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Multilevel simulation of deformation and fracture of brittle porous materials in the method of movable cellular automata
Ig.S. Konovalenko, A.Yu. Smolin and S.G. Psakhie / Physical Mesomechanics 13 12 (2010) 47–53
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Multilevel simulation of deformation and fracture of brittle porous materials in the method of movable cellular automata Ig.S. Konovalenko1*, A.Yu. Smolin1,2 and S.G. Psakhie1,2,3 1
Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia 2 Tomsk State University, Tomsk, 634050, Russia 3 Tomsk Polytechnic University, Tomsk, 634050, Russia
In the framework of the method of movable cellular automata, an approach is proposed to multilevel description of deformation and fracture of brittle porous media with a single maximum in the pore size distribution histogram. In the approach, the effective response function of a cellular automaton is defined through direct simulation of the representative volume of a porous medium. A hierarchical model is developed for the mechanical behavior of ZrO 2-based ceramics with the pore size comparable with the average grain size under shear and uniaxial compression. The developed approach is analyzed for feasibility of taking into account the spatial distribution nonuniformity of strength properties of porous media by varying the automata interaction parameters in stochastically chosen directions. It is shown that this method of considering the nonuniformity holds much promise for multilevel description of porous media with a hierarchical pore space structure. Keywords: multilevel approach, porous materials, dynamic loading, deformation and fracture, method of movable cellular automata
1. Introduction Now, many areas of human activity use porous materials, in particular nanostructured porous ceramics [1]. These materials have a rather complex porous structure; pores of several scales and in various proportions and spatial distributions can be found in them depending on manufacturing techniques. For example, the walls of macropores (cells) can contain smaller pores [2, 3]. Detailed experimental data on the structure and mechanical characteristics of these materials are sometimes hard to access. Moreover, experimental research often provides macroscale description without considering other structural levels of the material properties reliable data on which are critical, in particular, for estimation of their contribution to deformation and fracture. In this context, numerical simulation can be used to advantage to solve the above problems. In single-level simulation, explicit consideration of the structural properties and behavior of the materials at each scale level is impossible. This is due to the time-consuming computations (a great quantity of computational elements) in explicit definition * Corresponding author Dr. Igor S. Konovalenko, e-mail: [email protected] Copyright © 2010 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2010.03.006
of a model porous structure of the material and to the structural hierarchy of the ceramics pore space [2, 3]. The latter fact implies by itself the use of a multilevel approach for description of this type of systems. Thus, the objective of the work is to develop a multilevel approach and an appropriate hierarchical model based on the method of movable cellular automata (MCA) for description of deformation and fracture of nanostructured porous ceramics under shear and uniaxial compression [4]. The choice of the MCA method was dictated by its efficiency in examination of mechanical behavior of brittle porous media from the nucleation of first damages to the point of fracture [58]. The calculations were performed for a model material with the properties of sintered ZrO 2 ceramics, average pore size comparable with the grain size, and single maximum in the pore size distribution [2, 3]. The hierarchical model of the material under study was constructed in several steps. At the first stage, the effective response functions of movable automata of macroscopic structural level were defined through calculations with explicit consideration of the material structure at the microlevel under various mechanical loads; and the representative volume of this hierarchical level was determined. At the second stage, we simulated continuous specimens at the
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Ig.S. Konovalenko, A.Yu. Smolin and S.G. Psakhie / Physical Mesomechanics 13 12 (2010) 47–53
macrolevel and took into account the porous structure of the lower levels in the effective response functions defined at the first stage. However, this model medium was characterized by uniform spatial distribution of mechanical properties and was free from stress concentrators and heterogeneities peculiar to porous materials at the chosen scale level. One of the ways to consider the macroscale nonuniformity typical of porous materials is to vary the mechanical parameters, e.g., strength, of automata. In this work, we vary the rupture criterion of inter-element bonds for stochastically chosen automata in stochastically chosen directions. At the third stage, the developed model was verified and possible ways of its improvement were examined through implicit consideration of the stochastic nonuniformity of the strength properties of porous materials. 2. Representative volume and effective response functions of automata at the microscale level In the model proposed, the microscale representative volume was determined by analyzing the convergence of the elastic and strength characteristics of the model porous specimens with an increase in their size. For this purpose, the mechanical behavior of six groups of 2D porous ceramic specimens under shear and uniaxial compression was modeled. The specimens of each group featured the same size, but differed in spatial pore distribution. The groups contained five specimens each. The specimens under study were squares with a side h of 6, 12, 36, 60, 90 and 120 Pm. It was assumed all pores in the model material are alike and are spherical. Their size, according to the maximum on the distribution histogram of the real ceramics, was 1.8 Pm [2, 3]. The size of a cellular automaton was chosen to match the average grain size and was 0.6 Pm. The pore structure of the specimens was specified by removing single automata and their six nearest neighbors in a random fashion. The porosity of the specimens was 15 %. The initial structure of one of the model specimens is shown in Fig. 1. In shear, the mechanical load was applied by assigning the same horizontal velocity to the automata of the upper a
specimen layer; the automata of the lower layer were rigidly fixed (Fig. 1(a)). Initially, the velocity of the automata of the upper layer was increased by the sinusoidal law from 0 to 1 m/s, and then it remained constant (Fig. 1(c)). This pattern was used to eliminate artificial dynamic effects and to provide gradual and fast establishment of quasisteady deformation of the specimen. Periodic boundary conditions was assigned along the horizontal axis. In uniaxial compression, the vertical velocity of the automata of the upper layer was also increased by the sinusoidal law (up to 1 m/s), and that of the automata of the lower level was zero. The automata of the upper and lower specimen layers were allowed for horizontal displacements and the lateral specimen surfaces were free (Fig. 1(b)). The problems were solved for plane deformation. The response function of the automata was linear and was appropriate to the diagram of the model ceramics of porosity 2 % [2, 3]. The shear modulus G of a cellular automaton was 30.8 GPa, and Poissons ratio was Q = 0.3. The rupture criterion of bonds between automata was the fracture criterion in terms of tangential stress intensity [9]. In uniaxial compression, the representative volume was determined in two steps: for nonporous and porous ceramics; in shear, only for porous ceramics. For the nonporous model material, the representative volume was determined on six square solid specimens of size corresponding to that of the porous specimens. As the specimen size was increased, the convergence of the effective elastic modulus Eeff of the model specimen and the corresponding elastic modulus E0 of a cellular automatonspecified in the model was analyzed. The result of analysis was the specimen size (the representative volume) beginning with which the deviation of Eeff from E0 is no greater than 3 %, which was taken appropriate for the problem solving. At the second stage, the representative volume was determined for the porous ceramics on groups of porous specimens of size equal to or larger than that of the representative volume for the continuous medium (all specimens in a group were of like size, but were different in spatial pore distribution). b
c
Fig. 1. Initial structure of the model specimen with a side h = 60 Pm and pattern of load application in shear (a) and uniaxial compression (b); law of variation in the velocity of automata of the upper specimen layer (c)
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teristics change but little with an increase in specimen size. This is due to the periodic conditions imposed on the specimen along the horizontal axis. The results of calculations that reflect the relative variation of the effective elastic properties of solid (continuous) specimens in relation to their size are presented in Fig. 2. It can be seen that Eeff nonlinearly converges to E0 , and with a specimen size of t 60 Pm, Eeff E0 is no greater than 1.5 %. This accuracy is more than sufficient for the task of our work, and hence the solid specimen with a side l = 60 Pm can be considered as the representative volume for the continuous model medium under study. Figure 3 depicts the results of calculation that reflect the relative variation of the effective elastic and strength properties of the specimens (compression Eeff and shear G eff moduli, compressive Vc and shear Wc strengths) in relation to their sizes for the porous model material under various mechanical loads. These characteristics also reveal a nonlinear convergence in the range of the specimen size under study. The relative deviations of Eeff , Geff , Vc and Wc from the corresponding group-averages Eeff , Geff , Vc , W c are 1.85, 0.94, 13.7 and 9.3 %. These values are no greater than the required threshold (3 and 20 %) and are sufficient for the problem solving in our work. Thus, it is shown that the porous specimens with a side h t 60 Pm are the microscale representative volumes of the model medium under study. The obtained results made it possible to continue the research at the higher level macroscale; in so doing, the group-averages Eeff and
Fig. 2. Deviation of the effective elastic modulus Eeff of the model specimen from the elastic modulus E0 of a cellular automaton in relation to the specimen size
The deviation of the effective elastic modulus Eeff and strength Vc (determined from the calculated loading diagram) of the model specimen from the corresponding groupaverages Eeff , Vc was analyzed. The result of analysis was the specimen size (representative volume) beginning with which the deviation of Eeff and Vc from Eeff and V c is no greater than the value acceptable for the problem solving. In our work, this value was 3 % for Eeff and 20 % for V c which is acceptable for heterogeneous media of porosity greater than 10 %. The preliminary analysis allows a near threefold reduction of the computation time and memory space. Similar analysis for nonporous ceramics under shear makes no sense, since its elastic characa
b
c
d
Fig. 3. Deviation of the elastic modulus Eeff , shear modulus Geff , compressive strength V c and shear strength Wc of the model specimens from the corresponding group-averages Eeff , Geff , Vc and Wc for uniaxial compression (a, b) and shear (c, d )
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Vc were taken as the parameters of the response functions of automata at this scale level. For plane deformation, the change-over from the elastic modulus E PSS determined from the calculated loading diagram to Youngs modulus relied on E E PSS (1 Q 2 ) [10]. 3. Macrolevel calculations with implicit consideration of the strength nonuniformity For macrolevel calculations, we used 2D square continuous specimens with a side h = 6 mm. According to the determined representative volume, the diameter of an automaton was 60 Pm. For the micro-to-macroscale transfer of the data on the porous structure and corresponding effective strength and elastic properties of the material, the response functions defined for the representative volumes of the microlevel were applied to automata of the macrolevel. The response function of the macrolevel automata was linear, and its parameters (maximum specific resistance force and Youngs modulus) were 298 MPa and 46.8 GPa for shear and 474 MPa and 45.23 GPa for uniaxial compression respectively. The pattern of mechanical load application and the assumption of the stress state are similar to those used in the representative volume problem. In the model, the nonuniformity of the strength properties of the material at the macrolevel due to the presence of defects and to the nonuniform pore distribution in the specimen at this hierarchical level was taken into account by varying the strength criterion for two of six bonds of stochastically chosen automata in stochastically chosen directions (Fig. 4). The parameters for this macrolevel nonuniformity are (i) the percentage of automata [ in the model specimen for which the threshold of the rupture strength V of automaton bonds is varied, (ii) the relative variation M of this threshold for a stochastically chosen bond of an individual automaton, and (iii) the number of bonds N for this automaton. In the general case, N = 1 6. It should be noted that the parameters M and N are more local than [, since they refer to only individual elements of the specimen (cellular automata), and [ determines the number of these elements and hence characterizes the specimen as a whole.
Fig. 4. Variation of the bond parameters for two of six neighbors of one automaton
We modeled four groups of specimens for which [ was 1, 5, 10 and 15 %, and N = 2. The quantity M was varied from 15 to +50 %. A group comprised three subgroups of five specimens and with the same range of V each. Thus, we considered three combinations of M: 10 and 10 %, 10 and 10 %, 15 and 50 % for the first and second stochastically chosen bonds of automata. For convenience, these subgroups are further referred to as the first, second and third subgroup. The maximum and minimum values of M in the above ranges were chosen such that in the first subgroup, the total strength of the specimen for all bonds was the least, and in the third group, it was the highest; this corresponded to three different materials with fixed [. For correct examination of the relationship between the degree of nonuniformity [ of the strength properties and their macroscale response, it is necessary to compare the behavior of specimens of one material. In terms of our model, this corresponds to comparison of the mechanical behavior of a series of specimens with the same values of M, but different values of [. 4. Model verification Verification is an essential step in construction of any model of mechanical behavior of actual materials. In our study, the model is considered adequate to the examined porous ceramics if the results of model calculations satisfy the following criteria: (i) the loading diagram of the model specimens is linear, which is characteristic of porous brittle materials; (ii) the fracture patterns of equidimensional macroscopic specimens with explicit and implicit consideration of porous structure agree qualitatively; (iii) the strength characteristics of the specimens fall on a certain interval found from approximate estimates. Loading diagrams typical of all model specimens under uniaxial compression and shear are shown in Fig. 5. The diagrams reveal several characteristic portions. The first portion is a linear portion corresponding to elastic deformation of the specimen. Next, there is a short ascending portion with stress drops (generation and development of individual damages in the specimen bulk) and a descending portion (development of a macrocrack or system of macrocracks, and generation of individual multiple damages). In the shear diagrams (Fig. 5(b)), the above portions are followed by an ascending portion with stress drops and next by one more descending portion. Because of constrained deformation, it is to these latter portions that the development of the macrocrack system in the specimen corresponds, and the first descending portion corresponds to the one-time generation of damages throughout the specimen and their development without the formation of the macrocrack system. Comparison of the above diagrams with appropriate diagrams of brittle porous solids in shear and uniaxial com-
Ig.S. Konovalenko, A.Yu. Smolin and S.G. Psakhie / Physical Mesomechanics 13 12 ((2010)) 47–53 g y a
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b
Fig. 5. Loading diagrams of the model specimens with a side h = 6 mm in uniaxial compression (a) and in shear (b); [ = 15 %
pression [2, 8, 11] shows a good qualitative agreement. Thus, the first macroscopic criterion of model adequacy the general form of the loading diagram is fulfilled. Direct model verification for the second criterion is now impossible. This is associated with time- and memory-consuming computations (a large number of computational elements and a small time step) required for explicit consideration of the porous structure in the model. Therefore in comparison, we used the simulation results with explicit consideration of the porous structure obtained for brittle porous macroscopic specimens of smaller size in [58]. Figure 6 shows fracture patterns (inter-element bonds) typical of the model specimens at the instant macrocracks appear in the specimens. Under uniaxial compression (Fig. 6(a, b)), the specimens are fractured due to the deve-
lopment of an asymmetric system of oblique cracks in the conjugate directions; and under shear, of oblique and horizontal cracks (Fig. 6(c, d )). In both cases, individual multiple damages are generated near the cracks in the specimens. The character of the formed damages and macrocracks in the model material with an implicitly considered porosity of 15 % and nonuniform spatial strength distribution corresponds to that found in both actual [2, 3] and model [8] brittle porous media of low porosity (510 %) specified explicitly. It should be noted that a complete agreement between the fracture patterns of the examined model specimens and specimens with explicit consideration of the porous structure is impossible to obtain at all. The differences in the internal structure of these specimens determine to a
a
b
c
d
Fig. 6. Fracture patterns (macrocracking) of the specimens with the same combinations of M values (for specimens (a, b) and (c, d )) in uniaxial compression (a, b) and shear (c, d). [ = 15 (a, b), 5 % (c, d )
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great extent the distinctions in the processes evolved in them under loading. With explicit consideration of pores, the model contains a free volume and hence local distribution gradients of mass, elastic and strength properties. Moreover, mass transfer occurs and a pore works as both a concentrator and a mechanism of stress relaxation conductive to crack retardation and arrest. With implicit consideration of pores and variation in the strength parameters of automata interaction for individual elements, the specimens are a material with uniform distribution of mass and elastic properties in which mass transfer is absent and the varied bonds between automata work only as stress concentrators. In view of the foregoing, the qualitative agreement between the fracture of the model macrospecimens and that of brittle porous media suggests that the second criterion of model adequacy is fulfilled. In model verification for the third criterion, the range of the strength V c for the groups of specimens with the same values of [, M and N was determined from the following. The upper limit of the Vc max range corresponded to the strength V0 of the macrospecimen with intact (unvaried) bonds between automata and was 500 and 166 MPa for uniaxial compression and shear, respectively. With [ % of automata deprived of inter-element bond and with consideration of possible variation in the strength of the representative volume by T %, the lower limit of the Vc min range is determined by the expression:
Vc min
Vc max (1 [ 100)(1 T 100).
Because the bonds between automata are not removed in the model and only their strength is decreased by M %, the specimen strength Vc is bound to vary in the range Vc min d Vc d Vc max . The sought-for Vc min and Vc max and the range of the specimen strength V c obtained in modeling at [ = 5 and 15 % for shear and uniaxial compression are presented in Fig. 7. It can be seen that for all specimens, V c falls within the above range and hence the last criterion of adequacy of the constructed model is fulfilled. 5. Features of the developed approach Using the verified model, we studied the developed approach for feasibility of implicit consideration of different degrees of stochastic nonuniformity of the strength properties [ in porous media. The results of calculations show that with implicit consideration of the spatial distribution nonuniformity of the strength properties in the model, the effective elastic properties Eeff , Geff are near invariant with [. The effective strength properties Vc , Wc do vary with [, but the strength range for the groups of specimens with the same combinations of M values and different [ are mainly overlapped (Fig. 7). Thus, among the variety of specimens with the same
Fig. 7. Ranges of the strength of the model specimens with different values of [ and M, and their upper and lower analytical estimates under various mechanical loads
M and different [, we can always find specimens with equal or very nearly equal values of Vc and Wc . The effect of the parameter [ on the character of fracture of the examined specimens was studied by analyzing their corresponding inter-element bonds at the instant a system of macrocracks is formed in them (Fig. 6). For more accurate analysis of these interactions, specimens with low (5 %) and high (15 %) values of [ were considered. It is found that as [ increases, unmerged local damages and edge-to-edge macrocracks in the specimen increase in number and their spatial distribution becomes more uniform. In this case, the number of cracks that propagate deep into the specimen and fail to develop into main crack decreases. As the parameter [ is decreased, the reverse situation is observed. Moreover, with small values of [, the shear of the specimens gives rise to horizontal macrocracks inside them, whereas with high values of [, horizontal macrocracks are formed mainly at the surfaces of load application (upper surface) and fixation (lower surface). Thus, the variation in one of the model parameters [ responsible for the degree of the spatial distribution nonuniformity of the strength characteristics changes the fracture patterns of the model specimens while the strength and the elastic modulus remain near constant. It should be noted that the specimen strengths Vc and Wc depend in a complicated manner on many factors, in particular, on the distribution of the varied bonds between automata over the specimen, degree of their variation M, their number N for one automaton, and on the orientation of these bonds about the loading direction. For example, the specimens with the same values of M, N and [, but with different spatial distributions of the varied bonds of automata feature different strength properties Vc and Wc . Thus, the interplay between these parameters and the behavior of the system is a rather intricate and requires additional study.
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However, analysis of the influence of even one of these parameters, [, on the mechanical behavior of the system gives interesting results and indicates that the developed MCA-based approach holds promise for construction of ceramics models with implicit consideration of the porous structure. 6. Conclusion Thus, the obtained results allows the conclusion the multilevel approach developed based on the method of movable cellular automata and appropriate hierarchical model adequately describe deformation and fracture of porous media under mechanical loading. It is shown that the method used in the approach to consider the spatial distribution uniformity of the strength properties hold much promise for multilevel description of porous mediums with a hierarchical pore space structure. The work was supported by RFBR grants Nos.07-0812179-ofi, 08-08-12055-ofi. The authors are thankful to E.V. Shilko, S.N. Kulkov and S.P. Buyakova for the attention paid to the work and useful comments. References [1] Global Roadmap for Ceramics: Proceedings of 2nd International Congress on Ceramics (ICC2), Ed. by A. Belosi and G.N. Babini, Institute
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of Science and Technology for Ceramics, National Research Council, Verona, 2008. [2] S.P. Buyakova, Properties, Structure, Phase Composition and Mechanisms of the Formation of Porous ZrO 2-Based Nanosystems: Doct. Degree Book (Engng.), ISPMS SB RAS, Tomsk, 2008 (in Russian). [3] S.N. Kulkov, S.P. Buyakova and V.I. Maslovskii, Structure, Phase composition and mechanical properties of ZrO2-based ceramics, Vestnik TGU, No. 13 (2003) 34 (in Russian). [4] S.G. Psakhie, G.P. Ostermeyer, A.I. Dmitriev, E.V. Shilko, A.Yu. Smolin and S.Yu. Korostelev, Method of movable cellular automata as a new trend of discrete computational mechanics. I. Theoretical description, Phys. Mesomech., 3, No. 2 (2000) 5. [5] Ig.S. Konovalenko, A.Yu. Smolin and S.G. Psakhie, Deformation and fracture features in brittle porous media with various pore morphology, Izv. Vuzov, Fizika, 48, No. 6 (2005) 25 (in Russian). [6] A.Yu. Smolin, Ig.S. Konovalenko, S.N. Kulkov and S.G. Psakhie, Quasi-viscous fracture of brittle media with stochastic pore distribution, Techn. Phys. Lett., 32, No. 9 (2006) 738. [7] A.Yu. Smolin, Ig.S. Konovalenko, S.N. Kulkov and S.G. Psakhie, Simulation of fracture in brittle porous media with various internal structure, Izv. Vuzov, Fizika, 49, No. 3, Supplement (2006) 70 (in Russian). [8] Ig.S. Konovalenko, Theoretical Study of Deromation and Fracture of Porous Materials for Medicine and Biomechanical Structures: Cand. Degree Book (Phys. & Math.), ISPMS SB RAS, Tomsk, 2007 (in Russian). [9] L.M. Kachanov, The Foundations of Fracture Mechanics, Nauka, Moscow, 1974 (in Russian). [10] L.I. Sedov, A Course in Continuum Mechanics, Groningen, WoltersNoordhoff, 1971. [11] G.A. Gogotsi, The problem of the classification of low-deformation materials based on the features of their behavior under load, Strength of Materials, 9, No. 1 (1977) 77.
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Multilevel simulation of deformation and fracture of brittle porous materials in the method of movable cellular automata Ig.S. Konovalenko1*, A.Yu. Smolin1,2 and S.G. Psakhie1,2,3 1
Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia 2 Tomsk State University, Tomsk, 634050, Russia 3 Tomsk Polytechnic University, Tomsk, 634050, Russia
In the framework of the method of movable cellular automata, an approach is proposed to multilevel description of deformation and fracture of brittle porous media with a single maximum in the pore size distribution histogram. In the approach, the effective response function of a cellular automaton is defined through direct simulation of the representative volume of a porous medium. A hierarchical model is developed for the mechanical behavior of ZrO 2-based ceramics with the pore size comparable with the average grain size under shear and uniaxial compression. The developed approach is analyzed for feasibility of taking into account the spatial distribution nonuniformity of strength properties of porous media by varying the automata interaction parameters in stochastically chosen directions. It is shown that this method of considering the nonuniformity holds much promise for multilevel description of porous media with a hierarchical pore space structure. Keywords: multilevel approach, porous materials, dynamic loading, deformation and fracture, method of movable cellular automata
1. Introduction Now, many areas of human activity use porous materials, in particular nanostructured porous ceramics [1]. These materials have a rather complex porous structure; pores of several scales and in various proportions and spatial distributions can be found in them depending on manufacturing techniques. For example, the walls of macropores (cells) can contain smaller pores [2, 3]. Detailed experimental data on the structure and mechanical characteristics of these materials are sometimes hard to access. Moreover, experimental research often provides macroscale description without considering other structural levels of the material properties reliable data on which are critical, in particular, for estimation of their contribution to deformation and fracture. In this context, numerical simulation can be used to advantage to solve the above problems. In single-level simulation, explicit consideration of the structural properties and behavior of the materials at each scale level is impossible. This is due to the time-consuming computations (a great quantity of computational elements) in explicit definition * Corresponding author Dr. Igor S. Konovalenko, e-mail: [email protected] Copyright © 2010 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2010.03.006
of a model porous structure of the material and to the structural hierarchy of the ceramics pore space [2, 3]. The latter fact implies by itself the use of a multilevel approach for description of this type of systems. Thus, the objective of the work is to develop a multilevel approach and an appropriate hierarchical model based on the method of movable cellular automata (MCA) for description of deformation and fracture of nanostructured porous ceramics under shear and uniaxial compression [4]. The choice of the MCA method was dictated by its efficiency in examination of mechanical behavior of brittle porous media from the nucleation of first damages to the point of fracture [58]. The calculations were performed for a model material with the properties of sintered ZrO 2 ceramics, average pore size comparable with the grain size, and single maximum in the pore size distribution [2, 3]. The hierarchical model of the material under study was constructed in several steps. At the first stage, the effective response functions of movable automata of macroscopic structural level were defined through calculations with explicit consideration of the material structure at the microlevel under various mechanical loads; and the representative volume of this hierarchical level was determined. At the second stage, we simulated continuous specimens at the
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macrolevel and took into account the porous structure of the lower levels in the effective response functions defined at the first stage. However, this model medium was characterized by uniform spatial distribution of mechanical properties and was free from stress concentrators and heterogeneities peculiar to porous materials at the chosen scale level. One of the ways to consider the macroscale nonuniformity typical of porous materials is to vary the mechanical parameters, e.g., strength, of automata. In this work, we vary the rupture criterion of inter-element bonds for stochastically chosen automata in stochastically chosen directions. At the third stage, the developed model was verified and possible ways of its improvement were examined through implicit consideration of the stochastic nonuniformity of the strength properties of porous materials. 2. Representative volume and effective response functions of automata at the microscale level In the model proposed, the microscale representative volume was determined by analyzing the convergence of the elastic and strength characteristics of the model porous specimens with an increase in their size. For this purpose, the mechanical behavior of six groups of 2D porous ceramic specimens under shear and uniaxial compression was modeled. The specimens of each group featured the same size, but differed in spatial pore distribution. The groups contained five specimens each. The specimens under study were squares with a side h of 6, 12, 36, 60, 90 and 120 Pm. It was assumed all pores in the model material are alike and are spherical. Their size, according to the maximum on the distribution histogram of the real ceramics, was 1.8 Pm [2, 3]. The size of a cellular automaton was chosen to match the average grain size and was 0.6 Pm. The pore structure of the specimens was specified by removing single automata and their six nearest neighbors in a random fashion. The porosity of the specimens was 15 %. The initial structure of one of the model specimens is shown in Fig. 1. In shear, the mechanical load was applied by assigning the same horizontal velocity to the automata of the upper a
specimen layer; the automata of the lower layer were rigidly fixed (Fig. 1(a)). Initially, the velocity of the automata of the upper layer was increased by the sinusoidal law from 0 to 1 m/s, and then it remained constant (Fig. 1(c)). This pattern was used to eliminate artificial dynamic effects and to provide gradual and fast establishment of quasisteady deformation of the specimen. Periodic boundary conditions was assigned along the horizontal axis. In uniaxial compression, the vertical velocity of the automata of the upper layer was also increased by the sinusoidal law (up to 1 m/s), and that of the automata of the lower level was zero. The automata of the upper and lower specimen layers were allowed for horizontal displacements and the lateral specimen surfaces were free (Fig. 1(b)). The problems were solved for plane deformation. The response function of the automata was linear and was appropriate to the diagram of the model ceramics of porosity 2 % [2, 3]. The shear modulus G of a cellular automaton was 30.8 GPa, and Poissons ratio was Q = 0.3. The rupture criterion of bonds between automata was the fracture criterion in terms of tangential stress intensity [9]. In uniaxial compression, the representative volume was determined in two steps: for nonporous and porous ceramics; in shear, only for porous ceramics. For the nonporous model material, the representative volume was determined on six square solid specimens of size corresponding to that of the porous specimens. As the specimen size was increased, the convergence of the effective elastic modulus Eeff of the model specimen and the corresponding elastic modulus E0 of a cellular automatonspecified in the model was analyzed. The result of analysis was the specimen size (the representative volume) beginning with which the deviation of Eeff from E0 is no greater than 3 %, which was taken appropriate for the problem solving. At the second stage, the representative volume was determined for the porous ceramics on groups of porous specimens of size equal to or larger than that of the representative volume for the continuous medium (all specimens in a group were of like size, but were different in spatial pore distribution). b
c
Fig. 1. Initial structure of the model specimen with a side h = 60 Pm and pattern of load application in shear (a) and uniaxial compression (b); law of variation in the velocity of automata of the upper specimen layer (c)
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teristics change but little with an increase in specimen size. This is due to the periodic conditions imposed on the specimen along the horizontal axis. The results of calculations that reflect the relative variation of the effective elastic properties of solid (continuous) specimens in relation to their size are presented in Fig. 2. It can be seen that Eeff nonlinearly converges to E0 , and with a specimen size of t 60 Pm, Eeff E0 is no greater than 1.5 %. This accuracy is more than sufficient for the task of our work, and hence the solid specimen with a side l = 60 Pm can be considered as the representative volume for the continuous model medium under study. Figure 3 depicts the results of calculation that reflect the relative variation of the effective elastic and strength properties of the specimens (compression Eeff and shear G eff moduli, compressive Vc and shear Wc strengths) in relation to their sizes for the porous model material under various mechanical loads. These characteristics also reveal a nonlinear convergence in the range of the specimen size under study. The relative deviations of Eeff , Geff , Vc and Wc from the corresponding group-averages Eeff , Geff , Vc , W c are 1.85, 0.94, 13.7 and 9.3 %. These values are no greater than the required threshold (3 and 20 %) and are sufficient for the problem solving in our work. Thus, it is shown that the porous specimens with a side h t 60 Pm are the microscale representative volumes of the model medium under study. The obtained results made it possible to continue the research at the higher level macroscale; in so doing, the group-averages Eeff and
Fig. 2. Deviation of the effective elastic modulus Eeff of the model specimen from the elastic modulus E0 of a cellular automaton in relation to the specimen size
The deviation of the effective elastic modulus Eeff and strength Vc (determined from the calculated loading diagram) of the model specimen from the corresponding groupaverages Eeff , Vc was analyzed. The result of analysis was the specimen size (representative volume) beginning with which the deviation of Eeff and Vc from Eeff and V c is no greater than the value acceptable for the problem solving. In our work, this value was 3 % for Eeff and 20 % for V c which is acceptable for heterogeneous media of porosity greater than 10 %. The preliminary analysis allows a near threefold reduction of the computation time and memory space. Similar analysis for nonporous ceramics under shear makes no sense, since its elastic characa
b
c
d
Fig. 3. Deviation of the elastic modulus Eeff , shear modulus Geff , compressive strength V c and shear strength Wc of the model specimens from the corresponding group-averages Eeff , Geff , Vc and Wc for uniaxial compression (a, b) and shear (c, d )
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Vc were taken as the parameters of the response functions of automata at this scale level. For plane deformation, the change-over from the elastic modulus E PSS determined from the calculated loading diagram to Youngs modulus relied on E E PSS (1 Q 2 ) [10]. 3. Macrolevel calculations with implicit consideration of the strength nonuniformity For macrolevel calculations, we used 2D square continuous specimens with a side h = 6 mm. According to the determined representative volume, the diameter of an automaton was 60 Pm. For the micro-to-macroscale transfer of the data on the porous structure and corresponding effective strength and elastic properties of the material, the response functions defined for the representative volumes of the microlevel were applied to automata of the macrolevel. The response function of the macrolevel automata was linear, and its parameters (maximum specific resistance force and Youngs modulus) were 298 MPa and 46.8 GPa for shear and 474 MPa and 45.23 GPa for uniaxial compression respectively. The pattern of mechanical load application and the assumption of the stress state are similar to those used in the representative volume problem. In the model, the nonuniformity of the strength properties of the material at the macrolevel due to the presence of defects and to the nonuniform pore distribution in the specimen at this hierarchical level was taken into account by varying the strength criterion for two of six bonds of stochastically chosen automata in stochastically chosen directions (Fig. 4). The parameters for this macrolevel nonuniformity are (i) the percentage of automata [ in the model specimen for which the threshold of the rupture strength V of automaton bonds is varied, (ii) the relative variation M of this threshold for a stochastically chosen bond of an individual automaton, and (iii) the number of bonds N for this automaton. In the general case, N = 1 6. It should be noted that the parameters M and N are more local than [, since they refer to only individual elements of the specimen (cellular automata), and [ determines the number of these elements and hence characterizes the specimen as a whole.
Fig. 4. Variation of the bond parameters for two of six neighbors of one automaton
We modeled four groups of specimens for which [ was 1, 5, 10 and 15 %, and N = 2. The quantity M was varied from 15 to +50 %. A group comprised three subgroups of five specimens and with the same range of V each. Thus, we considered three combinations of M: 10 and 10 %, 10 and 10 %, 15 and 50 % for the first and second stochastically chosen bonds of automata. For convenience, these subgroups are further referred to as the first, second and third subgroup. The maximum and minimum values of M in the above ranges were chosen such that in the first subgroup, the total strength of the specimen for all bonds was the least, and in the third group, it was the highest; this corresponded to three different materials with fixed [. For correct examination of the relationship between the degree of nonuniformity [ of the strength properties and their macroscale response, it is necessary to compare the behavior of specimens of one material. In terms of our model, this corresponds to comparison of the mechanical behavior of a series of specimens with the same values of M, but different values of [. 4. Model verification Verification is an essential step in construction of any model of mechanical behavior of actual materials. In our study, the model is considered adequate to the examined porous ceramics if the results of model calculations satisfy the following criteria: (i) the loading diagram of the model specimens is linear, which is characteristic of porous brittle materials; (ii) the fracture patterns of equidimensional macroscopic specimens with explicit and implicit consideration of porous structure agree qualitatively; (iii) the strength characteristics of the specimens fall on a certain interval found from approximate estimates. Loading diagrams typical of all model specimens under uniaxial compression and shear are shown in Fig. 5. The diagrams reveal several characteristic portions. The first portion is a linear portion corresponding to elastic deformation of the specimen. Next, there is a short ascending portion with stress drops (generation and development of individual damages in the specimen bulk) and a descending portion (development of a macrocrack or system of macrocracks, and generation of individual multiple damages). In the shear diagrams (Fig. 5(b)), the above portions are followed by an ascending portion with stress drops and next by one more descending portion. Because of constrained deformation, it is to these latter portions that the development of the macrocrack system in the specimen corresponds, and the first descending portion corresponds to the one-time generation of damages throughout the specimen and their development without the formation of the macrocrack system. Comparison of the above diagrams with appropriate diagrams of brittle porous solids in shear and uniaxial com-
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b
Fig. 5. Loading diagrams of the model specimens with a side h = 6 mm in uniaxial compression (a) and in shear (b); [ = 15 %
pression [2, 8, 11] shows a good qualitative agreement. Thus, the first macroscopic criterion of model adequacy the general form of the loading diagram is fulfilled. Direct model verification for the second criterion is now impossible. This is associated with time- and memory-consuming computations (a large number of computational elements and a small time step) required for explicit consideration of the porous structure in the model. Therefore in comparison, we used the simulation results with explicit consideration of the porous structure obtained for brittle porous macroscopic specimens of smaller size in [58]. Figure 6 shows fracture patterns (inter-element bonds) typical of the model specimens at the instant macrocracks appear in the specimens. Under uniaxial compression (Fig. 6(a, b)), the specimens are fractured due to the deve-
lopment of an asymmetric system of oblique cracks in the conjugate directions; and under shear, of oblique and horizontal cracks (Fig. 6(c, d )). In both cases, individual multiple damages are generated near the cracks in the specimens. The character of the formed damages and macrocracks in the model material with an implicitly considered porosity of 15 % and nonuniform spatial strength distribution corresponds to that found in both actual [2, 3] and model [8] brittle porous media of low porosity (510 %) specified explicitly. It should be noted that a complete agreement between the fracture patterns of the examined model specimens and specimens with explicit consideration of the porous structure is impossible to obtain at all. The differences in the internal structure of these specimens determine to a
a
b
c
d
Fig. 6. Fracture patterns (macrocracking) of the specimens with the same combinations of M values (for specimens (a, b) and (c, d )) in uniaxial compression (a, b) and shear (c, d). [ = 15 (a, b), 5 % (c, d )
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great extent the distinctions in the processes evolved in them under loading. With explicit consideration of pores, the model contains a free volume and hence local distribution gradients of mass, elastic and strength properties. Moreover, mass transfer occurs and a pore works as both a concentrator and a mechanism of stress relaxation conductive to crack retardation and arrest. With implicit consideration of pores and variation in the strength parameters of automata interaction for individual elements, the specimens are a material with uniform distribution of mass and elastic properties in which mass transfer is absent and the varied bonds between automata work only as stress concentrators. In view of the foregoing, the qualitative agreement between the fracture of the model macrospecimens and that of brittle porous media suggests that the second criterion of model adequacy is fulfilled. In model verification for the third criterion, the range of the strength V c for the groups of specimens with the same values of [, M and N was determined from the following. The upper limit of the Vc max range corresponded to the strength V0 of the macrospecimen with intact (unvaried) bonds between automata and was 500 and 166 MPa for uniaxial compression and shear, respectively. With [ % of automata deprived of inter-element bond and with consideration of possible variation in the strength of the representative volume by T %, the lower limit of the Vc min range is determined by the expression:
Vc min
Vc max (1 [ 100)(1 T 100).
Because the bonds between automata are not removed in the model and only their strength is decreased by M %, the specimen strength Vc is bound to vary in the range Vc min d Vc d Vc max . The sought-for Vc min and Vc max and the range of the specimen strength V c obtained in modeling at [ = 5 and 15 % for shear and uniaxial compression are presented in Fig. 7. It can be seen that for all specimens, V c falls within the above range and hence the last criterion of adequacy of the constructed model is fulfilled. 5. Features of the developed approach Using the verified model, we studied the developed approach for feasibility of implicit consideration of different degrees of stochastic nonuniformity of the strength properties [ in porous media. The results of calculations show that with implicit consideration of the spatial distribution nonuniformity of the strength properties in the model, the effective elastic properties Eeff , Geff are near invariant with [. The effective strength properties Vc , Wc do vary with [, but the strength range for the groups of specimens with the same combinations of M values and different [ are mainly overlapped (Fig. 7). Thus, among the variety of specimens with the same
Fig. 7. Ranges of the strength of the model specimens with different values of [ and M, and their upper and lower analytical estimates under various mechanical loads
M and different [, we can always find specimens with equal or very nearly equal values of Vc and Wc . The effect of the parameter [ on the character of fracture of the examined specimens was studied by analyzing their corresponding inter-element bonds at the instant a system of macrocracks is formed in them (Fig. 6). For more accurate analysis of these interactions, specimens with low (5 %) and high (15 %) values of [ were considered. It is found that as [ increases, unmerged local damages and edge-to-edge macrocracks in the specimen increase in number and their spatial distribution becomes more uniform. In this case, the number of cracks that propagate deep into the specimen and fail to develop into main crack decreases. As the parameter [ is decreased, the reverse situation is observed. Moreover, with small values of [, the shear of the specimens gives rise to horizontal macrocracks inside them, whereas with high values of [, horizontal macrocracks are formed mainly at the surfaces of load application (upper surface) and fixation (lower surface). Thus, the variation in one of the model parameters [ responsible for the degree of the spatial distribution nonuniformity of the strength characteristics changes the fracture patterns of the model specimens while the strength and the elastic modulus remain near constant. It should be noted that the specimen strengths Vc and Wc depend in a complicated manner on many factors, in particular, on the distribution of the varied bonds between automata over the specimen, degree of their variation M, their number N for one automaton, and on the orientation of these bonds about the loading direction. For example, the specimens with the same values of M, N and [, but with different spatial distributions of the varied bonds of automata feature different strength properties Vc and Wc . Thus, the interplay between these parameters and the behavior of the system is a rather intricate and requires additional study.
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However, analysis of the influence of even one of these parameters, [, on the mechanical behavior of the system gives interesting results and indicates that the developed MCA-based approach holds promise for construction of ceramics models with implicit consideration of the porous structure. 6. Conclusion Thus, the obtained results allows the conclusion the multilevel approach developed based on the method of movable cellular automata and appropriate hierarchical model adequately describe deformation and fracture of porous media under mechanical loading. It is shown that the method used in the approach to consider the spatial distribution uniformity of the strength properties hold much promise for multilevel description of porous mediums with a hierarchical pore space structure. The work was supported by RFBR grants Nos.07-0812179-ofi, 08-08-12055-ofi. The authors are thankful to E.V. Shilko, S.N. Kulkov and S.P. Buyakova for the attention paid to the work and useful comments. References [1] Global Roadmap for Ceramics: Proceedings of 2nd International Congress on Ceramics (ICC2), Ed. by A. Belosi and G.N. Babini, Institute
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of Science and Technology for Ceramics, National Research Council, Verona, 2008. [2] S.P. Buyakova, Properties, Structure, Phase Composition and Mechanisms of the Formation of Porous ZrO 2-Based Nanosystems: Doct. Degree Book (Engng.), ISPMS SB RAS, Tomsk, 2008 (in Russian). [3] S.N. Kulkov, S.P. Buyakova and V.I. Maslovskii, Structure, Phase composition and mechanical properties of ZrO2-based ceramics, Vestnik TGU, No. 13 (2003) 34 (in Russian). [4] S.G. Psakhie, G.P. Ostermeyer, A.I. Dmitriev, E.V. Shilko, A.Yu. Smolin and S.Yu. Korostelev, Method of movable cellular automata as a new trend of discrete computational mechanics. I. Theoretical description, Phys. Mesomech., 3, No. 2 (2000) 5. [5] Ig.S. Konovalenko, A.Yu. Smolin and S.G. Psakhie, Deformation and fracture features in brittle porous media with various pore morphology, Izv. Vuzov, Fizika, 48, No. 6 (2005) 25 (in Russian). [6] A.Yu. Smolin, Ig.S. Konovalenko, S.N. Kulkov and S.G. Psakhie, Quasi-viscous fracture of brittle media with stochastic pore distribution, Techn. Phys. Lett., 32, No. 9 (2006) 738. [7] A.Yu. Smolin, Ig.S. Konovalenko, S.N. Kulkov and S.G. Psakhie, Simulation of fracture in brittle porous media with various internal structure, Izv. Vuzov, Fizika, 49, No. 3, Supplement (2006) 70 (in Russian). [8] Ig.S. Konovalenko, Theoretical Study of Deromation and Fracture of Porous Materials for Medicine and Biomechanical Structures: Cand. Degree Book (Phys. & Math.), ISPMS SB RAS, Tomsk, 2007 (in Russian). [9] L.M. Kachanov, The Foundations of Fracture Mechanics, Nauka, Moscow, 1974 (in Russian). [10] L.I. Sedov, A Course in Continuum Mechanics, Groningen, WoltersNoordhoff, 1971. [11] G.A. Gogotsi, The problem of the classification of low-deformation materials based on the features of their behavior under load, Strength of Materials, 9, No. 1 (1977) 77.