PSI - Issue 80

Xinpeng Tian et al. / Procedia Structural Integrity 80 (2026) 451–461 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 (d) the magnitude of electric field intensity =√ 1 2 + 22 + 32 .

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(c) (d) Fig. 5. Comparisons of physical fields near the penny-shaped crack for various tensile loadings along the predefined path ( x 1 = x 3 = 0, 0 < x 2 < R ) 4.2 The flexoelectric effect around the penny-shaped crack for different length scale parameters According to the strain gradient elastic theory, the material length scale parameter l serves as a key indicator of the strain gradient effect in materials. Variations in l can significantly influence the spatial distribution of the strain gradient field, thereby affecting the flexoelectric response in structures. Accordingly, this subsection investigates the influence of the material length scale parameter l on the flexoelectric response near the crack tip with fixed tensile loading 9 a 110 P  =  and flexoelectric coefficient 9 1 110 C/m f − =  . Fig. 6 presents the distribution profiles of displacement 3 u , Cauchy stress 33  , and the flexoelectric response near the crack tip for different values of the material length scale parameters 9 9 9 9 m , , {210 410 610,810} l − − − −      . Unlike the observations in Section 4.2, changes in l have a pronounced impact on the displacement distribution along the crack surface. As shown in Fig. 6 (a), when l increases from 2 nm to 8 nm, the maximum displacement on the crack surface decreases from 0.194 nm to 0.12 nm, a reduction of approximately 38%. This indicates that the material length scale parameter l can enhance the effective stiffness of the material, thereby contributing to improved local fracture toughness. Besides, an increase in l significantly reduces the stress concentration at the crack tip, leading to a smoother stress profile as can be seen in Fig. 6 (b). In terms of electrical response, Fig. 6 (c) shows the variation of electric potential near the crack tip with respect to different values of l . It is evident that the spatial distribution of the electric potential becomes more gradual as l increases. This implies that the electric field intensity induced by the flexoelectric effect in the vicinity of the crack tip weakens with increasing l , as depicted in Fig. 6 (d). These results