2.2 Energy density criterion At the continuum scale level, the sharp crack tip is assumed to lie within a macroscopic size core region with radius ro, Fig. 2. Mathematically speaking, the energy density function becomes unbounded as the crack tip is approached or as r → 0 since r S dV dW = (4) The relationship of eq. (4) is shown in Fig. 2 where S represents the area of the dW/dV versus r plot for a given level of dW/dV. The form of eq. (4) does not limit the criterion of energy density to linear elasticity even though the inverse square root of r stress singularity would correspond to 1/r for dW/dV. Note from Fig. 2 that r adopts a much more general interpretation since it is simply the linear distance measured from the crack tip. The following hypotheses applied to dW/dV are in general valid for any nonlinear constitutive relations, large deformation theories with or without dissipation. When applied to a local region ahead of the crack tip, they can be stated as [9]: • Hypothesis I: Location of crack initiation is assumed to coincide with the maximum of the minimum dW/dV or (dW/dV) . max min • Hypothesis II: The onset of stable crack growth is assumed to occur when (dW/dV) reaches a critical value (dW/dV) max min c. • Hypothesis III: Stable crack growth segments r1, r2, etc., are assumed to be governed by c c j j 2 2 1 1 c r S r S r S r S dV dW = = = = = = L L (5) The onset of rapid fracture is assumed to take place when c c c r S dV dW = (6) The ways with which Sc are related to K1c for Mode I crack extension depends on the constitutive relations and the kinetics of cracks under consideration. For a crack under static load applied to an isotropic, homogeneous and elastic body, it has been shown in [9] that o 2 1c c 2 E (1 )(1 2 )K π +ν − ν = S (7) in which ν is the Poisson’s ratio and Eo the Young’s modulus. For the PZT material considered in this work, the equivalent of eq. (7) takes the form 2 33 2 33 44 33 33 14 c 11 g A g A 2A β + β = + S (8) Here, A11 , A14 , A44 , g33 and β33 are complicated functions related to the elastic, piezoelectric and dielectric constants of the ferroelectric ceramics. The specific expressions can be found in [8]. Another set of Bij connected with specifying electric field E can be defined instead of Aij related specifying the electric displacements D. 2.3 Mode I crack extension Referring to Fig. 1, both the applied stress σ and the electric field E are such that the crack would extend straight ahead along the x1-axis where S possesses a relative minimum with reference to the angle θ in Fig.
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