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INFLUENCE OF GEOMETRICAL NONLINEARITIES ON THE SHAKEDOWN OF DAMAGED STRUCTURES D. Weichert* and A. Hachemi Institut 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 account geometrical eC128ects and plastic ductile damage. Numerical results illustrate the proposed method. # 1998 Elsevier Science Ltd. All rights reserved Keywords: A. plastic collapse, shakedown, B. constitutive behavior, elasticplastic material, finite strain. I. INTRODUCTION The development of numerical methods for the assessment of the long-time behavior, the usability and safety against failure of structures subjected to variable repeated loading is of great importance in mechanical and civil engineering. A particular kind of failure is caused by an unlimited accumulation of plastic strains during the loading process, leading to either incremental collapse or alternating plasticity. If, on the contrary, after some time plastic strains cease to develop further and the accumulated dissipated energy in the whole structure remains bounded such that the structure responds purely elastically to the applied 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 elastic perfectly plastic structures. Both criteria presume the existence of a convex yield surface andthevalidityofthenormalityrulefortheplasticstrainrates.Moreover,theinfluencesof material hardening, geometrical eC128ects and material damage are neglected. Consequently, extensions of the classical shakedown theorems have attracted much interest in the latter years. 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), where unlimited 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 eC128ects and Ko nig and Siemaszko (1988) considered the eC128ects of strainhardening in shakedown International Journal of Plasticity, Vol. 14, No. 9, pp. 891907, 1998 # 1998 Elsevier Science Ltd Pergamon Printed in Great Britain. All rights reserved PII: S0749-6419(98)00035-7 0749-6419/98 $see front matter 891 *Corresponding author. process. With the help of generalized standard material model introduced by Halphen and Nguyen (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- surfacematerialallowingforlimitedkinematicalhardeningandapplieditnumerically.The concept of internal variables for the representation of the hardening material behavior was in thesequelalso applied by Comiand Corigliano (1991) andPolizzotto et al. (1991).More generalnonlinearhardeninghasbeeninvestigatedbyMaier(1969)inthecontextofdiscrete systems and byStein et al. (1993), who usedforthis purposetheso-calledoverlay model. Other applications of internal parameters representation of changes of material properties can be found in Corigliano et al. (1995) and Pycko and Maier (1995). The geometrically nonlinear problem has been studied firstly by Maier (1973), who introduced a new class of shakedown problems for pre-stressed discrete structures and extended Melans and Koiters theorems as to include so-called second order geometric eC128ects by using piecewise linear yield conditions. Siemaszko and Ko nig (1985) showed the influence of geometrical eC128ects 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 eC128ects 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 deformation pattern are available. He assumed an additive strains decomposition and applied it to shell-like structures undergoing moderate rotations at small strains (Weichert, 1989). The same decomposition of total strain has been used by Gross-Weege (1990). He gives unified formulation of Melans theorem for structures subjected to a constant load, responsible for large displacements, and to small additional variable loads causing small additional displacement. 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 in the framework of large displacements. They studied the asymptotic response of the structure subjected to periodically variable loads in order to show the conditions under which 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 decomposition rule 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-Weeges shakedown 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 to reformulate shakedown theorems stating that shakedown occurs if there exist some real self-equilibrated residual state, which is dependent on the loading and unloading paths. More recently, Saczuk (1997) proposed a criterion of adaptation process, accounting for the influence of deformation path on the material properties based on the Finslerian continuum model within the theory of diC128erential inequalities. Influence of material damage on the formulation of the static shakedown theorems was first investigated in a thermodynamical framework from theoretical point of view by Hachemi and Weichert (1992) using the energy-based isotropic elastoplastic damage models given by Ju (1989). The material damage is taken into account by an internal scalar-valued parameter using the concept of eC128ective stresses following Lemaitre and 892 D. Weichert and A. Hachemi Chaboche (1985). Feng and Yu (1995) adopted this concept and applied it to thick-walled shells by using a mathematical programming method to calculate an upper bound of the ductile 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 to the 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 discrete structures taken into account nonlinear geometrical eC128ects, nonlinear hardening and ductile damage by using the material softening function by Perzyna (1984). Recently, Hachemi and Weichert (1997) proposed how to deal practically with the diC128erence between plasticity and material damage in relation with shakedown theory taken into account kinematical hardening according to the concept of the generalized standard material model by Halphen and Nguyen (1975). The authors show how to control the degree of material damage by imposing locally bounds on the evolution of ductile damage parameter as defined by Lemaitre (1985). The numerical solution technique as well as several numerical examples of axisymmetric structures under combined mechanical and thermal 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. All kinds of defects in elastoplastic material are considered as damage and may be pre-existing or developing during service. The elastoplastic damage behavior of materials is introduced through the concept of the eC128ective stress within the framework of continuum mechanics. The primary purpose of this paper is to extend Melans shakedown theorem to damaged structures accounting for geometrical eC128ects. This extension is a combination of the formulations proposed by Hachemi and Weichert (1992) and Tritsch and Weichert (1993). Only for simplicity we restrict our considerations to elasticperfectly plastic material behavior and isotropic damage. An extension of the presented formulation to hardening 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 eC128ects of geometrical changes due to deformation, the multiplicative decomposition of the total deformation gradient into an elastic and plastic parts as proposed by Lee (1969) is used for the development of the theoretical framework. Section II of this paper is devoted to the 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 an internal scalar variable. By the definition of the thermodynamic potential and from the second thermodynamics principle, we deduce the dissipative inequalities. A simple three- dimensional model of ductile plastic damage established by Lemaitre (1985) is used. This model is linear with equivalent plastic strain and quadratic with triaxiality ratio. In Section III, an extension of the static shakedown theorems is proposed, taking into account the influence of ductile plastic damage and geometrical nonlinearities. Strains are decomposed into elastic and plastic parts without using any simplifying assumptions. For that, 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 formulation however, delivers constructive methods for shakedown analysis, if additional assumptions on the deformation pattern are introduced (Weichert, 1986). Here, the most simple case is Geometrical nonlinearities and the shakedown of damaged structures 893 studied, where the considered body or structure is subjected to initial loads, inducing large displacements and initial damage suchthatitisin the reference configuration inequilibrium. The body is then subjected to additional variable in time or cyclic loading, causing small additional displacement, in comparison with the previous ones, and additional damage. In this case, the response of the reference configuration is calculated incrementally by using the NONSAP finite element program developed by Bathe et al. (1974). The lower bound of the load factor against failure due to non-shakedown or inadmissible damage is calculated by optimization program using the algorithm LPNLP developed by Pierre and Lowe (1975). In Section IV, results for axisymmetrical shells are presented showing the influence of finite displacements and damage on the shakedown behavior in comparison to results for undamaged materials and geometrical linear analysis. II. FORMULATION OF THE PROBLEM We consider the behavior of a three-dimensional elasticperfectly plastic bodyBunder the action of quasistatically varying external agencies a * consisting of surface tractions p * and surface displacements u * acting on the disjoint parts S p and S u of the surface S ofB, respectively, and volume forces f * . In the initial configuration C i at the time t 0, B occupies the volume V 0 . The motion ofBis described by the use of Cartesian coordinates, where the positions of the particles ofBin the undeformed and deformed state are given by the coordinates X X 1 ;X 2 ;X 3 and x x 1 ;x 2 ;x 3 , respectively. The actual config- uration C t ofBis 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 undeformed configuration is defined by: (i) Statical equations Div T f in V 0 n:T p on S p 2 with T FS 3 (ii) Kinematical equations u u on S u F I grad u in V 0 E 1 2 C I in V 0 4 with C F T F 5 Here, T and S are the unsymmetric first PiolaKirchhoC128 stress tensor and the symmetric second PiolaKirchhoC128 stress tensor whereas F and E are the deformation gradient and the GreenLagrange strain tensor, respectively. I denotes the metric tensor of second rank and n is the outer normal vector to S in C i . 894 D. Weichert and A. Hachemi Elasticplastic deformations are usually described by means of a fictitious intermediate configuration C , derived from the multiplicative decomposition of the deformation gra- dient F into an elastic part F e and a plastic part F p (Lee, 1969): F F e F p 6 where F e is obtained by unloading all infinitesimal neighborhoods of the body B. This decomposition provides the relation between elastic, plastic and total deformation valid for finites strains and leads to an additive decomposition of the GreenLagrange strain tensor E into a purely plastic part E p and an elastic part E e depending on the plastic deformation (Green and Naghdi, 1965): E E e E p 7 with E e 1 2 F p T F e T F e I F p and E p 1 2 F p T F p I 8 Here, the theory of thermodynamics with internal variables is used to derive the con- stitutive laws. For this, a local thermodynamic potential C9 is introduced assumed to be quadratic in E e and linear in 1 D (Chaboche, 1977, 1981; Lemaitre and Chaboche, 1985; Simo and Ju, 1987; Ju, 1989): C9 E E p ;D 1 D C9 0 E E p 9 with C9 0 1 2 0 L : E e : E e 10 where C9 0 is the energy function of the undamaged (virgin) material, L the tensor of elas- ticityand 0 theinitialmassdensity.Theoperator(:)standsfordoubletensorcontraction.Inthe formoftheClausiusDuheminequality,the2ndprincipleofthermodynamicsthenrequires S : _ E p 0 and Y _ D 0 11 with S 0 C9 E e 1 D L : E e 12 Y 0 C9 D 1 2 L : E e : E e 13 Hence, the thermodynamic force (Y) conjugate to the damage variable D is the energy function of the undamaged material C9 0 (Lemaitre and Chaboche, 1985). Superposed dots denote the rate of the considered quantity. To describe the plastic part of the material damage behavior, we assume the existence of a convex elastic domain, defined by the yield condition F S; F 0; 14 Geometrical nonlinearities and the shakedown of damaged structures 895 with S as second PiolaKirchhoC128 eC128ective stress tensor S S= 1 D and F as yield stress. In the sequel superposed tilda indicates quantities related to the damaged state of the material. Convexity of the yield surfaceF and validity of the normality rule can then be expres- sed by the maximum plastic work inequality S S s : _ E p 0 15 where S s is any safe state of stress defined by F S s ; F 1 and a time-independent state of eC128ective residual stresses S r , satisfying the following relations: Div F x S r 0 in V 0 n: F x S r 0 on S p F S e S r ; F t R the bodyBis submitted to additional variable loads a r* such that: a X;t a R X a r X;t 25 and occupies the actual configuration C t (see e.g. Weichert, 1986; Gross-Weege, 1990; Saczuk and Stumpf, 1990; Pycko and Ko nig, 1991). Since the actual configuration should also be an equilibrium configuration and the following equations hold: (i) Statical equations Div T R T r f R f r in V 0 n: T R T r p R p r on S p 26 with T R T r F r F R S R S r 27 (ii) Kinematical equations u u R u r in V 0 F F r F R I grad u R grad u r in V 0 E E R E r 1 2 C I in V 0 u u R u r on S u 28 with C F r F R T F r F R 29 where all quantities caused by the time-independent loads a R* are marked by a superscript ( R ), whereas the additional field quantities caused by the time-dependent loads a r* are marked by superscript ( r ). The additional field quantities caused by a r* have to satisfy the following equations: (i) Statical equations 898 D. Weichert and A. Hachemi Div T r f r inV 0 n:T r p r on S p 30 with T r H r F R S R F R S r H r F R S r 31 (ii) Kinematical equations F r I H r in V 0 E r 1 2 F R T H r T H r H r T H r F R in V 0 u r u r on S u 32 with H r grad R u r 33 Inthesequel,werestrictourconsiderationstoloadinghistoriescharacterizedbythemotionof a fictitious comparison bodyB (c) ,havingattimet R the same field quantities asBbutreacting, incontrasttoB,purelyelasticallytotheadditionaltime-dependentloadsa r* ,superimposedon a R* for tt R (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 bodyB (c) , no additional plastic strains and no damage can occur. The dif- ferences between the states inBandB (c) are then describedby the diC128erence fields: u u r u r c ; F F r F r c ; E E r E r c T T r T r c ; S S r S r c 34 and have to fulfill the following equations: Div T 0 in V 0 n: T 0 on S p 35 Geometrical nonlinearities and the shakedown of damaged structures 899 Fig. 2. Evolution of real bodyBand comparison bodyB (c) . and F H r H r c in V 0 E 1 2 F R T F T F F R 1 2 F R T H r T H r H r c T H r c F R in V 0 u 0 on S u 36 with T F F R S R F R S H r F R S r H r c F R S r c 37 In the following, we restrict our considerations to situations where the state of deforma- tion and the state of stress inBare 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 eC128ects induced by the additional time-dependent loads. Then the following extension of Melans theorem holds: If there exists a time-independent field of eC128ective residual stresses S such that the following relations hold: i Div F R S R f R inV 0 n: F R S R p R on S p u u R on S u 38 ii Div T 0 in V 0 n: T 0 on S p u 0 on S u 39 with T F F R S R F R S iii F S r c S R S ; F t R , then the original body B will shakedown under given program of loading a * . Material damage however by definition cannot grow indefinitely and is limited by local rupture. Therefore, it is necessary to control the degree of material damage by imposing locally bounds on the evolution of the damage parameter D r induced by additional vari- able loads a r* . For the considered approach (eqn (17), damage is basically generated by plastic strains, bounds for damage can be given in this special case by bounding of the equivalent plastic strains (for detail we refer to Hachemi, 1994; Hachemi and Weichert, 1997): p 1 V 0 1 1 D c F 1 V 0 1 2 S : L 1 : S dV 0 41 900 D. Weichert and A. Hachemi Then the safety factor against failure due to non-shakedown or inadmissible damage is defined by SD max _ T;D 42 with the subsidiary conditions Div _ T 0 in V 0 43 n: T 0 on S p 44 D R D r D c 0 in V 0 45 F S r c S R S ; F 0 in V 0 46 This is a problem of mathematical programming, with as objective function to be opti- mized with respect to T and D T F F R S R F R S ;D D R D r and with inequalities (45) and (46) as nonlinear constraints. The condition (45) assures structural safety against failure due to