A geometric algorithm for discrete element method to generate composite materials

Transcription

1 A geometric algorithm for discrete element method to generate composite materials J.F. Jerier, F.V. Donzé, D. Imbault & P. Doremus Laboratoire Sols, Solides, Structures, Risques Grenoble, France ABSTRACT The modelling of composite materials behaviour as reinforced concrete counts among the present and future numerical studies realised by the Discrete Element Method. To improve these different studies, a geometric algorithm is developed to create reinforced sphere packing. It generates sphere assemblies with reinforcements (fibers, steel bars) modelled by sphere clumps. The algorithm acted only on spheres belonging to initial packing which overlap the reinforcement spheres. Once, the initial sphere packing and the reinforcement spheres are created and superposed, the program is executed. Thus, the overlaps are detected and the overlapping spheres are replaced due to a geometric function called the geometric inversion. The interest of the method is to prove an efficient tool in order to help all existing geometric algorithm to generate sphere packing with reinforcements. INTRODUCTION The composite materials are actually integrated in all the manufacturing (aeronautic, building, mechanical engineering, medical, ). Their utilization is necessary to solve problems of reliability, durability but also to reduce the fabrication cost. In the coming years, the development of materials reinforced by fibers or steel bars will be important. Their validation and their characterization should be realized in majority by reliable computation code based on finite element method (Roque et al, 2002) or on discrete element method (DEM) (Cundall et al, 1979). The difficulty for these codes is to reproduce the complex nonlinear behaviour of composite materials by example for the reinforced concrete with steel bars. Nonlinearities arise mainly from two major material factors: plasticity of reinforcements and compressive concrete, and cracking of concrete. Other nonlinearities are due to bond-slip interaction between reinforcements and concrete, aggregate interlock of cracked concrete, dowel action of reinforcements, concrete creep and shrinkage etc. A 1

2 For the finite element method, the modelling of reinforced concrete by steel bar is realized with smeared crack model or discrete crack model (matrix behaviour) and by Bond-slip model to reproduce the behaviour between the bar and the concrete (Yang et al, 2005). The results obtained with this method on a reinforced beam in 2D are really encouraging but it remains to realize progresses. The major aim to reach with this method is the modelling problems with multiple distributed cracks in the materials. Actually modelling composite materials with a continuum-type method (finite element method) makes tough the prediction of the crack propagation process and the final crack pattern. Because cracks are only allowed to initiate at nodes and cracks cannot initiate at two nodes in one element. For the discrete element method, the most of composite materials reinforced with fibers or bars can be modelled through sphere packing. This method integrates different local behaviour laws between spheres in contact (microscale) in order to reproduce physical phenomenons at the macroscale. The DEM has been used by numerous authors to simulate concrete fracture, brittle rock mass. The discontinuous medium created by the reinforcements furthers the study of composite s properties by the discrete element method in order to relate them to the macroscopic behaviour. In the literature, among the authors who have studied the composite materials in 2D and in 3D by the DEM, we can cite Magnier et al (1998), Shiu et al (2008) and Hentz et al (2005). These different authors have simulated successfully an impact on a reinforced concrete slab in order to study the damaging and the material behaviour submitted at a strong stress. However, the DEM simulations need a lot of time to be realized and even more in the case of composite materials. Because of supplementary spheres (reinforcement spheres) are fixed in the space, only algorithms very slow as dynamic algorithm can generate these types of packing. In the next section, we present the different algorithms are adapted to generate composite materials by sphere packing and in same time we develop our tool. Its aim is to replace in the assembly the spheres overlap the reinforcement spheres. Once, the initial sphere assembly and the reinforcement spheres are superposed, the program detects the overlapping spheres and their neighbors in order to replace them in respecting a criterion replacement. GENERAL PRINCIPLE In 3D, few algorithms are programmed to generate sphere packing with reinforcement spheres and the challenge to program the Algorithm for the initial sphere packing generation is still open. On the other hand, the studies on materials reinforced by fibers or bars in DEM interest more and more researchers and industrials for arming, medical, technology of plastics, etc. Therefore, it is necessary to create an efficient algorithm to further and to develop theses types of simulations. Until now, the dynamic method has used to generate sphere packing with reinforcements. In a first time the approach consists in representing the bars or the fibers by sphere clumps are fixed in the domain. Next, the initial sphere packing which represents the matrix is realized by deposition (Kadushnikov et al, 2001), radius expansion (Lubachevsky et al, 1990) or collective rearrangement (Jullien et al, 2000) method. However, the computation time enormously increase due to: The multiplicity collisions between the fixed spheres and the others. The large-size assembly necessaries to have a representative sphere packing in comparison to real material. A 2

3 Another algorithm based on the Jodrey-Tory s approach (Jodrey et al, 1985) has been programmed and integrated in freewares (SDEC and YADE). The interest of this algorithm compared to dynamic algorithms is that the Jodrey-Tory s algorithm is a geometric algorithm. It is normally faster than the others, because there are not dynamic equations to solve for the generation of assembly. In a first time, this algorithm places randomly overlapping spheres in a box to create the intial assembly. Next, the reinforcement spheres are placed and superposed at the initial packing. To finish, a collective rearrangement step is realised in order to delete all overlaps. This relaxation process consists in moving spheres along a specific direction (i.e., the direction is defined by the line going through the center of two spheres in overlapping). But, this type of rearrangement is again too slow time consuming and keeps residual overlaps inside final packing. The proposed tool to create reinforced sphere packing is inspired of the rearrangement method. The principle is very simple and can be presented in three stages: Step1: An initial sphere packing is realised in an area with a geometric algorithm Sep2: The reinforcement spheres are positioned in the same area. Step3: The spheres overlap the reinforcements are detected with their neighbours. Next, the concerned spheres are replaced in contact with the four nearest spheres tanks to a geometric function called the geometric inversion. Geometric inversion The inversion function (Borkovec et al) is an important function integrated in the geometric algorithm which places a sphere in contact with four spheres already placed. This function realises a special homothetic transformation on points composing geometry elements as: lines, circles, spheres, planes. It is defined in 1D (see figure 1a) and in 2D (see figure 1b) with the following examples: Figure 1. Geometric inversion: A) Inversion of point B, B) Inversion of points A and B placed on the line (d) In 1D, the inversion (inversion noted I) of center O and radius k transforms a point B into inverse point B. Such as B and B are on the same ray going through O, and OB times OB are equal the squared radius (see figure 1A). To resume, B = I(B) with B the inverse point of B and with O, B, B are aligned: OB. OB ' = k² In 2D, if the inversion is applied on a line (d) or more precisely on all points being part of the line (d). With the inversion center O, a distinct point compared with (d) and the radius k. We obtain inverse points which defines a circle, with this circle going through the point O (see A 3

4 figure 1B). If the inversion with center O and the radius k is applied on two points (A and B) are parts of the line (d). The resulting inverse points (A and B ) are placed on a circle defined from points O and B (see figure 1B). More, the inversion function is an involutive function (I I = Id). Therefore, a circle can be transformed into a line, if the center O is placed on the circle. In 3D, the inversion works in the same way as in 2D. A sphere can be created from a plane (see figure 2 D and E) and inversely due to the involutive property. The developed tool integrates this function to create a composite material modelled by non-overlapping spheres. Reinforced packing Our method consists in detecting the overlapping spheres and replacing them. To replace these spheres, the algorithm integrates a routine based on the geometric inversion (see figure 2). This routine can be developed in six steps: Step 1: Once the sphere P5 to replace is detected. The algorithm records the sphere P5 number and the spheres neighbours number. Next, the sphere P5 is deleted. Step 2: Four spheres (P1, P2, P3, P4) are chosen among the neighbours. The sphere radius is decreased by a value equals to r=min{r, R1, R2, R3} (see figure 2B). Step 3: The algorithm chose the sphere with a radius null (P1) as inversion center (O). Next, the inversion is applied on the spheres (P2, P3, P4) with an inversion radius k (k is a positive integer). Three inverse spheres are obtained: P'1(O'1, R"1), P'2(O'2, R"2) and P'3(O'3, R"3) (see figure 2C). Step 4: A plane put on the spheres (P'1, P'2, P'3) must be created (see figure 2D). If this plane exists ( i.e, if the spheres are not aligned), then the sphere P5 can be replaced in contact with the four initial spheres (P,P1,P2,P3). Step 5: Once the plane has been created, the inversion of center O and radius k is applied on the inverse spheres (P'1, P'2, P'3) and on the plane. Due to the involutive property, the spheres P1(O1, R'1), P2(O2, R'2), P3(O3, R'3) and a new sphere P5(O4, R'4) are obtained (see figure 2E). Step 6: The radius of spheres (P, P1, P2, P3) are increased by a value r, whereas the radius of sphere P5 is decreased by a value r (see figure 2F). The effectiveness procedure is based on the fact that: the inversion is an involutive function and two objects in contact remain in contact after inversion. This developed geometric procedure is very important in the algorithm's approach because it is the major element. The others functions executed by the algorithm are to list the neighbours and to apply the routine on different groups of four spheres (spheres nearby the sphere P5) until the new position of the sphere P5 generates not overlaps with others spheres and respects the size distribution. In order to know, how this method works on large sphere packing, reinforced concrete samples are realised in discrete element. A 4

5 Figure 2. The substitution of sphere by a geometric procedure SIMULATION OF REINFORCED PACKING In this section, different packing are realised to verify the efficiency geometric algorithm. Three beams (0.5x0.25x0.25) in square section reinforced by five bars are created; the beams are represented by sphere packing with a size distribution varies between: (0.0025, ), so the ratio (min(radius)/max(radius)) value is equal to 3. These beams are different by their relative density; three values are realised: 0.5, 0.6 and The interest to generate these three packing is to know, if the geometric algorithm is capable of repositioning spheres inside a reinforced assembly more and more dense. Every initial packing are created with the same geometric algorithm (Jerier et al, 2008), this geometric algorithm generates random dense sphere packing without overlaps from a tetrahedral mesh. It densifies the packing up to obtain a number of spheres required or the relative density value wanted (see figure 3). The five reinforcement bars are modelled aside by 325 spheres with radii equal to Once the reinforcements and the packing are realised, they are superposed and the developed algorithm is executed. Figure 3. Left) Five reinforcement bars Center) The initial sphere packing Rigth) The reinforced assembly with bars and initial packing superposed The results for each packing are listed in the table 1, these results show the number of overlapping spheres to replace inside the packing and the number of spheres are replaced. The overlapping spheres not replaced are deleted because the obtained final reinforced packing has not overlaps. A 5

6 Table 1: Results concerning the overlapping spheres repositioned inside the packing. Reinforced packing Number of spheres Relative density Number of spheres to replace Number of spheres replaced Packing Packing Packing The results show that in the worse case one sphere in five to replace is deleted, this value decreases when the ratio value increases or the relative density is lower. In order to have a point of view some reinforcement spheres in the assembly, you can study the figure 4 which represents a view inside the packing 3 and you can observe the spheres are in contact with the reinforcements. Figure 4. The reinforcement spheres in contact with the spheres appertaining to the initial sphere packing With this tool, it is possible to create large reinforced assemblies as them used in the Shiu et al s simulations (see figure 5). The reinforced assembly is composed of spheres generated by a geometric algorithm (Jerier et al) and reinforcement spheres have been positioned due to the developed tool. The generation time plus the rearrangement time is 12 hours with spheres replaced in spheres to replace. Figure 5 Concrete slab entirely reinforced by steel bars A 6

7 CONCLUSION The aim of our algorithm is to create reinforced sphere packing from reinforcement spheres and initial sphere packing generated by another algorithm. The developed method is based on a geometric routine which replaces the spheres are overlapping the reinforcement spheres inside the assembly. The spheres are repositioned due to the inversion function in respecting the sphere size distribution of assembly and the sphere neighbours. The developed algorithm has a first advantage which is its adaptability at all algorithms capable generating only initial sphere packing. The second advantage is the calculation time very short due to the geometric function integrated in this routine. REFERENCES Borkovec M & de Paris W 1994, The fractal dimension of the apollonian sphere packing. Fractals Vol 2 pp Cundall P.A. & Strack O.D.L. 1979, A discrete numerical model for granular assemblies. Géotechnique Vol 29 pp Hentz S & Donzé F.V. & Daudeville L. 2005, Discrete elements modelling of a reinforced concrete structure submitted to a rock impact, Italian Geotechnical Journal, XXXIX, n 4, Jerier J-F. & Imbault D. & Donzé F.V & Doremus P. 2008, A geometric algorithm based on tetrahedral mesh to generate dense polydisperse sphere packing (submitted) Jodrey W & Tory E 1985, Computer simulation of close random packing of equal spheres. Physical Review A Vol. 32:pp Jullien R & Meakin P 2000 Computer simulations of steepest descent ballistic deposition. Colloids and Surfaces A: Physicochemical and Engineering Aspects Vol.165:pp Kadushnikov R & Nurkanov E 2001 Investigation of the density characteristics of three-dimensional stochastic packs of spherical particles using a computer model. Powder Metallurgy and Metal Ceramics Vol. 40:pp Lubachevsky & B, Stillinger F 1990, Geometric properties of random disk packings. Statistical Physics Vol. 60:pp Magnier S.A. & Donzé. F.V Numerical simulations of impacts using a discrete element method. Mech. Cohes.-Frict. Mater. 3: Roque L.S.P & Wolnei I.S 2002, Object oriented implementation of reinforcement and bond-slip finite element models for nonlinear analysis of reinforced concrete structures. Mecanica Computacional Vol XXI pp Shiu W & Donzé F.V & Daudeville Compaction process in concrete during missile impact: a Dem analysis (submitted) Yang Z.J & Chen J Finite element modelling of multiple cohesive discrete crack propagation in reinforced concrete beams. Engineering Fracture Mechanics Vol 72 pp A 7

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