The finite element method has been demonstrated previously to provide an effective means for the analysis of two dimensional elastic and plastic continua. The method is applied herein to the axially symmetric, solid, and is extended to fracture analysis. The numerical analysis may be broken into three parts. The first involves the role of linear elasticity, the second deals with the elastic-plastic deformations, and the third is concerned with the interaction between such deformation and the fracture process. For the finite solid element method, the crack is thought to initiate below the surface of a notch, approximately in the region of highest triaxiality of the stress σ_ii under the ultimate load, rather than at the root of the notch. Thus the maximum value of σ_ii at ultimate load becomes the assumed criterion for the first phase of the fracture; i.e., the brittle fracture. The crack propagates inward and outward until the octahedral shear stress 𝜏_C, in the remainder of the cross section are above the triaxial stresses. At this stage, the specimen fails in shear and a shear lip forms at the root of the notch, provided it is not exceptionally sharp, and at the inner circumference of the hole of the hollow notched specimens. Thus 𝜏_C becomes the criterion of fracture during this second stage. For the lattice analogy method, the fracture is assumed to progress as each critically stresses member reaches its ultimate and is deleted from the remaining assembly. Both the finite solid element method and the lattice analogy method are used to analyze numerically the solid and hollow notched tensile specimens with the ultimate load held constant during fracture. The finite solid element analysis was also made with this load decreased so as to keep σ_ii and/or 𝜏_C constant at their original maximum values. Comparisons with analytical and exp7rimental results are made and found to be satisfactory. / Ph. D.