## 2 Literature Review  
### Theoretical Background  
Cracks prefer to grow perpendicular to the normal stress direction. However, cracked structures subjected to multiaxial loading can display three fracture modes [12]: mode I (opening), mode II (in-plane shear), and mode III (out-of-plane shear), as shown in Figure 1. Most of the solutions for SIFs are numerical, with only a few analytical approaches, such as [PERSON] series, based on a complex Airy stress function, as shown in Equation (1) [25], for an infinite cracked plate, as seen in Figure 1, with the coordinate system located at the crack tip. The independent displacements \(U\), \(V\), and \(W\) are represented as an infinite sequence of \(n\) terms. Polar coordinates (\(r\), \(\theta\)) describe the positions of the opposite-to-the-crack points \(A\) and \(B\) from the origin located at the crack tip. \[ \begin{split} u&=\sum\limits_{n=1}^{\infty}\frac{r^{ n/2}}{2G}\left(\begin{array}{c}a_{n}\bigg{\{}\Big{[}k+\frac{n}{2}+(-1)^{n} \Big{]}Cos\frac{n\theta}{2}-\frac{n}{2}Cos\bigg{(}\frac{n-4}{2}\bigg{)}\theta \bigg{\}}-\\ b_{n}\bigg{\{}\Big{[}k+\frac{n}{2}-(-1)^{n}\Big{]}Sin\frac{n\theta}{2}-\frac {n}{2}Sin\bigg{(}\frac{n-4}{2}\bigg{)}\theta\bigg{\}}\end{array}\right)\\ v&=\sum\limits_{n=1}^{\infty}\frac{r^{n/2}}{2G}\left(\begin{array}{c}a_{n} \bigg{\{}\Big{[}k-\frac{n}{2}+(-1)^{n}\Big{]}Sin\frac{n\theta}{2}+\frac{n}{2} Sin\bigg{(}\frac{n-4}{2}\bigg{)}\theta\bigg{\}}+\\ b_{n}\bigg{\{}\Big{[}k-\frac{n}{2}+(-1)^{n}\Big{]}Cos\frac{n\theta}{2}+\frac{n}{2} Cos\bigg{(}\frac{n-4}{2}\bigg{)}\theta\bigg{\}}\end{array}\right)\\ w&=\sum\limits_{n=1}^{\infty}\frac{2r^{n-\frac{1}{2}}}{G}\bigg{\{}c_{n}Sin \bigg{(}n-\frac{1}{2}\bigg{)}\theta\bigg{\}}\end{split}  \tag{1}\]where \(u\) is parallel-to-crack displacements, \(v\) is perpendicular-to-crack displacements, \(w\) is perpendicular-to-plane displacements, \(a_{1}=K_{I}/\sqrt{2\pi}\), \(b_{1}=K_{II}/\sqrt{2\pi}\), \(c_{1}=K_{III}/\sqrt{2\pi}\), \(a_{2}=\sigma_{\text{ox}}/4\), \(r\) and \(\theta\) are the radius and angle of an arbitrary point, \(G\) is shear modulus, \(n\) is the numbers of terms in the expansion series, and \(k\) is the Kolosov constant given by Equation (2). \[k=\left\{ \begin{array}{l}3-4v;\epsilon_{pl}\\ \frac{3-v}{1+v^{*}};\sigma_{pl}\end{array} \right. \tag{2}\]  
Using Equation (1), [PERSON] and [PERSON] [26] computed SIF by expressing Williams' displacement fields as Equation (3) using experimentally obtained mixed-mode data (I and II). \[ \begin{array}{l}u=\sum\limits_{n=1}^{N}a_{n}f_{ln(r,\theta)}-\sum\limits_{n= 1}^{N}b_{n}f_{1 ln(r,\theta)}+T_{x}-Ry_{n}\\ v=\sum\limits_{n=1}^{N}a_{n}g_{ln(r,\theta)}-\sum\limits_{n=1}^{N}b_{n}g_{1 ln( r,\theta)}+T_{y}+Rx_{n}\end{array}  \tag{3}\]  
where \(N\) is the number of terms in [PERSON] series, \(T_{x}\) and \(T_{y}\) are the rigid body components in \(x\) and \(y\), respectively, and \(Ry_{n}\) and \(Rx_{n}\) are the rigid body rotation around \(X\) and \(Y\), respectively. Equation (3) can be rewritten in matrix form, as shown in Equation (4), where [\(a(r,\,\theta)\)] and [\(b(r,\,\theta)\)] is a matrix made up of functions described in Equation (5) that are explained in Equation (6), \(\{\Delta\}\) is the unknown vector coefficients of the displacement functions, and \(\{h\}\) is the concatenated vector of known displacements. The Least Squares Method (LSM) [27] can solve the resulting over-determined system. Furthermore, [PERSON] and [PERSON] [28] checked the accuracy of the LSM to calculate SIF. This over-deterministic method has been implemented, among others, for modes I and II [29]. \[\{h\}=[b]\{\Delta\} \tag{4}\]  
Figure 1: Notation for the coordinate system and displacements around a cracked domain showing parallel-to-crack \(u\)-displacement, perpendicular-to-crack \(v\)-displacement, and out-of-plane \(w\)-displacement. \[h= \begin{bmatrix}u_{1}\\ \vdots\\ u_{M}\\ v_{1}\\ \vdots\\ v_{M}\end{bmatrix} ;\Delta= \begin{bmatrix}a_{1}\\ \vdots\\ a_{M}\\ b_{1}\\ \vdots\\ b_{M}\end{bmatrix} ;b=