(Solved): Mechanical Behavior of Materials A large plate is subjected to an applied stress \\( \\boldsymbol{S} ...

A large plate is subjected to an applied stress \\( \\boldsymbol{S} \\) in the y- direction, and another applied stress (0.5) \\( S \\) in the x-direction. This plate has an inclined crack in it. If \\( \\boldsymbol{S} \\) keeps increasing, the inclined crack is risking mixed-mode fracture (Mode I + Mode II). A handbook gives the stress intensity factors \\( K_{I} \\) and \\( K_{I I} \\) for a crack of half-length \\( a \\) inclined at an angle \\( \\theta \\) to the x-axis in a large plate subjected to a stress \\( S \\) applied in in the y-direction and another stress \\( \\boldsymbol{\\alpha} \\boldsymbol{S} \\) applied in the x-direction (where \\( \\boldsymbol{\\alpha} \\) is a number such as \\( 0.5,1.0,2.0, \\ldots \\) etc.): \\[ \\begin{array}{l} K_{I}=S \\sqrt{\\pi a}\\left(\\cos ^{2} \\theta+\\alpha \\sin ^{2} \\theta\\right) \\\\ K_{I I}=S \\sqrt{\\pi a}(1-\\alpha) \\sin \\theta \\cos \\theta \\end{array} \\] The angle of inclination \\( \\theta \\) is the angle formed by the \\( \\mathrm{x} \\)-axis and the crack line and is measured counterclockwise from the positive half of the \\( x \\)-axis. The Mixed-Mode Facture Criterion of the material is: \\[ \\left(\\frac{K_{I}}{K_{I C}}\\right)^{2}+\\left(\\frac{K_{I I}}{K_{I I C}}\\right)^{2}=1 \\quad\\left(K_{I C}=30 \\mathrm{MPa} \\sqrt{m}, K_{I I C}=\\frac{K_{I C}}{3}\\right), \\] For an inclined crack with \\( a=0.003 \\mathrm{~m} \\) and angle of inclination \\( \\theta=45^{\\circ} \\), determine the magnitude of \\( S \\) that gives a safety factor against mixed-mode fracture of \\( X_{K}=4 \\)