On Modeling Of Constrained Piezoelectric Thin Films For Structural Health Monitoring
The behaviour of a free-standing thin film differs from that of a film surface-bonded or embedded due to the boundary constraints. A general dearth of analytical models, in regard to prediction of the operational competence of a constrained Piezoelectric thin film, prevails. In conventional design of miniaturized thin film devices, several non classical effects, for instance the effect of boundary constraints, are not considered. To warrant the design and performance optimisation of thin film sensors, such effect must be taken into account in a forethoughtful manner. This thesis is an attempt to achieve such optimisation through modeling of thin films. The coupled problem of a film on a substrate is solved semi-analytically in theoretical cases; and by finite element analysis in realistic cases for damage identification in the host structure. We first propose a two-dimensional analytical model of a constrained Piezoelectric thin film embedded in a host. Analytical expressions of capacitance and voltage across the electrodes are obtained by assuming first order shear deformation across the film thickness. The bonding layer between the film and the substrate, which is assumed to be an equivalent single layer including electrodes, insulation layer, adhesive layer etc., is modeled by taking into account its viscoelastic property. Residual stress is incorporated in the constitutive model through equivalent residual strain. Simulations on 10 m thick PVDF and 100 mPZT films are conducted. They illustrate the dependence of voltage response and capacitance on the applied stress, as well as on the residual stress. A maximum percentage variation in capacitance, as compared to the conventional estimate, is about 2% in a PVDF film and +75% to-65% in a PZT film for various combinations of tensile stresses applied at the ends of the film. Effect of residual stress is also exemplified via comparative response of a 1 m PZT film deposited on Pt/Ti/Si(0 0 1), with and without residual stress. For this case, an almost +50% increase in the voltage and an equivalent drop in the capacitance is observed. Next, we look into the voltage response profile of this model by employing it as a sensor to identify a finite mode I and mode II sub-surface cracks in a finite size host. To model the embedded crack, additional perturbation functions in the displacement field due to linear elastic crack tips in an infinite solid under plane strain condition are introduced to accommodate the stress free conditions at its surfaces. The film model requires the interfacial displacement and traction conditions, which are obtained from the analysis of the host. The combined analysis of the film and crack models brings forth the voltage gradient along the film span as a direct indicator of the location of crack in the axial direction, whereas the voltage magnitude represents the size of the crack. Following this analysis, a quasi three-dimensional(3-D) model of a Piezoelectric thin film surface-bonded to the host structure is proposed. With due consideration of restriction on the thickness of the film, here the model is based on a reduced 3-D continuum mechanics approach. The displacement field in the film is assumed to vary according to the third-order shear deformation theory; and the electrical and mechanical boundary conditions on the surfaces of the film are accommodated in a consistent manner. The formulation yields a governing inhomogeneous system of second-order Partial Differential Equations(PDEs), which is dependent on the displacement field at the film-host interface through force terms. Semi-analytical expressions of potential difference and capacitance are obtained. This system is solved numerically for two unknown rotations about X and Y axes of the film by finite element method. A maximum variation of about 2.5% is obtained in the capacitance of a 10 m PVDF film, as compared to its conventional estimate. The operational performance of this model is assessed in terms of its voltage response over the film area for various displacement fields. Conformation of this response to the input displacement field attests to its mathematical integrity. Next, we ascertain the versatility of this model in its role as a sensor for Structural Health Monitoring. To deal with cracks in the host plate, finite size rectangular surfaces are introduced as crack faces. The film domain and the host domain are discretized with an a posteriori h-refinement strategy and compatible interfacial nodes at the film-host interface via finite element interpolation. The resulting coupled problem is solved by static finite element analysis. The nature of the voltage pattern over the film surface is peculiar to the mode of crack, and is a qualitative portrayal of its presence. To correlate the electric potential(voltage) –a distributed parameter – to the geometry and orientation of the crack, as well as to quantify it, electrostatic measures in terms of integrated potential difference and its spatial gradients on the film surface are proffered. The numerical implications of these measures are elicited through simulation results of various crack sizes in damaged and healthy hosts under identical conditions of stress and boundary. The pattern of these measures in a damaged host becomes oscillatory as compared to straight lines observed in a healthy host. Furthermore, the reduced 3-D model is extended to perform dynamic analysis with the inclusion of inertial terms in the governing equilibrium equations. Subsequently, the acceleration terms appear in the governing inhomogeneous system of PDEs in the force terms. Finite element analyses of this extended film model on an isotropic beam with surface and sub-surface cracks, and on a composite plate with delamination, are then performed in the time domain. In all cases, an excellent conformation of the voltage profile at any point in the film domain to the velocity profile at the corresponding point in the film-host interface is observed. Again, to quantify the extent of damage in the host, we proffer electrical measures based on the Lpnorm, of second order, of the voltage and its directional derivatives. We exemplify the numerical implications of these measures in the time domain through sensitivity analysis in regard to the defected areas, and their region of occurrence relative to the film sensor. The response of the film model educes that the relatively flat curves after the first incident pulse in a healthy structure shoots off to a monotonic pattern in damaged hosts. The measures depict high degree of sensitivity in regard to the variation in the area of damage of any nature. An apposition of the static and dynamic analyses is elaborated towards the end of this dissertation. It proves to be very insightful in the damage assessment of the host structure, for it shows the utility of the dynamic model to sense the location of the damage occurrence, whereas a more in-depth assessment on its nature and mode of the crack would demand a static analysis in its proximal regions. To sum up, in light of these models and the proposed measures, this thesis establishes salient justifications pertaining to their pragmatic significance. We believe that these results represent an important contribution towards the ongoing research on understanding the role of boundary constraints in mechanically thin Piezoelectric films.