Identification of crack parameters and stress intensity factors in finite and semi-infinite plates solving inverse problems of linear elasticity

Abstract

A method for the detection of a single or multiple straight cracks in finite and semi-infinite plane structures is presented. This allows both the identification of crack parameters such as length, position, and inclination angles with respect to a reference coordinate system and the calculation of stress intensity factors (SIFs). The method is based on strains measured at different locations on the surface of a structure and the application of the dislocation technique of linear elasticity. Cracks and boundaries are modelled by continuous distributions of dislocation densities. This approach gives a set of singular integral equations with Cauchy kernels, which are solved using Gauss–Chebyshev numerical quadrature, and, in contrast to e.g. a finite element calculation, spares a discretization of the structure. Once knowing the dislocation densities, the strain at an arbitrary point can be calculated. The crack parameters as well as external loads are determined by solving the inverse problem with a genetic and a simulated annealing algorithm. Once knowing loading and crack parameters, the SIFs are subsequently calculated. With the presented approach, the unknown parameters can be determined very accurately, being aware of some restrictions, which are thoroughly investigated.

Appendices

Appendix A. Dislocation influence functions

The dislocation influence functions relating infinitesimal dislocations at source points $(\xi ,\eta )$ to stresses in an elastic half-plane at field points (x, y) and possessing the unit 1/m are defined as

$$\begin{aligned} G_{xxx}(x,y;\xi ,\eta )= & {} G^{s}_{xxx}+G^{r}_{xxx}\nonumber \\= & {} (\,y-\eta )\left\{ -\frac{1}{r_1^2}-\frac{2x_1^2}{r_1^4}+\frac{1}{r_2^2}+\frac{2x_2^2}{r_2^4}-\frac{4\xi x_2}{r_2^4}+\frac{4\xi ^2}{r_2^4}+\frac{16\xi x_2^3}{r_2^6}-\frac{16\xi ^2x_2^2}{r_2^6}\right\} ,\nonumber \\ G_{xyy}(x,y;\xi ,\eta )= & {} G^{s}_{xyy}+G^{r}_{xyy}\nonumber \\= & {} (\,y-\eta )\left\{ -\frac{1}{r_1^2}+\frac{2x_1^2}{r_1^4}+\frac{1}{r_2^2}-\frac{2x_2^2}{r_2^4}+\frac{2\xi x_2}{r_2^4}-\frac{4\xi ^2}{r_2^4}-\frac{16\xi x_2^3}{r_2^6}+\frac{16\xi ^2x_2^2}{r_2^6}\right\} ,\nonumber \\ G_{xxy}(x,y;\xi ,\eta )= & {} G^{s}_{xxy}+G^{r}_{xxy}\nonumber \\= & {} -\frac{x_1}{r_1^2}+\frac{2x_1^3}{r_1^4}+\frac{x_2}{r_2^2}-\frac{2 \xi }{r_2^4}-\frac{2x_2^3}{r_2^4}+\frac{16\xi x_2^2}{r_2^4}-\frac{12 \xi ^2 x_2}{r_2^4}-\frac{16\xi x_2^4}{r_2^6}+\frac{16\xi ^2 x_2^3}{r_2^6},\nonumber \\ G_{yxx}(x,y;\xi ,\eta )= & {} G^{s}_{yxx}+G^{r}_{yxx}\nonumber \\= & {} -\frac{x_1}{r_1^2}+\frac{2x_1^3}{r_1^4}+\frac{x_2}{r_2^2}-\frac{2 \xi }{r_2^4}-\frac{2x_2^3}{r_2^4}+\frac{8\xi x_2^2}{r_2^4}+\frac{12 \xi ^2 x_2}{r_2^4}+\frac{16\xi x_2^4}{r_2^6}-\frac{16\xi ^2 x_2^3}{r_2^6},\nonumber \\ G_{yyy}(x,y;\xi ,\eta )= & {} G^{s}_{yyy}+G^{r}_{yyy}\nonumber \\= & {} \frac{3x_1}{r_1^2}+\frac{2x_1^3}{r_1^4}+\frac{3x_2}{r_2^2}-\frac{2 \xi }{r_2^4}+\frac{2x_2^3}{r_2^4}+\frac{16\xi x_2^2}{r_2^4}-\frac{12 \xi ^2 x_2}{r_2^4}-\frac{16\xi x_2^4}{r_2^6}+\frac{16\xi ^2 x_2^3}{r_2^6},\nonumber \\ G_{yxy}(x,y;\xi ,\eta )= & {} G^{s}_{yxy}+G^{r}_{yxy}\nonumber \\= & {} (\,y-\eta )\left\{ -\frac{1}{r_1^2}+\frac{2x_1^2}{r_1^4}+\frac{1}{r_2^2}-\frac{2x_2^2}{r_2^4}-\frac{4\xi x_2}{r_2^4}+\frac{4\xi ^2}{r_2^4}+\frac{16\xi x_2^3}{r_2^6}-\frac{16\xi ^2x_2^2}{r_2^6}\right\} \nonumber \\ \end{aligned}$$

(A.1)

where terms including $r_1$ and $x_1$ are associated with the Green’s functions $G^{s}_{ijk}$, and those comprising $r_2$ and $x_2$ are related to $G^{r}_{ijk}$, see Eq. (2). The $r_1$, $r_2$, $x_1$, and $x_2$ are further defined as

$$\begin{aligned} r_1&=\sqrt{x_1^2+(y-\eta )^2},\,\,\,r_2=\sqrt{x_2^2+(y-\eta )^2}\\ x_1&=x-\xi ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x_2=x+\xi . \end{aligned}$$

Equation (A.1) refers to the formulation of Eq. (4), where the coordinates $x,\,y,\,\xi , \eta$ of Eq. (1) have been replaced by ${{\hat{x}}},\,{{\hat{y}}},\,{\hat{\xi }},\, {\hat{\eta }}$, introducing a local coordinate system.

Appendix B. Equations for parameter identification

The following equations are the basis of the inverse problems, seeking crack parameters and boundary loads.

Finite plate with internal cracks:

(B.1)

Semi-infinite plate with two boundaries and internal cracks:

(B.2)

Semi-infinite plate with one boundary and an edge crack:

$$\begin{aligned} \begin{aligned}&\begin{bmatrix} \varepsilon ^*_{{x}{x}}(x,y)\\ \varepsilon ^*_{{y}{y}}(x,y)\\ \gamma ^*_{xy}(x,y) \end{bmatrix} =\frac{1}{E} \begin{bmatrix} \sigma _{{x}{x}}^{\infty }-\nu \sigma _{{y}{y}}^{\infty }\\ \sigma _{{y}{y}}^{\infty }- \nu \sigma _{{x}{x}}^{\infty }\\ (1+\nu )\sigma _{{x}{y}}^{\infty } \end{bmatrix} +\frac{2\mu }{E \pi \left( \kappa +1 \right) } \left[ a_e\sum _{i=1}^{N} \frac{2(1+s_i^e)}{2N+1}\right. \\&\quad \left. .\begin{bmatrix} G_{{x}{x}{x}}^{IJ}(x,y;{s}_i^e)- \nu G_{{x}{y}{y}}^{IJ}(x,y;{s}_i^e)&G_{{y}{x}{x}}^{IJ}(x,y;{s}_i^e)-\nu G_{{y}{y}{y}}^{IJ}(x,y;{s}_i^e)\\ G_{{x}{y}{y}}^{IJ}(x,y;{s}_i^e)- \nu G_{{x}{x}{x}}^{IJ}(x,y;{s}_i^e)&G_{{y}{y}{y}}^{IJ}(x,y;{s}_i^e)-\nu G_{{y}{x}{x}}^{IJ}(x,y;{s}_i^e) \\ (1+\nu ) G_{{x}{x}{y}}^{IJ}(x,y;{s}_i^e)&(1+\nu ) G_{{y}{x}{y}}^{IJ}(x,y;{s}_i^e) \end{bmatrix} \, \begin{bmatrix} \phi _{{x}}({s}_i^e)\\ \phi _{{y}}({s}_i^e) \end{bmatrix} \right. \\&\qquad \left. +\sum _{J=1}^{{\tilde{M}}}a_J\sum _{i=1}^{N} \frac{1}{N}\right. \\&\quad \left. .\begin{bmatrix} G_{{x}{x}{x}}^{IJ}(x,y;{s}_i^J)-\nu G_{{x}{y}{y}}^{IJ}(x,y;{s}_i^J)&G_{{y}{x}{x}}^{IJ}(x,y;{s}_i^J)-\nu G_{{y}{y}{y}}^{IJ}(x,y;{s}_i^J)\\ G_{{x}{y}{y}}^{IJ}(x,y;{s}_i^J)-\nu G_{{x}{x}{x}}^{IJ}(x,y;{s}_i^J)&G_{{y}{y}{y}}^{IJ}(x,y;{s}_i^J)-\nu G_{{y}{x}{x}}^{IJ}(x,y;{s}_i^J)\\ (1+\nu ) G_{{x}{x}{y}}^{IJ}(x,y;{s}_i^J)&(1+\nu ) G_{{y}{x}{y}}^{IJ}(x,y;{s}_i^J) \end{bmatrix} \, \begin{bmatrix} \phi _{{x}}({s}_i^J)\\ \phi _{{y}}({s}_i^J) \end{bmatrix} \right] ,\\&\quad I=1,\ldots ,{\tilde{M}}+1. \end{aligned} \end{aligned}$$

(B.3)

The influence functions $G_{ijk}^{IJ}$ depend on the unknown crack lengths, while the function $\phi$ depends on the crack lengths, positions, and inclination angles as well as on the external loads.