Equilibrium path bifurcation due to strain-softening localization in ellipsoidal region

A preceding study of the loss of stability of a homogeneous strain state in infinite homogeneous solid due to localization of strain into an ellipsoidal region is complemented by determining the condition of bifurcation of equilibrium path due to ellipsoidal localization mode. The bifurcation occurs when the tangential moduli matrix becomes singular, which coincides with Hill's classical bifurcation condition for localization into an infinite layer. The bifurcation is normally of Shanley type, occurring in absence of neutral equilibrium while the controlled displacements at infinity increase. During the loading process with displacement increase controlled at infinity, this type of bifurcation precedes the loss of stability of equilibrium due to an ellipsoidal localization mode, except when the tangential moduli change suddenly (which happens, e.g., when the slope of the stress-strain diagram is discontinuous, or when temperature is increased.