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Structure and fracture property relation for silicon nitride on the microscale
Computational Materials Science 64 (2012) 234–238
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Structure and fracture property relation for silicon nitride on the microscale Johannes Wippler ⇑, Thomas Böhlke Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT), Kaiserstraße 10, 76131 Karlsruhe, Germany
a r t i c l e
i n f o
Article history: Received 25 October 2011 Received in revised form 14 February 2012 Accepted 29 February 2012 Available online 5 April 2012 Keywords: Silicon nitride Fracture Finite element simulation Homogenization R-curve Fracture toughness
a b s t r a c t The fracture toughness of silicon nitride ceramics is closely related to its microstructure, which is characterized by a bimodal distribution of partially large and elongated grains. In the first stage of fracture, these grains act as elastic bridges, which avoid a quick crack propagation. In order to improve the understanding of this R-curve effect, a unit cell approach for multiscale finite element simulation has been chosen. For the examination of the morphology influence on fracture stress and toughness unit cells with different mean aspect ratios have been utilized for the fracture simulations. As results, on the one hand effective stress–strain curves have been applied for the computation of the effective R-curves. On the other hand, these results have been compared with observations of the stress and strain fields on the local level. Findings on the local level support the concept of elastic bridging grains. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Silicon nitride is an important engineering material for challenging applications like tool tips for turning with gray cast iron or for forming rolls for the production of high-strength steel wires, see, e.g. Souza et al. [17] and Kailer and Hollstein [9]. The key properties are the excellent combination of strength, toughness and thermal shock resistance, which can be related to the microstructure. The morphology of silicon nitride is characterized by the elongated b-Si3N4 grains of a specific size–shape distribution and a matrix, which is formed from sinter additives. Such additives are used for the dense-sintering and for the adjustment of specific grain lengths and shape distributions. As additives, often rare-earth oxides like yttria in combination with magnesia or alumina are used. Fundamental work on this field has been done by Sun et al. [18] and Becher et al. [1]. Kruzic et al. [11] enlightened the relevance of a careful selection of the grain boundary toughness for an optimal constellation of strength and stiffness. Additionally to the well-known fact, that overly strong grain interfaces lead to a strong but brittle material without the toughening mechanisms like grain bridging, the disadvantageous influence of too weak interfaces has been pointed out. This results are low strength and shallow R-curves. Hence, a good combination of structural morphology and the corresponding interface strength and toughness is crucial for a reliable tool material. ⇑ Corresponding author. Tel.: +49 (0)721 608 48852; fax: +49 (0)721 608 44187. E-mail addresses: [email protected] (J. Wippler), thomas.boehlke@kit. edu (T. Böhlke). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.02.042
The glassy phase, which fills the space between the grains (12 vol.%) exists as glass pockets at grain-free material volume and forms intergranular films between the grains. The glass pockets are relevant for the residual stresses in the material due to the thermal mismatch and is important for the formation of bridging grains, as it is pointed out in [16]. The intergranular films with relatively low strength lead to toughening due to crack path deflection. This means that cracks have to propagate around the stronger grains [1,3], such that the crack propagation consumes more energy. A tougher material is the consequence. In order to capture this complex behavior, a comprehensive constitutive model has to be created for each fracture mechanism on the microscale. This has been implemented into a unit cell framework, which incorporates all features of importance. The microscopic geometry is modeled according to experimental observations [10] by an enhanced sequential adsorption technique [22]. For thermoelastic properties on the microscale experimental observations [20,7,6] have been implemented [24]. The fracture, which occurs is often intergranular. This is described by a cohesive zone model that combines the concepts of Govindjee et al. [4] and Wei and Anand [21] and has been introduced in Wippler and Böhlke [23]. For the transgranular fracture of the grains and of the glassy phase pockets, constitutive assumptions will be proposed in this work. The toughness of the material can be assessed on the macroscopic level by R-curve measurements [14,2]. This concept describes the relation between fracture toughness and crack extension. In general, a rising R-curve indicates a tough material. For the evaluation of the unit cell simulation, a simplified fracture
J. Wippler, T. Böhlke / Computational Materials Science 64 (2012) 234–238
mechanics concept has been used in order to prove the significance of the proposed model on the macrolevel. In order to give further insight into the acting mechanisms, this work will examine the influence of different mean aspect ratios on both the effective fracture toughness as well as on the microscopic fracture processes. 2. Fracture of phases 2.1. Glassy phase The estimation of the strength of brittle materials on the microscale is challenging due to a significant volume effect. This means that the probability of a larger flaw decreases with decreasing volume, which leads to an increase of the strength at microscopic dimensions after the weakest-link theory. In order to account for this fact, the Weibull theory has been used to estimate the microscopic strength from macroscopic experiments. The rela1=m M tion rM Cð1 þ 1=mÞ provides a relation bec ðVÞ ¼ rc ðV 0 ÞðV 0 =VÞ tween the fracture strength at different volumes [5]. Let the volume of the experimentally assessed sample be V0. The volume of the portion, the strength of which should be estimated is V. The Weibull modulus, which is a measure for the scatter in the experimental data is denoted by m. As macroscopic strength values, the glass fiber data for SiCaAlON glass from Iba et al. [8] is used. This type of glass is not the same, as it is present in silicon nitride with Al and Y oxide composition. However, the strength and the Weibull modulus are expected to be similar enough to be a useful substitute for the YAlSiON glass. The glass fibers with a diameter d0 = 20 lm resisted stresses between 3 and 5 GPa. Together with an assumed fiber length of l0 = 30 mm, the 2 fiber volume is V 0 ¼ p=4d0 l0 ¼ 9:42 103 mm3 . For the Weibull modulus of glass fibers [12] provides a wide range between 10 and 30. An estimation of a glass pocket is possible, if its shape is assumed to be a regular tetrahedron with an edge length of 1 lm. pffiffiffi 3 Then the volume is V ¼ l =4 3 ¼ 1:44 1010 mm3 . Putting these assumptions together, the strength values reach from 5.36 to 28.7 GPa. The upper value from the Weibull consideration has to be compared with the upper bound for material strength, i.e., the theoretical strength. Based on the assumed Young’s modulus of 133 GPa from Hampshire et al. [6], this physical limit can reach from E/8 . . . E/p = 16.6 . . . 42.3 GPa for brittle materials, see, e.g. Mecholsky [13]. Hence, the assumption of 10 GPa appears as a realistic choice. The fracture behavior of the glassy matrix phase M is assumed to be caused by the maximum tensile stress. They are obtained from the eigenvalue problem rppr = rprppr with the principal stresses rpr and the corresponding normalized eigenvectors ppr. The fracture criterion is formulated as ‘p-norm on the tensile part of the principal stresses, where 1 6 N 6 3 is the number of different eigenvalues of the stress tensor r and rM c is the cracking stress of the matrix material. Thus, criterion can be denoted P M p 1=p N pr 1. =rc /M ¼ n¼1 max 0; rn 2.2. b-Si3N4-grains The fracture criterion for the b-grain phase G is motivated by the transverse-isotropy of the elastic stiffness tensor. Here, the po sitive eigenvalues max 0; rpr and the corresponding projectors n pr pr + P pr n ¼ pn pn are used to construct the tensile stress tensor r pr PN pr þ in the spectral representation r ¼ n¼1 max 0; rn P n . The normal direction of the grain faces are used for the construction of the projectors P Gf ¼ nGf nGf ; f ¼ 1 . . . 4, which map the tensile stress
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tensor. Using the ‘p-norm, the fracture criterion for a grain takes P 1=p 4 G G p þ 1. Fig. 1a shows the the form /G ¼ f¼1 jP f r =rc;f j three-dimensional contour of the fracture criterion /G ¼ 0. Motivated by the ab initio simulation results of [15], the selected parameters are rGc;1...3 ¼ 1923 MPa and rGc;4 ¼ 2000 MPa and p = 20. In Fig. 1b, the influence of the orientation on the cracking direction is visualized. The notched sample is thought to be cut out of a single crystal and loaded by displacements on the front faces. The crystal directions are indicated by the small coordinate system. The upper row shows the distribution of the fracture criterion /G . Green dyeing indicates high stress level of approximately 800–1200 MPa around the notch tip shortly before cracking. The effects of anisotropy are obvious, because the orientation of the contour lines corresponds to the crystal orientation. The lower row shows the cracks. It is interesting to observe the final fracture never occurs in the direction of the notch. The fracture criteria are implemented as user subroutine VUMAT for Abaqus/Explicit. In integration points, where the fracture criteria /M=G are fulfilled, stress is set to zero. 3. Determination of the effective fracture properties For the examination of the structure–property relation, three periodic unit cells with 64 grains and an edge length of 3 lm, in the further context designated with I, II and III have been used. The aspect ratio distribution of the grains in the unit cells can be characterized by the mean aspect ratio and the standard deviation. Due to the limited size of the statistical ensembles, higher-order measures are not expected to deliver deeper insight. The mean aspect ratios of the grains in the cells are hAI i ¼ 3:07; hAII i ¼ 4:03 and hAIII i ¼ 4:93. The corresponding standard deviations are rðAI Þ ¼ 1:10; rðAII Þ ¼ 1:41 and rðAI Þ ¼ 2:18. The grain volume fraction is 88%. The cells are created after Wippler and Böhlke [22] and can be seen in Fig. 2. The difference in the mean grain shapes is obvious. The cells are deformed uniaxially with a free transverse contraction under periodic boundary conditions Wippler et al. [24]. In order to examine the influence of the structural properties on the effective behavior a consideration of the relations between effective stress hri and strain e in tensile direction have been considered [24,19]. Those two quantities allow for the determination of the fracture toughness, when certain assumptions are made. The fracture toughness of a material is characterized by the energy release rate G. Following the basic relations of the linear fracture mechanics, as it is outlined in detail in [14], the energy release rate is given by G ¼ 1=2F 2 dC=dA. Here, F is assumed to be the force on the surface of the unit cube F: = hriw2. The compliance is denoted by C = u/F with the boundary displacement u ¼ ew. The cracked area A has to be determined under the assumption that fracture uniaxial tensile load occurs due to mode I. An equivalent description of the fracture toughness of a brittle material is provided by the stress intensity factor. There are two representations, which can be exploited for the determination of the cracked surface. First, it is a relation between the critical stress hri or the related force F pffiffiffi pffiffiffi and the crack length a and is given by K R ¼ hri aY ¼ F=w2 aY. The geometry factor Y can be assumed to have a constant value of 1 for a basic understanding. The second relation is between the energy release rate and the stress intensity factor. Given the effective uniaxial stress state, which is a special case of plane stress, the relation K 2R ¼ GE, where E is Young’s modulus for undamaged silicon nitride holds. The aforementioned relations allow for the determination of the crack surface increment and delivpffiffiffi ers dA ¼ 1=2w4 E=ð AY 2 ÞdC. From this relation, an incremental calculation for an equivalent crack surface is possible.
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Fig. 1. (a) Contour plot of the failure criterion for the b-grain /G ¼ 0 in the principal stress space; planes indicate vanishing stresses; (b) examination of the fracture behavior on a simplified compact tension specimen with different grain orientations; arrows with u indicate the prescribed displacement; the crystal orientation is visualized by the small coordinate system; c represents the axial and a the basal direction.
Fig. 2. Three unit cells with different mean aspect ratios: cell I ðhAI i ¼ 3:07Þ, cell II ðhAII i ¼ 4:03Þ and cell III ðhAIII i ¼ 4:93Þ; the coordinate system defines the loading directions in the further context.
4. Results In Fig. 3, the effective stress–strain curve and the fracture toughness are compiled for representative cases. Fig. 3a shows that the higher the aspect ratio, the higher the fracture strength. This trend can be attributed to the higher content of large and elongated grains, which are transferring higher local loads. Subfigure (b) shows the fracture toughness for the fracture mode I based on the aforementioned assumptions, which correspond to the effective stress–strain curves in subfigure (a). The crack length is normalized on the unit cell size w. The most obvious feature is the general rising trend of the curves, which corresponds with experimental observations by Fett et al. [2]. The significant influence of the grain shape is of special relevance in the context of the work. The clear trend: The higher the aspect ratio, the higher and steeper is the R-curve. It is natural that an effect of the loading direction can be observed. This is, basically, due to the limited size of the considered unit cells. However, the general trend towards increasing fracture strength and resistance with increasing aspect ratios is observable, as it can be seen in Fig. 4. The influence of the geometry is present
in all cases and becomes more pronounced with increasing aspect ratio. The microscopic reasons for the differences in the macroscopic behaviors can be observed in Fig. 5. Here, the corresponding displacement (left) and stress fields (right) for the microstructure II with the medium mean aspect ratio is gathered for the two loading directions with the biggest difference in the macroscopic behavior. The values of the displacement field are minimum and maximum values at the present state. The stress field color code is starting with blue color ( ) at 100 MPa, which represents the compressive stresses in the grains due to thermal strains and is ending in red color ( ) at 2000 MPa, which is the fracture strength of the grains in axial direction. The high values of strength and fracture toughness, which have been measured for the loading direction e3 can be related to the presence of a large and elongated grain, which is aligned mainly in this direction. The right side of Fig. 5a shows the displacement field. The jump in the field is next to the grain, which indicates, that the crack has to propagate around this grain. Fig. 5b shows the corresponding stress field. In the region with the separation, the stress field is almost vanishing due to the post-failure unloading. The load is carried by the elastic bridging grain, where loads up to 2000 MPa are localized. For geometry II, a considerable outlier for toughness but not for strength is observed, when loaded in e2-direction (Fig. 4( )). This decreased fracture toughness has its origin in the presence of an unreinforced plane in the unit cell, which can be seen in Fig. 5c and d. Due to the relatively homogeneous stress distribution, the fracture strength is not affected considerably. 5. Discussion The results confirm the experimental observations, where long and elongated grains lead to higher fracture toughness.
(a)
Fig. 3. Influence of the mean aspect ratio on different effective properties; hAI i ( crack length (normalized on the edge length of the unit cell w).
(b)
), hAII i (
), hAIII i (
); (a) effective stress–strain curve, (b) stress intensity factor KR over
J. Wippler, T. Böhlke / Computational Materials Science 64 (2012) 234–238
(a)
237
(b)
Fig. 4. (a) Maximum crack stress and (b) maximum fracture resistance values for the three considered geometries under loading direction e1 ( ), e2 ( ) and e3 ( ), mean values () and adjustment by linear regression ( ); loading directions correspond to definition in Fig. 2.
The general trend: The R-curve starts for very short cracks of single microns at a level, which is approximately one half of the plateau level like it can be seen in Fig. 3d. Hence, the ratio between the initial and peak unit cell fracture toughness is lower than it was measured in Fünfschilling et al. [3]. This difference is, as well as the deviation from the absolute experimental values are likely to be caused by the size of the unit cell and the simplicity of the chosen framework. The main features of the complex fracture process in silicon nitride can be captured by the unit cell approach with structure and the related constitutive assumptions. It has to be emphasized that the model material for the glassy phase (SiCaAlON [8]) is different to the additive compositions in the material, which have been examined in Fett et al. [2] and Fünfschilling et al. [3]. However, the simple assumptions are able to capture the observations of Becher et al. [1], where large and elongated grains are supporting high fracture toughness and strength. 6. Conclusions
Fig. 5. Displacement and stress fields for the unit cell II with A ¼ 4:03; (a) displacement and (b) stress at the onset of fracture with bridging grain (loading direction e3), (c) displacement and (d) stress field after fracture without structural reinforcement (loading direction e2).
Hence it can be concluded that it is possible to capture the main features of the microscopic fracture behavior of silicon nitride by the proposed finite element simulation framework. Furthermore, a determination of the effective properties is possible under certain assumptions, which should be developed further in the future. The sensitivity of the microscopic models on the morphology makes the application of the model a feasible option for the improvement of the understanding and the further development of silicon nitride. Acknowledgement
The result for the unit cell II has shown the effect of missing denticulation of the structure is: A significant decrease in the fracture resistance. The estimated values of the unit cell fracture toughness (KR = 0.6 . . . 1.2 MPa m1/2) underestimate the fracture toughness, which can be observed experimentally for silicon nitride. So, a range from 6 to 11 MPa m1/2 has been reported by Becher et al. [1] for different chemical compositions and the related differences in the microstructure. Fett et al. [2] presents values between 2 and 7 MPa m1/2. Here, the lower initial value is caused by a special consideration of extremely short crack lengths. The discrepancy of the unit cell based values to the experimental data is not unexpected, due to the simplicity of the chosen fracture mechanics framework, which was solely meant to give an idea of the fracture toughness on the microlevel. Nevertheless, the results confirm the picture, which is given by experimental observations, even for different additive compositions. Examples are the findings of Fett et al. [2], where the additive composition in the examined silicon nitride grade is 5 wt.% Y2O3 and 2 wt.% MgO, and Fünfschilling et al. [3], where several rare earth oxides are used together with MgO or alumina.
This work was funded by the German Research Foundation (DFG) within the Center of Excellence SFB 483 C8 (Mechanismbased micromechanical simulation of crack propagation in in-situ-reinforced high performance ceramics) and by the European Commission within the 7th Framework Programme Project ROLICER (Enhanced reliability and lifetime of ceramic components through multi-scale modelling of degradation and damage). References [1] P. Becher, E. Sun, K. Plucknett, K. Alexander, C. Hsueh, H.-T. Lin, S. Waters, C.-G. Westmoreland, Journal of Crystal Growth 81 (1998) 2821–2830. [2] T. Fett, S. Fünfschilling, M. Hoffmann, R. Oberacker, Journal of the American Ceramic Society 91 (2008) 3638–3642. [3] S. Fünfschilling, T. Fett, M. Hoffmann, R. Oberacker, T. Schwind, J. Wippler, T. Böhlke, H. Özcoband, G. Schneider, P. Becher, J. Kruzic, Acta Materialia 59 (2011) 3978–3989. [4] S. Govindjee, G. Kay, J. Simo, International Journal for Numerical Methods in Engineering 38 (1995) 3611–3633. [5] D. Gross, T. Seelig, Bruchmechanik, Springer Berlin Heidelberg, 2007. [6] S. Hampshire, E. Nestor, R. Flynn, P. Goursat, M. Sebai, D. Thompson, K. Liddell, Journal of the European Ceramic Society 14 (1994) 261–273.
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[7] C. Henderson, D. Taylor, Transactions and Journal of the British Ceramic Society 74 (1975) 49–53. [8] H. Iba, T. Naganuma, K. Matsumara, Y. Kagawa, Journal of Material Science 34 (1999) 5701–5705. [9] A. Kailer, T. Hollstein, Walzen mit Keramik, Fraunhofer IRB-Verlag, Stuttgart, 2004. [10] M. Krämer, M. Hoffmann, G. Petzow, Acta Metallurgica 41/10 (1993) 2939– 2947. [11] J. Kruzic, R. Satet, M. Hoffmann, R. Cannon, R. Ritchie, Journal of the American Ceramic Society 91 (2008) 1986–1994. [12] E. Le Bourhis, Glass – Mechanics and Technology, Wiley-VHC, 2008. [13] J.J. Mecholsky, Material Letters 60 (2006) 2485–2488. [14] D. Munz, T. Fett, Ceramics – Mechanical Properties, Failure Behaviour, Materials Selection, Springer, 1999. [15] S. Ogata, N. Hirosaki, Y. Shibutani, Acta Materialia 52 (2004) 233–238. [16] I. Peterson, T.-Y. Tien, Journal of the American Ceramic Society 78 (1995) 2345–2352.
[17] J. Souza, M. Nono, M. Ribeiro, J. Machado, O. Silva, Materials & Design 30 (2009) 2715–2720. [18] E. Sun, P. Becher, K. Plucknett, C.-H. Hsueh, K. Alexander, S. Waters, Journal of the American Ceramic Society 81 (1998) 2831–2840. [19] C. Verhoosel, J. Remmers, A. Miguel, R. de Borst, International Journal for Numerical Methods in Engineering 83 (2010) 1155–1179. [20] R. Vogelgesang, M. Grimsditch, J. Wallace, Applied Physics Letters 76/8 (2000) 982–984. [21] Y. Wei, L. Anand, Journal of the Mechanics and Physics of Solids 52 (2004) 2587–2616. [22] J. Wippler, T. Böhlke, Journalof the European Ceramics Society (2011), http:// dx.doi.org/10.1016/j.jeurceramsoc.2011.10.001. [23] J. Wippler, T. Böhlke, Journal of the European Ceramic Society 32 (2012) 589– 602. [24] J. Wippler, S. Fünfschilling, F. Fritzen, T. Böhlke, M. Hoffmann, Acta Materialia 59 (2011) 6029–6038.
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Structure and fracture property relation for silicon nitride on the microscale Johannes Wippler ⇑, Thomas Böhlke Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT), Kaiserstraße 10, 76131 Karlsruhe, Germany
a r t i c l e
i n f o
Article history: Received 25 October 2011 Received in revised form 14 February 2012 Accepted 29 February 2012 Available online 5 April 2012 Keywords: Silicon nitride Fracture Finite element simulation Homogenization R-curve Fracture toughness
a b s t r a c t The fracture toughness of silicon nitride ceramics is closely related to its microstructure, which is characterized by a bimodal distribution of partially large and elongated grains. In the first stage of fracture, these grains act as elastic bridges, which avoid a quick crack propagation. In order to improve the understanding of this R-curve effect, a unit cell approach for multiscale finite element simulation has been chosen. For the examination of the morphology influence on fracture stress and toughness unit cells with different mean aspect ratios have been utilized for the fracture simulations. As results, on the one hand effective stress–strain curves have been applied for the computation of the effective R-curves. On the other hand, these results have been compared with observations of the stress and strain fields on the local level. Findings on the local level support the concept of elastic bridging grains. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Silicon nitride is an important engineering material for challenging applications like tool tips for turning with gray cast iron or for forming rolls for the production of high-strength steel wires, see, e.g. Souza et al. [17] and Kailer and Hollstein [9]. The key properties are the excellent combination of strength, toughness and thermal shock resistance, which can be related to the microstructure. The morphology of silicon nitride is characterized by the elongated b-Si3N4 grains of a specific size–shape distribution and a matrix, which is formed from sinter additives. Such additives are used for the dense-sintering and for the adjustment of specific grain lengths and shape distributions. As additives, often rare-earth oxides like yttria in combination with magnesia or alumina are used. Fundamental work on this field has been done by Sun et al. [18] and Becher et al. [1]. Kruzic et al. [11] enlightened the relevance of a careful selection of the grain boundary toughness for an optimal constellation of strength and stiffness. Additionally to the well-known fact, that overly strong grain interfaces lead to a strong but brittle material without the toughening mechanisms like grain bridging, the disadvantageous influence of too weak interfaces has been pointed out. This results are low strength and shallow R-curves. Hence, a good combination of structural morphology and the corresponding interface strength and toughness is crucial for a reliable tool material. ⇑ Corresponding author. Tel.: +49 (0)721 608 48852; fax: +49 (0)721 608 44187. E-mail addresses: [email protected] (J. Wippler), thomas.boehlke@kit. edu (T. Böhlke). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.02.042
The glassy phase, which fills the space between the grains (12 vol.%) exists as glass pockets at grain-free material volume and forms intergranular films between the grains. The glass pockets are relevant for the residual stresses in the material due to the thermal mismatch and is important for the formation of bridging grains, as it is pointed out in [16]. The intergranular films with relatively low strength lead to toughening due to crack path deflection. This means that cracks have to propagate around the stronger grains [1,3], such that the crack propagation consumes more energy. A tougher material is the consequence. In order to capture this complex behavior, a comprehensive constitutive model has to be created for each fracture mechanism on the microscale. This has been implemented into a unit cell framework, which incorporates all features of importance. The microscopic geometry is modeled according to experimental observations [10] by an enhanced sequential adsorption technique [22]. For thermoelastic properties on the microscale experimental observations [20,7,6] have been implemented [24]. The fracture, which occurs is often intergranular. This is described by a cohesive zone model that combines the concepts of Govindjee et al. [4] and Wei and Anand [21] and has been introduced in Wippler and Böhlke [23]. For the transgranular fracture of the grains and of the glassy phase pockets, constitutive assumptions will be proposed in this work. The toughness of the material can be assessed on the macroscopic level by R-curve measurements [14,2]. This concept describes the relation between fracture toughness and crack extension. In general, a rising R-curve indicates a tough material. For the evaluation of the unit cell simulation, a simplified fracture
J. Wippler, T. Böhlke / Computational Materials Science 64 (2012) 234–238
mechanics concept has been used in order to prove the significance of the proposed model on the macrolevel. In order to give further insight into the acting mechanisms, this work will examine the influence of different mean aspect ratios on both the effective fracture toughness as well as on the microscopic fracture processes. 2. Fracture of phases 2.1. Glassy phase The estimation of the strength of brittle materials on the microscale is challenging due to a significant volume effect. This means that the probability of a larger flaw decreases with decreasing volume, which leads to an increase of the strength at microscopic dimensions after the weakest-link theory. In order to account for this fact, the Weibull theory has been used to estimate the microscopic strength from macroscopic experiments. The rela1=m M tion rM Cð1 þ 1=mÞ provides a relation bec ðVÞ ¼ rc ðV 0 ÞðV 0 =VÞ tween the fracture strength at different volumes [5]. Let the volume of the experimentally assessed sample be V0. The volume of the portion, the strength of which should be estimated is V. The Weibull modulus, which is a measure for the scatter in the experimental data is denoted by m. As macroscopic strength values, the glass fiber data for SiCaAlON glass from Iba et al. [8] is used. This type of glass is not the same, as it is present in silicon nitride with Al and Y oxide composition. However, the strength and the Weibull modulus are expected to be similar enough to be a useful substitute for the YAlSiON glass. The glass fibers with a diameter d0 = 20 lm resisted stresses between 3 and 5 GPa. Together with an assumed fiber length of l0 = 30 mm, the 2 fiber volume is V 0 ¼ p=4d0 l0 ¼ 9:42 103 mm3 . For the Weibull modulus of glass fibers [12] provides a wide range between 10 and 30. An estimation of a glass pocket is possible, if its shape is assumed to be a regular tetrahedron with an edge length of 1 lm. pffiffiffi 3 Then the volume is V ¼ l =4 3 ¼ 1:44 1010 mm3 . Putting these assumptions together, the strength values reach from 5.36 to 28.7 GPa. The upper value from the Weibull consideration has to be compared with the upper bound for material strength, i.e., the theoretical strength. Based on the assumed Young’s modulus of 133 GPa from Hampshire et al. [6], this physical limit can reach from E/8 . . . E/p = 16.6 . . . 42.3 GPa for brittle materials, see, e.g. Mecholsky [13]. Hence, the assumption of 10 GPa appears as a realistic choice. The fracture behavior of the glassy matrix phase M is assumed to be caused by the maximum tensile stress. They are obtained from the eigenvalue problem rppr = rprppr with the principal stresses rpr and the corresponding normalized eigenvectors ppr. The fracture criterion is formulated as ‘p-norm on the tensile part of the principal stresses, where 1 6 N 6 3 is the number of different eigenvalues of the stress tensor r and rM c is the cracking stress of the matrix material. Thus, criterion can be denoted P M p 1=p N pr 1. =rc /M ¼ n¼1 max 0; rn 2.2. b-Si3N4-grains The fracture criterion for the b-grain phase G is motivated by the transverse-isotropy of the elastic stiffness tensor. Here, the po sitive eigenvalues max 0; rpr and the corresponding projectors n pr pr + P pr n ¼ pn pn are used to construct the tensile stress tensor r pr PN pr þ in the spectral representation r ¼ n¼1 max 0; rn P n . The normal direction of the grain faces are used for the construction of the projectors P Gf ¼ nGf nGf ; f ¼ 1 . . . 4, which map the tensile stress
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tensor. Using the ‘p-norm, the fracture criterion for a grain takes P 1=p 4 G G p þ 1. Fig. 1a shows the the form /G ¼ f¼1 jP f r =rc;f j three-dimensional contour of the fracture criterion /G ¼ 0. Motivated by the ab initio simulation results of [15], the selected parameters are rGc;1...3 ¼ 1923 MPa and rGc;4 ¼ 2000 MPa and p = 20. In Fig. 1b, the influence of the orientation on the cracking direction is visualized. The notched sample is thought to be cut out of a single crystal and loaded by displacements on the front faces. The crystal directions are indicated by the small coordinate system. The upper row shows the distribution of the fracture criterion /G . Green dyeing indicates high stress level of approximately 800–1200 MPa around the notch tip shortly before cracking. The effects of anisotropy are obvious, because the orientation of the contour lines corresponds to the crystal orientation. The lower row shows the cracks. It is interesting to observe the final fracture never occurs in the direction of the notch. The fracture criteria are implemented as user subroutine VUMAT for Abaqus/Explicit. In integration points, where the fracture criteria /M=G are fulfilled, stress is set to zero. 3. Determination of the effective fracture properties For the examination of the structure–property relation, three periodic unit cells with 64 grains and an edge length of 3 lm, in the further context designated with I, II and III have been used. The aspect ratio distribution of the grains in the unit cells can be characterized by the mean aspect ratio and the standard deviation. Due to the limited size of the statistical ensembles, higher-order measures are not expected to deliver deeper insight. The mean aspect ratios of the grains in the cells are hAI i ¼ 3:07; hAII i ¼ 4:03 and hAIII i ¼ 4:93. The corresponding standard deviations are rðAI Þ ¼ 1:10; rðAII Þ ¼ 1:41 and rðAI Þ ¼ 2:18. The grain volume fraction is 88%. The cells are created after Wippler and Böhlke [22] and can be seen in Fig. 2. The difference in the mean grain shapes is obvious. The cells are deformed uniaxially with a free transverse contraction under periodic boundary conditions Wippler et al. [24]. In order to examine the influence of the structural properties on the effective behavior a consideration of the relations between effective stress hri and strain e in tensile direction have been considered [24,19]. Those two quantities allow for the determination of the fracture toughness, when certain assumptions are made. The fracture toughness of a material is characterized by the energy release rate G. Following the basic relations of the linear fracture mechanics, as it is outlined in detail in [14], the energy release rate is given by G ¼ 1=2F 2 dC=dA. Here, F is assumed to be the force on the surface of the unit cube F: = hriw2. The compliance is denoted by C = u/F with the boundary displacement u ¼ ew. The cracked area A has to be determined under the assumption that fracture uniaxial tensile load occurs due to mode I. An equivalent description of the fracture toughness of a brittle material is provided by the stress intensity factor. There are two representations, which can be exploited for the determination of the cracked surface. First, it is a relation between the critical stress hri or the related force F pffiffiffi pffiffiffi and the crack length a and is given by K R ¼ hri aY ¼ F=w2 aY. The geometry factor Y can be assumed to have a constant value of 1 for a basic understanding. The second relation is between the energy release rate and the stress intensity factor. Given the effective uniaxial stress state, which is a special case of plane stress, the relation K 2R ¼ GE, where E is Young’s modulus for undamaged silicon nitride holds. The aforementioned relations allow for the determination of the crack surface increment and delivpffiffiffi ers dA ¼ 1=2w4 E=ð AY 2 ÞdC. From this relation, an incremental calculation for an equivalent crack surface is possible.
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Fig. 1. (a) Contour plot of the failure criterion for the b-grain /G ¼ 0 in the principal stress space; planes indicate vanishing stresses; (b) examination of the fracture behavior on a simplified compact tension specimen with different grain orientations; arrows with u indicate the prescribed displacement; the crystal orientation is visualized by the small coordinate system; c represents the axial and a the basal direction.
Fig. 2. Three unit cells with different mean aspect ratios: cell I ðhAI i ¼ 3:07Þ, cell II ðhAII i ¼ 4:03Þ and cell III ðhAIII i ¼ 4:93Þ; the coordinate system defines the loading directions in the further context.
4. Results In Fig. 3, the effective stress–strain curve and the fracture toughness are compiled for representative cases. Fig. 3a shows that the higher the aspect ratio, the higher the fracture strength. This trend can be attributed to the higher content of large and elongated grains, which are transferring higher local loads. Subfigure (b) shows the fracture toughness for the fracture mode I based on the aforementioned assumptions, which correspond to the effective stress–strain curves in subfigure (a). The crack length is normalized on the unit cell size w. The most obvious feature is the general rising trend of the curves, which corresponds with experimental observations by Fett et al. [2]. The significant influence of the grain shape is of special relevance in the context of the work. The clear trend: The higher the aspect ratio, the higher and steeper is the R-curve. It is natural that an effect of the loading direction can be observed. This is, basically, due to the limited size of the considered unit cells. However, the general trend towards increasing fracture strength and resistance with increasing aspect ratios is observable, as it can be seen in Fig. 4. The influence of the geometry is present
in all cases and becomes more pronounced with increasing aspect ratio. The microscopic reasons for the differences in the macroscopic behaviors can be observed in Fig. 5. Here, the corresponding displacement (left) and stress fields (right) for the microstructure II with the medium mean aspect ratio is gathered for the two loading directions with the biggest difference in the macroscopic behavior. The values of the displacement field are minimum and maximum values at the present state. The stress field color code is starting with blue color ( ) at 100 MPa, which represents the compressive stresses in the grains due to thermal strains and is ending in red color ( ) at 2000 MPa, which is the fracture strength of the grains in axial direction. The high values of strength and fracture toughness, which have been measured for the loading direction e3 can be related to the presence of a large and elongated grain, which is aligned mainly in this direction. The right side of Fig. 5a shows the displacement field. The jump in the field is next to the grain, which indicates, that the crack has to propagate around this grain. Fig. 5b shows the corresponding stress field. In the region with the separation, the stress field is almost vanishing due to the post-failure unloading. The load is carried by the elastic bridging grain, where loads up to 2000 MPa are localized. For geometry II, a considerable outlier for toughness but not for strength is observed, when loaded in e2-direction (Fig. 4( )). This decreased fracture toughness has its origin in the presence of an unreinforced plane in the unit cell, which can be seen in Fig. 5c and d. Due to the relatively homogeneous stress distribution, the fracture strength is not affected considerably. 5. Discussion The results confirm the experimental observations, where long and elongated grains lead to higher fracture toughness.
(a)
Fig. 3. Influence of the mean aspect ratio on different effective properties; hAI i ( crack length (normalized on the edge length of the unit cell w).
(b)
), hAII i (
), hAIII i (
); (a) effective stress–strain curve, (b) stress intensity factor KR over
J. Wippler, T. Böhlke / Computational Materials Science 64 (2012) 234–238
(a)
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(b)
Fig. 4. (a) Maximum crack stress and (b) maximum fracture resistance values for the three considered geometries under loading direction e1 ( ), e2 ( ) and e3 ( ), mean values () and adjustment by linear regression ( ); loading directions correspond to definition in Fig. 2.
The general trend: The R-curve starts for very short cracks of single microns at a level, which is approximately one half of the plateau level like it can be seen in Fig. 3d. Hence, the ratio between the initial and peak unit cell fracture toughness is lower than it was measured in Fünfschilling et al. [3]. This difference is, as well as the deviation from the absolute experimental values are likely to be caused by the size of the unit cell and the simplicity of the chosen framework. The main features of the complex fracture process in silicon nitride can be captured by the unit cell approach with structure and the related constitutive assumptions. It has to be emphasized that the model material for the glassy phase (SiCaAlON [8]) is different to the additive compositions in the material, which have been examined in Fett et al. [2] and Fünfschilling et al. [3]. However, the simple assumptions are able to capture the observations of Becher et al. [1], where large and elongated grains are supporting high fracture toughness and strength. 6. Conclusions
Fig. 5. Displacement and stress fields for the unit cell II with A ¼ 4:03; (a) displacement and (b) stress at the onset of fracture with bridging grain (loading direction e3), (c) displacement and (d) stress field after fracture without structural reinforcement (loading direction e2).
Hence it can be concluded that it is possible to capture the main features of the microscopic fracture behavior of silicon nitride by the proposed finite element simulation framework. Furthermore, a determination of the effective properties is possible under certain assumptions, which should be developed further in the future. The sensitivity of the microscopic models on the morphology makes the application of the model a feasible option for the improvement of the understanding and the further development of silicon nitride. Acknowledgement
The result for the unit cell II has shown the effect of missing denticulation of the structure is: A significant decrease in the fracture resistance. The estimated values of the unit cell fracture toughness (KR = 0.6 . . . 1.2 MPa m1/2) underestimate the fracture toughness, which can be observed experimentally for silicon nitride. So, a range from 6 to 11 MPa m1/2 has been reported by Becher et al. [1] for different chemical compositions and the related differences in the microstructure. Fett et al. [2] presents values between 2 and 7 MPa m1/2. Here, the lower initial value is caused by a special consideration of extremely short crack lengths. The discrepancy of the unit cell based values to the experimental data is not unexpected, due to the simplicity of the chosen fracture mechanics framework, which was solely meant to give an idea of the fracture toughness on the microlevel. Nevertheless, the results confirm the picture, which is given by experimental observations, even for different additive compositions. Examples are the findings of Fett et al. [2], where the additive composition in the examined silicon nitride grade is 5 wt.% Y2O3 and 2 wt.% MgO, and Fünfschilling et al. [3], where several rare earth oxides are used together with MgO or alumina.
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