As for the model of kinetic theory of fatigue crack propagation it may be classified into the following two types:Thefirst type: The model in which a microcrack nucleates in the neighbourhood of the tip of the main crack through the mechanism of separation of atoms caused by stress concentration due to the notch effect [1--4] (Fig. la) or due to slip (Fig. lc) or the mechanism of the condensation of vacancies [5] (Fig. lb), and the main crack join this, the process being repeated. The second type: The model in which the main crack itself extends by the mechanism of separation of atoms [6] (Fig. l d) or the mechanism of offset by slip (Fig. le) without nucleating a microcrack and joining it in front of the tip of the main crack.With respect to the model of the first type, some theoretical approaches have been attempted in the previous papers [1][2][3][4][5]. In the present article, from the viewpoint of the second type an attempt has been made based on dislocation dynamics.In the present article it is assumed that an irreversible slip occurs at the tip of crack every L W--° (a) (e) Figure 1. Typical examples of models in kinetic theory of fatigue crack propagation. half cycle under tension, but not during half cycle under compression. As far as this respect is concerned, the model is not inconsistent with those of McEvily et al. [7], Tomkins [8] and Laird et al. [9]. The number Z of Frank-Read sources along the crack tip edge of the length L~ of the main crack may be denoted by L 1 P~, that is Z = L1 p½, where p = the dislocation density at the concerned edge area. The relative offset ~ of the two parts of the crack at the edge between the distance L~ may be written as ~= Z n b = L l p * n b , where n=the number of dislocations emitted from the crack tip, b is Burgers vector. Then fatigue crack propagation rate, dc/dN is written as :where c = half length of crack and N is repeated cycles and 0~ is geometrical factor. For instance, in the case of tension-compression loading as shown in Fig. 2, ~ is 2 -½. lnt, Journ. of Fracture, 10 (1974)467-470
The electroelastic field concentration due to circular electrodes in piezoelectric
ceramics has been discussed through theoretical, numerical and experimental
characterizations. This paper consists of two parts. In the first part, the problem of a
surface electrode attached to a semi-infinite piezoelectric solid was formulated by means
of Hankel transforms and the solution was solved exactly. The displacements
and electric potential were expressed in closed form. In the second part, finite
element analysis was carried out to study electroelastic fields in piezoelectric disks
containing circular electrodes of different radii by introducing a model for polarization
switching in local areas of electroelastic field concentration. A nonlinear behavior
induced by localized polarization switching was observed between the strain and the
voltage applied to the electrode. Experiments were also conducted to study the
strain state near the electrode tip. Comparison of the predictions by the present
model with experimental data is conducted and pertinent conclusions are drawn.
This work is the first attempt to obtain nonlinear electroelastic fields around a
circular electrode in piezoelectric ceramics and validate the polarization switching
model.
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