- Email: [email protected]
An effective model for the elastic behavior of two-phase materials based on structural recognition
Solid State Communications 298 (2019) 113639
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
An effective model for the elastic behavior of two-phase materials based on structural recognition
T
Xin-Wei Tanga, Huai-Qiu Zhenga, Yi-Chao Gaob,∗, Yuan-De Zhouc,∗∗ a
School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, People's Republic of China College of Civil Engineering, Huaqiao University, Xiamen 361021, People's Republic of China c Department of Hydraulic Engineering, Tsinghua University, Beijing 100086, People's Republic of China b
A R T I C LE I N FO
A B S T R A C T
Communicated by L. Brey
The elastic behavior of materials is dominated by the microstructure, which will evolve with the temperature in the fabricating process. Recently, we have proposed an efficient and simple algorithm for generating spatiallycorrelated random fields, reflecting more realistic microstructures. Using a mesoscale Finite Element Model, we have studied the elastic behavior of materials with the pores of various concentrations and spatial correlations. Considering the distributions of pore sizes, we are able to predict the modulus of elasticity with the relative deviation of 5%–10%. Based on structural recognition and Monte-Carlo tree search, we have determined the main characteristic of microstructure which is dominant to the elastic properties, where the relative deviation of modulus prediction can be further reduced to be around 3%–6%. Our finding will provide a better understanding and prediction of mechanical behavior based on the microstructures of materials.
Keywords: A. Composites D. Elastic properties E. Simulations
1. Introduction The properties of material often depend sensitively on the microstructures, e. g. the component concentrations and atomic distributions, which can be modulated by the cooling rate and the depth of quench [1–3]. For example, the distribution of voids in three-dimensional nanocrystalline palladium can influence the mechanical behavior during elastic and plastic deformations [4], since the annealing shrinkage of the voids leads to an increase of the Young's modulus during plastic deformation. For the mesoscale material with given microstructures, the elastic properties of two-phase materials can be obtained through solving the relevant governing equations based on the popular finiteelement technique [5,6]. To reflect the realistic microstructures of the materials, the spatial correlation is introduced by various methods, such as the Local Average Subdivision Method [7], the simulated annealing and genetic algorithm [8]. In our previous work [9], we have proposed an iterative algorithm which is simple and practical to generate the spatially correlated random field, describing the microstructures with the spatial correlated length and the anisotropy parameter. In microscale, the structural stabilities and electronic properties of materials can be obtained when the atomic distributions are determined. According to the lattice theories [10,11], the total energies of
∗
given system can be expressed by the summation of component clusters. Similarly, the optical and thermodynamic properties of a number of semiconductor alloys can be represented by the Special Quasi-random Structures [12,13], based on the pair-correlation functions. However, there is a lack of studies whether the elastic properties can be estimated using the component clusters or the multisite correlation functions. In addition, we should reduce the number of model parameters [14] and find the main characteristic of microstructure which dominates the corresponding properties. For the elastic behavior of two-phase materials, the concentration of components is the most common and simplest descriptor, which often fails to give a good estimation. The elastic behavior of materials might also depend on the spatial correlations, the domain sizes and shapes. Thus, it is still a challenge to search the proper descriptor for typical structures, which is helpful for understanding the diversity of behaviors attributed to the microstructures. To predict the effective mechanical properties of composites, various theoretical models have been proposed [15], such as the Voigt and Reuss models for the upper and lower bounds, Hashin Shtrikman model with a narrower and improved bounds, Mori-Tanaka model based on dilute dispersion approximation, Self-consistent model and Differential scheme based models. For complex porous rocks, there are micro-inhomogeneous units with randomly distributed inclusions and aligned oriented inclusions, where the rock's dispersive behaviors due to the
Corresponding author. Corresponding author. E-mail addresses: [email protected] (Y.-C. Gao), [email protected] (Y.-D. Zhou).
∗∗
https://doi.org/10.1016/j.ssc.2019.05.010 Received 11 March 2019; Received in revised form 7 May 2019; Accepted 17 May 2019 Available online 23 May 2019 0038-1098/ © 2019 Elsevier Ltd. All rights reserved.
Solid State Communications 298 (2019) 113639
X.-W. Tang, et al.
visco-elastic characteristics of the pore fluids can be simulated by the dynamic stress-strain method [16]. For the fluid-filled porous materials [17], a multi-scale framework is developed by utilizing a two-dimensional micro-mechanical model with porosity at two scales, focusing on the effects of fluid diffusion and the geometric arrangement of pores on the evolution of effective properties in fluid-filled porous materials. To establish comprehensive structure-property linkages, a machine learning technique of Gaussian process regression is used to estimate the elastic properties of given microstructures, based on a small number of full-field simulation with reliable data for training [18]. In this paper, we have firstly generated the materials with various pores and obtained the elastic properties with a mesoscale Finite Element Model. To fit the elastic properties, the microstructures are represented as the distributions of pore sizes, which are further decomposed by a series of substructures based on structural recognition. To optimize the fitting, we have determined the key substructures which are dominant to the elastic properties, varying with the length of spatial correlation. Our finding provides an effective avenue to describe the elastic properties with the crucial characteristic of microstructures. 2. Methods The spatially-correlated random fields are firstly generated from that without spatial correlation, by the weighted average according to the spatially-correlated length and anisotropy parameter [9]. For simplicity, the microstructures possess the correlation functions of exponential decaying, since Ising model has demonstrated the spatial correlation in phase transitions is temperature-dependent following the exponential decay. We use 1 and − 1 to represent the components of a two-phase material, where the spatial correlation is defined as:
S (d ) = 〈ai aj 〉 =
1 n
n
d
∑ ai aj = exp ⎛− L ⎞, i=1
⎝
⎠
(1)
where d is the distance between ai and aj , L is the spatial correlation length, n is the number of elements in the material. Herein, the mesoscale finite elements in the context of continuum mechanics is used, and the mechanical behavior of cement mortar is simulated as an elastic model with E = 28.3 GPa and ν = 0.2. In our simulations, the samples with the same pore concentration have different elastic properties. Thus, we should consider the sizes and shapes of pores to model the mechanic properties, whose distributions depend on the spatial correlation of microstructures. In order to demonstrate the fitting results, we adopt the R2 statistic as the objective function. The possible substructures in microstructures are distinguished by the structural recognition, based on the Minimum Vertextype Sequence Indexing [19], in which only one integer is needed for each particle to label its vertex type and the sequence is orientation independent. As the pore size increases, the number of substructures grows exponentially. To reduce the number of model parameters and find the main characteristic of microstructure, the key substructures are determined by maximizing the R2 statistic (E) in the fitting, based on the MonteCarlo tree search which balances the search of depth and width [20]. The main idea of Monte Carlo tree search to screen the key substructures is select → expand → select. We firstly select a substructure to fit the elastic properties and the R2 statistic might be very small. Then, we can add another substructure for fitting to increase the R2 statistic. The possible combinations of substructures will increase exponentially and we have to cut the branches according to the R2 statistic (E). The probability of choosing a structure is determined by its E, where the selection probability (P) of searching for larger E can be written as: P = exp(βE ) , where β is a positive constant which herein only provides a strategy to acquire large E structures and does not pledge detailed balance to reach the equilibrium state. To improve the searching efficiency, we can adjust this constant β in the simulation to
Fig. 1. (a) Typical microstructures with various spatially-correlated lengths. (b) The maximum and minimum of elastic modulus for L = 30.
balance the depth and breadth of the search. 3. Results and discussions As shown in Fig. 1a, we consider the two-dimensional microstructures of material generated by our approach on a 120 × 120 grid system, where the pore concentration x ranges from 0 to 0.1 with various lengths for the spatial correlation function (L = 0.1, 0.5, 5 and 30). The size of pore becomes larger as the spatial correlation length increases, with more inhomogeneous distributions. The elastic modulus is calculated with a mesoscale Finite Element Model, where 1000 samples are obtained for the given pore concentration and spatial correlation length. The averaged mechanical properties and the relative deviation as a function of the pore concentration. The elastic modulus will decrease as the concentration increases, where the relative deviation will increase with the spatial correlation length, due to the more inhomogeneous distributions of pores with larger sizes. As shown in Fig. 1b, the max modulus is around two times larger than that of minimum for the same pore concentration, indicating that the detailed microstructures will play an important role in the elastic properties. Thus, we should consider the characteristic of microstructures to 2
Solid State Communications 298 (2019) 113639
X.-W. Tang, et al.
Fig. 2. (a) The determination of pore size distribution of given structures. (b) The distribution of pore sizes for structures with L = 0.5 and 30.
establish the structure-property linkage and only the pore concentration is not enough to describe the elastic properties. According to the microstructures in Fig. 1, the pore size will vary with the spatially-correlated lengths. For given microstructures, we firstly use the distributions of pore sizes to describe the elastic properties. In our model, the units with zero elastic modulus are pores and the connected units are considered as an individual pore. Thus, the size distributions of pore size are obtained by the structural recognition. Fig. 2a shows the size distribution of pores for L = 30 as an example and the pores are sorted by the size and divided into four groups, where the size of pores increases gradually, marked by light blue, green, yellow and red respectively. For each group, we will obtain the average sizes of pores and each sample can be expressed by a vector with four parameters, which can be used to fit the elastic properties. Practically, we divide the pores in every sample into 10 groups and express the microstructures by the size distribution of pores (a vector with 10 parameters) for the fitting of elastic properties. For L = 0.5, most pores contain the units less than 10 while the pores contain 30–80 units for L = 30. As a result, the R2 statistic are found to be 0.906, 0.864, 0.681, 0.714, for L = 0.1, 0.5, 5, 30 respectively. The relative of variance between the predicted value and exact ones are 5.5%, 5.7%, 11.0%, 10.6%, respectively. Compared to the pore concentration, the size distributions of pores exhibit more information of the microstructures and thus fit the elastic properties to better extent. In order to improve the fitting, the detailed geometry of pores is considered as follows, where the possible substructures in microstructures are distinguished by the structural recognition, based on the Minimum Vertex-type Sequence Indexing [19]. As shown in Fig. 3a, there are 2 structures for the pore with 3 units, 5 structures for 4 units, and 12 structures for 5 units. That the number of substructures will grow exponentially with the increasing size. Herein, we have considered the possible substructures with sizes less than 9 units with over
Fig. 3. (a) The structural recognition of substructures with various sizes. (b) The fitting with the size distribution and structural recognition with L = 30. (c) The comparison for the samples with L = 0.1, 0.5, 5.
3
Solid State Communications 298 (2019) 113639
X.-W. Tang, et al.
samples by various threshold values of relative deviation. For example, there is around 40% samples with the relative deviation less than 3% when fitting with substructures, while there is only 20% when fitting with the size distribution. As shown in Fig. 3c, the fitting is also improved for the cases with L = 0.1, 0.5, 5. Thus, we have shown that the detailed configuration by structural recognition will effectively improve the fitting, compared to the pore concentration or the size distributions of pores. In general, the fitting will be improved as the number of parameters increases. In the following, we have taken the R2 statistic as the objective function and employed the Monte-Carlo tree search to screen the key substructures which play important roles in the mechanical properties. As shown in the inset of Fig. 4a, there is a crucial point of the parameter number where the fitting will be improved dramatically. The R2 statistic and relative deviation did not vary linearly with the number of parameters (shown in Fig. 4a), therefore, we can find out the key substructures according to the R2 statistic. Fig. 4b shows the fitting with 9 key parameters for L = 30 is as good as the one with 532 parameters, which is much better than the fitting with 10 parameters of size distribution. The key substructures are shown in the inset. Similarly, we have obtained the key substructures for L = 0.1, 0.5, 5, where the fitting is also significantly improved. Thus, the improved fitting by the structural recognition is attributed to the selection of key substructures which is dominant to the elastic properties, instead of increasing the number of parameters. 4. Conclusion In summary, we have modeled the elastic behavior of two-phase material with the spatial correlation lengths. Only the pore concentration is not enough to describe the elastic properties, since the max modulus is around two times larger than that of minimum for the same pore concentration. Firstly, we divide the pores in every sample into 10 groups and obtain the average sizes of pores for each group as the distributions of pore sizes, describing the microstructure of each sample by a vector with 10 parameters. Fitting elastic modulus with the distributions of pore sizes, the relative deviation is found to be within 5%–10% for various spatial correlation lengths. Secondly, we have distinguished the substructures in microstructures by the structural recognition, where the relative deviation of modulus prediction can be further reduced to be around 3%–6%. Notably, we have screened the key substructures by the Monte-Carlo tree search, where the fitting is improved while the number of parameters is reduced. Thus, the detailed microstructures should be considered to establish the structureproperty linkage. Our method will provide a good description of elastic behavior based on the microstructures, which can be further extended to the multi-phase material. Acknowledgments
Fig. 4. The number of parameters and key substructures for fitting the elastic modulus: (a) the R2 statistic and the distribution of variance; (b) the fitting with key substructures for L = 30; (c) the key substructures for L = 0.1, 0.5, 5.
The authors acknowledge the support by the National Natural Science Foundation of China under Grant No. 51109083, by the open fund of the State Key Laboratory of Hydro Science and Engineering of China, Tsinghua University (Grant No.sklhse-2017-c-03) and partial support by Fundamental Research Funds for the Central Universities under Grant No.2017ZD019.
500 configurations. To describe the properties with the microstructures, each sample can be expressed by various spatial correlations [14]:
E ({ai}) = J0 +
∑ Ji ai + ∑ sites
pairs
Jij ai aj +
∑ triplets
Jijk ai aj ak + …, (2)
Appendix A. Supplementary data
where {ai} denotes the microstructures of the given sample, the parameters {J} represent the contribution of each substructure to E ({ai}) . Considering over 500 substructures with sizes less than 9 units, we improve the R2 statistic to be 0.9684, 0.9679, 0.9221, 0.9123 for the cases with L = 0.1, 0.5, 5, 30 respectively. As shown in the inset of Fig. 3b, the relative deviation shows a shallow distribution for the fitting with over 500 substructures (marked as Shape) compared to the one with the size distribution. In Fig. 3b, we count the number of
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ssc.2019.05.010. References [1] M. Rao, S. Sengupta, Phys. Rev. Lett. 91 (2003) 045502. [2] V.R. D'Costa, Y.-Y. Fang, J. Tolle, J. Kouvetakis, J. Mene´ndez1, Phys. Rev. Lett. 102
4
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[12] [13] [14] [15] [16] [17]
A. Zunger, S.H. Wei, L.G. Ferreira, J.E. Bernard, Phys. Rev. Lett. 65 (1990) 353. S.H. Wei, L.G. Ferreira, J.E. Bernard, A. Zunger, Phys. Rev. B 42 (1990) 9622. G.L.W. Hart, V. Blum, M.J. Walorski, A. Zunger, Nat. Mater. 4 (2005) 391. B. Raju, S.R. Hiremath, D.R. Mahapatra, Compos. Struct. 204 (2018) 607–619. W. Zhu, L. Zhao, R. Shan, Geo. Phys. 82 (2017) 163–174. M. Liu, J. Wu, Y. Gan, D. AH Hanaor, C.Q. Chen, Int. J. Solids Struct. 162 (2019) 36–44. [18] J. Jung, J.I. Yoon, H.K. Park, J.Y. Kim, H.S. Kim, Comput. Mater. Sci. 156 (2019) 17–25. [19] L.G. Liao, Y.J. Zhao, Z.X. Cao, X.B. Yang, Sci. Rep. 7 (2017) 392. [20] Z.P. Cao, Y.J. Zhao, J.H. Liao, X.B. Yang, RSC Adv. 7 (2017) 37881.
(2009) 107403. [3] J. Lin, L. Wang, Z. Hu, X. Li, H. Yan, Solid State Commun. 251 (2017) 98–103. [4] D.V. Bachurin, Solid State Commun. 275 (2018) 43–47. [5] C.A. Tang, Y.B. Zhang, Z.Z. Liang, T. Xu, L.G. Tham, P.-A. Lindqvist, S.Q. Kou, H.Y. Liu, Phys. Rev. E 73 (2006) 056120. [6] G.A. Fenton, E.H. Vanmarcke, ASCE J. Engrg. Mech. 116 (1990) 1733. [7] C.L.Y. Yeong, S. Torquato, Phys. Rev. E 57 (1998) 495. [8] H. Lee, M. Brandyberry, A. Tudor, K. Matouš, Phys. Rev. E 80 (2009) 061301. [9] X.W. Tang, X.B. Yang, Y.D. Zhou, Solid State Commun. 182 (2014) 30–33. [10] U. Brandt, Z. Phys. B 53 (1983) 283. [11] J. Stolze, Phys. Rev. Lett. 64 (1990) 970.
5
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
An effective model for the elastic behavior of two-phase materials based on structural recognition
T
Xin-Wei Tanga, Huai-Qiu Zhenga, Yi-Chao Gaob,∗, Yuan-De Zhouc,∗∗ a
School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, People's Republic of China College of Civil Engineering, Huaqiao University, Xiamen 361021, People's Republic of China c Department of Hydraulic Engineering, Tsinghua University, Beijing 100086, People's Republic of China b
A R T I C LE I N FO
A B S T R A C T
Communicated by L. Brey
The elastic behavior of materials is dominated by the microstructure, which will evolve with the temperature in the fabricating process. Recently, we have proposed an efficient and simple algorithm for generating spatiallycorrelated random fields, reflecting more realistic microstructures. Using a mesoscale Finite Element Model, we have studied the elastic behavior of materials with the pores of various concentrations and spatial correlations. Considering the distributions of pore sizes, we are able to predict the modulus of elasticity with the relative deviation of 5%–10%. Based on structural recognition and Monte-Carlo tree search, we have determined the main characteristic of microstructure which is dominant to the elastic properties, where the relative deviation of modulus prediction can be further reduced to be around 3%–6%. Our finding will provide a better understanding and prediction of mechanical behavior based on the microstructures of materials.
Keywords: A. Composites D. Elastic properties E. Simulations
1. Introduction The properties of material often depend sensitively on the microstructures, e. g. the component concentrations and atomic distributions, which can be modulated by the cooling rate and the depth of quench [1–3]. For example, the distribution of voids in three-dimensional nanocrystalline palladium can influence the mechanical behavior during elastic and plastic deformations [4], since the annealing shrinkage of the voids leads to an increase of the Young's modulus during plastic deformation. For the mesoscale material with given microstructures, the elastic properties of two-phase materials can be obtained through solving the relevant governing equations based on the popular finiteelement technique [5,6]. To reflect the realistic microstructures of the materials, the spatial correlation is introduced by various methods, such as the Local Average Subdivision Method [7], the simulated annealing and genetic algorithm [8]. In our previous work [9], we have proposed an iterative algorithm which is simple and practical to generate the spatially correlated random field, describing the microstructures with the spatial correlated length and the anisotropy parameter. In microscale, the structural stabilities and electronic properties of materials can be obtained when the atomic distributions are determined. According to the lattice theories [10,11], the total energies of
∗
given system can be expressed by the summation of component clusters. Similarly, the optical and thermodynamic properties of a number of semiconductor alloys can be represented by the Special Quasi-random Structures [12,13], based on the pair-correlation functions. However, there is a lack of studies whether the elastic properties can be estimated using the component clusters or the multisite correlation functions. In addition, we should reduce the number of model parameters [14] and find the main characteristic of microstructure which dominates the corresponding properties. For the elastic behavior of two-phase materials, the concentration of components is the most common and simplest descriptor, which often fails to give a good estimation. The elastic behavior of materials might also depend on the spatial correlations, the domain sizes and shapes. Thus, it is still a challenge to search the proper descriptor for typical structures, which is helpful for understanding the diversity of behaviors attributed to the microstructures. To predict the effective mechanical properties of composites, various theoretical models have been proposed [15], such as the Voigt and Reuss models for the upper and lower bounds, Hashin Shtrikman model with a narrower and improved bounds, Mori-Tanaka model based on dilute dispersion approximation, Self-consistent model and Differential scheme based models. For complex porous rocks, there are micro-inhomogeneous units with randomly distributed inclusions and aligned oriented inclusions, where the rock's dispersive behaviors due to the
Corresponding author. Corresponding author. E-mail addresses: [email protected] (Y.-C. Gao), [email protected] (Y.-D. Zhou).
∗∗
https://doi.org/10.1016/j.ssc.2019.05.010 Received 11 March 2019; Received in revised form 7 May 2019; Accepted 17 May 2019 Available online 23 May 2019 0038-1098/ © 2019 Elsevier Ltd. All rights reserved.
Solid State Communications 298 (2019) 113639
X.-W. Tang, et al.
visco-elastic characteristics of the pore fluids can be simulated by the dynamic stress-strain method [16]. For the fluid-filled porous materials [17], a multi-scale framework is developed by utilizing a two-dimensional micro-mechanical model with porosity at two scales, focusing on the effects of fluid diffusion and the geometric arrangement of pores on the evolution of effective properties in fluid-filled porous materials. To establish comprehensive structure-property linkages, a machine learning technique of Gaussian process regression is used to estimate the elastic properties of given microstructures, based on a small number of full-field simulation with reliable data for training [18]. In this paper, we have firstly generated the materials with various pores and obtained the elastic properties with a mesoscale Finite Element Model. To fit the elastic properties, the microstructures are represented as the distributions of pore sizes, which are further decomposed by a series of substructures based on structural recognition. To optimize the fitting, we have determined the key substructures which are dominant to the elastic properties, varying with the length of spatial correlation. Our finding provides an effective avenue to describe the elastic properties with the crucial characteristic of microstructures. 2. Methods The spatially-correlated random fields are firstly generated from that without spatial correlation, by the weighted average according to the spatially-correlated length and anisotropy parameter [9]. For simplicity, the microstructures possess the correlation functions of exponential decaying, since Ising model has demonstrated the spatial correlation in phase transitions is temperature-dependent following the exponential decay. We use 1 and − 1 to represent the components of a two-phase material, where the spatial correlation is defined as:
S (d ) = 〈ai aj 〉 =
1 n
n
d
∑ ai aj = exp ⎛− L ⎞, i=1
⎝
⎠
(1)
where d is the distance between ai and aj , L is the spatial correlation length, n is the number of elements in the material. Herein, the mesoscale finite elements in the context of continuum mechanics is used, and the mechanical behavior of cement mortar is simulated as an elastic model with E = 28.3 GPa and ν = 0.2. In our simulations, the samples with the same pore concentration have different elastic properties. Thus, we should consider the sizes and shapes of pores to model the mechanic properties, whose distributions depend on the spatial correlation of microstructures. In order to demonstrate the fitting results, we adopt the R2 statistic as the objective function. The possible substructures in microstructures are distinguished by the structural recognition, based on the Minimum Vertextype Sequence Indexing [19], in which only one integer is needed for each particle to label its vertex type and the sequence is orientation independent. As the pore size increases, the number of substructures grows exponentially. To reduce the number of model parameters and find the main characteristic of microstructure, the key substructures are determined by maximizing the R2 statistic (E) in the fitting, based on the MonteCarlo tree search which balances the search of depth and width [20]. The main idea of Monte Carlo tree search to screen the key substructures is select → expand → select. We firstly select a substructure to fit the elastic properties and the R2 statistic might be very small. Then, we can add another substructure for fitting to increase the R2 statistic. The possible combinations of substructures will increase exponentially and we have to cut the branches according to the R2 statistic (E). The probability of choosing a structure is determined by its E, where the selection probability (P) of searching for larger E can be written as: P = exp(βE ) , where β is a positive constant which herein only provides a strategy to acquire large E structures and does not pledge detailed balance to reach the equilibrium state. To improve the searching efficiency, we can adjust this constant β in the simulation to
Fig. 1. (a) Typical microstructures with various spatially-correlated lengths. (b) The maximum and minimum of elastic modulus for L = 30.
balance the depth and breadth of the search. 3. Results and discussions As shown in Fig. 1a, we consider the two-dimensional microstructures of material generated by our approach on a 120 × 120 grid system, where the pore concentration x ranges from 0 to 0.1 with various lengths for the spatial correlation function (L = 0.1, 0.5, 5 and 30). The size of pore becomes larger as the spatial correlation length increases, with more inhomogeneous distributions. The elastic modulus is calculated with a mesoscale Finite Element Model, where 1000 samples are obtained for the given pore concentration and spatial correlation length. The averaged mechanical properties and the relative deviation as a function of the pore concentration. The elastic modulus will decrease as the concentration increases, where the relative deviation will increase with the spatial correlation length, due to the more inhomogeneous distributions of pores with larger sizes. As shown in Fig. 1b, the max modulus is around two times larger than that of minimum for the same pore concentration, indicating that the detailed microstructures will play an important role in the elastic properties. Thus, we should consider the characteristic of microstructures to 2
Solid State Communications 298 (2019) 113639
X.-W. Tang, et al.
Fig. 2. (a) The determination of pore size distribution of given structures. (b) The distribution of pore sizes for structures with L = 0.5 and 30.
establish the structure-property linkage and only the pore concentration is not enough to describe the elastic properties. According to the microstructures in Fig. 1, the pore size will vary with the spatially-correlated lengths. For given microstructures, we firstly use the distributions of pore sizes to describe the elastic properties. In our model, the units with zero elastic modulus are pores and the connected units are considered as an individual pore. Thus, the size distributions of pore size are obtained by the structural recognition. Fig. 2a shows the size distribution of pores for L = 30 as an example and the pores are sorted by the size and divided into four groups, where the size of pores increases gradually, marked by light blue, green, yellow and red respectively. For each group, we will obtain the average sizes of pores and each sample can be expressed by a vector with four parameters, which can be used to fit the elastic properties. Practically, we divide the pores in every sample into 10 groups and express the microstructures by the size distribution of pores (a vector with 10 parameters) for the fitting of elastic properties. For L = 0.5, most pores contain the units less than 10 while the pores contain 30–80 units for L = 30. As a result, the R2 statistic are found to be 0.906, 0.864, 0.681, 0.714, for L = 0.1, 0.5, 5, 30 respectively. The relative of variance between the predicted value and exact ones are 5.5%, 5.7%, 11.0%, 10.6%, respectively. Compared to the pore concentration, the size distributions of pores exhibit more information of the microstructures and thus fit the elastic properties to better extent. In order to improve the fitting, the detailed geometry of pores is considered as follows, where the possible substructures in microstructures are distinguished by the structural recognition, based on the Minimum Vertex-type Sequence Indexing [19]. As shown in Fig. 3a, there are 2 structures for the pore with 3 units, 5 structures for 4 units, and 12 structures for 5 units. That the number of substructures will grow exponentially with the increasing size. Herein, we have considered the possible substructures with sizes less than 9 units with over
Fig. 3. (a) The structural recognition of substructures with various sizes. (b) The fitting with the size distribution and structural recognition with L = 30. (c) The comparison for the samples with L = 0.1, 0.5, 5.
3
Solid State Communications 298 (2019) 113639
X.-W. Tang, et al.
samples by various threshold values of relative deviation. For example, there is around 40% samples with the relative deviation less than 3% when fitting with substructures, while there is only 20% when fitting with the size distribution. As shown in Fig. 3c, the fitting is also improved for the cases with L = 0.1, 0.5, 5. Thus, we have shown that the detailed configuration by structural recognition will effectively improve the fitting, compared to the pore concentration or the size distributions of pores. In general, the fitting will be improved as the number of parameters increases. In the following, we have taken the R2 statistic as the objective function and employed the Monte-Carlo tree search to screen the key substructures which play important roles in the mechanical properties. As shown in the inset of Fig. 4a, there is a crucial point of the parameter number where the fitting will be improved dramatically. The R2 statistic and relative deviation did not vary linearly with the number of parameters (shown in Fig. 4a), therefore, we can find out the key substructures according to the R2 statistic. Fig. 4b shows the fitting with 9 key parameters for L = 30 is as good as the one with 532 parameters, which is much better than the fitting with 10 parameters of size distribution. The key substructures are shown in the inset. Similarly, we have obtained the key substructures for L = 0.1, 0.5, 5, where the fitting is also significantly improved. Thus, the improved fitting by the structural recognition is attributed to the selection of key substructures which is dominant to the elastic properties, instead of increasing the number of parameters. 4. Conclusion In summary, we have modeled the elastic behavior of two-phase material with the spatial correlation lengths. Only the pore concentration is not enough to describe the elastic properties, since the max modulus is around two times larger than that of minimum for the same pore concentration. Firstly, we divide the pores in every sample into 10 groups and obtain the average sizes of pores for each group as the distributions of pore sizes, describing the microstructure of each sample by a vector with 10 parameters. Fitting elastic modulus with the distributions of pore sizes, the relative deviation is found to be within 5%–10% for various spatial correlation lengths. Secondly, we have distinguished the substructures in microstructures by the structural recognition, where the relative deviation of modulus prediction can be further reduced to be around 3%–6%. Notably, we have screened the key substructures by the Monte-Carlo tree search, where the fitting is improved while the number of parameters is reduced. Thus, the detailed microstructures should be considered to establish the structureproperty linkage. Our method will provide a good description of elastic behavior based on the microstructures, which can be further extended to the multi-phase material. Acknowledgments
Fig. 4. The number of parameters and key substructures for fitting the elastic modulus: (a) the R2 statistic and the distribution of variance; (b) the fitting with key substructures for L = 30; (c) the key substructures for L = 0.1, 0.5, 5.
The authors acknowledge the support by the National Natural Science Foundation of China under Grant No. 51109083, by the open fund of the State Key Laboratory of Hydro Science and Engineering of China, Tsinghua University (Grant No.sklhse-2017-c-03) and partial support by Fundamental Research Funds for the Central Universities under Grant No.2017ZD019.
500 configurations. To describe the properties with the microstructures, each sample can be expressed by various spatial correlations [14]:
E ({ai}) = J0 +
∑ Ji ai + ∑ sites
pairs
Jij ai aj +
∑ triplets
Jijk ai aj ak + …, (2)
Appendix A. Supplementary data
where {ai} denotes the microstructures of the given sample, the parameters {J} represent the contribution of each substructure to E ({ai}) . Considering over 500 substructures with sizes less than 9 units, we improve the R2 statistic to be 0.9684, 0.9679, 0.9221, 0.9123 for the cases with L = 0.1, 0.5, 5, 30 respectively. As shown in the inset of Fig. 3b, the relative deviation shows a shallow distribution for the fitting with over 500 substructures (marked as Shape) compared to the one with the size distribution. In Fig. 3b, we count the number of
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