معادله انتگرال و تجزیه و تحلیل حساسیت رفتار خزشی برای فیلم های نازک PVDF

This paper presents an analytic study of a linear viscoelasticity constitutive equation involving stress, strain and creep compliance while simultaneously correcting a previously reported investigation [Vinogradov, A.M., Schmidt, V.H., Tuthill, G.F., Bohannan, G.W., (2004). Damping and electromechanical energy losses in the piezoelectric polymer PVDF. Mechanics of Materials 36, 1007–1016]. The constitutive equation is presented as a linear, weakly-singular Volterra integral equation of the second kind in the stress variable. An analytic solution is developed, using the Laplace transform technique, for acquiring the stress history based on a specified creep compliance function and input strain. The time-dependent stress solution is expressed in terms of an infinite series involving the provided strain history. An example is studied involving constant strain input. This example permits an aposteriori error estimate for the stress based on the truncated series. Finally, a novel first-order sensitivity analysis is presented to assist in developing experiments for estimating the parameters associated with the compliance function. Using the proposed first-order sensitivity analysis, it is possible to investigate how the uncertainty associated with these parameters propagate into the stress history.

The study of mechanical creep and relaxation of polyvinylidene fluoride (PVDF) has emerged to be of considerable importance for use in microphones, transducer and sensors. This piezoelectric polymer material is commercially available as thin films ranging in thickness from 10 to 760 μm (Vinogradov et al., 2004). Constitutive equation development for quantifying the behavior of such materials is highly important for understanding the underlying physics of these materials. Vinogradov et al. (2004) have suggested that the classical theory of piezoelectricity has significant limitations in terms for representing the electromechanical properties of PVDF’s. This brief paper revisits the recent work of Vinogradov et al. (2004) in order to correct numerous typographical and graphical errors that permeate in the paper. Additionally, this paper presents a first-order sensitivity analysis that is highly useful to the experimentalist and analyst for (a) designing experiments; and (b) propagating parameter uncertainty onto the stress history.

This brief paper serves to clarify and correct a previous work using a formal derivation; to provide some rigor into error estimation; and to provide insight into the practical application and use of sensitivity analysis. The proposed integral equation method for estimating the sensitivity parameters based on a simple numerical method has been demonstrated and shown to be both accurate and insightful. A consistent numerical method can be imposed without undue complication that solves both the nominal stress history and sensitivity functions. Both the analyst and experimentalist can use Fig. 5a, Fig. 5b, Fig. 5c and Fig. 6 for guiding experimental studies and view times where data acquisition are critical for estimating system parameters. Additionally, in some instances higher-order sensitivity analyses may be required.