Page 304 - Introduction to Continuum Mechanics
P. 304

288 Stress Concentration due to a Smalt Circular Hole in a Plate under Tension

        consider the region between two concentric circles: r = a and r = b. The surface r = a is
        traction-free, i.e.,



























                                             Fig. 5.18




        If b is much larger than a, then the effect of the small hole will be negligible on points lying on
        the surface r = b so that the state of stress at r = b as a/b~*Q will be that due to the uniaxial
        tensile stress a in the absence of the hole. In Cartesian coordinates, this state of stress is
        TH = o with all other stress components zero. In cylindrical coordinates this same state of
        stress has the following nonzero stress components




        Thus, the stress vector acting on the surface r = b has the r-component and Q—component
        given by




        and




        Therefore, the solution to the problem at hand can be obtained as follows: Find the elastos-
        tatically possible equilibrium plane stress field which satisfies the boundary conditions: (i) at
   299   300   301   302   303   304   305   306   307   308   309