Natarajan Shankar

A Fully Coupled Constitutive Model for Electrostrictive Ceramic Materials

A three-dimensional, electromechanical constitutive law has been formulated for electrostrictive ceramic materials. This fully coupled, phenomenological model relates the key state variables of stress, strain, electric field, polarization and temperature in a set of compact nonlinear equations. The direct and converse electrostrictive effects are modeled by assuming that the electri cally induced strain depends on second-order polarization terms. In addition, a simple empirical relationship for the dielectric behavior is used to model the saturation of the induced polarization with increasing electric field. Unlike previous electrostrictive constitutive laws based on polynomial expansions, this consti tutive law depends on a manageable number of material constants. As an example, material constants for the model were determined from induced strain and dielectic data for a relaxor-ferroelectric based on lead magnesium niobate, Pb(Mg1/3Nb2/3)O 3-PbTiO3-BaTiO3 (PMN-PT-BT). Finally, pre dictions of the materials mechanical behavior under constant electric field and its electrical behavior under constant applied stress are made.

A finite element method for electrostrictive ceramic devices

Abstract A nonlinear, static finite element technique is developed and implemented for electrostrictive ceramic solids. This numerical method is based on Toupins elastic dielectric theory and models full electromechanical coupling in the solid via the Maxwell stress and constitutive equations [Toupin, R. A. (1956). The elastic dielectric. J. Rational Mech. Anal.5, 849–915; Toupin, R. A. (1963). A dynamical theory of elastic dielectrics. Int. J. Engng Sci.1, 101–126]. The formulation incorporates the constitutive model of Hom and Shankar [(1994). A fully coupled constitutive model for electrostrictive ceramic materials. J. Intell. Mater. Syst. Struct.5, 795–801]. This model simulates polarization saturation at high electric fields and nonlinear coupling of the mechanical and electric field variables. The finite element technique is demonstrated by solving the problem of a multilayered actuator constructed from a lead-magnesium-niobate electrostrictor. Both the electric field and stress state are computed near the tip of an internal electrode. The results show that the nonlinear dielectric behavior significantly alters the electric field near the tip to form a stress singularity. An analytical solution of the internal electrode problem is presented and compared with the finite element predictions for verification. The comparison shows a good qualitative agreement between the two solutions. Finally, the numerical results are used to examine crack nucleation and growth from the electrode tip.

Milling machine for the 21st century: goals, approach, characterization, and modeling

Ingersolls Octahedral Hexapod--a milling machine for the future--is described. The specific target applications and the performance goals for an enhanced version of the machine are illustrated. The approach to achieving the goals by incorporating of advanced composites and active chatter and vibration control using smart materials is discussed. The machine characterization performed on an existing machine, the FE models developed and the plans to use the characterization and the validated models in designing an enhanced machine are described.

A dynamics model for nonlinear electrostrictive actuators

This paper examines the nonlinear vibration of an electrostrictive ceramic rod actuator excited by a harmonic voltage source. A frequency-domain model was developed using the nonlinear constitutive law for electrostriction. The results predict harmonic distortion of the devices displacement due to the ceramics nonlinear behavior. AC voltage signal and DC voltage bias were studied to determine the optimum power source parameters for minimizing distortion. The calculations show that the rods resonance frequency and amplitude depend on the electromechanical coupling strength and differ greatly for large AC voltages from the equivalent linear piezoelectric results. The nonlinear analysis relates the devices electromechanical coupling coefficient to the computed resonance and antiresonance frequencies. This important result could provide the basis for future measurement of the electrostrictive coupling coefficient using resonance techniques.

Modeling nonlinearity in electrostrictive sonar transducers

Electrostrictive driver materials with large strain capability hold great promise for the advancement of sonar projector technology. However, the nonlinear induced strain of these materials can create acoustic distortion in transducers through higher-order harmonics. Electrostrictors also possess complicated prestress and temperature dependencies, and an elastic modulus that depends strongly on electric field. This investigation examined these issues with a nonlinear, frequency domain model for a flextensional transducer powered by an electrostrictive stacked actuator. A simple, linear lumped-parameter model of a flextensional shell and its surrounding acoustic medium were combined with a nonlinear model of the electrostrictive driver. This model accounted for the material’s nonlinear dependencies and behavior. Predictions of the device’s acoustic/electric response during operation compared favorably with experiments performed on a single element flextensional transducer. The model’s results showed that p...

Active chatter control in a milling machine

The use of active feedback compensation to mitigate cutting instabilities in an advanced milling machine is discussed in this paper. A linear structural model delineating dynamics significant to the onset of cutting instabilities was combined with a nonlinear cutting model to form a dynamic depiction of an existing milling machine. The model was validated with experimental data. Modifications made to an existing machine model were used to predict alterations in dynamics due to the integration of active feedback compensation. From simulations, subcomponent requirements were evaluated and cutting enhancements were predicted. Active compensation was shown to enable more than double the metal removal rate over conventional milling machines.

Advanced reconfigurable machine for Flexible Fabrication

In a recently awarded ARPA program to advance the state of the art of parallel actuated next-generation machine tools, a vertically integrated team led by Martin Marietta is applying recent advanced in electroceramic smart materials and advanced composites to achieve leapfrog advanced in precision, flexibility, and speed of machine tools. Specific approached to achieve these advanced include active vibration cancellation, improved control technology, and design optimization using advanced structural and dynamic models. In this program, the team will integrate large high-force actuators, composites, and active vibration control with the Ingersoll Milling Machine Companys innovative Octahedral Hexapod machine to develop to Advanced Reconfiguration Machine for Flexible Fabrication. The enhanced Octahedral Hexapod machine will provide new levels of machining flexibility while still retaining precision and low cost. This technology will have widespread impact on the flexible fabrication of materials--especially those that are tough to machine traditionally--in several industries, e.g., aerospace, defense, aircraft, and automotive.

An acoustic/thermal model for self-heating in PMN sonar projectors

Dielectric hysteresis and a strong material temperature dependence uniquely couple the acoustic output and temperature of a sonar projector driven by electrostrictive Pb(Mg1/3, Nb2/3)O3 (PMN). Both the source level and the source of self-heating, i.e., dielectric hysteresis, dramatically decrease as the PMN driver heats. The final temperature delineates outstanding PMN transducers from mediocre PMN transducers, so accurate acoustic performance prediction requires accurate transducer temperature prediction. This study examined this self-heating phenomenon by combining an electro-acoustics model for a PMN flextensional transducer with a thermal finite element model. The sonar model calculated the source level and heat generation rate for the PMN driver as a function of temperature. This computed source level varied 12 dB over a 75 degrees C temperature range solely due to the temperature dependent ceramic. The heat transfer model used the computed heat rate to predict the transducers transient thermal response. The results clearly demonstrate that the transducer reached a steady-state equilibrium temperature, where the heat generated by the PMN driver balanced the heat dissipated. While the transducer model predicted a significant temperature rise, the corresponding acoustic output still surpassed the output of an equivalent Pb(Zr,Ti)O3 (PZT) transducer by 8 dB. Good agreement with experiments made on a PMN flextensional transducer validated the model.

Constitutive and failure models for relaxor ferroelectric ceramics

A non-linear constitutive model for relaxor ferroelectrics developed by Hom and Shankar is examined and verified with electromechanical experiments. This model links polarization and strain to the electric field and stress in an electrostrictive material. A set of tests were performed to study the quasi-static electrical behavior of PMN-PT-BT materials under prestress. Another set of tests investigate the effect of DC electric field on the elastic modulus of the material. The results show excellent correlation between the predicted behavior of the model and the experiments. Failure models for electrostrictive ceramic materials are presented which address the issues of actuator reliability. The constitutive model of Hom and Shankar is incorporated into a nonlinear finite element code. A new finite element technique for computing the J-Integral for cracks in electromechanical materials is developed. This technique is based on the domain integral method and computes both the mechanical and electrical contributions to the energy release rate. The finite element code and the J-Integral computation are used to study crack growth in multilayered electrostrictive ceramic actuators.

Finite element modeling of multilayered electrostrictive actuators

Nonlinear, quasi-static finite element calculations are performed for multilayered, electrostrictive, ceramic actuators. Both a stand-alone device and an array of devices embedded in a 1 - 3 composite are studied. The numerical model is based on a fully coupled constitutive law for electrostriction which uses strain and polarization as independent state variables. This law accounts for the stress dependency of ceramics dielectric behavior and simulates polarization saturation at high electric fields. Two-dimensional plane strain computations are done for a single actuator constructed from Pb(Mg1/3Nb2/3)O3- PbTiO3-BaTiO3 (PMN-PT-BT). The stress state near an internal electrode tip is computed and a fracture mechanics analysis is performed to assess the devices reliability. The effect of compressive prestress on the actuators induced strain response is also predicted. In a second problem, a 1 - 3 composite embedded with an array of PMN-PT-BT multilayered actuators is studied with a plane stress version of the finite element technique. A unit cell model is used to compute the surface displacements of the composite.