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Fracture patterns of quartz particles in glass feldspar matrix
Materials Letters 72 (2012) 148–152
Contents lists available at SciVerse ScienceDirect
Materials Letters journal homepage: www.elsevier.com/locate/matlet
Fracture patterns of quartz particles in glass feldspar matrix F.A. Gilabert a,⁎, M. Dal Bó b, V. Cantavella a, E. Sánchez a a b
Instituto de Tecnología Cerámica-Asociación de Investigación de las Industrias Cerámicas (ITC-AICE), Campus Riu Sec, 12006, Castellón, Spain Universidade Federal de Santa Catarina (UFSC), 88040-900 Florianópolis, SC, Brazil
a r t i c l e
i n f o
Article history: Received 10 August 2011 Accepted 17 December 2011 Available online 24 December 2011 Keywords: Micromechanical modeling Fracture Heterostructures Ceramic matrix composites Material Point Method
a b s t r a c t We report a comparison between experimental evidences and simulation results of the micro cracking in ceramic materials provoked by the cooling process. The solid model and the fracture process are carried out using the Material Point Method. This model, where cracks may initiate and propagate, consists of a quartz particle embedded in a glassy phase. We focus our interest on the key role of the particle morphology on the micro fracture patterns. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Quartz is the most abundant crystalline phase in porcelain ware compositions, and many studies have emphasized its role when it comes to define the mechanical properties of the final product [1–5]. It is known that the difference between the shrinkage of quartz and glass matrix produces a reinforcement effect, since the matrix is subjected to compressive residual stresses caused during the cooling stage in the industrial firing process [2]. In opposition to this reinforcement, it has been observed that these residual stresses produce microscopic cracks around the quartz particles, leading to a partial relaxation of stresses and a noteworthy increase of microstructural damage, what severely affects the global mechanical behavior [6,7]. The features of this kind of damage in the ceramic microstructure has barely been addressed in the literature. Few works study the interaction between the microscopic residual stresses and the presence of microcracks in composites consisting of a glass matrix with inclusions [8–10]. In these studies, the main aim has been to establish the effect of damaged microstructure on the global fracture behavior when the ceramic-in-use is subjected to mechanical loads. Particularly, in references [11–13] the differential thermal shrinkage between spherical particles and a matrix has been studied. In this problem, it was assumed that a pre-existent circumferential crack with known length was present in the material interface. Under this situation, they showed that the existent crack would be able to propagate if the size of the particle reached certain critical value.
⁎ Corresponding author. E-mail address: [email protected] (F.A. Gilabert). 0167-577X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2011.12.074
This paper focuses on understanding where and how multiple microcracks may spontaneously appear and develop during the cooling process. In particular, the results concern the role played by the quartz particle morphology on the micro fracture pattern, in comparison with the fracture pattern found from the experimental observations. To carry out this study, we have developed a simulation code based on the Material Point Method (MPM), which has been already successfully used to model a wide variety of material types under different conditions [14–18]. 2. Experimental In this work we prepared two types of compositions. The first, as material reference (MR), was a dense glass matrix made of sodium feldspar. The second, as material study (MS), consisted of the same matrix to which a specific amount of quartz particles was added (detailed below). A set of ten specimens of the MR and MS each were prepared. The schematic description of the preparation process was as follows: The initial size of the natural feldspar powder particles was reduced in acetone using a planetary mill during 30 min with alumina balls for high energy milling. Once the feldspar was dried, it was granulated using 8 wt.% aqueous solution with concentration 5 wt.% PVA (Polyvinyl alcohol). The resultant material was compacted using a pressing pressure of 35 MPa. Sinterization process was done using an electric furnace Pirometrol R-series, with a first heating rate of 3.5 °C/s between 25 and 500 °C, followed by a heating rate of 0.4 °C/s up to reach 1200 °C. This temperature was hold during 6 min, followed by a quenching process. This cooling consisted in a rapid extraction of the specimens from the furnace and forced ventilation using 1 bar of air-pressure.
F.A. Gilabert et al. / Materials Letters 72 (2012) 148–152
149
(Expert System Solutions). Finally, the fracture toughness of the feldspar glass and the crystal quartz were taken from [20] and [21,22], respectively. As detailed later, these material properties are required for modeling. 3. Modeling
Fig. 1. Scanning electron microscopy (SEM) image showing the typical fracture patterns after fast cooling of a feldspar matrix containing particles of crystal quartz.
The average measured water absorption in the specimens was lower than 0.5%. Using two pyrometers, the top and bottom surface temperatures of the specimen were recorded. The average cooling rate was 10 °C/s. The volumetric fraction of the composition for MS was: (i) 64.8 vol.% albite glass, (ii) 26.2vol.% quartz particles and (iii) a residual 9.1 vol.% crystalline albite. Quartz particle size powder exhibited an average particle size of 31.3 μm. Fig. 1 shows the typical micrography of the MS at room temperature using a scanning electron microscopy (SEM) Philips XL30 CP. On the other hand, for the MR, albeit an unavoidable residual amount of quartz was detected, it was four times less than in the MS. The average size of the quartz particles was also smaller, in the range 3–5 μm. MR has been used to obtain the elastic and thermal properties of the matrix, namely, it is assumed as a pure matrix free-of-quartz. Its vitreous nature allows to treat the mechanical behavior as elastic and isotropic. Its Young's modulus was determined within the temperature range 22–700 °C with an impulse excitation technique device (Grindosonic), and the result exhibited a negligible temperature dependency. On the other hand, the elastic properties of quartz were taken from [19]. The linear thermal expansion curves (CTE) for both crystal quartz and the matrix were determined using an optical dilatometer Misura HSM
The general procedure in MPM, as well as the complete details about the mathematical formalism can be found in [14]. In particular, a recent work shows how this method can be applied to ceramic materials [23]. The model of solid under the MPM methodology consists of a set of material points located over a background static mesh. The properties of the material such as density, stiffness, fracture toughness or the coefficient of thermal expansion are carried by the points, whereas the mesh is used to perform dynamic calculations of the system along the time. This mesh, composed by elements, is carried out in the same way as in the Finite Element Method. The elements, or cells in our description, are connected through nodes. This work is limited to a two-dimensional strain plane analysis, and for the sake of simplicity, a regular square mesh is chosen. Although a 2D case may involve a biased view of a material model, this approach can be very useful to find out qualitatively the main features of the fracture due to the particle shrinkages. At the beginning of the computation, the properties and the dynamic state of the solid reside in the points. This information is transferred to the mesh nodes, where the equations of motions are solved. To carry out this extrapolation point-to-nodes, the finite element shape function interpolation is used [24]. Once the dynamic magnitudes of the nodes are updated (velocities, forces, strains and stresses), a reversal interpolation node-to-point is done: the material points retrieve the whole updated information from the cell nodes. This procedure is then repeated every time step. A fracture is a discontinuity of the solid, therefore, under this methodology, a crack can be represented by a set of massless material points. On the other hand, each constituent material presents a natural resistance to crack initiation and propagation. The material property that measures this resistance is Gc, usually referred to as critical fracture energy. To model an initial material failure mechanism, we have used a deliberately simple procedure based on principal stresses calculation on each cell, namely a local Rankine criterion, which is specially suited for brittle material. We calculate the principal stresses in every cell center, extrapolated from their four closest nodes. A small flaw of size a may be generated in mode-I of fracture if the maximum principal stress reaches the value σrup = (2GcE*/πa) 1/2, being E* = E/(1 − ν 2), where E and ν are Young's modulus and Poisson's coefficient of the cell material, respectively. The length scale a corresponds to the closest distance
Fig. 2. Detailed view of a portion of a crack in the solid.
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Table 1 E — Young's modulus, ν — Poisson's ratio, Gc — fracture energy, CTE — coefficient of thermal expansion and ρ — density. Material
E (MPa)
ν
Gc (J/m2)
CTE (°C− 1)
ρ (kg/m3)
Matrix Inclusion
53.1 90.0
0.21 0.08
58.3 5.4
77 · 10− 7 300 · 10− 7
2419 2650
since X-ray diffraction measurements revealed a negligible quantity of dissolved quartz in the matrix, the interface incorporation was obviated. 4. Results and discussion The schematic setup of the model comprises a single particle embedded in the matrix. This last can be considered as a quasi-infinite
between two material points belonging to one cell. This model takes into account that adjacent small flaws may break at the same time, in which case they can be merged in only one crack. Fig. 2 represents a schematic view of an arbitrary crack, which consists of a central path and two lips. The central path represents the material discontinuity in itself, and it is useful to track which material points are above or below the path. The upper and the lower edges represent the lips of the crack, with which we can calculate the opening or closure of the discontinuity. The bounding points of the central path represent the crack tips, as it is indicated in Fig. 2. At each crack tip, we calculate the energy release rate G, which is the change rate of potential energy associated with an incremental crack extension along the crack path. Thus, if G > Gc then the crack will propagate. The value of G at the tip can be calculated by means of the concept given by the J-integral [25], whose physical meaning is the energy density that flows to the crack tip. Let us consider a local system of reference at the crack tip, then we can write that G = JI + JII, where JI and JII are the energy release rates associated with the fracture modes I and II, respectively. The component Jα (with α = I, II) is given by
J α ¼ ∫Γ
! → ∂u ðW þ K Þnα − n ⋅σ⋅ ds; ∂xα →
a
b
ð1Þ
where Γ is the path of the integral contour, ds is the elemental arc length, W and K are the strain and kinetic energy densities, respectively, → n is the unitary normal outward vector on Γ, σ is the stress tensor, and → ∂ u =∂xα is the α-component of the displacement gradient in relation to the tip local system of reference. The advantages of using this procedure to obtain G are numerous, and in references [26,27] much more exhaustive analyses can be found. However there are three important features worth mentioning: (i) the material can be non-linear, inelastic, anisotropic and heterogeneous, (ii) the result is independent of the path integral, and (iii) the computation can be done efficiently since only few points representing the contour are needed to obtain accurate results. For this last issue, we considered a circular integration path (see Fig. 2) adaptable as a function of the crack length. Direction of crack growth follows the maximum hoop stress criterion, which assumes that the growth direction is perpendicular to that of the maximum principal stress [28]. The crack tip angle is a function that involves the magnitudes of the relative displacements at the crack tip, or equivalently, the stress intensity factors [27]. Our case study considers a heterogeneous solid, although in a first approximation, each constituent material (matrix and quartz) has been assumed linearly elastic, isotropic and non-dependent of temperature. The material properties required for the model are shown in Table 1. These properties correspond average values close to the instant when the quartz transition takes place. We are aware of these deliberate simplifications, however, we must emphasize that our interest resides on the specific effect of the mismatch between the material properties. The most critical assumptions are probably the isotropy and temperature independence of quartz fracture toughness. Nevertheless, experimental measurements supported by studies in [21,22,29,30] have allowed to assume such a simplification. Another simplification concerns the matrix/inclusion interface. This model would allow to implement such interface as a particle coating material, whose properties (E, ν, Gc and CTE) should be determined. However,
c
d
Fig. 3. (a) Cracks around a rounded particle. (b) Multiple transversal cracking within an elongated shaped particle. (c) Single transversal crack within a rectangular particle. (d) Cracks close to corner regions in an irregular quartz particle.
F.A. Gilabert et al. / Materials Letters 72 (2012) 148–152
medium, in such a way that its four free bounds are far enough from the inclusion. Taking as a reference the actual microstructure shown in Fig. 1, we have selected four types of representative shapes, which are depicted in the right part of Fig. 3. Three of them are simple geometric shapes: circular, elliptical and rectangular. The last is an irregular shape extracted from the micrography by means of digitalization techniques. As we commented above, once the specimen is extracted from the kiln, a fast cooling process is initiated. This situation can be emulated in the model by means of setting an adequate strain rate to each material point, namely ε_ i ¼ α k T_ , where ε_ i is the thermal deformation rate applied to the i-point that represents the k-type material, and T_ is the experimental cooling rate. Once the temperature reaches 573 °C, crystal quartz suffers the allotropic transition. In real materials, this transition involves two effects: (i) an intense and sudden profile of stress in the particle caused by the fact that crystal quartz decreases in volume abruptly [31], and (ii) a notorious CTE change of the quartz, which varies from zero to a finite value about one order of magnitude bigger than the CTE of the matrix. The simulation process starts just when transition takes place. This means that abovementioned change of quartz volume is implicitly considered in the model. Strictly speaking, exactly at the transition point, the CTE of the quartz would theoretically reach an infinite value, although experimentally a finite value was measured. Such a value is considered within the average value used for calculations presented in the previous section. Numerical results have effectively shown that the mismatch of expansion coefficients leads to a tensional state strong enough so as to break the quartz particle. It is worth mentioning that although the crystal stiffness is higher than that of a glassy material, its fracture toughness is lower than that of the vitreous phase. For this reason the model predicts the initial cracks within the quartz particle domain. Fig. 3 presents the crack patterns for each particle shape: circular, elliptical, rectangular and irregular, respectively. The left side images show real quartz particles from the microstructure of the Fig. 1, while the right side images present the fracture pattern obtained with the model. As it can be seen, the agreement between the simulation and the real patterns is quite good. A rounded particle exhibits a clear circumferential crack (Fig. 3a), according to our experimental observations, as well as the reported literature [3,11,32]. A predominant radial stress profile induces small initial circumferential cracks close to the interface, although producing the failure within the quartz (with lower fracture toughness). As the crack growths, due to the shrinkage, the
151
path goes along the edges of the particle, as Fig. 4 shows. This deflection mechanism is originated by dissimilar elastic properties of the particle and the matrix [33–35]. In particles with elongated shapes, the maximum stress is reached along the direction parallel to the largest dimension (Fig. 3b and c), causing visible transversal cracks. Moreover, if the particle shape also presents sharp regions, such as tips or corners, the tensile stress also tends to produce cracks there (Fig. 3d). These results correspond with a microstructural scenario of isolated particles, or equivalently, when there is not interaction between nearby particles. Thus, resultant patterns are consequence of morphology, material properties and the mechanical conditions imposed by quenching. The way particle size affects crack initiation and crack propagation, referred by [11,12], will be studied deeply in the next stage of this model. 5. Conclusions This letter intends to advance towards a resolution of the controversy in the literature with regard to the final effect of quartz on the mechanical behavior of porcelain tile compositions. The results of the simulations have revealed the important role played by the quartz particle morphology on the micro fracture pattern in a ceramic material during the fast cooling. The most remarkable features predicted by the model were: • Rounded particles tend to generate circumferential initial flaws close to the matrix–quartz interface. Crack propagation in this configuration may lead to a complete detach of the particle from the matrix. • Elongated particles present transversal cracks perpendicular to the direction of the largest dimension. Multiple transversal cracks may take place if its shape is thin enough. • Irregular particles tend to develop the highest tensile stresses close to their sharp regions. These features have been successfully compared with real fracture patterns found in experimental observations using SEM. These numerical results have been possible thanks to a simulation model based on the Material Point Method. This method has been adapted to incorporate both the initiation and the propagation of cracks in a heterogeneous solid, in which each material constituent may have different elastic, thermal and fracture properties.
Fig. 4. Snapshots of the sequence of crack growth around a circular quartz particle.
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Further research aims at predicting and designing those decisive microstructural aspects that may improve the macroscopic behavior of complex ceramic materials. Acknowledgments This work was financially supported by the Ministry of Science and Innovation, within the National Programme for Fundamental Research Projects (BIA2009-10692), and by the project CAPES-DGU (BEX 6505/10-4). References [1] Kato E, Satoh T, Ohira O, Kobayashi Y. Effect of quartz on the sintering and bending strength of the porcelain bodies in quartz–feldspar–kaolin system. J Cer Soc Jap Int 1994;102(1):99–104. [2] Carty WM, Senapati U. Porcelain-raw materials, processing, phase evolution, and mechanical behavior. J Am Ceram Soc 1998;81(3):3–20. [3] Ohya Y, Takahashi Y, Murata M, Nakagawa Ze, Hamano K. Acoustic emission from a porcelain body during cooling. J Am Ceram Soc 1999;82(2):445–8. [4] Carty WM, Pinto BM. Effect of filler size on the strength of porcelain bodies; chap. 23. John Wiley & Sons, Inc; 2008. p. 95–105. [5] Ftikos C, Stournaras CJ, Ekonomakou A, Stathis G. Effect of firing contidion, filler grain size and quarz content on bending strength and physical properties of sanitaryware porcelain. J Eur Ceram Soc 2004;24(8):2357–66. [6] Leonelli C, Bondioli F, Veronesi P, Romagnoli M, Manfredini T, Pellacani GC, et al. Enhancing the mechanical properties of porcelain stoneware tiles: a microstructural approach. J Eur Ceram Soc 2001;21(6):785–93. [7] Noni AD, Hotza D, Cantavella V, Sánchez E. Influence of macroscopic residual stresses on the mechanical behavior and microstructure of porcelain tile. J Eur Ceram Soc 2008;28(13):2463–9. [8] Cannillo V, Pellacani GC, Leonelli C, Boccaccini AR. Numerical modelling of the fracture behaviour of a glass matrix composite reinforced with alumina platelets. Composites Part A 2003;34(1):43–51. [9] Cannillo V, Leonelli C, Manfredini T, Montorsi M, Veronesi P, Minay EJ, et al. Mechanical performance and fracture behaviour of glass–matrix composites reinforced with molybdenum particles. Compos Sci Technol 2005;65(7–8):1276–83. [10] Xu C. Effects of particle size and matrix grain size and volume fraction of particles on the toughening of ceramic composite by thermal residual stress. Ceram Int 2005;31(4):537–42. [11] Davidge RW, Green TJ. The strength of two-phase ceramic-glass materials. J Mater Sci 1968;3(6):629–34. [12] Ito YM, Rosenblatt M, Cheng LY, Lange FF, Evans AG. Cracking in particulate composites due to thermalmechanical stress. Int J Fracture 1981;17(5):483–91. [13] Mastelaro VR, Zanotto ED. Residual stresses in a soda-lime-silica glass-ceramic. J Non-Cryst Solids 1996;194(3):297–304. [14] Sulsky D, Chen Z, Schreyers HL. A particle method for history-dependent materials. Comput Methods Appl Mech Engrg 1994;118(1–2):179–96.
[15] Bardenhagen SG, Brackbill JU, Sulsky D. The material-point method for granular materials. Comput Methods Appl Mech Engrg 2000;187(3–4):529–41. [16] Nairn JA. Material point method calculations with explicit cracks. Comput Model Eng Sci 2003;4(6):649–63. [17] Wieckowski Z. The material point method in large strain engineering problems. Comput Meth Appl Mech Eng 2004;193(39–41):4417–38. [18] Huang P, Zhang X, Ma S, Huang X. Contact algorithms for the material point method in impact and penetration simulation. Int J Numer Methods Eng 2011;85(4): 498–517. [19] Ohno I, Harada K, Yoshitomi C. Temperature variation of elastic constants of quartz across the alpha-beta transition. Phys Chem Minerals 2006;33(1):1–9. [20] Fracture mechanics for ceramics, rocks, and concrete. In: Freiman SW, Fuller ER, editors. American Society for Testing and Materials, 745. ASTM Special Technical Publication; 1981. Philadelphia;. [21] Harding DS, Oliver WC, Pharr GM. Cracking during nanoindentation and its use in the measurement of fracture toughness. Materials Research Society Symposium Proceedings, 356. The Materials Research Society; 1994. p. 663–9. [22] Guzzo PL, Raslan AA, Mello JDBD. Ultrasonic abrasion of quartz crystals. Wear 2003;225(1–6):67–77. [23] Gilabert FA, Cantavella V, Sánchez E, Mallol G. Modelling fracture process in ceramic materials using the material point method. Europhys Lett 2011;96(2) 24002(1–6). [24] Reddy JN. An introduction to the finite element method. New York: McGraw-Hill Education; 2006. [25] Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 1968;35:379–86. [26] Nishioka T, Atluri SN. A numerical study of the use of path independent integrals in elasto-dynamic crack propagation. Eng Fract Mech 1983;18(1):23–33. [27] Nishioka T. Computational dynamic fracture mechanics. Int J Fracture 1997;86(1–2): 127–59. [28] Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear, transactions of the ASME Series D. J Basic Eng 1963;85(4):519–27. [29] Ball A, Payne BW. The tensile fracture of quartz crystals. J Mater Sci 1976;11(4): 731–40. [30] Iwasa M, Bradt RC. Cleavage of natural and synthetic single crystal quartz. Mat Res Bull 1987;22(9):1241–8. [31] Tsuneyuki S, Aoki H, Tsukada M, Matsui Y. Molecular-dynamics study of the alpha to beta structural phase transition of quartz. Phys Rev Lett 1990;64(7):776–9. [32] Braganca SR, Bergmann CP, Hubner H. Effect of quartz particle size on the strength of triaxial porcelain. J Eur Ceram Soc 2006;26(16):3761–8. [33] He MY, Hutchinson JW. Crack deflection at an interface between dissimilar elastic materials. Int J Solids Struct 1989;25(9):1053–67. [34] Kitey R, Phan AV, Tippur HV, Kaplan T. Modeling of crack growth through particulate clusters in brittle matrix by symmetric-galerkin boundary element method. Int J Fracture 2006;141(1–2):11–25. [35] Lei J, Zhang C, Yang Q, Wang YS. Dynamic effects of inclusions and microcracks on a main crack. Int J Fract 2010;164(2):271–83.
Contents lists available at SciVerse ScienceDirect
Materials Letters journal homepage: www.elsevier.com/locate/matlet
Fracture patterns of quartz particles in glass feldspar matrix F.A. Gilabert a,⁎, M. Dal Bó b, V. Cantavella a, E. Sánchez a a b
Instituto de Tecnología Cerámica-Asociación de Investigación de las Industrias Cerámicas (ITC-AICE), Campus Riu Sec, 12006, Castellón, Spain Universidade Federal de Santa Catarina (UFSC), 88040-900 Florianópolis, SC, Brazil
a r t i c l e
i n f o
Article history: Received 10 August 2011 Accepted 17 December 2011 Available online 24 December 2011 Keywords: Micromechanical modeling Fracture Heterostructures Ceramic matrix composites Material Point Method
a b s t r a c t We report a comparison between experimental evidences and simulation results of the micro cracking in ceramic materials provoked by the cooling process. The solid model and the fracture process are carried out using the Material Point Method. This model, where cracks may initiate and propagate, consists of a quartz particle embedded in a glassy phase. We focus our interest on the key role of the particle morphology on the micro fracture patterns. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Quartz is the most abundant crystalline phase in porcelain ware compositions, and many studies have emphasized its role when it comes to define the mechanical properties of the final product [1–5]. It is known that the difference between the shrinkage of quartz and glass matrix produces a reinforcement effect, since the matrix is subjected to compressive residual stresses caused during the cooling stage in the industrial firing process [2]. In opposition to this reinforcement, it has been observed that these residual stresses produce microscopic cracks around the quartz particles, leading to a partial relaxation of stresses and a noteworthy increase of microstructural damage, what severely affects the global mechanical behavior [6,7]. The features of this kind of damage in the ceramic microstructure has barely been addressed in the literature. Few works study the interaction between the microscopic residual stresses and the presence of microcracks in composites consisting of a glass matrix with inclusions [8–10]. In these studies, the main aim has been to establish the effect of damaged microstructure on the global fracture behavior when the ceramic-in-use is subjected to mechanical loads. Particularly, in references [11–13] the differential thermal shrinkage between spherical particles and a matrix has been studied. In this problem, it was assumed that a pre-existent circumferential crack with known length was present in the material interface. Under this situation, they showed that the existent crack would be able to propagate if the size of the particle reached certain critical value.
⁎ Corresponding author. E-mail address: [email protected] (F.A. Gilabert). 0167-577X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2011.12.074
This paper focuses on understanding where and how multiple microcracks may spontaneously appear and develop during the cooling process. In particular, the results concern the role played by the quartz particle morphology on the micro fracture pattern, in comparison with the fracture pattern found from the experimental observations. To carry out this study, we have developed a simulation code based on the Material Point Method (MPM), which has been already successfully used to model a wide variety of material types under different conditions [14–18]. 2. Experimental In this work we prepared two types of compositions. The first, as material reference (MR), was a dense glass matrix made of sodium feldspar. The second, as material study (MS), consisted of the same matrix to which a specific amount of quartz particles was added (detailed below). A set of ten specimens of the MR and MS each were prepared. The schematic description of the preparation process was as follows: The initial size of the natural feldspar powder particles was reduced in acetone using a planetary mill during 30 min with alumina balls for high energy milling. Once the feldspar was dried, it was granulated using 8 wt.% aqueous solution with concentration 5 wt.% PVA (Polyvinyl alcohol). The resultant material was compacted using a pressing pressure of 35 MPa. Sinterization process was done using an electric furnace Pirometrol R-series, with a first heating rate of 3.5 °C/s between 25 and 500 °C, followed by a heating rate of 0.4 °C/s up to reach 1200 °C. This temperature was hold during 6 min, followed by a quenching process. This cooling consisted in a rapid extraction of the specimens from the furnace and forced ventilation using 1 bar of air-pressure.
F.A. Gilabert et al. / Materials Letters 72 (2012) 148–152
149
(Expert System Solutions). Finally, the fracture toughness of the feldspar glass and the crystal quartz were taken from [20] and [21,22], respectively. As detailed later, these material properties are required for modeling. 3. Modeling
Fig. 1. Scanning electron microscopy (SEM) image showing the typical fracture patterns after fast cooling of a feldspar matrix containing particles of crystal quartz.
The average measured water absorption in the specimens was lower than 0.5%. Using two pyrometers, the top and bottom surface temperatures of the specimen were recorded. The average cooling rate was 10 °C/s. The volumetric fraction of the composition for MS was: (i) 64.8 vol.% albite glass, (ii) 26.2vol.% quartz particles and (iii) a residual 9.1 vol.% crystalline albite. Quartz particle size powder exhibited an average particle size of 31.3 μm. Fig. 1 shows the typical micrography of the MS at room temperature using a scanning electron microscopy (SEM) Philips XL30 CP. On the other hand, for the MR, albeit an unavoidable residual amount of quartz was detected, it was four times less than in the MS. The average size of the quartz particles was also smaller, in the range 3–5 μm. MR has been used to obtain the elastic and thermal properties of the matrix, namely, it is assumed as a pure matrix free-of-quartz. Its vitreous nature allows to treat the mechanical behavior as elastic and isotropic. Its Young's modulus was determined within the temperature range 22–700 °C with an impulse excitation technique device (Grindosonic), and the result exhibited a negligible temperature dependency. On the other hand, the elastic properties of quartz were taken from [19]. The linear thermal expansion curves (CTE) for both crystal quartz and the matrix were determined using an optical dilatometer Misura HSM
The general procedure in MPM, as well as the complete details about the mathematical formalism can be found in [14]. In particular, a recent work shows how this method can be applied to ceramic materials [23]. The model of solid under the MPM methodology consists of a set of material points located over a background static mesh. The properties of the material such as density, stiffness, fracture toughness or the coefficient of thermal expansion are carried by the points, whereas the mesh is used to perform dynamic calculations of the system along the time. This mesh, composed by elements, is carried out in the same way as in the Finite Element Method. The elements, or cells in our description, are connected through nodes. This work is limited to a two-dimensional strain plane analysis, and for the sake of simplicity, a regular square mesh is chosen. Although a 2D case may involve a biased view of a material model, this approach can be very useful to find out qualitatively the main features of the fracture due to the particle shrinkages. At the beginning of the computation, the properties and the dynamic state of the solid reside in the points. This information is transferred to the mesh nodes, where the equations of motions are solved. To carry out this extrapolation point-to-nodes, the finite element shape function interpolation is used [24]. Once the dynamic magnitudes of the nodes are updated (velocities, forces, strains and stresses), a reversal interpolation node-to-point is done: the material points retrieve the whole updated information from the cell nodes. This procedure is then repeated every time step. A fracture is a discontinuity of the solid, therefore, under this methodology, a crack can be represented by a set of massless material points. On the other hand, each constituent material presents a natural resistance to crack initiation and propagation. The material property that measures this resistance is Gc, usually referred to as critical fracture energy. To model an initial material failure mechanism, we have used a deliberately simple procedure based on principal stresses calculation on each cell, namely a local Rankine criterion, which is specially suited for brittle material. We calculate the principal stresses in every cell center, extrapolated from their four closest nodes. A small flaw of size a may be generated in mode-I of fracture if the maximum principal stress reaches the value σrup = (2GcE*/πa) 1/2, being E* = E/(1 − ν 2), where E and ν are Young's modulus and Poisson's coefficient of the cell material, respectively. The length scale a corresponds to the closest distance
Fig. 2. Detailed view of a portion of a crack in the solid.
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Table 1 E — Young's modulus, ν — Poisson's ratio, Gc — fracture energy, CTE — coefficient of thermal expansion and ρ — density. Material
E (MPa)
ν
Gc (J/m2)
CTE (°C− 1)
ρ (kg/m3)
Matrix Inclusion
53.1 90.0
0.21 0.08
58.3 5.4
77 · 10− 7 300 · 10− 7
2419 2650
since X-ray diffraction measurements revealed a negligible quantity of dissolved quartz in the matrix, the interface incorporation was obviated. 4. Results and discussion The schematic setup of the model comprises a single particle embedded in the matrix. This last can be considered as a quasi-infinite
between two material points belonging to one cell. This model takes into account that adjacent small flaws may break at the same time, in which case they can be merged in only one crack. Fig. 2 represents a schematic view of an arbitrary crack, which consists of a central path and two lips. The central path represents the material discontinuity in itself, and it is useful to track which material points are above or below the path. The upper and the lower edges represent the lips of the crack, with which we can calculate the opening or closure of the discontinuity. The bounding points of the central path represent the crack tips, as it is indicated in Fig. 2. At each crack tip, we calculate the energy release rate G, which is the change rate of potential energy associated with an incremental crack extension along the crack path. Thus, if G > Gc then the crack will propagate. The value of G at the tip can be calculated by means of the concept given by the J-integral [25], whose physical meaning is the energy density that flows to the crack tip. Let us consider a local system of reference at the crack tip, then we can write that G = JI + JII, where JI and JII are the energy release rates associated with the fracture modes I and II, respectively. The component Jα (with α = I, II) is given by
J α ¼ ∫Γ
! → ∂u ðW þ K Þnα − n ⋅σ⋅ ds; ∂xα →
a
b
ð1Þ
where Γ is the path of the integral contour, ds is the elemental arc length, W and K are the strain and kinetic energy densities, respectively, → n is the unitary normal outward vector on Γ, σ is the stress tensor, and → ∂ u =∂xα is the α-component of the displacement gradient in relation to the tip local system of reference. The advantages of using this procedure to obtain G are numerous, and in references [26,27] much more exhaustive analyses can be found. However there are three important features worth mentioning: (i) the material can be non-linear, inelastic, anisotropic and heterogeneous, (ii) the result is independent of the path integral, and (iii) the computation can be done efficiently since only few points representing the contour are needed to obtain accurate results. For this last issue, we considered a circular integration path (see Fig. 2) adaptable as a function of the crack length. Direction of crack growth follows the maximum hoop stress criterion, which assumes that the growth direction is perpendicular to that of the maximum principal stress [28]. The crack tip angle is a function that involves the magnitudes of the relative displacements at the crack tip, or equivalently, the stress intensity factors [27]. Our case study considers a heterogeneous solid, although in a first approximation, each constituent material (matrix and quartz) has been assumed linearly elastic, isotropic and non-dependent of temperature. The material properties required for the model are shown in Table 1. These properties correspond average values close to the instant when the quartz transition takes place. We are aware of these deliberate simplifications, however, we must emphasize that our interest resides on the specific effect of the mismatch between the material properties. The most critical assumptions are probably the isotropy and temperature independence of quartz fracture toughness. Nevertheless, experimental measurements supported by studies in [21,22,29,30] have allowed to assume such a simplification. Another simplification concerns the matrix/inclusion interface. This model would allow to implement such interface as a particle coating material, whose properties (E, ν, Gc and CTE) should be determined. However,
c
d
Fig. 3. (a) Cracks around a rounded particle. (b) Multiple transversal cracking within an elongated shaped particle. (c) Single transversal crack within a rectangular particle. (d) Cracks close to corner regions in an irregular quartz particle.
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medium, in such a way that its four free bounds are far enough from the inclusion. Taking as a reference the actual microstructure shown in Fig. 1, we have selected four types of representative shapes, which are depicted in the right part of Fig. 3. Three of them are simple geometric shapes: circular, elliptical and rectangular. The last is an irregular shape extracted from the micrography by means of digitalization techniques. As we commented above, once the specimen is extracted from the kiln, a fast cooling process is initiated. This situation can be emulated in the model by means of setting an adequate strain rate to each material point, namely ε_ i ¼ α k T_ , where ε_ i is the thermal deformation rate applied to the i-point that represents the k-type material, and T_ is the experimental cooling rate. Once the temperature reaches 573 °C, crystal quartz suffers the allotropic transition. In real materials, this transition involves two effects: (i) an intense and sudden profile of stress in the particle caused by the fact that crystal quartz decreases in volume abruptly [31], and (ii) a notorious CTE change of the quartz, which varies from zero to a finite value about one order of magnitude bigger than the CTE of the matrix. The simulation process starts just when transition takes place. This means that abovementioned change of quartz volume is implicitly considered in the model. Strictly speaking, exactly at the transition point, the CTE of the quartz would theoretically reach an infinite value, although experimentally a finite value was measured. Such a value is considered within the average value used for calculations presented in the previous section. Numerical results have effectively shown that the mismatch of expansion coefficients leads to a tensional state strong enough so as to break the quartz particle. It is worth mentioning that although the crystal stiffness is higher than that of a glassy material, its fracture toughness is lower than that of the vitreous phase. For this reason the model predicts the initial cracks within the quartz particle domain. Fig. 3 presents the crack patterns for each particle shape: circular, elliptical, rectangular and irregular, respectively. The left side images show real quartz particles from the microstructure of the Fig. 1, while the right side images present the fracture pattern obtained with the model. As it can be seen, the agreement between the simulation and the real patterns is quite good. A rounded particle exhibits a clear circumferential crack (Fig. 3a), according to our experimental observations, as well as the reported literature [3,11,32]. A predominant radial stress profile induces small initial circumferential cracks close to the interface, although producing the failure within the quartz (with lower fracture toughness). As the crack growths, due to the shrinkage, the
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path goes along the edges of the particle, as Fig. 4 shows. This deflection mechanism is originated by dissimilar elastic properties of the particle and the matrix [33–35]. In particles with elongated shapes, the maximum stress is reached along the direction parallel to the largest dimension (Fig. 3b and c), causing visible transversal cracks. Moreover, if the particle shape also presents sharp regions, such as tips or corners, the tensile stress also tends to produce cracks there (Fig. 3d). These results correspond with a microstructural scenario of isolated particles, or equivalently, when there is not interaction between nearby particles. Thus, resultant patterns are consequence of morphology, material properties and the mechanical conditions imposed by quenching. The way particle size affects crack initiation and crack propagation, referred by [11,12], will be studied deeply in the next stage of this model. 5. Conclusions This letter intends to advance towards a resolution of the controversy in the literature with regard to the final effect of quartz on the mechanical behavior of porcelain tile compositions. The results of the simulations have revealed the important role played by the quartz particle morphology on the micro fracture pattern in a ceramic material during the fast cooling. The most remarkable features predicted by the model were: • Rounded particles tend to generate circumferential initial flaws close to the matrix–quartz interface. Crack propagation in this configuration may lead to a complete detach of the particle from the matrix. • Elongated particles present transversal cracks perpendicular to the direction of the largest dimension. Multiple transversal cracks may take place if its shape is thin enough. • Irregular particles tend to develop the highest tensile stresses close to their sharp regions. These features have been successfully compared with real fracture patterns found in experimental observations using SEM. These numerical results have been possible thanks to a simulation model based on the Material Point Method. This method has been adapted to incorporate both the initiation and the propagation of cracks in a heterogeneous solid, in which each material constituent may have different elastic, thermal and fracture properties.
Fig. 4. Snapshots of the sequence of crack growth around a circular quartz particle.
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Further research aims at predicting and designing those decisive microstructural aspects that may improve the macroscopic behavior of complex ceramic materials. Acknowledgments This work was financially supported by the Ministry of Science and Innovation, within the National Programme for Fundamental Research Projects (BIA2009-10692), and by the project CAPES-DGU (BEX 6505/10-4). References [1] Kato E, Satoh T, Ohira O, Kobayashi Y. Effect of quartz on the sintering and bending strength of the porcelain bodies in quartz–feldspar–kaolin system. J Cer Soc Jap Int 1994;102(1):99–104. [2] Carty WM, Senapati U. Porcelain-raw materials, processing, phase evolution, and mechanical behavior. J Am Ceram Soc 1998;81(3):3–20. [3] Ohya Y, Takahashi Y, Murata M, Nakagawa Ze, Hamano K. Acoustic emission from a porcelain body during cooling. J Am Ceram Soc 1999;82(2):445–8. [4] Carty WM, Pinto BM. Effect of filler size on the strength of porcelain bodies; chap. 23. John Wiley & Sons, Inc; 2008. p. 95–105. [5] Ftikos C, Stournaras CJ, Ekonomakou A, Stathis G. Effect of firing contidion, filler grain size and quarz content on bending strength and physical properties of sanitaryware porcelain. J Eur Ceram Soc 2004;24(8):2357–66. [6] Leonelli C, Bondioli F, Veronesi P, Romagnoli M, Manfredini T, Pellacani GC, et al. Enhancing the mechanical properties of porcelain stoneware tiles: a microstructural approach. J Eur Ceram Soc 2001;21(6):785–93. [7] Noni AD, Hotza D, Cantavella V, Sánchez E. Influence of macroscopic residual stresses on the mechanical behavior and microstructure of porcelain tile. J Eur Ceram Soc 2008;28(13):2463–9. [8] Cannillo V, Pellacani GC, Leonelli C, Boccaccini AR. Numerical modelling of the fracture behaviour of a glass matrix composite reinforced with alumina platelets. Composites Part A 2003;34(1):43–51. [9] Cannillo V, Leonelli C, Manfredini T, Montorsi M, Veronesi P, Minay EJ, et al. Mechanical performance and fracture behaviour of glass–matrix composites reinforced with molybdenum particles. Compos Sci Technol 2005;65(7–8):1276–83. [10] Xu C. Effects of particle size and matrix grain size and volume fraction of particles on the toughening of ceramic composite by thermal residual stress. Ceram Int 2005;31(4):537–42. [11] Davidge RW, Green TJ. The strength of two-phase ceramic-glass materials. J Mater Sci 1968;3(6):629–34. [12] Ito YM, Rosenblatt M, Cheng LY, Lange FF, Evans AG. Cracking in particulate composites due to thermalmechanical stress. Int J Fracture 1981;17(5):483–91. [13] Mastelaro VR, Zanotto ED. Residual stresses in a soda-lime-silica glass-ceramic. J Non-Cryst Solids 1996;194(3):297–304. [14] Sulsky D, Chen Z, Schreyers HL. A particle method for history-dependent materials. Comput Methods Appl Mech Engrg 1994;118(1–2):179–96.
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