Analytical Axisymmetric Coupled Piezo-Elastodynamic Models for Guided-Wave Structural Health Monitoring

1 With guided-wave approaches becoming prevalent for structural health monitoring, the need has arisen for more effective models employing these methods using piezoelectric transducers. This paper describes an analytical formulation to couple transducer dynamics with axisymmetric guided-wave excitation models for isotropic plates. The finite-dimensional piezo-actuator dynamics are modeled using coupled piezoelectricity-elasticity equations, assuming the actuators thickness is small compared to its in-plane dimensions. The surface-bonded actuator is assumed to cause a shear traction on the structural substrate along the actuator’s edge. The amplitude of the exerted shear traction is computed by matching the traction and displacement at the actuator’s edge with that of the structure, taking into consideration the structural contributions from all possible guided-wave modes. The structural guided-wave response is computed using axisymmetric elasticity based model developed earlier with the piezo-actuator’s computed shear traction as the excitation function. The theoretical predictions from this approach are compared with results from corresponding finite element simulations for various frequencies and actuator thicknesses. INTRODUCTION Guided-wave (GW), also known as Lamb-wave, approaches have emerged as a promising solution for structural health monitoring (SHM) in recent years. Structures are excited with high frequency GWs, the difference in their response with respect to a baseline signal are processed, and algorithms can be trained to detect and characterize damage. The main advantages of GW methods over other Christopher T. Dunn, Metis Design Corporation, 10 Canal Park, Cambridge, MA 02141 Ajay Raghavan, Metis Design Corporation, 10 Canal Park, Cambridge, MA 02141 Seth S. Kessler, Metis Design Corporation, 10 Canal Park, Cambridge, MA 02141 methods are their ability to interrogate large structural areas and locate/characterize damage on demand with a sparse network of transducers. Most commonly, piezoelectric transducers (piezos) are used for GW excitation and sensing. A detailed description of GW SHM, its basic principles and a summary of efforts by other researchers have been presented in Raghavan and Cesnik [1]. Reliable models for GW excitation/sensing are important for efficient design of GW SHM systems and algorithms. This is because the interpretation of GW signals can be complicated by their dispersive, multimodal nature. Earlier efforts in this direction for piezoelectric actuators have largely examined reduced structural theories or 2-D plane strain models [2]. Raghavan and Cesnik developed 3-D elasticity models for GW excitation and sensing, however their work assumes uncoupled piezo-structural dynamics [3,4]. This implicitly assumes that the shear traction exerted by the piezo on the structure is independent of frequency. It also implies that at least one calibration experiment/coupled finite element simulation is needed for each piezostructural material combination to estimate this traction. Only some limited mathematical work in the literature has examined coupled piezo-elastodynamic models for GW excitation, which assume plane strain [5]. Thus, the developed models in the open literature are restricted in their applicability for transducer design, either leaving out transducer dynamics or the third dimension. The present paper seeks to bridge this gap by proposing an analytical formulation to couple the finite-dimensional piezoelectric transducer dynamics with earlier axisymmetric GW excitation models for isotropic plates. The piezo-actuator dynamics are modeled using coupled piezoelectricity-elasticity equations, assuming its thickness is small compared to its in-plane dimensions. This is a reasonable assumption for most practical structural health monitoring applications. The piezoactuator is assumed to cause a shear traction on the structural substrate surface along the actuator’s edge. The amplitude of the exerted shear traction is computed by matching the traction and displacement at the piezo’s edge with that of the structure, taking into consideration the structural contributions from all possible GW modes. The structural GW response is computed using axisymmetric models developed earlier with the piezo-actuator’s computed shear traction as the excitation function [5]. The results of this approach are compared with those from corresponding finite element simulations. Finally, the validity of some of the model’s assumptions is examined and the use of these models for designing more effective structural health monitoring systems is discussed. PROBLEM FORMULATION Consider a piezoelectric disk actuator bonded to an infinite plate structure, as shown in Figure 1. The r, θ, z coordinate system is a cylindrical coordinate system with the r axis pointing in the plane of the structure, and the z axis along the out-ofplane direction. The structure is of thickness 2b, and actuator is of thickness Act h and radius a respectively. The actuator is poled in the z direction and is electroded on the z surfaces. The actuator is driven using a voltage VAct(t), and a current iAct(t) is induced. As can be seen, the actuator voltage and current follow the passive sign convention. The actuator and structure is joined at r = a, as will be described in a later section of this paper. Figure 1. Axisymmetric view of a piezoelectric actuator bonded to an infinite structure MODELS The following is a discussion of the closed-form structural and actuator models and the approximations used to couple the two together. The finite element model of the actuator/structure system is also discussed. Structure The structural response of an isotropic plate of thickness 2b with a surface bonded circular actuator of radius a exerting shear traction of magnitude Str τ , can be found in Raghavan and Cesnik [5]. The problem is solved by applying the Hankel transform along the radial axis r to the 3-D elasticity equations and using residue calculus to recover the spatial domain solution. The harmonic surface radial displacement at radius r can be shown to be: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥