Measurement of the Total Energy Release Rate for Cracks in PZT Under Combined Mechanical and Electrical Loading

Schematic geometrical four-point-bending setup with electromechanical load

Schematic drawing and photograph of the displacement controlled four-point-bending device. The insertion in the photograph shows the mechanical arrangement to measure the displacement of the upper support. The two V-shaped rods are movably connected by an axis and transfer the displacement of the upper support rollers to the position encoder. The electrical insulation at the high voltage side of the specimen is achieved by a coating with thermoplastics.

Raw data of the load-displacement diagram for poled PZT. Two (out of six) unloading cycles are shown. The measurement is performed with the 5Hz modulation.

Load-displacement curves of Fig. 3, including all six unloading cycles. The dotted “small signal” lines and the displacement are corrected according to Eqs. 1,5 in the Appendix . The starting point on the displacement axis is shifted arbitrarily to the origin (as also shown in Fig. 3).

Schematic load-displacement diagrams for crack advance in ferroelectric PZT. (i) Energies during a complete unloading cycle at crack initiation and after a certain crack advance, (ii) energies during crack growth from (c) to (e) without unloading.

Schematic crack and process zone area. (a) Initial notch. (b) The specimen is loaded to a value just before the crack starts to grow, which creates the frontal process zone. (c) The crack has grown. (d) The specimen is completely unloaded. (e) The crack is loaded again and has grown by an amount Δa. The diagonally hatched area (F) in (e) shows the process zone area of remanently switched domains, corresponding to the crack extension Δa. The vertical arrows indicate tensile stress. The capital letters (A), (C), and (F) correlate to the corresponding areas in Fig. 5. The panels (a) to (e) correspond to the points (a) to (e) in Fig. 5.

Intrinsic toughness for poled PZT without applied electric field. The compliance curve for calculating the energy release rate is fitted for crack extensions between 0mm and 1.5mm. The dashed line represents 12J∕m2.

(a) Mechanical and (b) piezoelectric compliance as well as (c) capacitance as a function of the crack length for different electric fields and corrected according to Eqs. 1,2. For a better comparison the crack length is used, being the sum of notch depth (between 0.98mm and 1.07mm) and crack extension.

(a) and (b) Compliances and (c) capacitance as a function of the crack extension for an electric field of 500V∕mm and corrected as described before. The fitted analytical functions are used to differentiate the experimental data with respect to the crack surface area. (For the discontinuity of Cp, see text and compare with Fig. 8).

Measured energy release rates GmV, GeF, and Gp for an electric field of 500V∕mm during stable crack advance

Intrinsic energy release rates averaged between 0.5mm and 1.5mm crack extension. The linear and parabolic functions shown are valid for the idealized case, i.e., that the derivatives of Cp and CeF are independent of the electric field (see text). The data points are measured with only one sample for each electric field, except for the one at zero field, where two samples are averaged. The open circles connected by the gray line are the sum of all three contributions, where additionally GeF has been set to zero. Thus, actually it is the sum of GmV and Gp.

Fracture resistance curves for external electric fields between −500 and 500V∕mm. The initial increase and also the main level are similar for all fields with the exception of −250V∕mm. Here, the curve proceeds slightly lower than the other ones.

Critical mechanical energy release rates, especially linear contribution (black points) and mechanical work of fracture (open points) for different applied electric fields (−500, 0, and +500V∕mm). The latter energy release rate comprises energies due to linear elastic processes as well as remanent energies.

Comparison of the “intrinsic” mechanical energy release rate GmV (lower curve, as in Fig. 1) with Gcmech (upper curve) including remanent switching processes during crack advance. When calculating the standard deviation for the data points between 0.5mm and 1.5mm crack extension (Fig. 1), which gives a rough estimate for the error of the data, we get for the lower curve around ±1J∕mm2 and for the upper curve about ±10J∕mm2 (the latter number seems a little bit high.)

Schematic arrangement of mechanical compliances of the specimen and the equipment with Cm01⪡Cm⪡Cm02. The parameters dΔ1 as well as dΔ2 are the displacements, belonging to Cm and Cm01. The latter one represents the compliance of the mechanical frame plus the compression compliance of the specimen and Cm02 corresponds to small elastic deformations in the mounting and to inertia effects of the upper movable support.

Schematic drawing of the mechanical load configuration: (a) bending and compression of the specimen as well as loading of the device, (b) compression of the specimen and loading of the device. The difference (a)−(b) yields the pure bending compliance of the specimen.

(a) Mechanical and (b) piezoelectric compliance measured with the 20∕20mm supports (Fig. 1).

Displacement as a function of the force, measured with equal support distances as in Fig. 1. As described before, this effect corresponds to the compliance of the mechanical support and to the compression of the sample. The zero position on the displacement axis is arbitrary. The slightly different paths for loading and unloading are fitted by the same curve (Eq. 5).