Application of elastoplastic damage models within the boundary element method U. Herding, G. Kuhn Universitdt Erlangen-Nurnberg, Lehrstuhlfur Technische Mechanik, 91058 Erlangen, Germany Abstract The process of crack initiation and growth can be numerically simulated by means of damage models. In this work the GURSON model for elastoplastic material behaviour is investigated. It is based on micro mechanics and describes the failure mechanism through initiation, growth and coalescence of voids. Within the framework of thermodynamics of irreversible processes and the concept of internal variables the GURSON model is applied within the boundary element method. Kinematic and isotropic hardening, isotropic damage and arbitrary uniaxial stress-strain relations are used to describe material behaviour. The constitutive equations will be formulated in 3D and applied to two-dimensional and axisymmetric problems at small strain. 1 Introduction In the past decade, CDM was established to close the gap between continuum and fracture mechanics. The most common models can be divided into two groups. The first one is based on KACHANOV's ideas [1], who first introduced the damage concept. He investigated creep in uniaxial tension tests and found that tertiary creep was not included in standard formulations. He therefore introduced a phenomenological variable to characterize that part of a cross section which is still able to carry load. This approach of phenomenological damage, which ignores micro mechanics like e.g. initiation and growth of voids or cracks, is used in a large number of damage models. These models are founded on KACHANOV's idea and contain an arbitrary tensor as a damage variable. Scalar variables are most commonly used, because the problems of damage evolution and physical interpreta-
668 Localized Damage tion increase considerably as the number of components grows. In most models, a damage variable is added to the constitutive equations following the concept of effective stress (or strain). The second group of damage models is based on micromechanical reflections. It wasfirstinvestigated by McCLINTOK [2], RlCE & TRACY [3]. Later on, GURSON [4] and TvERGAARD & NEEDLEMAN [5] as well as several other authors developed them further. The main idea is to characterize damage by voids which can nucleate and grow. Representative defects are examined to attain constitutive equations. By means of averaging techniques, a constitutive theory is developed which follows standard continuum mechanic formulations. In this work the GURSON model and its modifications by TVERGAARD & NEEDLEMAN are investigated. As shown by NAPPI [6] for elastic damage, a boundary element formulation with a domain integral describing the influence of damage can be found. The formulation presented here is applicable to both elastic and elastoplastic damage models. As the deformation can be considered strain driven, the displacement gradients are introduced as the main unknowns. Whenever stress is needed in calculation, the stressstrain relation is evaluated separately. Thus it is easy to implement new constitutive models and to deal with material behaviour that changes its stress-strain relation during loading. The available boundary element code is able to handle two-dimensional and axisymmetric problems, where verification via experiments is done more easily for the later ones. Calculations for both will be shown and compared qualitatively with results of NEEDLEMAN [7]. Throughout this work tensorial notation is used for convenience. The symbol (g) denotes the tensorial product. Colons are used for the double contracted product of two tensors. Boldface symbols denote for capital letters system matrices and for small letters vectors, which consist of values for all boundary or cell nodes. 2 Gurson model The formulation of damage due to GURSON is based on a special yield surface. According to this model, the plastic flow of porous media is influenced by a single damage parameter 0 < / < 1, which represents the void volume fraction. In the course of micromechanical investigations [8] [9] of materials with periodically distributed voids TVERGAARD introduced some modifications to achieve better correspondence with experimental results. Thus the yield function 0 is given by where the parameters qi = 1.5, #2 = 1 and #3 = q^ are used, /i denotes the hydrostatic stress invariant, J? the mises stress and K the actual yield stress of the matrix material. To improve the numerical results for large values of
/, the modified damage parameter /*(/) Localized Damage 669 f /,for/</c was introduced in the yield function i-ftr' = o (3) by TVERGAARD & NEEDLEMAN [10]. Thus coalescence of two neighbouring voids due to slip planes or necking may be modelled. The parameter // denotes the final relative void volume, /* the final modified relative void volume for material failure. The above mentioned effect is furthermore made dependent on a limiting factor, the critical relative void volume /c. In order to get the evolution equations the principle of maximum plastic dissipation is used in combination with the consistency condition, which is introduced, as a restriction, via the LAGRANGE multiplier 7. Thus, for the evolution of plastic strains, the relation is obtained. The evolution of the internal variables / for damage and 6% for the accumulated plastic strain follow the equations ^ = -^f/)%7=#7 (5) and with / = f growth + f nucleation = Df ~f (6) =(!-/) t A k -f B ^p- if nucleation condition is met v ' / i.. 0 if nucleation condition is not met. For nucleation two main mechanisms are known. One is governed by the accumulated plastic strain, which leads due to CHU & NEEDLEMAN [11] to where 5 is the standard deviation of the normal distribution for void nucleation around the mean value CN. The fraction of void nucleating particles is denoted fjy. The second one due to ARGON et al. [12] is driven by the
670 Localized Damage hydrostatic stress. Thus the factors A and J3, which are formulated analogously via a normal distribution, read Together with the nucleation condition that no unloading takes place and the consistency parameter (9) the constitutive theory for damage in combination with elastoplastic material behaviour is described completely. 3 Boundary Element Formulation To deal with elastoplastic material behavior in the boundary element method afieldboundary integral formulation has to be used. The well known boundary integral equation is modified to <=( )«( ) = / (U(t, x) *(*) - T(, x) «(*)) dt-j W(, x):c:»'(*) d$l, where U and T are the KELVIN fundamental solutions. The components of matrix c are integral free terms and depend only on the geometry of the boundary. With this relation the displacements u( ) at a source point located at the boundary can be calculated depending on all displacements u(x) and tractions t(x) at the boundary and the plastic strain e**(x) in the domain. As the plastic strains depend on the current stresses and thus on the current displacement derivatives, a method for calculation of these derivatives is necessary. In the boundary element method an integral representation formula - V; n for source points located inside the domain is set up. When differentiating the domain integral, it becomes strongly singular. Among the various existing techniques to deal with this, the regularization method shown in [13] is used. For source points at the boundary a hypersingular representation formula is used. To solve the above mentioned equations discretization of the boundary and the domain in that region where plastic flow appears has to be done. The tractions and displacements on the boundary as well as the plastic
Localized Damage 671 strains in the domain are approximated via shape functions. By making use of the collocation method, for the field boundary equation a system of equations can be set up and arranged in matrix form A - u = B -1 - E (13) where the matrices A, B and E contain the results of integration over F, and ftj. The vectors u and t consist of all nodal displacement and traction values at the boundary, e** contains the plastic strain at all nodes of the domain discretization. Taking into account the boundary conditions and applying the above described procedure to get an algebraic system of equations for (12), a nonlinear set of equations may be set up, which has to be solved with an iterative incremental procedure. As an advantage only those regions have to be discretized with a domain net where plastic flow and thus damage occurs. Due to this it is possible to start with a small domain net and apply an algorithm of net growing, whenever the zone of damage reaches the boundary of the domain net. This is of importance when numerical simulation of crack propagation is implemented, because no pre defined path of growth via an existing domain net is present. 4 Example For the 2D example a plate of dimension ^ = 3 with a circular notch of radius f* = 0.5 (Fig. 1) was taken. As loading condition a uniform displacement of f = 0.0112 was used. Due to double symmetry only a quarter of the plate was modelled. The geometry of the axisymmetric example may be generated out of the 2D contour via rotation round the y-axis. Figure 1: notched plate Calculations for plain strain and axisymmetry were done with the boundary element program BEATAX (Boundary Element Analysis of Twodimensional and AXisymmetric Systems), which was developed at the Lehrstuhl fur Technische Mechanik in Brian gen. Elastic material properties and
672 Localized Damage values of the uni-dimensional stress-strain relation are shown in the two following tables. In BEATAX the uniaxial stress-strain relation, which is used for calculation of isotropic hardening, is approximated via splines. This is of advantage when derivatives of this curve are needed. elastic material behaviour Table 1: elastic material data YOUNG'S modulus 210000 MPa Table 2: plastic material data plastic material behaviour cp' ^m 0.00000 0.01879 0.02805 0.05902 0.12449 0.29700 0.57569 1.99425 2.99353 POISSON'S 0.3 <r in M Pa 460.00 472.81 521.39 628.96 736.30 837.41 905.83 1208.0 1360.0 ratio Material parameters of strain and stress controlled void nucleation are listed in the table below. Table 3: material data of the GURSON model strain controlled void nucleation 0.9 0.1 0.03 0.00025 0.15 0.25 0.66 nucleation germs standard deviation mean value mean value starting void volume critical void volume final void volume mod. final void v. fn s SN <TN h fc // K stress controlled void nucleation 0.04 46 1012 0.00025 0.15 0.19 0.66 In Figure 2 the relative modified void volume fraction /* at the begin of void nucleation and at the final load are shown qualitatively for both plane strain and axisymmetric calculations. Different scales were chosen so that for every calculation all colours are used. Although calculations were done for small deformations, results for strain controlled calculations in 2D and axisymmetry and for stress controlled calculations in 2D largely correspond to those obtained by NEEDLEMAN [7] (Fig. 3). As during calculation the void volume fraction reaches nearly its ultimate value /*, it is necessary to implement crack initiation and propagation into the boundary element code. For this purpose new boundaries have to be created and a simple element stiffness degradation technique, as know from thefiniteelement method, can not be applied.
Localized Damage 673 begin final load begin final load Figure 2: relative void volume fraction (BEM calculations) -o JD s 4-* c 2D AX "O JQ> begin final load begin final load 2D O O CO begin final load Figure 3: relative void volume due to Needleman 5 Conclusions The GURSON model as a main representative of elasto plastic damage models was applied within the boundary element method. Especially for local damage this seems to be very promising, as the possibilities of growing domain discretization and not pre defined directions of damage and crack
674 Localized Damage growth via the domain net are a great advantage arising out of the boundary element method. The model was implemented in a 2D and an axisymmetric boundary element code. The latter one opens the possibility to check numerical results and to derive the material parameters of the GURSON model via experiments, as 2D conditions (plain strain or stress) are hard to achieve in testing. The presented model should be extended to large deformations, to which, according to FOERSTER [14], the boundary domain element method is applicable. Having done this one has a numerical tool for calculations of crack initiation and growth for a wide class of materials. References [1] Kachanov, L. M., On failure time under conditions of creep, Akademlja Nauk SSSR, Izvestlja/Otdelenle Technicesklch Nauk, 1958, 26-31. [2] McClintok, F. A., Ductile Rupture by the Growth of Holes, J. Appl. Mec6., 1968, 35-36. [3] Rice, J. R. & Tracey, D. M., On the Ductile Enlargement of Voids in Triaxial Stress Fields, J. Mech. Phys. Solids, 1969, 17-201. [4] Gurson, A. L., Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I - Yield Criteria and Flow Rules for Porous Ductile Media, Transactions of the ASME, 1977, 2-15. [5] Needleman, A. & Tvergaard, V., An Analysis of Ductile Rupture in Notched Bars, J. Mech. Phys. Solids, 1984, 461-490. [6] Rajgelj, S. & Amadio, C. & Nappi, A., Application of damage mechanics concepts to the boundary element method, Boundary Element Technology VII, ed. C. A. Brebbia & M. S. Ingber, Elsevier Applied Science, London, 1992, 617-634. [7] Needleman, A. & Tvergaard, V., An analysis of ductile rupture in notched bars, J. Mech. Phys. Solids, 1984, 32, 461-490. [8] Tvergaard, V., Influence of voids on shear band instabilities under plane strain conditions, Int. J. Fracture, 1981, 17, 389-407. [9] Tvergaard, V., On localization in ductile materials containing spherical voids, Int. J. Fracture, 1982, 18, 237-252. [10] Tvergaard, V. & Needleman, A., Analysis of the cup-cone fracture in a round tensile bar, Acta MetalL, 1984, 157-169. [11] Chu, C. C. & Needleman, A., Void nucleation effects in biaxially stretched sheets, J. Eng. Materials Techno!., 1980, 102, 249-256. [12] Argon, A. S. & Im, J. & Safoglu, R., Cavity Formation from Inclusions in Ductile Fracture, Metallurgical Transactions, 1975, 6A, 825. [13] Dallner, R. & Kuhn, G., Efficient evaluation of volume integrals in the boundary element method, Comp. Meth. Appl. Mech. Engng., 1993, 109, 95-109. [14] Foerster, A. & Kuhn, G., A Field Boundary Element Formulation for Material Nonlinear Problems at Finite Strains, Int. J. Solids Structures, 1994, 31, 1777-1792.