INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2006; 68:338–380 Published online 3 April 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1711 Elasto-plastic ﬁnite element analysis of shells with damage due to microvoids Pawel Woelke1, ‡, ¶ , George Z. Voyiadjis2, ∗, † and Piotr Perzyna3, § 1Weidlinger Associates, Inc., Applied Science Department, 375 Hudson Street, 12 FL, New York, NY 10014-3656, U.S.A. 2 Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803-6405, U.S.A. 3 Institute of Fundamental Technological Research, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland SUMMARY This paper presents a non-linear ﬁnite element analysis for the elasto-plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The reﬁned shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41:3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C 0 quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the reﬁned theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto-plastic model for shells presented by Voyiadjis and Woelke (General non-linear ﬁnite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek-Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD-Vol. 183/MD-50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic ﬂow. The anisotropic damage effects, the inﬂuence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34:1089–1104) is used to derive the large rotation, elasto-plastic-damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the reﬁned theory for thick spherical ∗ Correspondence to: George Z. Voyiadjis, Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803-6405, U.S.A. † E-mail: voyiadjis@eng.lsu.edu ‡ E-mail: woelke@wai.com § E-mail: pperzyna@ippt.gov.pl ¶ Previously at Louisiana State University, Baton Rouge, LA 70803-6405, U.S.A. Contract/grant sponsor: Air Force Institute of Technology, WPAFB; contract/grant number: F33601-01-P-0343 Copyright 䉷 2006 John Wiley & Sons, Ltd. Received 18 September 2005 Revised 7 February 2006 Accepted 13 February 2006 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 339 shells. 2006, submitted). In addition, a non-layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto-plastic-damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright 䉷 2006 John Wiley & Sons, Ltd. KEY WORDS: thick plates and shells; elasto-plastic analysis; kinematic hardening; large displacements; ductile damage analysis; microvoids; isotropic damage 1. INTRODUCTION 1.1. Motivation and scope Shells are very important structures for both onshore and offshore engineering applications. Analysis and design of these is therefore continuously of interest to the scientiﬁc and engineering communities. Accurate and conservative assessments of the maximum load carried by the structure, as well as the equilibrium path in both elastic and inelastic ranges are therefore of paramount importance. Although still there is room for improvement, the elastic behaviour of shells has been very thoroughly investigated, mostly by means of the ﬁnite element method. Inelastic analysis on the other hand, especially accounting for damage effects, received much less attention from researchers [1–3]. One of the major difﬁculties in computations for the inelastic behaviour of structures is the fact that they are based on incremental and/or iterative algorithms, which may require prohibitively large storage in the computer. Thus, computational efﬁciency needs special attention in non-linear modelling of shells. Voyiadjis and Woelke [4] presented a general, accurate and very efﬁcient ﬁnite element model for the elasto-plastic analysis of thick/thin, isotropic plates and shells, including geometric non-linearities, isotropic and kinematic hardening rules and featuring an explicit form of the tangent stiffness matrix. The objective of the present work is to extend the work of Voyiadjis and Woelke [4] to account for the damage effects due to growth of microvoids. Damage is modelled here as an isotropic process induced by the plastic ﬂow. Static loading conditions are considered, with both plasticity and damage treated as rate-independent processes. 1.2. Shell theory The problem of shell constitutive equations could be avoided by following a layered approach, also referred to as ‘through-the-thickness-integration’ [5–8]. This procedure is conceptually close to the analysis of shells by means of three-dimensional, solid ‘brick’ elements, and is best suited for composite plates and shells. In the case of isotropic materials, the layered approach, although accurate, is unnecessarily complicated and computationally expensive. A non-layered ﬁnite element approach, on the other hand, requires a general and accurate shell theory. Voyiadjis and Woelke [4] presented a reﬁned theory for thick shells, which was developed based on analytical closed-form solutions for thick containers. This theory proves to be very efﬁcient in the treatment of both thin and thick shells of general shape, and is therefore adopted in this work. It accounts for the effect of transverse shear deformation and distribution of radial stresses. These are very important features for the thick shells. In addition, the initial curvature effect is included, which not only contributes to the stress resultants and stress couples, but also leads to non-linear distribution of the in-plane stresses across the thickness Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 340 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA of the shell. The resulting constitutive equations offer very good approximations for extremely thick (R/t = 3) and very thin shells of general shape, as well as for plates and beams. A brief outline of the theory is provided in Section 2. 1.3. Shell ﬁnite element The numerical implementation of the aforementioned theory provides a shell ﬁnite element [9, 10], which is a very effective tool for the elastic analysis of the structures under consideration. The quasi-conforming technique given in Reference [11] is an extension of the assumed strain ﬁelds method [12, 13], and it has been successfully applied to overcome the shear and membrane locking phenomena [14, 15]. The appropriate choice of the strain ﬁelds provides an adequate representation of the rigid-body modes and allows one to avoid spurious energy modes. The biggest advantage of this technique, when compared with the most widely used selective integration technique [16–20] is the fact that the stiffness matrix of the element is given explicitly. Thus, this method is very attractive for non-linear calculations because the element matrices are evaluated many times during the analysis. Moreover, the selective integration technique requires an explicit segregation of transverse shear terms from bending and membrane terms, which is not possible when a coupling between these exists, as is mostly the case for non-linear analysis [21]. This problem was solved by a generalization of the selective integration procedure [16]. The quasi-conforming technique is chosen here for its simplicity and low computational cost. As a result of that, and application of the non-layered approach, numerical integration is not performed in the present procedure at any stage of the analysis. All the integrals are calculated analytically, and later introduced into a computer code. This makes the current formulation consistent mathematically and extremely efﬁcient from the point of view of computer time and power. 1.4. Geometrical non-linearities Geometrical non-linearities are crucial in the elasto-plastic and damage modelling of shell [2]. Displacements at the regions of the structure, which undergo inelastic deformations, can be very large. Thus, to achieve the desired accuracy, geometric non-linearities must be considered. The Updated Lagrangian description, which has proven to be a very effective method [4, 6, 22–25] is adopted here. The element local co-ordinates and the local reference frame are continuously updated during the deformation. We consider large rotations and rigid translations here, but small strains with the total rotations decomposed into large rigid rotations and moderate relative rotations. The relative rotations and the derivatives of the in-plane displacements from two consecutive conﬁgurations may be considered small [24, 25]. Consequently, the quadratic terms of the derivatives of the in-plane displacement are negligible. We therefore have a non-linear analysis with large displacements and rotations, but small strains. The transformation matrix given in Reference [26] is employed to handle large rigid rotations. The assumed strain ﬁnite element with an explicit form of the stiffness matrix, as described above, provides the linear part of the element tangent stiffness matrix. 1.5. Material non-linearities—plasticity The experimental results [27–29] show that the degradation of material properties of ductile metals in the elastic range due to the damage effects is negligible. Hence, the damage is Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 341 considered here as a phenomenon induced by the plastic strain. A reliable elasto-plastic procedure is needed in order to perform damage. In the layered plastic model a plate or a shell is divided into layers where stresses are calculated and the yield condition is checked at each layer separately. The forces and moments are then calculated by integration through the thickness. Although this method can give very accurate results, it can also be very demanding in terms of computational power. If, on the other hand, a ‘non-layered’ approach is adopted, the yield function is expressed in terms of the stress resultants and couples. Numerical integration of the stresses is not necessary in this case, which makes the ‘non-layered’ formulation much cheaper computationally. Voyiadjis and Woelke developed a very accurate, non-layered elasto-plastic model for shells with an isotropic and a new kinematic hardening rule [4, 10]. An Iliushin’s yield function [30] is employed in this work, modiﬁed to account for the progressive development of the plastic curvatures across the thickness of the shell, as shown in Reference [31], and the transverse shear forces, which may signiﬁcantly affect the plastic behaviour of both thick and, for certain loading conditions, thin shells. The evolution of damage is directly linked to the plastic strain, hence all the factors that affect plastic behaviour, are also very important for the damage analysis. Both isotropic and kinematic hardening rules are deﬁned, with the latter correctly representing the rigid translation of the yield surface during the non-elastic deformation in the stress resultant space, and thus capturing the Bauschinger effect. The stiffness matrix in Reference [4] is derived by means of the principle of virtual work and the plastic node method [32], which considers the inelastic deformations to be concentrated in the plastic hinges. This method originates from the analytical limit analysis of structures performed under the assumption of elastic-perfectly plastic behaviour of the material [33–35]. Following the work of Shi and Voyiadjis [31, 3] the plastic node method is adopted here to derive the elasto-plastic-damage stiffness matrix of the element. The explicit form of the stiffness matrix is therefore preserved. 1.6. Damage analysis A ductile metal or structure is capable of undergoing large inelastic deformations. The plastic strains can induce changes in the microstructure of the material, leading to its softening. These changes in the microstructure of the material are irreversible thermodynamic processes and result in the progressive degradation of the material properties [3]. The experimental investigations [36–38] show that the softening of the material triggered by inelastic strains is mainly due to the nucleation, growth and coalescence of microvoids and microcracks (sometimes thermal effects are also pronounced) [39, 40]. This process is called ductile plastic damage. Damage in the elastic region is mostly negligible in ductile materials. Modelling of damage is aimed at the assessments of the inﬂuence of microvoids, microcracks and other microdamages on the degradation of the material properties. The investigations of the damage accumulation and evolution can be carried out following a micromechanical approach (micromechanical damage models) or continuum damage theory (phenomenological damage model). The latter approach is based on the pioneering work of Kachanov [41], who introduced the effective stress concept, as well as a scalar damage variable representing the effective surface density of microdamages per unit volume [42, 43]. The effective stress concept involves comparison of the actual damaged conﬁguration with the ﬁctitious undamaged conﬁguration [41, 44]. Many authors used a phenomenological approach as a basis for modelling of damage [44–56]. Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 342 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA An isotropic scalar damage parameter, based on the concept of Kachanov [41], was used by many authors [48, 57, 58]. In this method, the stiffness of the material is reduced according to the same relation in all the directions. For a better description of the anisotropic effects, a second-order damage tensor, capable of representing different levels of material degradation in different directions is often employed [43–45, 50–57, 59–63]. An anisotropic damage variable poses however the problem which is not often addressed. For the appropriate depiction of directional dependency of the evolution of damage, it is necessary to determine the material constants, which deﬁne the evolution laws in different directions. Extensive experimental data are needed to calibrate these constants with sufﬁcient accuracy and consistency. The isotropic damage formulation requires determination of fewer constants (two in the case of the current analysis), while at the same time it delivers very accurate results for a variety of structural applications. Moreover, it would be unrealistic to include in the investigation of structures all effects observed experimentally on the level of the material behaviour. Constitutive modelling is understood as a reasonable choice of effects, which are the most important for explanation of the phenomenon described [40]. For the current work, concerning the investigation of the behaviour of isotropic plates and shells, the use of the isotropic scalar parameter in the representation of damage is deemed satisfactory. The effects of anisotropy of damage are not accounted for here. The validity of these assumptions is veriﬁed by the discriminating numerical examples. Micromechanical damage models are based on the observations of the material at the microscale. The observations of ductile fracture in metals [64] lead to a conclusion that this process can involve the generation of considerable porosity through nucleation and growth of voids [65]. Gurson developed a mathematical model [65, 66] describing the damage effects through the evolution of porosity, which was incorporated into the yield function. He investigated a yield criterion and ﬂow rule for porous ductile materials. Various modiﬁcations of Gurson’s formulation appeared later in the literature [67], as well as the articles discussing the model [68, 69]. Further investigations of ductile fracture aimed at explanation of the formation of white-etching bands, commonly referred to as shear bands. A general conclusion from the experimental results by Giovanola [70] is that the thermomechanical strain localization and microdamage mechanisms become the main co-operative phenomena responsible for adiabatic shear band formation and localized fracture [40]. Based on the microscopic observations of the shear bands [71], it was found that the fracture preceded by the shear band formation, occurred through nucleation, growth and coalescence of voids. An extensive study of the shear bands and fracture phenomena, followed by the development of microdamage model by means of the porosity function, was performed by Duszek-Perzyna et al. [72–78], and Perzyna [40, 79–88]. Duszek-Perzyna and Perzyna presented a theoretical formulation for the description of the intrinsic microdamage process through evolution of the isotropic scalar damage variable, i.e. the porosity parameter [76]. Similar to Gurson’s model [65, 66], the porosity was incorporated directly into the yield function, obtaining a consistent and convenient procedure for the elasticviscoplastic damage analysis of ductile solids, with a coupling between plasticity and damage. The evolution of porosity reduced to a rate-independent case, consisted of three terms responsible for the cracking of the second-phase particles, debonding of the second-phase particles from the matrix material, and the void growth assumed to be controlled only by plastic ﬂow phenomena. The ﬁrst term (cracking of the second-phase particles) was only dependent on the stress, which allowed for variation of damage, even without occurrence of the plastic ﬂow. This made the formulation universal and capable of describing correctly the material behaviour under all loading conditions, including the hydrostatic stress. In the present paper, the elastic Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 343 damage is regarded negligible. In addition, we consider only damage resulting from microvoids, neglecting the effects of microcracks. The porosity parameter deﬁned by Duszek-Perzyna and Perzyna [76] is used here to describe damage effects in shells. We only consider a rate-independent case here, and the evolution of porosity, which as previously mentioned, accounts for the cracking of the second-phase particles, debonding of the second-phase particles from the matrix material, and void growth is reduced to represent void growth only, since we investigate isotropic plates and shells. The effects of the microcracks are not considered in this work. Since void growth is a phenomenon induced by the plastic deformation, elastic damage is neglected here. The yield function given in Reference [76], which could be directly related to Gurson’s model [65, 66], is expressed in terms of the stress resultants and stress couples, similar to Iliushin’s yield function [30], following the procedure outlined by Bieniek and Funaro [89]. The yield surface derived here is very similar to the one presented by Voyiadjis and Woelke [4], with kinematic hardening parameters in the form of residual normal and shear forces, and residual bending moments. It is however enhanced here to account also for damage effects, leading to the reduction of the stiffness, by means of the porosity parameter. The current formulation is an attempt to deliver a very simple and convenient way of a detailed analysis of shells. It is, at the same time, mathematically consistent and produces accurate results. One of the biggest advantages of this work is its simplicity and computational efﬁciency. The stiffness matrix is given here explicitly, and calculated without performing numerical integration. This is due to the application of the quasi-conforming technique in derivation of the elastic stiffness matrix, where all the integrations are computed analytically. In this non-linear analysis, the non-layered and plastic node methods are employed, with the yield surface deﬁned in the stress resultant space, and the damage parameter incorporated into a yield function. This approach is very advantageous from the point of view of structural analysis. The validity of the assumptions and the derivation presented here is veriﬁed through a series of discriminating examples. We solve a plate and a spherical shell problem focusing mainly on the representation of damage as this is the most important feature of the current work. This paper is divided into eight sections. After the Introduction, the shell constitutive equations are brieﬂy introduced. In Section 3, we present shell kinematics. Section 4 is devoted to the linear stiffness matrix of the shell element. Section 5 gives a description of material non-linearities, with the deﬁnition of the porosity function as a scalar damage parameter, the yield surface, the ﬂow and hardening rules. The elasto-plastic-damage stiffness matrix of the element is derived in Section 6. In Section 7 we present an outline of a numerical procedure and discriminating examples, demonstrating that the current computational model gives good results for a variety of problems in elasto-plastic and damage analysis of shells and plates. Finally, in Section 8 we summarize the results and draw the conclusions. 2. SHELL CONSTITUTIVE EQUATIONS Voyiadjis and Woelke [4] presented a detailed derivation of the shell constitutive equations adopted for the ﬁnite element formulation. Only the ﬁnal set of relations is given here for self-completeness. The reﬁned theory accounts for the effect of transverse shear deformation, distribution of radial stresses and initial curvature of the shell, which results in a non-linear distribution of the in-plane stresses across the thickness of the shell. Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 344 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA The main features of the shell equations are the following: (1) assumed out-of-plane stress components that satisfy the given traction boundary conditions. These are due to a closed-form elasticity solution for thick walled spherical containers under internal and/or external uniform pressure, obtained by Lame [90]; (2) three-dimensional elasticity equations with an integral of the equilibrium equations; and (3) stress resultants and stress couples acting on the middle surface of the shell together with average displacements along a normal of the middle surface of the shell and the average rotations of the normal [91]. The membrane strains and curvatures in a rectangular co-ordinate system (x, y, z) are given by Equations (1)–(6). x = *u w + R *x (1) *v w + R *x 1 *u *v + xy = 2 *y *x * *w *x u = − xz − x = R *x *x *x y = y = *y = * *y *w v − yz − R *y *y *y 1 *x + xy = 2 *y *x (2) (3) (4) (5) (6) where x , y , xy are normal and shear strains and x , y , xy are curvatures at the midsurface in planes parallel to the xz, yz and xy planes, respectively; u, v, w are the displacements along the x, y, z axes, respectively (Figures 2, 3); xz , yz are transverse shear strains in xz and yz planes (Figure 1); x , y are angles of rotations of the cross-sections that were normal to the midsurface of the undeformed shell; R is a radius of a shell. The stress resultants and couples Mx , My , Mxy , Nx , Ny , Nxy , Qx , Qy shown in Figure 1, can be expressed in terms of the strains given above as follows: Mx = D[x + y ] (7) My = D[y + x ] (8) Mxy = D(1 − )xy (9) Nx = S[x + y ] (10) Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 345 Figure 1. Stress resultants on a shell element. Figure 2. Local co-ordinate system and normal vector es3 . Figure 3. Incremental degrees of freedom of the shell element in local co-ordinates. Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 346 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA Ny = S[y + x ] (11) Nxy = S(1 − )xy (12) Qx = T xz (13) Qy = T yz (14) where D= Eh3 , 12(1 − 2 ) S= Eh , (1 − 2 ) T= 5 Eh 12 (1 + ) (15) and E is the Young’s Modulus, h is the thickness of the shell, and is the Poisson’s ratio. These constitutive equations reduce to those given by Flugge [92] when the shear deformation and radial effects are neglected. We use the above equations to formulate the coupled strain energy density and derive the corresponding stiffness matrix of the element. 3. SHELL KINEMATICS As in the case of the plastic analysis of shells by Voyiadjis and Woelke [4], the Updated Lagrangian method is employed in the present study of large displacements and rotations of the shell element. The co-ordinates of the nodal points are continuously updated during the deformation. The rotations are additively decomposed into large rigid rotations and moderate relative rotations [25]. The structure under consideration is deﬁned in the global, ﬁxed co-ordinate system X. We also have the local co-ordinate system x, surface co-ordinates at any nodal point xs , and base co-ordinates, which serve as a reference frame for the global degrees of freedom (Figure 2). • In order to obtain the unit vector in the direction normal to the plane of the element, in → → the local co-ordinate system, we ﬁrst deﬁne two vectors, 41 and 42 connecting the origin of the co-ordinate system (point 4) to points 1 and 2, respectively. The cross-product of these two vectors, divided by its length, gives e3 , as shown in Figure 3 and given by Equation (16): → → 41 × 42 e3 = → → | 41 × 42| (16) The unit vector e2 can be similarly obtained as a cross-product of e3 and e1 . We can now determine the relation between the global co-ordinates X and element local co-ordinates in conﬁguration k: k e = k RE (17) where k e is the unit base vector of the local co-ordinates in conﬁguration k, E is the unit base vector of the global co-ordinates; R is a transformation matrix from local to global co-ordinates. Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 347 • The surface co-ordinate system xS originates at each node of the element. As deﬁned by Shi and Voyiadjis in Reference [25], the position and direction of this system are functions of rotations. Surface co-ordinates translate and rigidly rotate with the element. Consequently, xS3 is always normal to the surface of the element. The ﬁnite rigid-body rotation vector V is given by ⎡ ⎤ 1 ⎢ ⎥ ⎥ (18) V= ⎢ ⎣ 2 ⎦ 3 where 1 , 2 , 3 are rigid-body rotations around x, y, z axes, respectively. The transformation matrix of large rotations T , given by Argyris [26] is used here: ˜ T = exp() (19) with: ˜ = ˜ij = eij k k , k = 1, 2, 3 (20) where ˜ is a skew symmetric matrix and eij k is the permutation tensor. In the above equation, the indicial notation is used with Einstein’s summation convention. The transformation of the surface co-ordinates is therefore V = T V (21) where V is a rigid-body rotation vector transformed into a new position. Similarly, we can write a transformation of the surface co-ordinates for a given rotation vector j resulting from conﬁguration k − 1 to k at node j : k es = Tk−1 j es (22) where k es are the unit base vectors of the surface co-ordinates at conﬁguration k. Deﬁning the transformation between E and k es as k es = k R s E (23) we can rewrite Equation (22) as follows: k k k Tk k k es = Tk−1 j Rs E = Rs R e = Sj e (24) where k RT is the transpose of k R deﬁned in Equation (17) and k Sj is a transformation matrix from local to the surface co-ordinate system. It is worthwhile to note that 0 Rs is a 3 × 3 identity matrix for a ﬂat plate. • The base co-ordinates as deﬁned as by Horrigmoe and Bergan [23] are adopted here as a common reference frame to which all element properties are transformed, prior to the assembly of the stiffness matrices. The base co-ordinates are deﬁned by the combination of the ﬁxed global and base co-ordinates. Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 348 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA The global degrees of freedom at node j are the incremental translations: Uj , Vj , Wj in directions of global co-ordinates X, Y, Z and rotations xj , yj around xS , yS . The local degrees of freedom at node j are the incremental translations uj , vj , wj in directions of local co-ordinates x, y, z and rotations xj , yj around x, y, respectively. The transformation of the increments of the displacements at node j from the local co-ordinate system qej , to the corresponding base co-ordinates, qbj can be written as ⎧ ⎧ ⎫ ⎫ uj ⎪ Uj ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v Vj ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k T ⎨ ⎬ ⎬ R 0 ⎨ w W (25) qbj = = = k Tbj qej j j k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 s j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ xj ⎪ xj ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ yj yj in which k sj is the upper left 2 × 2 submatrix of k Sj deﬁned in Equation (24). The transformation matrix for the nodal displacement vector can be written as qb = k Tb qe (26) where k Tb is composed of k Tbj with j = 1, 2, 3, 4. The vector of the local increments of nodal displacements is shown in Figure 3 and is given by Equation (27): qej = {uj , vj , wj , xj , yj }T j = 1, 2, 3, 4 (27) 4. LINEAR ELEMENT STIFFNESS MATRIX An accurate and efﬁcient shell ﬁnite element was presented by Woelke and Voyiadjis [9]. It is an assumed strain type of element, free from locking and spurious energy modes. The quasi-conforming technique [11] was used which gives an explicit form of the stiffness matrix, as integrations are carried out directly. The strain ﬁelds in the element are interpolated as follows: • Linear bending strain ﬁeld: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ *x ⎫ ⎧ *x x ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ *y y b = = ⎪ ⎪ ⎪ *y ⎪ ⎪ ⎭ ⎪ ⎪ ⎩ ⎪ ⎪ 2xy ⎪ ⎪ *y * ⎪ ⎪ ⎩ x + *y *x ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎡ 1 x y xy ⎢ =⎢ ⎣ 0 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎭ 0 0 0 0 1 x y xy 0 0 0 0 ⎧ ⎫ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎤⎪ ⎪ ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎥ 3 ⎬ 0 0 0⎥ = P b b ⎦⎪ ⎪ ⎪ ···⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 x y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 10 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 11 (28) Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS • Stretch strain ﬁeld: ⎧ ⎫ *u w ⎪ ⎪ ⎪ ⎪ ⎫ ⎪ ⎪ *x + R ⎪ ⎪ ⎧ ⎡ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ *v ⎨ w⎬ ⎢ + y m = = =⎢ ⎣0 ⎪ ⎪ ⎪ R ⎪ *y ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ 0 2xy ⎪ ⎪ ⎪ *u *v ⎪ ⎪ ⎪ ⎩ ⎭ + *y *x y 0 0 0 1 x 0 0 0 • Constant transverse shear strain: ⎧ ⎫ u *w ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ *x − x − R ⎪ ⎬ xz 1 s = = = ⎪ ⎪ *w v⎪ yz ⎪ 0 ⎪ ⎩ ⎭ − y − ⎪ R *y 0 1 ⎧ ⎫ 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎤⎪ ⎪ ⎪ 13 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎥⎨ ⎥ 0 ⎦ 14 = Pm m ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ 15 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 16 17 18 349 (29) = P s s (30) where 1 , 2 , . . . , 18, are the undetermined strain parameters. Let P be the trial function for the assumed strain ﬁeld, i.e.: = P (31) and N, the corresponding test function. We multiply both sides by the test function and integrate over the element domain: T N d = NT P d (32) The strain parameter is determined from the quasi-conforming technique as follows: = A−1 Cq (33) where q is the element nodal displacement vector given by Equation (27), and NT P d A= (34) Cq = NT d (35) We may now express the strain ﬁeld in terms of the nodal displacements as follows: = P = PA−1 Cq = Bq (36) It is convenient to take P = N in order to obtain a symmetric stiffness matrix. This is the case adopted in this formulation. Both matrices A and C can be easily evaluated explicitly. Illustration of this procedure is given in References [9–11]. We therefore obtain b = Pb A−1 b Cb q = Bb q Copyright 䉷 2006 John Wiley & Sons, Ltd. (37) Int. J. Numer. Meth. Engng 2006; 68:338–380 350 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA m = Pm A−1 m Cm q = Bm q (38) 1 Cb q = Bs q (39) s = where Bb , Bm , Bs are the strain displacement matrices related to bending, stretch and transverse shear deformation, respectively. The element stress resultants and stress couples given by Equations (7)–(14) can be rewritten in terms of the strain ﬁelds b , m , s : ⎧ ⎫ ⎡ Mx ⎪ 1 ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ M = My =D⎢ ⎣ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 Mxy ⎧ ⎫ ⎡ Nx ⎪ 1 ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ N = Ny =S ⎢ ⎣ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 Nxy Q= Qx Qy =T 0 1 0 0 1 − /2 0 1 0 0 1 − /2 1 0 0 1 ⎤ ⎥ ⎥ b = Db ⎦ (40) ⎤ ⎥ ⎥ m = Sm ⎦ (41) s = Ts (42) where D, S, T are ﬂexural, membrane and shear rigidities, respectively. In order to determine the stiffness matrix of the element we make use of the strain energy density, expressed as follows: U = 21 (Mx x + My y + 2Mxy xy + Nx x + Ny y + 2Nxy xy + Qx xz + Qy yz ) (43) Substituting Equations (1)–(14) into the above expression we obtain the following: U = Ub + Um + Us (44) where Ub , Um , Us are, respectively: the bending component of the strain energy density function (quadratic function of curvatures), the membrane component (quadratic function of membrane strains) and the transverse shear component of the strain energy. Using Equations (1)–(15) and (28)–(30) we may write the strain energy quantities Ub , Um , Us in the matrix forms as follows: ⎡ 1 ⎢ 1 Ub = Tb D ⎢ ⎣ 2 0 Copyright 䉷 2006 John Wiley & Sons, Ltd. 0 1 0 0 1 − /2 ⎤ ⎥ ⎥ b = 1 T Db ⎦ 2 b (45) Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS ⎡ 1 ⎢ 1 Um = Tm S ⎢ ⎣ 2 0 1 Us = Ts T 2 0 1 0 0 1 − /2 1 0 0 1 ⎤ ⎥ ⎥ m = 1 T Sm ⎦ 2 m 1 s = Ts Ts 2 The total strain energy e in the element domain may be written as 1 (T Db + Tm Sm + Ts Ts ) d e = 2 b or using Equations (37)–(39): 1 e = qT 2 351 (46) (47) (48) (BTb DBb + BTm SBm + BTs TBs ) dq (49) which leads to e = 21 qT [Kb + Km + Ks ]q (50) where Kb , Km , Ks are the element stiffness matrices related to bending, stretch, and transverse shear deformation, given by Kb = BTb DBb d (51) Km = BTm SBm d (52) BTs TBs d (53) Ks = The elastic element stiffness matrix is then given by K = Kb + Km + Ks (54) 5. YIELD CRITERION AND HARDENING RULE As discussed in the Introduction, a yield criterion for porous metals, expressed in terms of the stress resultants and couples is used here, similar to the Iliushin’s yield function modiﬁed to account for the shear forces, as given in Reference [31], the progressive development of the plastic curvatures, and damage caused by growth of voids. The Iliushin’s yield function F can be written as follows: F= M2 N2 1 |MN | Y (k) + 2+√ − 2 =0 2 M N M0 N0 0 3 0 0 Copyright 䉷 2006 John Wiley & Sons, Ltd. (55) Int. J. Numer. Meth. Engng 2006; 68:338–380 352 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA or F= |M| N 2 Y (k) + 2 − 2 =0 M0 N0 0 (56) where 2 N 2 = Nx2 + Ny2 − Nx Ny + 3Nxy (57) 2 M 2 = Mx2 + My2 − Mx My + 3Mxy (58) 2 MN = Mx Nx + My Ny − 21 Mx Ny − 21 My Nx + 3Mxy M0 = 0 h2 , 4 N0 = 0 h (59) (60) and 0 is the uniaxial yield stress, Y (k) is a material parameter, which depends on isotropic hardening parameter k, h is the thickness of the shell, and |.| denotes the absolute value. The form of the yield condition given by Equation (55), can be easily derived from the von Mises function and the deﬁnition of normal stresses at the top and the bottom surfaces of the shell, as is shown in Reference [89]. Instead we use in this work the yield criterion for porous ductile metals as originally proposed by Gurson [65, 66], and later modiﬁed by Perzyna [80] and Dornowski and Perzyna [93]. Although it is of a form similar to von Mises equation, it accounts for the isotropic damage effects through the dependence of the ﬁrst invariant of stress and the evolution of porosity. The plastic potential function deﬁned by Dornowski and Perzyna [93] can be written as 3 (61) Sij Sij + n 2ii , i, j = 1, 2, 3 f= 2 where Sij is deviatoric stress tensor given by Sij = ij − 13 kk (62) ij ij is a stress tensor given by ij = Nij 6Mij ± 2 h h (63) where Nij are normal forces; Mij are bending moments, h is a thickness of the shell and ij is a Kronecker delta. The parameter n in Equation (61) is a material constant, determined (for ductile metals) by Perzyna [80]: n = 1.2587. The symbol in Equation (61) is a porosity parameter given by Gurson [65, 66] and modiﬁed by Duszek-Perzyna and Perzyna [76]: p p = k1 ii + k2 ij ij + k3 ii (64) where k1 , k2 , k3 denote the material constants, and p are the increments of stress and plastic strain, respectively. The ﬁrst two terms in the above equation are responsible for nucleation due to the cracking of the second-phase particles, and debonding of the second-phase Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 353 particles from the matrix material, respectively. The third term depicts the growth of voids, and is controlled only by the plastic ﬂow. The main term in the current work is the growth term. We may assume that from the metallurgical investigations of the isotropic materials comprising a plate or a shell, we can determine the initial porosity (t = 0) = 0 , and we shall consider only the growth term in the evolution of porosity, i.e.: p = k3 ii (65) It is very important to determine or, in the absence of sufﬁcient experimental data, to assume the initial level of porosity in the virgin material. If nucleation is accounted for in the description of damage, then it is possible to assume 0 = 0, which corresponds to a situation in which there are no pores in the virgin material. Even though it is not a very realistic assumption, since certain level of porosity exists in the undeformed material, through the representation of nucleation in the damage model we could recognize the opening of voids at a certain level of stress. The expansion of voids leading to localization and fracture may be approximated by means of the growth term. In the current work however, the damage representation is reduced to void growth only, hence the initial ﬁnite value of porosity must be determined or assumed. Equations (61)–(65) are written using the indicial notation and a summation convention. Rewriting Equation (65) in engineering notation yields: p p p = k3 (x + y + z ) p p (66) p where x ; y ; z are increments of the normal plastic strains due to both membrane and p p bending actions in the x, y, z directions, respectively. x and y may be written as follows: p p p p p p p p p p x = mx + bx = mx + zx (67) y = my + by = my + zy p p where mx and my are the increments of plastic strains due to the membrane action only, p p in the x, y directions; bx and by are the increments of plastic strains due to the bending action only, in the x, y directions; z is the distance from the mid-plane to the plane under p p consideration; and x , y are the increments of plastic curvatures at the midsurface in planes parallel to the xz, yz planes, respectively. The maximum normal plastic strain caused by bending will occur at z = h/2 which leads to p p p p h p x 2 h p + y 2 x = mx + y = my (68) p Substituting Equations (68) into Equation (66) and neglecting z we obtain h p p p p = k3 mx + my + (x + y ) 2 (69) We now proceed to the determination of the plastic potential function expressed in terms of the stress resultants and couples. For the purpose of conciseness, we neglect radial and Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 354 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA transverse shear stresses in the current derivation. However, transverse shear forces are later introduced into the yield condition. Equation (61) can be written using the engineering notation: 1 (70) f = √ [(x − y )2 + 2x + 2y + 6 2xy + n (x + y )2 ] 2 where x , y are the normal stresses in the x, y directions, respectively, and stress on the xy plane. We can deﬁne the yield condition as follows: 1 √ [(x − y )2 + 2x + 2y + 6 2xy + n (x + y )2 ] = 0 2 xy is a shear (71) where 0 denotes the uniaxial yield stress. Substituting Equations (63) into (71) and performing some mathematical manipulations result in the following relation: N2 M2 NM + ±2 =1 2 2 N N0 M0E 0 M0E (72) where 2 N 2 = 1 + 21 n (Nx2 + Ny2 ) − (1 − n )Nx Ny + 3Nxy 2 M 2 = 1 + 21 n (Mx2 + My2 ) − (1 − n )Mx My + 3Mxy (73) 1 N M = 1 + 21 n (Nx Mx + Ny My ) − (1 − n )(My Nx + Mx Ny ) + 3Nxy Mxy 2 and N0 = 0 h, M0E = 0 h2 6 (74) Both the top and the bottom surfaces of the shell should be considered to obtain the larger value of the term ±2(N M/N0 M0E ). We can ensure representation of the most negative effect by writing Equation (72) in the following form [89, 94]: N2 M2 |NM| + +2 =1 2 2 N0 M0E N0 M0E (75) The yield surface given above is very similar to Iliushin’s yield function [30] given by Equation (55). In order to derive Equation (55) we follow the procedure outlined by Bieniek and Funaro [89], which is essentially the surface ﬁtting approach. We write Equation (75) as follows: a M2 |NM| N2 + b +c =1 2 2 N0 M0E N0 M0E Copyright 䉷 2006 John Wiley & Sons, Ltd. (76) Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 355 We determine the parameters a, b, c by considering the special loading cases separately. If we account for membrane forces only, we see that for a = 1 we obtain the exact limit condition. Similarly, if we take a pure bending case, Equation (76) will produce exact results for 2 /M 2 . To ﬁnd c we investigate the loading case corresponding to the maximum value b = M0E 0 of the ratio N M/N0 M0E which occurs if Nx = Ny , Mx = My and Nxy = Mxy = 0. The stress distribution in the cross-section in this case is as shown in Figure 4. Based on the stress distribution in Figure 4, we can calculate the normal force: Nx = h/2 −h/2 x dz = √ −h/2 3 −h/2 −0 dz+ h/2 −h/2 √ 0 h 0 dz = √ 3 3 (77) Using Equation (74) we may write Nx2 1 = N02 3 (78) Similarly, we may obtain 4M02 M2 = 2 2 M0E 9M0E and √ M0 NM =2 3 N0 M0E 9M0E (79) Substitution of Equations (78)–(79) and previously determined parameters a = 1 and 2 /M 2 into Equation (76), yields b = M0E 0 2 4M 2 √ M0 1 M0E 0 + 2 3c =1 + 2 2 3 9M0E M0 9M0E (80) M0E c= √ 3M0 (81) which leads to Substituting the parameters a, b, c into Equation (76) we arrive at the limit yield surface as deﬁned by Iliushin: F= N2 1 |MN | M2 + 2+√ =1 2 M0 N0 3 M0 N0 (82) The stress intensities are given by Equation (73) and unlike the original Iliushin yield function, they account for the damage effects. Voyiadjis and Woelke [4] introduced several other modiﬁcations to the Iliushin yield surface for a better description of the plastic behaviour of shells. The damage variable is a function of the plastic ﬂow here, which makes the accuracy of the representation of plastic behaviour very important. The same modiﬁcations of the yield function are therefore adopted in this work. We can include the transverse shear forces Qx , Qy by expanding one of the stress intensities given in Equation (73), cf. References [4, 31]: 2 N 2 = 1 + 21 n (Nx2 + Ny2 ) − (1 − n )Nx Ny + 3(Nxy + Q2x + Q2y ) (83) Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 356 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA √ Figure 4. Stress distribution corresponding to maximum NM/N0 M0E ( = h/2 3). It was shown previously [4, 31] that the inﬂuence of the shear forces on the plastic behaviour of thick plates and shells may be very important. For a bending dominant situation, according to Equation (55) or (56), the structure will deform linearly until the whole cross-section is plastic, i.e. the plastic hinge has formed. In reality however, the plastic curvature develops progressively from the outer ﬁbres of the shell or plate and the material behaves non-linearly as soon as the outer ﬁbres start to yield. To account for the development of plastic curvature across the thickness, Crisﬁeld [95] introduced a plastic curvature parameter (¯ p ), into Equation (82): F= M2 N2 1 |MN| Y (k) + +√ − 2 =0 2 2 2 M N M0 N0 0 3 0 0 (84) |M| N2 Y (k) + 2 − 2 =0 M0 N0 0 (85) or F= where was chosen such that M0 follows the uniaxial moment–plastic curvature relation = 1 − 13 exp − 83 ¯ p (86) and ¯ p = Eh p p p p p ¯ p = √ ((x )2 + (y )2 + x y + (xy )2 /4)1/2 30 p p (87) p The symbol ¯ p is the equivalent plastic curvature, and x , y and xy are the increments of the plastic curvatures. We note that for ¯ p = 0, = 2/3 and we obtain M0 = 0 t 2 /6 which represents ﬁrst ﬁbre yielding. If, on the other hand, ¯ p = ∞, = 1 and we obtain a fully plastic cross-section. Therefore, through the introduction of the plastic curvature parameter we account for the progressive development of the plastic curvatures and correctly predict the ﬁrst yield. We note that a material parameter Y (k), was employed in Equations (84)–(85) which depends on the isotropic hardening parameter k, similar to Equations (55)–(56). Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 357 To model the elasto-plastic behaviour of shells subjected to reversing loads, one needs a reliable kinematic hardening rule. Bieniek and Funaro [89] introduced residual bending moments (‘hardening parameters’), allowing for the description of the Bauschinger effect. These were later successfully applied to dynamic [94] and viscoplastic dynamic analysis of shells [96, 97]. To determine correctly the rigid translation of the yield surface in the stress resultant space, we need not only residual bending moments, but also residual normal and shear forces. Voyiadjis and Woelke [4] presented a new kinematic hardening rule for shells, with residual bending moments and residual normal and shear forces as kinematic hardening parameters, related directly to the backstress given by Armstrong and Frederick [98], and representing the centre of the yield surface in the stress resultant space. Adopting that same hardening rule in the current paper, we express the yield surface as follows: F∗ = |M ∗ | (N ∗ )2 Y (k) + − 2 =0 M0 N02 0 (88) where (N ∗ )2 = 1 + 21 n [(Nx − Nx∗ )2 + (Ny − Ny∗ )2 ] − (1 − n )(Nx − Nx∗ )(Ny − Ny∗ ) ∗ 2 ) + (Qx − Q∗x )2 + (Qy − Q∗y )2 ] + 3[(Nxy − Nxy (89) (M ∗ )2 = 1 + 21 n [(Mx − Mx∗ )2 + (My − My∗ )2 ] ∗ 2 − (1 − n )(Mx − Mx∗ )(My − My∗ ) + 3(Mxy − Mxy ) (90) ∗ , N ∗ , N ∗ , N ∗ , Q∗ , Q∗ are the above-described residual bending moments, and Mx∗ , My∗ , Mxy x y xy x y normal and shear forces, respectively. It is worthwhile to mention that by setting the porosity parameter to zero, i.e. = 0, the yield surface given by Equations (88)–(90) reduces to the one given by Voyiadjis and Woelke [4], where the damage effects are not considered. Detailed derivation of the kinematic hardening parameters is presented in Reference [4]. We only brieﬂy discuss the concept in the present paper. For the purpose of conciseness, we use the indicial notation in the derivation, and only the ﬁnal result is given employing the engineering notation. Armstrong and Frederick’s evolution of the backstress ij is given by ij p = cij − a p ij eq where a and c are constants and the equivalent plastic strain increment is given by p p p eq = 23 ij ij (91) (92) The backstress tensor represents the centre of the translated yield surface in the stress space. It has the same dimension as the stress tensor. To compute the stress resultants we need to integrate the stresses over the thickness of the shell. We use the same deﬁnition here to derive the hardening parameters, which represents the centre of the yield surface in the stress resultant Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 358 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA space. We therefore need to integrate the backstress over the thickness of the plate or shell, in order to obtain the residual normal and shear forces, and the bending moments. The deﬁnitions of the increments of the hardening parameters are as follows [4]: Nij∗ = Mij∗ = h/2 −h/2 h/2 −h/2 ij dz (93) ij z dz (94) Substituting Equation (91) into Equations (93)–(94) and after some mathematical manipulations we obtain the deﬁnition of the increments of the kinematic hardening parameters in the engineering notation as follows: If F ∗ = 1 and ∇F ∗ > 0 (plastic loading) N0 1 p p Nx∗ = 1 (1 − F ) x − Nx∗ eq 0 h N0 1 p p Ny∗ = 1 (1 − F ) y − Ny∗ eq 0 h N0 1 ∗ p p ∗ xy − Nxy eq Nxy = 1 (1 − F ) 0 h N0 1 ∗ p p ∗ xz − Qx eq Qx = 1 (1 − F ) 0 h N0 1 ∗ p p ∗ yz − Qy eq Qy = 1 (1 − F ) 0 h M0 6 ∗ p p ∗ Mx = 2 (1 − F ) x − 2 Mx eq 0 h M0 6 ∗ p p ∗ My = 2 (1 − F ) y − 2 My eq 0 h M0 6 ∗ p p ∗ xy − 2 Mxy eq Mxy = 2 (1 − F ) 0 h (95) (96) If F ∗ < 1 and ∇F ∗ 0 (unloading or neutral loading) (97) ∗ Nx∗ = Ny∗ = Nxy = Q∗x = Q∗y ∗ = Mx∗ = My∗ = Mxy =0 The parameters 1 and 2 in the above formulation control the membrane-force–membranestrain and moment–curvature relations. A value 1 = 2 = 2.0 was found to be of sufﬁcient accuracy in the representation of shells. Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 359 Figure 5. Yield surface on Nx Mx plane—interpretation of kinematic hardening parameters O is the centre of the translated yield surface. We therefore arrive at a ﬁnal form of the yield function for ductile porous metals, given by Equations (88)–(90) and (95)–(97), expressed in terms of the stress resultants and couples, with both isotropic and kinematic hardening rules. This is a very convenient form of the yield surface for the analysis of shells accounting for the damage effects through the evolution of porosity. A graphic representation of the yield surface on the Nx Mx plane with = 1 and Y = 20 is shown in Figure 5. Point O denotes the transferred centre of the yield surface. 6. EXPLICIT TANGENT STIFFNESS MATRIX The plastic node method is employed here in the derivation of the stiffness matrix, i.e. the plastic deformations and damage are considered to be concentrated in the plastic hinges. The yield function is only checked at each node of the ﬁnite elements. If the combination of stress resultants satisﬁes the yield condition, that node is considered to be plastic, which triggers the void growth, as the porosity is a function of the plastic ﬂow. Thus, in this method the inelastic deformations are only considered at the nodes, while the interior of the element remains always elastic. When node i of the element becomes plastic, the yield function takes the form Fi∗ (Ni , Qi , Mi , Ni∗ , Q∗i , Mi∗ , ki , i ) = 0 (98) where ⎧ ⎫ Nx ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ Ni = Ny ; ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ Nxy Copyright 䉷 2006 John Wiley & Sons, Ltd. Qi = Qx Qy ; ⎫ ⎧ Mx ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ Mi = My ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ Mxy Int. J. Numer. Meth. Engng 2006; 68:338–380 360 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA ⎧ ∗⎫ Nx ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ∗ ∗ N ; Ni = y ⎪ ⎪ ⎪ ⎭ ⎩ ∗ ⎪ Nxy Q∗i = Q∗x Q∗y ⎧ ∗⎫ Mx ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ∗ ∗ M Mi = y ⎪ ⎪ ⎪ ⎭ ⎩ ∗ ⎪ Mxy (99) At the same time the stress resultants must remain on the yield surface, i.e. the consistency condition must be satisﬁed: *F ∗ *F ∗ *Fi∗ *Fi∗ *Fi∗ *Fi∗ ∗ ∗ dMi + i dNi + i dQi + dN + dQ∗i ∗ dMi + i *Mi *Ni *Qi *Ni∗ *Q∗i *M i + *Fi∗ *F ∗ dki + i di = 0 *ki *i (100) We assume an additive decomposition of strains into elastic and plastic parts: = e + p (101) The associated ﬂow rule is used here to determine the increments of plastic strains: p x = NPN i=1 i *Fi∗ *Mxi and p x = NPN i=1 i *Fi∗ *Nxi (102) where NPN is the number of plastic nodes in the element and di is a plastic multiplier. The remaining increments of the plastic strains are obtained in the same way. The plastic strain ﬁelds are interpolated as in the linear elastic analysis (Equations (28)–(30)) given here in the incremental form: ⎧ ⎧ p ⎫ p ⎫ x ⎪ x ⎪ ⎪ ⎪ p ⎪ ⎪ ⎪ ⎪ ⎬ ⎬ ⎨ ⎨ xz p p p p p b = (103) , m = , s = y y p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ yz ⎩ ⎩ p ⎭ p ⎭ 2xy 2xy The evolution of the porosity parameter representing damage is given by Equation (69) repeated here for convenience: h p p p p (104) = k3 x + y + (x + y ) 2 The assumption of an additive decomposition of strains can be extended to displacements provided that the strains are small [31, 32]. Although geometric non-linearities are taken into account in the current work, we only consider large rigid rotations and translations, but small strains. Thus, we may write q = qe + qp (105) Following the work of Shi and Voyiadjis [31] we approximate the increments of plastic p displacements by the increments of plastic strains. The plastic rotation x is a function of p p both x and xy , as can be deduced from Equation (28). Assuming that the increment of Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 361 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS p plastic nodal rotation xi is proportional to the increment of elastic nodal rotation xi we can express the former as 2 p p p xi xi = lim x + 2xy dx dy →0 2xi + 2yi i = i *Fi∗ *Fi∗ 22xi + 2 2 *Mxi xi + yi *Mxyi (106) where i represents the inﬁnitesimal neighbourhood of node i. The vector of the incremental nodal plastic displacements of the element at node i can be then expressed as follows: p qi = ai i (107) with ai given by aiT = *Fi∗ *Fi∗ *Fi∗ *Fi∗ *Fi∗ *Fi∗ + pu ; + pv ; + ; *Nxi *Nxyi *Nyi *Nxyi *Qxi *Qyi *Fi∗ *Fi∗ *Fi∗ *Fi∗ + p x ; + py *Mxi *Mxyi *Myi *Mxyi pu = 2u2i u2i + vi2 ; pv = 2vi2 u2i + vi2 ; px = (108) 22xi 2xi + 2yi ; py = 22yi 2xi + 2yi Equations (107) and (108) indicate that the plastic displacements at the nodes are only functions of the stress resultants at this node [31]. Therefore, we may write the vector of the increments of the nodal plastic displacements, as follows: ⎫ ⎤⎧ ⎡ 0 1 ⎪ a1 0 ⎪ ⎪ ⎪ ⎬ ⎥⎨ ⎢ ⎥ 0 a 0 qp = ⎢ = a (109) i i ⎦⎪ ⎣ ⎪ ⎪ ⎪ ⎩ ⎭ 0 0 aNPN NPN In order to determine the tangent stiffness matrix of the element we deﬁne b , m , s as virtual elastic bending, membrane and transverse shear strains, respectively ( -virtual), and M, N, Q as stress couples and stress resultants of the element. We also make use of the linearized equilibrium equations of the system at conﬁguration k + 1 in the Updated Lagrangian formulation, expressed by the principle of the virtual work, which in ﬁnite element modelling takes the form: ( Tb Db + Tm Sm + Ts Ts ) dx dy + Tk F dx dy =k+1 R − ( Tb k M + Tm k N + Ts k Q) dx dy Copyright 䉷 2006 John Wiley & Sons, Ltd. (110) Int. J. Numer. Meth. Engng 2006; 68:338–380 362 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA where k+1 R is the total external virtual work at step k + 1 and is the slope vector and k F is a membrane stress resultant matrix at step k given as follows: ⎧ ⎫ *w ⎪ ⎪ ⎪ ⎪ k ⎪ ⎨ *x ⎪ ⎬ Nx k Nxy k = (111) F= , k k ⎪ *w ⎪ ⎪ ⎪ N N xy y ⎪ ⎪ ⎩ ⎭ *y The slope ﬁeld is evaluated in a similar way to the strain ﬁelds, using the quasi-conforming technique [11]. A bilinear interpolation is used as in Reference [25] to approximate the slope ﬁeld: ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ 1 x y xy 0 0 0 0 ⎨ 3 ⎪ = = P (112) .. ⎪ ⎪ ⎪ 0 0 0 0 1 x y xy ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 7⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ 8 with P denoting the trial function matrix and is a vector of undetermined parameters, calculated in the same way as the vectors of strain parameters used to approximate the strain ﬁelds (Equations (28)–(30)): = A−1 Cqe , A = PT P dx dy, Cqe = PT dx dy (113) The slope ﬁeld is therefore expressed in terms of the slope–displacement matrix G: = PA−1 Cqe = Gqe (114) The cubic interpolation of w along the boundary of the elements, given by Hu [99] is used here to evaluate the C matrix: w(s) = [1 − + ( − 32 + 23 )]wi + [ − 2 + ( − 32 + 22 )] lij si 2 + [ − ( − 32 + 23 )]wj + [− + 2 + ( − 32 + 22 )] = s ; lij 0slij ; 01; 1 = D 1 − 12 T L2 lij sj 2 (115) where lij is the distance between nodes i and j , si , sj are tangential rotations at nodes i and j , respectively, and D, T are ﬂexural and transverse shear rigidities. The inﬂuence of the parameter is explained in References [9, 10, 99]. Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 363 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS Using Equation (114), the virtual work principle given by (110) may now be rewritten as follows: T ( Tb Db + Tm Sm + Ts Ts ) dx dy + qe Kg qe = k+1 R− ( Tb k M + Tm k N + Ts k Q) dx dy (116) where Kg is the initial stress matrix deﬁned as Kg = GTk FG dx dy (117) Substituting Equations (37)–(39) on to the right-hand side of Equation (116), we may write: ( Tb k M + Tm k N + Ts k Q) dx dy = qT f (118) where f is the internal force vector resulting from the unbalanced forces in conﬁguration k and is expressed as follows: f= (BTb k M + BTm k N + BTs k Q) dx dy (119) We may now rewrite Equation (116) using Equations (40)–(42), (101) and (119) as follows: T T T T pT pT pT [( eb + b )M + ( em + m )N + ( es + s )Q] dx dy + qe Kg qe = k+1 R − qT f (120) Re-arranging terms and writing the above equation in incremental form we obtain T T T ( eb M + em N + es Q) dx dy + pT pT pT ( b M + m N + s Q) dx dy + qe Kg qe = k+1 R − qT f T (121) Substituting Equations (102) into Equation (121) we obtain T T T ( eb M + em N + es Q) dx dy + NPN i=1 i *Fi∗ *Fi∗ *Fi∗ T dMi + dNi + dQi + qe Kg qe = *Mi *Ni *Qi k1 R − qT f (122) Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 364 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA Making use of Equations (40)–(42), (48)–(50), as well as the consistency condition given by Equation (100), we may write eT q (K + Kg )q − e NPN i=1 i *Fi∗ *Fi∗ *Fi∗ *Fi∗ *Fi∗ ∗ ∗ ∗ dM + dN + dQ + dk + di i i i i *Mi∗ *Ni∗ *Q∗i *ki *i = k+1 R − qT f (123) where K is the linear elastic stiffness matrix given by Equation (54). Similar to Equation (109) we deﬁne ⎡ abT = ⎢ *Fi∗ ⎢ = ⎢ 0 *Mi∗ ⎣ 0 ⎡ asT T ab1 T as1 *F ∗ ⎢ ⎢ = i∗ = ⎢ 0 ⎣ *Qi 0 ⎤ 0 0 T abi 0 0 T abNPN 0 0 T asi 0 0 T asNPN ⎥ ⎥ ⎥, ⎦ ⎡ T am = ⎤ ⎥ ⎥ ⎥, ⎦ T am1 *Fi∗ ⎢ ⎢ =⎢ 0 *Ni∗ ⎣ 0 ⎡ a 1 *F ∗ ⎢ a = i =⎢ ⎣ 0 *i 0 T ⎤ 0 0 T ami 0 0 T amNPN 0 a 0 0 0 i a ⎥ ⎥ ⎥ ⎦ ⎤ (124) ⎥ ⎥ ⎦ NPN and ∗ ∗ ∗ *F *F *F i i i T = abi ∗ ; *M ∗ ; *M ∗ *Mxi yi xyi *Fi∗ *Fi∗ *Fi∗ T ami = ∗ ; *N ∗ ; *N ∗ *Nxi yi xyi *Fi∗ *Fi∗ T = ; asi *Q∗xi *Q∗yi a i= (125) *Fi∗ * i Substituting Equations (102) into (95) and (96) we obtain dMx∗ = Mx∗ ⎤ ⎡ ! ∗ 2 ∗ ∗ 2 ∗ 2 !2 M0 *F 6 *F *F *F ⎦ = 2 (1 − F ) ⎣ − M ∗" + + 0 *Mx h2 x 3 *Mx *My *Mxy Copyright 䉷 2006 John Wiley & Sons, Ltd. (126) Int. J. Numer. Meth. Engng 2006; 68:338–380 365 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS and similarly for the remaining hardening parameters. The vectors of the hardening parameters therefore yield: ⎧ ∗ ⎫ Nxi ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ∗ ∗ dNi = Nyi = Am ; ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ∗ Nxyi dQ∗i = Q∗xi Q∗yi = As ; ⎧ ∗ ⎫ Mxi ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ∗ ∗ = Ab dMi = Myi ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ∗ Mxyi (127) where Am , As , Ab are given by ⎡ 0 0 Ami 0 0 AmNPN ⎤ 0 ⎥ 0 ⎥ ⎦ Am1 ⎢ Am = ⎢ ⎣ 0 0 ⎡ 0 As1 ⎢ As = ⎢ ⎣ 0 Asi 0 0 ⎤ ⎥ ⎥, ⎦ ⎡ Ab1 ⎢ Ab = ⎢ ⎣ 0 0 0 0 Abi 0 0 AbNPN ⎤ ⎥ ⎥ ⎦ (128) AsNPN and ⎧ ⎪ ⎪ N0 ⎪ ⎪ 1 (1 − F ) ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ N Ami = 1 (1 − F ) 0 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ N0 ⎪ ⎪ ⎪ 1 (1 − F ) ⎪ ⎩ 0 Asi = ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ! ⎪ 2 2 2 ⎬ ∗ ∗ ∗ ∗ ! *Fi *F *F *F 1 2 i i i ∗ " ⎣ ⎦ − Nyi + + ⎪ h 3 *Nyi *Nxi *Nyi *Nxyi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎤ ⎡ ⎪ ⎪ ! ⎪ 2 2 2 ∗ ∗ ∗ ∗ ⎪ !2 *F *Fi *F *F 1 i i i ∗ ⎪ " ⎦⎪ ⎣ ⎪ − Nxyi + + ⎪ ⎭ h 3 *Nxyi *Nxi *Nyi *Nxyi ⎤ ! ∗ 2 ∗ 2 ∗ 2 !2 *F *F *F *Fi∗ 1 i i i ∗" ⎦ ⎣ − Nxi + + h 3 *Nxi *Nxi *Nyi *Nxyi ⎡ ⎧ ⎡ ⎤⎫ ! 2 2 ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎪ ! ⎪ ⎪ *Fi *Fi N0 ⎣ *Fi 1 ∗ "2 ⎪ ⎦⎪ ⎪ ⎪ (1 − F ) − + Q ⎪ ⎪ 1 xi ⎪ ⎪ 0 *Qxi h 3 ⎪ ⎪ *Qxi *Qyi ⎨ ⎬ ⎡ ⎪ ⎤ ⎪ ! ⎪ ⎪ ∗ ∗ 2 ∗ 2 ⎪ ⎪ ! ⎪ ⎪ *F *F *F N0 ⎣ i 1 ∗ "2 ⎪ ⎪ i i ⎪ ⎪ ⎦ (1 − F ) − + Q ⎪ 1 ⎪ yi ⎪ ⎪ ⎩ ⎭ 0 *Qyi h 3 *Qxi *Qyi Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 366 Abi P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA ⎧ ⎪ ⎪ M0 ⎪ ⎪ 2 (1 − F ) ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ M = 2 (1 − F ) 0 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ M0 ⎪ ⎪ ⎪ (1 − F ) ⎪ ⎩ 2 0 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ! ⎪ 2 2 2 ⎬ ∗ ∗ ∗ ∗ ! *Fi *F *F *F 6 2 i i i ∗" ⎣ ⎦ − 2 Myi + + ⎪ 3 h *Myi *Mxi *Myi *Mxyi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ! ⎪ 2 2 2 ∗ ∗ ∗ ∗ ⎪ ! *Fi *F *F *F 6 2 ⎪ i i i ∗ " ⎪ ⎣ ⎦ ⎪ − 2 Mxyi + + ⎪ ⎭ 3 h *Mxyi *Mxi *Myi *Mxyi ⎤ ! ∗ 2 ∗ 2 ∗ 2 !2 *F *F *F *Fi∗ 6 i i i ∗" ⎦ ⎣ − 2 Mxi + + 3 h *Mxi *Mxi *Myi *Mxyi ⎡ (129) The evolution equation for the porosity parameter may be written by substituting Equations (102) into (104): *Fi∗ *Fi∗ *Fi∗ h *Fi∗ d i = i = k3 i (130) + + + = A i i 2 *Mxi *Nxi *Nyi *Myi As previously done, we apply the plastic node method to derive the matrix form of the above equation: di = i = A where: ⎡ A ⎢ A =⎢ ⎣ 0 0 1 0 A 0 0 0 i A ⎤ ⎥ ⎥ ⎦ and A i = k3 (131) *Fi∗ *Fi∗ h + + 2 *Nxi *Nyi *Fi∗ *Fi∗ + *Mxi *Myi (132) NPN Following the work of Shi and Voyiadjis [31] we also deﬁne the isotropic hardening parameter as ⎧ ⎫ *F1∗ ⎪ ⎪ ⎪ ⎪ ⎪ dk1 ⎪ ⎪ ⎪ ⎫ ⎪ ⎪ ⎡ ⎤⎧ *k1 ⎪ ⎪ ⎪ ⎪ H1 0 0 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎨ ⎬ ∗ ⎢ ⎥ *F i ⎥ 0 H 0 (133) H = ⎢ = − dki i i ⎣ ⎦⎪ ⎪ ⎪ ⎪ *k ⎪ ⎪ ⎪ ⎪ i ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 HNPN NPN ⎪ ⎪ ∗ ⎪ *FNPN ⎪ ⎪ ⎪ ⎪ dkNPN ⎪ ⎩ ⎭ *kNPN We may now substitute Equations (124), (127), (131) and (133) into (123) to obtain T qe (K + Kg )qe + T [H − abT Ab − am Am − asT As − aT A ] = k+1 R − qT f T (134) Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 367 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS Using Equations (105) and (107) in Equation (134) we may write: T T Am − asT As − aT A ] ( qT − qp )(K + Kg )qe + T [H − abT Ab − am − k+1 R + qT f= qT [(K + Kg )qe − k+1 R ∗ + f] + T [−aT (K + Kg )qe T + (H − abT Ab − am Am − asT As − aT A )] = 0 (135) with k+1 R = k+1 R ∗ q (136) By the virtue of the variational method equation (135) we obtain (K + Kg )qe − k+1 R ∗ + f = 0 T −aT (K + Kg )qe + (H − abT Ab − am Am − asT As − aT A ) = 0 (137) Substituting (105) and (107) into the above equations we get (K + Kg )qe − k+1 R ∗ + f = (K + Kg )(q − a) = k+1 R ∗ − f (138) T −aT (K + Kg )(q − a) + (H − abT Ab − am Am − asT As − aT A ) = 0 (139) Equation (139) leads to the following expression: T Am − asT As − aT A )]−1 aT (K + Kg )q = [aT (K + Kg )a + (H − abT Ab − am (140) Equation (138) becomes Kepdg q = k+1 R ∗ − f (141) where Kepdg is the elasto-plastic damage large displacement stiffness matrix of the element, given by Kepdg = (K + Kg ){I − a[aT (K + Kg )a T Am − asT As − aT A )]−1 aT (K + Kg )} + (H − abT Ab − am (142) The tangent stiffness matrix given by Equation (142) is similar to the one presented by Shi and Voyiadjis [31]. The present formulation accounts for large displacements. Consequently, the stiffness matrix of the element contains the initial stress matrix Kg . The above-derived stiffness matrix describes not only the isotropic hardening, by means of the parameter H, but also the kinematic hardening, through parameters Ab , Am , As , which are not determined by curve ﬁtting, but derived explicitly from the evolution equation of the backstress given by Armstrong and Frederick [98]. The most important characteristic of the current work is a consistent and convenient incorporation of the damage effects into the yield condition and stiffness matrix, by means of the A matrix. We therefore have a non-layered ﬁnite element formulation with shell constitutive equations, a yield condition for porous ductile metals, the ﬂow and hardening Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 368 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA rules expressed in terms of the membrane and shear forces and the bending moments. All the variables used here, namely the porosity function, the stress resultants and couples, as well as the residual stress resultants and couples, representing the centre of the yield surface, are derived in a very rigorous manner. A very important feature of the derived tangent stiffness is its explicit form. The linear elastic stiffness matrix and initial stress matrix are determined by the quasi-conforming technique, which allows all the integrations to be performed analytically. The hardening parameters are also given explicitly. In addition, the through-the-thickness-integration is not employed here either, since the current model is the non-layered model with the yield condition expressed in terms of stress couples and resultants. The beneﬁts of an explicit stiffness is low computational cost. Numerical integration of the stiffness matrix using a standard Gaussian Quadrature method requires a number of ﬂoating-point operations of the order O(p 6 ) where is the order of the interpolation functions (p = 3 in the current model). The Vector Quadrature technique [100] performs fewer ﬂoating-point operations than standard Gaussian Quadrature but has the same level of complexity [101]. In the current model, these operations are not performed at all, since the stiffness matrix is given explicitly. This leads to substantial savings in computer time. Actual time savings are dependent on the implementation method, the problem solved and a processing unit. In the non-linear simulations, where the stiffness matrix is evaluated many times during the analysis, the improvement in computational efﬁciency is even more apparent. 7. NUMERICAL EXAMPLES For the purpose of computational implementation of the proposed model, a ﬁnite element computer program previously developed for the elasto-plastic analysis of shells using the programming language Fortran 95 is enhanced to account for the damage effects. The modiﬁed Newton–Raphson technique is employed to solve a system of non-linear, incremental equations. In order to overcome a singularity problem appearing at the limit point, the arc-length method [102] is adopted to determine the local load increment for each iteration. The return to the yield surface algorithm is also implemented [102]. The numerical results delivered by the current model are computed using a personal computer. Some of the reference solutions obtained with the layered approach (ABAQUS) are determined using a Silicon Graphics Onyx 3200 system. The accuracy of the description of the elasto-plastic and damage behaviour of shells is veriﬁed through a series of discriminating numerical examples. This paper is a continuation of the previous work of the authors [4, 9], where linear elastic and elasto-plastic formulations are given. The most important novel feature of the present algorithm is the description of isotropic damage in plates and shells. Thus, the examples presented here are selected to challenge mainly the representation of the evolution of damage in shells and the associated reduction in stiffness. Table I lists the references used here, and their corresponding abbreviations used later in the text. 7.1. Clamped square plate subjected to a central point load In this example we consider a square plate with all the edges ﬁxed, with an aspect ratio of L/ h = 20, where L is the length of the plate and h is the thickness. The plate is subjected Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 369 Table I. Listing of the models used with abbreviations. Name Description W&V-E W&V-EP W&V-EPD The present formulation—elastic analysis The present formulation—elasto-plastic analysis The present formulation—elasto-plastic, damage analysis 2 s n 2 Figure 6. Clamped square plate subjected to a central point load—geometry and material properties. to a central point load. Only a quarter of the plate needs to be examined due to symmetry. This problem was analysed by Shi and Voyiadjis [3], by means of the 4 × 4 mesh of ﬁnite elements. The same mesh of 4 × 4 elements per quarter of the plate is employed here. The geometry of the problem and the material properties are given in Figure 6. The equilibrium path is a universal curve providing most of the information regarding the functioning of the model independently of whether the deformation of the structure is governed by the bending, membrane or shear strains. Thus, we study the equilibrium path for the problem described above. The material parameters n and k3 appearing in Equations (61) and (65) are: n = 1.2587, k3 = 0.09, as determined by Perzyna [80]. The central deﬂection of the plate as a function of the applied load is given in Figure 7. Shi and Voyiadjis [3] solved this problem using a phenomenological damage model with isotropic damage parameter deﬁned by Lemaitre [49]. The ultimate load for this problem, given in Reference [3], without the inﬂuence of damage is Pc = 10M0 . Shi and Voyiadjis also showed the substantial reduction in stiffness of the structure when damage was considered. The ultimate load of the damaged plate was about Pc = 8M0 . The current model is based on micromechanical observations of the material and features more physically sound interpretation of the damage variable, i.e. the porosity. The result of the current analysis with the inﬂuence of damage considered, yields Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 370 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA 10 9 8 7 P/M0 6 5 4 W&V-E 3 W&V-EP 2 W&V-EPD 1 0 0 2 4 6 8 2 10 12 2 wD/M0L x10 Figure 7. Clamped square plate subjected to a central point load—load–displacement curve. approximately the same ultimate load Pc = 8M0 . For the current example the ultimate load carried by the structure, calculated through the elasto-plastic investigations is 20% higher than the one calculated through the application of the elasto-plastic-damage model. We therefore conclude that neglecting the damage effects can result in unsafe design of the structures. As expected, we notice that the damage variable only becomes signiﬁcant when the structure deforms plastically. This is because the evolution of damage is neglected in the elastic zone in ductile materials. The current formulation shows a robust performance in this test. A 4 × 4 mesh is used in this problem for comparison purposes with the reference result. A ﬁner mesh would not alter the quality of the results signiﬁcantly. A higher number of plastic hinges could lead to additional softening, which would however not substantially alter the estimate of the maximum load carried by the structure. Mesh sensitivity investigations would be imperative for analysis of postcritical behaviour, which in turn would require accounting for large strains and strain gradients. Such a study is very important and will be the subject of future work of the authors. It is however beyond the scope of this paper. 7.2. Spherical dome subjected to a ring of pressure The problem of a spherical dome with an 18◦ hole at the top, subjected to a ring of pressure was investigated in the ﬁrst two authors’ previous paper [4]. It is an important engineering problem, as well as a discriminating test for accuracy of the ﬁnite element representation of the behaviour of shells. It was shown in Reference [4] that the stress-resultant-based shell model with the kinematic hardening rule given by Equations (95)–(97) is capable of correctly predicting the elasto-plastic behaviour of shells, including the Bauschinger effect. In this paper, we revisit the problem of the spherical dome subjected to a ring of pressure, in order to Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 371 R=10 in t=0.04 in E=6.82x107 psi ν =0.3 σ 0=125 psi Figure 8. Spherical dome with an 18◦ cut-out; geometry and material properties. establish the performance of the current formulation in approximating damage due to microvoids. As previously, the structure is loaded into a plastic zone, and subsequently the pressure is reversed. We examine the elasto-plastic load–displacement curve and compare these results with that of the curve obtained when the inﬂuence of damage taken into account, in order to test the functioning and accuracy of the proposed yield surface for ductile porous metals, deﬁned in the stress resultant space. The material parameters n and k3 are the same as in example 7.1: n = 1.2587, k3 = 0.09. The geometrical and material data are shown in Figure 8 and the resulting load–displacement curves are plotted in Figure 9. Through the introduction of the porosity function, which characterizes damage into the yield function we obtain a strong coupling between plasticity and damage. The damage variable is dependent on the plastic deformation. Therefore, through the application of the robust kinematic hardening rule, we may model the evolution of damage in the dome that is loaded into the plastic zone in tension and subsequently the load is reversed, in compression (Figure 9). The lowered yield point due to the Bauschinger effect is again correctly approximated. The reduction in stiffness that is caused by damage initiated by the inelastic strains is signiﬁcant in this case. It is very important to note that the damage curve (W&V-EPD in Figure 9) as well as the porosity curve (Figure 10) indicate unloading at the last increment of load before the load is reversed. This means that considerable porosity has been developed (Figure 10) and the ultimate load-carrying capacity of the structure has been reached. If the loading force P were kept at the same level, the porosity would grow even further, leading to localization and fracture and ultimately the collapse of the structure. Based on the value of the porosity, the fracture criterion could be postulated. It cannot, however, be performed consistently using the current formulation. This is because the strains will reach very high values at the localized area of deformation. This theory is a large displacement but small strain theory, and it may not be applied to model localization problems. Furthermore, the gradients of deformation around the localization area are very high, which can only be approximated by the gradient plasticity [103] or viscoplasticity theories with damage variables [80–88]. Thus, proposing a fracture criterion requires further study of the microstructural material characterization. This is beyond the scope of this work. Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 372 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA 80 60 40 Load 20 0 -1.E-04 -8.E-05 -6.E-05 -4.E-05 -2.E-05 0.E+00 2.E-05 4.E-05 6.E-05 8.E-05 1.E-04 -20 W&V-EP -40 W&V-EPD -60 -80 Vertical Displacement Figure 9. Spherical dome with an 18◦ cut-out—load–displacement curve. 70 60 Load 50 40 30 20 W&V-EPD - porosity 10 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Porosity Figure 10. Spherical dome with an 18◦ cut-out—porosity as a function of load. Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 373 The plasticity curve (W&V-EP in Figure 9) reaches a plateau at a load about 8% lower than in the case of the damage analysis. The elasto-plastic analysis is therefore leading to the overprediction of the ultimate load. It is a very substantial factor from the point of view of engineering analysis of important structures. Moreover, the theory of plasticity alone does not provide reliable information about the behaviour of the material or structure after the ultimate load has been reached. Although the present elasto-plastic-damage model is limited to rate-independent and small strain problems, it compares much better with the experimental investigations, showing softening of the material after reaching the ultimate load level. Figure 10 presents a plot of the porosity as a function of load. The initial porosity in the virgin material is assumed to be 0 = 0.01. Since the damage variable is a function of the plastic strains, we only see growth of voids when plasticity occurs. As discussed above at the ultimate load level of approximately P = 63 lbf/in we observe softening of the material. The collapse will therefore occur at a load level approximately 8% lower that the load predicted by the elasto-plastic analysis. The present approach provides a good approximation of the evolution of damage of the modelled structure, proving the validity of the original assumptions. The elasto-plastic-damage model presented here is a very signiﬁcant advancement over the elasto-plastic formulation. The computational cost of the performed calculations using the proposed formulation is much lower than in the case of the shell elements with a layered approach. In order to substantiate this statement with appropriate data, we compare the number of ﬂoating point operations (FLOPS) per iteration and total CPU time necessary to solve the above problem by means of 800 elements based on the current elasto-plastic formulation (without damage), and a layered approach with Gaussian Quadrature method used for numerical integration of the stiffness matrix. The layered method with 10 layers across the thickness of each element and reduced-selective integration requires 8.96 × 107 FLOPS per iteration and the total CPU time of 38.44 s. The current model requires 8.04 × 107 FLOPS per iteration and the total CPU time of 32.38 s. This means that, based on the current example, the current formulation is about 16% cheaper than a layered method with implicit stiffness matrix. The savings of computational effort are even more signiﬁcant in the case of elasto-plastic-damage model. This is due to the explicit form of the stiffness matrix, and the application of the single load surface with the damage variable incorporated. It is noteworthy that a three-dimensional analysis with solid elements would be even more expensive. In problems with a complicated geometry, the computational cost of the ﬁnite element procedure may be decisive. 8. CONCLUSIONS The current work presents a mathematically consistent ﬁnite element model for the elasto-plastic, large rotation analysis of thin/thick shells incorporating the inﬂuence of damage due to microvoids. An accurate set of thick/thin shell constitutive equations [104], previously developed by the ﬁrst two authors is adopted here, along with its ﬁnite element implementation [9]. The non-layered yield surface with the stress-resultant-based kinematic hardening rule is used in this work in order to model the elasto-plastic behaviour of shells, including the Bauschinger effect. Iliushin’s yield function expressed in terms of stress resultants and couples, modiﬁed to account for the progressive development of the plastic deformation and transverse shear forces is used. Similar to the case of elastic considerations all the integrals are calculated analytically, Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 374 P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA which makes the current formulation extremely efﬁcient as the numerical integration is not employed at any stage of the computations. The tangent stiffness matrix is obtained explicitly. Large rotations and displacements that are often associated with inelastic deformation are described using the Updated Lagrangian method. The most important and novel feature of this paper is a simple and convenient, yet accurate description of the damage evolution in plates and shells. Since this work concerns the study of thick, homogenous isotropic and ductile shells, damage is modelled here as an isotropic, rate-independent process, caused by the growth of microvoids only. This can be regarded as a limitation of the current model, since the inﬂuence of nucleation due to microcracks is very important for certain applications. However, the current model is based on the evolution of porosity deﬁned by Duszek-Perzyna and Perzyna [76], who reported excellent results in modelling ductile metals. According to Reference [76], the inﬂuence of microcracks is very important when analysing metal matrix composites because of the cracking of the reinforcing ﬁbres. In the case of homogenous and isotropic shells, the void growth is decisive, and thus it is the only damage-causing phenomenon described in this work. The evolution of porosity is introduced into the yield function leading to a strong coupling between plasticity and damage. The initial porosity is evolving due to the presence of the inelastic strains, which means that the elastic damage is disregarded in this work. Only two additional material parameters need to be determined to account for damage here, as opposed to some higher-order approximations, where sometimes tremendous experimental data are necessary to calibrate all the required material constants. This would be the case if, for example, a second-order damage tensor were used. Furthermore, while a more advanced procedure would be needed to model the elasto-plastic and damage behaviour in anisotropic materials, the accuracy of the current analysis from the point of view of practical structural analysis is satisfactory. The reliability of the presented concepts was evaluated through example problems. Unfortunately, there is very limited amount of data in references regarding the evolution of damage in plates and shells. Moreover, it is unrealistic to verify the damage formulation based on comparisons to other results obtained by approximate methods. The robustness of this algorithm should be tested against experimental results, particularly for the case of damage characterization. References providing information about damage in structures that are based on experimental results are even more difﬁcult to ﬁnd than the numerical estimates. Nevertheless, based on the limited references as well as the fact that the results presented here show the expected pattern of the reduction of the stiffness caused by the evolution of damage, it can be concluded that the current formulation provides extremely valuable information about damage in plates and shells. The results of the examples given in Sections 7.1 and 7.2 proved that neglecting damage in the analysis of plates and shells leads to the overprediction of the ultimate load carried by the structure. In the example presented in Section 7.2 both the equilibrium path as well as the plot of porosity versus load showed softening of the structure at a certain load level. Softening will be followed by localization, fracture and ultimately collapse of the structure. These phenomena could not be observed if only the elasto-plastic analysis is performed. It is necessary to account for damage in the model in order to better approximate the behaviour of the structure after its ultimate load has been reached. Although addressing the problems of localization and postulating a fracture criterion require considering large strains, and using viscoplasticity or gradient plasticity theory with damage as a regularization tool, the formulation Copyright 䉷 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 68:338–380 ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS 375 presented in this paper provides a powerful tool for the comprehensive modelling of shells. The current elasto-plastic-damage model is a very signiﬁcant advancement over the elasto-plastic analysis. This paper is a continuation of the previous work by the ﬁrst two authors. The original objective was to formulate a simple and efﬁcient computational model for the detailed analysis of plates and shells. Traditionally, shell elements follow a layered approach when non-linear calculations are performed. This is because most of the yield functions featuring reliable isotropic and kinematic hardening rules and accounting for the damage effects are expressed in terms of the stresses. A layered approach requires however discretization of the structure through the thickness, and calculating a yield function for each layer separately. This is conceptually close to the three-dimensional analysis by means of the solid elements, and the use of the degenerated shell elements loses its advantage. A non-layered method, on the other hand, seems to be a natural consequence of the shell element development, as the system of non-linear equations is expressed in terms of forces and bending moments, and solved without discretization of the shell through the thickness. A reliable yield surface expressed in terms of the stress resultants would allow for the analysis of shells without ‘through-the-thickness-integration’, which is simpler and much more efﬁcient. It also uses the shell element concepts and shell constitutive equations, not only for the linear calculations, but also for the elasto-plastic, damage and geometrically non-linear analysis. The current work provides a veracious loading surface, deﬁned in the stress resultant space, featuring both isotropic and kinematic hardening rules, as well as strong coupling between plasticity and the porosity function, describing damage due to microvoids. 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