PSI - Issue 80

Xinpeng Tian et al. / Procedia Structural Integrity 80 (2026) 451–461 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Under external mechanical loading, pronounced flexoelectric fields are induced in the vicinity of the crack tip as a result of stress concentration. To systematically elucidate the underlying mechanisms of multi-field coupling, the effects of tensile loading  and material length scale parameter l on the distribution of flexoelectric fields are studied in the following. 4.1 The flexoelectric effect around the penny-shaped crack for different tensile loadings Previous studies have demonstrated that tensile loading intensity is a key parameter in affecting the flexoelectric response near crack tips (Tian et al. 2023). Therefore, the regulation mechanism of tensile loadings on the flexoelectric effect near the penny-shaped crack are studied in this section. Fig. 4 illustrates the spatial distribution of the out-of plane displacement 3 u , electric potential  and electric field intensity 2 2 2 1 2 3 v E E E E = + + around penny-shaped cracks for different uniformly tensile loadings 9 8 8 8 {3 10 5 10 7 10 1 10}Pa , , ,       . It can be observed from Fig. 4 that the displacement 3 u increases with the increase of tensile loadings, which is consistent with the conclusions of classical fracture mechanics. Under external mechanical loadings, significant stress concentration occurs around the crack tip, accompanied by a pronounced flexoelectric response (see in Fig. 4(c)-(d)). Moreover, as the tensile loading increases, the stress concentration effect near the crack tip becomes more prominent, resulting in a corresponding enhancement of the electric field intensity induced by the flexoelectric effect, as shown in Figs. 4(c)-(d). Figure 4 also reveals the spatial distribution characteristics of various physical fields. On isotropic planes perpendicular to the cylinder axis, the distributions of physical fields exhibit central symmetry with respect to the center of the cylinder, whereas significant spatial non-uniformity can be observed along the height of the cylinder. In addition, numerical results indicate that a more pronounced flexoelectric response exists near regions closer to the crack surface (with 3 0 z x = = ). Thus, to quantitatively analyze the regulatory effect of external loadings on the flexoelectric responses, Fig. 5 presents the spatial distributions of displacement, Cauchy stress, electric potential, and electric field intensity along the path of maximum flexoelectric response (highlighted by the red solid line in Fig. 3(b)). As shown in Fig. 5(a), the 3 u on the crack surface increases with the increase of loadings, and the stress profile becomes steeper, which indicates a more significant stress concentration effect near the crack tip. In contrast to the classical fracture mechanics, the Cauchy stresses are finite at the crack tip in this theory. Consequently, the electric potential reaches a peak value near the crack tip, and with increasing loadings, the potential profile becomes increasingly steep. This leads to a significantly enhanced electric field response, as shown in Fig. 5(d), thereby confirming the crucial role of tensile loadings in modulating the flexoelectric effect.

Fig. 4. Distributions of physical fields around the penny-shaped crack for different tensile loadings  . (a) The out of-plane displacement u 3 , (b) the stress σ 33 , (c) the electric potential ϕ ,