Polyvinylidene fluoride (PVDF) is tested using single edge notch tensile (SENT) specimens under mode I and plane stress conditions. The novel aspect of the testing is the use of electrodes attached close to the crack tip which measured an electric potential. The electric potential was analyzed with a flexoelectric model that related the electric potential with the mechanical dilatation due to the stresses close to the crack tip.
In our previous study (Part I), the anti-plane steady state hyperbolic mode III fracture of a magneto-flexoelectric material was solved for the displacement, the polarization and the magnetic fields. The solution, however, was based on the assumption of the development of strain discontinuities, and the propagation of the crack-tip was related to a critical shear strain. However, in the current study, the asymptotic details of the fields close to the crack-tip were investigated. The asymptotic analysis assumes strain continuity at the crack-tip (discontinuity in the strain gradients), and reveals the existence of a positive dynamic J-integral. The asymptotic analysis was performed not only for hyperbolic but also for elliptic conditions, and the energy release rate was calculated as a function of the crack-tip velocity in both regimes. These results are very different from those predicted by classical singular elastodynamics, where the dynamic J-integral is zero when super-shear is attained and there can be only an elliptic solution. Moreover, the results are very useful for couple stress elastodynamics where equivalent length scales are present due to the analogy with flexoelectricity.
This work examines the sub-shear and super-shear steady state growth of mode III fractures in flexoelectric materials, non the less, exhibiting Mach type shock wave patterns that resemble reported lattice dynamics results and three-dimensional calculations and experiments. Our mathematical models provide weak discontinuous solutions of the steady state dynamic equations. In flexoelectric solids, super-shear rupture is possible with Mach lines appearing at sub-shear as well as super-shear crack rupture velocities. This is contrary to classical singular elastodynamics, where the notions of super-shear growth and hyperbolicity coincide. The results show that the deformation near the crack-tip agrees with studies based on lattice dynamics. In the first part of this work, a novel finite element approach has been developed where the problem is decomposed in two prestressed plates which are interconnected, resulting into the predicted radiation patterns and Mach lines. The polarization field is obtained from the calculated displacement field and is used in turn to calculate the magnetic and the electric fields. The analysis offers an analogy to the co-seismic magnetic fields encountered during mode III dominated earthquake rupture events.
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