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INFLUENCE OF GEOMETRICAL NONLINEARITIES ON THESHAKEDOWN OF DAMAGED STRUCTURESD. Weichert* and A. HachemiInstitut fu r Allgemeine Mechanik, RWTH-Aachen Templergraben 64, 52056 Aachen, Germany(Received in final revised form 18 March 1998)AbstractA generalization of Melans shakedown theorem is presented taking into accountgeometrical e?ects and plastic ductile damage. Numerical results illustrate the proposed method.# 1998 Elsevier Science Ltd. All rights reservedKeywords: A. plastic collapse, shakedown, B. constitutive behavior, elasticplastic material, finitestrain.I. INTRODUCTIONThe development of numerical methods for the assessment of the long-time behavior, theusability and safety against failure of structures subjected to variable repeated loading isof great importance in mechanical and civil engineering. A particular kind of failure iscaused by an unlimited accumulation of plastic strains during the loading process, leadingto either incremental collapse or alternating plasticity. If, on the contrary, after some timeplastic strains cease to develop further and the accumulated dissipated energy in the wholestructure remains bounded such that the structure responds purely elastically to theapplied variable loads, one says that the structure shakes down.The foundations of these theories have been given by Melan (1936) and Koiter (1960),who derived su?cient criteria for shakedown and non-shakedown, respectively, of elasticperfectly plastic structures. Both criteria presume the existence of a convex yield surfaceand the validity of the normality rule for the plastic strain rates. Moreover, the influences ofmaterial hardening, geometrical e?ects and material damage are neglected. Consequently,extensions of the classical shakedown theorems have attracted much interest in the latteryears. Reviews of former investigations can be found for example in Gokhfeld and Cher-niavsky (1980), Ko nig and Maier (1981), Ko nig (1982, 1987) and Mro z et al. (1995).Material hardening has been addressed in the pioneering work by Melan (1938), whereunlimited linear kinematical hardening was taken into account in the framework of con-tinuum mechanics. On the basis of this concept further results have been obtained by Neal(1950), Ponter (1975) and Zarka and Casier (1981). For discretized structures and piece-wise linear yield function, Maier (1972) investigated linear hardening and softening e?ectsand Ko nig and Siemaszko (1988) considered the e?ects of strainhardening in shakedownInternational Journal of Plasticity, Vol. 14, No. 9, pp. 891907, 1998# 1998 Elsevier Science LtdPergamonPrinted in Great Britain. All rights reservedPII: S0749-6419(98)00035-70749-6419/98 $see front matter891*Corresponding cess. With the help of generalized standard material model introduced by Halphen andNguyen (1975), Mandel (1976) gives a simple and pertinent formulation of Melans theo-rem for hardening materials. By imposing limits to the evolution of the internal para-meters in this model, Weichert and Gross-Weege (1988) interpreted it as a simplified two-surface material allowing for limited kinematical hardening and applied it numerically. Theconcept of internal variables for the representation of the hardening material behavior wasin the sequel also applied by Comi and Corigliano (1991) and Polizzotto et al. (1991). Moregeneral nonlinear hardening has been investigated by Maier (1969) in the context of discretesystems and by Stein et al. (1993), who used for this purpose the so-called overlay model.Other applications of internal parameters representation of changes of material propertiescan be found in Corigliano et al. (1995) and Pycko and Maier (1995).The geometrically nonlinear problem has been studied firstly by Maier (1973), whointroduced a new class of shakedown problems for pre-stressed discrete structures andextended Melans and Koiters theorems as to include so-called second order geometrice?ects by using piecewise linear yield conditions. Siemaszko and Ko nig (1985) showedthe influence of geometrical e?ects on the stability of the deformation process for parti-cular structures under certain assumptions on the deformation modes. Weichert (1983,1986), investigated the problem of geometrical e?ects in several papers within the frame-work of continuum mechanics and gave an extension of Melans theorems which is prac-tically applicable to situations where informations about the expected deformationpattern are available. He assumed an additive strains decomposition and applied it toshell-like structures undergoing moderate rotations at small strains (Weichert, 1989). Thesame decomposition of total strain has been used by Gross-Weege (1990). He gives unifiedformulation of Melans theorem for structures subjected to a constant load, responsiblefor large displacements, and to small additional variable loads causing small additionaldisplacement. The same concept was used by Pycko and Ko nig (1991). Recently, Poliz-zotto and Borino (1996), give an extension of Melans and Koiters shakedown theorem inthe framework of large displacements. They studied the asymptotic response of thestructure subjected to periodically variable loads in order to show the conditions underwhich there may exist a stabilized long term response. In order to overcome the restric-tions of an additive decomposition of total strains, the multiplicative strain decompositionrule was used by Saczuk and Stumpf (1990), Tritsch (1993), Tritsch and Weichert (1993)and Stumpf (1993). In Saczuk and Stumpf (1990), an extension of the Gross-Weegesshakedown formulation (Gross-Weege, 1990) to more general nonlinear problems is pro-posed. In Tritsch (1993) and Tritsch and Weichert (1993) a su?cient Melan-type state-ment for shakedown and a comparative study with previous works are given. Stumpf(1993) employed the multiplicative decomposition of total strains and attempted toreformulate shakedown theorems stating that shakedown occurs if there exist some realself-equilibrated residual state, which is dependent on the loading and unloading paths.More recently, Saczuk (1997) proposed a criterion of adaptation process, accounting forthe influence of deformation path on the material properties based on the Finsleriancontinuum model within the theory of di?erential inequalities.Influence of material damage on the formulation of the static shakedown theorems wasfirst investigated in a thermodynamical framework from theoretical point of view byHachemi and Weichert (1992) using the energy-based isotropic elastoplastic damagemodels given by Ju (1989). The material damage is taken into account by an internalscalar-valued parameter using the concept of e?ective stresses following Lemaitre and892D. Weichert and A. HachemiChaboche (1985). Feng and Yu (1995) adopted this concept and applied it to thick-walledshells by using a mathematical programming method to calculate an upper bound of theductile damage parameter. In a similar sense, Polizzotto et al. (1996) proposed an exten-sion of Melans shakedown theorem for elasticplastic damage, or elastic damage,material endowed with a general free energy potential. This methods has been applied tothe example of pinned bars by using the damage model of Ju (1989). Siemaszko (1993)presented a step-by-step method of inadaptation analysis for elasticplastic discretestructures taken into account nonlinear geometrical e?ects, nonlinear hardening andductile damage by using the material softening function by Perzyna (1984). Recently,Hachemi and Weichert (1997) proposed how to deal practically with the di?erencebetween plasticity and material damage in relation with shakedown theory taken intoaccount kinematical hardening according to the concept of the generalized standardmaterial model by Halphen and Nguyen (1975). The authors show how to control thedegree of material damage by imposing locally bounds on the evolution of ductile damageparameter as defined by Lemaitre (1985). The numerical solution technique as well asseveral numerical examples of axisymmetric structures under combined mechanical andthermal variable loading can be found in Hachemi and Weichert (1998).In the most general case, the degradation of ductile material is related to the initiation,growth and coalescence of microcracks or microvoids induced by large plastic strains. Allkinds of defects in elastoplastic material are considered as damage and may be pre-existingor developing during service. The elastoplastic damage behavior of materials is introducedthrough the concept of the e?ective stress within the framework of continuum mechanics.The primary purpose of this paper is to extend Melans shakedown theorem todamaged structures accounting for geometrical e?ects. This extension is a combination ofthe formulations proposed by Hachemi and Weichert (1992) and Tritsch and Weichert(1993). Only for simplicity we restrict our considerations to elasticperfectly plasticmaterial behavior and isotropic damage. An extension of the presented formulation tohardening material behavior in the sense of the work of Weichert and Gross-Weege (1988)is immediately possible (see e.g. Hachemi and Weichert, 1992, 1997). To model e?ects ofgeometrical changes due to deformation, the multiplicative decomposition of the totaldeformation gradient into an elastic and plastic parts as proposed by Lee (1969) is usedfor the development of the theoretical framework. Section II of this paper is devoted tothe constitutive equations and general assumptions by assuming finite transformations.The adopted formulation is based on the thermodynamic concept of irreversible pro-cesses, which constitutes a necessary basis to describe the damage phenomena by aninternal scalar variable. By the definition of the thermodynamic potential and from thesecond thermodynamics principle, we deduce the dissipative inequalities. A simple three-dimensional model of ductile plastic damage established by Lemaitre (1985) is used. Thismodel is linear with equivalent plastic strain and quadratic with triaxiality ratio.In Section III, an extension of the static shakedown theorems is proposed, taking intoaccount the influence of ductile plastic damage and geometrical nonlinearities. Strains aredecomposed into elastic and plastic parts without using any simplifying assumptions. Forthat, a global intermediate configuration is introduced in the deformation process corre-sponding to a state of deformation satisfying the compatibility conditions. This config-uration contains elastic and plastic residual deformations. This general formulationhowever, delivers constructive methods for shakedown analysis, if additional assumptionson the deformation pattern are introduced (Weichert, 1986). Here, the most simple case isGeometrical nonlinearities and the shakedown of damaged structures893studied, where the considered body or structure is subjected to initial loads, inducing largedisplacements and initial damage such that it is in the reference configuration in equilibrium.The body is then subjected to additional variable in time or cyclic loading, causing smalladditional displacement, in comparison with the previous ones, and additional damage. Inthis case, the response of the reference configuration is calculated incrementally by using theNONSAP finite element program developed by Bathe et al. (1974). The lower bound of theload factor against failure due to non-shakedown or inadmissible damage is calculated byoptimization program using the algorithm LPNLP developed by Pierre and Lowe (1975).In Section IV, results for axisymmetrical shells are presented showing the influence offinite displacements and damage on the shakedown behavior in comparison to results forundamaged materials and geometrical linear analysis.II. FORMULATION OF THE PROBLEMWe consider the behavior of a three-dimensional elasticperfectly plastic body B underthe action of quasistatically varying external agencies a*consisting of surface tractions p*and surface displacements u*acting on the disjoint parts Spand Suof the surface S of B,respectively, and volume forces f*. In the initial configuration Ciat the time t ? 0, Boccupies the volume V0. The motion of B is described by the use of Cartesian coordinates,where the positions of the particles of B in the undeformed and deformed state are givenby the coordinates X ? ?X1;X2;X3? and x ? ?x1;x2;x3?, respectively. The actual config-uration Ctof B is then defined by the displacement function u:u?X;t? ? x?X;t? ? X?1?Under this assumption the boundary value problem referred to the initial undeformedconfiguration is defined by:(i) Statical equationsDiv?T? ? ?f?in V0n:T ? p?on Sp?2?withT ? FS?3?(ii) Kinematical equationsu ? u?on SuF ? I ? grad?u? in V0E ?12?C ? I? in V0?4?withC ? FTF?5?Here, T and S are the unsymmetric first PiolaKirchho? stress tensor and the symmetricsecond PiolaKirchho? stress tensor whereas F and E are the deformation gradient andthe GreenLagrange strain tensor, respectively. I denotes the metric tensor of second rankand n is the outer normal vector to S in Ci.894D. Weichert and A. HachemiElasticplastic deformations are usually described by means of a fictitious intermediateconfiguration C?, derived from the multiplicative decomposition of the deformation gra-dient F into an elastic part Feand a plastic part Fp(Lee, 1969):F ? FeFp?6?where Feis obtained by unloading all infinitesimal neighborhoods of the body B. Thisdecomposition provides the relation between elastic, plastic and total deformation validfor finites strains and leads to an additive decomposition of the GreenLagrange straintensor E into a purely plastic part Epand an elastic part Eedepending on the plasticdeformation (Green and Naghdi, 1965):E ? Ee? Ep?7?withEe?12?Fp?T?Fe?T?Fe? ? I?Fp? and Ep?12?Fp?T?Fp? ? I?8?Here, the theory of thermodynamics with internal variables is used to derive the con-stitutive laws. For this, a local thermodynamic potential ? is introduced assumed to bequadratic in Eeand linear in ?1 ? D? (Chaboche, 1977, 1981; Lemaitre and Chaboche,1985; Simo and Ju, 1987; Ju, 1989):?E ? Ep;D? ? ?1 ? D?0?E ? Ep?9?with?0?12?0L : Ee: Ee?10?where ?0is the energy function of the undamaged (virgin) material, L the tensor of elas-ticityand?0theinitialmassdensity.Theoperator(:)standsfordoubletensorcontraction.IntheformoftheClausiusDuheminequality,the2ndprincipleofthermodynamicsthenrequiresS :_Ep? 0 and Y_D ? 0?11?withS ? ?0?Ee? ?1 ? D?L : Ee?12?Y ? ?0?D?12L : Ee: Ee?13?Hence, the thermodynamic force (Y) conjugate to the damage variable D is the energyfunction of the undamaged material ?0? (Lemaitre and Chaboche, 1985). Superposeddots denote the rate of the considered quantity.To describe the plastic part of the material damage behavior, we assume the existence ofa convex elastic domain, defined by the yield conditionF?S;?F? ? 0;?14?Geometrical nonlinearities and the shakedown of damaged structures895withS as second PiolaKirchho? e?ective stress tensorS ? S=?1 ? D? and ?Fas yieldstress. In the sequel superposed tilda indicates quantities related to the damaged state ofthe material.Convexity of the yield surface F and validity of the normality rule can then be expres-sed by the maximum plastic work inequality?S ? S?s? :_Ep? 0?15?where S?s?is any safe state of stress defined byF?S?s?;?F? 1 and a time-independent state of e?ective residualstressesS?r?, satisfying the following relations:Div?F?x?S?r? ? 0 in V0n:?F?x?S?r? ? 0 on SpF?S?e?S?r?;?F? tRthe body B is submittedto additional variable loads ar*such that:a?X;t? ? aR?X? ? ar?X;t?25?and occupies the actual configuration Ct(see e.g. Weichert, 1986; Gross-Weege, 1990;Saczuk and Stumpf, 1990; Pycko and Ko nig, 1991). Since the actual configuration shouldalso be an equilibrium configuration and the following equations hold:(i) Statical equationsDiv?TR? Tr? ? ?fR? fr?in V0n:?TR? Tr? ? pR? pr?on Sp?26?withTR? Tr? ?FrFR?SR? Sr?27?(ii) Kinematical equationsu ? uR? urin V0F ? FrFR? I ? grad?uR? ? grad?ur?in V0E ? ER? Er?12?C ? I?in V0u ? uR? ur?on Su?28?withC ? ?FrFR?T?FrFR?29?where all quantities caused by the time-independent loads aR*are marked by a superscript(R), whereas the additional field quantities caused by the time-dependent loads ar*aremarked by superscript (r). The additional field quantities caused by ar*have to satisfy thefollowing equations:(i) Statical equations898D. Weichert and A. HachemiDiv?Tr? ? ?fr?inV0n:Tr? pr?on Sp?30?withTr? HrFRSR? FRSr? HrFRSr?31?(ii) Kinematical equationsFr? I ? Hrin V0Er?12?FR?T?Hr?T? Hr? ?Hr?THr?FR? in V0ur? ur?on Su?32?withHr? gradR?ur?33?Inthesequel,werestrictourconsiderationstoloadinghistoriescharacterizedbythemotionofa fictitious comparison body B(c), having at time tRthe same field quantities as B but reacting,incontrasttoB,purelyelasticallytotheadditionaltime-dependentloadsar*,superimposedonaR*for t tR(Fig. 2) (cf. Weichert, 1986; Gross-Weege, 1990; Saczuk and Stumpf, 1990).Henceforth, all quantities related to this comparison problem are indicated by superscript (c).Obviously, eqns (30)(33) also hold for the comparison problem with the only exception that,in the comparison body B(c), no additional plastic strains and no damage can occur. The dif-ferences between the states in B and B(c)are then described by the di?erence fields:?u ? ur? ur?c?;?F ? Fr? Fr?c?;?E ? Er? Er?c?T ? Tr? Tr?c?;?S ? Sr? Sr?c?34?and have to fulfill the following equations:Div?T? ? 0in V0n:?T? ? 0on Sp?35?Geometrical nonlinearities and the shakedown of damaged structures899Fig. 2. Evolution of real body B and comparison body B(c).and?F ? Hr? Hr?c?in V0?E ?12?FR?T?F?T? ?F?FR? ?12?FR?T?Hr?T?Hr? ? ?Hr?c?T?Hr?c?FR?in V0?u ? 0on Su?36?with?T ? ?F?FRSR? FR?S? ? HrFRSr? Hr?c?FRSr?c?37?In the following, we restrict our considerations to situations where the state of deforma-tion and the state of stress in B are subjected to small variations in time (Weichert, 1986;Gross-Weege, 1990). Consequently, we neglect in the governing eqns (34)(37) all terms,which are nonlinear in the time-dependent additional field quantities marked by a super-script (r). This excludes to study buckling e?ects induced by the additional time-dependentloads. Then the following extension of Melans theorem holds:If there exists a time-independent field of e?ective residual stresses ?S?such that thefollowing relations hold:?i?Div?FRSR? ? ?fR?inV0n:?FRSR? ? pR?on Spu ? uR?on Su?38?ii?Div? T? ? 0in V0n:? T? ? 0on Sp? u? 0on Su?39?with? T? ?F?FRSR? FR?S?iii?F?Sr?c?SR? ?S?;?F? tR, then the original body B will shakedown under given program ofloading a*.Material damage however by definition cannot grow indefinitely and is limited by localrupture. Therefore, it is necessary to control the degree of material damage by imposinglocally bounds on the evolution of the damage parameter Drinduced by additional vari-able loads ar*. For the considered approach (eqn (17), damage is basically generated byplastic strains, bounds for damage can be given in this special case by bounding of theequivalent plastic strains (for detail we refer to Hachemi, 1994; Hachemi and Weichert,1997):p ?1V01?1 ? Dc?F? ? 1?V0?12? S? : L?1: ? S?dV0?41?900D. Weichert and A. HachemiThen the safety factor against failure due to non-shakedown or inadmissible damage isdefined by?SD? max?_T;D?42?with the subsidiary conditionsDiv?_T? ? 0 in V0?43?n:? T? ? 0 on Sp?44?DR? Dr? ? Dc 0 in V0?45?F?Sr?c?SR? ?S?;?F? 0 in V0?46?This is a problem of mathematical programming, with ? as objective function to be opti-mized with respect to ? T?and?D?T? ? F?FRSR? FR?S?;D ? DR? Dr? and withinequalities (45) and (46) as nonlinear constraints. The condition (45) assures structuralsafety against failure due to material damage and the condition (46) assures that safestates of stresses are never outside the loading surface.IV. NUMERICAL EXAMPLESThe concepts developed in the preceding sections for a continuous medium have beenapplied to axisymmetric shells subjected to finite displacements and moderate rotations.The foregoing presented approach has been implemented into a finite element programinitially developed by Gross-Weege (1988, 1990), based on the principle of complementaryenergy (Morelle and Nguyen Dang Hung, 1983). The program uses a nonlinear optimi-zation algorithm with an augmented Lagrangian formulation (Pierre and Lowe, 1975).Lower bounds of the safety factor are derived taking into account the influence of thereference state (see e.g. Gross-Weege, 1988; Tritsch and Weichert, 1995). To determine thestate of stresses SR, displacement uRand damage parameter DRin the reference config-uration CRunder the time-independent load aR*, a modified version of the NONSAPstep-by-step finite element program (Bathe et al., 1974) of Kreja et al. (1992, 1993) to takeinto account ductile damage model has been used. The symmetry of shell and loadsis assumed which allows to limit calculations to half of the shell. Sensitivity analysis showedthat the study of the investigated cases does not need a greater number of elements than 10.Damage parameter and the yield criterion are checked in three points per element.IV.1. Example 1The first investigated structure is a short clamped cylindrical shell studied byGross-Weege (1988, 1990) with sandwich cross-section, length L, radius R and wall-thickness h, where L=R ? 1=10 and h=R ? 1=200 (Fig. 3). The shell is subjected to internalpressure p:p ? pR? prGeometrical nonlinearities and the shakedown of damaged structures901902D. Weichert and A. HachemiFig. 3. Cylindrical shell under internal pressure.Fig. 4. Shakedown load domain for cylindrical shell under internal pressure.Fig. 5. Cylindrical shell under internal pressure and bending moment.where pRis a time-independent reference load and pr? ?t?pRis a time-dependent addi-tional load, where ?t? varies between fixed bounds ?and ?, so that ? ? ? ?.This is a typical problem in process engineering, when fluctuations of pressure in theneighborhood of the nominal pressure in pipelines and other pressure vessels may occur.For this example, the following values for material properties are adopted: Youngsmodulus E ? 2:1 ? 105MPa, Poissons ratio ? ? 0:3 and damage properties Dc? 0:42,R? 0:25 and D? 0:0. It should be noted that here and in the following, the choice ofthe threshold value D? 0:0, although somewhat artificial, was made in order to empha-size and better visualize the influence of material damage on the shakedown process in thepresented examples. The shakedown load domains for di?erent values of the uniaxialyield stress are shown in Fig. 4. For the sake of comparison, the results of undamagedshell are represented as well as geometrical linear solution using the von Mises sandwichyield condition.IV.2. Example 2The cylindrical shell described above with the following dimensions L=R ? 1=10 andh=R ? 1=100 is loaded as shown in Fig. 5 by two independent loads, the internal pressure0 ? p ? p?and a bending moment 0 ? M ? M?. The following values of material propertiesE ? 1:6:105MPa, ?F? 360 MPa, ? ? 0:3, Dc? 0:42, R? 0:29, D? 0:0 and initial loadsM0? 10:39:103N and p0? 4:4 MPa have been adopted in the numerical analysis.Geometrical nonlinearities and the shakedown of damaged structures903Fig. 6. Shakedown load domain for cylindrical shell under internal pressure and bending moment.904D. Weichert and A. HachemiFig. 7. Conical shell under internal pressure and axial ring load.Fig. 8. Shakedown load domain for conical shell under internal pressure and axial ring load.For this example, the Ilyuschin yield condition is used (Ilyuschin, 1956). The obtainedshakedown domain is shown in Fig. 6, where also the results of undamaged behavior arepresented. Curves (a) and (b) correspond to the shakedown limit load due to incrementalcollapse for uniform cross-sections. We observe in the major part of the loading space asignificant di?erence between geometrical linear, curves (a), and geometrical nonlinear,curves (b), analysis and between damaged and undamaged behavior.IV.3. Example 3A simply supported conical shell with smaller radius Ri, external radius Re, wall-thick-ness h and angle is considered (Fig. 7). This kind of mechanical structure is found invalve systems. The shell is submitted to an axial ring load at the large radius edge andinternal pressure. Both internal pressure and axial ring load can vary independently withinthe bounds 0 ? Q ? Q?and 0 ? p ? p?. The following magnitudes of geometric dimen-sions were adopted in the numerical analysis: Ri=Re? 3=4, h=R ? 1=200 and ? 60?withthe same mechanical properties as example 2. The following initial values of loads areconsidered: Q0? 1:2:103N mm?1and p0? 1:6 MPa.It can be seen in Fig. 8 that the shakedown domain does not increase significantly dueto the consideration of geometric e?ects and damage has no great influence when the axialforce Q acts in the same axial direction as pressure p. However, the e?ects are moreimportant when these loads act in the opposite direction.V. CONCLUSIONThe presented extension of Melans statical shakedown theorem accounting for geo-metrical nonlinearities and plastic ductile damage and its numerical applications is meantto contribute to the improvement of global methods for the assessment of structuresunder variable loads beyond the elastic limit. Although the methodology of the extensionof the theorem is quite general, the class of materials considered in this paper is ratherrestricted: in particular, anisotropic damage evolution and more general models of inter-action between plasticity and damage seem to be important issues for further research.In the applications to thin-walled shells subjected to finite displacements and moderaterotations, analogously to Weichert (1986) and Gross-Weege (1990), only special loadingcases were considered, where external loads may vary randomly within a given range inthe neighborhood of a prescribed service-load. More general loading should be consideredin the future. The obtained results show in general a reduction of shakedown loads due toductile damage compared to the undamaged behavior. It appears that the considerationof geometrical nonlinearities can have a stabilizing or destabilizing e?ect. The combina-tion of an incremental analysis with the shakedown analysis seems to be promising; so oneobserves in the first of the presented examples that for certain loading cases, the defor-mations exceed the limit of moderate rotations so that the validity of the DonnelMush-tariVlasov theory becomes questionable.Altogether, the presented results indicate that this method is well adopted for thenumerical determination of the safety factor with respect to non-shakedown and inad-missible damage. Nevertheless, quite a number of open questions remains to be answered.Among them, a sound experimental validation of the proposed method is of majorimportance. Unfortunately, the authors observed a considerable lack of adapted andGeometrical nonlinearities and the shakedown of damaged structures905reliable experimental data in literature. Therefore, in the future, special research e?ortshould be put on experimental shakedown analysis in conjunction with theoretical andnumerical modeling.REFERENCESBathe, K. J., Wilson, E. L. and Iding, R. H. (1974) NONSAP: a structural analysis program for static anddynamic response of nonlinear systems. Report No. 31 UGSESM 74-3, University of California, Berkeley,CA.Chaboche, J. L. (1977) Sur lutilisation des variables de tat internes pour la description du comportement visco-plastique et de la rupture par endommagement. Symp. Franco-Polonais de Rhe ologie et de Me canique,Krakow.Chaboche, J. L. (1981) Continuous damage mechanicsa tool to describe phenomena before crack initiation.Nucl. Engng. Design 64, 233.Comi, C. and Corigliano, A. (1991) Dynamic shakedown in elastoplastic structures with general internal variableconstitutive laws. Int. J. Plasticity 7, 679.Corigliano, A., Maier, G. and Pycko, S. (1995) Dynamic shakedown analysis and bounds for elastoplasticstructures with nonassociative internal variable constitutive laws. Int. J. Solids Struct. 32, 3145.Feng, X. Q. and Yu, S. W. (1995) Damage and shakedown analysis of structures with strain-hardening. Int. J.Plasticity 11, 237.Gokhfeld, D. A. and Cherniavsky, O. F. (1980) Limit Analysis of Structures at Thermal Cycling. Sijtho? andNoordho?, Leyden, The Netherlands.Green, A. E. and Naghdi, P. M. (1965) A general theory of an elasticplastic continuum. Arch. Rat. Mech.Analys. 18, 251.Gross-Weege, J. (1988) Zum Einspielverhalten von Fla chentragwerken. IfM-Report, no. 58, Ruhr-Universita t,Bochum.Gross-Weege, J. (1990) A unified formulation of statical shakedown criteria for geometrically nonlinear pro-blems. Int. J. Plasticity 6, 433.Hachemi, A. (1994) Contribution a lAnalyse de lAdaptation des Structures Ine lastiques avec Prise en Comptede lEndommagement. Ph. 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