For the problem of mechanical properties of heterogeneous dissimilar metal welded joints, when analyzed by the finite element method, it is usually simplified into a “sandwich” material structure model. However, the mechanical properties of materials in different regions of the “sandwich” material mechanics model are different, and there will be mutations at the material interface. In order to accurately describe the mechanical properties of welded joints, the constitutive equations of dissimilar metal welded joint materials were compiled, and the constitutive equations of inhomogeneous materials whose material mechanical properties were continuously changed with space coordinates were established. The ABAQUS software was used to establish the “sandwich” model and the continuous transition model. The model is used to compare and analyze the crack tip stress distribution of different yield strength mismatch coefficients. The results show that the continuous transition material model eliminates the mutation of the “sandwich” model at the material interface and achieves the continuous change of the mechanical properties of the material. For the longitudinal crack, under the influence of different mismatch coefficients, the crack tip stress field of the transitional material model is deflected toward the low yield strength side. The compilation of constitutive equations for continuous transition materials of dissimilar metal welded joints provides a basis for the safety evaluation of dissimilar metal welded joints.