3D computer simulation of mechanical behavior of a brittle porous material under uniaxial compression is considered. The movable cellular automaton method, which is a representative of particle methods in solid mechanics, is used for computation. In an initial structure the automata are positioned in FCC packing. The pores are set up explicitly by removing single automata from the initial structure. The computational results show that the curves of dependence of strength and elastic properties of the modeled specimens on porosity have a break at the porosity value about 20 %, i.e. percolation threshold. The obtained results are in close agreement with available experimental data

Konovalenko, Igor S.

Loginova, Darya S.

Psakhie, G.

Roman, Nikita V.

Smolin, Alexey Yu.

English

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Influence of porosity percolation on mechanical properties of ceramic materials. 3D simulation using movable cellular automata

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Xuelong LU,Hiromasa KITAKAWA,Takuya MUKAI and Yuzuru SAKAI.   12 5 CONCLUSIONS Authors addressed the problem of make a model from medical images directly, and simulate in high velocity impact of human head crash by SPH method. From now, authors need to make more precise human head model to assessment of human body injury (Include: fracture, retraction, amputate ). And also authors must work for to apply SPH simulation and real time rending to three-dimensional virtual surgery (Include: dissection, ablation, bleeding). Of course, in a traffic accident, head impact will come from any direction, authors need to calculate not only a head but also a man in the car or in other situation. 6 ACKNOWLEDGEMENT Thank here, the CT/MRI was taken under the cooperation of teacher Tokyo Medical & Dental University Tsuruda Jun. REFERENCES  [1] Lucy,L.B,,”a numerical approach to the testing of the fission hypothesis”Astron,J, 82,(1977),1013-1024 [2] Gingold,R.A,Monaghan,J.J.,”Smoothed particle hydrodynamics: theory and application to non-spherical stars”,Monthly Notices of the Royal Astronomical Society 181(1977),375-389 [3] W.Benz, Smooth particle hydrodynamics: A review, Harvard-Smithsonian Center for Astrophysics, Preprint 2884,1989 [4] Monaghan,J.J.,Smoothed particle hydrodynamics, Annu Rev. astron. Astrophys. 30(1992) 543-574 [5] L.D. Libersky and A.G. Petschek, Smooth particle hydrodynamics with strength of materials, Advances in the free Lagrange Metho , Lecture Notes in Physics 395(1990) 248-257 [6] Johnson., G. R, A.G. Petscheck and R.A. Stryk, Incorporation of an SPH option in the EPIC code for a wide range of high velocity impact computations, Int.J. Impact Engrg. 14 (1993) 385-394 [7] A.G. Petschek and L.D Libersky, Cylindrical smoothed particle hydrodynamics, J. Cmput.Phys. 109 76-83 (1993) [8] Swegle, J.W., Attaway, S.W., Heinstein, M.W., Mello, M.W., Hicks, D.L., “An Analysis of Smoothed Particle Hydrodynamics”, Report No. SAND93-2513-UC-705, Sandia National Laboratory, Albuquerque, NM, (1994). [9] S. Koshizuka, H. Tamako and Y. Oka , A Particle Method for Incompressible Viscous Flow with fluid Fragmentation, Computational Fluid Dynamics Jounal, 4, 1, (1995),pp.29-46. [10] Johnson., G. R., “Artificial viscosity effects for SPH impact computations”, International Journal of Impact Engineering, Volume 18, Number 5, July 1996, pp. 477-488. [11] Nick Foster and Dimitri Metaxas. Realistic animation of liquids. Graph.Models Image Process., Vol. 58, No. 5, pp. 471–483, 1996. [12] Song moo seop, Sakai Yuzuru,Yamasita Akihiko “Elastic-plastic Analysis by SPH method:1st report,Small displacement problem in 2-Dim ” JSME,A 68(669) pp.772-778(20020525) [13] Sakai Yuzuru, Yang zongyi, Jung youngguan,Imcompressible viscous flow analysis by SPH,JSME,B70,666,(2004-8) [14] Tanaka, Analysis of flow between Red Cell and Fluid with MPS method, The 18th CFD symposium C2-3(2004) [15] Mark Carlson, Peter J. Mucha, and Greg Turk. Rigid fluid: animating the interplay between rigid bodies and fluid. ACM Trans. Graph., Vol. 23, No. 3, pp. 377–384, 2004. [16] KoukPoh Tang, Yuzuru Sakai, Nobuki Yamagata: Heat conduction and Heat convection Analysis by SPH Method, JSME 18th Computational Mechanics Conference [No.05-2] pp.585-586,(2005) [17] Y. Zhu and R. Bridson: Animating Sand as a Fluid, SIGGRAPH'05, pp. 965-972 (2005) [18] Jos Stam. Stable fluids. In Proceedings of the 26th annual conference on Computer graphics and interactive techniques, pp. 121–128. ACM Press/Addison-Wesley Publishing Co., 1999. [19] Oh-Young Song, Hyuncheol Shin, and Hyeong-Seok Ko. Stable but nondissipative water. ACM Trans. Graph., Vol. 24, No. 1, pp. 81–97, 2005. [20] Zanuttigh, B., Lamberti, A.: Experimental analysis of the impact of dry avalanches on structures and consequences for debrisflows. J. Hydraul. Res. 44(4), 522–534 (2006) [21] Chin G L, Lam K Y, Liu G R. Advances in Meshfree and X-FEM Methods ［M］. Liu G R Ed. World Scientific, Singapore , 2002. [22] Johnson., G. R. and S.R Beissel, Normalized smoothing functions for SPH impact computations, Int. J. Numer . Methods Engrg., Accepted for publication [23] S.W Attaway, M.W. Heinstein and J.W. Swegle, Coupling of smooth particle hydrodynamics with the finite element method, Nucl. Engrg. De. 150 199-205 (1994) II International Conference on Particle-based Methods - Fundamentals and ApplicationsPARTICLES 2011E. Oñate and D.R.J. Owen (Eds)INFLUENCE OF POROSITY PERCOLATION ONMECHANICAL PROPERTIES OF CERAMIC MATERIALS.3D SIMULATION USING MOVABLE CELLULARAUTOMATAALEXEY YU. SMOLIN∗†, NIKITA V. ROMAN†, DARYA S. LOGINOVA†,IGOR S. KONOVALENKO∗ AND SERGEY G. PSAKHIE∗†∗Institute of Strength Physics and Materials Science (ISPMS)Siberian Branch of Russian Academy of Sciencespr. Akademicheskiy 2/4, 634021, Tomsk, Russiae-mail: root@ispms.tomsk.ru, www.ispms.ru/†Tomsk State University (TSU)pr. Lenina 36, 634050, Tomsk, Russiae-mail: rector@tsu.ru, www.tsu.ruKey words: Porous Materials, Mechanical Properties, Computer Simulation, MovableCellular AutomataAbstract. 3D computer simulation of mechanical behavior of a brittle porous mate-rial under uniaxial compression is considered. The movable cellular automaton method,which is a representative of particle methods in solid mechanics, is used for computation.In an initial structure the automata are positioned in FCC packing. The pores are setup explicitly by removing single automata from the initial structure. The computationalresults show that the curves of dependence of strength and elastic properties of the mod-eled specimens on porosity have a break at the porosity value about 20%, i.e. percolationthreshold. The obtained results are in close agreement with available experimental data.1 INTRODUCTIONIt is well known that porosity dependence of strength and elastic properties of porousmaterials is determined by the pore morphology. If porosity is small, each pore is closedand isolated form the others. With porosity increasing the number of pores becomesgreater and the distance among pores decreases. When porosity reaches percolationthreshold the pores are not closed anymore and consequently form a new morphology.Thus, dependence of strength and elastic properties on porosity should be changed whenpercolation threshold is exceeded.1249Alexey Yu. Smolin, Nikita V. Roman, Darya S. Loginova, Igor S. Konovalenko, Sergey G. PsakhieNote, it is very difficult to study the problem experimentally, because of stochasticnature of the pore morphology in real materials. For example, it is impossible to fabricatespecimens with various porosity value and the same porous structure. At the same timepotentialities of the modern computational mechanics allow to study in detail not onlydeformation of complex heterogeneous materials but also fracture of such materials.2 COMPUTATIONAL MODEL FOR POROUS CERAMICSFor 3D computer simulation of mechanical behavior of a porous material under uni-axial compression we used the movable cellular automaton method (MCA) [1]. MCA isa method in discrete computational mechanics which allows modeling deformation andfracture of heterogeneous material with taking into account the structure of the materialexplicitly. Modeled material is represented as a set of elements of finite size (automata)interacting with each another according to some pre-determined rules. Mechanical interac-tion among automata are defined by so called response function, which can be interpretedas a local stress-strain diagram. Equation of translation motion and rotation of the au-tomata are written in many-body approach. This means that interacting force betweena couple of automata depends not only on position and state of these automata but alsoon position and state of their neighboring automata. This allows modeling as solid alsoas granular material behavior within the framework of one MCA method.Response function of automata used in this study corresponded to ZrO2 ceramics withaverage size of pores commensurable with the grain size of the material. According tothe pore distribution diagram of this material [2] the automaton size in the computationswas 1µm. All the modeled specimens where bricks with a square base. The loading wasapplied by assignment of vertical velocity for automata of the top layer of a specimen.At the same time the substrate automata were rigidly fixed. At the initial stage of theloading velocity of the upper automata increased by a sinusoidal law from 0 to 1 cm/sand then was constant.Pores were generated by removing single automata from the basic structure. The poros-ity values were varied from 0 to 50 %. Note, that the maximal porosity on the assumptionof pore disconnection in closed packing of elements for 2D is 1/2 = 50 % (Fig. 1,a) whilefor 3D it is 1/4 = 25 % (in Fig. 1,b one closed packing plane is shown for FCC). In this(a) (b)Figure 1: Periodic pores (black) in closed packing2250Alexey Yu. Smolin, Nikita V. Roman, Darya S. Loginova, Igor S. Konovalenko, Sergey G. Psakhiecase the pores are situated regularly.The maximal value of closed porosity for random removing of single automata fromFCC packing is 17.6 %. The percolation limit for 3D FCC packing is 19.8%. Consequently,all the specimens used in this study with porosity greater than 20 % contained clusters ofinterconnecting pores. For the specimens with porosity greater than 25% their porositywas interconnected.In [3] it is shown that porosity structure defines not only elastic, but also strengthproperties of a material. The clusters of interconnecting pores present a new element ofstructure in addition to single closed pores. Thus, with porosity increasing the changes ofelastic and strength properties of a material are expected after transition of the percolationlimit.3 DETERMINING THE REPRESENTATIVE VOLUMEAt the first stage it is required to determine the representative volume of the modeledmaterial. For material without pores the representative volume was determined based onfour solid specimens. For this purpose the convergence of effective elastic modulus of thespecimen Eeff to the automaton elastic modulus E0 with increasing the specimen size wasanalyzed.The convergence was assumed to be satisfied if Eeff differed from E0 not greater then3 %. The computations showed the representative volume was equal 10µm of the brickbase size. The fracture pattern (cracks) in this specimen is shown in Fig. 2. The cracks aregenerated from the stress concentrators and propagate along maximum tangential stressdirection.(a) (b)Figure 2: Fracture pattern of a solid ceramic specimen under compression: a) lines connect linkedautomata, b) lines connect unlinked automataIf a material contains pores the cracks may initiate not in the corners but in the bulk3251Alexey Yu. Smolin, Nikita V. Roman, Darya S. Loginova, Igor S. Konovalenko, Sergey G. Psakhieregion of the highest local porosity. The direction of cracks propagation in such materialis defined by the porous topology. Fig. 3 shows the first cracks in porous specimens withvarious values of porosity. One can see that the failure behavior of a porous material withlarge porosity and the failure of a solid material differs dramatically. In particular, thepath of the crack propagation in porous material is rather crinkly.Figure 3: First cracks in specimens with various values of porosity C under compressionThe fracture pattern of the modeled brittle porous 3D specimens qualitative corre-sponds to 2D simulation results [3]. The quantitative difference is defined by the fact thatporosity in 2D specimen is “interconnected” in the normal direction to the computationalplane.To analyze the dependence of elastic and strength properties of the modeled materialon its porosity it is necessary to determine the size of representative specimen (volume).It is obvious that the representative volume for a porous material is different for differentvalues of porosity. The simulations conducted show that for porosity value less then 15 %the representative volume is equal 20 µm (Fig. 4). For porosity value from 15 up to 35 %the size of representative volume iss found to be 30 µm (Fig. 4,b). For porosity variedfrom 35 up to 50 % — 40 µm (Fig. 4,c).4252Alexey Yu. Smolin, Nikita V. Roman, Darya S. Loginova, Igor S. Konovalenko, Sergey G. PsakhieFigure 4: Convergency of effective elastic moduli of the modeled specimens on their basis size for variousporosity: a) 10%, b) 25 %, c) 50 %4 STUDYING ELASTIC AND STRENGTH PROPERTIES OF POROUSCERAMICSThe loading diagrams of 3D modeled specimens are shown in Fig. 5. They are in closequalitative agreement with 2D simulation results [3, 4]. A quantitative difference between2D and 3D results could be explained by the different influence of porosity on effectiveporous characteristics in 2D and 3D tasks.Figure 5: Loading diagrams for ceramics with various porosity C = n·10 %First, it has to be noted that 3D specimens with porosity stochastically distributed inspace can demonstrate quasi-viscous regime of fracture like 2D models [4]. To be able tofailure in this regime the porosity of 3D specimens has to be greater than 40 % (Fig. 5).Let us consider dependence of the effective elastic modulus of 3D specimens on itsporosity. In Fig. 6,a each square represents the value averaged on five representativespecimens with various pore distribution in space. This plot obviously can be divided intotwo characteristic parts connected with porous structure: the first corresponds to closed5253Alexey Yu. Smolin, Nikita V. Roman, Darya S. Loginova, Igor S. Konovalenko, Sergey G. Psakhiepores (5. . . 20 %), the second corresponds to clusters of interconnected pores (20. . . 50 %).The experimental data taken from [2] are presented in Fig. 6,b for comparison. One cansee the computational results are in close qualitative agreement with the experimentaldata. However, the inclinations of approximating lines showing degree of the influence ofporosity on elastic characteristics of a material within the intervals are little bit differentfor the experimental and computational data. It can be explained by stochastic choiceof elements for pore generation in the modeled specimens, and hence by the orientationsof clusters of interconnected pores, which leads to reduction of porous structure influenceon mechanical properties of a material. Besides, according to [2] percolation transitionsin ceramics ZrO2 cause microstructure change. In particular, internal stresses in ceramicswith continuous porous structure restrain grain growth. Thus, simulation of uniaxialcompression of brittle porous 3D specimens by the movable cellular automaton methodshowed that percolation transition from the closed pores to interconnected pores in aporous material leads to change in the dependence of its elastic properties on porosity.(a) (b)Figure 6: Dependences of elastic modulus of ceramics on its porosity in logarithmic coordinates: a) sim-ulation; b) experimental dataIt is necessary to notice that this result could be obtained only in three-dimensionalsimulation because two-dimensional specimens with interconnected porosity are not topo-logically connected and do not resist to the mechanical loading. Besides, the porosity valueat which formation of interconnected clusters occurs in a two-dimensional case (22.4 %)is considerably less than the corresponding percolation limit (69.6 %).Fig. 7 shows the simulated and experimental dependencies of ceramics strength on itsporosity. Again it can be divided into two characteristic parts connected with porousstructure.6254Alexey Yu. Smolin, Nikita V. Roman, Darya S. Loginova, Igor S. Konovalenko, Sergey G. Psakhie(a) (b)Figure 7: Dependences of ceramics strength on porosity in logarithmic coordinates: a) simulation, b) ex-perimental data5 CONCLUSIONSThe simulations performed by the movable cellular automaton method show that thecurves of dependence of strength and elastic properties of the brittle porous specimens onporosity have a break at the porosity value about 20 %, which corresponds to percolationthreshold. The obtained results are in close agreement with available experimental data.In summary it is necessary to notice that for the purpose of the further applicationof the proposed 3D MCA model of porous ceramics it is necessary to extend the modelwith possibility of describing nonlinear properties of materials (degradation of elasticproperties, relaxation at high-speed loading, etc.)ACKNOWLEDGEMENTSThis work was supported by the interdisciplinary integration projects No 32 and No 113of the Siberian Branch of the Russian Academy of Sciences.REFERENCES[1] Psakhie, S.G. Horie, Y. Ostermeyer, G.P. et al. Theor. Appl. Fract. Mech. (2001) 37:311–334.[2] Kulkov, S.N. Buyakova, S.P Maslovsky, V.I. Bulletin of Tomsk State University (2003)13:34–57. [in Russian][3] Konovalenko, Ig.S. Theoretical research of deformation and failure of porous materialsfor medical and biomechanical design. PhD thesis, Tomsk (2007). [in Russian][4] Smolin, A.Yu. Konovalenko, Ig.S. Kul’kov, S.N. and Psakhie, S.G. Tech. Phys. Lett.(2006). 32(17):7–14.7255

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Influence of porosity percolation on mechanical properties of ceramic materials. 3D simulation using movable cellular automata

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