Broadband Electromechanical Spectroscopy: characterizing the dynamic mechanical response of viscoelastic materials under temperature and electric field control in a vacuum environment


The viscoelasticity of a variety of active materials is controllable, e.g., by the application of electric or thermal fields. However, their viscoelastic behavior cannot be fully explored by current methods due to limitations in their control of mechanical, electrical, and thermal fields simultaneously. To close this gap, we introduce Broadband Electromechanical Spectroscopy (BES). For the specific apparatus developed, specimens are subjected to bending and torsional moments with frequencies up to 4 kHz and amplitudes up to 10−4 Nm (the method is sufficiently general to allow for higher and wider frequency ranges). Deflection/twist is measured and moments are applied in a contactless fashion to minimize the influence of the apparatus compliance and of spurious damping. Electric fields are applied to specimens via surface electrodes at frequencies up to 10 Hz and amplitudes up to 5 MV/m. Experiments are performed under vacuum to remove noise from the surrounding air. Using BES, the dynamic stiffness and damping in bending and torsion of a ferroelectric ceramic, lead zirconate titanate, were measured at room temperature, while applying large, cyclic electric fields to induce domain switching. Results reveal large increases of the specimen’s damping capacity and softening of the modulus during domain switching. The effect occurs over wide ranges of mechanical frequencies and permits lowering of the resonance frequencies. This promises potential for using ferroelectrics for active vibration control beyond linear piezoelectricity. More generally, BES helps improve current understanding of microstructure kinetics (such as during domain switching) and how it relates to the macroscopic viscoelastic response of materials.

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The authors gratefully acknowledge financial support from United Technologies Research Center as well as from the Caltech Innovation Initiative (CI2).

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Solution of the dynamic Euler–Bernoulli beam

For bending experiments, the beam deflection \(w(y,t)\) can be approximated by a dynamic Euler–Bernoulli beam with the governing equation

$$\begin{aligned} EI_z\frac{\partial ^{4}w}{\partial y^{4}}(y,t)=-\rho A \ddot{w}(y,t), \end{aligned}$$

with boundary conditions

$$\begin{aligned}&w(0,t)=0, \quad \frac{\partial w}{\partial y}(0,t)=0, \quad EI_z\frac{\partial ^3w}{\partial y^3}(L,t)=m\,\ddot{w}(L,t), \nonumber \\&\quad EI_z\frac{\partial ^2w}{\partial y^2}(L,t)=M_z(t), \end{aligned}$$

where \(E\) is Young’s modulus, \(I_z\) is the bending moment of inertia, \(\rho \) is the mass density, \(A\) denotes the cross-sectional area, \(m\) represents the end mass due to the clamped magnet, and \(M_z(t)\) is the applied moment. Assuming harmonic motion of the beam at steady state (i.e., \(w(y,t)=\hat{w}(y)e^{i\omega t}\) and \(M_z(t)=\hat{M}_ze^{i\omega t}\)) with frequency \(\omega \), Eqs. (33) and (34) become

$$\begin{aligned} EI_z\frac{\partial ^{4}\hat{w}}{\partial y^{4}}(y)=\rho \omega ^2 A \hat{w}(y) \end{aligned}$$


$$\begin{aligned}&\hat{w}(0)=0, \quad \frac{\partial \hat{w}}{\partial y}(0)=0, \quad EI_z\frac{\partial ^3\hat{w}}{\partial y^3}(L)=-m\,\omega ^2\hat{w}(L), \nonumber \\&\quad EI_z\frac{\partial ^2\hat{w}}{\partial y^2}(L)=\hat{M}_z, \end{aligned}$$

respectively. The solution of (35) with boundary conditions (36) yields the amplitude of the beam deflection as

$$\begin{aligned} \hat{w}(y)&= \frac{\hat{M}_z}{2EI_z\lambda ^2}[\cosh {(\xi (1-y'))}+r\xi \sinh {(\xi (1-y'))}\nonumber \\&\quad +\cosh {(y'\xi )}(\cos \xi -r\xi \sin \xi ) \nonumber \\&\quad +\sinh {(y'\xi )}(\sin \xi +r\xi \cos \xi ) \nonumber \\&\quad -\cos {(y'\xi )}(\cos \xi +\cosh \xi -r\xi (\sin \xi -\sinh \xi )) \nonumber \\&\quad -\sin {(y'\xi )}(\sin \xi -\sinh \xi +r\xi (\cos \xi -\cosh \xi ))] \nonumber \\&\quad /\left[ 1+\cos {\xi }\cosh {\xi }+r\xi \left( \cos {\xi }\sinh {\xi }-\sin {\xi }\cosh {\xi }\right) \right] , \end{aligned}$$

where \(\lambda ^4 = \rho A\omega ^2/(EI_z)\), \(y'=y/L\), \(\xi =\lambda L\), and \(r=m/(\rho A L)\). Using this result, the angle at the end of the beam \(\hat{\theta }_z=(\partial \hat{w}/\partial y)(L)\) is,

$$\begin{aligned} \hat{\theta }_z=\frac{\hat{M}_z\left[ \cosh \xi \left( r\xi \cos \xi +\sin \xi \right) +\cos \xi \sinh \xi -r\xi \right] }{EI_z\lambda \left[ \cosh \xi \left( \cos \xi -r\xi \sin \xi \right) +r\xi \cos \xi \sinh \xi +1\right] }. \end{aligned}$$

Solution of the dynamic torsion of a bar

The derivation for the solution of the dynamic torsion of a bar with an attached end mass with rotational inertia \(I_m\) is given in [34] and is repeated here for convenience. The governing equation for the twisting angle \(\alpha (y,t)\) along the bar is

$$\begin{aligned} G\frac{\partial ^2\alpha }{\partial y^2}(y,t)=-\rho \ddot{\alpha }(y,t), \end{aligned}$$

with boundary conditions,

$$\begin{aligned} \alpha (0,t)=0, \quad GJ_y\frac{\partial \alpha }{\partial y}(L,t)=M_y(t)-I_m\ddot{\alpha }(L,t). \end{aligned}$$

Assuming harmonic motion at steady state (\(\alpha (y,t)=\hat{\alpha }(y)e^{i\omega t}\) and \(M_y(t)=\hat{M}_ye^{i\omega t}\)), the governing equation and boundary conditions become

$$\begin{aligned} G\frac{\partial ^2\hat{\alpha }}{\partial y^2}(y)=\rho \omega ^2\hat{\alpha }(y), \end{aligned}$$


$$\begin{aligned} \hat{\alpha }(0)=0, \quad GJ_y\frac{\partial \hat{\alpha }}{\partial y}(L)=\hat{M}_y+I_m\omega ^2\hat{\alpha }(L), \end{aligned}$$

respectively. Solving Eq. (41) with boundary conditions (42) gives the twisting angle

$$\begin{aligned} \hat{\alpha }(y)=\frac{\hat{M}_y\sin (\Lambda y)}{GJ_y\Lambda \cos (\Lambda L)+I_m\omega ^2\sin (\Lambda L)}. \end{aligned}$$

Finally, evaluating (43) at the free end yields the twisting angle \(\hat{\theta }_y=\hat{\alpha }(L)\), specifically

$$\begin{aligned} \hat{\theta }_y=\frac{\hat{M}_y}{GJ_y\Lambda \left[ -(I_m/J_y)\Lambda /\rho +\cot (\Lambda L)\right] }, \end{aligned}$$

where \(\Lambda =\omega /\sqrt{G/\rho }\).

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le Graverend, JB., Wojnar, C.S. & Kochmann, D.M. Broadband Electromechanical Spectroscopy: characterizing the dynamic mechanical response of viscoelastic materials under temperature and electric field control in a vacuum environment. J Mater Sci 50, 3656–3685 (2015).

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  • Loss Tangent
  • Dynamic Mechanical Analysis
  • Coercive Field
  • Electric Displacement
  • Dynamic Stiffness