Contents lists available at ScienceDirect Ceramics International journal homepage: www.elsevier.com/locate/ceramint Modelling of multilayer actuator layers by homogenisation technique using Digimat software Tomasz Trzepieciński a, ⁎ , Grażyna Ryzińska a , Mojtaba Biglar a , Magdalena Gromada b a Rzeszow University of Technology, Department of Materials Forming and Processing, Al. Powstancow Warszawy 8, 35-959 Rzeszów, Poland b Institute of Power Engineering, Ceramic Department CEREL, ul. Techniczna 1, 36-040 Boguchwała, Poland ARTICLE INFO Keywords: Barium titanate Ceramic Digimat Homogenisation Microstructure ABSTRACT This paper is concerned with the microstructure of adhesive joints composed of BaTiO 3 ceramic material and two kinds of adhesives, with a special emphasis on the numerical modelling of grain size, shape and material porosity. BaTiO 3 powder for use in stacked-disk multilayer actuator production is manufactured using the solid- state technique. The properties of BaTiO 3 material at each stage of its fabrication that influence its application for the stacked-disk multilayer actuator are presented and discussed. The parameters characterising the properties of the fabricated pellets are also described. Implicit time-discretisation and consistent tangent operators are employed. A 3D microstructure model of BaTiO 3 using Digimat-FE software is generated. Then, effective elastic constants of BaTiO 3 and epoxy adhesive composites are calculated by numerical simulation. The Mori-Tanaka homogenisation scheme for representative volume elements of BaTiO 3 and epoxy adhesives composites is carried out using Digimat-MF software in order to obtain failure characteristics of the composite material. 1. Introduction The compound barium titanate (BaTiO 3 ) is the most intensively studied perovskite material due to its wide use in the ceramic and electronic industries [1–4]. Two decades ago, piezoelectric monolithic transducers (PZT) and actuators operated at driving voltages in the order of 10 2 –10 3 V, with resultant mechanical strains of 0.1%. The requirements of the electronic industry forced the development of devices that reach the same strain at lower driving voltages. This was achieved by making thin, individually electroded piezoelectric elements that could be stacked on the top of each other to produce multilayer actuators [5]. The principle of a multilayer actuator is that thin layers of piezoelectric ceramic material are electrically connected in parallel (Fig. 1). To join the ceramic layers, a bonding using many kinds of electrically conductive adhesives is mainly used. Upon actuation the active part of the multilayer element expands in the polarisation direction, thus contracting in the perpendicular direction (Fig. 1). The protective top layer and the inactive parts of the multilayer elements themselves partly constrain this lateral compression, which leads to shear stresses at this interface and tensile stresses in the poling direction in the inactive region near the corners of the multilayer element [6]. The dielectric properties of BaTiO 3 are controlled by purity and microstructure, which are dependent on the methods of preparation [7,8]. BaTiO 3 powders are mostly manufactured at high temperatures by solid-state reaction. The advantages of the solid-state reaction used in mass production are its simplicity, precise stoichiometric control and low cost, with the main disadvantage being that the high calcining temperature results in very large and non-uniform grain sizes [9]. Controlling the phase, composition homogeneity, particle size and monodispersity are other concerns in developing techniques for synthesising BaTiO 3 . Thermoelastic properties in porous ceramics are simply determined by a rule of mixtures or by the respective properties of the solid phase (e.g., density, heat capacity and the coefficient of thermal expansion) [10]. On the other hand, thermal conductivity and elastic coefficients are strongly related to microstructure [11]. Material properties of ideal, single-phase materials can be obtained either by measurements or from numerical modelling. However, measurements cannot be used if the range of material properties is to be estimated for multi-phase materials with novel microstructures [12]. Modelling material behaviour in 3D requires accurate representa- tions of microstructure if models electrical and/or mechanical systems are to produce realistic property predictions [13,14]. Ondrack [15] developed the first model concept to calculate effective thermal, elastic or electric properties of multi-phase materials with simple structures: ellipsoidal particles embedded in a matrix phase or two interpenetrat- http://dx.doi.org/10.1016/j.ceramint.2016.11.157 Received 9 October 2016; Received in revised form 20 November 2016; Accepted 22 November 2016 ⁎ Corresponding author. E-mail address: tomtrz@prz.edu.pl (T. Trzepieciński). Ceramics International 43 (2017) 3259–3266 Available online 23 November 2016 0272-8842/ © 2016 Elsevier Ltd and Techna Group S.r.l. All rights reserved. MARK