Research Program Michel Cnrs

Research Program Dr. Thomas Michelitsch University of Sheffield for application at CNRS

December 2006

Topic Self Consistent Modelling of the dynamic effective properties of piezoelectric composites.

General Area

Mathematical Physics, Mechanics of Materials, Applied Mathematics, Analytical Dynamics

Summary The modelling of the effective properties of composite materials leads in the dynamic case to a multiple-scattering propblem. In this project piezoelectric composite materials are considered due to their technological importance in smart structures and materials. Self consistent methods were invented in Quantum Mechanics by Kohn and Sham. The class of these methods have in common that they reduce untractable multiple particle problems to much simpler effective one particle problems. Self consistent approaches constitute superior techniques useful to deduce a powerful approximate scheme to model the effective dynamic characteristics in piezoactive materials such as mean field wave spead, effective wave vector and the electroelastic effective moduli. The goal of the research program is to develop and extend the class of self consistent approaches to model dynamic effective properties of piezoelectric composite materials.

Background I started working on the field of mathematical modelling of piezoelectric materials since 1996, when I was a postdoc at the Max Planck Unit “Mechanics of heterogeneous solids” in Dresden, Germany, and continued to do so when leading the research group “Smart Materials” in the department “Theory of Mesocopic Phenomena” at Max Planck Institute in Stuttgart, Germany in 2001 (details, see my enlosed CV). At Max Planck Institute, I attracted sucessfully funding from the German Research Council (Deutsche Forschungsgemeinschaft – DFG) on behalf of the head of department, Professor Huajian Gao for the project ” Mathematical Modelling of the Dynamical Characteristics in Piezocomposites”, DFG project no. GA 732/1-1 for part I of the project and DFG project no. GA 732/1-2 for part II of the project (referenced in all publications performed within the framework of this project). I managed to attract funding for the full length of a maximum of three years for the project. The topic proposed for the present project is mainly based on the results achieved in the DFG project. These achievements are reflected by 12 research papers

which were published in the framwork of this project [1-13]. In the present proposal we want to tackle those questions which were not resolved in the DFG proposal. The main issue which was not yet addressed in the DFG project was the development of an appropriate self-consistent method which allows it to tackel the effective dynamic homogeneisation problem in piezoelectric micro-inhomogeneous materials. In this sense the present proposal can be conceived as a successing proposal to the DFG proposal. The motivation to continue this work is that on the one hand essential questions could not be answered in the time of three years given for the DFG project, on the other hand we achieved remarkable results which should enable us to answer many of these questions.

Possibly a research group devoted to the topic of the present suggested proposal is to be established, as well as the collaboration with the DFG project team (Michelitsch, Levin, Wang) is to be maintained.

Introduction Piezoelectric composites are an important class of engineering materials with a high potential for applications in so called Smart Structures. Piezoelectric composites are being used to numerous only recently emerging technologies taking advantage of the superior properties of this material class. Some of the most important applications include: microphones, high frequency loudspeakers, acoustic modems; pressure switches, impact detectors, flow meters and load cells. These recent new applications especially focus on dynamic processes (wave propagation) effects having been key subject of this project. Future sophistication of these new applications is expected to raise an enormous market potential. The exploitation of these processes for the above mentioned new applications is only in the beginning stage. A fundamental understanding of the dynamics in piezoelectric (electro-elastic coupled) composite materials is inevitable to improve this exploitation especially the lack of understanding in the analytical description how the microstructure (inclusions) affects the macroscopic dynamic response and observable quantities, among them effective dynamic electroelastic moduli, the mean field wave speed and wave vector. The main goal of the project is to establish analytical approaches which describe the effective

dynamic responses of piezoelectric composite materials and to create analytical benchmark solutions. As a key problem to be resolved for the mathematical modeling of the dynamic behavior of micro-inhomogeneous piezoelectric medium such as piezo-composites we identified the Dyanmic Variant of Eshelby’s inclusion problem in both, the purely isotropic, i.e. piezo-passive medium and in the piezoelectric medium. In the analytical modelling considerable progress could be achieved in the DFG project [1-9]. To this end the development of appropriate dynamic Green’s functions were a crucial prerequisite in order to tackle dynamic Eshelby problems. Thus the analytical development of dynamic Green’s functions was one of the central issues in which the project will be based on. In the present project, for the aim of analytical feasibility either a three-dimensional elastic (piezo-passive) or a quasiplane two dimensional piezoelectric medium (equivalent to a fiber reinforced piezo-composite material) is to be considered. By using the solutions of the dynamic variant of Eshelby inclusion problem to be developed together with the Green’s functions to be derived, the Scattering Problem of a piezo-composite material was to be tackled and to be modeled as a piezoactive matrix containing a random set of piezoelectric inclusions (of other material characteristics as the matrix) subjected to an incoming electro-acoustic wave field. The goal is to obtain analytical expressions describing the macroscopic (effective) dynamic response of this model system such as the effective dynamic moduli, the mean wave speed and the mean field wave vector. All these observables are to be related to the characteristics (piezoelectric moduli of the constituents, concentration of inclusions, geometric characteristics of inclusions such aspect ratios etc) of the microstructure. For a real material system this problem is a highly complex multiple scattering problem which cannot be solved in closed form due to the huge set of unknowns. Instead, a powerful class of approximations, namely the self consistent methods was to be utilized, strictly speaking the Kanaun-Levin variant of the Effective Field Method (EFM) was identified to be highly promising to tackle this problem and to yield expressions for the dynamic characteristics in a reasonable approximation. This was indicated by the success of the EFM in the modeling effective properties in the framework of statics. As a first step the main focus in the project is to apply the analytical solutions for the dynamic variants of Eshelby inclusion problem, and closely related to utilize solutions Helmholtz equation which we call “Dynamic Potentials” (frequency domain

solutions) to establish a wave equation for the effective retarded potentials representing the causal time domain solutions. This approach will take advantage by the analytical results deduced in [8]. A second step will be focused on developing time domain solutions and to develop numerical tools (codes) for the transformation between time domain and frequency domain solutions by using approaches deveoped in [9]. Moreover it is intended to develop further time domain solutions for Eshelby inclusion problems based on [7] and retarded potentials [8]. The dynamic potential functions were identified to represent fundamental quantities to generate the dynamic Eshelby tensor function. The wave and Helmholtz equations were solved for source regions which represent the space occupied by the inclusions in the DFG project [2-5]. We allready considered spatial homogeneous and inhomogeneous source distributions with Dirac-delta type time profile [8]. In most cases the inclusions were assumed to be ellipsoidal. As “spin off” results some fundamental problems of Mathematical Physics were solved analytically in closed form in [5] in terms of one-dimensional quadratures, similar to Elliptic Integrals, namely the problem of determining the solution of the Helmholtz equation for ellipsoidal shells. The static limiting cases recovered some classical Newtonian Potentials due to Ferrers and Rahman. Moreover, by using a similar approach, the dynamic variant of Eshelby’s inclusion problem for an isotropic three dimensional infinite medium was considered containing a homogeneous ellipsoidal inclusion, i.e. the complete analogue problem to the classical Eshelby problem [3].

With the approaches to be deveoped in the proposed project, various problems of materials science (Dynamic Eshelby inclusion problem in the frequency and time domain and their expression in terms of relatively simple scalar functions (“Dynamic Potentials”) become accessible. The analytical approaches to be developed in this project are generally applicable in a wide range of fields in engineering sciences, mechanics of materials and physics.

Previous Work In the following paragraph we describe the essence of the outcomes of previous work relevant for the present project in more details. First of all, the propagation of electroacoustic waves in a piezoelectric medium containing a statistical ensemble of cylindrical fibers was considered. Both the matrix and the fibers consist of piezoelectric transversely isotropic material with symmetry axis parallel to the fiber axes. Special emphasis was given on the propagation of an electroacoustic axial shear wave polarized parallel to the axis of symmetry propagating in the direction normal to the fiber axis. The scattering problem of one isolated continuous (‘‘one-particle scattering problem’’) was considered. By means of a Green’s function approach a system of coupled integral equations for the electroelastic field in the medium containing a single inhomogeneity was solved in closed form in the long-wave approximation. The total scattering cross-section obtained in closed form is in accordance with the electroacoustic analogue of the optical theorem. The solution of the one-particle scattering problem was used to solve the homogenization problem for a random set of fibers by means of the self-consistent scheme of effective field method (EFM). Closed form expressions for the dynamic characteristics such as total cross-section, effective wave velocity and attenuation factor were obtained in the long-wave approximation [1]. The dynamic potentials of a two-dimensional (2D) quasi-plane piezoelectric infinite medium of transversely isotropic symmetry containing an inclusion of arbitrary shape was derived in terms of scalar solutions of the 2D Laplace and Helmholtz equations. Closed-form expressions for the interior and exterior space were obtained for the case when the spatial source distribution is characterized by a region occupied by a circular inclusion embedded in a quasi-plane transversely isotropic matrix (see (5.18)-(5.23) in paper [2]). The results are used to solve the dynamic Eshelby problem of a circular inclusion (6.1)ff. in [2] (plane region with the same material characteristics as the matrix) undergoing uniform eigenstrain and eigenelectric field. In contrast to the static case, the dynamic electroelastic fields inside the circular inclusion are non-uniform in the space-frequency representation. The derived dynamic piezoelectric potentials represent basic quantities for the description of the dynamic properties of microinhomogeneous quasi-plane piezoelectric material systems (e.g. fiber-reinforced

piezocomposites) [2]. In paper [5] the dynamic Eshelby inclusion problem for an ellipsoidal inclusion in a three dimensional infinite elastic isotropic medium was considered. The dynamic Eshelby tensor was expressed in terms of solution of the Helmholtz equation (Helmholtz potentials) of the form of eq. (2.14) in that paper. A compact formulation for the components of the dynamic Eshelby tensor was derived for the interior region of the inclusion (eqs. (5.1)-(5.9) in [5]). The approach is straight-forward to be extended to derive expressions of the dynamic Eshelby tensor also for the exterior region (by employing in (5.1) the exterior dynamic potentials (3.24) instead of the interior potentials (3.18)). For spheroidal inclusions, relations (5.16)-(5.25) together with integrals (5.14) (5.15). We also recovered closed-form expressions in cases such as spheres and continuous fibers in accordance with those given in 1990 by Mikata & Nemat-Nasser. The formulation (5.7)-(5.13) for the interior dynamic Eshelby tensor is in complete analogy to Eshelby’s 1957 paper, only the integrals (5.14) (5.15) are not elliptic in the dynamic case. This formulation is especially convenient to perform the static limit of vanishing frequency (eqs. (5.26)ff.). The static limiting case recovered Eshelby’s classical 1957 result (eqs. (5.31)-(5.33) in [5]). A quasiplane piezoelectric medium of transversely isotropic symmetry with continuous fiber inclusion parallel to the axis of symmetry was considered in [4,6,7]. The problem is equivalent to a two-dimensional quasi-plane piezoelectric medium containing a 2D inclusion. The inclusion was assumed to undergo a spatially uniform

δ (t ) -type time domain transformation. The continuous fiber has elliptical, circular and arbitrary cross-sections. The solutions of the inclusion problem were expressed by scalar dynamic potentials. In the time domain two of these functions correspond to the retarded potential integrals of the inclusion. Their frequency domain representation which we shall call the dynamic potentials of the inclusion were also considered. Integral formulae were derived for continuous fiber inclusions with elliptical crosssections. Known closed form solutions were reproduced for circular fibers as limiting cases in [4,6,7] which confirmed our analytical approaches and closed form solutions. For fibers with arbitrary cross-sections a numerical method based on Gauss quadrature was applied in order to obtain their dynamic potentials in the frequency and time domain. High accuracy and efficiency of the numerical method was

confirmed by comparing numerical solutions with analytical cases solutions for dynamical potentials. Characteristic observable superposition and runtime effects for the time domain wave fields generated by dynamically transforming inclusions were found (see eqs. (17)ff. in paper [8]). Closed form solutions for retarded potentials of spheres were obtained (eqs. (21)-(24) in [8]) and cubic and prismatic source regions were treated numerically with above Gauss quadrature method (Sec. 3 of paper [8]). The solutions of many dynamical problems as wave propagation in electrodynamics, acoustics or elasticity very often require the solution of inhomogeneous Helmholtz equations (determination of dynamic potentials) for ellipsoidal source regions. As in the case of the static (Newtonian) potentials a compact representation of the dynamic potentials in terms of one-dimensional integrals was highly desirable. Due to the mathematical complexity of the problem for ellipsoids such a representation was not reported in the literature so far. In paper [5] we close this gap for the dynamic potential of an ellipsoidal shell for internal spacepoints. The solution derived (eqs. (3.27) and alternatively (3.38) in [5]) is a 1D integral representation for the real part of the dynamic potential of a homogeneous shell. The classical limit yields the well classical result for a Newtonian potential due to Ferrers (eq (3.28)). The imaginary part of the dynamic potential of a homogeneous shell was obtained in terms of an infinite series (4.6) which is related to the imaginary part of the Helmholtz Green’s function which is a homogeneous solution of the Helmholts equation. In the static limiting case the imaginary part of the Helmholtz potential is vanishing. The derived solution of the inside region can easily be used to find the solution for the outside region by applying Ivory's theorem. In the static limit classical results of Ferrers and Dyson for the Newtonian potential of inhomogeneous ellipsoids were reproduced. In paper [8] analyzed the three-dimensional infinite space solutions of the threedimensional inhomogeneous wave equation (the ‘retarded potentials’ or ‘causal propagators’) for ellipsoidal sources and for sources of arbitrary shapes. The ‘shorttime characteristics’ of the retarded potential for a spatially inhomogeneous source density of δ-shaped time profile was considered. It was found that, the short-time characteristics is governed by the spatial inhomogeneity of the source density in the immediate vicinity of a spacepoint, more precisely the time expansion of the retarded potential is governed by the derivatives of the source function. A spatial derivative of order m of the source generated a retarded potential of order 2m+1 in the immediate vicinity of the source-point (see eq. (44) in [8]).

In paper [9] a simple formula for Laplace inverse transform was derived based on the wavelet theory. The efficiency and robustness are demonstrated through numerical examples. This numerical approach was successfully in the numerical computations involved in the project. The high accuracy of the approach was confirmed by comparing analytical results and reproducing them with the numerical method [9]. All results of the project were continuously presented at renowned international conferences (see paragraph “Conference papers”) and have attracted considerable interest by the audience.

Outlook and on future applications A fundamental understanding of the dynamics in piezoelectric composite materials is inevitable to improve the technological exploitation of piezo-compposites in smart structures. To reduce the so far considerable gap in understanding, especially in the modeling of the effective macroscopically observable dynamic characteristics of these materials, is among the main goals of this project. The results of the present project are to be contributed to basic research which is inevitable for the progress in science and for the development of human society and culture. More precisely, the developed mathematical techniques in this project are useful for a wide range of fundamental problems in mathematical physics and mechanics of materials. The results deduced in in [2] have already attracted a wide attention in the community and also are to be used in the present suggested project. The approach developed in [2] already is extensively used and currently being extended to magnetic materials by Chen and Shen (“Dynamic Potentials and Green’s Functions of a quasiplane magneto-electro-elastic medium with inclusion”, to appear in: International Journal of Engineering Science.). The basic research and the development of numerical codes are to be expected to highly benefit from the analytical approaches further developed in this project.

Analytical Approaches and problems to be addressed in the present project The elastic and electric as well as geometrical characteristics of piezoelectric composites are random function of coordinates. The problem of these fields determination in such materials is not even tractable by using the most advanced computer techniques. But in most practically important cases it is not necessary to know these fields in details. Very often it is quite enough to have information about the average response of the microinhomogeneous medium on the external actions. To solve this problem it is only necessary to estimate the mean values of these fields under the deterministic external loading. The solution of this problem would allow us to find the effective (or overall) properties of the composite material and replace the initial inhomogeneous medium by a homogeneous one with known deterministic constitutive law and response on the external actions. The determination of the elastic and electric fields in such inhomogeneous materials is a classical problem of the coupled theory of electroelasticity. The construction of a mathematical model of a homogeneous medium which behaves macroscopically equivalent to the real composite material is the central problems of the mechanics and physics of the media with microstructure, the so-called homogenization problem. The solution of the homogenization problem allows not only to establish the macroscopic response of the composite material on the external loading but also to connect its overall characteristics with the details of the microstructure (the components properties, sizes and shapes of inclusions and their spatial distribution in the matrix). In the case of a stochastic microstructure as in composite materials, the homogenization problem can be solved only approximately. The main difficulties are how to account the many-particle-interactions between the randomly placed inclusions. There is a group of methods in theoretical physics known as self-consistent methods introduced by Kohn and Sham to model Multi-electron systems (density functional theory), which allows solving the many-particle problem approximately. As a rule self-consistent methods reduce the many-particle problem to an effective one-particle problem and thus provide the opportunity for the effective solution of the homogenization problem. The idea to apply self-consistent methods to problems in Mechanics of Materials was due to Kanaun and Levin (the latter being involved in

this project as cooperation partner). Most elementary dynamical problems of reralistic material systems are considered not being analytically accessible due to these considerable mathematical difficulties. In view of this situation, the main goal of this project is it to contribute to close the existing gap in dynamical modelling and to provide appropriate analyticalmathematical approaches. Especially the following problems were identified to be addressed in the present project to make the dynamical modelling of the dynamic characteristics of piezo-composites more tractable:


Solution

of

the

one-particle

diffraction

problem

for

the

coupled

electroelasticity (patically done in the DFG project);
Development of mathematical approaches for the dynamic variant of Eshelby’s inclusion problem in both the frequency and time domain (application of closed-form results achieved in the DFG project).
Application of this solution in the self-consistent scheme in the Kanaun-Levin formulation which was already successfully applied to problems in statics (EFM, first of all in the quasicrystaline approximation) for description of the effective dynamic properties for different inclusion species for statistically isotropic inclusion distributions;
Construction and analysis of the dynamic characteristics of the composite material such as dispersion relation for the electroacoustic waves, attenuation factor, mean field wave speed, total scattering cross section, and effective electroelastic moduli (i.e. their frequency dependence).
Addressing corresponding causal time domain problems (a statistical ensemble of inclusions is subjected to an incoming wave package)
Extension of the EFM beyond the quasi-crystalline approximation to tackle statistically inhomogeneous inclusion distributions

Due to the extensive work on Eshelby inclusion problems and associated Dynamic Potentials in two and three dimensions for purely elastic media and for quasiplane piezoelectric materials, above tasks could not be resolved in the DFG project in the given project time of three years. A further open question so far remains to extend the EFM to cases beyond the long-wave approximation and the extension of the EFM beyond the quasicrystalline approximation. However the derived approaches were

crucial cornerstones necessary to be resolved in the present project, and crucial for proceeding towards the determination of the effective characteristics by using the EFM. However, due to the considerbale experience gained in the DFG project which is reflected last but not least by all in all 13 research papers including 9 journal papers among them such renowned ones as the Proceedings of the Royal Society [2,3] and Quarterly Journal of Mechanics and Applied Mathematics [5]. As the mathematical approaches developed in the DFG project are completely general, it is to be expected that the present project highly benefits from them as well as other scientists will adopt them to their fields. The dynamical Eshelby inclusion problem is a problem occurring in any physical and material sciences in different contexts, where the approaches developed in the DFG project will be directly applicable. It will be also a central goal in the present project to develop elegant mathematical-analytical approaches applicable in a wider range of physical and engineering problems.

Interdiciplinary Approaches The mathematical approach developed in [2] which focused on piezoelectric materials is being extended to magnetic materials by other scientists (Chen P., and Shen Y., “Dynamic Potentials and Green’s Functions of a quasi-plane magnetoelectro-elastic medium with inclusion”, to appear in: International Journal of Engineering Science) As the mathematical approaches developed in the project represent key solutions of general problems of Mathematical Physics and the Mechanics of Materials, it is to be expected that they will be widely utilized and further developed. Related problems are to be continuously identified and if useful included into the present project. In this sence the direction of impact of the present project is supposed to be far beyond only the modelling of the dynamics of piezoelectric materials.

Own publications relevant to the above suggested project  Journal Papers 1) Levin, V. M., Michelitsch, T.M. and Gao, H., 2002, Propagation of Electroacoustic Waves in the Transversely Isotropic Piezoelectric Medium Reinforced by Randomly Distributed Cylindrical Inhomogeneities. International Journal of Solids and structures, 39, 5013-5051.

2) Michelitsch, T.M., Levin, V.M. and Gao, H., 2002. Dynamic Potentials and Green's Functions of a Quasiplane Piezoelectric Medium with Inclusion. Proceedings of the Royal Society of London A, 458, 2393-2415.

3) Michelitsch, T.M., Gao, H., Levin, V.M., 2003. Dynamic Eshelby Tensor and Potentials for Ellipsoidal Inclusions, Proceedings of the Royal Society of London A, 459, 863-890.

4) Wang, J., Michelitsch, T.M., Gao, H, 2003. Dynamic Fiber Inclusions with Ellipsoidal and Arbitrary Cross-Sections and Related Retarded Potentials in a QuasiPlane Piezoelectric Medium. (Special Volume, M. Kachanov guest Ed.) International Journal of Solids and Structures, 40, 6307-6333.

5) Michelitsch, T.M., Gao, H. Levin, V.M., 2003, On the Dynamic Potentials of Ellipsoidal Shells. Quarterly Journal of Mechanics and Applied Mathematics, 56 (4), 629-648.

6) Michelitsch, T. M., Gao, H., 2003, Dynamic Eshelby Inclusion Problem of a Quasiplane Transversely Isotropic Piezoelectric Medium, Chinese Journal of Mechanics-Series A, 19 (1), 113-118.

7) Wang, J., Michelitsch, T.M., Gao, H., Levin, V.M., 2005, On the solution of the dynamic Eshelby problem for inclusions with ellipsoidal and arbitrary shapes. International Journal of Solids and Structures, 42, 353-363.

8) Michelitsch, T.M., Wang, J., Gao, H., Levin, V.M., 2005, On the retarded potentials of inhomogeneous ellipsoids and sources of arbitrary shapes in the threedimensional infinite space. International Journal of Solids and Structures, 42, 51-67.

9) Wang, J. and Gao, H., 2005, A simplified formula of Laplace inversion based on wavelet theory. Communications in Numerical methods in Engineering, 21, 527-530.

 Conference papers Refereed:

10) Levin, V. M., Michelitsch, T.M. and Gao, H., 2002. Modeling of the Effective Dynamic Characteristics of Fiber Reinforced Transversely Isotropic Piezoelectric Materials. Proceedings of the SPIE, 4699, 103-113.

11) Michelitsch, T. M., Wang, J., Gao, H. and Levin, V.M., 2004. On the Solution of the Inhomogeneous Helmholtz Wave Equation for Ellipsoidal Sources, in: Continuum Models and Discrete Systems, eds. D. Bergman et al., Kluwer Academic Publishers, The Netherlands, 115-122.

Non-refereed:

12) Michelitsch, T. M., Gao, H. Levin, V.M., 2003. Solution of the inhomogeneous Helmholtz equation for an ellipsoidal source region. International Conference on the Mechanical Behaviour of Materials (ICM9), Geneva May 25-29, 2003.

13) Wang, J. , Michelitsch, T.M. Gao., 2003. Numerical Solution of the Dynamic Eshelby Problem for Inclusions with Arbitrary Shapes. International Conference on the Mechanical Behaviour of Materials (ICM9), Geneva May 25-29, 2003.

Further new research interests and future research vision: Lattice Dynamics and statistical mechanics in self-similar (fractal) gaskets. A new field which one might call ``fractal analysis'' or ``physics on self similar material systems'' has been developing. This new field requires a new kind of mathematics which has been emerging recently based on a pioneering work of Kigami [J. Kigami, Japan J. Appl. Math. 8 (1989), 259-290.}]. Basic equations of physics need to be defined and solved on a fractal gasket (such as for instance the Laplace equation).

Beside the above introduced research program I have the goal to devote future research efforts in this new challenging direction, i.e. investigating aspects of selfsimilarity and its influence on dynamical characteristics such as vibration spectra of self-similar lattices. Once having defined these quantities on a fractal, concepts of statistical physics yield physical observables including the partition function with the complete statistical information about the fractal system. My experience in lattics dynamics and group theory which I collected during my Diploma thesis is useful to tackle this class of problems.

An interesting question I intend to investigate is the following: Is there a Noether's theorem for self-similarity, i.e. does the symmetry of self-similarity implicate a new so far unknown conservation law? If so what quantity is conserved in a physical system with self-similar symmetry?

Recently I started an attempt to deduce the dynamic lattice Green's function on the Sierpinski gasket. I believe there is an enormous interdisciplinary research- and application potential of the fractal approach. My vision is, beside the above described research program, to establish a strong interdiciplinary research group which is devoted to investigate related aspects of the theory of self-similar (fractal) systems.