On the Plastic Strains in Slabs With Cutouts

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Abstract
The complete elastic-plastic problem is set up for a circular slab with a central circular cutout, subjected to uniform external tension. The analysis is carried out under the assumption of generalized plane stress but with possibly finite deformations. The material is assumed to be isotropic, homogeneous, and incompressible, to yield according to the Tresca yield condition, and to satisfy Hooke’s law in the elastic domain and the plastic potential law in the plastic domain. The equations are solved by a perturbation method based on the ratio of the maximum shear stress to the shear modulus, in which each of the significant quantities, stress, displacement, and slab thickness, is expanded in a power series in this ratio. A similar technique is employed to solve the fully plastic flow problem. It is shown that for cutout radii greater than 1 per cent of the radius of the slab, the “first approximations” obtained by neglecting all but the leading term in each series are satisfactory, up to loads at which the slab becomes wholly plastic. The ratio of the maximum strain in the just fully plastic slab to that in the completely elastic slab is computed. It is found that this ratio is less than about 6 if the cutout radius is at least 1 per cent of the radius of the slab. Some observations are advanced on the case where the cutout is very small compared to the slab.

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