Modelling and Simulation of Core Shell Precipitates under Tensial Loading Using the Cellular Automata Method

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1 Mechanics and Mechanical Engineering Vol. 21, No. 3 (2017) c Lodz University of Technology Modelling and Simulation of Core Shell Precipitates under Tensial Loading Using the Cellular Automata Method Anna Staszczyk M Debyser Jacek Sawicki Institute of Materials Science and Engineering Lodz University of Technology Stefanowskiego 1/15, Lódź, Poland anna.staszczyk@p.lodz.pl Received (20 June 2017) Revised (26 July 2017) Accepted (18 August 2017) The equivalent truss method has been known for years as a numerical model used in problems with structural optimization. It is often implemented in computational algorithms based on cellular automation (CA). This method (CA) is highly versatile and allows the modelling of phenomena which occur in multiple dimensional scales, including material engineering issues. This paper describes a numerical model used to simulate the stress caused by the external load in a multi-phase material. The authors propose applying the algorithm in modelling the behavior of alloy materials following a thermal treatment which leads to precipitate hardening. Keywords: Precipitation hardening, stress state, numerical modeling, core shell. 1. Introduction Heterogeneous materials account for a large portion of engineering materials. A material can have a heterogeneous structure (e.g. a polycrystal) or can be mechanically heterogeneous, composed of phases of various elastic properties. The presence of foreign phases can be accomplished by purposefully introducing particles of different properties e.g. by making a composite, or by inducing changes in the material structure effected by external factors, for example, in materials subjected to thermal treatment. Examples of such materials include construction alloys of aluminum after precipitation hardening [1][2]. An analysis based on solutions of interactions for various shapes, sizes and properties of heterogeneities can significantly improve the predictability of the effective properties of multi-phase materials. A mathematical model is sought that will allow the determination of stress fields around precipitates with real shapes on the microscopic scale.

2 444 Staszczyk, A., Debyser, M. and Sawicki, J. Despite a lot of attempts to find mathematical models, effective algorithms that could be used to calculate the states of stress in multi-phase microstructures by methods other than the Finite Element Method (FEM) are lacking. Although applied widely, FEM has a number of limitations, especially on smaller-dimension scales [3]. Other iterative methods, such as cellular automata (CA), the phase field method or the Monte Carlo method, are more often applied to these problems on the micro or nanoscale. Shen et al. [4] developed a mathematical model of a heterogeneous material with phases of various structures and elastic properties, which they subsequently implemented in the phase field method. The stress field around ellipsoidal gaps in various materials was determined by Chiang [5]. The CA method in combination with the FEM was applied by Pourian et al. [6] to model the microstructure of titan α alloys, taking into account the varied crystallographic orientation of grains. The state of stress was calculated for each cell and the simulation result was compared to the calculations made with the FEM. Montheillet and Gilormini [7], using the CA method, derived a general solution to calculate local stresses in two-phase materials in the flat state of strain. Bocher et al. [8] also applied the CA method to analyze the behavior of a two-phase titanium alloy during thermoplastic treatment. The cellular automata method was first described by mathematician John von Neumann as a simplified mathematical model of actual physical processes over time [9]. A cellular automaton is a discrete model which consists of a finite number of cells. The cells can pass into a finite number of states and time takes the form of a discrete computation step. A cell changes its state in the next passage based on its previous state and the states of the adjacent cells. This change of state is described by the rule of passage [10]. Cellular automata are an important and often-used tool to simulate local phenomena occurring in the microstructure of a material. The CA method has a number of advantages including a short calculation times of 2D lattices, uncomplicated discretization of the computational domain and the general nature of the results, which can be easily implemented in other models. The numerical model presented in this paper combines the mathematical model of an equivalent truss with the cellular automata computation method. The equivalent truss model has long been used to solve problems optimizing the topology of structures. Optimization involves seeking the maximum or the minimum function value, or the functional equation while meeting a certain number of restricting conditions. In this study, the elastic potential energy in a material is the sought minimum value. In this case, the equivalent truss model is based on the assumption that a solid body, which is a continuous medium, can be regarded as a lattice of a specific geometry. Cellular automata are an often applied method of implementation of the lattice model for numerical algorithms. Their application for this purpose has been described by Gurdal and Abdalla [11, 12], Kita and Toyoda [13], Gurdal and Tatting [14, 15] as well as by Abdalla and Zakhama [16]. However, so far, the lattice method has not been applied in cellular automata to determine the internal stress in multi-phase materials.

3 Modelling and Simulation of Core Shell Precipitates Numerical model The numerical algorithm developed in the study operates as shown in Fig. 1. The simulation process consists of several stages. First, a user enters input data, such as the geometry of the material phases and their mechanical properties. This data is used by the program to generate the domain of the cellular automaton as a 200x200 pixel bitmap showing a single heterogeneity (precipitation) in the material matrix. Examples of such geometries are shown in Fig. 2. Every pixel on the bitmap represents a single automata cell surrounded by eight adjacent cells. After the graphic representation is generated, the value of the Young module is assigned to every cell based on the cell belonging to a precipitate or to the matrix. The values are assigned based on the pixel color: the grey pixels belong to a precipitate and the black ones to the matrix. Figure 1 A block diagram of the numerical algorithm

4 446 Staszczyk, A., Debyser, M. and Sawicki, J. Figure 2 Examples of the geometry of the precipitates generated by the program The problem to be solved is determining the state of stress for a material which is stretched along its axis in a flat state of strain. The pre-defined strain was always 0.05%, which is the conventional limit of proportionality for elastic materials.

5 Modelling and Simulation of Core Shell Precipitates Mathematical model Each cell of a body is regarded as a node in a lattice connected with eight adjacent cells in the Moore neighborhood with rods of specific cross-section and material constants. The height and width of each cell is the same and equal to 1 in a nonstrained state. It is a two-dimensional system therefore the thickness of the cells is virtual and is assumed to be Figure 3 Representation of a solid as an equivalent truss In order to transform a solid into an equivalent truss, a condition must be met that the elastic potential energy at a point in the solid body should be equal to the elastic potential energy of the corresponding lattice node [15]: [ U = E 1 ν 2 (ε2 x + ε 2 y + 2νε x ε y ) + Gγ 2 xy U = [A 0 ε 2 h + A 0 ε 2 v + 2A d (ε 2 d1 + ε 2 d2)] El 2 ] l 2 t 2 where: E Young s modulus, ε x, ε y directional strain in a cell, ν Poisson s ratio, G shear modulus, γ xy shear strain in a cell, l cell height, t cell thickness, A 0 cross-sectional area of a vertical or horizontal member, A d - cross-sectional area of a diagonal member, ε h strain in a horizontal member, ε v strain in a vertical member, ε d1, ε d2 strain in diagonal members. Comparing the two equations enables one to calculate the cross-sections of the lattice rods with which the condition can be met. The cross-section areas of the horizontal and vertical rods are different than the cross-section of diagonal rods and they are, respectively: A 0 = 3 4 lt A d = 3 2 lt (3) 8 There is force acting on each rod, expressed by the equation: (1) (2) F = E A ε + F i (4) where: A cross-sectional area of a particular member, ε strain in that member, F i external force.

6 448 Staszczyk, A., Debyser, M. and Sawicki, J. Figure 4 Forces acting on a lattice node Force is associated with the load applied to the whole body during the stretching process. For a state of equilibrium to be reached, the minimum potential energy of elasticity, the sum of forces in both directions should approach zero. ΣF x = 0 ΣF y = 0 (5) ΣF = E 1 A d ε 1 + E 2 A 0 ε E 8 A 0 ε 8 (6) When this condition is met with the assumed accuracy (condition of convergence), the final values of strain are calculated. ε k = L k L 0 L 0 (7) where: L k deformed height of a cell, L 0 nundeformed length of a cell. The established strain difference between the current and the previous time step is the criterion of convergence. It was set at 10-8 for the simulations performed by the authors. The Gauss-Seidel iterative model was used in the algorithm due to its computational efficiency [15]. After convergence of the system is achieved, strains are converted to direction-oriented stress and displayed as a map. 4. Results Simulations were made for various shapes and sizes of precipitates formed in Al 2024 alloys after precipitation hardening. In individual cases where two-step thermal treatment - ageing - is applied, two-phase precipitates are formed which consist of a hard core and softer shell, so-called core-shell precipitates. Individual material phases differ by mechanical properties, including the Young module. The Young module for the matrix 73,000 MPa for the precipitate shell 200,000 MPa for the core 240,000 MPa.

7 Modelling and Simulation of Core Shell Precipitates A numerical analysis was conducted on precipitate models with a circular or elliptical shape. An analysis was conducted on the position of elliptical precipitation in the matrix relative to the horizontal direction for various inclination angles relative to the horizontal axis ranging from 0 to 90. The horizontal direction is perpendicular to the direction of stretching and the angle is defined between this direction and the major axis which determines the shape of the elliptical precipitate. The area of various phases of core precipitates was the same in all of the cases analyzed. The results of stress calculations along the stretch direction are shown in Fig. 5. Figure 5 Stress Maps along the stretching direction calculated for precipitates: a) circular, b) elliptical at an angle of 0 relative to the horizontal direction, c) elliptical at an angle of 15 relative to the horizontal direction, d) elliptical at an angle of 30 relative to the horizontal direction, e) elliptical at an angle of 45 relative to the horizontal direction, f) elliptical at an angle of 90 relative to the horizontal direction Additionally, the simulation results were used to compare the profiles of stress along the direction of stretching running through the geometric center of the system. Examples of their profiles and a comparison for all the shapes studied are shown in Fig. 6.

8 450 Staszczyk, A., Debyser, M. and Sawicki, J. Figure 6 Stress profiles along the stretching direction in the vertical axis running through the center of the precipitate: a) circular, b) elliptical at an angle of 0 relative to the horizontal direction, c) elliptical at an angle of 45 relative to the horizontal direction, d) elliptical at an angle of 90 relative to the horizontal direction, and pooled comparison of stress profiles for all the precipitates studied Figure 7 A comparison of the average stress values along the stretching direction inside the precipitate cores for all the studied

9 Modelling and Simulation of Core Shell Precipitates In order to determine the effect of the shape and position of elliptical precipitates relative to the stretching direction, the average values of directional stress inside the precipitate core(s) were compared. A comparison of the results is shown in Fig Conclusions A numerical algorithm based on CA has been proposed also based on the usergenerated geometry and elasticity coefficients of the precipitate phases and the matrix. It will be able to calculate the stress in the defined state of flat strains. An analysis was conducted on examples of precipitates with a circular and elliptic shape, placed within the material matrix at various angles relative to the direction, perpendicular to the direction of stretching. The cross-section area of the core and the shell was the same in all the precipitates analyzed in the simulations. The results provided grounds for the conclusion that the greater the inclination angle (relative to the horizontal axis for elliptic precipitates), the greater the stress inside the core. The largest value was obtained for an elliptic precipitate at an angle of 90 relative to the horizontal direction i.e. positioned parallel to the direction of stretching. The smallest value was obtained for an elliptic precipitate at an angle of 0 relative to the horizontal direction i.e. positioned perpendicular to the direction of stretching. The value of stress obtained for the circular precipitate was different to any elliptic precipitates. References [1] Kaczmarek, L., Stegliński, M., Radziszewska, H., Ko lodziejczyk, L., Sawicki, J., Szymański, W., Atraszkiewicz, R. and Świniarski, J.: Effect of double-phase segregations formed due to two-stage aging on the strength properties of alloy, PN-EN 2024, Met. Sci. Heat Treat., 54, 9-10, , [2] Kaczmarek, L., Kula, P., Sawicki, J. Armand, S., Castro, T., Kruszyński, P. and Rochel, A.: New possibilities of applic ations aluminum alloys in transport, Arch. Metall. Mater., 54, 4, , [3] Sawicki, J., Dudek, M., Kaczmarek, L., Wiȩcek, B., Światczak, T. and Olbrycht, R.: Numerical analysis of thermal stresses in carbon films obtained by RF PECVD method on surface of cannulated screw, Arch. Metall. Mater., 58, 1, 77 81, [4] Shen, Y., Li, Y., Li, Z., Wan, H. and Nie, P.: An improvement on the threedimensional phase-field microelasticity theory for elastically and structurally inhomogeneous solids, Scr. Mater., 60, 10, , [5] Chiang, C.: Stress concentration around a triaxial ellipsoidal cavity, Arch. Appl. Mech., 85, , [6] Hamid, M., Pilvin, P., Bridier, F. and Bocher, P.: Modeling the elastoplastic behaviors of alpha Ti-alloys microstructure using Cellular Automaton and finite element methods, Comput. Mater. Sci., 99, 33 42, [7] Montheillet, F. and Gilormini, P.: Predicting the mechanical behavior of twophase materials with cellular automata, Int. J. Plast., 12, 4, , [8] Bastajib, R., Boutana, N., Bocher, P. and Jahazi, M.: Application of cellular automata method to simulate the hot deformation behavior of a dual phase titanium alloy, Tech. Mech., , 1 14, 2010.

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