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bUniversit Europenne de Bretagne, Laboratoire de Gnie Civil et Gnie, Mcanique (LGCGM) IUT de Rennes, 3, rue du clos Courtel, 35704 Rennes cedex, Franceevolution problem for plastic softening models. More precisely, he highlighted the impossibility of the plastic softening beamto ow, if the plastic curvature is assumed to be a continuous function in space, a phenomenon sometimes called Woodsparadox. Hardeningsoftening plasticity models may concern a wide class of structural mechanics problems. Such simpliedmodels can be useful for the fundamental understanding of bending of structural members at their ultimate state (reinforced0020-7225/$ - see front matter 2009 Elsevier Ltd. All rights reserved.* Corresponding author.E-mail addresses: noel.challamel@insa-rennes.fr (N. Challamel), christophe.lanos@univ-rennes1.fr (C. Lanos), charles.casandjian@insa-rennes.fr(C. Casandjian).International Journal of Engineering Science 48 (2010) 487506Contents lists available at ScienceDirectInternational Journal of Engineering Sciencedoi:10.1016/j.ijengsci.2009.12.002Material time derivative distinguished, especially for moving elastoplastic boundaries. It is recommended to usethe material time derivative in the rate-format of the boundary value problem. 2009 Elsevier Ltd. All rights reserved.1. IntroductionThis paper is focused on the propagation of localization in hardeningsoftening plasticity media, and more specically inan elementary beam model. The propagation of plasticity along a bending beam is studied for a piecewise hardeningsoft-ening momentcurvature relationship. Historically, momentcurvature relationships with softening branch were rst intro-duced for reinforced concrete beams [1]. Wood [1] did point out some specic difculties occurring during the solution of thea r t i c l e i n f oArticle history:Received 12 August 2009Received in revised form 24 November 2009Accepted 12 December 2009Available online 12 January 2010Communicated by M. KachanovKeywords:BeamHardeningSofteningGradient plasticityNon-local plasticityLocalizationCantileverPropagationVariational principleBoundary conditionsa b s t r a c tThis paper is focused on the propagation of localization in hardeningsoftening plasticitymedia. Using a piecewise linear plasticity hardeningsoftening constitutive law, we lookat the 1D propagation of plastic strains along a bending beam. Such simplied modelscan be useful for the understanding of plastic buckling of tubes in bending, the bendingresponse of thin-walled members experiencing softening induced by the local bucklingphenomenon, or the bending of composite structures at the ultimate state (reinforcedconcrete members, timber beams, composite members, etc.). The cantilever beam isconsidered as a structural paradigm associated to generalized stress gradient. An inte-gral-based non-local plasticity model is developed, in order to overcome Woods paradoxwhen softening prevails. This plasticity model is derived from a variational principle, lead-ing to meaningful boundary conditions. The need to introduce some non-locality in thehardening regime is also discussed. We show that the non-local plastic variable duringthe softening process has to be strictly dened within the localized softening domain.The propagation of localization is theoretically highlighted, and the softening region growsduring the softening process until a nite length region. The pre-hardening response has noinuence on the propagation law of localization in the softening regime. It is also shownthat the material time derivative and the partial time derivative have to be explicitlyOn the propagation of localization in the plasticity collapseof hardeningsoftening beamsNol Challamel a,*, Christophe Lanos b, Charles Casandjian aaUniversit Europenne de Bretagne, Laboratoire de Gnie Civil et Gnie, Mcanique (LGCGM) INSA de Rennes, 20, Avenue des Buttes de Cosmes, 35043Rennes cedex, Francejournal homepage: www.elsevier .com/locate / i jengsci488 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506concrete members, timber beams, composite members, etc. [1,2] or [3]). The plastic buckling of tubes in bending, can be alsomodelled with a hardeningsoftening momentcurvature relationship ([49]). The bending response of thin-walled mem-bers can also experience a softening phenomenon induced by the local buckling phenomenon [10]. The localization processin these hardeningsoftening structural members is analysed in detail in this paper.Woods paradox is met for local softening momentcurvature relationship. A non-local (gradient) momentcurvatureconstitutive relation was introduced in [11] to overcome the Woods paradox. Non-local models at the beam scale abandonthe classical assumption of locality, and admit that the bending moment depends not only on the state variables (curvature,plastic curvature) at that point. Non-local inelastic models (damage or plasticity models) were successfully used as a local-ization limiter with a regularization effect on softening structural response in the 1980s. The non-local character of the con-stitutive law, generally introduced through an internal length, is restricted to the loading function (damage loading functionor plasticity loading function). Pijaudier-Cabot and Bazant [12] rst elaborated a non-local damage theory, based on theintroduction of the non-locality in the damage loading function. This theory has the advantage to leave the initial elasticbehaviour unaffected, and to control the localization process in the post-peak regime. It is worth mentioning that this ideawas already used before to model shear bands [13,14]. Gradient plasticity models (also called explicit gradient plasticitymodels) and integral plasticity models may be distinguished. In case of explicit gradient plasticity models [15,16], theplasticity loading function depends on the plastic strain and its derivative, whereas for integral plasticity models, theplasticity loading function is expressed from an integral operator of the plastic strain (see for instance [17,18]). Moreover,it can be shown as in case of non-local elastic models [19], that some relevant integral plasticity models can be cast in adifferential form (Engelen et al. [20,21]). These models are called implicit gradient plasticity models, but can be viewedas particular cases of integral plasticity models with specic weight functions dened as Greens function of the differentialoperator.More recently, an implicit gradient plasticity model was used at the beam scale to solve Woods paradox in beams withmoment gradient and without hardening range [22,23]. Localization is controlled by a non-local softening plasticity model,based on a combination of the local and the non-local plastic variables (as suggested by [24] see also [18] or [25]). Themodel postulated in [22] or [23] is different from the ones generally considered for implicit gradient plasticity models, inthe sense that the boundary conditions have to be necessarily postulated at the boundary of the elastoplastic zone. Thesehigher-order boundary conditions may be obtained from a variational principle, as for explicit gradient plasticity models.It has been shown on simple structural examples that the softening evolution problem was well-posed with this non-localconstitutive law. In particular, the uniqueness of the evolution problem is clearly obtained in presence of gradient moment,typically for a cantilever beam solicited by a vertical force. Note that this uniqueness result of the evolution problem wouldnot be obtained for homogeneous structures with constant generalized stress (constant moment) (see [16] for gradient plas-ticity models, or more recently for the non-local beam problem [23]). The same kind of results (loss of uniqueness with uni-form state of stress) has been also recently noticed by [26] for damage problems. Introduction of some heterogeneities canrestore the uniqueness property for these non-local damage problems [27]. Most of the presented theoretical results dealwith softening media without hardening range. Hence, up-to-now, very few results are available for hardeningsofteningnon-local plasticity media, even if this conguration is of fundamental importance from an engineering point of view.The localization process studied in this paper is restricted to the unidimensional softening constitutive law. However, it isworth mentioning that other phenomena may lead to localization, such as the non-associative nature of the plastic ow rulefor two-dimensional or three-dimensional media (see for instance [28,29] or [3]). Furthermore, the methodology presentedin this paper is inspired by an engineering approach, based on a macroscopic bending moment curvature constitutive lawvalid for various physical problems. The same fundamental softening constitutive law is effectively used to cover both thegeometrically-induced softening phenomenon of thin-walled members, and the microcracking-induced softening of themicrostructured composite beam. It is clear for the writers that the basic phenomena behind the unied presentation of thispaper are rmly different. The characteristic length associated to each phenomenon has to be scaled in a relevant way foreach model.Some open questions remain to be solved. The rst point to be investigated is the fundamental understanding of the evo-lution of localization process. How does the localization zone evolve during the softening process? The propagation of shearlocalization has been recently studied numerically from a second grade models in [30] for instance, but the particular prop-erty of the localization front in presence of stress gradient still merits some theoretical investigations. It has been recentlyshown in [22] or [23] that the plastic zone grows during the softening process until an asymptotic limited value, whichdepends on the characteristic length of the section. A related question is to know if it is necessary to introduce a variablecharacteristic length in the model (see [31,32]), although the model with constant characteristic length is able to reproducea variable localization zone during the softening process. A comparison of non-local or gradient plasticity models can befound in [33] or [34], where the softening localization process is specically characterized for uniform stress state. Thethermodynamics background and the physically meaning of generic gradient plasticity models are also analysed in[35,36]. Despite the numerous models devoted to the plastic localization phenomenon, the inuence of the hardening phaseon the localization process has not been specically addressed. A second point is related to the relevancy of higher-orderboundary conditions in presence of hardening plasticity. Finally, the possible decoupling of local/non-local models betweeneach hardening/softening domain will be discussed at the end of the paper. Some answers will be given for these difcultquestions from the simplest structural example exhibiting moment gradient, namely the cantilever beam loaded by a ver-tical force at its extremity.2. Local softening constitutive law: Woods paradoxThe homogeneous cantilever beam of length L is loaded by a vertical concentrated load P at its end (Fig. 1). One recognizesthe Galileos cantilever beam previously solved by Galileo himself (15641642) using equilibrium, strength and dimensionalarguments [3741]. The cantilever beam loaded by a concentrated force can be viewed as a typical case of plastic beams withnon-constant bending moment. The axial and transversal coordinates are denoted by x and y, respectively, and the trans-verse deection denoted by w. The symmetrical section has a constant second moment of area denoted by I (about the z-axis). We assume that plane cross sections remain plane and normal to the deection line and that transverse normal stres-ses are negligible (EulerBernoulli assumption). According, the curvature v is related to the deection through:v x w00 x 1where a prime denotes a derivative with respect to x. The problem being statically determinate, equilibrium equations di-rectly give the moment distribution along the beam:M x P L x with P P 0 and x 2 0; L 2At the end of the beam, the displacement v = w(L) of concentrated force P is used to control the loading process. The localmomentcurvature relationship (M,v) considered is bilinear with a linear elastic part and a linear softening part (Fig. 2). Thismodel is rst considered in a local form, i.e. standard plasticity model with negative hardening. The non-local extension willbe investigated later in the paper. Mp is the limit elastic moment, and vY is the limit elastic curvature, related through Mp/vY = EI where E is the Young modulus of the homogeneous beam. In practice, the curvature cannot increase indenitely andis limited by vu (the ultimate admissible curvature). However such limitation is not taken into account in the present study.The elastoplastic model represented in Fig. 2 is a standard plasticity model with negative hardening (softening). The yieldfunction f is given by:f M;M Mj j Mp M 3l0x P O N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506 489EIO Y p u Fig. 2. Elasticplastic softening momentcurvature law.pM ( )kEIkEI +/My Fig. 1. The cantilever beam.L whereusingAccurvato bementbendiY Ydomadomaplasticw+ isand thThlocal scally([1,2,2Forequat490 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506vp l2cvp 00 vp 13the implicit gradient plasticity model, the non-local plastic curvature vp is dened as the solution of the differentialion:law (see [22,23]).3. Non-local hardening/softening constitutive law, a variational principleno reasonable phenomenon of failure with zero dissipation. This paradox is well documented in the literature2,23,44,45]). A possible way to overcome Woods paradox is to introduce a non-local plastic softening constitutivealso be interpreted as the appearing of plastic curvature increments localized into one single section, leading to the physi-is additional assumption gives the Wood paradox. The unloading elastic solution is the only possible solution of theoftening problem, if the plastic curvature is assumed to be a continuous function in space (Fig. 3). This paradox canP L l0 MpPL 6 Mp) l0 0 12e plastic domains). Enforcing that vp is also a continuous function of x vp l0 0 leads to: (w 0 0w0 0 0andw l0 w l0 w0 l0 w0 l0 (11The deectionw(x) and the rotation w0(x) must be continuous functions of x (in particular at the intersection of the elastic x 2 l0 ; L : EIw x P L x 10the deection in the elastic region. The boundary conditions can be summarised as:x 2 0; l0 :EI w00 x vp x P L x vp x P Lx Mpk8 0 6M* is an additional moment variable which accounts for the loading history. The plastic curvature rate _vp is obtainedthe normality rule:Thgen inlutionrate fodeveloN. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506 491can be chosen:W w;vph iZ L012EI w00 vp 2Mpvp k2v2p k2f 1 vp vp 2 k2l2c f 1 vp 0 2dx Pw L 15wherealso uelasticMoboundThwith ts) including the associated boundary conditions can be obtained from a variational principle, as already obtained inrm for gradient plasticity [15]. The extension to non-local plasticity is inspired by the miromorphic approach recentlyped for elastic and inelastic media, in a consistent thermodynamic framework [46]. The following energy functionalspatial weighted average of the variable vp. This spatial weighted average is calculated on the plastic domain:vp x Z l00G x; y vp y dy 14where the weighting function G(x,y) is the Greens function of the differential system with appropriate boundary conditions.The non-local hardening/softening constitutive law of modulus k (k = k+ for hardening evolutions, k = k for softening evo-erefore, a characteristic length lc is introduced in the denition of the non-local plastic curvature vp. As shown by Erin-1983 for non-local elasticity [19], this differential equation clearly shows that the non-local plastic curvature vp is aFig. 3. Woods paradoxlocal softening plasticity models.f is a dimensionless parameter that appears in the hardening/softening evolution law. Following a classical proceduresed for explicit gradient plasticity models (see [15,16]), the overall domain can be divided into a plastic domain and anone. The rst variation of this functional leads to the extremal condition:dW w;vph iZ L0EI w00 vp dw00dxZ l00EI w00 vp dvp Mpdvp k fvp 1 f vp dvpdx k f 1 Z l00vp vp l2cvp 00 dvp k2l2c f 1 vp 0dvph il00 Pdw L 016reover, following Green-type identity associated to the self-adjoint property of the regularized operator for relevantary conditions, and accordingly to the denition of the non-local plastic curvature, the following identity holds:Z l00vp vp l2cvp 00 dvpdx Z l00vp vp l2cvp 00 dvpdx 0 17erefore, the rst variation of the energy functional can be also simplied as:dW w;vph iZ L0Mdw00dxZ l00M Mp M dvpdxk2l2c f 1 vp 0dvph il00 Pdw L 0 18he associated constitutive law for the elastic, and the non-local hardening/softening law:M EI w00 vp and M k fvp 1 f vp 19ThL L Thpthe boundary of the elastoplastic domain. Considering the higher-order boundary conditions at the elastoplastic boundaryhas thover tphysicnon-locurvadomaTh2 00ary co~ ~Sucing ev 2 00 2 00492 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506M lcM k vp flcvp 29h a combination of local and non-local plastic variables was initially proposed by Vermeer and Brinkgreve for soften-olutions [24] (see also [18]). In the present case, this model can be also written in a differential format:h iM kvp with vp fvp 1 f vp vp flcvp 28 2 00The non-local plastic constitutive law appearing from the variational principle is based on a combination of the local plas-tic curvature and the non-local plastic curvature.vp 0 0 vp 0 l0 0 27The proof is based on the calculation extracting the non-local plastic terms of Eq. (24):W vph i k2Z l00vpvp fl2cv0pvp 0dx k2Z l00vp vp l2cvp 00 fl2cv0pvp 0dx k2Z l00v2p f 1 l2cv0pvp 0dxk2l2cvpvp0h il0025Finally, it can be shown that the non-local plastic terms of Eq. (15) are obtained:W vph i k2Z l00v2p f 1 vp vp 2 l2c f 1 vp 0 2 dx k2l2c vpvp 0h il00 k2f 1 l4c vp 0vp 00h il0026even if the boundary terms are not strictly equivalent, but are reducing to the same nal result:nditions:W w;vph iZ L012EI w00 vp 2 k2fl2cvp 0v0p Mpvp k2vpvpdx Pw L 24 k2 0vp vp lcvp dx 23The introduction of the Lagrange multipliers for constrained variables has been already used for gradient media (see forinstance [49]). A similar discussion on independent or dependent variables can be found in [50] for gradient media, or in [51]for the coupling of internal variables in local media.Note that a different functional was considered in [22] leading to the same constitutive behaviour with the same bound-W w;vp;vp 012EI w00 vp2Mpvp k2v2p k2f 1 vp vp2 k2l2c f 1 vp 02dx Pw L kZ L n o2by a Lagrange multiplier k added in the functional energy Eq. (15) such as:h i Z L al boundary of the solid would lead to different results, as detailed in the Appendix A (see also [48]). Note that thecal plastic curvature does not necessarily vanish at the boundary of the elastoplastic domain, whereas the plasticture is a continuous variable of the spatial coordinate and vanishes at the boundary between the elastic and the plasticin.e same constitutive equations would be obtained by considering two independent internal variables vp and vp linkeddynamics background of integral-based non-local plasticity models). For instance, a uniform plastic variable in the plasticdomain would lead to a non-local variable that is identical. Introduction of the higher-order boundary conditions at thee advantage to be variationnally and physically motivated. In this case, the non-local plastic variable is calculatedhe plastic domain (see Eq. (14) as for most integral-based non-local plasticity models see also [47] for the thermo-The high-order boundary conditions of the non-local plasticity model are included in these equations, and are applied atM L 0; M0 L P; w0 w00 0; vp 0 0 vp 0 l0 vp l0 0 22with the natural boundary conditions:e extremal condition leads to the equilibrium equation and the yield function:M00 0 and M M M 210Mdw00dx Mdw0 L0 M0dw L0 0M00dwdx with M EI w00 vp 20e following integration by part can be considered for the deection:Z ZInticityp c pHothe usual gradient plasticity models dealing with only the derivative of the plastic curvatures. Eq. (29) is the plasticity gen-of lattTh+the chstructtutiveathe model, and leads to a well-posed evolution problem. In fact, it is not necessary to introduce some non-locality inthe hduringa awith tN. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506 493vp l0 0; vp 0 l0 0 and vp 0 0 0 38khe boundary conditions obtained from the variational principle:The general solution of this differential equation is written as (see also [23]):x 2 0; l0 : vp x A coshx B sinh x P L x Mp 37vp a2vp 00 P L x Mpk36model, M is related to the combined non-local plastic curvature variable ~vp through the linear model (see for instanceEq. (28)):M k~vp with ~vp vp a2vp 00 35Introducing the combined non-local plastic curvature into the loading function leads to a differential equation:ardening range from a mathematical point of view. However, it is also possible to introduce some non-localityhardening, to introduce some scale effects in the hardening range. For the non-local hardening plasticity*M l2cM00 k vp a2v00p with f lc 34A relevant choice often assumed in the softening constitutive behaviour is to assume that a is equal to lc (f = 1)(see also [22] or [23]). In the following, a local hardening momentcurvature relationship will be incorporated inlaw:h i 2M lcM EI v a v or p lc p k0 y a y 33Interestingly, the momentcurvature (M,v) constitutive model Eq. (33) has been proposed for applications in compositebeams with imperfect connections between the two elements (such as steel-concrete composite structures, timber-concreteelements, layered wood systems with interlayer slip) [5557]. Note the similarity with the non-local bending constitutivelaw recently studied for elastic problems [42,43]. As recently shown in [58], models of elastic foundation can also involvesome non-locality. In fact, the model of Reissner [59,60] is also based on the differential equation Eq. (33) where p and yare the foundation reaction and the deection. The model of Pasternak in 1954 is recognized when the parameter lc is van-ishing (lc = 0), which is the analogous of a gradient elasticity model.On the opposite, for softening evolutions, f has to take negative values [23], leading to the non-local softening consti-aracteristic length lc leads to an innite value of f . The differential format Eq. (32) has been already used in the past inural engineering for some specic applications:2 00 2 00 2 00 2 00 shown in [23]). Typically, f can be understood as a regularization parameter. For hardening evolutions, f has to be positive,leading to the non-local hardening constitutive law:M l2cM00 k vp a2v00ph iwith f alc 232This model comprises the purely non-local plastic softening model (a = 0), and the gradient plasticity model for hardeningevolution (lc = 0) (see [54] for hardening gradient plasticity models). According to the notation of Eq. (32), the vanishing of+ice models.e sign of f controls the well-posedness of the plasticity evolution problem for both hardening/softening behaviours (aswhere r and e are the uniaxial stress and the uniaxial strain. Eq. (31) gives satisfactory results for dispersive wave equationr l2cr00 E e fl2c e00h i31eralization of the mixed elastic constitutive law investigated for a one-dimensional non-local elastic bar [53]:wever, the boundary conditions written in Eq. (27) for the non-local plastic curvatures are different from the ones ofM00 0 ) M00 0 ) M k v fl2v00h i30the case of the cantilever beam, it is worth mentioning that the non-local differential format looks like a gradient plas-model (see [52] for the comparison of non-local and gradient plasticity models):Thf. The494 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506Eq. (32)):Thfor f lening4. NoForthe linIntdiffereThb n 1 ffcosh n 1sinh n41e plastic zone n versus the loading parameter b is shown in Fig. 4 and is parameterised by the dimensionless parametergradient plasticity model (in the hardening range) is recovered from this relationship as an asymptotic law (lc? 0 inleading to the localization relation:b 1 PYP LlcP 0 and n l0lcP 0 40The following dimensionless parameters may be introduced as:Aa sinhl0lc Ba coshl0lc Pk 0Ba Pk 0>>>:39The non-linear system of three equations with three unknowns A, B and l0 is nally obtained:A 1 lca 2 cosh l0a B 1 lca 2 sinh l0a P Ll0 Mpk 08>>> 0.f!1 ) b n cosh n 1sinh n42e width of the plastic zone associated with the non-local models is larger than the reference width of the local model,arger than unity, whereas this width is smaller than the one of the local model for f smaller than unity. The local hard-plastic zone relation is obtained by setting f = 1.n-local softening constitutive law: application to the cantilever beamthe non-local softening plasticity model,M* is related to the combined non-local plastic curvature variable ~vp throughear model (see for instance Eq. (28)):M k~vp with ~vp vp a2vp 00 43roducing the combined non-local plastic curvature (with a = lc see [23]) into the loading function leads to a linearntial equation:vp l2cvp 00 P L x Mpk44e general solution of this differential equation is written as (see also [23]):x 2 0; l0 : vp x A cosxlc B sin xlc P L x Mpk45with the boundary conditions obtained from the variational principle:vp l0 0; vp 0 l0 0 and vp 0 0 0 46An important difference with the implicit gradient plasticity model presented in [20,21] or [25], however, is that the extraboundary conditions are valid over the plastic domain, rather than over the entire domain (see Appendix A for a discussionon higher-order boundary conditions). The non-linear system of three equations with three unknowns A, B and l0 is nallyobtained:2A cos l0lc 2B sin l0lc P Ll0 Mpk 0 Alc sinl0lc Blc cosl0lc Pk 0Blc Pk 08>>>>>:47The following dimensionless parameters may be introduced as:b 1 PYP Llc6 0 and n l0lcP 0 48and the loadplastic zone relationship is nally written as:b n 21 cos nsin nfor sin n 0 49N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506 495Fig. 5. Evolution of the plastic zone n versus the loading parameter b. Non-local softening plasticity model.In other words, Woods paradox is overcome for the non-local softening cantilever case and uniqueness prevails for thesoftening evolution considered in the paper. Fig. 5 shows the evolution of the plastic zone n in term of the positive dimen-sionless parameter j bj. The parameter jbj varies between 0 and tends towards an innite value when P tends towards zero.Moreover, the size of the plastic zone tends towards an asymptotic value for large values of jbj (and sufciently small valuesof P). n0 = p is the limiting value of the maximum width of the localization zone. The plastic zone evolves from a transitoryregime towards a material (or section) scale that does not depend anymore on the loading range. The results reveal that theevolution tends towards one unique solution with a nite energy dissipation that depends only on the characteristic length.The maximum width of the localization zone l0 directly depends on the characteristic length of the non-local model via therelation l0 plc (for the cantilever beam). The determination of the characteristic length lc (or the maximum width of thelocalization zone l0 is related to the question of the nite length hinge model, a central question of the present non-localmodel. Wood [1] inspired by the works of Barnard and Johnson [61] suggested the term of discontinuity length. Many papershave been published on the experimental or theoretical investigation of such a length ([1,2,6167]) for reinforced concretebeams. It is generally acknowledged that the value of lc (or the maximum localization zone l0 must be related to the depth ofthe cross section h. The rigid body moment-rotation mechanism detailed in [65] or [66] may be used to calibrate this char-acteristic length for reinforced concrete beams. Therefore, it is recommended that the maximum width of the localizationzone lbeaminvest496 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506The deection in the elastic zone x 2 l0 ; L is derived from the continuity condition given by Eq. (13):w x PLx22EI Px36EI w0 l0 PLl0EI P l0 22EI" #x w l0 l0 w0 l0 PL l0 22EI P l0 33EI" #51Figthe pe5. LocInening(M,v)piecewwas aThMpmaximand jk+ is petersw x PLEI PLMpkx22 PEI Pkx36 2 Pl3ckcos 0lc 1sin l0lc cos xlc 1 2 Pl3cksinxlc xlc50 The deection in the plastic zone x 2 0; l0 is obtained by integrating twice the elastic curvature: l 0 is chosen in the order of magnitude of the depth of the cross section h (see also [64]). This implies for the cantileverthat the characteristic length lc is in the order of magnitude of h/p. This theoretical aspect certainly merits some furtherigations. However, the existence of the nite size fracture process zone leads to the specic structural size effect.Fig. 6. Response of the elastoplastic non-local softening beam; EIk 5; lcL 0:1.. 6 shows the resolution of Woods paradox with the non-local softening plastic model considered in the paper. Afterak load, the softening plasticity propagates along the beam, leading to the global softening phenomenon.al hardeningsoftening constitutive law: Woods paradoxthis section, the effect of a pre-hardening range is studied for the cantilever beam. It is rst assumed that both the hard-and the softening part of the constitutive behaviour are ruled by a local law. The local momentcurvature relationshipconsidered is tri-linear with a linear elastic part coupling to a linear hardening and softening part (Fig. 7). This is aise linear hardeningsoftening plasticity model. This hardening/softening plasticity model presented in this sectionlready studied in [11] at the beam scale, for a beam with uniform bending moment.e hardening/softening rule associated to the yield function written by Eq. (3) is now given in the following form:M vp kvpifvp 2 0;jc M vp k vp jc mMpD EMpifvp R 0;jc 8>: with jc m 1 Mpkandk P 0k 6 0(52is the limit elastic moment, and vY is the limit elastic curvature, related throughMp/vY = EI.m is the ratio between theum moment reached during positive hardening and the limit elastic moment (m is necessarily greater than unity),c is the plastic curvature reached before the initialization of the non-local softening process. The hardening modulusositive whereas the softening modulus k is negative. The simple relation is obtained between the constitutive param-in the hardening range:kEI m 1v m andjvvY v m with v vvvY53increa0 p ThN. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506 497The displacement eld in the plastic zone is obtained using the boundary conditions (clamped beam):ThinterfaThFigrelatioremarEq. (5hardeb n with b 1 PYP LlcP 0 and n l0lcP 0 57p 0 L PNote that the propagation of the local hardening process zone is equivalent to the linear relationship between previouslyused dimensionless parameters:e continuity of the plastic curvature at the elasticplastic interface leads to the plastic zoneload relationship:v l 0 ) l0 1 PY 56ksing:x 2 0; l : v x 1 P L x Mp 55For increasing value of the load outside the elastic domain, the plastic regime starts and the beam can be split into anelastic and a plastic domain. The size of the plastic zone is denoted by l0 . In the plastic zone, the plastic curvature is linearlyP 2 PY ;mPY with PY MpL 54vv is the curvature value associated to the softening process. In the hardening range, the load is increasing such as:Fig. 7. Elasticplastic hardeningsoftening momentcurvature law.1 Y Y Yv u 1 pMm Mx 2 0; l0 : EIw x 1 EIk Px36 1 EIk PLMp EIk x2258e displacement in the elastic zone is obtained by enforcing the continuity of the displacement and the rotation at thece:x 2 l0 ; L : EIw x P x36 PL x22 EIkPl0 22x EIkPl0 3659e loaddeection relationship is nally deduced in the hardening range from:vvY PPY 3 EIkPPY12l0L 2 16l0L 3" #withl0L 1 PYP60. 8 shows the hardening range for the cantilever beam. It can be easily checked from Eq. (60) that the loaddeectionnship is not linear, even for the local hardening constitutive behaviour considered in this paragraph (see also Fig. 8). Akable result is that the plastic curvature distribution depends on the hardening law but the propagation law Eq. (56) or7) does not depend on the model of hardening law. In fact, whatever the hardening model (even in case of non-linearning), the same equality is valid:vp l0 0) M l0 0 ) M l0 P L l0 Mp 616. Loc498 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506Woods paradox for the hardeningsoftening beam can be solved by using a non-local softening momentcurvature law,as in the case of the elastic-softening beams. Once the bending moment in the clamped section M(x = 0) = PL reaches theyield seningThnon-loas weplasticBydomawith trequirThal hardening and non-local softening constitutive lawThe plastic softening response may start once the load P reaches the maximum value mPY. Enforcing that vp is a contin-uous function of x vp l0 jc leads to:P L l0 mMpPL 6 mMp() l0 0 62This additional assumption gives the new Wood paradox for hardeningsoftening local constitutive relationship. Theunloading elastic solution is the only possible solution of the local softening problem (Fig. 8). In this case again, the paradoxcan also be interpreted as the appearing of plastic curvature increments localized into one single section, leading to the phys-ically no reasonable phenomenon of failure with zero dissipation.Fig. 8. Woods paradoxlocal hardening/softening plasticity models; m 54; EIk 11.trength mMp, the softening zone can propagate from the clamped section, whereas unloading is observed in the hard-plastic zone and in the elastic zone. The local hardening and non-local softening constitutive relationship are given by:M vp kvpifvp 2 0;jc M vp k ~vp jc mMp Mpifvp R 0;jc 8>: with ~vp vp l2cvp 00 63e problem of the continuity requirement between both hardening and softening constitutive law (a local one and acal one) will be implicitly solved by the fact that the length of the softening zone at the yield strengthmMp will vanishwill see (as for the elastic softening problem). Furthermore, the non-local plastic variable is integrated over the activedomain, i.e. the softening zone, as the hardening zone is in unloading during this nal process.considering the yield function in the softening area, the linear differential equation is obtained in the softeningin:x 2 0; l0 : vp l2cvp 00 P L x mMpk jc 64he boundary conditions, associated to the higher-order boundary conditions of the non-local model and the continuityement of the plastic curvature:vp l0 jc; vp 0 l0 0 and vp 0 0 0 65e general solution of the differential equation Eq. (64) is written as:x 2 0; l0 : vp x A cosxlc B sin xlc P L x mMpk jc 66ThleadinN. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506 499depend on the hardening range. In other words, the hardening modulus (or the material history in the hardening domain)does nThThcurvatThGeeninghardee following dimensionless parameters may be introduced as:b 1mPYP Llc6 0 and n l0lcP 0 68g to the localization relation of Eq. (49). A remarkable result is that the plastic diffusion in the softening range does notBlc Pk 0:2A cos lc 2B sin lc k 0 Alc sinl0lc Blc cosl0lc Pk 0>>>> 67The non-linear system of three equations with three unknowns A, B and l0 is nally obtained:l0 l0P Ll0 mMp8>k L 4 kFig. 9. Response of the elastoplastic hardeningnon-local softening beam;EI 5; lc 0:1; m 5; EI 11.ot affect the localization process, from a qualitative point of view.e deection in the plastic zone x 2 0; l0 is obtained by integrating twice the elastic curvature:w x PLEI PLmMpk jc x22 PEI Pk x36 2 Pl3ckcos l0lc 1sin l0lc cos xlc 1 2 Pl3cksinxlc xlc 69e deection in the elastic zone x 2 l0 ; l0 is derived from the continuity condition given by Eq. (13), whereas the plasticure distribution is constant in the unloading phase:w1 x PLEI m 1 Mpk x22 PEImPYk x36 w0 l0 PLl0EI P l0 22EI m 1 Mpl0kmPY l0 22k" #x w l0 l0 w0 l0 PL l0 22EI P l0 33EI m 1 Mp l0 22kmPYkl0 33" #70e deection in the elastic zone x 2 l0 ; L is derived from the continuity condition given by Eq. (11):w2 x PLx22EI Px36EI w01 l0 PLl0EI P l0 22EI" #x w1 l0 l0 w01 l0 PL l0 22EI P l0 33EI" #withl0 m 1mL 71nerally speaking, the plastic zone growth in the hardening range until the maximum load, then a more localized soft-zone arises from the clamped section and controls the mode of collapse. The global softening is then observed after thening behaviour (Fig. 9).7. On the law of propagation of localizationIn this part, the fundamental question of the localization process in the softening range is investigated from the shape ofthe softening law. A non-linear softening law is studied and compared to the linear model, as characterized in the main partof the paper. The analysis is restricted to the elastic softening beam, without pre-hardening stage.M K~vpqwith ~vp fvp 1 f vp 72where the parameter K is negative for softening models. In the particular case f = 1, it can be observed that the non-localplastic curvature may be also dened as:f 1 ) ~vp vp l2cvp 00 MK 2 P L x MpK 273The general solution of this differential equation is written as:x 2 0; l0 : vp x A cosxlc B sin xlc P L x Mp 2 2l2c P2K 274The boundary conditions are expressed by Eq. (46) for the elastic-softening beam model. The plastic zone n versus theloading parameter b is nally obtained from these boundary conditions:b2 b 2n 41 cos nsin n n2 4 4n cos nsin n 0 for sin n 0 75the global softening response depends on the softening model considered, i.e. linear or non-linear softening models. This re-sult isstrongboth plasticity models studied in this paper. A numerical comparison of numerous non-local softening models can be found500 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506Fig. 10. Evolution of the plastic zone n versus the loading parameter b. Non-local softening plasticity model; comparison of softening models.in [33] or [34], in case of homogeneous state of stress.quite similar to the result highlighted in [22] where the loading mode (concentrated force or distributed loading) has ainuence on the propagation of localization, even if the localization zone converges towards a nite length zone forwhose softening solution is given by:b n 21 cos nsin nn 21 cos nsin n 2 n2 4 4n cos nsin n s76Fig. 10 shows the comparison of the two non-local softening models. The width of the localization zone grows faster incase of linear softening non-local model than for the non-linear softening model. In both cases, the localization zone in-creases until a nite plasticity length. This could be considered as a strong difference with the non-local damage model stud-ied in [27] for a damageable beam, where the localization zone is growing without any threshold. In any cases, it is clear thatInmodeto thedenotOf course, these two localization zones are strongly different l l . For the non-local models studied in this paper, thethe hadid nostoodThthe haN. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506 501@tx 0; l0 ; t 0; @l0x 0; l0 ; t 0 and @x x 0; l0 ; t lck sin n ) dt x 0; l0 ; t 085@vp 0 @vp 0 @vp 0 P cos n 1 dvp 0 dt @x @x dtIt would be possible to use the partial time derivative in the loading function, but the exact boundary conditions areexpressed in rate-form as:dvp 0dtx 0; l0 ; t 0; dvp 0dtx l0 ; l0 ; t 0 and dvpdtx l0 ; l0 ; t 0 84The rst rate boundary condition Eq. (84) can be obtained from:d @2vp2" #@22dvp 830According to Eq. (82), Eq. (80) is not a linear second-order differential equation with respect to the rate of non-local plas-tic curvature, as:dvpdtx; l0 ; t @vp@tx; l0 ; t _l0 @vp@l x; l0 ; t _x @vp@xx; l0 ; t 82_f M;vp 0 )d vpdt l2cd vp 00dt ddtP L x k80It is difcult to solve this differential equation for the rate-problem. In fact, from Eqs. (45) and (47), the exact expressionof the non-local plastic curvature is given by:x 2 0; l0 : vp x; l0 ; t P t lckcos n l0 1sin n l0 cos xlc P t lcksinxlc P t L x Mpk81One has to take care to distinguish the material time derivative, and the partial time derivative, that are not identical.The use of material time derivative instead of partial time derivative is rigorously developed in case of boundary elementmethods [68], and more recently for interface tracking [69], or from thermodynamics point of view [70].e rate-form of the non-local problem is sometimes preferred to solve the propagation of the localization process alongrdeningsoftening beam. Indeed, during the softening process, the stationarity of the loading function implies that: 8. On the rate-form of the non-local equationst depend on the hardening stage. This distinction between the active and the passive plastic zone can be clearly under-in an incremental time-formulation.In fact, during the softening process, a part of the hardening zone is in unloading and can be therefore considered as apassive plastic zone. During the softening localization process, this passive plastic zone does not inuence the propagationof this localized plastic zone associated to the collapse of the beam. We have shown that the load-plastic zone propagationlcL6 1p1 1m ) l0 6 l0 79where the hardening zone has been calculated from the local hardening model. Therefore, the softening plastic zone is nec-essarily smaller than the hardening plastic zone for sufciently small characteristic length, i.e.:rdening and the softening plastic zones are given by:l0L 1 1mandl0L6 p lcL780 0plastic variable is integrated on an active plastic domain. In particular, during the softening process, the non-local plasticvariable is integrated on the localization length l0 , even if the plasticity zone is generally much larger l0 > l0 . In fact,hardening domain is propagating along the beamwithout any material limits, whereas the softening localization zone,ed by l0 is increasing during the softening process, until a nite length which depends on the material-section model. The plasticity zones in both regimes appear to be signicantly different (see also [67] for the same conclusions). l0 relatedconclusions, the hardening non-local model (with hardening modulus k+) can be compared to the softening non-locall (with softening modulus k):M l2cM00 k vp a2v00ph i M l2cM00 k vp a2v00ph i77Thpointsical cbendinon-lorigoroWependswell ktural snoticemodelocalizThin botout an502 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506during the softening process, until a characteristic nite length. Some similar conclusions have been recently numericallyreached in [67] with a variable inelastic end zone model (except that the softening length is xed in [67]). For the non-localls. The softening model and the loading mode, have a strong inuence on the propagation of localization, even if theation zone converges towards the same nite length zone for the plasticity models studied in this paper.e inuence of the hardening phase on the localization process has also been specically addressed. The plasticity zonesh regimes appear to be signicantly different. l0 related to the hardening domain is propagating along the beam with-y material limits (except the length of the beam), whereas the softening localization zone, denoted by l0 is increasingat least for the appearance of this specic feature. The beam response is studied for linear and non-linear non-local softeningshow that the localization zone evolves during the softening process, until an asymptotic limited value, which de-on the characteristic length of the section. This nite character of the localization propagation can be related to thenown concept of nite length region. The existence of this nite size fracture process zone leads to the specic struc-ize effect. As a consequence of this model, the plastic length evolves during the loading process, a phenomenon oftend in structural design. Therefore, it is not necessary to introduce a variable characteristic length in the non-local model,we look at the 1D propagation of plastic strains along the bending beam. It is concluded that the mode of collapse is rmlya non-local phenomenon.cal plasticity model is developed, in order to overcome Woods paradox when softening prevails. This model can beusly derived from a variational principle. Using a piecewise linear plasticity hardeningsoftening constitutive law,ing of composite structures at the ultimate state (reinforced concrete members, timber beams, composite members, etc.).The cantilever beam is considered as a structural paradigm associated to generalized stress gradient. An integral-basedantilever beam. Such simplied models can be useful for the understanding of plastic buckling of tubes in bending, theng response of thin-walled members experiencing softening induced by the local buckling phenomenon, or the bend-structures.9. ConclusionsThis paper questions the mode of collapse of some simple hardeningsoftening structural systems, comprising the clas-dvpdtx l0 ; l0 ; t @vp@tx l0 ; l0 ; t 0 anddvpdtx l0 ; l0 ; t @vp@tx l0 ; l0 ; t 090erefore, it is recommended to use the material time derivative in the rate-format of the boundary value problem. Thishas certainly to be rigorously taken into account in a numerical time-integration format applied to more complexThe second rate boundary condition Eq. (84) can be checked from:@vp 0@tx l0 ; l0 ; t 0; @vp 0@l0x l0 ; l0 ; t Plckcos n 1sin nand@vp 0@xx l0 ; l0 ; t Plckcos n 1sin n) dvp0dtx l0 ; l0 ; t 0 86whereas the last boundary condition Eq. (84) is conrmed by:@vp@tx l0 ; l0 ; t _PlckLlc n 21 cos nsin n ;@vp@l0x l0 ; l0 ; t 2Pkcos n 1sin2 ncos n and@vp@xx l0 ; l0 ; t Pk) dvpdtx l0 ; l0 ; t 0 87Note that the boundary conditions cannot be expressed in rate form using the partial time derivative if:@vp 0@tx l0 ; l0 ; t 0 or@vp@tx l0 ; l0 ; t 0 88However, the boundary condition at the clamped end was easier to derive, as this xed point does not move:dvp 0dtx 0; l0 ; t @vp 0@tx 0; l0 ; t 0 89It has to be outlined that it is difcult to use the rate form to solve the exact differential equations, in case of moving elas-toplastic boundaries. The same remark can be formulated for usual gradient plasticity models expressed in rate form. Such amathematical difculty does not arise in case of a localization zone with constant width, as observed for beams or bars with-out any stress gradient (non-moving elastoplastic boundaries) [23]. In fact,_l0 0 )dvp 0dtx 0; l0 ; t @vp 0@tx 0; l0 ; t 0;0 0models studied in this paper, the plastic variable is integrated on an active plastic domain. In particular, during the softeningprocess, the non-local plastic variable is integrated on the localization length l0 , even if the global plasticity zone is generallymuch larger l0 > l0 . In fact, during the softening process, a part of the hardening zone is unloaded and can be thereforeconsidered as a passive plastic zone. During the softening localization process, this passive plastic zone does not inuencethe propagation of this localized plastic zone associated to the collapse of the beam. We show that the non-local plastic var-iable has to be dened strictly within the localized softening domain (this is also valid for the higher-order boundary con-ditions). A fundamental property is that the load-plastic zone propagation in the softening stage did not depend on thehardening stage. This distinction between the active and the passive plastic zone can be clearly understood in an incrementaltime-formulation. It is also shown that the material time derivative and the partial time derivative have to be explicitlydistinguished, especially for moving elastoplastic boundaries. It is recommended to use the material time derivative in therate-format of the boundary value problem. Finally, we mention at this stage the possible coupling between non-local elas-ticity, non-local hardening plasticity and non-local softening plasticity.AppenA.1. HWeThe linear softening law is written as:the plH lc p 1 lc p 1 2Byat theN. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506 503jl02 r r0H A pp 1 cosl02lcp 1p 0 A:5In the case of higher-order boundary conditions postulated at the elastoplastic interface ([22,23]), as adopted in this pa-per, the last boundary condition is written as:j0l02 0 ) l02lcp 1p np with n 1 A:6x L y F l0F Fig. 11. The tension bar.virtue of symmetry, we have j 0 0 leading to B = 0. Furthermore, the plastic variable is assumed to be continuouselastoplastic interface: 0j r r0 A cos xp B sin xp for x 2 0; l0 A:4H 2The general solution in term of non-local variable is written as:astic zone:j p 1 l2c j00 r r0 for x 2 0; l0 A:3For the tension bar considered in Fig. 11, the stress is uniform, and the yield condition leads to the differential equation inr r0 H~j with ~j fj 1 f j j fl2c j00 j p 1 l2c j00 and p 1 f A:2j l2c j00 j A:1igher-order boundary conditions at the elastoplastic interfacestudy the implicit gradient plasticity model based on the non-local plastic strain j (see Eq. (13)):In this Appendix, we investigate the specic effect of higher-order boundary conditions on the response of a non-localsoftening tension bar under uniform stress state.dix A. Inuence of higher-order boundary conditionsEqtion inl l r r xThplasticwherestrainA.2. Hwith Thelastonon-lo>504 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487506j0 l02 j0 l02 ) C sinh l02lc D coshl02lc 1p rr0Hp 1ptan l02lcp1p>>>>:j0 L2 0 ) C sinh L2lc D cosh L2lc 0j l02 j l02 ) C cosh l02lc D sinhl02lc 1p rr0H>>>>>