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

Alberto Franchi; Pietro Crespi; Manuela Scamardo; Helen Miranda; Rejnalda Golemaj

doi:10.3390/math14071206

journal article Open Access Apr 03, 2026

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

Alberto Franchi Pietro Crespi

Manuela Scamardo

Helen Miranda

Rejnalda Golemaj

Mathematics Vol. 14 No. 7 pp. 1206 · MDPI AG

View at Publisher Save 10.3390/math14071206

Abstract

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.

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References

27

[1]

de Borst, R., Crisfield, M.A., Remmers, J.J.C., and Verhoosel, C.V. (2012). Nonlinear Finite Element Analysis of Solids and Structures, John Wiley & Sons, Ltd.. [2nd ed.]. 10.1002/9781118375938

[2]

Karihaloo "Tensile Response of Quasi-Brittle Materials" Pure Appl. Geophys. (1991) 10.1007/bf00879045

[3]

Bazant "Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories" J. Appl. Mech. (1993) 10.1115/1.2900839

[4]

Gastebled "Bifurcation in the Numerical Simulation of Softening Mechanisms" Comput. Struct. (2000) 10.1016/s0045-7949(00)00059-6

[5]

Dawud "Stability Analysis of a Competitive Ecological System in a Polluted Environment" Commun. Math. Biol. Neurosci. (2022)

[6]

Folker, H.W. (1995). Holonomic and Nonholonomic Simulations of Quasi-Brittle Fracture: A Comparative Study of Mathematical Programming Approaches. Proceedings of the Fracture Mechanics of Concrete Structures, Proceedings FRAMCOS-2, AEDIFICATIO Publishers.

[7]

Maier "A Quadratic Programming Approach for Certain Classes of Non Linear Structural Problems" Meccanica (1968) 10.1007/bf02129011

[8]

Best, M.J., and Ritter, K. (1985). Linear Programming: Active Set Analysis and Computer Programs, Prentice-Hall.

[9]

Maier "A Matrix Structural Theory of Piecewise Linear Elastoplasticity with Interacting Yield Planes" Meccanica (1970) 10.1007/bf02133524

[10]

Lemke "On Complementary Pivot Theory" Mathematics of the Decision Sciences: Part 1 (1968)

[11]

Hillerborg "Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements" Cem. Concr. Res. (1976) 10.1016/0008-8846(76)90007-7

[12]

Hillerborg, A. (1989, January 20–24). Mixed Mode Fracture in Concrete. Proceedings of the 7th International Conference On Fracture (ICF7), Houston, TX, USA. 10.1016/b978-0-08-034341-9.50232-1

[13]

Cohn, M.Z., and Maier, G. (1979). Historical Elastic-Plastic Analysis by PLCP and RBLP. Proceedings of the Engineering Plasticity by Mathematical Programming, Proceedings of the NATO Advanced Study Institute, Pergamon Press Inc.

[14]

Colonnetti "Elastic Equilibrium in the Presence of Permanent Set" Q. Appl. Math. (1950) 10.1090/qam/33732

[15]

Maier "Historical Deformation Analysis of Elastoplastic Structures as a Parametric Linear Complementarity Problem" Meccanica (1976) 10.1007/bf02133868

[16]

Hodge "Automatic Piecewise Linearization in Ideal Plasticity" Comput. Methods Appl. Mech. Eng. (1977) 10.1016/0045-7825(77)90072-x

[17]

Cottle "Complementary Pivot Theory of Mathematical Programming" Linear Algebra Appl. (1968) 10.1016/0024-3795(68)90052-9

[18]

Cottle "Monotone Solutions of the Parametric Linear Complementarity Problem" Math. Program. (1972) 10.1007/bf01584990

[19]

Scamardo "A Non-Standard Numerical Method for Finite Element Modelling of Tensile Cracks in Quasi-Brittle Material" Comput. Struct. (2022) 10.1016/j.compstruc.2021.106664

[20]

Bolzon "On Multiplicity of Solutions in Quasi-Brittle Fracture Computations" Comput. Mech. (1997) 10.1007/s004660050201

[21]

Franchi "On the Determination of Bifurcation and Limit Points" J. Eng. Mech. (1998) 10.1061/(asce)0733-9399(1998)124:8(866)

[22]

Bassi "A Linear Complementarity Formulation of Rate-Independent Finite-Strain Elastoplasticity. Part II: Calculation of Bifurcation and Limit Points" Eur. J. Mech. A/Solids (2012) 10.1016/j.euromechsol.2011.10.003

[23]

Tseng "Efficient Computation of Multiple Solutions in Quasibrittle Fracture Analysis" Comput. Methods Appl. Mech. Eng. (2003) 10.1016/s0045-7825(02)00645-x

[24]

Franchi "A Numerical Scheme for Integrating the Rate Plasticity Equations with an “a Priori” Error Control" Comput. Methods Appl. Mech. Eng. (1987) 10.1016/0045-7825(87)90138-1

[25]

Cottle "Polyhedral Sets Having a Least Element" Math. Program. (1972) 10.1007/bf01584992

[26]

Analysis of concrete fracture in “direct” tension

Jan G. Rots, René de Borst

International Journal of Solids and Structures 1989 10.1016/0020-7683(89)90107-8

[27]

Rodigari "A Linear Complementarity Approach to the Time Integration of Dynamic Elastic–Plastic Structural Problems" Meccanica (2019) 10.1007/s11012-019-01028-w

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Published: Apr 03, 2026
Vol/Issue: 14(7)
Pages: 1206
License: View

Authors

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Alberto Franchi