A rate-dependent incremental variational formulation of ferroelectricity

Abstract

This paper presents a variational-based modeling and computational implementation of the non-linear, rate-dependent response of piezoceramics under electro-mechanical loading. The point of departure is a general internal variable formulation that describes the hysteretic electro-mechanical response of the material as a standard dissipative solid. Consistent with this type of dissipative continua, we develop a variational formulation of the coupled electro-mechanical boundary-value-problem based on incremental potentials for the stresses and the electric displacement. We specify the variational formulation to a model that describes time-dependent, electric polarizations accompanied by remanent strains. It is governed by a dual dissipation function formulated in terms of the internal driving forces. The model reproduces experimentally observed dielectric and butterfly hystereses, which are characteristic for ferroelectric materials. It accounts for the rate-dependency of the hystereses and the macroscopically non-uniform distribution of the polarization in the solid. An important aspect of our treatment is the numerical implementation of the coupled problem. The monolithic discretization of the two-field problem appears, as a consequence of the proposed variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of a spectrum of benchmark problems.

Introduction

Increased demands for high performance control design in combination with recent advances in material science have produced a class of material systems termed smart, intelligent or adaptive. We refer to Smith (2005) for a recent review. Typical examples are piezoelectric materials, magnetostrictive materials, shape-memory alloys, electro-rheological fluids, electrostrictive materials and optical fibers. Among these, piezoelectrics are most widely used due to their fast electro-mechanical coupling capability, relatively low power requirements, and high generative forces. Piezoelectricity is the property of many non-centrosymmetric ceramics, polymers, and other biological systems. The common structure of materials with piezoelectric properties, such as ${\text{BaTiO}}_3$ and PZT, is polycrystalline. Jaffe et al., 1971, Lines and Glass, 1977, Moulson and Herbert, 1990 give details of the material science of piezoceramics. Ferroelectricity is the property of certain dielectrics that exhibit spontaneous electric polarization, i.e. separation of the center of positive and negative electric charges that makes one side of the crystal positively charged and the opposite side negatively charged. The spontaneous polarization can be reversed in direction by applying an appropriate electric field. This orientation process is known as poling and is the prerequisite to get a material with piezoelectric properties. The poling process is typically dissipative and rate-dependent, accompanied by hysteretic effects in cyclic loading programs as visualized in Fig. 1. The goal of this work is the development of a variational-based framework for ferroelectrics that allows the prediction of these effects in simulations of macroscopic technical applications of these materials.

Piezoceramics are exploited in industrial applications as sensors and actuators. Detection of pressure in the form of sound is one of the most common sensor applications that exploit the so-called direct piezoelectric effect. In a piezoelectric microphone, sound waves bend the material creating a changing voltage. They are especially used at high frequencies in ultrasonic transducers for medical imaging. Furthermore, they can be used in acoustic-electrical applications in noise analysis or acoustic emission spectroscopes. In addition, they are well known in mechanic-electrical applications as igniters or accelerometers. The inverse piezoelectric effect is used in actuators, or piezoelectric motors for micro-and nano-positioning, laser tuning, active vibration damping et cetera. Everyday life applications are ink jet printers, where piezoelectric crystals are used to control the flow of ink from the ink jet head to the paper. In automotive engineering piezoceramics have been utilized in a new generation of common-rail piezo inline injectors. This system reduces not just exhaust emission of the diesel engine but also its operating noise and fuel consumption. In comparison with solenoidal valves, piezo injectors can be controlled much more precisely so that they can inject fuel during an engine cycle with much more accuracy. Piezoceramic materials are also particularly suitable for precise micro-and nano-deformation of sensors and actuators. Nowadays, actuators and sensors operate at high stress and electric fields, where they show strong non-linearities due to hystereses and rate effects induced by domain switching processes. There is a strong need for the construction of predictive models, which describes these phenomena.

The theoretical foundations for the analysis of electro-mechanical interactions in solids are summarized, for example, in Truesdell and Toupin, 1960, Landau et al., 1967, Pao, 1978, Hutter et al., 2006, Maugin, 1988, Eringen and Maugin, 1990a, Eringen and Maugin, 1990b and the recent treatments McMeeking and Landis, 2005, McMeeking et al., 2007. The existing approaches may be divided into two main categories: microscopically motivated and purely phenomenological models. A review in this regard is given by Kamlah, 2001, Landis, 2004, Huber, 2005. Microscopically motivated material models are presented by Chen and Lynch, 1998, Huber et al., 1999, Huber and Fleck, 2001. These models are concerned with the constitutive behavior of single crystals and employ energy arguments as switching criteria. Other models describe the poling and domain evolution on the microscale by phase-field approaches, such as Zhang and Bhattacharya, 2005a, Zhang and Bhattacharya, 2005b, Su and Landis, 2007, Schrade et al., 2007. The overall material behavior of a ceramic polycrystal is then obtained by a homogenization-type averaging of an aggregate of oriented crystallites. The goal of reduction of the number of internal variables motivates phenomenological macroscopical models. Bassiouny et al., 1988a, Bassiouny et al., 1988b, Bassiouny and Maugin, 1989a, Bassiouny and Maugin, 1989b outline thermodynamically-consistent formulations of the macroscopic electro-mechanical hysteresis. One of the first applications of these notions for algorithmic model development can be found in Cocks and McMeeking (1999), where a phenomenological constitutive law for piezoceramics is formulated in analogy to incremental plasticity. The models proposed by Kamlah and Tsakmakis, 1999, Kamlah, 2001, Kamlah and Böhle, 2001 assume a one-to-one relationship between the remanent strain and the irreversible polarization. A multiaxial, thermodynamically-consistent description is given by Landis (2002). The use of switching surfaces and associated evolution equations guarantee a positive dissipation during switching. In this paper, as well as in Huber and Fleck (2001), a cross-coupling into the switching function is introduced in order to account for the generation of remanent strain by the application of an electric field and remanent polarization by stress. This model was simplified in McMeeking and Landis (2002), where the remanent strain was assumed to be a function of the remanent polarization. A consistent model in this spirit, including the formulation of constitutive update algorithms and finite element implementation was suggested by Schröder and Romanowski (2005). However, in this model, the polarization direction is assumed to be constant. Instead of splitting the electric displacement in a reversible and irreversible part, Klinkel (2006) introduces an irreversible electric field, which serves, instead of the remanent polarization, as internal variable. The above mentioned models are all rate-independent. However, it is experimentally observed (see Zhou et al., 2001, Viehland and Chen, 2000), that the dissipative ferroelectric response is rate-dependent as illustrated in Fig. 1. Only few rate-dependent models exist in the literature. Belov and Kreher (2005) simulate non-linear hysteretic phenomena in polycrystalline ferroelectric ceramics based on an Arrhenius-type kinetic evolution equation based on the thermal activation theory. Arockiarajan, Delibas, Menzel, and Seemann (2006) proposed a model for rate-dependent switching effects of piezoelectric materials based on an incremental formulation of a linear kinetic theory that uses the reduction of free energy as a criterion for the onset of the domain switching. However, a unified macroscopic formulation of the rate-dependent dissipative response, associated incremental update algorithms and finite element formulations embedded into variational principles are missing in the literature.

The first goal of this paper is the formulation of incremental variational principles, which govern both the local constitutive response as well as the global boundary-value-problem of solids under dissipative, electro-mechanically-coupled actions. These principles turn out to provide the most natural and compact guidance to the theoretical and computational modeling of dissipative electro-mechanics under quasi-static conditions. The variational structure is related to a canonical constitutive modeling approach, often addressed to standard dissipative materials. These are governed by only two scalar constitutive functions: the energy storage and the dissipation functions. Standard dissipative solids are well established in the purely mechanical context, see for example Biot, 1965, Ziegler and Wehrli, 1987, Germain, 1973, Halpen and Nguyen, 1975, Maugin, 1992, Miehe, 2002. The associated field equations may be obtained as the Euler equations of suitably defined rate-type incremental variational principles. We refer to the treatments of plasticity by Ortiz and Stainier, 1999, Miehe, 2002, Miehe et al., 2002, Carstensens et al., 2002. Generalizations of these variational concepts are outlined in Mielke and Timofte (2006) for electro-mechanics and in Miehe, Rosato, and Kiefer (2010) for fully coupled electro-magneto-mechanics. This work extends these concepts to rate-dependent electro-mechanics and investigates details for the variational-based modeling and algorithmic implementation. The point of departure is a general internal variable formulation that determines the hysteretic response of electro-mechanically coupled materials in terms of an energy storage function and a dissipation potential function. For these materials we construct incremental potentials for the stresses and the electric displacement by exploiting local constitutive optimization problems. These appear in the form of minimum principles, when the dissipation function is explicitly modeled in terms of the rate of the internal variables. We obtain local saddle-point problems, when the dissipation potential is defined by a maximum dissipation concept in terms of the internal forces dual to the internal variables, which govern threshold, yield- or switching-functions. The underlying basic approach is the optimization of a path of the internal variables in a finite increment of time that minimizes a generalized incremental work expression, taking into account both the energy stored and dissipated. Using the discrete counterpart of this generalized potential, we are able to update the local, coupled electro-mechanical response within a typical time interval. Next, the solution of the coupled boundary value problem is obtained by a global saddle-point principle, enforcing the stationarity of an incremental energy–enthalpy functional. This functional is based on the incremental potential for the stresses and the electric displacement mentioned above. The numerical solution of this global variational problem is obtained by a finite element method. This is achieved by a straightforward discretization of a global incremental potential. As a consequence of this underlying variational structure, the discretization of the electro-mechanical two-field problem results into a symmetric matrix format for the monolithic problem. This canonical setting allows for the application of algebraic solvers for symmetric matrices, which provide a superior accuracy at large time steps as compared to staggered schemes. Section 2 outlines the variational formulation, first for the local constitutive modeling and then within the global treatment of the multi-field boundary-value-problem. This includes details of the algorithmic implementation within a finite element method.

The proposed variational-based setting provides a guidance to the formulation and numerical solution of boundary-value-problems in dissipative electro-mechanics. In particular, this scheme is employed in this paper for the modeling of electro-mechanical systems incorporating piezoceramics, which show a hysteretic material response. Hence, the second goal of the paper is the construction of a variational-based constitutive model, that describes the rate-dependent, hysteretic response of ferroelectrics depicted in Fig. 1. Here, an important aspect of our treatment is a descriptive, step-wise build-up of the ingredients which enter this model. To this end, we start in Section 3 with the construction of constitutive functions for piezoelectricity. They are obtained on the basis of a minimum number of thought experiments, resulting in simple structures, which describe the most relevant phenomena. Here, the most intuitive approach is based on deformation experiments and results in a convex free enthalpy function, which is subsequently Legendre-transformed to a convex-concave mixed-energy–enthalpy function. The next step concerns the modeling of the dissipative effects. Section 4 makes the mixed energy–enthalpy function dependent on internal variables that describe the macroscopic polarization and remanent strains. Again, we proceed in a descriptive and step-wise build-up of the necessary ingredients based on thought experiments. Furthermore, a dual dissipation function in the space of the internal driving forces is defined in terms of a threshold function. The time-dependence of the polarization process is assumed to be governed by the norm of the internal over-force that exceeds the threshold. Both, the energy–enthalpy function and the dissipation function define the incremental potential. The prosed variational-based modeling of dissipative ferroelectrics is summarized in Box 1. The model reproduces experimentally observed dielectric and butterfly hystereses, which are characteristic for ferroelectric materials. It accounts for the rate-dependency of the hystereses and the macroscopically non-uniform distribution of the polarization in the solid. This is demonstrated in Section 5 by representative numerical examples which show the capacity of the proposed formulation.