Comptes Rendus
Short paper
Path-following methods for unstable structural responses induced by strain softening: a critical review
Comptes Rendus. Mécanique, Volume 350 (2022), pp. 205-236.

Path-following methods for describing unstable structural responses induced by strain-softening are discussed. The main ingredients of the formalisms introduced by Riks and Crisfield for arc-length methods for geometrical non-linearities are presented. A link between two ways (monolithic and partitioned) of solving the resulting augmented equilibrium problem is discussed based on the Sherman–Morrison formula. The original monolithic approach assumes that the path-following constraint equation is differentiable with respect to the unknown displacement field and load factor. However, when dealing with material non-linearities, it is often preferred to consider constraint equations controlling the maximum of a field defined on the computational domain (e.g., a scalar strain measure, the rate of variation of an internal variable of the constitutive model). In that case, differentiability cannot be guaranteed due to the presence of the maximum operator. This makes only the partitioned formulation usable. Several path-following constraint equations from the literature are presented, and the corresponding implementations in the finite element method are discussed. The different formulations are compared based on a simple two-dimensional test case of damage localization in a beam submitted to tension. A test case involving multiple snap-backs is illustrated, finally, to show the robustness of the considered formulations.

Des techniques de pilotage indirect du chargement pour décrire des réponses structurelles instables induites par l’adoucissement en déformation sont discutées. Après avoir rappelé les principaux ingrédients des formalismes introduits par Riks et Crisfield pour les méthodes de longueur d’arc pour le traitement de non-linéarités géométriques, un lien entre les deux façons (monolithique et partitionnée) de résoudre le problème d’équilibre augmenté qui en résulte est établi en se servant de la formule de Sherman–Morrison. L’utilisation de solveurs monolithiques repose sur l’hypothèse de différentiabilité des équations de pilotage par rapport aux inconnues du problème, le champ de déplacement et le facteur de chargement. Cependant, lorsqu’il s’agit de décrire des instabilités d’origines matérielles, des phénomènes locaux sont souvent responsables de la réponse instable obtenue au niveau global. Par conséquent, il est parfois préférable d’introduire des équations de pilotage permettant de contrôler le maximum d’un champ défini sur le domaine de calcul (e.g., la variation d’une mesure scalaire de déformation, le taux de variation d’une variable interne du modèle constitutif). Dans ce cas, la différentiabilité de l’équation de pilotage n’est plus garantie en raison de la présence de l’opérateur de maximum, ce qui rend utilisable uniquement la formulation partitionnée. Plusieurs exemples issus de la littérature scientifique pour les deux classes d’équations de pilotage sont présentés. Ensuite, on dérive les formulations d’éléments finis correspondantes et on décrit les avantages et inconvénients pour chacune d’elles. Une comparaison détaillée entre les différentes formulations est présentée sur la base d’un cas test bidimensionnel simple simulant la localisation de l’endommagement dans une poutre soumise à une sollicitation de traction. Un cas de test impliquant l’apparition de plusieurs snap-backs dans la courbe de réponse globale est enfin illustré pour montrer la robustesse des formulations considérées.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmeca.112
Keywords: Path-following methods, Monolithic and partitioned formulations, Dissipative constraint equations, Non-dissipative constraint equations, Material non-linearities
Mot clés : Méthodes de pilotage indirect du chargement, Formulations monolithique et partitionnée, Méthodes dissipatives, Méthodes non dissipatives, Non-linéarités matériaux

Giuseppe Rastiello 1; Hugo Luiz Oliveira 1; Alain Millard 1

1 Université Paris-Saclay, CEA, Service d’études mécaniques et thermiques, 91191, Gif-sur-Yvette, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMECA_2022__350_G2_205_0,
     author = {Giuseppe Rastiello and Hugo Luiz Oliveira and Alain Millard},
     title = {Path-following methods for unstable structural responses induced by strain softening: a critical review},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {205--236},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {350},
     year = {2022},
     doi = {10.5802/crmeca.112},
     zbl = {07526206},
     language = {en},
}
TY  - JOUR
AU  - Giuseppe Rastiello
AU  - Hugo Luiz Oliveira
AU  - Alain Millard
TI  - Path-following methods for unstable structural responses induced by strain softening: a critical review
JO  - Comptes Rendus. Mécanique
PY  - 2022
SP  - 205
EP  - 236
VL  - 350
PB  - Académie des sciences, Paris
DO  - 10.5802/crmeca.112
LA  - en
ID  - CRMECA_2022__350_G2_205_0
ER  - 
%0 Journal Article
%A Giuseppe Rastiello
%A Hugo Luiz Oliveira
%A Alain Millard
%T Path-following methods for unstable structural responses induced by strain softening: a critical review
%J Comptes Rendus. Mécanique
%D 2022
%P 205-236
%V 350
%I Académie des sciences, Paris
%R 10.5802/crmeca.112
%G en
%F CRMECA_2022__350_G2_205_0
Giuseppe Rastiello; Hugo Luiz Oliveira; Alain Millard. Path-following methods for unstable structural responses induced by strain softening: a critical review. Comptes Rendus. Mécanique, Volume 350 (2022), pp. 205-236. doi : 10.5802/crmeca.112. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.112/

[1] G. A. Wempner Discrete approximations related to nonlinear theories of solids, Int. J. Solids Struct., Volume 7 (1971) no. 11, pp. 1581-1599 | DOI | Zbl

[2] E. Riks The application of Newton’s method to the problem of elastic stability, J. Appl. Mech., Volume 39 (1972) no. 4, pp. 1060-1065 | DOI | Zbl

[3] E. Riks An incremental approach to the solution of snapping and buckling problems, Int. J. Solids Struct., Volume 15 (1979) no. 7, pp. 529-551 | DOI | MR | Zbl

[4] M. Crisfield An arc-length method including line searches and accelerations, Int. J. Numer. Methods Eng., Volume 19 (1983) no. 9, pp. 1269-1289 | DOI | MR | Zbl

[5] S. Skatulla; C. Sansour On a path-following method for non-linear solid mechanics with applications to structural and cardiac mechanics subject to arbitrary loading scenarios, Int. J. Solids Struct., Volume 96 (2016), pp. 181-191 | DOI

[6] K. Schweizerhof; P. Wriggers Consistent linearization for path following methods in nonlinear fe analysis, Comput. Methods Appl. Mech. Eng., Volume 59 (1986) no. 3, pp. 261-279 | DOI | Zbl

[7] E. Ramm Strategies for tracing the nonlinear response near limit points, Nonlinear Finite Element Analysis in Structural Mechanics (W. Wunderlich; E. Stein; K.-J. Bathe, eds.), Springer, Berlin, Heidelberg, 1981, pp. 63-89 | DOI | Zbl

[8] M. Ritto-Corrêa; D. Camotim On the arc-length and other quadratic control methods: Established, less known and new implementation procedures, Comput. Struct., Volume 86 (2008) no. 11, pp. 1353-1368 | DOI

[9] R. De Borst Computation of post-bifurcation and post-failure behavior of strain-softening solids, Comput. Struct., Volume 25 (1987) no. 2, pp. 211-224 | DOI | Zbl

[10] R. De Borst; M. A. Crisfield; J. J. C. Remmers; C. V. Verhoosel Nonlinear Finite Element Analysis of Solids and Structures, John Wiley & Sons, 2012 | DOI

[11] J. Napoleão; F. A. Elwi; D. Murray An eigenvector-based strategy for the analysis of inelastic structures, Comput. Struct., Volume 42 (1992) no. 5, pp. 833-848 | DOI

[12] Z. Chen; H. L. Schreyer A numerical solution scheme for softening problems involving total strain control, Comput. Struct., Volume 37 (1990) no. 6, pp. 1043-1050 | DOI

[13] M. G. D. Geers Enhanced solution control for physically and geometrically non-linear problems. Part I—the subplane control approach, Int. J. Numer. Methods Eng., Volume 46 (1999) no. 2, pp. 177-204 | DOI | Zbl

[14] T. Pohl; E. Ramm; M. Bischoff Adaptive path following schemes for problems with softening, Finite Elem. Anal. Des., Volume 86 (2014), pp. 12-22 | DOI | MR

[15] E. Lorentz; P. Badel A new path-following constraint for strain-softening finite element simulations, Int. J. Numer. Methods Eng., Volume 60 (2004) no. 2, pp. 499-526 | DOI | Zbl

[16] N. Singh; C. Verhoosel; R. de Borst; E. van Brummelen A fracture-controlled path-following technique for phase-field modeling of brittle fracture, Finite Elem. Anal. Des., Volume 113 (2016), pp. 14-29 | DOI | MR

[17] A. Stanić; B. Brank A path-following method for elasto-plastic solids and structures based on control of plastic dissipation and plastic work, Finite Elem. Anal. Des., Volume 123 (2017), pp. 1-8 | DOI | MR

[18] E. Barbieri; F. Ongaro; N. M. Pugno A J-integral-based arc-length solver for brittle and ductile crack propagation in finite deformation-finite strain hyperelastic solids with an application to graphene kirigami, Comput. Methods Appl. Mech. Eng., Volume 315 (2017), pp. 713-743 | DOI | MR | Zbl

[19] M. A. Gutiérrez Energy release control for numerical simulations of failure in quasi-brittle solids, Commun. Numer. Methods Eng., Volume 20 (2004) no. 1, pp. 19-29 | DOI | Zbl

[20] C. V. Verhoosel; J. J. C. Remmers; M. A. Gutiérrez A dissipation-based arc-length method for robust simulation of brittle and ductile failure, Int. J. Numer. Methods Eng., Volume 77 (2009) no. 9, pp. 1290-1321 | DOI | Zbl

[21] A. Fayezioghani; B. Vandoren; L. Sluys A posteriori performance-based comparison of three new path-following constraints for damage analysis of quasi-brittle materials, Comput. Methods Appl. Mech. Eng., Volume 346 (2019), pp. 746-768 | DOI | MR | Zbl

[22] G. Rastiello; F. Riccardi; B. Richard Discontinuity-scale path-following methods for the embedded discontinuity finite element modeling of failure in solids, Comput. Methods Appl. Mech. Eng., Volume 349 (2019), pp. 431-457 | DOI | MR | Zbl

[23] G. Rastiello; C. Giry; F. Gatuingt; R. Desmorat From diffuse damage to strain localization from an Eikonal Non-Local (ENL) continuum damage model with evolving internal length, Comput. Methods Appl. Mech. Eng., Volume 331 (2018), pp. 650-674 | DOI | MR | Zbl

[24] F. Thierry; G. Rastiello; C. Giry; F. Gatuingt One-dimensional Eikonal Non-Local (ENL) damage models: Influence of the integration rule for computing interaction distances and indirect loading control on damage localization, Mech. Res. Commun., Volume 110 (2020), 103620 | DOI

[25] K. Moreau; N. Moës; N. Chevaugeon; A. Salzman Concurrent development of local and non-local damage with the Thick Level Set approach: Implementation aspects and application to quasi-brittle failure, Comput. Methods Appl. Mech. Eng., Volume 327 (2017), pp. 306-326 | DOI | MR | Zbl

[26] G. Alfano; M. A. Crisfield Solution strategies for the delamination analysis based on a combination of local-control arc-length and line searches, Int. J. Numer. Methods Eng., Volume 58 (2003) no. 7, pp. 999-1048 | DOI | Zbl

[27] D. Bellora; R. Vescovini Hybrid geometric-dissipative arc-length methods for the quasi-static analysis of delamination problems, Comput. Struct., Volume 175 (2016), pp. 123-133 | DOI

[28] P. Massin; G. Ferté; A. Caron; N. Moës Pilotage du chargement en formulation X-FEM : application aux lois cohésives, 10e colloque national en calcul des structures (2011), p. Clé-USB

[29] J. Oliver; A. Huespe; J. Cante An implicit/explicit integration scheme to increase computability of non-linear material and contact/friction problems, Comput. Methods Appl. Mech. Eng., Volume 197 (2008) no. 21, pp. 1865-1889 | DOI | Zbl

[30] B. Brank; A. Stanić; A. Ibrahimbegovic A path-following method based on plastic dissipation control, Computational Methods for Solids and Fluids, Springer, Cham, 2016, pp. 29-47 | DOI

[31] F. Cazes; G. Meschke; M.-M. Zhou Strong discontinuity approaches: An algorithm for robust performance and comparative assessment of accuracy, Int. J. Solids Struct., Volume 96 (2016), pp. 355-379 | DOI

[32] A. Brencich; A. Carpinteri Interaction of a main crack with ordered distributions of microcracks: a numerical technique by displacement discontinuity boundary elements, Int. J. Fract., Volume 76 (1996) no. 4, pp. 373-389 | DOI

[33] A. Carpinteri; I. Monetto Snap-back analysis of fracture evolution in multi-cracked solids using boundary element method, Int. J. Fract., Volume 98 (1999) no. 3, pp. 225-241 | DOI

[34] H. L. Oliveira; G. Rastiello; A. Millard Partitioned path-following strategy for nonlinear structural analyses using the boundary element method, Comput. Methods Appl. Mech. Eng., Volume 394 (2022), 114875 | MR | Zbl

[35] J.-L. Batoz; G. Dhatt Incremental displacement algorithms for nonlinear problems, Int. J. Numer. Methods Eng., Volume 14 (1979) no. 8, pp. 1262-1267 | DOI | MR | Zbl

[36] M. Bonnet; A. Frangi; C. Rey The Finite Element Method in Solid Mechanics, McGraw Hill Education, New York, 2014

[37] I. Babuška The finite element method with Lagrangian multipliers, Numer. Math., Volume 20 (1973) no. 3, pp. 179-192 | DOI | MR | Zbl

[38] B. Richard; G. Rastiello; C. Giry; F. Riccardi; R. Paredes; E. Zafati; S. Kakarla; C. Lejouad CastLab: an object-oriented finite element toolbox within the Matlab environment for educational and research purposes in computational solid mechanics, Adv. Eng. Softw., Volume 128 (2019), pp. 136-151 | DOI

[39] H. L. Oliveira; G. Rastiello; A. Millard; I. Bitar; B. Richard Modular implementation framework of partitioned path-following strategies: Formulation, algorithms and application to the finite element software Cast3M, Adv. Eng. Softw., Volume 161 (2021), 103055 | DOI

[40] P. Verpeaux; A. Millard; T. Charras; A. Combescure A modern approach of large computer codes for structural analysis, IASMiRT, Proc. SMiRT 10 Conf., Anaheim, CA, USA, 1989

[41] J. Pellet Dualisation of the boundary conditions, 2011 (Code_Aster Open Source-General FEA Software)

[42] P. Pegon; A. Anthoine Numerical strategies for solving continuum damage problems with softening: application to the homogenization of masonry, Comput. Struct., Volume 64 (1997) no. 1–4, pp. 623-642 | DOI | Zbl

[43] S. May; J. Vignollet; R. de Borst A new arc-length control method based on the rates of the internal and the dissipated energy, Eng. Comput., Volume 33 (2016) no. 1, pp. 100-115 | DOI

[44] Z. Chen; H. Schreyer A numerical solution scheme for softening problems involving total strain control, Comput. Struct., Volume 37 (1990) no. 6, pp. 1043-1050 | DOI

[45] J. Lemaitre; R. Desmorat Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures, Springer-Verlag, Berlin, Heidelberg, 2005

[46] N. Moës; N. Chevaugeon Lipschitz regularization for softening material models: the Lip-field approach, C. R. Mécanique, Volume 349 (2021) no. 2, pp. 415-434 | DOI

[47] J. Mazars Application de la mécanique de l’endommagement au comportement non linéaire et à la rupture du béton de structure, Ph. D. Thesis, Université Pierre et Marie Curie-PARIS 6 (1984) (Thèse de docteur es sciences)

[48] J.-J. Marigo Formulation d’une loi d’endommagement d’un materiau élastique, C. R. Acad. Sci. Sér. II, Volume 292 (1981) no. 19, pp. 1309-1312 | MR | Zbl

[49] Z. Chen; H. Schreyer Secant structural solution strategies under element constraint for incremental damage, Comput. Methods Appl. Mech. Eng., Volume 90 (1991) no. 1, pp. 869-884 | DOI

[50] J. G. Rots; R. De Borst Analysis of concrete fracture in “direct” tension, Int. J. Solids Struct., Volume 25 (1989) no. 12, pp. 1381-1394 | DOI

[51] G. Rastiello; J.-L. Tailhan; P. Rossi; S. Dal Pont Macroscopic probabilistic cracking approach for the numerical modelling of fluid leakage in concrete, Ann. Solid Struct. Mech., Volume 7 (2015) no. 1, pp. 1-16 | DOI

[52] H. B. Hellweg; M. A. Crisfield A new arc-length method for handling sharp snap-backs, Comput. Struct., Volume 66 (1998) no. 5, pp. 704-709 | DOI | Zbl

[53] L. F. Paullo Muñoz; D. Roehl A Continuation method with combined restrictions for nonlinear structure analysis, Finite Elem. Anal. Des., Volume 130 (2017), pp. 53-64 | DOI

[54] A. Fayezioghani; B. Vandoren; L. Sluys Performance-based step-length adaptation laws for path-following methods, Comput. Struct., Volume 223 (2019), 106100 | DOI | Zbl

[55] M. J. Clarke; G. J. Hancock A study of incremental-iterative strategies for non-linear analyses, Int. J. Numer. Methods Eng., Volume 29 (1990) no. 7, pp. 1365-1391 | DOI

[56] M. G. D. Geers Enhanced solution control for physically and geometrically non-linear problems. Part II—comparative performance analysis, Int. J. Numer. Methods Eng., Volume 46 (1999) no. 2, pp. 205-230 | DOI | Zbl

[57] J. Sherman; W. J. Morrison Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Statist., Volume 21 (1950) no. 1, pp. 124-127 | DOI | MR | Zbl

Cited by Sources:

Comments - Policy