}_{yy}$$ . We show that if V* vanishes then there is no solution. When V* ≠ 0, it is possible to construct weakly singular solutions satisfying all stick-slip conditions except over a narrow zone at transition points: the shear stress is singular over a small zone and the normal contact stress exhibits a positive singularity over a very small zone in the slip region which implies a separation near the singular transition.

This article considers the frictional interface waves generated by the flutter instability of the sliding steady state for an elastic tube in frictional contact with a rigid and rotating shaft. According to the values of the contact pressure, the rotation velocity and the friction coefficient, several periodic dynamical responses can be found under the form of travelling surface waves. Such a periodic solution may be interesting in the study of a possible dynamic transition from the sliding steady state in the spirit of Andronov–Hopf bifurcation. Examples of stick-slip, stick-slip-separation and stick-slip-separation-reverse-slip waves propagating along the contact surface, obtained by various semi-analytical and numerical approaches, are reported here. Some results on the stability of these travelling waves are also indicated. To cite this article: Q.S. Nguyen et al., C. R. Mecanique 336 (2008).

The dynamical problem of a brake-like mechanical system composed of an elastic cylindrical tube with Coulomb's friction in contact with a rigid and rotating cylinder is considered. This model problem enables us to give an example of non-trivial periodic solutions in the form of stick-slip or stick-slip-separation waves propagating on the contact surface. It is shown in particular that the stick-slip-separation solutions would occur for small normal pressures or high rotational speeds. The existence and the characteristics of stick-slip and stick-slip-separation waves are discussed numerically.

One considers an interface crack between an elastic half-plane and a rigid half-plane, in the presence of Coulomb's friction. There exists two singular solutions at the semi-infinite crack tip, which correspond physically to the push-in and the pull-out of a rigid fiber in an elastic matrix. The exact solutions show that only the mode II is present and there is no oscillatory behavior of the stress and the displacement near the crack tip, but an oscillatory behavior is observed at the transition point with the stress free zone. There exists four types of singular solutions for finite crack in bounded medium, but only one in an infinite medium.; On étudie une fissure à l'interface d'un massif élastique semi-infini avec un demi-plan rigide, en présence de frottement de Coulomb. Il existe deux solutions singulières à la tête d'une fissure semi-infinie correspondant physiquement aux deux problèmes d'enfoncement et d'arrachement de la fibre indéformable dans une matrice élastique. Les solutions exactes montrent que seul le mode II est présent et qu'il n'y a pas de comportement oscillant près de la pointe de la fissure. La solution pour une fissure d'interface de longueur finie montre qu'il n'y a pas d'oscillations près des pointes de la fissure, mais qu'il y en a plus loin à la transition avec la zone décollée. Il y a quatre types de solutions singulières pour une fissure finie dans un solide borné, mais une seule solution dans un milieu infini.