On the relationship between crack initiation angle and loading equivalent angle for asymmetric Three-Point Bend Specimens under mixed-mode I/II loading based on GMPZR criterion

The multifarious nature of the real loading condition ahead of the crack tip for service-manufacturing components under combined (Mode I & Mode II) conditions is linked to the estimation of crack/fracture initiation [1]. Preventing catastrophic failure in industrial components requires the precise measurement and prediction of the crack initiation angle and direction [2]. The predictions of crack initiation for structures/components were analyzed using several fracture criteria [3] based on the fracture parameters such as the stress intensity factor, J-integral, and crack tip opening displacement [4, 5]. The researchers [6] proposed various fracture criteria by considering a constant core radius around the crack tip, known as the plastic zone (PZ), which occurs from the localization of stress amplitude for analyses of fracture initiation/direction. However, the PZ was not constant near the crack tip, and Breitbarth et al. reported that the fracture initiation occurs where the PZ is smallest [7]. Later, based on the PZ size and shape near the crack tip derived from the von Mises yield criterion [8], Bian and Kim [9] proposed the minimum plastic zone radius (MPZR) criterion. They [9, 10] showed that the crack initiation angle ( ${\theta }_0$${\theta }_0$theta_(0)\theta_{0}${\theta }_0$ ) for mixed-mode loading occurs at the point where the PZ radius attains a
global minimum ahead of the crack tip. Subsequently, detailed finite element analyses were performed to measure the crack initiation based on the MPZR criterion ahead of the crack tip under mixed-mode (I/II) loading. Accordingly, researchers [11] numerically analyzed the estimation of crack initiation by considering the shape and size of a PZ ahead of the crack tip derived from the von Mises yield criterion. Furthermore, many investigators [12, 13] used a non-dimensional applied load parameter, the loading equivalent angle ( ${\beta }_{\text{eq}}={\tan }^{-1}\left(K_I/K_{II}\right)$${\beta }_{\text{eq }}={\tan }^{-1}\left(K_I/K_{II}\right)$beta_("eq ")=tan^(-1)(K_(I)//K_(II))\beta_{\text {eq }}=\tan ^{-1}\left(K_{I} / K_{I I}\right)${\beta }_{\text{eq }}={\tan }^{-1}\left(K_I/K_{II}\right)$ ), to determine crack initiation under mixed-mode (I/II) loading. Next, improvements in crack-prediction models for structures/components can be achieved by considering the second term in the stress field around the crack tip. Related to the second term, or non-singular parameter ( T -stress), many researchers [12, 14] have recognized that the effect of the second term of the Williams Series on the plastic core and on the crack growth direction/initiation is significant. Shahani and Tabatabaei [12] numerically extracted T-stress from the FE software tool for an asymmetric four-point bend fracture specimen and suggested that the T -stress would affect the different fracture criteria [15]. However, the theoretical fracture criteria do not include the T-stress term in the analysis. Ayatollahi et al. [16] demonstrated a generalized maximum tensile stress criterion by considering T -stress for predicting the crack initiation under combined tensile and shear (mixed mode) loading. The authors [16] numerically analyzed various asymmetric specimens and found that the generalized maximum tensile stress criterion yields better results than the maximum tensile stress criterion. In addition, Nazarali and Wang [17] studied the effects of T-stress on the plastic zone ahead of the crack tip under mixed-mode (I/II) loading conditions and suggested that T-stress is an important parameter for fatigue and fracture analyses to identify defects in structures. Miao et al. [18] numerically analyzed the effect of T-stress on plastic zone for various mixed-mode (I/II) fracture specimens and recommended that the T -stress plays a significant role in crack initiation angle and plastic zone around the crack tip, which in turn can affect the fracture of the specimen. However, the examination of fracture initiation criteria based on PZ ahead of the crack tip, by considering stress intensity factors and Tstress to estimate the crack initiation direction under mixed-mode (I/II) loading conditions, was not available. Hence, the objectives of the present work are as follows i) A GMPZR criterion were carried out from stress intensity factors and Tstress parameters to estimate fracture initiation, which is extension of MPZR criterion, ii) the FE analyses were carried out on asymmetric TPB specimen for various $a/W$$a/W$a//Wa / W$a/W$ ratios and ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ to analyze the size and shape of crack-tip PZ, iii) The crack initiation angle estimated from FE analyses was compared with the proposed GMPZR fracture criterion, other theoretical fracture criteria and available experimental results, iv) from the FE analyses an attempt is been made to estimate approximate formulation for GMPZR and crack initiation angles based on stress intensity factors under mixed-mode (I/II) loading, and v) finally, the effects of T-stress on crack initiation angle estimated from GMPZR criterion were also studied

Ageneralized minimum plastic zone radius (GMPZR) criterion is an extension of the MPZR [9, 19] criterion. It states that the direction of crack initiation coincides with the direction of minimum plastic zone radius evaluated from the von Mises yield criterion. The radius of PZ ahead at the crack tip contains stress intensity factors and Tstress. The crack initiation can be determined by minimizing $r$$r$rr$r$ [9, 19]:

The stresses near the crack tip expressed in terms of Cartesian co-ordinates ( $\mathrm{x},\mathrm{y}$$\mathrm{x},\mathrm{y}$x,y\mathrm{x}, \mathrm{y}$\mathrm{x},\mathrm{y}$ ) are given as:

where $K_{\mathrm{I}}$$K_{\mathrm{I}}$K_(I)K_{\mathrm{I}}$K_{\mathrm{I}}$ and $K_{\mathrm{II}}$$K_{\mathrm{II}}$K_(II)K_{\mathrm{II}}$K_{\mathrm{II}}$ are the stress intensity factors in Mode I and Mode II, respectively. The Von Mises yield criteria for a 3D object can be written in the following form

One can substitute the singular stress field of Eq. (2) in the above yield criterion Eq. (4) and solve for the PZ radius r. It can be expressed as follows.

By substituting Eq. (5) in Eq. (1), we get,

Eqn. (6) is used to determine the crack initiation / direction ( ${\theta }_0$${\theta }_0$theta_(0)\theta_{0}${\theta }_0$ ) according to the GMPZR criterion.

The sequence of 2D stress analyses for the asymmetric TPB specimen was performed using ABAQUS. The asymmetric TPB specimen dimensions for the FE analyses are taken from the work of Shahani and Tabatabaei [12], as shown in Fig. 1. During FE analyses, the loading and boundary conditions for the asymmetric TPB specimen were similar to those reported by Ma et al. [20]. Due to asymmetric loading, the full specimen geometry is considered in the FE analyses. The meshing of the asymmetric TPB specimen is done using quadratic 2D Isoparametric solid elements. Researchers [16] utilized the quadratic 2D Isoparametric solid elements to extract $K$$K$KK$K$ and $T$$T$TT$T$ for the various specimens in the FE analyses. Fig. 2 shows a 2D FE mesh of the asymmetric TPB specimen used for the analyses. The asymmetric TPB specimen is analyzed for central loading position of $a/W=0.4-0.7$$a/W=0.4-0.7$a//W=0.4-0.7a / W=0.4-0.7$a/W=0.4-0.7$ ratios with an increment of 0.1 and for 6 different crack positions $(s/L=0-1$$(s/L=0-1$(s//L=0-1(s / L=0-1$(s/L=0-1$, where, s and $L$$L$LL$L$ are shown in Fig. 1). The number of elements for these six different crack positions of the asymmetric TPB specimen are varies from 1500 to 2000. The ABAQUS post-processor has extracted the magnitudes of $K$$K$KK$K$ and $T$$T$TT$T$. The procedure for computing $K$$K$KK$K$ and $T$$T$TT$T$ in the ABAQUS FE software tool uses the J-integral method [21] which is documented. Interstitial free steel (IF steel) was considered for the linear elastic FE analysis, which is used in car bodies due to its high formability [22]. The tensile properties of the material for FE analyses were extracted from earlier works [22], and they are: elastic modulus $(E)=197\mathrm{GPa}$$(E)=197\mathrm{GPa}$(E)=197GPa(E)=197 \mathrm{GPa}$(E)=197\mathrm{GPa}$, yield strength $\left({\sigma }_{\mathrm{y}}\right)=155\mathrm{MPa}$$\left({\sigma }_{\mathrm{y}}\right)=155\mathrm{MPa}$(sigma_(y))=155MPa\left(\sigma_{\mathrm{y}}\right)=155 \mathrm{MPa}$\left({\sigma }_{\mathrm{y}}\right)=155\mathrm{MPa}$, Poisson's ratio $(\mathcal{U})=0.3$$(\mathcal{U})=0.3$(U)=0.3(\mathcal{U})=0.3$(\mathcal{U})=0.3$.

The analytical formulation for the asymmetric three-point bend specimen is not available in the literature. The formulations proposed by He and Hutchinson [23] for estimating $K$$K$KK$K$ for an asymmetric four-point bend specimen are used to modify for the asymmetric 3PB specimen as under:

where, $3Qs/h{W}^2$$3Qs/h{W}^2$3Qs//hW^(2)3 Q s / h W^{2}$3Qs/h{W}^2$ is the applied bending stress in the case of the 3 PB specimen
Similarly,

In the case of the 3 PB specimen, $Q=F$$Q=F$Q=FQ=F$Q=F$
$Q=$$Q=$Q=Q=$Q=$ Shear Force
where, $b$$b$bb$b$ is the thickness of the specimen, $a$$a$aa$a$ is the crack length, and $W$$W$WW$W$ is the width of the specimen. The $Fi=(a/W)$$Fi=(a/W)$Fi=(a//W)F i=(a / W)$Fi=(a/W)$ polynomial equations by Murakami [24]:

Using the above equations, a C program has been developed to compute $K_I$$K_I$K_(I)K_{I}$K_I$ and $K_{II}$$K_{II}$K_(II)K_{I I}$K_{II}$ for various applied loads.

To estimate $\mathrm{K},\mathrm{T}$$\mathrm{K},\mathrm{T}$K,T\mathrm{K}, \mathrm{T}$\mathrm{K},\mathrm{T}$, and to accomplish the contour and magnitude of PZ ahead of the crack-tip, various load steps were applied on an asymmetric TPB specimen of different $a/W$$a/W$a//Wa / W$a/W$ and ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$. For different load steps and ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$, the $K$$K$KK$K$ and $T$$T$TT$T$ for the asymmetric TPB specimen have been computed using the ABAQUS postprocessor. The analyses of $K_I$$K_I$K_(I)K_{I}$K_I$ and $K_{II}$$K_{II}$K_(II)K_{I I}$K_{II}$ for each ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ will be handled using the effective stress intensity factor ( $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ ), given by $K_{\text{eff}}=\sqrt{K_I^2+K_{II}^2}$$K_{\text{eff }}=\sqrt{K_I^2+K_{II}^2}$K_("eff ")=sqrt(K_(I)^(2)+K_(II)^(2))K_{\text {eff }}=\sqrt{K_{I}^{2}+K_{I I}^{2}}$K_{\text{eff }}=\sqrt{K_I^2+K_{II}^2}$. In the analyses, the $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ will be normalized by yield stress and the square root of the thickness $\left({\sigma }_{\mathrm{y}}{\mathrm{h}}^{1/2}\right)$$\left({\sigma }_{\mathrm{y}}{\mathrm{h}}^{1/2}\right)$(sigma_(y)h^(1//2))\left(\sigma_{\mathrm{y}} \mathrm{h}^{1 / 2}\right)$\left({\sigma }_{\mathrm{y}}{\mathrm{h}}^{1/2}\right)$. The premeditated values of normalized $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ for $a/W=0.4-0.7$$a/W=0.4-0.7$a//W=0.4-0.7a / W=0.4-0.7$a/W=0.4-0.7$ in asymmetric TPB specimens, typically for a 1 kN applied load, are plotted against ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ in Fig. 3. The finite element analysis (FEA) results and theoretical results of normalized $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ vs. ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ show good agreement. The analytical values of $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ for various load steps, $a/W$$a/W$a//Wa / W$a/W$, and ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ of the asymmetric TPB specimen have been calculated from Eqns. (7) and (8). The percentage error between the FEA and theoretical results is less than $3.5\mathrm{\%}$$3.5\mathrm{\%}$3.5%3.5 \%$3.5\mathrm{\%}$. This error may be attributed to the varied $s/L$$s/L$s//Ls / L$s/L$ ratio and bend loading condition of the asymmetric TPB specimen. Fig. 3 demonstrates that the normalized $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ increases as ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ changes from $0^{\circ }$$0^{\circ }$0^(@)0^{\circ}$0^{\circ }$ (Mode II) to ${90}^{\circ }$${90}^{\circ }$90^(@)90^{\circ}${90}^{\circ }$ (Mode I), and for the same applied load, the normalized $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ is almost 20 times higher for ${\beta }_{eq}={90}^{\circ }$${\beta }_{eq}={90}^{\circ }$beta_(eq)=90^(@)\beta_{e q}=90^{\circ}${\beta }_{eq}={90}^{\circ }$ than for ${\beta }_{eq}=0^{\circ }$${\beta }_{eq}=0^{\circ }$beta_(eq)=0^(@)\beta_{e q}=0^{\circ}${\beta }_{eq}=0^{\circ }$. Also, Fig. 3 shows that normalized $K_{eff}$$K_{eff}$K_(eff)K_{e f f}$K_{eff}$ depends on $a/W$$a/W$a//Wa / W$a/W$ and the applied load. The T-stress will be evaluated using the dimensionless biaxiality ratio (B) $B=T\sqrt{\pi a}/K_{\text{eff}}$$B=T\sqrt{\pi a}/K_{\text{eff }}$B=Tsqrt(pi a)//K_("eff ")B=T \sqrt{\pi a} / K_{\text {eff }}$B=T\sqrt{\pi a}/K_{\text{eff }}$ and $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ [12, 25]. The variation of $B$$B$BB$B$ vs. $a/W$$a/W$a//Wa / W$a/W$ is shown in Fig. 4 for different ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$. From Fig. 4, the computed B exactly matches Sham's [26] results for the TPB specimen with ${\beta }_{eq}={90}^{\circ }$${\beta }_{eq}={90}^{\circ }$beta_(eq)=90^(@)\beta_{e q}=90^{\circ}${\beta }_{eq}={90}^{\circ }$ (Mode I). $B$$B$BB$B$ is negative for ${\beta }_{eq}=0^{\circ }$${\beta }_{eq}=0^{\circ }$beta_(eq)=0^(@)\beta_{e q}=0^{\circ}${\beta }_{eq}=0^{\circ }$ and it is positive for the remaining ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$ values, for $a/W=0.4$$a/W=0.4$a//W=0.4a / W=0.4$a/W=0.4$ to 0.7 . The Fig. 4 clearly shows that B is almost constant for ${\beta }_{\text{eq}}={75}^{\circ }$${\beta }_{\text{eq }}={75}^{\circ }$beta_("eq ")=75^(@)\beta_{\text {eq }}=75^{\circ}${\beta }_{\text{eq }}={75}^{\circ }$ to ${86}^{\circ }$${86}^{\circ }$86^(@)86^{\circ}${86}^{\circ }$ with $a/W$$a/W$a//Wa / W$a/W$ ratios of 0.4 to 0.7 ratios.

The studies on the magnitudes and contours of the PZ ahead of a crack tip were considered for various ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$ values in an asymmetric TPB specimen. The contours of the PZ were determined by plotting iso-contours based on the von Mises criterion, using the materials yield stress. Since an elastic constitutive model is used for the material, contour plots of the von Mises stress show the PZ shape. This differs from the actual PZ, which is affected by stress redistribution during plastic flow. Fig. 5(a-c) shows the chronological growth of PZ ahead of the crack tip for ${\beta }_{eq}=0$${\beta }_{eq}=0$beta_(eq)=0\beta_{e q}=0${\beta }_{eq}=0$ (Mode II), ${\beta }_{eq}=84$${\beta }_{eq}=84$beta_(eq)=84\beta_{e q}=84${\beta }_{eq}=84$ (mixed-mode (I/II)), and ${\beta }_{\text{eq}}={90}^{\circ }$${\beta }_{\text{eq }}={90}^{\circ }$beta_("eq ")=90^(@)\beta_{\text {eq }}=90^{\circ}${\beta }_{\text{eq }}={90}^{\circ }$ (Mode-I) of the asymmetric TPB specimen, subjected to a range of applied loads. The crack-tip PZ shapes for each load step are merged in Figs. 5(a-c). The displacement scaling of asymmetric TPB specimens is set to zero in Fig. 5(a-c) to analyze the magnitudes and shape of the crack-tip PZ. From Fig. 5(a), it is clearly seen that the development
of PZ contours takes place in the vertical direction (i.e., along the ligament) for ${\beta }_{eq}=0^{\circ }$${\beta }_{eq}=0^{\circ }$beta_(eq)=0^(@)\beta_{e q}=0^{\circ}${\beta }_{eq}=0^{\circ }$. In this figure, the shape of PZ at the lower right of the crack is found to deviate from the expected normal shape due to the loading point at the crack tip. Fig. 5(b) reveals that the growth of the PZ takes place at an angle of stretch ( $\theta$$\theta$theta\theta$\theta$ to crack plane) between $0^{\circ }$$0^{\circ }$0^(@)0^{\circ}$0^{\circ }$ and ${90}^{\circ }$${90}^{\circ }$90^(@)90^{\circ}${90}^{\circ }$. For ${\beta }_{\text{eq}}={90}^{\circ }$${\beta }_{\text{eq }}={90}^{\circ }$beta_("eq ")=90^(@)\beta_{\text {eq }}=90^{\circ}${\beta }_{\text{eq }}={90}^{\circ }$ (Mode I), the PZ grows horizontally (i.e., perpendicular to the ligament) as shown in Fig. 5(c). From Fig. 5(a) for ${\beta }_{eq}=0^{\circ }$${\beta }_{eq}=0^{\circ }$beta_(eq)=0^(@)\beta_{e q}=0^{\circ}${\beta }_{eq}=0^{\circ }$ (Mode II), it is clearly seen that the PZ contours are more dominating at the loading rather than at the crack-tip. Hence, the applied load on the asymmetric TPB specimen under Mode II loading will not be distributed at the crack tip. This is because, for Mode II loading, the crack is exactly at the roller of the asymmetric TPB specimen.

The growth and alteration of the crack-tip PZ for various ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$, and $a/W$$a/W$a//Wa / W$a/W$, typically for $K_{\text{eff}}=900{\mathrm{MPamm}}^{1/2}$$K_{\text{eff }}=900{\mathrm{MPamm}}^{1/2}$K_("eff ")=900MPamm^(1//2)K_{\text {eff }}=900 \mathrm{MPamm}^{1 / 2}$K_{\text{eff }}=900{\mathrm{MPamm}}^{1/2}$ in an asymmetric TPB specimen, are shown in Fig. 6, with the contours are superimposed. The plastic contours that are superimposed in Fig. 6 for ${\beta }_{eq}={90}^{\circ }$${\beta }_{eq}={90}^{\circ }$beta_(eq)=90^(@)\beta_{e q}=90^{\circ}${\beta }_{eq}={90}^{\circ }$ (Mode I) to $0^{\circ }$$0^{\circ }$0^(@)0^{\circ}$0^{\circ }$ (Mode II) of the asymmetric TPB specimen are Von-Mises stress distribution, typically for yield stress 155 MPa [20] of the material. The contour lines appear to cross due to the superposition of multiple plastic zone boundaries for different loading angles ( ${\beta }_{\text{eq}}=0^{\circ },{75}^{\circ },{82}^{\circ },{84}^{\circ },{86}^{\circ }$${\beta }_{\text{eq }}=0^{\circ },{75}^{\circ },{82}^{\circ },{84}^{\circ },{86}^{\circ }$beta_("eq ")=0^(@),75^(@),82^(@),84^(@),86^(@)\beta_{\text {eq }}=0^{\circ}, 75^{\circ}, 82^{\circ}, 84^{\circ}, 86^{\circ}${\beta }_{\text{eq }}=0^{\circ },{75}^{\circ },{82}^{\circ },{84}^{\circ },{86}^{\circ }$, and ${90}^{\circ }$${90}^{\circ }$90^(@)90^{\circ}${90}^{\circ }$ ) plotted. For the same value of $K_{\text{eff}}=900$$K_{\text{eff }}=900$K_("eff ")=900K_{\text {eff }}=900$K_{\text{eff }}=900$ MPamm ${}^{1/2}$${}^{1/2}$^(1//2){ }^{1 / 2}${}^{1/2}$ from Fig. 6, one can observe that the size of PZ is larger for ${\beta }_{eq}=0^{\circ }$${\beta }_{eq}=0^{\circ }$beta_(eq)=0^(@)\beta_{e q}=0^{\circ}${\beta }_{eq}=0^{\circ }$ (Mode II) compared to larger ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$, and it will be smaller PZ for ${\beta }_{\text{eq}}={90}^{\circ }$${\beta }_{\text{eq }}={90}^{\circ }$beta_("eq ")=90^(@)\beta_{\text {eq }}=90^{\circ}${\beta }_{\text{eq }}={90}^{\circ }$ (Mode I). The shapes of the crack tip PZ for various ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ of an asymmetric TPB specimen obtained from FE analysis will closely match with the theoretical PZ shapes presented in Eqn. (5), as shown in Fig. 7, typically for $K_{\text{eff}}=900{\mathrm{MPamm}}^{1/2}$$K_{\text{eff }}=900{\mathrm{MPamm}}^{1/2}$K_("eff ")=900MPamm^(1//2)K_{\text {eff }}=900 \mathrm{MPamm}^{1 / 2}$K_{\text{eff }}=900{\mathrm{MPamm}}^{1/2}$. Earlier work demonstrated that the size of the crack tip PZ depends on the materials fracture toughness [9, 19]. So, based on the magnitude of the PZ form Fig. 6 and Fig. 7, the result clearly reveals that the loading for ${\beta }_{eq}={90}^{\circ }$${\beta }_{eq}={90}^{\circ }$beta_(eq)=90^(@)\beta_{e q}=90^{\circ}${\beta }_{eq}={90}^{\circ }$ is more dangerous than other ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$ for the asymmetric TPB specimen. In the present work, the major crack-tip PZ parameters will concentrate on i) the measurement of GMPZR and (ii) the crack initiation angle ${\theta }_o$${\theta }_o$theta_(o)\theta_{o}${\theta }_o$ at which GMPZR occurs, are schematically exemplified in Fig. 7.

For the analysis of GMPZR with normalized $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$, the GMPZR will be normalized by thickness ( $h$$h$hh$h$ ). The plot of normalized GMPZR vs. normalized $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ is shown in Fig. 8. This Figure demonstrates that the variation is linear, and the slope estimated will be nearly equal across different $a/W$$a/W$a//Wa / W$a/W$ and ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$. Form Fig. 8, the normalized GMPZR vs. normalized $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ shows a proportional relationship for all $a/W$$a/W$a//Wa / W$a/W$ and ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ of the asymmetric TPB specimen. Hence, one can establish the proportionality constant by fitting a straight-line equation to all the normalized GMPZR data. The estimated slope for the linear fit is 0.0035 for the asymmetric TPB specimen of various $a/W$$a/W$a//Wa / W$a/W$ and ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$. Mathematically, the relation between normalized GMPZR $\left(r_{\text{min}}\right)$$\left(r_{\text{min }}\right)$(r_("min "))\left(r_{\text {min }}\right)$\left(r_{\text{min }}\right)$ and normalized $K_{\text{eff}}$$K_{\text{eff }}$K_("eff ")K_{\text {eff }}$K_{\text{eff }}$ can be given as:

For an asymmetric TPB specimen with any $a/W$$a/W$a//Wa / W$a/W$ and ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$, if $K_{\text{eff}},{\sigma }_y$$K_{\text{eff }},{\sigma }_y$K_("eff "),sigma_(y)K_{\text {eff }}, \sigma_{y}$K_{\text{eff }},{\sigma }_y$, and $b$$b$bb$b$ are known in Eqn. (11), one can estimate the value of GMPZR, or vice versa. Consequently, the obtained Eqn. (11) can be of great use in GMPZR crack initiation criteria:

The crack initiation angle ( ${\theta }_0$${\theta }_0$theta_(0)\theta_{0}${\theta }_0$ ) is studied as a function of the equivalent loading ( ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ ) angle. The variation of ${\theta }_0$${\theta }_0$theta_(0)\theta_{0}${\theta }_0$ vs. ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ obtained in this analysis for various applied loads is compared with the theoretical GMPZR criterion (Eq. (5)), similar results of Khan and Khraisheh [19], Bian and Kim [9], MTS criterion [27], and GMTS criteria [16] in Fig. 9.
Fig. 9 shows that the FEA-measured value of ${\theta }_o$${\theta }_o$theta_(o)\theta_{o}${\theta }_o$ for the asymmetric TPB specimen decreases from ${83}^{\circ }$${83}^{\circ }$83^(@)83^{\circ}${83}^{\circ }$ to $0^{\circ }$$0^{\circ }$0^(@)0^{\circ}$0^{\circ }$ as ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$ varies from $0^0$$0^0$0^(0)0^{0}$0^0$ to ${90}^0$${90}^0$90^(0)90^{0}${90}^0$. From Fig. 9, it is clear that the FEA-measured ${\theta }_o$${\theta }_o$theta_(o)\theta_{o}${\theta }_o$ vs. ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ for the asymmetric TPB specimen is independent of the applied loads. Due to measurement of ${\theta }_o$${\theta }_o$theta_(o)\theta_{o}${\theta }_o$ difficulties in FEA analysis for an asymmetric TPB specimen, one can observe a slight variation in ${\theta }_o$${\theta }_o$theta_(o)\theta_{o}${\theta }_o$ for various ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$. The FE results of ${\theta }_o$${\theta }_o$theta_(o)\theta_{o}${\theta }_o$ vs. ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$ shown in Fig. 9 are in good agreement with the theoretical results estimated by Eq. (5). In Fig. 9, the measured values of ${\theta }_o$${\theta }_o$theta_(o)\theta_{o}${\theta }_o$ from FE analysis of asymmetric TPB specimen are closely matched with the analytical results of Khan and Khraisheh [19], and about $5^{\circ }-{10}^{\circ }$$5^{\circ }-{10}^{\circ }$5^(@)-10^(@)5^{\circ}-10^{\circ}$5^{\circ }-{10}^{\circ }$ higher than that of Bian and Kim [9] results for ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$ between ${50}^{\circ }-{70}^{\circ }$${50}^{\circ }-{70}^{\circ }$50^(@)-70^(@)50^{\circ}-70^{\circ}${50}^{\circ }-{70}^{\circ }$. This deviation in results between the FE and the Bian and Kim [9] work may be attributed to the specimen geometry ( ${\theta }_o$${\theta }_o$theta_(o)\theta_{o}${\theta }_o$ in the earlier work [9] was obtained on a CTS specimen). In Fig. 9, the asymmetric TPB FE results for ${\theta }^o$${\theta }^o$theta^(o)\theta^{o}${\theta }^o$ vs. ${\beta }_{\text{eq}}$${\beta }_{\text{eq }}$beta_("eq ")\beta_{\text {eq }}${\beta }_{\text{eq }}$ were compared with the experimental results for a CTS mixed-mode (I/II) fracture specimen as reported in the work of Bian and Kim [9]. The nature of variation is similar, but due to differences in specimen geometry, the present FE results for the asymmetric TPB specimen deviate more from available experimental results [9]. In Fig. 9, it is clearly observed that the FE results of the asymmetric TPB specimen will deviate from the MTS criterion [27], and the GMTS criterion [16] for the variation of ${\theta }_o$${\theta }_o$theta_(o)\theta_{o}${\theta }_o$ vs. ${\beta }_{eq}$${\beta }_{eq}$beta_(eq)\beta_{e q}${\beta }_{eq}$. This dissimilarity between the present FE criterion and the MTS [27] and GMTS [16] criteria results from the measurement of ${\theta }_0$${\theta }_0$theta_(0)\theta_{0}${\theta }_0$. In the present FE results, the measurement of ${\theta }_0$${\theta }_0$theta_(0)\theta_{0}${\theta }_0$ is in accordance with the von-mises criterion, but in the MTS [27] and GMTS [16] criteria, the measurement of ${\theta }_0$${\theta }_0$theta_(0)\theta_{0}${\theta }_0$ is based on the presence of a maximum tangential stress ahead of the crack tip.