$4\mathrm{.}39$ An off$-$axis unidirectional lamina is loaded as shown in Fig. P$4\mathrm{.}39$ under uniaxial tension and in$-$plane shear at an angle to the fiber direction. Express the normal strains ${}_x$ and ${}_y$ in terms of the engineering properties $(E_x,E_y,G_{xy},v_{xy},v_{yx},{}_{xs},{}_{ys},{}_{sv},{}_{ss}).$ What relation must these engineering properties satisfy, so that for a certain ratio of $\frac{{}_s}{{}_x}$ the material behaves as an infinitely rigid one, that is$,{}_x={}_y=0?$ What is the stress ratio $k=\frac{{}_s}{{}_x}$ in that case?

Fig. P$4\mathrm{.}3$