We study the behavior of ferroelectric material (BaTiO3) for the design of a nano-generator to convert mechanical
into electrical energy. The investigations consider an electro-mechanical phase-field model with polarization as
state variable. This widely accepted model has its origins in the work of1-3 and is fully developed by Landis and
coworkers.4,5 We use a finite element model to simulate tetragonal regions of ferroelectric material sputtered on
substrate. Different geometries as well as various mechanical and electrical boundary conditions are considered.
The model parameters are normalized to achieve better computational conditions within the stiffness matrix.
The major objective of this contribution is the fundamental understanding of domain switching caused by a
cyclic electrical field. The corresponding hysteresis loops of the overall polarization cannot be achieved by using
a two-dimensional model because the domain topologies evolve in three dimensions. The three-dimensional
nature of the domain structure evolution is even true for flat regions or thin films.6 We show some examples of
three-dimensional domain topologies, which are able to break energetically unfavorable symmetries. Finally, the
computational model of a tetragonal nano-generator with dimensions 10 x 60 x 10 nm is presented. The specific
ratio of height to width and the mounting on substrate is essential for its performance and principle of energy
harvesting. We discuss the challenges and scopes of such a system.
A continuum thermodynamics framework is presented to model the evolution of domain structures in active/smart
materials. To investigate the consequences of the theories, fundamental defect interactions are studied. A principle of
virtual work is specified for the theory and is implemented to devise a finite element formulation. For ferroelectrics, the
theory and numerical methods are used to investigate the interactions of 180° and 90° domain walls with arrays of
charged defects and dislocations to determine how strongly domain walls are electromechanically pinned by the arrays of
defects. Additionally, the problems of nucleation and growth of domains from crack tips, and the propagation of domain
needles are studied. The importance of adaptive mesh refinement and coarsening is discussed in the context of this
modeling approach.
Domain switching near a stationary crack tip in a single crystal of ferroelectric material is investigated. The phase-field
approach applying the material polarization as the order parameter is used as the theoretical modeling framework, and
the finite element method is used for the numerical solution technique. The electromechanical form of the J-integral is
appropriately modified to account for the polarization gradient energy terms, and analyzed to illustrate the amount of
shielding, or lack thereof, due to domain switching at the crack tip.
A continuum thermodynamics formulation for micromagnetics coupled with mechanics is devised to model the
evolution of magnetic domain and martensite twin structures in ferromagnetic shape memory alloys. The theory falls
into the class of phase-field or diffuse-interface modeling approaches. In addition to the standard mechanical and
magnetic balance laws, a two sets of micro-forces their associated balance laws are postulated, one set for the
magnetization order parameter and one set for the martensite order parameter. The second law of thermodynamics is
analyzed to identify the appropriate material constitutive relationships. The general formulation does not constrain the
magnitude of the magnetization to be constant, allowing for the possibilities of spontaneous magnetization changes
associated with strain and temperature. The equations governing the evolution of the magnetization are shown to reduce
to the commonly accepted Landau-Lifshitz-Gilbert equations when the magnetization magnitude is constant. Numerical
solutions to the governing equations are presented to investigate the fundamental interactions between the magnetic
domain wall and the martensite twin boundary in ferromagnetic shape memory alloys. Calculations are performed to
determine under what conditions the magnetic domain wall and the martensite twin boundary can be dissociated,
resulting in a limit to the actuating strength of the material.
A continuum thermodynamics formulation for micromagnetics coupled with mechanics is devised to model the
evolution of magnetic domain structures in magnetostrictive materials. The theory falls into the class of phase-field or
diffuse-interface modeling approaches. In addition to the standard mechanical and magnetic balance laws, a set of
micro-forces their associated balance laws are postulated. Thereafter, the second law of thermodynamics is analyzed to
identify the appropriate material constitutive relationships. The general formulation does not constrain the magnitude of
the magnetization to be constant, allowing for the possibilities of spontaneous magnetization changes associated with
strain and temperature. The approach is shown to yield the commonly accepted Landau-Lifshitz-Gilbert equations for
the evolution of the magnetization when the magnetization magnitude is constant. Within the theory a form for the free
energy is postulated that can be applied to fit the general elastic and magnetic properties of a ferromagnetic shape
memory material near its spontaneously magnetized state.
A continuum thermodynamics framework for the diffuse interface or "phase field" approach to domain evolution is presented. The theory distinguishes the fundamental balance laws which are universal (i.e. mechanical equilibrium, Gauss' law, and a new micro-force balance) from the constitutive laws, which describe the behavior of a specific material. A finite element formulation based on a virtual work statement is implemented using mechanical displacements, electrical potential, and polarization components (the vector "order parameter") as nodal degrees of freedom. The finite element code is then applied to simulate the growth of a 180° domain needle through a parent domain.
The field-induced domain evolution is investigated in a single-domain ferroelectric solid undergoing spontaneous polarization and polarization reorientation. Domain wall velocities have been correlated with the driving force acting on the domain interface. This boundary driving force, which depends on the local electromechanical fields and local changes of the material properties due to reorientation of the crystal structure, has been associated with Eshelby's energy momentum tensor. Finite element analysis that implements an efficient computation method for Eshelby's energy momentum tensor is developed. The domain boundary's geometry is elastically updated during the finite element computation to capture the characteristic of the domain evolution. An analytical solution for the domain interface driving force will be given by implementing Eshelby's solution for piezoelectric inclusion. The numerical results are compared to the analytical solutions to conclude the validity of this approach. Finally, the finite element computation program is used in two domain switch simulations.
Mode I steady crack growth is analyzed to determine the toughening due to domain switching in poled ferroelectric ceramics. A multi-axial, electromechanically coupled, incremental constitutive theory is applied to model the material behavior of the ferroelectric ceramic. The constitutive law is then implemented within the finite element method to study steady crack growth. The effects of poling direction, either out of plane or in plane, and poling magnitude on the fracture toughness are investigated. Results for the predicted fracture toughness, remanent strain and remanent polarization distributions, and domain switching zone shapes and sizes are presented. Finally, the model predictions are discussed in comparison to experimental observations.
Nonlinear fracture mechanics concepts for ferroelastic materials are presented. A phenomenological constitutive law for ferroelastic domain switching is implemented within a steady state finite element formulation to determine the stress and strain fields near growing cracks in ferroelastic materials. Hutchinson's I-integral is applied to determine the relationship between the far field applied energy release rate and the local crack tip energy release rate. Computations are performed on both unpoled and "mechanically poled" materials with and without T-stress to quantitatively determine the toughening due to domain switching in these situations. Results are discussed in comparison to "transformations toughening" type analytical models.
Accurate phenomenological constitutive laws for ferroelectric ceramics that can be rapidly integrated are required for finite element models aiming to resolve the complex fields in ferroelectric devices. At best, phenomenological theories can provide a framework within which thermodynamic restrictions are satisfied and undetermined functions exist for "fitting" material behavior. The ultimate challenge in deriving a final constitutive law is to capture the physics of the material deformation and polarization processes within these undetermined functions. A number of micro-electromechanical models exist in the literature. These models are bases on domain switching events at the domain/grain level and then are generalized to polycrystalline behavior by averaging over many different domain/grain orientations. In this work it will be shown how information obtained from these micro-electromechanical models can be incorporated directly into the undetermined functions of a phenomenological theory.
A new finite element formulation for the solution of electromechanical boundary value problems is presented. As opposed to the standard formulation that utilizes a scalar electric potential as nodal variables, this new formulation implements a vector potential from which components of electric displacement are derived. For linear piezoelectric materials with positive definite material moduli, the resulting finite element stiffness matrix from the vector potential formulation is also positive definite. If the material is nonlinear in a fashion characteristic of ferroelectric materials, it is demonstrated that a straightforward iterative solution procedure is unstable for the standard scalar potential formulation, but stable for the new vector potential formulation. Finally, the method is used to compute fields around a crack tip in an idealized non-linear ferroelectric material, and results are compared to an analytical solution.
A class of fully coupled, symmetric, multi-axial, ferroelectric constitutive laws is presented. The foundation of the theory is an assumed form for the Helmholtz free energy of the material. Yield surfaces and associated flow rules are postulated in a modified stress and electric field space such that a positive dissipation rate during switching is guaranteed. The resulting tangent moduli relating increments of stress and electric field to increments of strain and electric displacement are symmetric since changes in the linear elastic, dielectric and piezoelectric properties of the material are included in the switching surface and flow law. Symmetry is further investigated with a simple one-dimensional loading situation comparing the thermodynamically consistent framework to a more ad hoc theory.
The energy release rates for dielectric, piezoelectric and ferroelectric strips are analyzed. Energy minimization is used to determine the electromechanical fields in the strip. Once these fields are computed conservation of energy is used to determine the energy release rate. Three different sets of assumptions are used to treat the void space and solid material. First, an impermeable void space assumption is analyzed with small deformation assumed to be valid in the solid. Next, the void space is assumed to have finite dielectric permittivity and aside form changes in the position of its boundary the solid is treated with small deformation theory. Finally, a large deformation formulation is used for the solid along with a permeable void space. By minimizing the energy of the system we are able to show that mechanical tractions act on the crack surfaces as a result of the finite dielectric permittivity of free space. Lastly, a piezoelectric strip with remanent polarization is analyzed and it is shown that the energy release rate for a poled piezoelectric is not equal to that of a material with identical elastic, dielectric and piezoelectric properties but no remanent polarization.
In this paper a self-consistent model for the switching of a ferroelectric polycrystal will be presented. We will assume that the polycrystal is made up of spherical grains or crystallites. Each crystallite is made up of two domain types separated by 180 degree domain walls. The linear dielectric properties of the domains are assumed to be isotropic. When the electric field in the crystallite reaches a critical level the domain walls are allowed to sweep through the crystallite. This gradual domain wall motion causes the tangential dielectric permittivity of the crystallite to be transversely isotropic. Each spherical crystallite is given a distinct orientation and embedded in a matrix material. The tangential dielectric permittivity of the matrix is taken to be consistent with the overall averaged incremental response of the crystallites taking into account the constraint or depolarization fields imposed by the matrix. Hysteresis loops and switching surfaces are presented for the polycrystal.
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