PSI - Issue 6

Anna Kudimova et al. / Procedia Structural Integrity 6 (2017) 301–308 Author name / Structural Integrity Procedia 00 (2017) 000–000

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crystallites can be added into ceramic composites. An original concept of microstructural designing of polymer-free polycrystalline composite materials with preliminary FEM modeling was suggested in [Rybianets et al. (2004)]. The development of theoretical and computational models for the homogenization of active composite materials has been going on for a long time starting from the works of Marutake and Ikeda (1956); Wersing et al. (1986) to name a few. However, due to the complexity and variety of di ff erent types of problems, research in this area is still actual [Iyer and Venkatesh (2014); Martinez-Ayuso et al. (2017); Nguyen et al. (2016)]. In this paper we discuss approaches to the simulation of piezoelectric materials implemented in the own devel oped finite element package ACELAN. We began the development of ACELAN (ACoustoELectric ANalysis) about 15 years ago. The package was initially intended for computer design of piezoelectric devices for two-dimensional plane and axisymmetric compound areas, as well as three-dimensional regions of generalized cylindrical shape. Basic methods and algorithms, underlying in the first version ACELAN, were described in [Belokon et al. (2002); Nasedkin (2010)]. Later releases of ACELAN package contained the models of elastic, piezoelectric and acoustic media, and were also supplemented with more complicated problems with connectivity of mechanical, electrical, magnetic and thermal fields [Kurbatova et al. (2017); Nasedkin et al. (2014); Skaliukh et al. (2015)]. The finite element technologies implemented in ACELAN are based on the generalized problem statements. Various numerical algorithms are used to maintain symmetrical structures of the finite element quasi-definite matrices (matrix structure for the problems with a saddle point), as well as the algorithms of the mode superposition methods. To increase the accuracy of calculations, especially for nanoscale and nonlinear problems, the ability of automatic transfer to the dimensionless problem state ments is provided. Other important features of ACELAN package include original models of irreversible processes of polarization and repolarization for polycrystalline ferroelectric materials [Kurbatova et al. (2017); Skaliukh et al. (2015)] and the homogenization models for composite active materials. The subject of this research is devoted to the simulation of two types of piezocomposites: piezoceramic – pores (porous piezoceramic) and piezoceramic – crystallite (elastic inclusions). We have introduced this kind of simulation in a special package ACELAN-COMPOS. In this software we perform the homogenization procedure using the ef fective moduli method. To find the e ff ective moduli of an inhomogeneous piezoelectric body of a general anisotropic class, we consider a representative volume and set nine static piezoelectric problems. These problems di ff er by the boundary conditions which are set on the representative volume surfaces. We derive special formulas to calculate the e ff ective moduli of piezoelectric media with arbitrary anisotropy. Using these formulas, we can find the full set of e ff ective moduli for porous and ceramic polycrystalline piezocomposites by the finite element method. As a representative volume, we consider a cube evenly divided into cubic eight-node or twenty-node finite elements. For a mixed two-phase composite such element would have piezoelectric properties for the material of the skeleton and elastic or porous properties for the material of inclusions or pores. We model the inclusions in the form of granules consisting of one or more structural elements not connected with other granules. The input user parameters are the minimum and maximum granule size, and the maximum percentage of inclusions in the representative volume. The random selection of the supporting element for the granule ensures the stochastic resulting distribution. The granule is being grown according to an algorithm that allows the granule to take on a shape as much close to a sphere as possible, while avoiding unnaturally elongated elements. Also, for a two-phase composite medium ACELAN-COMPOS has algorithms to form structures with di ff erent types of connectivity. Namely, for simulating such connectivity types as the 3–0 porous composite (with closed poros ity by the classification of [Newnham et al. (1978)]) the granule forming algorithm is used, and for simulationg the 3–3 porous composite (with open porosity by the classification [Newnham et al. (1978)]) full connectivity between both phases is generated. The composite materials with 3–3 connectivity are simulated using cubic elements (octants) in the representative volume. Initial volume is divided into octants using Octree algorithm, where the size and the number of the elements may vary depending on the physical properties of the material or the needed accuracy. The octants can easily be used as finite elements on a regular mesh. The structural elements of the same material are united into the composite component. Each component has a connectivity property: any element belonging to the component is reachable from any other component element from the transition between the adjacent elements. This approach allows to model 3–3 porous composites and materials with mixtures. To provide an example, we consider a porous piezoceramic and a polycrystalline piezoceramics with sapphire ( α – corundum) crystallites Al 2 O 3 as inclusions. The results of the calculations give the full set of the e ff ective moduli. The numerical experiments in ACELAN-COMPOS have shown that the structures of the representative volumes could