A cohesive boundary element approach to material degradation in three-dimensional polycrystalline aggregates I. Benedetti 1,2,a, M.H.

Transcription

1 A ohesive boundary element approah to material degradation in three-dimensional polyrystalline aggregates I. Benedetti 1,,a, M.H. Aliabadi 1 1 Department of Aeronautis, Imperial ollege London, South Kensington ampus,sw7 AZ, London, UK. Dipartimento di Ingegneria ivile, Ambientale, Aerospaziale e dei materiali, Università degli Studi di Palermo, Viale delle Sienze, Edifiio 8, 9018, Palermo, Italy a i.benedetti@imperial.a.u, a ivano.benedetti@unipa.it, b m.h.aliabadi@imperial.a.u Keywords: Polyrystalline materials, Mirostruture Modelling, Intergranular damage, ohesive laws, Anisotropi Boundary Element Method. Abstrat. A new three-dimensional grain-level formulation for intergranular degradation and failure in polyrystalline materials is presented. The polyrystalline mirostruture is represented as a Voronoi tessellation and the boundary element method is used to express the elasti problem for eah rystal of the aggregate. The ontinuity of the aggregate is enfored through suitable onditions at the intergranular interfaes. The grain-boundary model taes into aount the onset and evolution of damage by means of an irreversible linear ohesive law, able to address mixed-mode failure onditions. Upon interfae failure, a non-linear fritional ontat analysis is introdued for addressing the ontat between miro-ra surfaes. An inremental-iterative algorithm is used for traing the miro-degradation and raing evolution. The behavior of a polyrystalline speimen under tensile load has been performed, to show the apability of the formulation. Introdution For modern strutural appliations (aerospae, automotive, off-shore, et.), a deep understanding of materials degradation and failure is of ruial relevane. Frature modelling an be onsidered at different length sales: it is nowadays widely reognized that the marosopi material properties depend on the features of the mirostruture [1]. Polyrystalline materials (metals, alloys or eramis) are ommonly employed in engineering strutures. Their mirostruture is haraterized by features of the grains (morphology, size distribution, anisotropy and rystallographi orientation, stiffness and toughness mismath) and by physial and hemial properties of the intergranular interfaes. These aspets have a diret influene on the initiation and evolution of mirostrutural damage, whih is also sensitive to the presene of imperfetions, flaws or porosity. The mirostruture of polyrystalline materials, and its failure mehanisms, an be investigated using different experimental tehniques (see referenes in [] for a brief overview); these provide fundamental information and understanding but require sophistiated equipment, areful material manufaturing and preparation and ompliated postproessing, espeially whenever a truly three-dimensional (3D) haraterization is pursued. A viable alternative, or omplement, to the experimental effort is offered by the omputational Miromehanis [3]. Several investigations have been devoted to modelling of polyrystalline mirostrutures and their failure proesses [4] and there is urrently an interest for the development of truly 3D models. Until reently, this has been hindered by exessive omputational requirements. However, the present-day availability of heaper and more powerful omputational resoures and failities, namely high performane parallel omputing, is favoring the advanement of the subjet, espeially in the framewor of the Finite Element Method (FEM), see e.g. [5]. A popular approah for modelling both D and 3D frature problems in polyrystalline materials onsists in the use of ohesive surfaes embedded in a finite element (FE) representation of the mirostruture. In this way, initiation, propagation, branhing and oalesene of miroras stem as an outome of the simulation, without any assumptions. Several ohesive laws have been proposed in the literature [6].

2 An alternative to the FEM is the Boundary Element Method (BEM) that has proved effetive for a variety of physial and engineering problems [7,8]. A ohesive boundary element formulation for brittle intergranular failure in polyrystalline materials was proposed by Sfantos and Aliabadi [9]. A 3D grain boundary formulation has been reently developed by Benedetti and Aliabadi for the material homogenization of polyrystalline materials []. In this wor, a novel 3D grain-level model for the analysis of intergranular degradation and failure in polyrystalline materials is presented. The polyrystalline mirostruture is represented as a Voronoi tessellation and the formulation is based on a grain-boundary integral representation of the elasti problem for the anisotropi rystals, that have random orientation in the 3D spae. The integrity of the aggregate is restored by enforing suitable intergranular onditions. The onset and evolution of damage at the grain boundaries is modeled using an irreversible ohesive linear law. Upon interfae failure, a non-linear fritional ontat analysis is used, to address separation, sliding or stiing between miro-ra surfaes. An inremental-iterative algorithm is used for traing the degradation and miro-raing evolution. A numerial test is presented to demonstrate the apability of the formulation. Grain boundary formulation for polyrystalline aggregates Artifiial mirostruture. For polyrystalline materials, Voronoi tesselations are widely used for the generation of the mirostrutural models [10]. The assignation of a speifi orientation to eah rystal of the aggregate ompletes the mirostruture representation. Grain onstitutive modelling. Eah grain is modeled as a three-dimensional linear elasti orthotropi domain with arbitrary spatial orientation. This is not restritive, as the majority of single metalli and erami rystals present general orthotropi behavior. Grain boundary element formulation. Eah rystal is modeled using the BEM for 3D anisotropi elastiity [11]. The polyrystalline aggregate is seen as a multi-region problem, so that different elasti properties and spatial orientation an be assigned to eah grain []. Given a volume bounded by an external surfae and ontaining N g grains, two inds of grains an be distinguished: the boundary grains, interseting the external boundary, and the internal grains, ompletely surrounded by other grains. Boundary onditions are presribed on the surfae of the boundary grains lying on the external boundary, while interfae equations and equilibrium onditions are fored on interfaes between adjaent grains, to restore the integrity of the aggregate. The boundary integral equation for a generi grain G is written x u x T x,y u y db y U x,y t y db y (1) ij j i j j i j j BBN BBN where u j and t j represent omponents of displaements and trations of points belonging to the surfae of the grain G, the tilde refers to quantities expressed in a loal referene system set on the grain surfae, U ij and T ij are the 3D displaement and tration fundamental solutions for the anisotropi elasti problem. The integrals in Eq.(1) are defined over the surfae of the grain, that is generally given by the union of ontat interfaes B and external non-ontat surfaes B N. The model for the polyrystalline aggregate is obtained by writing Eq.(1) for eah grain and then omplementing the system so obtained with the boundary onditions u u or t t on B () i i i i N and with a set of suitable interfae equations, expressing the different possible states of an interfae. Interfae model. The interfae between two grains an be in three different possible states: pristine, when no damage is present and perfet bonding between the grains holds; damaged, when damage is present and intergranular trations and displaement jumps are lined through a ohesive tration-separation law; failed, when the grains are ompletely separated and the laws of the fritional ontrat mehanis hold. Let us onsider two adjaent grains G a and G b. When the interfae between them is in pristine state, the following interfae ontinuity equations hold

3 u ab u a u b 0 ( ontinuity) and t a t b ( equilibrium) (3) i i i i ii The previous equations express the absene of interfae displaement opening and the equilibrium of the interfae trations. The equilibrium equations always hold during the analysis, regardless the interfae state, so they are always assumed in the following, while the ontinuity equations are replaed by other laws expressing the interfae state during the interfae evolution. Damage is introdued at the interfae when the value of a suitable effetive tration overomes the interfae ohesive strength T max 1 eff n t max (4) t t t T In the previous equation, the loal trations are expressed in terms of loal normal and tangential ontribution, t n and t t. The parameters and give different weight to mode I and mode II loading. When the previous ondition is fulfilled, the following tration-separation laws are introdued at the interfae t 1 u 0 0 t u 1 1 d t Tmax 0 u 0 t u d (5) t u n u 3 where d [0,1] is a damage parameter given by 1 un u t d max { d} with d = Loading history un u t (6) where un and ut are the normal and tangential opening displaements at the interfae and u n and u t represent their ritial values in pure mode I and II respetively. The parameter d is the effetive opening displaement and the damage parameter is given by the maximum value reahed by the effetive displaement during the loading history. For d 0 the interfae is pristine, while d 1 expresses the failure of the interfae. Upon interfae failure, the tration-separation laws are replaed by the laws of the fritional ontat mehanis. In general, the miro-ra surfaes an be either separated or in ontat; moreover, two surfaes in ontat an either sti or slip over eah other. The equations of fritional ontat mehanis are not realled here, but the interested reader is referred to the literature on the subjet (see [8] and referenes therein). Disretization and numerial solution. The present formulation has the advantage that only meshing of the grain surfaes is required. The artifiial mirostruture is, in this ontext, a olletion of flat onvex polygonal faes. Plane triangular linear elements are used to disretize suh faes. Linear disontinuous triangular elements are implemented for representing the unnown boundary fields. Sine the Voronoi tessellations used for mirostruture modelling have stohasti nature, are must be taen to ensure mesh onsisteny and homogeneity to the greatest extent []. After disretization and lassial BEM treatment of the Eqs.(1), the following system an be written A1 0 0 y x 0 0 A (7) y Ng Ng x N g [ Ψ d ] ψ where the matrix blos A ontain olumns of the boundary element matries H and G orresponding y derive to the unnown displaements and trations of the -th grain, ontained in the vetor x, and

4 Ψ implements the varying oeffiients of the interfae equations, i.e. the ontinuity, ohesive and fritional ontat equations, for all the grain boundary interfaes. System (7) is sparse and speial solvers an be used for its solution. To tra the evolution o f a polyrystalline mirostruture, an inremental/iterative algorithm is employed. A load inrement is applied and the system solution is iterated until no violation of the interfae equations is deteted. At eah iteration, all the interfaes are heed, to assess whether any violation of the assumed interfae state is deteted. For example, if the effetive tration of a pristine interfae overomes the ohesive strength, then damage is initiated and the ontinuity equations are replaed by ohesive laws. Analogous hes are also done for interfaes in the ohesive or failed state. For the damaged interfaes, the ohesive law has to be updated if a loading ondition exists, while no update is required in the ases of unloading or reloading. When onvergene is reahed a new load inrement an be applied and the iterative searh is restarted. In this wor PARDISO [1] is used as iterative solver and a hybrid diret/iterative solution strategy is employed to speed up the numerial solution of the polyrystalline evolution problem. from the appliations of the nown boundary onditions. The blo d Numerial simulation of a Si miro-speimen under tensile load A prismati polyrystalline speimen subjeted to tensile load is onsidered. The speimen is omprised of N g 00 fully three-dimensional Si grains; an uniform displaement is applied over the bases and it is direted along the longer side, Fig.1. The material properties for rystalline Si are given in Table 1, the 3 6 grain size is ASTM G=1 (alulated number of grains per mm : n/ v [13]). The speimen's size is W W H with HW, its volume is V 8HW Ng Vgrain, where V grain is the estimated average grain volume. The mesh density is speified by dm 0.5 (see [] for further details about the meshing strategy). The properties of the interfaes are uniform and they are given in Table. Figure 1: Polyrystalline Si speimen with 00 grains subjeted to tensile load Table 1: Material onstant for single Si rystals.

5 Fig. 1 shows the mirostrutural ra pattern immediately before the omplete failure of the speimen. The orresponding marosopi stress-strain urve is shown in Fig.. The urve reports the value of the relevant omponent of the averaged stress tensor versus the nominal strain, obtained from the value of the applied displaement over the relevant speimen size. It is apparent how the speimen softens before the omplete failure. Table 3 reports some statistis about the onsidered mirostruture. Figure : Volume average stress omponent versus nominal strain for the onsidered speimen. T max GII G I Table : Seleted values for the interfae properties. N elements N DoFs interfaes T 17,031 7,709,660 ~5000s Table 3: Some statistis about the onsidered polyrystalline speimen. The time per load inrement was measured on 1-ore nodes. Summary A new three-dimensional formulation for the analysis of intergranular degradation and failure in polyrystalline materials has been developed. The polyrystalline mirostruture is represented as a threedimensional Voronoi tessellation, able to retain the main morphologial and rystallographi features of polyrystalline aggregates. The miromehanial model is expressed in terms of intergranular fields, namely displaement jumps and trations. The nuleation and evolution of intergranular damage has been followed using an irreversible ohesive law at the intergranular interfaes: this resulted partiularly straightforward, being the formulation itself expressed in terms of grain boundary variables. Upon omplete intergranular failure the fritional ontat analysis is introdued to follow the intergranular miro-raing proess, taing into aount separation, ontat and sliding between the miro-ra surfaes. A numerial test demonstrated the apability of the formulation to model the nuleation, evolution and oalesene of multiple damage and ras. For its nature, the developed formulation appears partiularly promising in the framewor of grain boundary engineering.

6 Anowledgements This researh was supported by a Marie urie Intra-European Fellowship within the 7th European ommunity Framewor Programme (Projet No 74161). Referenes [1] S. Nemat-Nasser, M. Hori, Miromehanis: overall properties of heterogeneous materials, North- Holland, Elsevier, The Netherlands, seond revised edition edition, (1999). [] I. Benedetti, M. H. Aliabadi, A three-dimensional grain boundary formulation for mirostrutural modelling of polyrystalline materials, omputational Materials Siene, 67, 49 60, (013). [3] T. I. Zohdi, P. Wriggers, An introdution to omputational miromehanis, Leture Notes in Applied and omputational Mehanis, vol. 0, Springer, Berlin, (005). [4] A. G. roer, P. E. J. Flewitt, G. E. Smith, omputational modelling of frature in polyrystalline materials, International Materials Reviews, 50, 99 14, (005). [5] I. Simonovsi, L. izelj, omputational multisale modeling of intergranular raing, Journal of Nulear Materials, 414, 43 50, (011). [6] N. handra, H. Li,. Shet, H. Ghonem, Some issues in the appliation of ohesive zone models for metal-erami interfaes, International Journal of Solids & Strutures, 39, , (00). [7] L.. Wrobel, M. H. Aliabadi, The boundary element method: appliations in thermo-fluids and aoustis., Vol. 1, John Wiley & Sons Ltd, England, (00). [8] M.H. Aliabadi, The boundary element method: appliations in solids and strutures., Vol., John Wiley & Sons Ltd, England, (00). [9] G. K. Sfantos, M. H. Aliabadi, A boundary ohesive grain element formulation for modelling intergranular mirofrature in polyrystalline brittle materials, International Journal for Numerial Methods in Engineering, 69, , (007). [10] S. Kumar, S. K. Kurtz, J. R. Banavar, M. G. Sharma, Properties of a three-dimensional Poisson- Voronoi tessellation: a Monte arlo study, Journal of Statistial Physis, 67, , (199). [11] R. B. Wilson, T. A. ruse, Effiient implementation of anisotropi three-dimensional boundaryintegral equation stress analysis, International Journal for Numerial Methods in Engineering, 1, , (1978). [1] O. Shen, K. Gartner, Solving Unsymmetri Sparse Systems of Linear Equations with PARDISO, Journal of Future Generation omputer Systems, 0, , (004). [13] ASTM E11-10, Standard Test Methods for Determining Average Grain Size. ASTM International. DOI: /E011-10, (010).