Stress concentrations due to pores and inclusions

Another important problem in elastic stress distributions, that relates to microstructures, is the disturbance of a stress field by an inclusion with different elastic properties, i.e., there is an elastic mismatch. Extreme examples occur when the inclusion is a pore or a rigid particle. Consider a plate under uniaxial tension that contains a circular hole. Fig. 4.20. As the hole surface is free from applied stresses, o rr-o re distances from the hole, the disturbance in 

This can be considered as the superposition of two stress fields, and where 

Each field must satisfy the boundary conditions at r=r, i.e., 

For the field o , try cos20, where f(r) is a function that depends only on 

The only stress at the surface of the hole is (Tgg=a l—2cos2d), which has a maximum value of 3cr at 0= -7t/2 and a minimum value of - tr at =0 and ir. In engineering design, stress concentrations in which applied stresses are amplified are very important, as failure is expected to initiate in such locations. It is important to clearly distinguish between residual stresses and stress concentrations, though both may be important in terms of failure processes. The stress field produced by the circular hole falls off fairly rapidly, primarily as 1/r and, thus, at r=3fp the stresses are approaching the applied values. 

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