The regularities of the effect of a complex stress state on the strength of an AlMg5/epoxy adhesive joint are experimentally studied at −50 and +23 °C in tension+shear and compression+shear tests with different normal-to-shear stress ratios. The tests use modified Arcan specimens and Brazil-nut-sandwich specimens, with the lateral faces of the adhesive layer having a shape of a mushroom-like “ridge” aimed at reducing stress concentration at the specimen edges. An original computational model of a selected microvolume including the interface together with the adjacent substrate and adhesive layers is used to process the experimental results. The attainment of the threshold value of strain energy density in the selected microvolume, W*, is used as the failure criterion. The effect of the hardener composition, the testing temperature, and the value of the phase angle β determining the proportion of normal and shear stresses at the adhesive interface on the threshold value W* is detected. W*(β) diagrams (fracture loci) are plotted and analytically described logarithmic functions. They can be used to make strength calculations for adhesive joints in structures and metal-polymer composites.