$4$ CONSIDER AN INFINITE SOUD, WHICH COUIAINS A SEMI$-$INFINITE CRACK ON THE $(x_1,x_3)$ PINNE. SUPPOSE THHT HE SOID DEFORMS IN PLANE STRAN

$(2)$

AND IS SUbJECTED TO BOWDEO STRESS AT INFINTY. THE STRESS FIEIO NEAR THE TIP OF THE CPAP CAN BE DERIVED FROM THEAIRY FUNCTON.

$=\frac{k_i}{3\sqrt[\phantom{2}]{2}}{r}^{\frac{3}{2}}(cos3\frac{}{2}+3cos\frac{}{2})-\frac{k11}{\sqrt[\phantom{2}]{2}}{r}^{\frac{3}{2}}(sin3\frac{}{2}+sin\frac{}{2})$

DERIEE THE STRESS EQUATIONS IN TERMS OF POLAR COORDINATES USING THE DEFINTIOUS ABOEE.

b$)$ CHECK IF THE ARY STRESS FUNCIONS SATSFY THE BHARMONIC CONOITION

$(\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}})+\frac{1}{r^2}\frac{de{l}^2}{\text{d}\text{e}\text{l}{}^2}{)}^2=0$