A comprehensive numerical homogenisation technique for calculating effective coefficients of uniaxial piezoelectric fibre composites

Transcription

1 Materials Science and Engineering A 412 (2005) A comprehensive numerical homogenisation technique for calculating effective coefficients of uniaxial piezoelectric fibre composites Harald Berger a,, Sreedhar Kari a, Ulrich Gabbert a, Reinaldo Rodríguez-Ramos b, Julián Bravo-Castillero b,raúl Guinovart-Díaz b a Institute of Mechanics, Otto-von-Guericke-University of Magdeburg, Universitaetsplatz 2, D Magdeburg, Germany b Facultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L, CP Vedado, Habana 4, Cuba Received in revised form 3 June 2005 Abstract This work deals with the modelling of periodic composites made of piezoceramic (lead zirconate-titanat) fibres embedded in a soft nonpiezoelectric matrix (polymer). The goal is to predict thective coefficients of such periodic transversely isotropic piezoelectric fibre composites by use of a representative volume element or unit cell. The solution is based on a numerical approach using finite element method (FEM). The necessary basic equations for the piezoelectric material are introduced and the special concept for definition of generalised periodic boundary conditions for the unit cell is explained. For a composite with square arrangements of cylindrical fibres the algorithm is demonstrated and the extension to other fibre arrangements is shown. For different fibre volume fractions the results are compared with analytical solutions Elsevier B.V. All rights reserved. Keywords: Homogenisation; Representative volume element; Piezoelectricity; Finite element method 1. Introduction Piezoelectric materials have the property of converting electrical energy into mechanical energy and vice versa. This reciprocity in the energy conversion makes piezoelectric ceramics such as lead zirconate-titanat (PZT) very attractive materials towards sensors and actuators applications. But bulk piezoelectric materials have several drawbacks, hence composite materials are often a better technological solution in the case of a lot of applications such as ultrasonic transducers, medical imaging, sensors, actuators, and damping. For the last 20 years, composite piezoelectric materials have been developed by combining piezoceramics with passive non-piezoelectric polymers. Superior properties have been achieved by these composites by taking advantage of most profitable properties of each constituents. Recently, due to the miniaturisation of piezoelectric composites and the use of PZT fibres instead of piezoelectric bars, homogenisation techniques are necessary to describe the behaviour of piezoelectric composites. Corresponding author. Tel.: ; fax: address: berger@mb.uni-magdeburg.de (H. Berger). Even if analytical and semi analytical models have been developed to homogenise piezoelectric composites, they are often reduced to specific cases. Numerical methods, such as the finite element method, seem to be a well-suited approach to describe the behaviour of these materials, because there are no restrictions to the geometry, the material properties, the number of phases in the piezoelectric composite, and the size. The prediction of the mechanical and electrical properties of piezoelectric fibre composites became an active research area in recent years. Micro mechanical methods provide an overall behaviour of piezoelectric fibre composites from known properties of their constituents (fibre and matrix) through an analysis of a periodic representative volume element (RVE) or a unit cell model. In the macro mechanical approach, on the other hand, the heterogeneous structure of the composite can then be replaced by a homogeneous medium with anisotropic properties. A number of methods have been developed to predict and to simulate the coupled piezoelectric and mechanical behaviour of composites. Basic analytical approaches have been reported [1,2], which are not capable of predicting the response to general loading. This restriction can be overcome by employing periodic micro field approaches (commonly referred to as unit cell models) where the fields are typically solved numerically with high resolution, e.g. by the finite element method [3]. Sometimes /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.msea

2 54 H. Berger et al. / Materials Science and Engineering A 412 (2005) special finite element formulations were introduced to solve the problem [4]. In such models, the representative unit cell and the boundary conditions are designed to capture a few special load cases, which are connected to specific deformation patterns [5,6]. But many of these approaches allow the prediction of only a few key material parameters and not the full set due to restrictions in load-boundary combinations for the numerical unit cell model. The aim of this paper is to predict the full set of material moduli, i.e. to determine the complete tensors associated with the overall elastic, dielectric, and piezoelectric behaviour. This means that the linear response to any mechanical and electrical load, or any combination of both, will be determined. In the present paper, the FEM based on a micro mechanical analysis is applied to uniaxial periodic piezoelectric fibre composites subjected to different loading conditions with different boundary T T 22 T T 23 T 31 = T 12 D 1 D 2 D 3 conditions to predict thective coefficients of transversely isotropic piezoelectric fibre composites (1 3 periodic). The advantage of this technique lies in using a standard FE code in connection with auxiliary routines written in FORTRAN for modelling of unit cells with the appropriate boundary and loading conditions and its possibility to extend it to complex composites with arbitrary types of inclusions and non-uniform distributions. The algorithm is demonstrated for a composite where the embedded fibres has a square arrangement. For verification of results various fibre volume fractions and different fibre arrangements are investigated and a comparison is done with values calculated by an analytical solution based on the asymptotic homogenisation method (AHM) reported in [7,8]. 2. Piezoelectricity and piezoelectric composites Coupled piezoelectric problems are those in which an electric potential gradient causes deformation (converse piezoelectric effect), while mechanical strains cause an electric potential gradient in the material (direct piezoelectric effect). The coupling between mechanical and electric fields is characterised by piezoelectric coefficients. Those materials respond linearly to changes in the electric field, the electrical displacements, or mechanical stresses and strains. These assumptions are compatible with the piezoelectric ceramics, polymers, and composites in current use [9]. Therefore, the behaviour of the piezoelectric medium is described by the following piezoelectric constitutive equations, which correlate stresses T, strains S, electric fields E, and electrical displacements D as follows: [ ] [ ][ ] T C e T S =, (1) D e ε E where C is the elasticity matrix, ε the permittivity matrix, and e is the piezoelectric strain coupling matrix. For a transversely isotropic piezoelectric solid, the stiffness matrix, the piezoelectric matrix, and the dielectric matrix simplify so that there remain independent coefficients. In the case of aligned fibres made of a transversely isotropic piezoelectric solid (PZT), embedded in an isotropic polymer matrix, the resulting composite is a transversely isotropic piezoelectric material too. Consequently, the constitutive Eq. (1) can be written as S S S S 23 S 31. (2) S 12 Ē 1 Ē C44 eff C66 eff ε eff ε eff ε eff In this matrix, the general variables of the coupled electromechanical problem were replaced by the appropriate values for the homogenised structure. So, Cij eff, eeff ij, εeff ij denote thective material coefficients and S ij, Ē i, T ij, D i denote average values. These relations represent the basis for the further considerations based on the unit cell. Ē 3 3. Representative volume element (unit cell) In general, the object under consideration is regarded as a large-scale/macroscopic structure. The common approach to model the macroscopic properties of 3D piezoelectric fibre composites is to create a representative volume element (RVE) or a unit cell that captures the major features of the underlying microstructure. The mechanical and physical properties of the constituent material is always regarded as a smallscale/microstructure. One of the most powerful tools to speed up the modelling process, both the composite discretisation and the computer simulation of composites in real conditions, is the homogenisation method. The main idea of the method is to find a globally homogeneous medium equivalent to the original composite, where the strain energy stored in both systems is approximately the same. In this paper, we limit ourselves to a quasistatic analysis of periodic (1 3) structures with perfectly bonded continuous fibres, which are aligned and poled along the x 3 axis as shown in Fig. 1. With help of symmetry, such a regular piezoelectric fibre composite may be analysed by using a representative volume element or unit cell. A unit cell is the smallest

3 H. Berger et al. / Materials Science and Engineering A 412 (2005) Fig. 1. Schematic diagram of a periodic 1 3 composite (a) and unit cell (b), picked from the original composite. part that contains sufficient information on the above-mentioned geometrical and material parameters at the microscopic level to deduce thective properties of the composite. Fig. 1 shows the unit cell that is picked from the periodic piezoelectric fibre composite. It has infinite lengths in all three directions. For demonstration of the algorithm we consider a composite with a square fibre arrangement. It is assumed that the material properties are the same in the first two directions (i.e. along x 1 and x 2 axis). All the fibres are assumed to be straight and poled in the third direction (i.e. along axis x 3 ). Fig. 1(b) shows the schematic diagram of the unit cell picked from the considered composite. 4. Numerical solution using finite element method 4.1. Periodic boundary conditions to RVE Composite materials can be represented as a periodical array of RVEs. Therefore, periodic boundary conditions must be applied to the RVE models. This implies that each RVE in the composite has the same deformation mode and there is no separation or overlap between the neighbouring RVEs. These periodic boundary conditions on the boundary of RVE are given in [10] u i = S ij x j + v i. (3) In the above Eq. (3) S ij are the average strains and v i is the periodic part of the displacement components u i on the boundary surfaces (local fluctuation), which is generally unknown and is dependent on the applied global loads. The indices i and j denote the global three-dimensional coordinate directions in the range from 1 to 3. A more explicit form of periodic boundary conditions, suitable for square RVE models can be derived from the above general expression. For the RVE as shown in Fig. 1(b), the displacements on a pair of opposite boundary surfaces (with their normal along the x j axis) are ui K+ = S ij xj K+ + v K+ i, u K i = S ij xj K + v K i (4) where index K + means along the positive x j direction and K means along the negative x j direction on the corresponding surfaces A /A +, B /B +, and C /C + (see Fig. 1(b)). The local fluctuations v K+ i and v K i around the average macroscopic value are identical on two opposing faces due to periodic conditions of RVE. So, the difference between the above two equations is the applied macroscopic strain condition and electric field condition, respectively u K+ i u K i = S ij (xj K+ xj K ), Φ K+ Φ K = Ē i (x K+ j x K j ). (5) It is assumed that the average mechanical and electrical properties of a RVE are equal to the average properties of the particular composite. The average stresses, strains, electric fields, and electrical displacements in a RVE are defined by S ij = 1 S ij dv, T ij = 1 T ij dv, V V V V Ē i = 1 E i dv, D i = 1 D i dv (6) V V V V where V is the volume of the periodic representative volume element Finite element modelling All finite element calculations were made with FE package ANSYS. For modelling of the RVE three-dimensional multifield eight node brick elements with displacement degrees of freedom (DOF) and additional electric potential (voltage) degree of freedom were used. These allows for fully coupled electromechanical analyses. To obtain the homogenised effective properties we apply the macroscopic boundary conditions (Eq. (5)) to the RVE by coupling opposite nodes on opposite boundaries. In order to apply these periodic boundary conditions in the FE analysis, the mesh on the opposite boundary surfaces must be the same. For each pair of displacement components at two corresponding nodes with identical in-plane coordinates on the two opposite boundary surfaces a constraint condition (periodic boundary condition Eq. (5)) is imposed. The prescription of these constraint conditions to all opposite nodes at opposite boundary surfaces directly in ANSYS is a very difficult task because of too many number of nodes. Consequently, we developed a FORTRAN program, which generates all required constraint conditions automatically. First the finite element mesh is created from the ANSYS preprocessor. Then based on the generated nodal coordinates the appropriate nodal pairs are selected by the FORTRAN program and a partial ANSYS input file is created containing the constraint conditions. Using this input file ANSYS continues with assigning the constraint equations and finally with solving the

4 56 H. Berger et al. / Materials Science and Engineering A 412 (2005) such a way that, except the strain in the x 3 direction ( S ), all other mechanical strains and gradients of electric potential (Ē i ) become zero. This can be achieved by constraining the normal displacements at all surfaces to zero except of surface C + (see Fig. 1(b)). At surface C + the periodic boundary condition corresponding to surface C must be applied. Due to applied zero displacements to surface C in x 3 direction (u C 3 = 0) the periodic boundary condition in this direction according to Eq. (5) simplifies to u C+ 3 = S (x3 C+ x3 C ). (7) Fig. 2. Periodic boundary conditions for a pair of nodes on opposite surfaces A and A +. problem. As an example Fig. 2 shows the constraint equations for a pair of nodes on opposite surfaces A and A +. For demonstration of calculation of effective coefficients we consider a piezoceramic (PZT-5) fibre embedded in a soft nonpiezoelectric material (polymer) with a square arrangement like shown in Fig. 1. We are assuming that the fibres and the matrix are ideally bonded and that the fibres are straight and parallel to the x 3 axis. The fibre cross section is circular and the unit cell is having a square cross section. The piezoelectric fibres are uniformly poled along the x 3 direction. To find thective coefficients special load cases with different boundary conditions must be constructed in such a way that for a particular load case only one value in the strain/electric field vector (see Eq. (2)) is non-zero and all others become zero. Then from one row in Eq. (2) the corresponding effective coefficient can be determined using the calculated average non-zero value in the strain/electric field vector and the calculated average values in the stress/electrical displacement vector. In the next chapters, the special models for calculating the different effective coefficients are explained in detail. But due to limitation of this paper only for one load case the FE model and deformed contour plots are shown. The used material properties of polymer and PZT-5 are listed in Table 1, where elastic properties, piezoelectric constants and permittivities are given in N/m 2, C/m 2, and F/m, respectively. 5. Numerical calculation of the different effective coefficients 5.1. Calculation of and Ceff For the calculation of thective coefficients and Ceff the boundary conditions have to be applied to the RVE in Because now this equation is independent of u C 3, instead of using constraint equations, an arbitrary constant prescribed displacement can be applied on surface C + to produce a strain in x 3 direction. To ensure that no charge transfer is between opposite surfaces the voltage degree of freedom on all surfaces is set to zero. For the calculation of the total average values S, T, and T according to Eq. (6) the integral was replaced by a sum over averaged element values multiplied by the respective element volume. Using these total average values the coefficients C eff and can be calculated from the matrix Eq. (2). Due to zero strains and electric fields, except S the first row becomes T = S. Then C44 eff can be calculated as the ratio of T / S. Similarly, can be evaluated as the ratio of T / S from the third row of matrix Eq. (2) Calculation of and Ceff 12 For the calculation of thective coefficients C eff and Ceff 12 we have similar conditions like for the calculation of thective coefficients and Ceff. But now a prescribed displacement in x 1 direction must be applied on surface A + and the normal displacements on all other surfaces must be set to zero. Also the electric potential DOF at all surfaces is set to zero. For the calculation of these coefficients the in-plane behaviour is relevant only. Using the total average strain S and stress T from first row of matrix Eq. (2) we get T = C eff S. From this relation C eff can be calculated as the ratio of T / S. Similarly, C12 eff can be found as the ratio of T 22 / S using the second row of matrix Eq. (2) Calculation of 66 and Ceff 44 To evaluate thective coefficient C66 eff the in-plane shear strain S 12 may have a non-zero value in strain/electric field vector Table 1 Material properties of the composite constituents fibre (PZT-5) and matrix (polymer) C (10 10 ) C 12 (10 10 ) C (10 10 ) C (10 10 ) C 44 (10 10 ) C 66 (10 10 ) e e e ε (10 9 ) ε (10 9 ) PZT Polymer

5 H. Berger et al. / Materials Science and Engineering A 412 (2005) Fig. 3. Finite element mesh with shear loads and out-of-plane supports. of Eq. (2) only. For the evaluation of C44 eff the out-of-plane shear strain S 23 or S 31 may have a non-zero value only. This can be achieved by applying appropriate shear forces to produce a pure shear stress state in the desired directions, which consequently, results in a pure shear strain state. Beside the applied shear forces the displacement boundary conditions must be chosen in the right way to get these states. That means that a symmetric deformation mode respective to the diagonal line of the shear plane must be ensured; see also Fig. 2, which visualizes such a pure shear stress state in a plane view. To receive such pure shear stress state a support normal to the diagonal at one edge of the finite element model was applied. To avoid rigid body movement the opposite edge was fixed in the shear plane. For calculation of C66 eff the x 1 x 2 plane was used as shear plane and for calculation of C44 eff the x 2 x 3 plane was chosen. Fig. 3 shows the finite element model with shear forces and the above-mentioned supports for the out-of-plane case. Furthermore, normal displacements at parallel surfaces to the shear plane must be set to zero to ensure zero strains perpendicular to the shear plane. Also the electric potential DOF on all surfaces must be set to zero. Using the calculated non-zero average strain and stress values from the first model, the sixth row in the matrix Eq. (2) becomes T 12 = C66 eff S 12, and consequently, C66 eff can be evaluated as the ratio of T 12 / S 12. With the second model C44 eff can be found as the ratio of T 23 / S 23. Figs. 4 and 5 show the calculated strains and stresses for this coefficient in deformed shape. Fig. 4. Strain distribution S 23. the electric potential difference in x 3 direction the fibre will try to expand and stresses as well as electric fields are produced in x 3 direction. Consequently, the total average stress and electrical displacement values as well as the total average electric field Ē 3 can be calculated. Then the third row of the matrix Eq. (2) reduces to T = Ē3, so that = T /Ē 3. Similarly, is the ratio of T /Ē 3. For the evaluation of ε eff the last row of matrix Eq. (2) is used which becomes D 3 = ε eff Ē3. From this equation ε eff can be calculated by εeff = D 3 /Ē Calculation of Thective coefficient can be calculated as the ratio of D 2 / S 23 from the seventh row of the matrix Eq. (2). In order to evaluate this coefficient, the boundary conditions should be applied in such a way that except of the strain S 23 in the strain/electric field vector in Eq. (2), all other strains and electric 5.4. Calculation of, eeff, and εeff In order to evaluate thective coefficients, eeff,and εeff, all surfaces must be constrained to have zero normal displacements. A zero electric potential is applied to the surface C and a non-zero electric potential is applied to the surface C +. The electric potential at all other surfaces is set to zero. Since all surfaces are constrained to have zero displacements and because of Fig. 5. Stress distribution T 23.

6 58 H. Berger et al. / Materials Science and Engineering A 412 (2005) Fig. 6. Hexagonal fibre arrangement and unit cell. Fig. 9. Variation of. Fig. 7. Random fibre arrangement and unit cell. fields are made to zero. Similarly, to the calculation of C44 eff outof-plane shear forces are applied in the x 2 x 3 plane. The electric potential in all directions is made to zero by constraining the voltage degree of freedom to zero. Consequently, in the matrix Eq. (2) the eighth row becomes D 2 = S 23. From this equation can be calculated by the ratio eeff = D 2 / S 23. Fig. 10. Variation of Calculation of ε eff ε eff can be evaluated as the ratio of D 2 /Ē 2. In order to solve these equations, all mechanical strains ( S ij ) should be made equal to zero. This can be achieved by constraining the displacements at all surfaces to zero. The electric potential difference should be applied in x 1 direction. Then the eighth row of the matrix Eq. (2) gives the ε eff coefficient. 6. Discussion of results The introduced method was applied to the calculation of piezoelectric fibre composites with square fibre arrangement shown in Fig. 1 and with hexagonal fibre arrangement shown in Fig. 6. Hexagonal arrangement is characterised by a constant distance between one fibre and all surrounding six fibres. So a repeatable part and consequently a unit cell can be found like shown in Fig. 6. A more complex composite with random fibre distribution and fibre diameter is shown in Fig. 7. The unit cell is here an appropriate repeatable part of the composite. Results Fig.. Variation of. are not presented in this paper. But the algorithm is in the same manner applicable like for the regular fibre arrangements. All effective coefficients have been calculated for 6 discrete fibre volume fractions. Fig. 8 demonstrates the used fractions for the square cell. Since only the volume fibre fraction has an influence on the results the size of the RVE was chosen with unit length in all directions. From these discrete fractions graphs were interpolated. For selected coefficients the behaviour related to the volume fraction is shown in the diagrams Figs as comparison between asymptotic homogenisation Fig. 8. Investigated ratios of fibre volume fraction shown for square arrangement.

7 H. Berger et al. / Materials Science and Engineering A 412 (2005) Fig. 12. Variation of 44. Fig. 16. Variation of ε eff. Fig.. Variation of 66. method (AHM) [6,7] and the introduced numerical method (FEM). FEM-Squ denotes the results for square arrangement and FEM-Hex for hexagonal arrangement. The coefficients, and εeff are not plotted because no difference is identifiable between the curves. The results show in general a good coincidence between calculation by AHM and FEM. Thective coefficients, Ceff, Fig. 14. Variation of.,, eeff, eeff, εeff, and εeff calculated by FEM show a very good agreement with the coefficients obtained by AHM in all range of the volume fraction. Qualitatively, the behaviour of the coefficients C eff, Ceff 12, Ceff 66, and Ceff 44 is the same although for a greater value of the volume fraction (0.4 and higher) the curves are not so close. For reliability the results were checked with the universal relation of Schulgasser []. It was satisfied with a maximum error less than 6%. Furthermore, the results were tested against the Hashin/Shtrikman lower and upper bounds [12]. It could be stated that the calculated coefficients lie inside these bounds over the full range of volume fraction. 7. Conclusions A numerical approach for predicting the homogenised properties of piezoelectric fibre composites has been presented. The numerical approach is based on the finite element method. Longitudinal and transversal elastic and piezoelectric effective coefficients have been calculated with the introduced finite element numerical approach and compared with analytical solutions based on the asymptotic homogenisation method. This permits us to estimate the range of validity of our approach. It can be stated that this comprehensive numerical homogenisation technique provides reliable results. In comparison to many other approaches our technique can be easily extended to determination of homogenized material data for more complex composites with arbitrary inclusions and arbitrary distribution (not only uniaxial fibres, also angle plied fibres, ellipsoids, spheres, etc.). The developed tool is based on using a standard FE package like ANSYS by interfacing with corresponding FORTRAN routines. It reduces the manual work and time and can be used as a template to determinective coefficients. Acknowledgement This work has been supported by DFG Graduiertenkolleg 828 Micro Macro Interactions in Structured Media and Particle Systems. References Fig.. Variation of. [1] M.L. Dunn, M. Taya, Int. J. Sol. Struct. 30 (1993) [2] P. Bisegna, R. Luciano, J. Mech. Phys. Solids 44 (1996)

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