A BEM implicit formulation applied to damage mechanics models

A BEM implicit formulation applied to damage mechanics models A. S. ~otta', W. S. ~enturini' & A. ~enallal~ 'S& Carlos School oj'engineering, University of $60 Paulo, Brazil 2 Laboratoire de Me'canique et Technologie, ENS de Cachan/CNRS/Universit2 Paris 6, France, Abstract In this work, the implicit BEM formulation, initially developed in the context of plasticity analysis is extended to incorporate damage mechanics models. The algebraic equations adopted for the formulation are obtained either using displacement or traction equations, for the boundary nodes, and stress (or strain) equations for the internal nodes. The consistent tangent operator has been obtained for situations: the conventional non-linear BEM system of equations. 1 Introduction Analysis of non-linear problems by means of BEM can be found since the end of seventies [l].the non-linear formulations used during a quite long time were all based on the initial stress and strain procedures, where a constant matrix scheme is employed. The consistent tangent operator (CTO), as proposed by Simo [2] for finite elements, has been introduced into elastoplas.tic BEM formulations only recently [3,4]. After that, the elastoplastic BEM formulations have been also extended to deal with localization problems in plasticity with emphasis to the regularization procedures [5,6]. In those works, the authors have shown the accuracy of the implicit formulation to compute very large deformation growing over a very

narrow and localized bandwish, comparing the results with solution obtained by using explicit models. They have also derived the complete implicit BEM formulation for gradient plasticity, adopted to avoid the mesh dependency. The results obtained using this formulation are accurate and clearly the mesh dependence problem is avoided. Modeling the mechanical behavior of brittle material structures is nowadays an important and interesting theme for research. Concrete, often assumed as a brittle material, is widely adopted in civil engineering construction, justifying therefore the interest on developing accurate numerical models in this context. This material exhibits a particular behavior characterized by the formation of micro-cracks resulting into losses of strength and rigidity of the structural members. This behavior is well represented by models based on the continuum damage mechanics [7,8]. So far, only limited applications of BEM to damage mechanics have been reported in the literature [g]. Damage mechanics is a complete different problem to be dealt with BEM. The Young's modulus is no longer constant, being its new value given by the adopted model that controls the degradation of the material. Besides that, using elastic prevision as in plasticity is not a convenient choice; the rigidity could be, at certain stage, so much deteriorated that elastic prevision may lead to an increasing number of iterations at each time step. In this paper, the non-linear BEM formulation is extended to solids governed by damage models, particularly proposed to deal with brittle material. First, the boundary algebraic equations are derived and then transformed appropriately to give an incremental solution scheme with tangent predictor. Some preliminary numerical results are shown to demonstrate the accuracy and stability of the developed model. 2 The continuum damage mechanics model Continuum Damage Mechanics (CDM) deals with the load carrying capacity of solids without majors cracks, but where the material itself is damaged due to the presence of microscopic defects such as microcracks and microvoids. The damage mechanics models, defined in the context of the thermodynamic of the irreversible processes, require the definition of internal variables to represent the energy dissipation process [7,8]. The internal variables are either scalar-valued ones, when isotropy is assumed (two variables can be assumed to represent the phenomenon differently when in tension or in compression), or tensor-valued ones, when anisotropic behaviors are represented. Without loss of generality, in this work, we decided to take a very simple isotropic damage model to derive BEM formulations in the context of elasticitybased damage models. The model is based on only one damage parameter D to reflect the amount of damage occurring in the material, either in compression or in tension. The corresponding constitutive relations are derived from the Helmoltz free energy y as follows,

Boundary Elrmcnt Tcrhnology XV 289 depending on the strain tensor E, and the damage variable D; p is the mass density and EUk+ is the forth-order tensor of the elastic sound material. The stress tensor is derived from eq. (l), resulting into, and the thermodynamically force Y, conjugated to the damage variable D, is Initiation and growth of damage is formulated through a damage criterion written in terms of the loading function f(y;d), as follows,. af D = h-(y;d) (4) ay The multiplier h, in equation (4) satisfies the Kuhn-Tucker conditions. From the consistency conditions, ' f = 0, one can write the following relation: In this work, we choose for the function f the very simple expression The history term k(d), in equation (7), measures the largest value that has been attained by the damage strain energy density rate Y. Accordingly, it is a nondecreasing function and grows only when f(y;d) = 0. An Example of k(d) is the local expression where M is a material constant. From the damage criterion f(y;d) = 0, one finds

3 Integral representations for non-linear problems Let us first consider an elastic body associated with many possible elastic states satisfying the Navier's equations, i.e.: where U, represents the components of the displacement field; p is the shear modulus and v is Poisson's ratio, while EIJkm stand for the elastic tensor. For a domain C2 with boundary r, standard integral representations are derived by applying Betti's principle (Green's second identity). Particularly, displacement integral representation is easily derived [l]: where uk and pk are the displacement and the traction components at boundary points, respectively, bk gives the body forces distributed over the domain, the free term C;, is the dependent upon the boundary geometry; while the symbol * indicates thiswell-known Kelvin fundamental solutions. For non-linear problems, Betti's principle cannot be directly applied. Moreover, the state variables for a non-linear model are usually history dependent. In this case, splitting the total strain into its elastic and non-linear components, i.e., ske =E& +E&, the Navier's operator applies only to the elastic part, so that equation (lo), now written in rates, becomes: -L..G.=bi-E.. 1~ J 11km km.j (12) where i:, is a tensor containing non-linear strain rates. In equation (12), the non-linear strain field acts therefore as fictitious body forces. This is referred as the initial stress method and usually the strain rate term is replaced by the initial stress rate term, given by '6; =. Thus, taking into account the last form of the Navier's equation (12), the Somigliana's identity (l l) now reads: where represents the non-linear stress rate components due to damage and is the strain filed given by the Kelvin's solution.

Boundary Elrmcnt Tcrhnology XV 29 1 As for elasticity, the integral representation of stress rates can be obtained by differentiating (13) at an internal point, with respect to space co-ordinates and applying Hooke's law. Thus, one obtains, where the new kernels indicated by * are obtained from the corresponding ones given in eq. (13) by applying the strain definition; bij(be) is a free-term resulting from the derivatives of the strong singular domain integral in equation (13) (e.g. Brebbia et alli [l]). In this work, we decided to not extend equation (14) to boundary points; as it will be discussed later, only internal points are used to approximate the strain corrector values. The alternative given by extending equation (14) to boundary points can be found elsewhere for the elastic case [10, 1 l], and for the non-linear problems [4,5]. Evaluating strains by mean of well-established relations: Cauchy's formula and the displacement derivatives with respect to the circumferential Cartesian coordinates (local coordinate system) are not reliable for non-linear analysis and therefore must be avoided. In this case, the cumulated errors during the iterations are enough significant to spoil the solution. For nonlinear applications exhibiting strain concentration or localization, this procedure always leads to numerical instability. In despite of computing strains only at internal points, equation (14) will be also used for outside collocations. In this case, the left hand side term vanishes leading to an expression relating only boundary values and the non-linear stress rate field, (rk. Equations (13) and (14) are exact equilibrium representations of the body under consideration. In the next section, they will be conveniently approximated to give algebraic relations written in terms of certain number of degrees of freedom to allow numerical solution of the problem. 4 Boundary element method In order to solve the boundary value problem described by the equilibrium equations (differential or integral equations) and the constrains given by the chosen damage criterion, discussed in section 2, a space discretization followed by displacement and strain field approximations must be adopted. The boundary l- is then discretized into elements, rs, along which displacements and tractions are approximated. For the examples presented in this work, only quadratic shape functions were used to describe those fields along each element. Discontinuities at some boundary points may be allowed when convenient; this is provided by defining collocation points inside the element or at any outside close points (they must be close to the boundary to assure the system stability). In order to have the integral equilibrium equations written in their algebraic forms, the strain (or stress) filed must be approximated in R. Internal cells 0, '

are defined, over which the strain components are approximated based on selected shape hnctions and nodal values. Triangular cells with linear approximation have been adopted with nodes defined at corner, but when convenient placed along the bisecting lines. This discretization is convenient to choose only internal nodes to approximate the strain field. When a cell has one or more nodes along the boundary, internal collocations are automatically defined. Thus, after selecting a collocation point, algebraic relations are obtained, performing properly the integrals over boundary elements r, and internal cells QC. Thus, the integral equations (13) and (14) for a single collocation point S, are transformed to: [W)]. (U)-[W)]. (*l- [~(s)]. bd]= 0 (15) where {U) and ffi} are vectors containing the boundary displacement and traction rates, respectively; (od] gives the damage stress rate vector defined at selected internal points, {b} is the stress rates computed at the collocation S; [I-~(s)], [~(s)], [~(s)],[~'(s)], [~'(s)], [Q($)] and [~'(s)] are matrices resulting by performing the appropriate integrals over boundary elements and internal cells. Note that equations (15) and (16) were obtained for any selected collocation point S; Collocations point may be placed at internal position, on the boundary (equation (14) must be derived for this situation) or external position. In order to have a closed system of equations to solve the problem, one must choose a number of relations equal the number of problem unknowns, i.e., boundary displacernents and tractions plus the stresses at the internal nodes defined by the discretization. Therefore, writing equation (16) for external or boundary collocation points give also independent relation, as follows: Using equation (17) to replace equation (15) defines an alternative scheme that has been experimented in this work to solve the proposed damage problem, However, for this particular case under analysis, using only stress equations, therefore only hypersingular equations, did not lead to a better performance of the method. The final results were very similar and no advantage was observed. After writing equations (15) and (16), or equations (17) and (16) if only hypersingular equations are chosen, in their incremental forms (time integration to be discussed in the next section), for a convenient number of collocations, the following matrix blocks of algebraic equations are found:

Boundaq Element Technologv XI/ 293 It is to be noted that the block of equations (18) can be written from (15), if the standard BEM is adopted, or from equation (17) if the hypersingular formulation is the chosen alternative. Equations (18) and (19) represent the approximated equilibrium of the solid during a time increment At. In the next section they will be used to derive an implicit BEM algorithm to deal with problem in the damage mechanics context. 5 Implicit formulation of BEM for damage mechanics The incremental forms of equations (18) and (19), after applying the boundary conditions are given by [AI. @n l = @n l+ [Q]. bo: / (20) where {AX, ] contains all boundary unknowns {~f, ] and {Agn) are given by the prescribed boundary values and a time increment At, has been considered. Then, equations (20) and (21) can be solved to give: where {Ah~l,f=[~]-'{~f~) and {AN~]=-[A'].[A]-'.(A~~)+{A~,~ (24a,b) [R]=[A]-'.[Q] and [S]=-[A'].[A]-'.[Q]+[Q1] (2w) It is important noting that the equilibrium equation is now reduced to equation (23). It can be rearranged to be expressed in terms of the strain increment AE, = &,+l - E, (note that As, = ~efl i- As f,, As:, damaged and elastic strain parts, respectively) and the increments of the n considered internal variables, i.e., Aakn =akn+l -akn, as ~O~~OWS,

294 Boundary Element Technologv XV where [S] = [S]+ [I 1, being [I ] the identity matrix. From the damage model constitutive equations, one may obtain explicitly the stress increment in terms of the strain increment, therefore, equation (26) reads, Equation (27) can now be solved by applying the Newton-Raphson scheme. An iterative process may be required to achieve the equilibrium. Then, from the iteration i the next try, (i+l), for the time increment At, is given by By linearizing the equilibrium equation, (27), using the fxst term of the Taylor's expansion, gives, Thus, the equilibrium is now expressed by where the algorithmic consistent tangent operator, ccto = ~ AcT/~AE is explicated. Although the presented scheme is similar to the one proposed for plasticity [5,6], there are some strong differences between them. In plasticity, the first try of each increment is always the elastic one, which makes the algorithm simpler. For damage, using the last tangent matrix is recommended, otherwise the number of iterations required for convergence increases too much. Rigidity at damaged regions is so severely reduced that elastic tries are always an inconvenient choice. In order to use reduced rigidity matrix, one has to modify the nth elastic increment that now reds and the equilibrium Then, the linearization follows the relations already presented, equations (2% (291, (30). Another important aspect shown by structures made with materials governed by damage laws is the presence of small and localized regions exhibiting severe reduction of rigidity. This fact usually induces the snap-

Boundaiy Element Technologv XV 295 back effects. The dissipation zone is so reduced that both referenced displacement and the total applied load are reduced simultaneously. In order to deal with that particular behavior of some damaged domains, we have incorporated into the implicit formulation presented in the previous sections the arc-length procedure. The formulation of this BEM arc-length will be presented in a separated paper [12]. The strategy for this class of methods is to assume as guide variable a specified length to go along the curve load X displacement. We define a priori the size of the chord to reach next point along the curve. With this technique the guide variable is neither the loading nor the displacement. Increments of the displacement X reaction curve chord are given as a prescribed load; the arc equation is added to the system of algebraic equations to complete the necessary relations. 6 Numerical examples In this section, two examples are presented to illustrate the applicability of the BEM to damaged domains. The first example is a one-dimensional test to show how the proposed method can model accurately the deteriorated (damaged) zone concentrated over a small region at the domain center. The geometry of the chosen thin domain (a X L) is presented in Figure la, together with the boundary conditions given by prescribing displacements along the left and right sides. The left side displacements are zero, while along the right side they are controlled. The discretization is shown in Figure lb, for which a small rectangle at the center will be assumed weaker, the region where the material will deteriorate (localized region). The damage model adopted is the one defined in section 2, for which the following parameters were adopted: Yo= 0.001667 and M= 0.025. For the weaker region the following values were assumed: Yo= 0.0015 and M= 0.0225. Figure 2 shows two curves captured for different sizes of the central deteriorated regions (dl=1/13 and a/l=1/27, respectively). This solution demonstrates clearly that the solution is not unique; it is dependent on the size of the dissipation zone. As can be seen for smaller weaker central region more pronounced is the snap-back effects. Using this simple example the mesh dependence of this problem can be easily observed. Dividing the weaker zone in several rows of cells, the damage region will localize only over one of them, leading therefore to different curves when the mesh is refmed. t- ---- I - ' 1 + 4 - -- 130 cm r-? \ 7 I \,,,jl l l l l 11111 -- - W, a - L Figure 1. one-dimensional domain. Geometry and discretization.

296 Boundary Element Technologv XV 1 Force X Displacement 1 0 2 4 6 8 displacement (cm) (X 0.01) Figure 2. Displacement X reaction curves for the 1D case The problem shown in the frst example exhibits the behavior of a onedimensional case. This analysis can be extended to see how the formulation behaves for problems where the two-dimension effects are clearly introduced. Then, we have chosen another rectangular domain that exhibits these characteristics, as shown in Figure 3 where the geometric data are specified. The loading is characterized by uniform distributed forces applied to the two opposite domain ends. The same parameters adopted for the first example have been used for this case. A square containing one node at the center of the rectangular domain has been assumed as a weaker zone to avoid capturing the uniform stiffness reduction solution. Two discretizations (Figure 4) are used to obtain the numerical solution to illustrate the mesh dependence for this problem. The energy dissipation occurs over the central row of cells as expected. As this area is very small, in comparison with the size of the domain, the snap-back effects are captured (see Figure 5). These effects are much more pronounced when the energy dissipation area is strongly reduced, which is the case observed for the fine mesh. The two solutions presented are only numerical solutions achieved for these specific discretizations. The solutions exhibit they particular characteristic, depending on the adopted mesh. The sequence of the damage appearance modifies abruptly when the discretization is little modified. Other solutions may be found for this case, depending on the approximations, positions of the internal nodes and other geometric data. In order to have single solution independent on the mesh, one must use any regularization technique. The complete formulation combining the implicit BEM presented here with a regularization technique is given in reference [ 121.

Boundaiy Element Technologv XV 297 Figure 3. 2D domain. weak points Figure 4. Discretizations and weak points. Force X Displacement -0- Figura 4a 0 2 4 6 8 10 12 displacement (cm)(x 0.01) Figure 5. Displacement X reaction curve for the 2D case. 7 Conclusions In this work, the boundary element method has been extended to analyze damaged domains. The BEM implicit formulation for a simple damage criterion

298 Boundary Element Technologv XV has been presented and used to show the accuracy of the method for this class of problems. The mesh dependence of the local model is clearly represented by the method. For unique solution a regularization technique must be incorporated. For brittle material, the snap-back effects are captured using the presented algorithm, which has incorporated the arc-length scheme. Acknowledgement To FAPESP - STio Paulo State Research Foundation. References [l] Brebbia, C.A., Telles, J.C.F & Wrobel, L.C. Boundary element techniques. Theory and applications in engineering, Springer-Verlag: Berlin and New York, 1984. [2] Sirno, J.C. & Taylor, R.L., Consistent tangent operators for rate-independent elastoplasticity. Comp. Meth. Appl. Mech. Engng, 48, pp. 10 1-1 18, 1985. [3] Bonnet, M. & Mukherjee, S., Implicit BEM formulation for usual and sensitivity problems in elasto-plasticity using the consistent tangent operator concept. Int. J. Solids and Structures, 33, pp.4461-4480, 1996. [4] Poon, H., Mukherjee, S. & Bonnet, M., Numerical implementation of a CTO-based implicit approach for solution of usual and sensitivity problems in elasto-plasticity. Engineering Analysis with Boundary Elements, 22, pp.257-269, 1998. [5] Fudoli, C.A., Venturini, W.S. & Benallal, A., BEM Formulations Applied to Gradient Plasticity. Boundary Elements 22, Eds. Brebbia, C.A. & Power,WIT Publ. Southampton, 2000. [6] Benallal, A., Fudoli, C.A. & Venturini, W.S., An implicit BEM formulation for Gradient plasticity and localization phenomena. Int. J Num. Meth. Eng., 53 pp. 1853-1869,2002. [7] Lemaitre, J., A course on damage mechanics, Springer-Verlag, 1992. [S] Lemaitre, J. & Chaboche J. L., Me'canique de mate'riaux solides, Dunod- Bordas, Paris, 1985. [9] Garcia R, Florez-Lopez J, Cerrolaza M., A boundary element formulation for a class of non-local damage models, Int. J. S. Struct., 36 pp. 3617-3638, 1999. [loiguiggiani, M. Hypersingular formulation for boundary stress evaluation. Engng. Analysis with Boundary Element, 13, pp. 169-179, 1994. [l l]cruse, T.A. &. Richardson, J.D. Non-singular Somigliana stress identities in elasticity. Int. J. Num. Meth. Engng., 39, pp. 3273-3304, 1996. [12]Benallal, A., Botta, A.S. & Venturini, W.S., On the description of localization and failure phenomena by the boundary element method. Int. J. Num. Meth. Eng., 2002 (in preparation).