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Experimental and numerical estimation of strength and fragmentation of different porosity alumina ceramics
Materials and Design 88 (2015) 1042–1048
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Materials and Design journal homepage: www.elsevier.com/locate/jmad
Experimental and numerical estimation of strength and fragmentation of different porosity alumina ceramics S.B. Sapozhnikov a,⁎, O.A. Kudryavtsev a, N.Yu. Dolganina b a b
South-Ural State University, Physics Department, Chelyabinsk, Russian Federation South-Ural State University, Computational Mathematics and Informatics Department, Chelyabinsk, Russian Federation
a r t i c l e
i n f o
Article history: Received 11 December 2014 Received in revised form 10 August 2015 Accepted 23 August 2015 Available online 1 September 2015 Keywords: Ceramics Strength Fragmentation Porosity Voxel numerical simulation
a b s t r a c t Ceramics are widely used in many applications as construction material. However, their strength and fragmentation details under tensile loading are not fully understood. In this work, the results of experimental and numerical strength investigations of hot-pressed alumina ceramics are presented. The disk-shape specimens with different porosities were subjected to bending test up to failure. Finite element analysis was performed for estimation of ceramic disk tensile strength. Elastic properties of porous ceramics for numerical simulations were determined with use of existing modulus–porosity relations and current experimental data. Three-dimensional models of test specimens with random distribution of strength and porosity based on voxel discretization were created by C++ program and implemented in LS-DYNA. Then they were used to simulate the specimen fracture and fragmentation. Obtained numerical data are in a good agreement with experimental results. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Ceramic materials possess many excellent mechanical properties like exceptional hardness, wear resistance, high compressive strength, low density, etc. They are widely used in aerospace and defense industries [1,2]. At the same time, low tensile strength (approximately one order of magnitude below the compressive strength) and inherently brittle behavior of ceramics are the main limitations for their extensive use in structural applications. Tensile strength is one of the main parameters that determines the ceramic's quality [3] (due to the presence of structural defects — pores and micro-cracks). There are several different approaches to take into account the details of the deformation and fracture of brittle materials with high hardness. Continuously damaged homogeneous material models [4] in different modifications [5] are the most popular models for high-velocity impact dynamic problems, primarily in the context of armor protection. Examples of the application of this model can be found in [6–12]. However, these models require a large number of expensive and complicated experiments like Edge-On-Impact (EOI) or Depth-Of-Penetration (DOP) tests to determine the material parameters for numerical analysis. Recently, the cohesive/volumetric finite element (CVFE) approach was used for ceramic fracture simulations [13–16]. The implementation of cohesive zone into numerical analysis was successful in many cases. In [17,18], the authors directly took into account the microstructure of ⁎ Corresponding author at: Lenin Ave., 76-355, Chelyabinsk, 454080, Russian Federation. E-mail address: [email protected] (S.B. Sapozhnikov).
http://dx.doi.org/10.1016/j.matdes.2015.08.117 0264-1275/© 2015 Elsevier Ltd. All rights reserved.
the ceramic material into fracture analysis to obtain a good agreement with experimental data. In addition, cohesive element approach was used to simulate ceramic fracture and cracks propagation under highvelocity impact conditions [19,20]. However, selection of cohesive elements parameters is still a labor-intensive task. Here we have to say that the cohesive approach leads to considerable computational cost due to increasing of the size of numerical problem in comparison with traditional LS-DYNA approach based on the death-of-element analysis. It should be noticed that the correct prediction of the target and projectile fragmentation is absolutely needed to estimate the damage of neighboring parts. In this paper, experimental and numerical tensile strength estimations of hot-pressed porous alumina ceramics were performed. Firstly, ceramic disk specimens were subjected to bending tests. Then we computed tensile strength of ceramic materials with different porosities with use of FEA and known modulus-porosity relations. These data were used in the novel low parametric model of ceramics based on the voxel [21,22] representation of material microstructure, stochastic distribution of local behaviors and the death-of-element analysis. This model allows us to obtain the fracture loads and pictures of fragmentation for porous ceramic specimens which are in good agreement with experimental data. 2. Specimens' fabrication An alumina powder produced by condensation of vapor on electrostatic filter was used as raw material. Fig. 1 shows the microstructure of this powder, average particle size is about 220 nm.
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Table 1 Properties of specimens for each group. Specimens group
Applied pressure (MPa)
Mass (g)
Height (mm)
Density ρ, g/cm3
Relative density (ρ/ρ0)
1 2 3 4
19.4 29 38.2 48.5
5.93 ± 0.03 5.94 ± 0.01 5.94 ± 0.02 5.92 ± 0.02
3.83 ± 0.05 3.32 ± 0.04 3.12 ± 0.03 3.05 ± 0.02
3.06 ± 0.04 3.53 ± 0.04 3.75 ± 0.03 3.82 ± 0.01
0.77 0.89 0.94 0.96
3. Elastic constants of porous alumina It is necessary to have mechanical properties at each density level for strength assessment of test specimens. The results of a brief review of some relations of ceramics elastic properties vs the porosity are presented below. In [23], the authors proposed relations of a shear modulus G and bulk modulus K with porosity for different materials in the porosity range of 0–40%: Fig. 1. Alumina powder (agglomerates).
All experiments were made with HP20-4560-20 hot-pressed system manufactured by the Thermal Technology LLC (Fig. 2). The graphite die with an external diameter of 100 mm and an internal diameter of 25.4 mm was filled with 5.95 ± 0.05 g of the powder and placed into the vacuum hot-press furnace with graphite heating elements. Hot pressing experiments were performed at different compression levels: 19.4, 29, 38.2 and 48.5 MPa to obtain ceramic specimens with different densities. The same heating rate of 20 °C/min, sintering temperature of 1200 °C, dwell times of 20 min were used for the sintering experiments. The vacuum of ~25 Pa was maintained during technological process. The cooling rate was 20 °C/min until a temperature 1000 °C was reached. The temperature control unit was off after that. Seven samples were manufactured for each compression level to get the density range between 3.06 and 3.82 g/cm3 (see Table 1). Density of pore-free material was assumed to ρ0 = 3.99 g/cm3 [2].
P ; K ¼ K max exp −q 1−P
ð1Þ
P ; G ¼ G max exp −g 1−P
ð2Þ
where P — porosity (dimensionless), Kmax an Gmax — bulk and shear moduli of pore-free material, g and q — material constants. For polycrystalline α-Al2O3 constants g and q are 3.960 and 1.617 respectively. Values g and q for other materials (such as iron or copper) can be found in [23]. The pore-free alumina elastic constants are presented in Table 2. The shear and bulk moduli were calculated for each level of the relative density (porosity) in accordance with Eqs. (1) and (2), see Table 3. The values of elastic modulus and Poisson's ratio were obtained by using the theory of isotropic elasticity.
Fig. 2. HP20-4560-20 hot-pressed system.
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Table 2 Elastic properties of pore-free polycrystalline α-Al2O3 at ambient temperature [2]. Elastic modulus Emax (GPa)
Bulk modulus Kmax (GPa)
Shear modulus Gmax (GPa)
Poisson's ratio νmax
402
251
163
0.233
Table 3 Elastic properties of porous alumina. Specimens group
Relative density
Elastic modulus E (GPa)
Bulk modulus K (GPa)
Shear modulus G (GPa)
Poisson's ratio ν
1 2 3 4
0.77 0.89 0.94 0.96
214 317 359 376
78 155 197 216
103 136 150 156
0.04 0.16 0.195 0.208
Table 4 Data for numerical simulations and corresponding average calculated tensile strength of the ceramic materials of disk specimens. Porosity
Elastic modulus E (GPa)
Poisson's ratio ν
Fracture load P (N)
Tensile strength (MPa)
0.23 0.11 0.06 0.04
205 300 345 363
0.199 0.222 0.229 0.230
2455 3115 3521 3902
182 299 377 429
As can be seen in Table 3, the value of Poisson's ratio at 23% porosity is 0.04 and becomes a negative with a further porosity increasing which is not physical. Now there are no adequate Poisson's ratio — porosity dependences for relative density of the material less than 0.88 [29].
Fig. 3. Comparison of different elastic modulus–porosity relations.
Experimental studies [27,29] have shown that the Poisson's ratio of αAl2O3 is weakly dependent on the porosity and slightly decreases with increasing porosity. In [29] Poisson's ratio was determined for α-Al2O3 with a porosity in the range 0–25% and these data were used in current work (see Table 4). There are several elastic modulus–porosity relations for ceramic [24–28]. It can be noticed, that reliable experimental data on the modulus of elasticity of porous alumina obtained by ultrasonic methods when the porosity less than 40%. A semi-analytical equation for the elastic modulus Е(Р) proposed in [28] is adequate if porosity less than 30%. At a higher porosity level, this function begins to increase which does not correspond to the experimental data. Other relations (see Fig. 3) are similar. In the further calculations, we are used Pabst–Gregorova relation [25] E ¼ E max ð1−P Þ 1−
P ; 0:684
ð3Þ
which gives the average results between the equations presented in [23,27]. Table 4 shows the values of the elastic modulus at different porosity levels obtained from Eq. (3). 4. Bending tests and strength assessment The disk-shape specimens were loaded in piston-on-ring bending tests in order to determine the tensile strength of porous ceramic material. In this case, the specimen was supported by a ring near external edge and equidistant from the center, and it was loaded through piston in the central region (Fig. 4). It should be noted that there are other methods for material biaxial testing such as piston-on-three ball and ring-on-ring test, see [30–32] for details. All bending tests were performed on universal testing machine Instron 5882. The area of maximum tension stresses appears at the center of bottom disk surface, where the stress is biaxial. Maximum principal stress failure criterion (Rankine or Coulomb criterion) can be used in this
Fig. 4. Diagram of disk subjected to piston-on-ring test.
S.B. Sapozhnikov et al. / Materials and Design 88 (2015) 1042–1048
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Fig. 5. Specimens after failure in ‘piston-on-ring’ tests.
case to determine tensile strength with sufficient accuracy for engineering applications. Ring with diameter of 2Rr = 19 mm and piston with diameter of 2Rp = 5 mm were used for experiments. The samples after the tests are shown in Fig. 5. Finite element analysis (FEA) was used to simulate ‘piston-on-ring’ tests. The contact between the experimental equipment and specimen was frictional with friction coefficient of 0.1. Materials are considered to be isotropic and linear elastic. The 1/4 model was implemented into FEA to reduce the computational cost because of the symmetry reasons. An example of the radial stress distribution with FE mesh shown in Fig. 6. The values of elastic constants and average experimental fracture loads for each group of specimens that were used for numerical simulations as well as corresponding calculated tensile strength of ceramic
materials (Rankine criterion) with different porosity levels are summarized in Table 4. One of the most commonly used forms of equations describing the effect of porosity on mechanical properties is the Duckworth equation [33] σ ¼ σ 0 expð−nPÞ;
ð4Þ
where σ — the tensile strength, P — the volume fraction of porosity, n — an empirical constant and the subscript 0 indicates zero porosity. Fig. 7 shows the experimental values of the tensile strengths of the tested ceramics versus porosity with approximating curve (Eq. (4)).
Fig. 6. Contours of radial stress for specimen with 4% of porosity.
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distribution). Voxel approach is relatively little used in the practice of engineering calculations, despite the high numerical efficiency (minimum computational cost or CPU time) [21,22]. Algorithm of the program includes the following steps: 1. Reading the coordinates of the boundaries of the surfaces from the input text file. 2. Border processing. 3. Placing the body in the parallelepiped, based on the calculated extreme points of the body: the minima and maxima for each axis. 4. Filling the resulting parallelepiped by voxels. 5. Determination of each voxel membership for the reconstructed body. From the current voxel's center the beams are directed along three axes, count the number of intersections with boundaries on the basis of the obtained data, determines the position of the finite element relative to the boundaries. 6. Removing of voxels that are not belonged to the body. 7. Determination of material properties for each voxel. 8. Recording the results in the output file to be used in the LS-DYNA software. Fig. 7. Variation of strength vs porosity.
Parameters σ0 = 510 MPa and n = 4629 are determined by least squares method. It is indicative that a slight increase of porosity up to 5–10% can lead to strength reduce of 25–30%. 5. Voxel generation To create a numerical model of the ceramic samples with a heterogeneous microstructure a program of reconstruction of inhomogeneous bodies by elementary volumes (voxels) [34] was created with using the language C++. Each elementary volumes (voxels) have the same size and different mechanical properties (random volume generation up to 10 groups of materials in accordance with the histogram of
Fig. 8 shows examples of the reconstruction of some bodies by elementary volumes (voxels) with two, three or four groups of materials. 6. Heterogeneity of ceramic microstructure LS-DYNA software was used along with the model to simulate the bending test up to a failure point, for the ceramic samples with 4, 6 and 11% porosity (death-of-element approach). Disk-shaped specimens were loaded according to Fig. 4: bottom surface of the ring was fixed and a force was linearly increased up to 5000 N and loaded to top surface of the piston. Calculation time was 1 ms and the explicit scheme of integration was used. In this case, kinetic energy of the system did not exceed 5% of total one (quasi-static regime). The specimen volume was stochastically reconstructed by the two different groups of voxels (finite elements) having the shape of a cube with a side of 0.5 mm by using special program [34]. The porosity of
Fig. 8. Voxel discretization: a) two groups of materials, b) three groups of materials, c) four groups of materials.
S.B. Sapozhnikov et al. / Materials and Design 88 (2015) 1042–1048 Table 5 Mechanical characteristics of porous ceramics.
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Table 6 Comparison of numerical and experimental loadings at failure.
Porosity, %
Density (kg/m3)
Elastic modulus, GPa
Poisson's ratio
Ultimate stress, MPa
Porosity, %
The mean value of the ultimate load (experiment), N
The value of the ultimate load (calculation), N
Difference, %
22 12 8
3112 3511 3671
215 293 327
0.200 0.220 0.226
195 300 340
11 6 4
3115 3521 3902
3102 3507 3852
0.40 0.39 1.27
the ceramic specimens in the numerical model takes into account the following: half of voxels had mechanical properties of porous-free ceramic (density ρ0 = 3990 kg/m3, elastic modulus Emax = 402 GPa, Poisson's ratio νmax = 0.233, ultimate tensile stress σu = 510 MPa); the second half of voxels had properties corresponding ceramic material with a doubled porosity and reduced density (8, 12 and 22% of porosity respectively) (Table 5). Thus, the overall mass and porosity of the specimen were equal to experimental ones. Voxel finite element mesh is shown in Fig. 9. In the calculations, for the piston and ring we used absolutely rigid material *MAT_RIGID. For ceramics there was material *MAT_ELASTIC with an option *MAT_ADD_EROSION. This option allows to consider failure (death of finite elements) when the first principal stress reached ultimate stress. Fragmentation of samples is shown in Fig. 10 for a time of 0.79 ms. Decreasing of material porosity leads to increasing of cracks and fragments both in experiment and in calculation (see Fig. 5). It should be noted that it was not possible to get a realistic fragmentation picture for homogeneous disk material because real brittle material always includes different defects which can create many starting points for cracks. For homogeneous material, there is only one finite element with maximum value of Rankine criterion, so we have only one starting point. Comparison of numerical and experimental failure loads is given in Table 6.
7. Conclusion In this work, disk-shaped samples were hot-press sintered from submicron alumina powder at 1200 °C. Samples had 4–23% of porosity by use of different sintering pressures. Tensile strength of these samples was varied from 180 up to 490 MPa. This information was obtained with the assistance of stress–strain calculations for disk-shaped sample under bending loading (‘piston-on-ring’ technique with assumption of homogeneity of disk material; Static Structural scheme, ANSYS Workbench software). Novel model (stochastic voxel technique, explicit dynamics, LSDYNA, death of finite elements) was devoted to estimation of fracture load and picture of ceramic disk fragmentation with direct implementation of inhomogeneity of material. Here we saw from three to eight visible pieces of disk after rupture with decreasing of porosity from 23 to 4%. Numerical data are in good agreement with experimental ones. The proposed stochastic voxel approach is numerically effective and can be used for the prediction of dynamic fracture and fragmentation picture of ceramic elements of high-level protective equipment.
Acknowledgments This work was carried out in South Ural State University (National Research University) with a financial support of the Russian Science Foundation (project No. 14-19-00327).
References
Fig. 9. The voxel mesh of finite elements.
[1] P.J. Hazell, Ceramic Armour: Design and Defeat Mechanisms, Argos Press, Canberra, 2006. [2] C.B. Carter, M.G. Norton, Ceramic Materials: Science and Engineering, 2nd ed. Springer, New York, 2007. [3] R. Danzer, On the relationship between ceramic strength and the requirements for mechanical design, J. Eur. Ceram. Soc. 34 (15) (2014) 3435–3460, http://dx.doi. org/10.1016/j.jeurceramsoc.2014.04.026. [4] G.R. Johnson, T.J. Holmquist, A computational constitutive model for brittle materials subjected to large strains, high strain rates, and high pressures, in: M.A. Meyers, L.E. Murr, K.P. Staudhammer (Eds.), Proceedings of EXPLOMET Conference, Marcel Dekker Inc., New York, San Diego 1992, pp. 1075–1081. [5] G.R. Johnson, Numerical algorithms and material models for high-velocity impact computations, Int. J. Impact Eng. 38 (2011) 456472, http://dx.doi.org/10.1016/j. ijimpeng.2010.10.017.
Fig. 10. Fragmentation pictures of disk-shaped ceramic samples with different porosities: a) 11%, b) 6%, c) 4%.
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[6] C.W. Ong, C.W. Boey, R.S. Hixson, J.O. Sinibaldi, Advanced layered personnel armor, Int. J. Impact Eng. 38 (2011) 369383, http://dx.doi.org/10.1016/j.ijimpeng.2010.12. 003. [7] D. Bürger, A.R. de Faria, S.F.M. de Almeida, F.C.L. de Melo, M.V. Donadon, Ballistic impact simulation of an armour-piercing projectile on hybrid ceramic/fiber reinforced composite armours, Int. J. Impact Eng. 43 (2012) 6377, http://dx.doi.org/10.1016/j. ijimpeng.2011.12.001. [8] S. Feli, M.R. Asgari, Finite element simulation of ceramic/composite armor under ballistic impact, Compos. Part B 42 (2011) 771–780, http://dx.doi.org/10.1016/j. compositesb.2011.01.024. [9] A. Prakash, J. Rajasankar, N. Anandavalli, M. Verma, N.R. Iyer, Influence of adhesive thickness on high velocity impact performance of ceramic/metal composite targets, Int. J. Adhes. Adhes. 41 (2013) 186–197, http://dx.doi.org/10.1016/j.ijadhadh.2012. 11.008. [10] P. Tan, Numerical simulation of the ballistic protection performance of a laminated armor system with pre-existing debonding/delamination, Compos. Part B 59 (2014) 50–59, http://dx.doi.org/10.1016/j.compositesb.2013.10.080. [11] T.J. Holmquist, G.R. Johnson, Response of boron carbide subjected to high-velocity impact, Int. J. Impact Eng. 35 (2008) 742–752, http://dx.doi.org/10.1016/j. ijimpeng.2007.08.003. [12] V.S. Deshpande, A.G. Evans, Inelastic deformation and energy dissipation in ceramics: a mechanism-based constitutive model, J. Mech. Phys. Solids 56 (2008) 3077–3100, http://dx.doi.org/10.1016/j.jmps.2008.05.002. [13] F. Zhou, J.F. Molinari, Stochastic fracture of ceramics under dynamic tensile loading, Int. J. Solids Struct. 41 (2004) 6573–6596, http://dx.doi.org/10.1016/j.ijsolstr.2004. 05.029. [14] R.C. Yu, G. Ruiz, A. Pandolfi, Numerical investigation on the dynamic behavior of advanced ceramics, Eng. Fract. Mech. 71 (2004) 897–911, http://dx.doi.org/10.1016/ S0013-7944(03)00016-X. [15] S. Maiti, K. Rangaswamy, P.H. Geubelle, Mesoscale analysis of dynamic fragmentation of ceramics under tension, Acta Mater. 53 (2005) 823–834, http://dx.doi.org/ 10.1016/j.actamat.2004.10.034. [16] S. Levy, J.F. Molinari, Dynamic fragmentation of ceramics, signature of defects and scaling of fragment sizes, J. Mech. Phys. Solids 58 (2010) 12–26, http://dx.doi.org/ 10.1016/j.jmps.2009.09.002. [17] J.D. Clayton, R.H. Kraft, R.B. Leavy, Mesoscale modeling of nonlinear elasticity and fracture in ceramic polycrystals under dynamic shear and compression, Int. J. Solids Struct. 49 (2012) 2686–2702, http://dx.doi.org/10.1016/j.ijsolstr.2012.05.035. [18] D. Wang, J. Zhao, Y.H. Zhou, X.X. Chen, A.H. Li, Z.C. Gong, Extended finite element modeling of crack propagation in ceramic tool materials by considering the microstructural features, Comput. Mater. Sci. 77 (2013) 236–244, http://dx.doi.org/10. 1016/j.commatsci.2013.04.045. [19] R. Radovitzky, A. Seagraves, M. Tupek, L. Noels, A scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method, Comput. Methods Appl. Mech. Eng. 200 (2011) 326–344, http://dx.doi.org/10. 1016/j.cma.2010.08.014.
[20] M. Lee, E.Y. Kim, Y.H. Yoo, Simulation of high speed impact into ceramic composite systems using cohesive-law fracture model, Int. J. Impact Eng. 35 (2008) 1636–1641, http://dx.doi.org/10.1016/j.ijimpeng.2008.07.031. [21] H.J. Kim, Swan CC voxel-based meshing and unit-cell analysis of textile composites, Int. J. Numer. Methods Eng. 56 (2003) 977–1006, http://dx.doi.org/10.1002/nme. 594. [22] S.D. Green, M.Y. Matveev, A.C. Long, D. Ivanov, S.R. Hallett, Mechanical modelling of 3D woven composites considering realistic unit cell geometry, Compos. Struct. 118 (2014) 284–293, http://dx.doi.org/10.1016/j.compstruct.2014.07.005. [23] A.P. Garshin, V.M. Gropoyanov, G.P. Zaytsev, S.S. Semenov, Ceramics for Engineering, Nauchtekhlitizdat, Moscow, 2003. [24] K.K. Phani, D. Sanyal, The relations between the shear modulus, the bulk modulus and Young's modulus for porous isotropic ceramic materials, Mater. Sci. Eng. A 490 (2008) 305–312, http://dx.doi.org/10.1016/j.msea.2008.01.030. [25] W. Pabst, E. Gregorova, G. Ticha, Elasticity of porous ceramics—a critical study of modulus–porosity relations, J. Eur. Ceram. Soc. 26 (2006) 1085–1097, http://dx. doi.org/10.1016/j.jeurceramsoc.2005.01.041. [26] R.A. Dorey, J.A. Yeomans, P.A. Smith, Effect of pore clustering on the mechanical properties of ceramics, J. Eur. Ceram. Soc. 22 (2002) 403–409, http://dx.doi.org/10. 1016/S0955-2219(01)00303-X. [27] Z. Zivcova, M. Cerny, W. Pabst, E. Gregorova, Elastic properties of porous oxide ceramics prepared using starch as a pore-forming agent, J. Eur. Ceram. Soc. 29 (2009) 2765–2771, http://dx.doi.org/10.1016/j.jeurceramsoc.2009.03.033. [28] D.T. Hristopulos, M. Demertzi, A semi-analytical equation for the Young's modulus of isotropic ceramic materials, J. Eur. Ceram. Soc. 28 (2008) 1111–1120, http://dx. doi.org/10.1016/j.jeurceramsoc.2007.10.004. [29] M. Asmani, C. Kermel, A. Leriche, M. Ourak, Influence of porosity on Young's modulus and Poisson's ratio in alumina ceramics, J. Eur. Ceram. Soc. 21 (2001) 1081–1086, http://dx.doi.org/10.1016/S0955-2219(00)00314-9. [30] C.W. Huang, C.H. Hsueh, Piston-on-three-ball versus piston-on-ring in evaluating the biaxial strength of dental ceramics, Dent. Mater. 27 (2011) 117–123, http://dx. doi.org/10.1016/j.dental.2011.02.011. [31] D. Zhang, J. Zhou, S. Zhang, L. Wang, S. Dong, Theoretical analyses and numerical simulations on the mechanical strength of multilayers subjected to ring-on-ring tests, Mater. Des. 51 (2013) 1–11, http://dx.doi.org/10.1016/j.matdes.2013.03.090. [32] A.A. Wereszczak, J.J. Swab, R.H. Kraft, Effects of machining on the uniaxial and equibiaxial flexure strength of CAP3 AD-995 Al2O3, Army Research Laboratory Technical Report, 2005 ((ARL-TR-3617). http://www.dtic.mil/cgi-bin/GetTRDoc?AD= ADA441313). [33] W. Duckworth, Discussion of Ryshkewitch paper by Winston Duckworth, J. Am. Ceram. Soc. 36 (1953) 68–69. [34] M.O. Kibel, N.Y. Dolganina, Development System for Reconstruction Inhomogeneous Bodies by Elementary Volumes on the Example of the Ceramics with Defects of Microstructure, Bulletin of the South Ural State University, Series “Computational Mathematics and Software Engineering”, vol. 3, no. 4 2014, pp. 109–115.
Contents lists available at ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/jmad
Experimental and numerical estimation of strength and fragmentation of different porosity alumina ceramics S.B. Sapozhnikov a,⁎, O.A. Kudryavtsev a, N.Yu. Dolganina b a b
South-Ural State University, Physics Department, Chelyabinsk, Russian Federation South-Ural State University, Computational Mathematics and Informatics Department, Chelyabinsk, Russian Federation
a r t i c l e
i n f o
Article history: Received 11 December 2014 Received in revised form 10 August 2015 Accepted 23 August 2015 Available online 1 September 2015 Keywords: Ceramics Strength Fragmentation Porosity Voxel numerical simulation
a b s t r a c t Ceramics are widely used in many applications as construction material. However, their strength and fragmentation details under tensile loading are not fully understood. In this work, the results of experimental and numerical strength investigations of hot-pressed alumina ceramics are presented. The disk-shape specimens with different porosities were subjected to bending test up to failure. Finite element analysis was performed for estimation of ceramic disk tensile strength. Elastic properties of porous ceramics for numerical simulations were determined with use of existing modulus–porosity relations and current experimental data. Three-dimensional models of test specimens with random distribution of strength and porosity based on voxel discretization were created by C++ program and implemented in LS-DYNA. Then they were used to simulate the specimen fracture and fragmentation. Obtained numerical data are in a good agreement with experimental results. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Ceramic materials possess many excellent mechanical properties like exceptional hardness, wear resistance, high compressive strength, low density, etc. They are widely used in aerospace and defense industries [1,2]. At the same time, low tensile strength (approximately one order of magnitude below the compressive strength) and inherently brittle behavior of ceramics are the main limitations for their extensive use in structural applications. Tensile strength is one of the main parameters that determines the ceramic's quality [3] (due to the presence of structural defects — pores and micro-cracks). There are several different approaches to take into account the details of the deformation and fracture of brittle materials with high hardness. Continuously damaged homogeneous material models [4] in different modifications [5] are the most popular models for high-velocity impact dynamic problems, primarily in the context of armor protection. Examples of the application of this model can be found in [6–12]. However, these models require a large number of expensive and complicated experiments like Edge-On-Impact (EOI) or Depth-Of-Penetration (DOP) tests to determine the material parameters for numerical analysis. Recently, the cohesive/volumetric finite element (CVFE) approach was used for ceramic fracture simulations [13–16]. The implementation of cohesive zone into numerical analysis was successful in many cases. In [17,18], the authors directly took into account the microstructure of ⁎ Corresponding author at: Lenin Ave., 76-355, Chelyabinsk, 454080, Russian Federation. E-mail address: [email protected] (S.B. Sapozhnikov).
http://dx.doi.org/10.1016/j.matdes.2015.08.117 0264-1275/© 2015 Elsevier Ltd. All rights reserved.
the ceramic material into fracture analysis to obtain a good agreement with experimental data. In addition, cohesive element approach was used to simulate ceramic fracture and cracks propagation under highvelocity impact conditions [19,20]. However, selection of cohesive elements parameters is still a labor-intensive task. Here we have to say that the cohesive approach leads to considerable computational cost due to increasing of the size of numerical problem in comparison with traditional LS-DYNA approach based on the death-of-element analysis. It should be noticed that the correct prediction of the target and projectile fragmentation is absolutely needed to estimate the damage of neighboring parts. In this paper, experimental and numerical tensile strength estimations of hot-pressed porous alumina ceramics were performed. Firstly, ceramic disk specimens were subjected to bending tests. Then we computed tensile strength of ceramic materials with different porosities with use of FEA and known modulus-porosity relations. These data were used in the novel low parametric model of ceramics based on the voxel [21,22] representation of material microstructure, stochastic distribution of local behaviors and the death-of-element analysis. This model allows us to obtain the fracture loads and pictures of fragmentation for porous ceramic specimens which are in good agreement with experimental data. 2. Specimens' fabrication An alumina powder produced by condensation of vapor on electrostatic filter was used as raw material. Fig. 1 shows the microstructure of this powder, average particle size is about 220 nm.
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Table 1 Properties of specimens for each group. Specimens group
Applied pressure (MPa)
Mass (g)
Height (mm)
Density ρ, g/cm3
Relative density (ρ/ρ0)
1 2 3 4
19.4 29 38.2 48.5
5.93 ± 0.03 5.94 ± 0.01 5.94 ± 0.02 5.92 ± 0.02
3.83 ± 0.05 3.32 ± 0.04 3.12 ± 0.03 3.05 ± 0.02
3.06 ± 0.04 3.53 ± 0.04 3.75 ± 0.03 3.82 ± 0.01
0.77 0.89 0.94 0.96
3. Elastic constants of porous alumina It is necessary to have mechanical properties at each density level for strength assessment of test specimens. The results of a brief review of some relations of ceramics elastic properties vs the porosity are presented below. In [23], the authors proposed relations of a shear modulus G and bulk modulus K with porosity for different materials in the porosity range of 0–40%: Fig. 1. Alumina powder (agglomerates).
All experiments were made with HP20-4560-20 hot-pressed system manufactured by the Thermal Technology LLC (Fig. 2). The graphite die with an external diameter of 100 mm and an internal diameter of 25.4 mm was filled with 5.95 ± 0.05 g of the powder and placed into the vacuum hot-press furnace with graphite heating elements. Hot pressing experiments were performed at different compression levels: 19.4, 29, 38.2 and 48.5 MPa to obtain ceramic specimens with different densities. The same heating rate of 20 °C/min, sintering temperature of 1200 °C, dwell times of 20 min were used for the sintering experiments. The vacuum of ~25 Pa was maintained during technological process. The cooling rate was 20 °C/min until a temperature 1000 °C was reached. The temperature control unit was off after that. Seven samples were manufactured for each compression level to get the density range between 3.06 and 3.82 g/cm3 (see Table 1). Density of pore-free material was assumed to ρ0 = 3.99 g/cm3 [2].
P ; K ¼ K max exp −q 1−P
ð1Þ
P ; G ¼ G max exp −g 1−P
ð2Þ
where P — porosity (dimensionless), Kmax an Gmax — bulk and shear moduli of pore-free material, g and q — material constants. For polycrystalline α-Al2O3 constants g and q are 3.960 and 1.617 respectively. Values g and q for other materials (such as iron or copper) can be found in [23]. The pore-free alumina elastic constants are presented in Table 2. The shear and bulk moduli were calculated for each level of the relative density (porosity) in accordance with Eqs. (1) and (2), see Table 3. The values of elastic modulus and Poisson's ratio were obtained by using the theory of isotropic elasticity.
Fig. 2. HP20-4560-20 hot-pressed system.
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Table 2 Elastic properties of pore-free polycrystalline α-Al2O3 at ambient temperature [2]. Elastic modulus Emax (GPa)
Bulk modulus Kmax (GPa)
Shear modulus Gmax (GPa)
Poisson's ratio νmax
402
251
163
0.233
Table 3 Elastic properties of porous alumina. Specimens group
Relative density
Elastic modulus E (GPa)
Bulk modulus K (GPa)
Shear modulus G (GPa)
Poisson's ratio ν
1 2 3 4
0.77 0.89 0.94 0.96
214 317 359 376
78 155 197 216
103 136 150 156
0.04 0.16 0.195 0.208
Table 4 Data for numerical simulations and corresponding average calculated tensile strength of the ceramic materials of disk specimens. Porosity
Elastic modulus E (GPa)
Poisson's ratio ν
Fracture load P (N)
Tensile strength (MPa)
0.23 0.11 0.06 0.04
205 300 345 363
0.199 0.222 0.229 0.230
2455 3115 3521 3902
182 299 377 429
As can be seen in Table 3, the value of Poisson's ratio at 23% porosity is 0.04 and becomes a negative with a further porosity increasing which is not physical. Now there are no adequate Poisson's ratio — porosity dependences for relative density of the material less than 0.88 [29].
Fig. 3. Comparison of different elastic modulus–porosity relations.
Experimental studies [27,29] have shown that the Poisson's ratio of αAl2O3 is weakly dependent on the porosity and slightly decreases with increasing porosity. In [29] Poisson's ratio was determined for α-Al2O3 with a porosity in the range 0–25% and these data were used in current work (see Table 4). There are several elastic modulus–porosity relations for ceramic [24–28]. It can be noticed, that reliable experimental data on the modulus of elasticity of porous alumina obtained by ultrasonic methods when the porosity less than 40%. A semi-analytical equation for the elastic modulus Е(Р) proposed in [28] is adequate if porosity less than 30%. At a higher porosity level, this function begins to increase which does not correspond to the experimental data. Other relations (see Fig. 3) are similar. In the further calculations, we are used Pabst–Gregorova relation [25] E ¼ E max ð1−P Þ 1−
P ; 0:684
ð3Þ
which gives the average results between the equations presented in [23,27]. Table 4 shows the values of the elastic modulus at different porosity levels obtained from Eq. (3). 4. Bending tests and strength assessment The disk-shape specimens were loaded in piston-on-ring bending tests in order to determine the tensile strength of porous ceramic material. In this case, the specimen was supported by a ring near external edge and equidistant from the center, and it was loaded through piston in the central region (Fig. 4). It should be noted that there are other methods for material biaxial testing such as piston-on-three ball and ring-on-ring test, see [30–32] for details. All bending tests were performed on universal testing machine Instron 5882. The area of maximum tension stresses appears at the center of bottom disk surface, where the stress is biaxial. Maximum principal stress failure criterion (Rankine or Coulomb criterion) can be used in this
Fig. 4. Diagram of disk subjected to piston-on-ring test.
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Fig. 5. Specimens after failure in ‘piston-on-ring’ tests.
case to determine tensile strength with sufficient accuracy for engineering applications. Ring with diameter of 2Rr = 19 mm and piston with diameter of 2Rp = 5 mm were used for experiments. The samples after the tests are shown in Fig. 5. Finite element analysis (FEA) was used to simulate ‘piston-on-ring’ tests. The contact between the experimental equipment and specimen was frictional with friction coefficient of 0.1. Materials are considered to be isotropic and linear elastic. The 1/4 model was implemented into FEA to reduce the computational cost because of the symmetry reasons. An example of the radial stress distribution with FE mesh shown in Fig. 6. The values of elastic constants and average experimental fracture loads for each group of specimens that were used for numerical simulations as well as corresponding calculated tensile strength of ceramic
materials (Rankine criterion) with different porosity levels are summarized in Table 4. One of the most commonly used forms of equations describing the effect of porosity on mechanical properties is the Duckworth equation [33] σ ¼ σ 0 expð−nPÞ;
ð4Þ
where σ — the tensile strength, P — the volume fraction of porosity, n — an empirical constant and the subscript 0 indicates zero porosity. Fig. 7 shows the experimental values of the tensile strengths of the tested ceramics versus porosity with approximating curve (Eq. (4)).
Fig. 6. Contours of radial stress for specimen with 4% of porosity.
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distribution). Voxel approach is relatively little used in the practice of engineering calculations, despite the high numerical efficiency (minimum computational cost or CPU time) [21,22]. Algorithm of the program includes the following steps: 1. Reading the coordinates of the boundaries of the surfaces from the input text file. 2. Border processing. 3. Placing the body in the parallelepiped, based on the calculated extreme points of the body: the minima and maxima for each axis. 4. Filling the resulting parallelepiped by voxels. 5. Determination of each voxel membership for the reconstructed body. From the current voxel's center the beams are directed along three axes, count the number of intersections with boundaries on the basis of the obtained data, determines the position of the finite element relative to the boundaries. 6. Removing of voxels that are not belonged to the body. 7. Determination of material properties for each voxel. 8. Recording the results in the output file to be used in the LS-DYNA software. Fig. 7. Variation of strength vs porosity.
Parameters σ0 = 510 MPa and n = 4629 are determined by least squares method. It is indicative that a slight increase of porosity up to 5–10% can lead to strength reduce of 25–30%. 5. Voxel generation To create a numerical model of the ceramic samples with a heterogeneous microstructure a program of reconstruction of inhomogeneous bodies by elementary volumes (voxels) [34] was created with using the language C++. Each elementary volumes (voxels) have the same size and different mechanical properties (random volume generation up to 10 groups of materials in accordance with the histogram of
Fig. 8 shows examples of the reconstruction of some bodies by elementary volumes (voxels) with two, three or four groups of materials. 6. Heterogeneity of ceramic microstructure LS-DYNA software was used along with the model to simulate the bending test up to a failure point, for the ceramic samples with 4, 6 and 11% porosity (death-of-element approach). Disk-shaped specimens were loaded according to Fig. 4: bottom surface of the ring was fixed and a force was linearly increased up to 5000 N and loaded to top surface of the piston. Calculation time was 1 ms and the explicit scheme of integration was used. In this case, kinetic energy of the system did not exceed 5% of total one (quasi-static regime). The specimen volume was stochastically reconstructed by the two different groups of voxels (finite elements) having the shape of a cube with a side of 0.5 mm by using special program [34]. The porosity of
Fig. 8. Voxel discretization: a) two groups of materials, b) three groups of materials, c) four groups of materials.
S.B. Sapozhnikov et al. / Materials and Design 88 (2015) 1042–1048 Table 5 Mechanical characteristics of porous ceramics.
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Table 6 Comparison of numerical and experimental loadings at failure.
Porosity, %
Density (kg/m3)
Elastic modulus, GPa
Poisson's ratio
Ultimate stress, MPa
Porosity, %
The mean value of the ultimate load (experiment), N
The value of the ultimate load (calculation), N
Difference, %
22 12 8
3112 3511 3671
215 293 327
0.200 0.220 0.226
195 300 340
11 6 4
3115 3521 3902
3102 3507 3852
0.40 0.39 1.27
the ceramic specimens in the numerical model takes into account the following: half of voxels had mechanical properties of porous-free ceramic (density ρ0 = 3990 kg/m3, elastic modulus Emax = 402 GPa, Poisson's ratio νmax = 0.233, ultimate tensile stress σu = 510 MPa); the second half of voxels had properties corresponding ceramic material with a doubled porosity and reduced density (8, 12 and 22% of porosity respectively) (Table 5). Thus, the overall mass and porosity of the specimen were equal to experimental ones. Voxel finite element mesh is shown in Fig. 9. In the calculations, for the piston and ring we used absolutely rigid material *MAT_RIGID. For ceramics there was material *MAT_ELASTIC with an option *MAT_ADD_EROSION. This option allows to consider failure (death of finite elements) when the first principal stress reached ultimate stress. Fragmentation of samples is shown in Fig. 10 for a time of 0.79 ms. Decreasing of material porosity leads to increasing of cracks and fragments both in experiment and in calculation (see Fig. 5). It should be noted that it was not possible to get a realistic fragmentation picture for homogeneous disk material because real brittle material always includes different defects which can create many starting points for cracks. For homogeneous material, there is only one finite element with maximum value of Rankine criterion, so we have only one starting point. Comparison of numerical and experimental failure loads is given in Table 6.
7. Conclusion In this work, disk-shaped samples were hot-press sintered from submicron alumina powder at 1200 °C. Samples had 4–23% of porosity by use of different sintering pressures. Tensile strength of these samples was varied from 180 up to 490 MPa. This information was obtained with the assistance of stress–strain calculations for disk-shaped sample under bending loading (‘piston-on-ring’ technique with assumption of homogeneity of disk material; Static Structural scheme, ANSYS Workbench software). Novel model (stochastic voxel technique, explicit dynamics, LSDYNA, death of finite elements) was devoted to estimation of fracture load and picture of ceramic disk fragmentation with direct implementation of inhomogeneity of material. Here we saw from three to eight visible pieces of disk after rupture with decreasing of porosity from 23 to 4%. Numerical data are in good agreement with experimental ones. The proposed stochastic voxel approach is numerically effective and can be used for the prediction of dynamic fracture and fragmentation picture of ceramic elements of high-level protective equipment.
Acknowledgments This work was carried out in South Ural State University (National Research University) with a financial support of the Russian Science Foundation (project No. 14-19-00327).
References
Fig. 9. The voxel mesh of finite elements.
[1] P.J. Hazell, Ceramic Armour: Design and Defeat Mechanisms, Argos Press, Canberra, 2006. [2] C.B. Carter, M.G. Norton, Ceramic Materials: Science and Engineering, 2nd ed. Springer, New York, 2007. [3] R. Danzer, On the relationship between ceramic strength and the requirements for mechanical design, J. Eur. Ceram. Soc. 34 (15) (2014) 3435–3460, http://dx.doi. org/10.1016/j.jeurceramsoc.2014.04.026. [4] G.R. Johnson, T.J. Holmquist, A computational constitutive model for brittle materials subjected to large strains, high strain rates, and high pressures, in: M.A. Meyers, L.E. Murr, K.P. Staudhammer (Eds.), Proceedings of EXPLOMET Conference, Marcel Dekker Inc., New York, San Diego 1992, pp. 1075–1081. [5] G.R. Johnson, Numerical algorithms and material models for high-velocity impact computations, Int. J. Impact Eng. 38 (2011) 456472, http://dx.doi.org/10.1016/j. ijimpeng.2010.10.017.
Fig. 10. Fragmentation pictures of disk-shaped ceramic samples with different porosities: a) 11%, b) 6%, c) 4%.
1048
S.B. Sapozhnikov et al. / Materials and Design 88 (2015) 1042–1048
[6] C.W. Ong, C.W. Boey, R.S. Hixson, J.O. Sinibaldi, Advanced layered personnel armor, Int. J. Impact Eng. 38 (2011) 369383, http://dx.doi.org/10.1016/j.ijimpeng.2010.12. 003. [7] D. Bürger, A.R. de Faria, S.F.M. de Almeida, F.C.L. de Melo, M.V. Donadon, Ballistic impact simulation of an armour-piercing projectile on hybrid ceramic/fiber reinforced composite armours, Int. J. Impact Eng. 43 (2012) 6377, http://dx.doi.org/10.1016/j. ijimpeng.2011.12.001. [8] S. Feli, M.R. Asgari, Finite element simulation of ceramic/composite armor under ballistic impact, Compos. Part B 42 (2011) 771–780, http://dx.doi.org/10.1016/j. compositesb.2011.01.024. [9] A. Prakash, J. Rajasankar, N. Anandavalli, M. Verma, N.R. Iyer, Influence of adhesive thickness on high velocity impact performance of ceramic/metal composite targets, Int. J. Adhes. Adhes. 41 (2013) 186–197, http://dx.doi.org/10.1016/j.ijadhadh.2012. 11.008. [10] P. Tan, Numerical simulation of the ballistic protection performance of a laminated armor system with pre-existing debonding/delamination, Compos. Part B 59 (2014) 50–59, http://dx.doi.org/10.1016/j.compositesb.2013.10.080. [11] T.J. Holmquist, G.R. Johnson, Response of boron carbide subjected to high-velocity impact, Int. J. Impact Eng. 35 (2008) 742–752, http://dx.doi.org/10.1016/j. ijimpeng.2007.08.003. [12] V.S. Deshpande, A.G. Evans, Inelastic deformation and energy dissipation in ceramics: a mechanism-based constitutive model, J. Mech. Phys. Solids 56 (2008) 3077–3100, http://dx.doi.org/10.1016/j.jmps.2008.05.002. [13] F. Zhou, J.F. Molinari, Stochastic fracture of ceramics under dynamic tensile loading, Int. J. Solids Struct. 41 (2004) 6573–6596, http://dx.doi.org/10.1016/j.ijsolstr.2004. 05.029. [14] R.C. Yu, G. Ruiz, A. Pandolfi, Numerical investigation on the dynamic behavior of advanced ceramics, Eng. Fract. Mech. 71 (2004) 897–911, http://dx.doi.org/10.1016/ S0013-7944(03)00016-X. [15] S. Maiti, K. Rangaswamy, P.H. Geubelle, Mesoscale analysis of dynamic fragmentation of ceramics under tension, Acta Mater. 53 (2005) 823–834, http://dx.doi.org/ 10.1016/j.actamat.2004.10.034. [16] S. Levy, J.F. Molinari, Dynamic fragmentation of ceramics, signature of defects and scaling of fragment sizes, J. Mech. Phys. Solids 58 (2010) 12–26, http://dx.doi.org/ 10.1016/j.jmps.2009.09.002. [17] J.D. Clayton, R.H. Kraft, R.B. Leavy, Mesoscale modeling of nonlinear elasticity and fracture in ceramic polycrystals under dynamic shear and compression, Int. J. Solids Struct. 49 (2012) 2686–2702, http://dx.doi.org/10.1016/j.ijsolstr.2012.05.035. [18] D. Wang, J. Zhao, Y.H. Zhou, X.X. Chen, A.H. Li, Z.C. Gong, Extended finite element modeling of crack propagation in ceramic tool materials by considering the microstructural features, Comput. Mater. Sci. 77 (2013) 236–244, http://dx.doi.org/10. 1016/j.commatsci.2013.04.045. [19] R. Radovitzky, A. Seagraves, M. Tupek, L. Noels, A scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method, Comput. Methods Appl. Mech. Eng. 200 (2011) 326–344, http://dx.doi.org/10. 1016/j.cma.2010.08.014.
[20] M. Lee, E.Y. Kim, Y.H. Yoo, Simulation of high speed impact into ceramic composite systems using cohesive-law fracture model, Int. J. Impact Eng. 35 (2008) 1636–1641, http://dx.doi.org/10.1016/j.ijimpeng.2008.07.031. [21] H.J. Kim, Swan CC voxel-based meshing and unit-cell analysis of textile composites, Int. J. Numer. Methods Eng. 56 (2003) 977–1006, http://dx.doi.org/10.1002/nme. 594. [22] S.D. Green, M.Y. Matveev, A.C. Long, D. Ivanov, S.R. Hallett, Mechanical modelling of 3D woven composites considering realistic unit cell geometry, Compos. Struct. 118 (2014) 284–293, http://dx.doi.org/10.1016/j.compstruct.2014.07.005. [23] A.P. Garshin, V.M. Gropoyanov, G.P. Zaytsev, S.S. Semenov, Ceramics for Engineering, Nauchtekhlitizdat, Moscow, 2003. [24] K.K. Phani, D. Sanyal, The relations between the shear modulus, the bulk modulus and Young's modulus for porous isotropic ceramic materials, Mater. Sci. Eng. A 490 (2008) 305–312, http://dx.doi.org/10.1016/j.msea.2008.01.030. [25] W. Pabst, E. Gregorova, G. Ticha, Elasticity of porous ceramics—a critical study of modulus–porosity relations, J. Eur. Ceram. Soc. 26 (2006) 1085–1097, http://dx. doi.org/10.1016/j.jeurceramsoc.2005.01.041. [26] R.A. Dorey, J.A. Yeomans, P.A. Smith, Effect of pore clustering on the mechanical properties of ceramics, J. Eur. Ceram. Soc. 22 (2002) 403–409, http://dx.doi.org/10. 1016/S0955-2219(01)00303-X. [27] Z. Zivcova, M. Cerny, W. Pabst, E. Gregorova, Elastic properties of porous oxide ceramics prepared using starch as a pore-forming agent, J. Eur. Ceram. Soc. 29 (2009) 2765–2771, http://dx.doi.org/10.1016/j.jeurceramsoc.2009.03.033. [28] D.T. Hristopulos, M. Demertzi, A semi-analytical equation for the Young's modulus of isotropic ceramic materials, J. Eur. Ceram. Soc. 28 (2008) 1111–1120, http://dx. doi.org/10.1016/j.jeurceramsoc.2007.10.004. [29] M. Asmani, C. Kermel, A. Leriche, M. Ourak, Influence of porosity on Young's modulus and Poisson's ratio in alumina ceramics, J. Eur. Ceram. Soc. 21 (2001) 1081–1086, http://dx.doi.org/10.1016/S0955-2219(00)00314-9. [30] C.W. Huang, C.H. Hsueh, Piston-on-three-ball versus piston-on-ring in evaluating the biaxial strength of dental ceramics, Dent. Mater. 27 (2011) 117–123, http://dx. doi.org/10.1016/j.dental.2011.02.011. [31] D. Zhang, J. Zhou, S. Zhang, L. Wang, S. Dong, Theoretical analyses and numerical simulations on the mechanical strength of multilayers subjected to ring-on-ring tests, Mater. Des. 51 (2013) 1–11, http://dx.doi.org/10.1016/j.matdes.2013.03.090. [32] A.A. Wereszczak, J.J. Swab, R.H. Kraft, Effects of machining on the uniaxial and equibiaxial flexure strength of CAP3 AD-995 Al2O3, Army Research Laboratory Technical Report, 2005 ((ARL-TR-3617). http://www.dtic.mil/cgi-bin/GetTRDoc?AD= ADA441313). [33] W. Duckworth, Discussion of Ryshkewitch paper by Winston Duckworth, J. Am. Ceram. Soc. 36 (1953) 68–69. [34] M.O. Kibel, N.Y. Dolganina, Development System for Reconstruction Inhomogeneous Bodies by Elementary Volumes on the Example of the Ceramics with Defects of Microstructure, Bulletin of the South Ural State University, Series “Computational Mathematics and Software Engineering”, vol. 3, no. 4 2014, pp. 109–115.