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On the propagation of localization in the plasticity collapse of hardening–softening beams Noël Challamel a, * , Christophe Lanos b , Charles Casandjian a a Université Européenne de Bretagne, Laboratoire de Génie Civil et Génie, Mécanique (LGCGM) INSA de Rennes, 20, Avenue des Buttes de Coësmes, 35043 Rennes cedex, France b Université Européenne de Bretagne, Laboratoire de Génie Civil et Génie, Mécanique (LGCGM) IUT de Rennes, 3, rue du clos Courtel, 35704 Rennes cedex, France article info Article history: Received 12 August 2009 Received in revised form 24 November 2009 Accepted 12 December 2009 Available online 12 January 2010 Communicated by M. Kachanov Keywords: Beam Hardening Softening Gradient plasticity Non-local plasticity Localization Cantilever Propagation Variational principle Boundary conditions Material time derivative abstract This paper is focused on the propagation of localization in hardening–softening plasticity media. Using a piecewise linear plasticity hardening–softening constitutive law, we look at the 1D propagation of plastic strains along a bending beam. Such simplified models can be useful for the understanding of plastic buckling of tubes in bending, the bending response of thin-walled members experiencing softening induced by the local buckling phenomenon, or the bending 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 inte- gral-based non-local plasticity model is developed, in order to overcome Wood’s paradox when 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 the hardening regime is also discussed. We show that the non-local plastic variable during the softening process has to be strictly defined within the localized softening domain. The propagation of localization is theoretically highlighted, and the softening region grows during the softening process until a finite length region. The pre-hardening response has no influence on the propagation law of localization in the softening regime. It is also shown that the ‘‘material” time derivative and the ‘‘partial” time derivative have to be explicitly distinguished, especially for moving elastoplastic boundaries. It is recommended to use the ‘‘material” time derivative in the rate-format of the boundary value problem. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction This paper is focused on the propagation of localization in hardening–softening plasticity media, and more specifically in an elementary beam model. The propagation of plasticity along a bending beam is studied for a piecewise hardening–soft- ening moment–curvature relationship. Historically, moment–curvature relationships with softening branch were first intro- duced for reinforced concrete beams [1]. Wood [1] did point out some specific difficulties occurring during the solution of the evolution problem for plastic softening models. More precisely, he highlighted the impossibility of the plastic softening beam to flow, if the plastic curvature is assumed to be a continuous function in space, a phenomenon sometimes called Wood’s paradox. Hardening–softening plasticity models may concern a wide class of structural mechanics problems. Such simplified models can be useful for the fundamental understanding of bending of structural members at their ultimate state (reinforced 0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.12.002 * Corresponding author. E-mail addresses: [email protected] (N. Challamel), [email protected] (C. Lanos), [email protected] (C. Casandjian). International Journal of Engineering Science 48 (2010) 487–506 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

On the propagation of localization in the plasticity collapse of hardening–softening beams

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Page 1: On the propagation of localization in the plasticity collapse of hardening–softening beams

International Journal of Engineering Science 48 (2010) 487–506

Contents lists available at ScienceDirect

International Journal of Engineering Science

journal homepage: www.elsevier .com/locate / i jengsci

On the propagation of localization in the plasticity collapseof hardening–softening beams

Noël Challamel a,*, Christophe Lanos b, Charles Casandjian a

a Université Européenne de Bretagne, Laboratoire de Génie Civil et Génie, Mécanique (LGCGM) INSA de Rennes, 20, Avenue des Buttes de Coësmes, 35043Rennes cedex, Franceb Université Européenne de Bretagne, Laboratoire de Génie Civil et Génie, Mécanique (LGCGM) IUT de Rennes, 3, rue du clos Courtel, 35704 Rennes cedex, France

a r t i c l e i n f o

Article history:Received 12 August 2009Received in revised form 24 November 2009Accepted 12 December 2009Available online 12 January 2010

Communicated by M. Kachanov

Keywords:BeamHardeningSofteningGradient plasticityNon-local plasticityLocalizationCantileverPropagationVariational principleBoundary conditionsMaterial time derivative

0020-7225/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.ijengsci.2009.12.002

* Corresponding author.E-mail addresses: noel.challamel@insa-rennes.

(C. Casandjian).

a b s t r a c t

This paper is focused on the propagation of localization in hardening–softening plasticitymedia. Using a piecewise linear plasticity hardening–softening constitutive law, we lookat the 1D propagation of plastic strains along a bending beam. Such simplified 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 Wood’s 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 defined within the localized softening domain.The propagation of localization is theoretically highlighted, and the softening region growsduring the softening process until a finite length region. The pre-hardening response has noinfluence 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 explicitlydistinguished, 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. Introduction

This paper is focused on the propagation of localization in hardening–softening plasticity media, and more specifically inan elementary beam model. The propagation of plasticity along a bending beam is studied for a piecewise hardening–soft-ening moment–curvature relationship. Historically, moment–curvature relationships with softening branch were first intro-duced for reinforced concrete beams [1]. Wood [1] did point out some specific difficulties occurring during the solution of theevolution problem for plastic softening models. More precisely, he highlighted the impossibility of the plastic softening beamto flow, if the plastic curvature is assumed to be a continuous function in space, a phenomenon sometimes called Wood’sparadox. Hardening–softening plasticity models may concern a wide class of structural mechanics problems. Such simplifiedmodels can be useful for the fundamental understanding of bending of structural members at their ultimate state (reinforced

. All rights reserved.

fr (N. Challamel), [email protected] (C. Lanos), [email protected]

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488 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506

concrete members, timber beams, composite members, etc. [1,2] or [3]). The plastic buckling of tubes in bending, can be alsomodelled with a hardening–softening moment–curvature relationship ([4–9]). 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 hardening–softening structural members is analysed in detail in this paper.

Wood’s paradox is met for local softening moment–curvature relationship. A non-local (gradient) moment–curvatureconstitutive relation was introduced in [11] to overcome the Wood’s 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] first 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 specific weight functions defined as Green’s function of the differentialoperator.

More recently, an implicit gradient plasticity model was used at the beam scale to solve Wood’s 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 hardening–softeningnon-local plasticity media, even if this configuration 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 flow 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 unified presentation of thispaper are firmly 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 first 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 specifically 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 influence of the hardening phaseon the localization process has not been specifically 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 difficultquestions from the simplest structural example exhibiting moment gradient, namely the cantilever beam loaded by a ver-tical force at its extremity.

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N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506 489

2. Local softening constitutive law: Wood’s paradox

The homogeneous cantilever beam of length L is loaded by a vertical concentrated load P at its end (Fig. 1). One recognizesthe Galileo’s cantilever beam previously solved by Galileo himself (1564–1642) using equilibrium, strength and dimensionalarguments [37–41]. 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 deflection 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 deflection line and that transverse normal stres-ses are negligible (Euler–Bernoulli assumption). According, the curvature v is related to the deflection through:

v xð Þ ¼ w00 xð Þ ð1Þ

where 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 xð Þ ¼ P L� xð Þ with P P 0 and x 2 0; L½ � ð2Þ

At the end of the beam, the displacement v = w(L) of concentrated force P is used to control the loading process. The localmoment–curvature relationship (M,v) considered is bilinear with a linear elastic part and a linear softening part (Fig. 2). Thismodel is first 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 indefinitely 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�ð Þ ¼ Mj j � Mp þM�� �ð3Þ

l0

x

P

O

L y

Fig. 1. The cantilever beam.

M

pM ( )kEIkEI +/

EI

O Yχ pχ uχ χ

Fig. 2. Elastic–plastic softening moment–curvature law.

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490 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506

where M* is an additional moment variable which accounts for the loading history. The plastic curvature rate _vp is obtainedusing the normality rule:

_vp ¼ _k@f@M

ð4Þ

As a rate-independent constitutive law is considered in this paper, it is equivalent to replace the time variable by a mono-tonic increasing variable such as the displacement at the tip of the beam v = w(L). The plastic multiplier _k must satisfy thecomplementary conditions:

_k P 0; f M;M�ð Þ 6 0; _kf M;M�ð Þ ¼ 0 ð5Þ

The hardening/softening being linear, the following relation holds for the ‘‘local” case:

M� vp

� �¼ k�vp with k� < 0 or M� vp

� �¼ kþvp with kþ > 0 ð6Þ

According to the sign of the plastic modulus k�, we can have softening for k� < 0. Using the decomposition of the totalcurvature v into an elastic part and a plastic part, the moment-elastic curvature relation gives:

M ¼ EI v� vp

� �ð7Þ

This elastic relationship is chosen here to be a local relationship, but formally non-local elastic behaviour is also possibleto be taken into account at this stage (see [42,43] for the bending of non-local elastic beams). The maximum bending mo-ment occurs at x = 0, where the beam is clamped. Plastic rotation starts as soon as the bending moment reach the plasticbending moment Mp. The maximum elastic displacement at the beam end vY and the corresponding load PY are given by:

vY ¼MpL2

3EIand PY ¼

Mp

Lð8Þ

For displacement v smaller than vY(v 6 vY), the beam remains elastic and the deflection can be computed using the elas-ticity solution. For v greater than vY(v P vY), the plastic regime starts and the beam can be split into an elastic and a plasticdomain. The size of the plastic domain is denoted by l0 6 L (see Fig. 2). We can introduce the notation of l�0 for the softeningdomain, and lþ0 for the hardening domain. Considering only the softening problem in this part, the governing equations in theplastic domain are:

x 2 0; l�0� �

:EI w�00 xð Þ � vp xð Þ� �

¼ P L� xð Þ

vp xð Þ ¼ P L�xð Þ�Mp

k�

8<: ð9Þ

where w� denotes the deflection in the plastic region. The elastic adjacent domain is governed by:

x 2 l�0 ; L� �

: EIwþ00 xð Þ ¼ P L� xð Þ ð10Þ

w+ is the deflection in the elastic region. The boundary conditions can be summarised as:

w� 0ð Þ ¼ 0w�0 0ð Þ ¼ 0

�and

w� l�0� �

¼ wþ l�0� �

w�0 l�0� �

¼ wþ0 l�0� �

(ð11Þ

The deflection w(x) and the rotation w0(x) must be continuous functions of x (in particular at the intersection of the elastic

and the plastic domains). Enforcing that vp is also a continuous function of x vp l�0� �

¼ 0� �

leads to:

P L� l�0� �

¼ Mp

PL 6 Mp

() l�0 ¼ 0 ð12Þ

This additional assumption gives the Wood paradox. The unloading elastic solution is the only possible solution of thelocal softening problem, if the plastic curvature is assumed to be a continuous function in space (Fig. 3). This paradox canalso be interpreted as the appearing of plastic curvature increments localized into one single section, leading to the physi-cally no reasonable phenomenon of failure with zero dissipation. This paradox is well documented in the literature([1,2,22,23,44,45]). A possible way to overcome Wood’s paradox is to introduce a non-local plastic softening constitutivelaw (see [22,23]).

3. Non-local hardening/softening constitutive law, a variational principle

For the implicit gradient plasticity model, the non-local plastic curvature vp is defined as the solution of the differentialequation:

vp � l2c vp

00 ¼ vp ð13Þ

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Fig. 3. Wood’s paradox–local softening plasticity models.

N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506 491

Therefore, a characteristic length lc is introduced in the definition of the non-local plastic curvature vp. As shown by Erin-gen in 1983 for non-local elasticity [19], this differential equation clearly shows that the non-local plastic curvature vp is aspatial weighted average of the variable vp. This spatial weighted average is calculated on the plastic domain:

vp xð Þ ¼Z l0

0G x; yð Þvp yð Þdy ð14Þ

where the weighting function G(x,y) is the Green’s 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-lutions) including the associated boundary conditions can be obtained from a variational principle, as already obtained inrate form for gradient plasticity [15]. The extension to non-local plasticity is inspired by the miromorphic approach recentlydeveloped for elastic and inelastic media, in a consistent thermodynamic framework [46]. The following energy functionalcan be chosen:

W w;vp

h i¼Z L

0

12

EI w00 � vp

� �2þMpvp þ

k2v2

p þk2

f� 1ð Þ vp � vp

� �2þ k

2l2c f� 1ð Þ vp

0� �2

dx� Pw Lð Þ ð15Þ

where f is a dimensionless parameter that appears in the hardening/softening evolution law. Following a classical procedurealso used for explicit gradient plasticity models (see [15,16]), the overall domain can be divided into a plastic domain and anelastic one. The first variation of this functional leads to the extremal condition:

dW w;vp

h i¼Z L

0EI w00 � vp

� �dw00dxþ

Z l0

0�EI w00 � vp

� �dvp þMpdvp

þ k fvp þ 1� fð Þvp

� �dvpdx� k f� 1ð Þ

Z l0

0vp � vp þ l2

cvp00

� �dvp þ

k2

l2c f� 1ð Þ vp0dvp

h il0

0� Pdw Lð Þ ¼ 0

ð16Þ

Moreover, following Green-type identity associated to the self-adjoint property of the regularized operator for relevantboundary conditions, and accordingly to the definition of the non-local plastic curvature, the following identity holds:

Z l0

0vp � vp þ l2

c vp00

� �dvpdx ¼

Z l0

0vp � vp þ l2

cvp00

� �dvpdx ¼ 0 ð17Þ

Therefore, the first variation of the energy functional can be also simplified as:

dW w;vp

h i¼Z L

0Mdw00dx�

Z l0

0M � Mp þM�� �

dvpdxþ k2

l2c f� 1ð Þ vp

0dvp

h il0

0� Pdw Lð Þ ¼ 0 ð18Þ

with the associated constitutive law for the elastic, and the non-local hardening/softening law:

M ¼ EI w00 � vp

� �and M� ¼ k fvp þ 1� fð Þvp

� �ð19Þ

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492 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506

The following integration by part can be considered for the deflection:

Z L

0Mdw00dx ¼ Mdw0½ �L0 � M0dw

� �L0 þ

Z L

0M00dwdx with M ¼ EI w00 � vp

� �ð20Þ

The extremal condition leads to the equilibrium equation and the yield function:

M00 ¼ 0 and M ¼ Mp þM� ð21Þ

with the natural boundary conditions:

M Lð Þ ¼ 0; M0 Lð Þ ¼ �P; wð0Þ ¼ w0ð0Þ ¼ 0; vp0 0ð Þ ¼ vp

0 l0ð Þ ¼ vp l0ð Þ ¼ 0 ð22Þ

The high-order boundary conditions of the non-local plasticity model are included in these equations, and are applied atthe boundary of the elastoplastic domain. Considering the higher-order boundary conditions at the elastoplastic boundaryhas the advantage to be variationnally and physically motivated. In this case, the non-local plastic variable is calculatedover the plastic domain (see Eq. (14) as for most integral-based non-local plasticity models – see also [47] for the thermo-dynamics 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 thephysical boundary of the solid would lead to different results, as detailed in the Appendix A (see also [48]). Note that thenon-local plastic curvature does not necessarily vanish at the boundary of the elastoplastic domain, whereas the plasticcurvature is a continuous variable of the spatial coordinate and vanishes at the boundary between the elastic and the plasticdomain.

The same constitutive equations would be obtained by considering two independent internal variables vp and vp linkedby a Lagrange multiplier k added in the functional energy Eq. (15) such as:

W w;vp;vp

h i¼Z L

0

12

EI w00 � vp

� �2þMpvp þ

k2v2

p þk2

f� 1ð Þ vp � vp

� �2þ k

2l2c f� 1ð Þ vp

0� �2

dx� Pw Lð Þ

þ kk2

Z L

0vp � vp þ l2

c vp00

n o2dx ð23Þ

The 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-ary conditions:

W w;vp

h i¼Z L

0

12

EI w00 � vp

� �2þ k

2fl2c vp

0v0p þMpvp þk2vpvpdx� Pw Lð Þ ð24Þ

The proof is based on the calculation extracting the non-local plastic terms of Eq. (24):

W� vp

h i¼ k

2

Z l0

0vpvp þ fl2

c v0pvp

0dx ¼ k2

Z l0

0vp vp þ l2

cvp00

� �þ fl2

cv0pvp

0dx

¼ k2

Z l0

0v2

p þ f� 1ð Þl2cv0pvp

0dxþ k2

l2cvpvp

0h il0

0ð25Þ

Finally, it can be shown that the non-local plastic terms of Eq. (15) are obtained:

W� vp

h i¼ k

2

Z l0

0v2

p þ f� 1ð Þ vp � vp

� �2þ l2

c f� 1ð Þ vp0

� �2�

dxþ k2

l2c vpvp

0h il0

0� k

2f� 1ð Þl4

c vp0vp00

h il0

0ð26Þ

even if the boundary terms are not strictly equivalent, but are reducing to the same final result:

vp0 0ð Þ ¼ vp

0 l0ð Þ ¼ 0 ð27Þ

The 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.

M� ¼ k~vp with ~vp ¼ fvp þ 1� fð Þvp ¼ vp � fl2cvp

00 ð28Þ

Such a combination of local and non-local plastic variables was initially proposed by Vermeer and Brinkgreve for soften-ing evolutions [24] (see also [18]). In the present case, this model can be also written in a differential format:

M� � l2c M�00 ¼ k vp � fl2

c v00ph i

ð29Þ

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N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506 493

In the case of the cantilever beam, it is worth mentioning that the non-local differential format looks like a gradient plas-ticity model (see [52] for the comparison of non-local and gradient plasticity models):

M00 ¼ 0 ) M�00 ¼ 0 ) M� ¼ k vp � fl2c v00p

h ið30Þ

However, the boundary conditions written in Eq. (27) for the non-local plastic curvatures are different from the ones ofthe usual gradient plasticity models dealing with only the derivative of the plastic curvatures. Eq. (29) is the plasticity gen-eralization of the mixed elastic constitutive law investigated for a one-dimensional non-local elastic bar [53]:

r� l2c r00 ¼ E e� fl2

c e00

h ið31Þ

where r and e are the uniaxial stress and the uniaxial strain. Eq. (31) gives satisfactory results for dispersive wave equationof lattice models.

The sign of f controls the well-posedness of the plasticity evolution problem for both hardening/softening behaviours (asshown 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� � l2c M�00 ¼ kþ vp � a2v00ph i

with fþ ¼ alc

�2

ð32Þ

This 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 ofthe characteristic length lc leads to an infinite value of f+. The differential format Eq. (32) has been already used in the past instructural engineering for some specific applications:

M � l2c M00 ¼ EI v� a2v00

� �or p� l2

c p00 ¼ k0 y� a2y00� �

ð33Þ

Interestingly, the moment–curvature (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) [55–57]. 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 deflection. 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-tutive law:

M� � l2c M�00 ¼ k� vp þ a2v00ph i

with f� ¼ � alc

�2

ð34Þ

A 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 moment–curvature relationship will be incorporated inthe model, and leads to a well-posed evolution problem. In fact, it is not necessary to introduce some non-locality inthe hardening range from a mathematical point of view. However, it is also possible to introduce some non-localityduring hardening, to introduce some scale effects in the hardening range. For the non-local hardening plasticitymodel, M* is related to the combined non-local plastic curvature variable ~vp through the linear model (see for instanceEq. (28)):

M� ¼ kþ~vp with ~vp ¼ vp � a2vp00 ð35Þ

Introducing the combined non-local plastic curvature into the loading function leads to a differential equation:

vp � a2vp00 ¼ P L� xð Þ �Mp

kþð36Þ

The general solution of this differential equation is written as (see also [23]):

x 2 0; lþ0� �

: vp xð Þ ¼ A coshxaþ B sinh

xaþ P L� xð Þ �Mp

kþð37Þ

with the boundary conditions obtained from the variational principle:

vp lþ0� �

¼ 0; vp0 lþ0� �

¼ 0 and vp0 0ð Þ ¼ 0 ð38Þ

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Fig. 4. Evolution of the plastic zone n versus the loading parameter b. Non-local hardening plasticity model: f > 0.

494 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506

The non-linear system of three equations with three unknowns A, B and lþ0 is finally obtained:

A 1� lca

� �2� �

cosh lþ0a þ B 1� lc

a

� �2� �

sinh lþ0a þ

P L�lþ0ð Þ�Mp

kþ¼ 0

Aa sinh lþ0

lcþ B

a cosh lþ0lc� P

kþ¼ 0

Ba � P

kþ¼ 0

8>>><>>>:

ð39Þ

The following dimensionless parameters may be introduced as:

b ¼ 1� PY

P

�Llc

P 0 and n ¼ lþ0lc

P 0 ð40Þ

leading to the localization relation:

b ¼ nþ 1� ff

cosh n� 1sinh n

ð41Þ

The plastic zone n versus the loading parameter b is shown in Fig. 4 and is parameterised by the dimensionless parameterf. The gradient plasticity model (in the hardening range) is recovered from this relationship as an asymptotic law (lc ? 0 inEq. (32)):

f!1 ) b ¼ n� cosh n� 1sinh n

ð42Þ

The width of the plastic zone associated with the non-local models is larger than the reference width of the local model,for f larger than unity, whereas this width is smaller than the one of the local model for f smaller than unity. The local hard-ening plastic zone relation is obtained by setting f = 1.

4. Non-local softening constitutive law: application to the cantilever beam

For the non-local softening plasticity model, M* is related to the combined non-local plastic curvature variable ~vp throughthe linear model (see for instance Eq. (28)):

M� ¼ k�~vp with ~vp ¼ vp þ a2vp00 ð43Þ

Introducing the combined non-local plastic curvature (with a = lc – see [23]) into the loading function leads to a lineardifferential equation:

vp þ l2c vp

00 ¼ P L� xð Þ �Mp

k�ð44Þ

The general solution of this differential equation is written as (see also [23]):

x 2 0; l�0� �

: vp xð Þ ¼ A cosxlcþ B sin

xlcþ P L� xð Þ �Mp

k�ð45Þ

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N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506 495

with the boundary conditions obtained from the variational principle:

vp l0ð Þ ¼ 0; vp0 l�0� �

¼ 0 and vp0 0ð Þ ¼ 0 ð46Þ

An 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 l�0 is finallyobtained:

2A cos l�0lcþ 2B sin l�0

lcþ P L�l�0ð Þ�Mp

k� ¼ 0

� Alc

sin l�0lcþ B

lccos l�0

lc� P

k� ¼ 0Blc� P

k� ¼ 0

8>>><>>>:

ð47Þ

The following dimensionless parameters may be introduced as:

b ¼ 1� PY

P

�Llc6 0 and n ¼ l�0

lcP 0 ð48Þ

and the load–plastic zone relationship is finally written as:

b ¼ n� 21� cos n

sin nfor sin nð Þ – 0 ð49Þ

In other words, Wood’s 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 infinite value when P tends towards zero.Moreover, the size of the plastic zone tends towards an asymptotic value for large values of jbj (and sufficiently 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 finite energy dissipation that depends only on the characteristic length.The maximum width of the localization zone l�0 directly depends on the characteristic length of the non-local model via therelation l�0 ¼ plc (for the cantilever beam). The determination of the characteristic length lc (or the maximum width of thelocalization zone l�0 Þ is related to the question of the finite 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 ([1,2,61–67]) for reinforced concretebeams. It is generally acknowledged that the value of lc (or the maximum localization zone l�0 Þmust 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 localization

Fig. 5. Evolution of the plastic zone n versus the loading parameter b. Non-local softening plasticity model.

Page 10: On the propagation of localization in the plasticity collapse of hardening–softening beams

Fig. 6. Response of the elastoplastic non-local softening beam; EIk� ¼ �5; lc

L ¼ 0:1.

496 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506

zone l�0 is chosen in the order of magnitude of the depth of the cross section h (see also [64]). This implies for the cantileverbeam that the characteristic length lc is in the order of magnitude of h/p. This theoretical aspect certainly merits some furtherinvestigations. However, the existence of the finite size fracture process zone leads to the specific structural size effect.

The deflection in the plastic zone x 2 0; l�0� �

is obtained by integrating twice the elastic curvature:

w� xð Þ ¼ PLEIþ PL�Mp

k�

�x2

2� P

EIþ P

k�

�x3

6� 2

Pl3c

k�cos l�0

lc

� �� 1

sin l�0lc

� � cosxlc

�� 1

� � 2

Pl3c

k�sin

xlc

�� x

lc

� ð50Þ

The deflection in the elastic zone x 2 l�0 ; L� �

is derived from the continuity condition given by Eq. (13):

wþ xð Þ ¼ PLx2

2EI� Px3

6EIþ w�0 l�0

� �� PLl�0

EIþ

P l�0� �2

2EI

" #xþ w� l�0

� �� l�0� �

w�0 l�0� �þ

PL l�0� �2

2EI�

P l�0� �3

3EI

" #ð51Þ

Fig. 6 shows the resolution of Wood’s paradox with the non-local softening plastic model considered in the paper. Afterthe peak load, the softening plasticity propagates along the beam, leading to the global softening phenomenon.

5. Local hardening–softening constitutive law: Wood’s paradox

In this section, the effect of a pre-hardening range is studied for the cantilever beam. It is first assumed that both the hard-ening and the softening part of the constitutive behaviour are ruled by a local law. The local moment–curvature relationship(M,v) considered is tri-linear with a linear elastic part coupling to a linear hardening and softening part (Fig. 7). This is apiecewise linear hardening–softening plasticity model. This hardening/softening plasticity model presented in this sectionwas already studied in [11] at the beam scale, for a beam with uniform bending moment.

The hardening/softening rule associated to the yield function written by Eq. (3) is now given in the following form:

M� vp

� �¼ kþvpifvp 2 0;jc½ �

M� vp

� �¼ k� vp � jc

� �þmMp

D E�Mpifvp R 0;jc½ �

8><>: with jc ¼ m� 1ð ÞMp

kþand

kþ P 0k� 6 0

(ð52Þ

Mp is the limit elastic moment, and vY is the limit elastic curvature, related through Mp/vY = EI. m is the ratio between themaximum moment reached during positive hardening and the limit elastic moment (m is necessarily greater than unity),and jc is the plastic curvature reached before the initialization of the non-local softening process. The hardening modulusk+ is positive whereas the softening modulus k� is negative. The simple relation is obtained between the constitutive param-eters in the hardening range:

EI¼ m� 1

v �mand

jv

vY¼ v �m with v ¼ vv

vYð53Þ

Page 11: On the propagation of localization in the plasticity collapse of hardening–softening beams

pM

M

m

1

1 Y

v

χχ

Y

u

χχ

Yχχ

Fig. 7. Elastic–plastic hardening–softening moment–curvature law.

N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506 497

vv is the curvature value associated to the softening process. In the hardening range, the load is increasing such as:

P 2 PY ; mPY½ � with PY ¼Mp

Lð54Þ

For 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 lþ0 . In the plastic zone, the plastic curvature is linearlyincreasing:

x 2 0; lþ0� �

: vp xð Þ ¼ 1kþ

P L� xð Þ �Mp� �

ð55Þ

The continuity of the plastic curvature at the elastic–plastic interface leads to the plastic zone–load relationship:

vp lþ0� �

¼ 0 ) lþ0L¼ 1� PY

Pð56Þ

Note that the propagation of the local hardening process zone is equivalent to the linear relationship between previouslyused dimensionless parameters:

b ¼ n with b ¼ 1� PY

P

�Llc

P 0 and n ¼ lþ0lc

P 0 ð57Þ

The displacement field in the plastic zone is obtained using the boundary conditions (clamped beam):

x 2 0; lþ0� �

: EIw� xð Þ ¼ � 1þ EIkþ

�P

x3

6þ 1þ EI

�PL�Mp

EIkþ

� x2

2ð58Þ

The displacement in the elastic zone is obtained by enforcing the continuity of the displacement and the rotation at theinterface:

x 2 lþ0 ; L� �

: EIwþ xð Þ ¼ �Px3

6þ PL

x2

2þ EI

kþP

lþ0� �2

2x� EI

kþP

lþ0� �3

6ð59Þ

The load–deflection relationship is finally deduced in the hardening range from:

vvY¼ P

PYþ 3

EIkþ

PPY

12

lþ0L

�2

� 16

lþ0L

�3" #with

lþ0L¼ 1� PY

Pð60Þ

Fig. 8 shows the hardening range for the cantilever beam. It can be easily checked from Eq. (60) that the load–deflectionrelationship is not linear, even for the local hardening constitutive behaviour considered in this paragraph (see also Fig. 8). Aremarkable result is that the plastic curvature distribution depends on the hardening law but the propagation law Eq. (56) orEq. (57) does not depend on the model of hardening law. In fact, whatever the hardening model (even in case of non-linearhardening), the same equality is valid:

vp lþ0� �

¼ 0) M� lþ0� �

¼ 0 ) M lþ0� �

¼ P L� lþ0� �

¼ Mp ð61Þ

Page 12: On the propagation of localization in the plasticity collapse of hardening–softening beams

Fig. 8. Wood’s paradox–local hardening/softening plasticity models; m ¼ 54; EI

kþ¼ 11.

498 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506

The 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 l�0

� �¼ jc

� �leads to:

P L� l�0� �

¼ mMp

PL 6 mMp

() l�0 ¼ 0 ð62Þ

This additional assumption gives the new Wood paradox for hardening–softening 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.

6. Local hardening and non-local softening constitutive law

Wood’s paradox for the hardening–softening beam can be solved by using a non-local softening moment–curvature law,as in the case of the elastic-softening beams. Once the bending moment in the clamped section M(x = 0) = PL reaches theyield strength mMp, the softening zone can propagate from the clamped section, whereas unloading is observed in the hard-ening plastic zone and in the elastic zone. The local hardening and non-local softening constitutive relationship are given by:

M� vp

� �¼ kþvpifvp 2 0;jc½ �

M� vp

� �¼ k� ~vp � jc

� �þmMp

� ��Mpifvp R 0; jc½ �

8><>: with ~vp ¼ vp þ l2c vp

00 ð63Þ

The problem of the continuity requirement between both hardening and softening constitutive law (a local one and anon-local one) will be implicitly solved by the fact that the length of the softening zone at the yield strength mMp will vanishas we will see (as for the elastic softening problem). Furthermore, the non-local plastic variable is integrated over the activeplastic domain, i.e. the softening zone, as the hardening zone is in unloading during this final process.

By considering the yield function in the softening area, the linear differential equation is obtained in the softeningdomain:

x 2 0; l�0� �

: vp þ l2c vp

00 ¼ P L� xð Þ �mMp

k�þ jc ð64Þ

with the boundary conditions, associated to the higher-order boundary conditions of the non-local model and the continuityrequirement of the plastic curvature:

vp l0ð Þ ¼ jc; vp0 l�0� �

¼ 0 and vp0 0ð Þ ¼ 0 ð65Þ

The general solution of the differential equation Eq. (64) is written as:

x 2 0; l�0� �

: vp xð Þ ¼ A cosxlcþ B sin

xlcþ P L� xð Þ �mMp

k�þ jc ð66Þ

Page 13: On the propagation of localization in the plasticity collapse of hardening–softening beams

Fig. 9. Response of the elastoplastic hardening–non-local softening beam; EIk� ¼ �5; lc

L ¼ 0:1; m ¼ 54; EI

kþ¼ 11.

N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506 499

The non-linear system of three equations with three unknowns A, B and l�0 is finally obtained:

2A cos l�0lcþ 2B sin l�0

lcþ P L�l�0ð Þ�mMp

k� ¼ 0

� Alc

sin l�0lcþ B

lccos l�0

lc� P

k� ¼ 0Blc� P

k� ¼ 0

8>>><>>>:

ð67Þ

The following dimensionless parameters may be introduced as:

b ¼ 1�mPY

P

�Llc6 0 and n ¼ l�0

lcP 0 ð68Þ

leading to the localization relation of Eq. (49). A remarkable result is that the plastic diffusion in the softening range does notdepend on the hardening range. In other words, the hardening modulus (or the material history in the hardening domain)does not affect the localization process, from a qualitative point of view.

The deflection in the plastic zone x 2 0; l�0� �

is obtained by integrating twice the elastic curvature:

w� xð Þ ¼ PLEIþ PL�mMp

k�þ jc

�x2

2� P

EIþ P

k�

�x3

6� 2

Pl3c

k�cos l�0

lc

� �� 1

sin l�0lc

� � cosxlc

�� 1

� � 2

Pl3c

k�sin

xlc

�� x

lc

� ð69Þ

The deflection in the elastic zone x 2 l�0 ; lþ0� �

is derived from the continuity condition given by Eq. (13), whereas the plasticcurvature distribution is constant in the unloading phase:

wþ1 xð Þ ¼ PLEIþ m� 1ð ÞMp

�x2

2� P

EIþm

PY

�x3

6þ w�0 l�0

� �� PLl�0

EIþ

P l�0� �2

2EI� m� 1ð ÞMpl�0

kþþm

PY l�0� �2

2kþ

" #x

þ w� l�0� �� l�0� �

w�0 l�0� �þ

PL l�0� �2

2EI�

P l�0� �3

3EIþ m� 1ð Þ

Mp l�0� �2

2kþ�mPY

kþl�0� �3

3

" #ð70Þ

The deflection in the elastic zone x 2 lþ0 ; L� �

is derived from the continuity condition given by Eq. (11):

wþ2 xð Þ ¼ PLx2

2EI� Px3

6EIþ wþ01 lþ0

� �� PLlþ0

EIþ

P lþ0� �2

2EI

" #xþ wþ1 lþ0

� �� lþ0� �

wþ01 lþ0� �þ

PL lþ0� �2

2EI�

P lþ0� �3

3EI

" #with

lþ0 ¼m� 1

mL ð71Þ

Generally speaking, the plastic zone growth in the hardening range until the maximum load, then a more localized soft-ening zone arises from the clamped section and controls the mode of collapse. The global softening is then observed after thehardening behaviour (Fig. 9).

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500 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506

7. On the law of propagation of localization

In 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.

Fi

M� ¼ K�ffiffiffiffiffi~vp

qwith ~vp ¼ fvp þ 1� fð Þvp ð72Þ

where 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 defined as:

f ¼ �1 ) ~vp ¼ vp þ l2cvp

00 ¼ M�

K�

�2

¼ P L� xð Þ �Mp

K�

�2

ð73Þ

The general solution of this differential equation is written as:

x 2 0; l�0� �

: vp xð Þ ¼ A cosxlcþ B sin

xlcþ

P L� xð Þ �Mp� �2 � 2l2

c P2

K�ð Þ2ð74Þ

The boundary conditions are expressed by Eq. (46) for the elastic-softening beam model. The plastic zone n versus theloading parameter b is finally obtained from these boundary conditions:

b2 þ b �2nþ 41� cos n

sin n

�þ n2 � 4þ 4n

cos nsin n

¼ 0 for sin nð Þ – 0 ð75Þ

whose softening solution is given by:

b ¼ n� 21� cos n

sin n�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� 2

1� cos nsin n

�2

� n2 � 4þ 4ncos nsin n

�sð76Þ

Fig. 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 finite 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 thatthe global softening response depends on the softening model considered, i.e. linear or non-linear softening models. This re-sult is quite similar to the result highlighted in [22] where the loading mode (concentrated force or distributed loading) has astrong influence on the propagation of localization, even if the localization zone converges towards a finite length zone forboth plasticity models studied in this paper. A numerical comparison of numerous non-local softening models can be foundin [33] or [34], in case of homogeneous state of stress.

g. 10. Evolution of the plastic zone n versus the loading parameter b. Non-local softening plasticity model; comparison of softening models.

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N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506 501

In conclusions, the hardening non-local model (with hardening modulus k+) can be compared to the softening non-localmodel (with softening modulus k�):

M� � l2c M�00 ¼ kþ vp � a2v00ph i

– M� � l2c M�00 ¼ k� vp þ a2v00p

h ið77Þ

The plasticity zones in both regimes appear to be significantly different (see also [67] for the same conclusions). lþ0 relatedto the hardening domain is propagating along the beam without any material limits, whereas the softening localization zone,denoted by l�0 is increasing during the softening process, until a finite length which depends on the material-section model.Of course, these two localization zones are strongly different lþ0 – l�0

� �. For the non-local models studied in this paper, the

plastic variable is integrated on an active plastic domain. In particular, during the softening process, the non-local plasticvariable is integrated on the localization length l�0 , even if the plasticity zone is generally much larger lþ0 > l�0

� �. In fact,

the hardening and the softening plastic zones are given by:

lþ0L¼ 1� 1

mand

l�0L6 p lc

Lð78Þ

where 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 sufficiently small characteristic length, i.e.:

lc

L6

1p

1� 1m

�) l�0 6 lþ0 ð79Þ

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 influence the propagationof this localized plastic zone associated to the collapse of the beam. We have shown that the load-plastic zone propagationdid not depend on the hardening stage. This distinction between the active and the passive plastic zone can be clearly under-stood in an incremental time-formulation.

8. On the rate-form of the non-local equations

The rate-form of the non-local problem is sometimes preferred to solve the propagation of the localization process alongthe hardening–softening beam. Indeed, during the softening process, the stationarity of the loading function implies that:

_f M;vp

� �¼ 0 )

d vp

� �dt

þ l2c

d vp00

� �dt

¼ ddt

P L� xð Þk�

ð80Þ

It is difficult 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; l�0� �

: vp x; l�0 ; t� �

¼ P tð Þlc

k�cos n l�0

� �� 1

sin n l�0� � cos

xlcþ P tð Þlc

k�sin

xlcþ P tð Þ L� xð Þ �Mp

k�ð81Þ

One 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].

dvp

dtx; l�0 ; t� �

¼@vp

@tx; l�0 ; t� �

þ _l�0@vp

@l�0x; l�0 ; t� �

þ _x@vp

@xx; l�0 ; t� �

ð82Þ

According 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:

ddt

@2vp

@x2

" #–

@2

@x2

dvp

dt

� ð83Þ

It would be possible to use the ‘‘partial” time derivative in the loading function, but the exact boundary conditions areexpressed in rate-form as:

dvp0

dtx ¼ 0; l�0 ; t� �

¼ 0;dvp

0

dtx ¼ l�0 ; l

�0 ; t

� �¼ 0 and

dvp

dtx ¼ l�0 ; l

�0 ; t

� �¼ 0 ð84Þ

The first rate boundary condition Eq. (84) can be obtained from:

@vp0

@tx ¼ 0; l�0 ; t� �

¼ 0;@vp

0

@l�0x ¼ 0; l�0 ; t� �

¼ 0 and@vp

0

@xx ¼ 0; l�0 ; t� �

¼ � Plck�

cos n� 1sin n

)dvp

0

dtx ¼ 0; l�0 ; t� �

¼ 0

ð85Þ

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502 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506

The second rate boundary condition Eq. (84) can be checked from:

@vp0

@tx ¼ l�0 ; l

�0 ; t

� �¼ 0;

@vp0

@l�0x ¼ l�0 ; l

�0 ; t

� �¼ � P

lck�cos n� 1

sin nand

@vp0

@xx ¼ l�0 ; l

�0 ; t

� �¼ P

lck�cos n� 1

sin n

)dvp

0

dtx ¼ l�0 ; l

�0 ; t

� �¼ 0 ð86Þ

whereas the last boundary condition Eq. (84) is confirmed by:

@vp

@tx ¼ l�0 ; l

�0 ; t

� �¼

_Plc

k�Llc� nþ 2

1� cos nsin n

�;

@vp

@l�0x ¼ l�0 ; l

�0 ; t

� �¼ 2P

k�cos n� 1

sin2 ncos n and

@vp

@xx ¼ l�0 ; l

�0 ; t

� �

¼ Pk�

)dvp

dtx ¼ l�0 ; l

�0 ; t

� �¼ 0 ð87Þ

Note that the boundary conditions cannot be expressed in rate form using the ‘‘partial” time derivative if:

@vp0

@tx ¼ l�0 ; l

�0 ; t

� �– 0 or

@vp

@tx ¼ l�0 ; l

�0 ; t

� �– 0 ð88Þ

However, the boundary condition at the clamped end was easier to derive, as this fixed point does not move:

dvp0

dtx ¼ 0; l�0 ; t� �

¼@vp

0

@tx ¼ 0; l�0 ; t� �

¼ 0 ð89Þ

It has to be outlined that it is difficult 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 difficulty 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,

_l�0 ¼ 0 )dvp

0

dtx ¼ 0; l�0 ; t� �

¼@vp

0

@tx ¼ 0; l�0 ; t� �

¼ 0;

dvp0

dtx ¼ l�0 ; l

�0 ; t

� �¼@vp

0

@tx ¼ l�0 ; l

�0 ; t

� �¼ 0 and

dvp

dtx ¼ l�0 ; l

�0 ; t

� �¼@vp

@tx ¼ l�0 ; l

�0 ; t

� �¼ 0

ð90Þ

Therefore, it is recommended to use the ‘‘material” time derivative in the rate-format of the boundary value problem. Thispoint has certainly to be rigorously taken into account in a numerical time-integration format applied to more complexstructures.

9. Conclusions

This paper questions the mode of collapse of some simple hardening–softening structural systems, comprising the clas-sical cantilever beam. Such simplified models can be useful for the understanding of plastic buckling of tubes in bending, thebending response of thin-walled members experiencing softening induced by the local buckling phenomenon, or the bend-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-basednon-local plasticity model is developed, in order to overcome Wood’s paradox when softening prevails. This model can berigorously derived from a variational principle. Using a piecewise linear plasticity hardening–softening constitutive law,we look at the 1D propagation of plastic strains along the bending beam. It is concluded that the mode of collapse is firmlya non-local phenomenon.

We show that the localization zone evolves during the softening process, until an asymptotic limited value, which de-pends on the characteristic length of the section. This finite character of the localization propagation can be related to thewell known concept of finite length region. The existence of this finite size fracture process zone leads to the specific struc-tural size effect. As a consequence of this model, the plastic length evolves during the loading process, a phenomenon oftennoticed in structural design. Therefore, it is not necessary to introduce a variable characteristic length in the non-local model,at least for the appearance of this specific feature. The beam response is studied for linear and non-linear non-local softeningmodels. The softening model and the loading mode, have a strong influence on the propagation of localization, even if thelocalization zone converges towards the same finite length zone for the plasticity models studied in this paper.

The influence of the hardening phase on the localization process has also been specifically addressed. The plasticity zonesin both regimes appear to be significantly different. lþ0 related to the hardening domain is propagating along the beam with-out any material limits (except the length of the beam), whereas the softening localization zone, denoted by l�0 is increasingduring the softening process, until a characteristic finite length. Some similar conclusions have been recently numericallyreached in [67] with a variable inelastic end zone model (except that the softening length is fixed in [67]). For the non-local

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N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506 503

models 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 l�0 , even if the global plasticity zone is generallymuch larger lþ0 > l�0

� �. In fact, during the softening process, a part of the hardening zone is unloaded and can be therefore

considered as a passive plastic zone. During the softening localization process, this passive plastic zone does not influencethe 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 defined 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.

Appendix A. Influence of higher-order boundary conditions

In this Appendix, we investigate the specific effect of higher-order boundary conditions on the response of a non-localsoftening tension bar under uniform stress state.

A.1. Higher-order boundary conditions at the elastoplastic interface

We study the implicit gradient plasticity model based on the non-local plastic strain �j (see Eq. (13)):

�j� l2c�j00 ¼ j ðA:1Þ

The linear softening law is written as:

r ¼ r0 þ H~j with ~j ¼ fjþ 1� fð Þ�j ¼ �j� fl2c�j00 ¼ �jþ p� 1ð Þl2

c�j00 and p ¼ 1� f ðA:2Þ

For the tension bar considered in Fig. 11, the stress is uniform, and the yield condition leads to the differential equation inthe plastic zone:

�jþ p� 1ð Þl2c�j00 ¼ r� r0

Hfor x 2 0;

l0

2

� ðA:3Þ

The general solution in term of non-local variable is written as:

�j ¼ r� r0

Hþ A cos

x

lc

ffiffiffiffiffiffiffiffiffiffiffiffip� 1

p þ B sinx

lcffiffiffiffiffiffiffiffiffiffiffiffip� 1

p for x 2 0;l0

2

� ðA:4Þ

By virtue of symmetry, we have �j0 0ð Þ ¼ 0 leading to B = 0. Furthermore, the plastic variable is assumed to be continuousat the elastoplastic interface:

jl0

2

�¼ r� r0

Hþ A

pp� 1

cosl0

2lc

ffiffiffiffiffiffiffiffiffiffiffiffip� 1

p ¼ 0 ðA:5Þ

In 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

�¼ 0 ) l0

2lc

ffiffiffiffiffiffiffiffiffiffiffiffip� 1

p ¼ np with n ¼ 1 ðA:6Þ

x

L

y

F

l0

F

Fig. 11. The tension bar.

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504 N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506

Eq. (A.6) gives the smallest softening plastic zone associated to the localization process. Therefore, in this case, the solu-tion in the plastic zone is finally written for the non-local plastic strain as:

x 2 � l0

2;l0

2

� : �j xð Þ ¼ r� r0

H1þ p� 1

pcos

x

lc

ffiffiffiffiffiffiffiffiffiffiffiffip� 1

p" #

ðA:7Þ

whereas the solution of the local plastic strain is:

x 2 � l0

2;l0

2

� : j xð Þ ¼ r� r0

H1þ cos

x

lc

ffiffiffiffiffiffiffiffiffiffiffiffip� 1

p" #

ðA:8Þ

Outside the plastic zone, there is no plastic strain as a definition:

x R � l0

2;l0

2

� : j xð Þ ¼ �j xð Þ ¼ 0 ðA:9Þ

The non-local plastic strain is also vanishing outside the plastic zone, as the non-local plastic strain is calculated over theplastic domain only (and not all the domain). The non-local plastic strain �j is the spatial weighted average of the variable j.In this paper, due to the choice of higher-order boundary conditions, the non-local plastic strain is calculated over the plasticdomain (see also [18] or [47]) :

�j xð Þ ¼Z l0

2

�l02

G x; yð Þj yð Þdy ðA:10Þ

where the weighting function G(x,y) is the Green’s function of the differential system. The particular case of uniform plasticstrain variable leads to:

j xð Þ ¼ j0 ) �j ¼ j0 ðA:11Þ

A.2. Higher-order boundary conditions at the boundary of the bar

In the case of the model of Engelen et al. [20] (see more recently [48]), we have to solve two differential equations:

�jþ p� 1ð Þl2c�j00 ¼ r� r0

Hfor x 2 0;

l02

� ; �j� l2c �j00 ¼ 0 for x 2 l0

2;L2

� ðA:12Þ

with five boundary conditions:

�j0 0ð Þ ¼ 0; jl�02

�¼ 0; �j

l�02

�¼ �j

lþ02

�; �j0

l�02

�¼ �j0

lþ02

�and �j0

L2

�¼ 0 ðA:13Þ

These boundary conditions express the symmetrical property of the problem, the continuity of the plastic variable at theelastoplastic interface, the continuity of the non-local plastic variable at the elastoplastic interface, the differentiability of thenon-local plastic variable, and finally the higher-order boundary condition at the bar extremity. The solution of the non-localplastic strain in the plastic zone is given by Eq. (A.4). Introducing the boundary conditions of Eqs. (A.5) and (A.6) leads to thenon-local plastic strain:

x 2 � l0

2;l0

2

� : �j xð Þ ¼ r� r0

H1� p� 1

p1

cos l02lc

ffiffiffiffiffiffip�1p

cosx

lcffiffiffiffiffiffiffiffiffiffiffiffip� 1

p264

375 ðA:14Þ

In the elastic zone, the model of Engelen et al. [20] would lead to the non-local plastic solution:

�j ¼ C coshxlcþ D sinh

xlc

for x 2 l0

2;L2

� ðA:15Þ

The three last boundary conditions of Eq. (A.13) are written as:

�j0 L2

� �¼ 0 ) C sinh L

2lcþ D cosh L

2lc¼ 0

�j l�02

� �¼ �j lþ0

2

� �) C cosh l0

2lcþ D sinh l0

2lc¼ 1

pr�r0

H

�j0 l�02

� �¼ �j0 lþ0

2

� �) C sinh l0

2lcþ D cosh l0

2lc¼ 1

pr�r0

H

ffiffiffiffiffiffiffiffiffiffiffiffip� 1

ptan l0

2lcffiffiffiffiffiffip�1p

8>>>>><>>>>>:

ðA:16Þ

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N. Challamel et al. / International Journal of Engineering Science 48 (2010) 487–506 505

A necessary condition for this system to have a solution is that:

ffiffiffiffiffiffiffiffiffiffiffiffip� 1

ptan

l0

2lcffiffiffiffiffiffiffiffiffiffiffiffip� 1

p ¼ tanhl0 � L

2lc

�ðA:17Þ

The case of an infinite bar is found from the asymptotic tendency:

Llc!1 ) l0

2lcffiffiffiffiffiffiffiffiffiffiffiffip� 1

p ! p� arctan1ffiffiffiffiffiffiffiffiffiffiffiffi

p� 1p ðA:18Þ

In the usual case of p = 2, the plastic domain is equal to:

p ¼ 2;Llc!1 ) l0

2lc! 3p

4ðA:19Þ

In the general case of Engelen et al.’s model, the plastic domain associated to the uniform stress case depends on thelength of the bar, a property that is not verified in the case where the higher-order boundary conditions are imposed atthe elastoplastic interface. Imposing the higher-order boundary conditions at the physical boundary of the domain meansthat the non-local plastic strain is calculated over the overall domain, as suggested by:

�j xð Þ ¼Z L

2

�L2

G x; yð Þj yð Þdy ðA:20Þ

In this case, one would obtain for a uniform plastic curvature field characterized by the length l0 with l0 < L:

j xð Þ ¼ j0 ) �j < j0 ðA:21Þ

Furthermore, the non-local plastic strain in the elastic zone is not vanishing and may evolve, a situation that may be ques-tionable from physical and thermodynamical point of view.

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