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This paper is devoted to the numerical simulations of elastoplastic plates at small strains by using the non-incremental Brezis-Ekeland-Nayroles (BEN) principle. This space-time variational principle is based on the energy dissipation and transforms the boundary value problem into a null minimization one under constraints. Unlike the standard incremental procedure, the BEN principle characterizes the entire trajectory and delivers the mechanical fields at all time steps simultaneously. Two kinds of plates: the thin plate Love-Kirchhoff and the thick Reissner-Mindlin circular plates undergoing uniform pressure are studied respectively. The description of the dicretization of the principle using the mixed finite element method and the implementation are detailed. The obtained results are compared against analytical solutions provided in the literature and numerical predictions derived by the classical incremental procedure.

Computational mechanics -, - (2021). doi:10.1007/s00466-021-01989-4, Published by Springer, Heidelberg

This paper is concerned with the energy-dissipation Brezis–Ekeland–Nayroles variational principle (BEN principle) for the numerical study of quasi-static elastoplastic and viscoplastic problems in small strains. This principle is based on the dissipation potential and its Fenchel transform and allows to have a consistent view of the whole evolution by computing the nonlinear response along the whole time history as a solution of a suitable minimization problem. In the present work, the BEN principle is applied to address the elastic perfectly plastic and viscoplastic thick hollow cylinder subjected to internal pressure. It turns out that the BEN variational formulation is based on a two-field functional, that leads naturally to discretize the displacement and stress fields. We present the detailing of the discretization and the numerical implementation of the minimization problem by using the mixed finite element method which is more efficient to enforce the yield condition. Computational accuracy and efficiency of the BEN principle is assessed by comparing the numerical results with the analytical ones and the simulations derived by the classical step-by-step finite element procedure.

The present work aims to optimize the mechanical brush finishing of Metal Inert Gas (MIG) welded joints on AISI316L thin steel sheet. The innovative methodology is based on the experimental design methodology of the Taguchi design (L9) and the analysis of variance (ANOVA) with objective function of microhardness in brushed layers. A “Signal to Noise” approach is adopted with the objective of “Large is Better”. The speed of rotation of the brush, the speed of advance of the sample, the number of brushing passes, and the depression ratio of the brush fibers are the four three-level factors of the L9 design. The study of the effects of these factors showed the major influence of the depression and the number of passes on the magnitude of hardening of the brushed layers in the various zones of the weld. Minor interactions, found in the regression model, are noted between the different factors. Measurements of microhardness in the depths of each zone reveal distinct cure rates from one zone to another. The melted zone of the weld undergone the minimum hardening, unlike the heat affected zone (HAZ), whose microhardness contribution reaches 30% compared with the non-brushed welded sample. It is revealed that the main results lie in the microhardness contribution, of the order of 30% at the joint, while keeping the same level of magnitude of the mechanical strength using the optimal parameters.

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.

91st Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM 2020, ; Proceedings in applied mathematics and mechanics : PAMM 20(1), 2021 e202000304 (2021). doi:10.1002/pamm.202000304 special issue: "Special Issue: 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) / Issue Edited by: D. Kuhl, A. Meister, A. Ricoeur, O. Wünsch", Published by Wiley-VCH, Weinheim [u.a.]

Most computer-aided engineering software provide a classical incremental computation procedure for nonlinear problems. Although little used in the literature, the Brezis-Ekeland-Nayroles (BEN) principle, an alternative step-by-step algorithm, based on the time integration of the sum of the dissipation potential and its Fenchel polar can have a global view of whole evolution. In short, the BEN principle converts a mechanical problem to a constrained optimization problem. Recently, Buliga and de Saxce have proposed a symplectic version of the BEN principle which generalizes the Hamiltonian inclusion formalism for the dissipative systems. In the present work, this formalism is specialized to the standard plasticity in small, finite strains, in statics and dynamics. We apply it numerically to solve the classical problem of a tube problem in plane strain subjected to an internal pressure in statics and dynamics. An excellent agreement is obtained between the numerical results obtained by the BEN approach and the reference numerical solution.

This paper is a survey of recent trends in the poroplasticity combined with Direct methods. Using the hollow sphere model as Reference Elementary Volume (REV) with a matrix obeying von Mises microscopic plastic yield criterion, a stress variational model (SVM), dual of Gurson’s one, has been proposed to find by the Limit Analysis a macroscopic criterion depending on the porosity. Remarkably, it depends on the third invariant \(J_3\) but only through its sign. Applying the normality law to the macroscopic criterion, the evolution of porosity with respect to the stress triaxiality exhibit clear discrepancies with Gurson’s one which is known to overestimate the variation of the porosity. Some extensions has been proposed to obtain a continuous dependence with respect to \(J_3\) through Lode’s angle, to improve the strength value for the pure deviatoric loading. Thanks to the bipotential formulation, a macroscopic yield criterion was also proposed for a non associated Drucker-Prager matrix. Using the Shakedown Analysis, the method has been extended to the repeated variable loadings to obtain a fatigue criterion for the porous materials. It depends on the porosity but also strongly on Poisson’s coefficient. The general case involving shear effects with any cyclic load fluctuations ranging from the pulsating load to the alternating one is considered. The macroscopic criteria depend on the first and second macroscopic stress invariants and the sign of the third one.

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.

The goal of the present work is the demonstration of the ability of the Symplectic Brezis–Ekeland–Nayroles (SBEN) principle for the numerical simulation of dynamic elastoplastic problems in small strains. This non-incremental approach is an alternative method to conventional step-by-step techniques, as it allows the computation of the whole evolution curve over the entire loading history. We show that the SBEN variational formulation yields to a constrained time–space minimization problem . The cost function depends on stress, displacements, and plastic multiplier fields, which, naturally, leads to a mixed finite element discretization . The details of the solution algorithm are illustrated through the numerical study of the elastoplastic response of the thin and thick pressurized tubes, including inertia effects. Moreover, the balance of momentum is handled using two approaches. Firstly, the Schaefer superposition technique is considered, where the balance of momentum is exactly satisfied. Secondly, the balance of momentum is considered as an optimization constraint and satisfied only at the Gauss integration point level of the finite element problem. The accuracy and efficiency of the SBEN principle are assessed by comparing the numerical results with the analytical solutions (for the thin tube) and the predictions derived by the conventional incremental finite element procedure.

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.

This paper revisits the sliding interface crack problem between elastic and rigid half-planes studied by Bui and Oueslati and provides an alternative method of derivation of the solution, which will then be extended to three-dimensional (3D) crack problems. Based upon the displacement continuation technique of complex potentials, an appropriate Green function for the isolated edge dislocation dipole at the interface is given. Then by considering the sliding condition along the interface crack, the field equations can be obtained for the two-dimensional (2D) problem. Furthermore, it is shown that the edge dislocation dipole in 2D appears to be a particular form of the fundamental Kupradze–Basheleishvili tensor in 3D, which provides a method for deriving the coupled nonlinear integral equations for the same frictional interface plane crack of an arbitrary shape. The present work describes how the 3D sliding interface crack is related to the same problem in 2D.

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.

International audience; This paper is devoted to the numerical study of the influence of the non-associativity of the Drucker–Prager model on the plastic limit load and the failure mechanism of the strip footings. The attention is mainly focused on the determination (estimation) of the mechanical fields and geometric characteristics of the collapse mechanism. Rough and smooth contact between the punch and the substrate are considered. Analyses were performed by incremental finite element simulations by using of the object oriented computer code Cast3m. The code is first validated against analytical solutions for two problems available in the literature. It is worth noting that during this validation, we provide new numerical results concerning the ultimate load of a pressurised pore with non-associate matrix. Then, the limit plastic load of the weightless soil is computed and the post-treatment of the numerical results permits one to select the relevant mechanical fields. The main result is that, for the Prandtl mechanism associated to the rough footing, the angle base of the triangular wedge under the footing is independent of the dilatancy angle. This property appears to be very interesting and useful when constructing analytical bounds of the plastic limit load within the framework of limit analysis.

International audience; This paper is concerned with a bipotential approach for estimating the plastic collapse loads of a half-space made with a non-associated Mohr–Coulomb material and indented by a rigid punch. In geotechnics, this problem is called the bearing capacity of shallow strip footing for which the analytical solution is derived by Prandtl (1920) [46] and Hill (1950) [35] in the context of associated plasticity. However, when the plastic model is not associated, no analytical methods have yet been developed. Here we explore this issue in a rigorous mathematical framework coupling the bipotential concept and limit analysis. First, the method proposed makes use of the method of characteristics to build a statically and plastically admissible stress field that enables a lower estimate of the plastic limit loads. Next, the extended kinematic theorem of limit analysis to non-standard plasticity is applied to derive an upper quasi-bound of the collapse loads. For this aim, the internal rate of plastic dissipation is obtained thanks to the bipotential functional depending on both a trial stress field and a Prandtl-like collapse mechanism. The analytic estimates are compared to the formulae and numerical results provided in literature.

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; The construction of an analytic solution of the problem of stick-slip waves crossing the interface between an elastic half-space and a tigid one under unilateral contact and Coulomb friction is considered. The method of solution is based on the analytic continuation method of Radok's complex potentials within the framework of steady elastodynamic problems. The governing equations combined with the boundary conditions are reduced to a Riemann-Hilbert problem with discontinuous coefficient, and closed-form expressions of the solution are derived. It is found that the existence of solutions depends on the additional velocity, which is related to the longitudinal elongation. If this velocity is ignored, there is no solution, if not, it is possible to construct weakly singular solutions satisfying all stick-slip conditions except over a narrow zone where the waves exhibit a crack-like behaviour.

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; This article discusses the reconstruction methods for the inversion of the residual stresses in rods under the Navier-Bernoulli or the Vlassov (containing warping) kinematic conditions. The proposed framework permits us to express the residual stresses in terms of the Airy stress potential expressed as the solution of an ordinary differential equation. The particular case of the Navier-Bernoulli kinematics leads to a closed-form solution.; Cet article présente les méthodes de reconstruction des contraintes résiduelles dans des poutres en considérant des hypothèses cinématiques de type Navier-Bernoulli ou Vlassov (avec gauchissment). La technique proposée permet d'exprimer les contraintes résiduelles à partir d'une fonction d'Airy, solution d'une équation différentielle ordinaire. La cinématique de Navier-Bernoulli conduit à une solution explicite de ce problème. Version française abrégée La détermination des contraintes résiduelles dans les structures est un sujet de grande importance puisque leur présence affecte considérablement la résistance à la fatigue et à la rupture des pièces mécaniques. Deux classes de méthodes d'identification peuvent être utilisées : les méthodes non destructives (mesures par diffraction des rayons X, par ultrasons, etc.) et destructives (enlèvement de la matière, perçage de trou, etc.). Ces dernières provoquent un relâchement de contraintes (et de déformation) conduisant ainsi à un problème inverse de reconstruction de champ de contraintes initial à partir de mesures effectuées sur la frontière du solide. (A. Constantinescu), Abdelbacet.Oueslati@univ-lille1.fr (A. Oueslati). 1 Dans cet article, nous présentons les méthodes de reconstruction des contraintes résiduelles après enlèvement de la matière dans le cas des barres élancées de section rectangulaire. L'hypothèse principale adoptée est celle d'une relaxation purement élastique des contraintes après enlèvement successif de couches d'épaisseurs h. Cette hypothèse permet de ramener la résolution du problème de reconstruction à l'étude de trois problèmes élastiques distincts (voir Fig. 2) : le premier problème P1 est celui de l'équilibre de la poutre Ω libre de charges extérieures et contenant le champ de contraintes résiduelles inconnu σ. Pour le deuxième problème noté P2, le solide est constitué par la barre Ω h obtenue à partir de Ω après enlèvement d'une couche d'épaisseur h, c'est sur cette configuration que la mesures des contraintes par diffraction des rayons X est effectuée sur la face supérieure. Enfin, P3 est le problème élastique auxiliaire traduisant l'équlibre de la barre Ω h , soumise aux tractions surfaciques (−σ · e z) sur sa face supé-rieure (σ étant le champ de contraintes résiduelles à identifier). L'application du principe de superposition à ces trois problèmes conduit à la relation (3). Il est impératif de noter que le problème de reconstruction possède une solution explicite si le système (2) et (3), ainsi que le problème auxiliaire ont des solutions explicites. Rappelons que pour le cas d'un demi-espace élastique, la solution du problème auxiliare P3 est fournie d'une manière explicite par la solution de Boussinesq-Cerutti et le système (2) est résolu après application de la transformée de Fourrier [2]. Pour le cas des barres de dimensions finies, une solution explicite analogue à celle de Boussinesq-Cerutti est difficille à établir. L'idée est de résoudre le problème auxiliaire dans le cadre de la théorie des poutres de Navier-Bernoulli puis celle de Vlasov et d'introduire le potentiel de contraintes d'Airy (13) pour résoudre les Éqs. (2) et (3). Des formules intégrales explicites (16) pour la reconstruction de champ de contraintes résiduelles sont obtenues pour la cinématique de Navier-Bernoulli. Elles constituent une généralisation des formules de Morre et Avans [3] établies pour les barres (plaques) de longueurs infinies.

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.

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. A semi-analytical analysis of stick-slip waves is obtained when the system of governing equations is reduced by condensation to a simpler system involving only the contact displacements. This reduced system, of only one space variable in addition to time, can be solved almost analytically and gives some interesting informations on the existence and the characteristics of stick-slip waves such as the wave numbers on the circumference, stick and slip proportions, wave celerities, tangential and normal forces. It is shown in particular that the stick-slip-separation solutions would occur for small normal pressures or high rotational speeds. Since the analytical discussion becomes cumbersome in this case, a second approach based on numerical analysis by the finite element method is performed. The existence and the characteristics of stick-slip and stick-slip-separation waves are discussed numerically.

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.

This paper is concerned with the elastic plastic response of a two-bar system with temperature-dependent elastic coefficients under cyclic thermomechanical loadings. Such materials are characterized by lack of results concerning the asymptotic behaviors and conditions for shakedown occurrence. This study shows that the considered simple structure is sufficiently complex to experience different periodic long term behaviors as in classical elastoplasticity. In order to understand how Melan-Koiter method works for such materials, the evolution of the structure’s response until the stabilization of the plastic strain (‘shakedown’) or the asymptotic dissipative behavior (‘alternating plasticity’ or ‘ratcheting’) is analytically addressed and the Bree diagram is then constructed. The main result of this work is that the residual stress and strain fields are time-dependent even when shakedown occurs. Incidently, we proved that Halphen’s conjecture (Halphen in C. R. Mec. 33:617–621, 2005) giving a sufficient condition for shakedown occurrence is not a necessary condition. Finally, numerical results performed by an incremental finite element procedure are presented.

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.

In this paper, an extension of the static shakedown theorem (Melan’s theorem) for elastic-plastic materials with temperature-dependent elastic modulus is presented. Cases of the decrease of yield function and the variation of coefficient of thermal expansion with temperature are included. The proposed extension leads to the introduction of a shakedown safety coefficient. To illustrate the statements of our theorem, step-by-step finite element procedure is applied to study a three-bar problem and a plate with the central hole subjected to thermo-mechanical cyclic loadings.

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.

International audience; A numerical study on the dynamic response of a mechanical system composed of two coaxial cylinders in relative rotation and unilateral contact with Coulomb friction is presented. This discussion complements the semi-analytical results on stick–slip waves given in Moirot, Nguyen, C. R. Acad. Sci. Paris Sér. IIb 328 (2000) 663–669, by some numerical simulations on the dynamic transition of the system from a given initial state to a surface wave. The case of stick–slip–separation waves is considered.; Il s'agit d'une étude numérique sur la réponse dynamique d'un système de deux cylindres coaxiaux, en rotation relative avec contact unilatéral frottant suivant la loi de Coulomb. On complète les résultats semi-analytiques antérieurs obtenus dans Moirot, Nguyen, C. R. Acad. Sci. Paris Sér. IIb 328 (2000) 663–669, sur les ondes adhérence–glissement en simulant par le calcul numérique la transition dynamique du système à partir d'un état initial quelconque vers la propagation d'une onde de surface. Le cas des ondes adhérence–glissement–décollement est examiné.