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Stability Test for Multiplicity of Solutions in Finite Element Analysis of Cracking Structures
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DOI：10.3390/math14071206.png)
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Abstract
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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 Delta alpha(center dot) 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.
Keywords:
Parametric Linear Complementarity Problems
stability analysis
bifurcation and multiplicity
tableau-based algorithms
nonlinear finite element method
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