Stability Test for Multiplicity of Solutions in Finite Element Analysis of Cracking Structures

Quasi-brittle structures modeled with softening constitutive laws may lose the uniqueness of equilibrium, producing bifurcation and multiple admissible crack evolutions even under symmetric loading. This paper develops a stability test and a constructive multiplicity procedure for finite element cracking analyses formulated as a Parametric Linear Complementarity Problem (PLCP) solved in tableau form. The approach exploits the pivot sequence of a complementary tableau to monitor stability by tracking the positive definiteness of the reduced active-mode Hessian A through a complement condition, without eigenvalue computations. A direct relationship between loss of positive definiteness and the sign of the incremental load factor Δα is established, providing an intrinsic indicator of transition to descending response. When degeneracy occurs, a “void pivot” mechanism is introduced to generate an alternative admissible tableau, enabling a systematic construction of multiple isolated solutions associated with competing crack patterns. The method is demonstrated on a two-notched direct tension specimen with cohesive softening, where symmetric and antisymmetric paths emerge at a critical step. The implementation is compatible with parallelized matrix operations and remains effective in the presence of non-holonomic constraints.