By their very nature, joints represent geometric and/or material discontinuities, giving rise to stress concentrations. In the case of lap joints, such as one shown in the Figure 1 above, points A and B are locations where elastic stress analysis indicates that stress components asymptotically approach infinity as r→ 0. The order of the singularity depends on the mismatch of elastic properties of the materials 1 and 2 being joined, and on the angle φ. For the special case when materials 1 and 2 are isotropic, Bogy [1] provides the asymptotic stress field solutions. Asymptotic analyses, such as that of Bogy, provide the order of the stress singularity as r→ 0. However, such analyses do not provide estimates of the length scale over which such singular stress fields are dominant. As in modern fracture mechanics, an estimate of the size of this singularity-dominant region is important in the assessing the role of the singular stress field on local failure mechanisms. The size of the singularity-dominant region can be estimated only by full-field analysis. In the present work, this is done for titanium-titanium, MMC-titanium, MMC-MMC lap joints using the finite element method.