Plastic ductile damage evolution and collapse of plates and shells

Transcription

1 Plastic ductile daage evolution and collapse of plates and shells I. Kreja 1 & R. Schidt 2 1 Technical University of Gdansk, Poland 2 Aachen University of Technology, Gerany Abstract This paper deals with odelling and siulation of plastic ductile daage evolution in thin-walled structures and its effect on the load-carrying behaviour in the geoetrically and physically non-linear range of deforation. In this context, gradual stiffness degradation, daage evolution, local failure initiation, and final collapse of the structure are probles of priary interest. The approach adopted accounts for elastic-plastic aterial behaviour, isotropic as well as kineatic hardening, aterial daage, finite displaceents and finite rotations. Finite eleent siulations illustrate the effect of aterial daage on the load carrying behaviour of thin-walled structures. 1 Introduction The accurate deterination of the ultiate load carrying capacity of thin-walled structures in the geoetrically and physically non-linear range of deforation is of crucial interest for safe design of structural coponents in any probles of advanced technology, e.g. in echanical, aerospace, and civil engineering. In this context recent results of aterial science in the field of daage echanics concerning the initiation and progression of aterial daage should be used for odelling and siulation of the load-carrying behaviour of structures. The coplex interaction of the different types of non-linearity, e.g. large deflections and rotations, elastic-plastic hardening aterial behaviour, and gradual stiffness degradation due to aterial daage evolution, poses a considerable proble for the nuerical siulation of structures close to failure. The ajority of papers on aterial daage evolution in structural ebers deal with uniaxial tension and copression of bars or plane probles of flat aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN

2 92 aage and Fracture Mechanics VIII panels subjected to in-plane loads, while only a relatively sall nuber of papers treat the load-carrying behaviour of plates and shells including the effects of aterial daage, see e.g. [1-4] for geoetrically linear, [5-8] for sall strain, geoetrically non-linear, and [9-14] for finite strain elastoplasticity approaches. In the present paper the plastic ductile daage odel of Leaitre and Chaboche [15,16] is adopted for the geoetrically non-linear odelling and FE siulation of plates and shells. The aterial odel accounts for von Mises-type plasticity with associated flow rule, isotropic and kineatic hardening, and isotropic daage. 2 Modelling of ductile daage and its evolution Gradual stiffness degradation is described by an internal variable easuring the reduction of the stress carrying surfaces by icroscopic daage of the aterial, s. [15-17]. In the particular case of isotropic daage considered in the present paper, icrocracks and volue cavities are assued to be uniforly distributed in all directions, so that a scalar paraeter can be used to describe the state of daage. The daage paraeter is defined as the ratio of daaged to total area of a surface eleent ~ d A d A =, (1) d A where d A denotes a surface eleent and d ~ A its effective stress carrying area, respectively, in any configuration C. Obviously, is equal to zero for ~ the undaaged virgin state of the aterial, i.e. d A = d A. At a critical value c, the initiation of acroscopic fracture is observed. The value = 1, i.e. ~ d A = 0, corresponds to a totally disconnected surface eleent. With the noinal Cauchy stress tensor τ ij in any configuration C being referred to the area eleents of a defored volue eleent, and an effective Cauchy stress tensor τ ij being referred to the actual resisting area of the volue eleent, Eq. (1) along with the relation between the Cauchy and second Piola-Kirchhoff stress tensors S ij leads to ij ij S S =. (2) 1 Hooke`s law along with Eq. (2) yield the uniaxial linear elastic law of the daaged aterial 11 S = Eε 1 11, (3) aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN

3 aage and Fracture Mechanics VIII 93 where E is Young`s odulus. Eq. (3) can be written as with ~ S = Eε, (4) ~ E = E (1 ). (5) This relation allows for an interpretation of E ~ as the elastic odulus of the daaged aterial. If Young`s odulus E is known, any easureent of E ~ in uniaxial loading-elastic unloading tests can be used to deterine the daage paraeter fro Eq. (5), which yields ~ E = 1. (6) E Generalizing Eq. (3) for the 3 case, the following constitutive law relating the stresses to the elastic part of the strains is adopted for the configuration C : S ij ( E) ( ) C ε ( E) 1 ijkl kl =. (7) Here, Cijkl ( E) is the usual elastic constitutive tensor of the undaaged, virgin aterial. For an isotropic aterial, we have E ijkl ( ) ( ) C = λ δ δ + µ δ δ + δ δ, (8) with λ and µ standing for Lae s coefficients ij kl ik jl il jk λ = Eν ( 1+ ν )( 1 2ν ), E µ = G =. (9) 2 1 ( + ν ) Fro Eq. (2) and Eq. (7) one gets the expression for the increent of the effective second Piola-Kirchhoff stress tensor as ~ 2 ~ 1 ~ ( E) ( E) S = S S = C ε. (10) ij ij ij The daage criterion can be written in ters of the accuulated equivalent (P) plastic strain ε EQ copared with the daage threshold strain ε. In a general, ultiaxial stress state such a coparison should take into account a correction ijkl kl aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN

4 94 aage and Fracture Mechanics VIII coefficient expressing the stress triaxiality. The following ductile plastic daage criterion has been used in this paper for any configuration C, s. [15,16]: EQ γ ε ε 0, (11) where the stress triaxiality coefficient γ is 2 ( 1 ) 3 ( 1 2 ) S H γ = + ν + ν. (12) 3 S EQ According to this odel, the daage paraeter is supposed to increase linearly with the plastic strain. This is well in accordance with the experiental results for any etallic aterials subjected to onotonic loading, [15,16]. In this case, the evolution of daage can be described by C ( γ ε ε ) γ d ε d = H EQ EQ. (13) ε ε Here, C and ε R stand for the critical value of and the strain, respectively, at the initiation of acroscopic fracture, and H denotes the Heaviside function. R 2 3 Yield criterion, flow rule, isotropic and kineatic hardening for the daaged aterial The plastic yield criterion of von Mises, odified to describe the plastic state for the daaged aterial in configuration C is used in the following for: F 1 ( σ, ) = σ ε : 1 ( σ ) 2 = 0 EQ 2 σ. (14) 3 S Here, σ denotes the deviatoric part of the reduced stress tensor in C given in ters of the effective second Piola-Kirchhoff stress tensor and the backstress tensor α as σ= S ~ α, (15) thus accounting for the effects of aterial daage and kineatic hardening, while σ is the actual yield stress. A ixed hardening rule consisting of a S aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN

5 aage and Fracture Mechanics VIII 95 linear cobination of isotropic and kineatic hardening, [18,19], is used, the partition of which is governed by the ixed hardening paraeter M: M = 0: kineatic hardening, 0 < M < 1: ixed hardening, (16) M = 1: isotropic hardening. The power-law for of the hardening function, [20], is adopted, where the actual yield stress in any configuration C is related to the effective plastic strain by 1/ M P y ( ) σ = σ S S ε EQ = σ + MK Y Y ε EQ. (17) Here, σ y stands for the initial yield stress, K Y and M are aterial Y coefficients. The Prager-Ziegler odel of kineatic hardening is used, with the rate of the back stress tensor α being defined as ( 1 M ) d σ d α = µ, (18) see e.g. [19]. The proportionality factor d µ depends on the deforation history represented by the value of the equivalent plastic strain ( ε ) dµ = dµ. (19) According to an associative flow law, the norality rule has been adopted to describe the evolution of the equivalent plastic strain during plastic flow, i.e. for F = 0 and df = 0 EQ F d ε = d Λ, (20) S where d Λ is the plastic ultiplyer to be deterined. Additionally, the hypothesis of plastic incopressibility has been assued: plastic strain occurs at constant volue and plastic flow is independent of the hydrostatic stress, i.e. Tr ( d ) = d ε = 0 With Eq. (21b), Eq. (20) can be written as F ε, = 0. (21) kk σ d ( ) σ H ε P = d Λ. (22) 1 aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN

6 96 aage and Fracture Mechanics VIII Based on the above assuptions in [21] the increental linearized constitutive relation for elastic-plastic daage analysis was deterined in the for θ EP S C ( ) : ε. (23) 4 Nuerical approach Based on the principle of virtual work a finite eleent algorith for geoetrically and physically nonlinear analysis of 2 shell probles in total Lagrangian description was developed (see [21] for details) for probles with axial syetry (e.g. shells of revolution), plane strain (e.g. slender cylinders or pipes) and plane stress (e.g. beas, fraes, arches). The finite eleent type used throughout this paper is the one-diensional 3- or 4-node degenerated isoparaetric eleent with quadratic or cubic interpolation, respectively, and corresponds to the odels considered e.g. in [22, 23]. The basic kineatic assuption of this approach is that lines noral to the id-surface reain straight but not noral to the defored id-surface. Reduced integration technique is applied to eliinate ebrane and shear locking. The Newton- Raphson ethod is applied for equilibriu iterations and the nonlinear equilibriu path is traced increentally using the Riks-Wepner arc-length control ethod. The finite rotation forulation based on the trigonoetric representation of rotational degrees of freedo results in an additional geoetric stiffness atrix, [24,25]. Follower-type pressure loading is included in the present algorith in a siilar way as in [26]. For details of the eleent forulation and solution strategy we refer to [27,28]. 5 Nuerical results Plastic ductile daage initiation and evolution as well as localisation of critical daage is deonstrated by the plane strain analysis of a long cylindrical shell with radius R = 120 and thickness 0,4. A segent of 20 is claped on both straight edges and subjected to a uniforly distributed hydrostatic pressure. The aterial paraeters for the steel are: Young s odulus E = MPa, Poisson s ratio ν =0.3, initial yield stress σ Y =138 MPa, hardening law coefficients K Y = 435 MPa, M Y = 4.55, critical value of the daage paraeter C = 0.24, strain at initiation of acroscopic fracture ε R =0.37, daage threshold strain ε =0, isotropic strain hardening, i.e. M = 1 is assued. The nuerical analysis was carried out with twenty 4-node degenerated isoparaetric eleents. Fig. 1c shows the load-displaceent diagra obtained by elastic-plastic ductile daage analysis. This predicts local failure initiation and final collapse of the structure in the region A which is conceptionally not included in any elastic- aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN

7 aage and Fracture Mechanics VIII 97 plastic (no daage) analysis. In Fig. 1b results of plastic ductile daage analysis perfored with four different schees of nuerical integration through the thickness of the shell are copared. First, calculations are perfored with five and seven Gauss integration points, respectively, across the thickness. Those integration points, where the daage paraeter reaches its critical value, are excluded fro the evaluation of the eleents stiffness atrix. Figure 1: Load deflection curve, crack initiation and failure of a long claped cylindrical shell subjected to onotonic pressure loading. aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN

8 98 aage and Fracture Mechanics VIII Figure 2: Localisation and evolution of critical daage in a long claped cylindrical shell subjected to hydrostatic pressure loading. In the odel adopted here, such points possessing zero stiffness indicate the onset of localised failure. One can observe two sharp drops in each of the loaddeflection curves. Each of these jups correspond to the failure of one of the integration points in the cross section close to the claped edge of the shell. In fact each of these sharp drops represents a jup to the load-deflection curve of the daaged shell. Based on these observations, one can expect that with a refined discretization in noral direction it should be possible to obtain a soother graph in the section of the equilibriu path with critical plastic ductile daage. It can be seen fro Fig. 1b that a discretization by sixteen integration layers is required to achieve a converged solution. The evolution of critical daage in the shell predicted on the basis of the sixteen-layers odel is shown in Fig. 2. Each of the depicted configurations shows a section of the defored shell in the vicinity of the claped boundary, aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN

9 aage and Fracture Mechanics VIII 99 the plastic zone, and points with critical daage in the respective load step. The first deforation profile corresponds to the last increental step before the initiation of a acroscopic crack. The first copletely daaged point appears in the next increental step accopanied by a sudden redistribution of plastically loaded and elastically unloaded zones in the vicinity of the crack onset. The axiu load level is obtained in step No. 24, when five integration points are copletely daaged. In the next step elastic unloading occurs in alost all undaaged integration layers followed by final collapse of the structure. References [1] Cocks, A.C.F.; Leckie, F.A. : ASME J. Appl. Mech. 55 (1988), [2] Bodnar, A. ; Chrzanowski, M. : J. of Theoretical and Applied Mechanics 32 (1994), [3] Kleiber, M.; Kollann, F.G. : Archives of Mechanics 45 (1993), [4] Feng, X.-Q.; Yu, S.-W. : Int. J. of Plasticity 11 (1995), [5] Krätzig, W.B.; Gruber, K.; Zahlten, W.; Zhuang, Y.: Proc. ICCES '91, Melbourne, [6] Altenbach, H.; Nauenko, K.: Proc. 1st Int. Conf. on Mechanics of Tie ependent Materials, Ljubljana 1995, eds. I. Eri, W.G. Knauss, SEM Inc., Ljubljana, [7] Altenbach, H.; Nauenko, K.: Coputational Mechanics 19 (1997), [8] Altenbach, H.; Morachkovsky, O.; Nauenko, K.; Sychov, A.: [9] Weichert,.; Hachei, A. : Int. J. of Plasticity 14 (1998), [10] Ki, S.J.; Ki, W..: ASME J. Appl. Mech. 61 (1994), [11] Zhu, Y.Y.; Cescotto, S.: in: Structures under Shock and Ipact II, ed. P.S. Bulson, , Coputational Mechanics Publications, Southhapton- Boston, [12] Zhu, Y.Y.; Cescotto, S.: Int. J. Solids Structures 32 (1995), [13] Fornefeld, W.: Mitteilungen aus de Institut für Mechanik Nr. 73, Ruhr- Universität Bochu, [14] Mittelbach, M.: Mitteilungen aus de Institut für Mechanik Nr. 100, Ruhr-Universität Bochu, [15] Leaitre, J.: Cop. Meth. Appl. Mech. Engng. 51 (1985), [16] Leaitre, J.; Chaboche, J.-L.: Mechanics of Solid Materials, Cabridge University Press, Cabridge 1990, translation of Mécanique des atériaux solides, Bordas, Paris [17] Kachanov, L.M.: Izv. Akad. Nauk. SSR, Otd. Tekh. Nauk., No. 8 (1958), [18] Hughes, T.J.R.: in: Theoretical Foundations for Large Scale Coputations of Nonlinear Material Behavior, 29-57, Northwestern University [19] Chen, W.F. Plasticity for Structural Engineers, Springer-Verlag, New York [20] Osgood, R.; Raberg, W.: NACA TN No. 902 (1947). aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN

10 100 aage and Fracture Mechanics VIII [21] Kreja, I.; Schidt, R.; Weichert,. : Archive of Applied Mechanics 71 (2001), [22] Hughes, T.J.R.; Liu, W.K. : Cop. Meth. Appl. Mech. Eng. 27 (1981), [23] Surana, K.S.: Int. J. Nu. Meth. Engng. 18 (1982), [24] Frey, F.; Cescotto, S.: Proc. Int. Conf. on Finite Eleents in Nonlinear Solid and Structural Mechanics, Geilo, Norway, 1977, vol. 1, [25] Ra, E.; Matzeniller, A.: Finite Eleent Methods for Plate and Shell Structures, vol. 1: Eleent Technology, eds. T.J.R. Hughes and E. Hinton, Pineridge Press Ltd., Swansea 1986, [26] Schweizerhof, K.; Ra, E.: Coputers & Structures 18 (1984), [27] Kreja, I.; Cywinski, Z.: Coputers & Structures 41 (1991), [28] Kreja, I.; Schidt, R.; Teyeb, M. ; Weichert,. : Cahiers de Mécanique 1/2-94, Laboratoire de Mécanique de Lille, aage and Fracture Mechanics VIII, C. A. Brebbia & A. Varvani-Farahani (Editors) 2004 WIT Press, ISBN