An adaptive multi-scale approach to the modelling of masonry structures. (English) Zbl 1183.74179
Summary: We present an adaptive multi-scale approach for predicting the mechanical behaviour of masonry structures modelled as dynamic frictional multi-body contact problems. In this approach, the iterative splitting of the contact problem into normal contact and frictional contact is combined with a semismooth Newton/primal-dual active-set procedure to calculate deformations and openings in the model structures. This algorithm is then coupled with a novel adaptive multi-scale technique involving a macroscopic scale, which is the size of the masonry structure, and a mesoscopic scale, which is the size of the constituents (bricks, stone-blocks), to predict appearance of dislocations and stress distribution in large-scale masonry structures. Comparisons of the numerical results with data from experimental tests and from practical observations illustrate the predictive capability of the multi-scale algorithm.
MSC:
| 74M15 | Contact in solid mechanics |
| 74M10 | Friction in solid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
Keywords:
dynamic contact; Coulomb friction; finite elements; optimization problems; linear elasticity; masonry structures; contact problem; adaptive algorithmReferences:
| [1] | Nečas, On the solution of the variational inequality to the Signorini problem with small friction, Bollettino dell’Unione Matematica Italiana 5 (17-B) pp 796– (1980) |
| [2] | Boieri, Existence, uniqueness and regularity results for the two-body contact problem, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics 15 pp 251– (1987) |
| [3] | Alart, Mixed formulation for frictional contact problems prone to Newton like solution methods, Computer Methods in Applied Mechanics and Engineering 92 pp 353– (1991) · Zbl 0825.76353 |
| [4] | Czekanski A. Novel nonlinear finite element analysis of dynamic contact problems using variational inequalities. Ph.D. Thesis, University of Toronto, 2001. · Zbl 1104.74320 |
| [5] | Hlaváček, Solution of Variational Inequalities in Mechanics (1988) · Zbl 0654.73019 · doi:10.1007/978-1-4612-1048-1 |
| [6] | Kikuchi, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods (1988) · Zbl 0685.73002 · doi:10.1137/1.9781611970845 |
| [7] | Eck, Unilateral Contact Problems-Variational Methods and Existence Theorems (2005) · Zbl 1079.74003 · doi:10.1201/9781420027365 |
| [8] | Acary, Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics (2008) · Zbl 1173.74001 |
| [9] | Lourenço, Computations on historic masonry structures, Progress in Structural Engineering and Materials 4 pp 301– (2002) |
| [10] | Lourenço PB. Computational strategies for masonry structures. Ph.D. Thesis, Faculty of Engineering, Delft University, The Netherlands, 1996. |
| [11] | Kuhn, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (1951) |
| [12] | Coulomb, Théorie des machines simples, en ayant égard au frottement de leurs parties et à la roideur des cordages, Mémoirs de Mathématique et de Physique, présentés à l’Académie Royale des Sciences par divers Savants, et lus dans ses assemblées 10 pp 163– (1785) |
| [13] | Heyman, Coulomb’s Memoir on Statics: An Essay in the History of Civil Engineering (1997) · Zbl 0268.01007 · doi:10.1142/p046 |
| [14] | Masiani, Masonry as structured continuum, Meccanica 30 pp 673– (1995) · Zbl 0840.73008 |
| [15] | Cecchi, Homogenization of masonry walls with a computational oriented procedure. Rigid or elastic block?, European Journal of Mechanics A-Solids 19 pp 535– (2000) · Zbl 0964.74050 |
| [16] | Alpa, Microstructural model for dry block masonry walls with inplane loading, Journal of the Mechanics and Physics of Solids 42 pp 1159– (1994) · Zbl 0806.73057 |
| [17] | Anthoine, Derivation of the in-plane elastic characteristics of masonry through homogenization theory, International Journal of Solids and Structures 32 pp 137– (1995) · Zbl 0868.73010 |
| [18] | Pegon, Numerical strategies for solving continuum damage problems with softening: application to the homogenisation of masonry, Computers and Structures 64 pp 623– (1997) · Zbl 0919.73004 |
| [19] | Luciano, Homogenization technique and damage model for old masonry material, International Journal of Solids and Structures 34 pp 3191– (1997) · Zbl 0942.74610 |
| [20] | Lourenço, A plane stress softening plasticity model for orthotropic materials, International Journal for Numerical Methods in Engineering 40 pp 4033– (1997) · Zbl 0897.73015 |
| [21] | Lourenço, Continuum model for masonry: parameter estimation and validation, Journal of Structural Engineering 124 (6) pp 642– (1998) |
| [22] | Alfano, A numerical strategy for finite element analysis of no-tension materials, International Journal for Numerical Methods in Engineering 48 pp 317– (2000) · Zbl 0991.74064 |
| [23] | Massart TJ. Multi-scale modelling of damage in masonry structures. Ph.D. Thesis, Technische Universiteit Eidhoven, The Netherlands, 2003. |
| [24] | Milani, Homogenised limit analysis of masonry walls. Part I: failure surfaces, Computers and Structures 84 pp 181– (2006) |
| [25] | Milani, Homogenised limit analysis of masonry walls. Part II: structural examples, Computers and Structures 84 pp 166– (2006) |
| [26] | Heyman, The Stone Skeleton-Structural Engineering of Masonry Architecture (1995) · doi:10.1017/CBO9781107050310 |
| [27] | Goodman, A model for the mechanics of jointed rock, Journal of Soil Mechanics and Foundations Division 94 pp 637– (1968) |
| [28] | Page, Finite element model for masonry, Journal of Structural Engineering 104 pp 1267– (1978) |
| [29] | Maier, A theory of no-tension discretized structural systems, Engineering Structures 12 pp 227– (1990) |
| [30] | Jean, The non-smooth contact dynamics method. Computational modeling of contact and friction, Computer Methods in Applied Mechanics and Engineering 177 (3-4) pp 235– (1999) |
| [31] | Giambanco, Numerical analysis of masonry structures via interface models, Computer Methods in Applied Mechanics and Engineering 190 pp 6493– (2001) · Zbl 1116.74341 |
| [32] | Acary V. Contribution à la modélisation mécanique et mumérique des édifices maçonnés. Mémoire de Thèse de Doctorat, Ecole Supérieure de Mécanique de Marseille, Université d’Aix-Marseille II, France, 2001. |
| [33] | Cundall, A discrete numerical model for granular assemblies, Geotechnique 29 pp 47– (1979) |
| [34] | Moreau, Some numerical methods in multibody dynamics: application to granular materials, European Journal of Mechanics A-Solids 13 (4) pp 93– (1994) · Zbl 0815.73009 |
| [35] | Ainsworth, Modeling and numerical analysis of masonry structures, Numerical Methods for Partial Differential Equations. An International Journal 23 (4) pp 798– (2007) · Zbl 1115.74036 |
| [36] | Oliveira DV. Experimental and numerical analysis of blocky masonry structures under cycling loading. Ph.D. Thesis, Department of Civil Engineering, University of Minho, Portugal, 2003. |
| [37] | Laursen, Continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems, International Journal for Numerical Methods in Engineering 36 (20) pp 3451– (1993) · Zbl 0833.73057 |
| [38] | Simo, Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms, International Journal for Numerical Methods in Engineering 26 pp 2161– (1988) · Zbl 0661.73058 |
| [39] | Ainsworth, A comparison of solvers for linear complementarity problems arising from large-scale masonry structures, Applications of Mathematics 51 (2) pp 93– (2006) · Zbl 1164.74355 |
| [40] | Duvaut, Inequalities in Mechanics and Physics (1976) · doi:10.1007/978-3-642-66165-5 |
| [41] | Glowinski, Numerical Analysis of Variational Inequalities (1981) · Zbl 0463.65046 |
| [42] | Chawla, Energy consistent algorithms for frictional contact problem, International Journal for Numerical Methods in Engineering 42 (5) pp 799– (1998) · Zbl 0916.73078 |
| [43] | Laursen, Design of energy conserving algorithms for frictionless dynamic contact problems, International Journal for Numerical Methods in Engineering 40 (5) pp 863– (1997) · Zbl 0886.73067 |
| [44] | Bajer, Dynamic contact/impact problems, energy conservation, and planetary gear trains, Computer Methods in Applied Mechanics and Engineering 191 (37-38) pp 4159– (2002) · Zbl 1083.74566 |
| [45] | Czekanski, Analysis of dynamic frictional contact problems using variational inequalities, Finite Elements in Analysis and Design 37 (11) pp 861– (2001) · Zbl 1104.74320 |
| [46] | Haslinger, On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction, Computer Methods in Applied Mechanics and Engineering 191 (21-22) pp 2262– (2002) |
| [47] | Dostál, Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique, Journal of Computational and Applied Mathematics 140 (1-2) pp 245– (2002) · Zbl 1134.74418 |
| [48] | Hintermüller, The primal-dual active set strategy as a semismooth Newton method, SIAM Journal on Optimization 13 (3) pp 865– (2003) · Zbl 1080.90074 |
| [49] | Hintermüller, The primal-dual active set method for a crack problem with non-penetration, IMA Journal of Applied Mathematics 69 pp 1– (2004) · Zbl 1084.49029 |
| [50] | Hintermüller, Generalized Newton methods for crack problems with nonpenetration condition, Numerical Methods for Partial Differential Equations. An International Journal 21 (3) pp 586– (2005) · Zbl 1086.74045 |
| [51] | Stadler G. Infinite-dimensional semi-smooth Newton and augmented Lagrangian methods for friction and contact problems in elasticity. Ph.D. Thesis, Institute for Mathematics, Karl Franzens University Graz, Austria, 2004. |
| [52] | Kunisch, Generalized Newton methods for the 2D-Signorini contact problem with friction in function space, Mathematical Modelling and Numerical Analysis 39 (4) pp 827– (2005) · Zbl 1330.74132 |
| [53] | Lourenço, Dry joint stone masonry walls subjected to in-plane combined loading, Journal of Structural Engineering 131 (11) pp 1665– (2005) |
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