Numerical Methods for Strong Nonlinearities in Mechanics E-Book
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- Herausgeber: John Wiley & Sons
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- Sprache: Englisch
Numerical Methods for Strong Nonlinearities in Mechanics deals with recent advances in the numerical treatment of contact/friction and damage phenomena. Although physically distinct, these phenomena both lead to a strong nonlinearity in the mechanical problem, therefore limiting the regularity of the problem, which is now non-differentiable.
This has two direct consequences: on the one hand, the mathematical characteristics of the problem deviate from wellestablished forms, requiring innovative discretization schemes; on the other hand, the low regularity makes it particularly difficult to solve the corresponding large-scale algebraic systems robustly and efficiently. In addition, neither the uniqueness, nor the existence of solutions, remain assured, resulting in bifurcation points, limit loads and structural instabilities, which are always tricky to overcome numerically.
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Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright Page
Preface
PART 1: Contact and Friction
1 Lagrangian and Nitsche Methods for Frictional Contact
1.1. Introduction
1.2. Small-strains frictional contact between two elastic bodies
1.3. Finite element approximation in small deformations
1.4. Large strain finite element approximation
1.5. Acknowledgments
1.6. References
2 High-performance Computing in Multicontact Mechanics: From Elastostatics to Granular Dynamics
2.1. Introduction
2.2. Multicontact in elastostatics
2.3. Diffuse non-smoothness in discrete structures: tensegrity
2.4. Granular dynamics
2.5. Conclusion
2.6. References
3 Numerical Methods in Micromechanical Contact
3.1. Introduction
3.2. Contact micromechanical problem
3.3. Finite element method
3.4. Application I: study of an isolated asperity
3.5. Application II: rough surface contact
3.6. Conclusion
3.7. References
PART 2: Damage and Cracking
4 Numerical Methods for Ductile Fracture
4.1. Introduction
4.2. Physical mechanisms of ductile fracture
4.3. Some ductile fracture models
4.4. Performing ductile fracture simulations with a finite elements code
4.5. Localization origin
4.6. Regularization methods
4.7. Conclusion
4.8. References
5 Quasi-brittle Fracture Modeling
5.1. What are the approaches for predicting quasi-brittle fracture?
5.2. Materials with internal lengths
5.3. Non-local formulations
5.4. Phenomenological aspects of quasi-brittle behavior
5.5. Numerical solving methods
5.6. Conclusion
5.7. References
6 Extended Finite Element (XFEM) and Thick Level Set (TLS) Methods
6.1. Introduction
6.2. Categorization of approaches to cracking
6.3. The XFEM method for cracking in non-softening media
6.4. XFEM-TLS for cracking in softening media
6.5. XFEM-TLS simulation examples
6.6. Conclusion
6.7. References
7 Damage-to-Crack Transition
7.1. Introduction
7.2. Localizing discontinuity
7.3. Inserting a discontinuity
7.4. Resuming computations after inserting a discontinuity
7.5. Conclusion
7.6. References
List of Authors
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1. Performance of EBE and ILU preconditioners (source: Alart et al. 20...
Table 2.2. 3D tensegrity grid with 144 subdomains – number of iterations to ac...
Table 2.3. Total number of FETI-NLGS iterations, average number of interface b...
Table 2.4. Comparison of the percentage of time spent in message exchanges for...
Chapter 4
Table 4.1. GTN model parameters used in calculations for structural steel
Chapter 7
Table 7.1. Operations required to obtain the fields on which the orientation c...
Table 7.2. Summary of input and output fields for the global method
Table 7.3. Summary of input and output fields for the Crack Path Field method
Table 7.4. Summary of input and output fields for the Marching Ridges method
Table 7.5. Summary of input and output fields for the method exploiting a poin...
Table 7.6. Summary of input and output fields for the projection maximum metho...
Table 7.7. Summary of input and output fields for the fitting method
Table 7.8. Characterization of the 3D surface construction methods according t...
List of Illustrations
Chapter 1
Figure 1.1. Two bodies with their respective potential contact boundaries.
Figure 1.2. Large-strain Lagrangian description
Figure 1.3. Illustration of projection and raytracing strategies
Figure 1.4. Example of discontinuity of the normal n2 with respect to x: (a) wh...
Figure 1.5. Deformation of the elastic half-ring coming into contact without (a...
Figure 1.6. Vertical displacement of the half-ring midpoint for different mesh ...
Figure 1.7. Geometry and mesh of hollow tubes in their undeformed configuration...
Figure 1.8. Von-Mises strain and stress of the two crossed tubes. Frictionless ...
Chapter 2
Figure 2.1. Subdomain decomposition of a collection of deformable hexagonal gra...
Figure 2.2. Honeycomb cell medium decomposition into one subdomain per cell (in...
Figure 2.3. Overall contact status at the internal contact interfaces to a subd...
Figure 2.4. Additional rigid modes associated with the overall status in Figure...
Figure 2.5. Numerical scalability of the different preconditioners for two stre...
Figure 2.6. Impact of the friction coefficient on the behavior of the solver fo...
Figure 2.7. Perfect substructuring-interface (nodes) – three macro displacement...
Figure 2.8. Shear of 10,000 pentagonal particles generating around 39,800 conta...
Figure 2.9. Shear of 10,000 pentagonal particles, study according to the number...
Figure 2.10. Convergence of CPG and NLGS algorithms on a time step for a two-di...
Figure 2.11. Convergence of CPG and NLGS algorithms on a time step for a two-d...
Figure 2.12. Convergence of CPG, PCPG (CPG with diagonal preconditioning), PG (...
Figure 2.13. “Box” distribution: application to a two-dimensional aggregate and...
Figure 2.14. Primal (a) and dual partitioning (b)
Figure 2.15. Different decompositions at their initial state for a biaxial test...
Figure 2.16. Granular flow: evolution of the average (a) and maximum (b) interp...
Figure 2.17. Granular flow: configuration of interpenetrations at the beginning...
Chapter 3
Figure 3.1. Contact problem between two bodies on the surface: (a) the macrosco...
Figure 3.2. Different geometries of natural surfaces
Figure 3.3. Hertz problem with friction
Figure 3.4. Finite element meshes for (a) 2D fatigue fretting contact (source: ...
Figure 3.5. Comparison of normal stresses σyy with (b) node-to-surface (NTS) an...
Figure 3.6. Cyclic loading of an elastoplastic “asperity” of the initial radius...
Figure 3.7. Stress under Hertzian contact
Chapter 4
Figure 4.1. Ductile fracture in pipeline steel (X100). (a) Fracture by nucleati...
Figure 4.2. Mesh geometry and examples (minimum size 200 μm): (a) circumferenti...
Figure 4.3. Comparison of stress opening fields at crack tip for a formulation...
Figure 4.4. Application of the Rice and Tracey criterion for NT and CT specimen...
Figure 4.5. Application of the GTN model for NT and CT specimens for mesh sizes...
Figure 4.6. Simulation of a Compact Tension (CT) specimen: opening stress at th...
Figure 4.7. Simulation of a Compact Tension (CT) specimen: crack tip damage dur...
Figure 4.8. Effect of mesh orientation on the crack path.
Figure 4.9. Geometry of a localization band
Figure 4.10. Damage profile, gradient and Laplacian
Figure 4.11. Effect of mesh orientation on the crack path, and nonlocal enhance...
Figure 4.12. Normalized force-diameter reduction response (S
0
initial section)....
Figure 4.13. Force-CMOD curves for the CT specimen.
Figure 4.14. Gaussian point values of opening stresses (σ
22
) and damage ...
Figure 4.15. Cross-section by synchrotron radiation tomography of an interrupte...
Figure 4.16. Schematic view of the transition flat fracture/slanted fracture in...
Figure 4.17. Refinement example of the mesh being computed. The aim here is to ...
Chapter 5
Figure 5.1. Damage localization (in red) in a uniaxial tensile bar.
Figure 5.2. Problems related to the definition of an elementary representative...
Figure 5.3. Reinforced concrete: distribution of cracks (in red and blue) due ...
Figure 5.4. Location band dependency: (a) on mesh size and (b) mesh orientatio...
Figure 5.5. Model problem for evaluating the effectiveness or the lack thereof...
Figure 5.6. Fracture process zone.
Figure 5.7. Simulation of localization bands during the excavation of a tunnel...
Figure 5.8. Simulation of a fracture test on a pre-stressed concrete wall unde...
Figure 5.9. Concrete notched specimens: orientation of the localization bands ...
Figure 5.10. Damage profile in a localization band which develops along a load...
Figure 5.11. Uniaxial tension: damage gradient model response.
Figure 5.12. Cohesive model asymptotic response when W
FPZ
→ 0.
Figure 5.13. Virtual pressurized cavity: damage gradient model versus cohesive...
Figure 5.14. Problem of tracking eigendirections over time (in blue, eigendire...
Figure 5.15. Limited effect of anisotropy in the presence of damage localizati...
Figure 5.16. Effect of the choice of unilateral model (from three variants) on...
Figure 5.17. Uniaxial tension: homogeneous and localized responses with a stra...
Figure 5.18. Uniaxial tension: homogeneous and localized responses with a dama...
Figure 5.19. Structure under opening and shear loading: compatible mesh adapta...
Figure 5.20. Schematic representation of total energy according to state varia...
Figure 5.21. Schematic response of a structure in the presence of instability.
Figure 5.22. Viscoelastic regularization
Figure 5.23. Different patterns of totally damaged finite elements and free de...
Chapter 6
Figure 6.1. A crack placed on a 2D mesh. Circled nodes denote the nodes enrich...
Figure 6.2. Local axes for polar coordinates at crack tips
Figure 6.3. Crack localization by two level functions (level sets). A series o...
Figure 6.4. Enrichment vector functions
G
1
and
G
2
.
Figure 6.5. Geometry of the Brokenshire torsion test
Figure 6.6. Deformation and stress level.
Figure 6.7. Experimental crack path
Figure 6.8. Damaged area emanating from a reentering corner. The skeleton is i...
Figure 6.9. A parabolic damage profile .
Figure 6.10. Dependency on volume D and surface damage d according to the dist...
Figure 6.11. View of a crack in deformed configuration in the TLS V2 approach
Figure 6.12. The different examples of double slicing for a tetrahedron.
Figure 6.13. Geometry and loading for the chalk problem under torsion (source:...
Figure 6.14. Final state of the chalk after fracture. The color indicates the ...
Figure 6.15. Localization of hollows in the cube, sizes in mm (source: Salzman...
Figure 6.16. Crack opening during cube cracking (source: Salzman et al. 2015).
Figure 6.17. Final damage state at yield. The blue color indicates the localiz...
Figure 6.18. Symmetric geometry of the three-point bending problem (source: Le...
Figure 6.19. Beam deformation with tensile forces (a) and enlargement of the d...
Chapter 7
Figure 7.1. Diagram illustrating the terminology associated with the represent...
Figure 7.2. Damage (left) and vertical displacement field (right) distribution...
Figure 7.3. Damage (left) and vertical displacement field (right) distribution...
Figure 7.4. Crack path obtained with the Marching Ridges technique presented i...
Figure 7.5. Damage (left) and vertical displacement field (right) distribution...
Figure 7.6. Damage (left) and vertical field displacement (right) distribution...
Figure 7.7. The orientation criteria use either a vector field (in blue) or a ...
Figure 7.8. Diagram summarizing the different methods for obtaining a continuo...
Figure 7.9. Schematic representation of the idea behind the global method to m...
Figure 7.10. Schematic representation of the idea on which the Crack Path Fiel...
Figure 7.11. Schematic representation of the idea on which the Marching Ridges...
Figure 7.12. Schematic diagram representing the discretization of a crack fron...
Figure 7.13. Diagram summarizing the different methods to obtain a three-dimen...
Figure 7.14. Diagram representing the construction of an explicit crack mesh, ...
Figure 7.15. Diagram illustrating crack advance in time based on an insertion ...
Figure 7.16. Initial mesh (a) and adapted mesh based on error estimation (b). ...
Figure 7.17. Iso-values of the vertical displacement field U
2
before remeshing...
Guide
Cover Page
Table of Contents
Title Page
Copyright Page
Preface
Begin Reading
List of Authors
Index
WILEY END USER LICENSE AGREEMENT
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SCIENCES
Mechanics, Field Director – Gilles Pijaudier-Cabot
Numerical Methods in Mechanics, Subject Head – Sylvain Drapier
Numerical Methods for Strong Nonlinearities in Mechanics
Contact and Fracture
Coordinated by
Jacques Besson
Frédéric Lebon
Eric Lorentz
First published 2024 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2024The rights of Jacques Besson, Frédéric Lebon and Eric Lorentz to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2024941794
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78945-081-1
ERC code:PE8 Products and Processes Engineering PE8_7 Mechanical and manufacturing engineering (shaping, mounting, joining, separation) PE8_8 Materials engineering (biomaterials, metals, ceramics, polymers, composites, etc.) PE8_9 Production technology, process engineering
Preface
Jacques BESSON1, Frédéric LEBON2 and Éric LORENTZ3
1 CNRS, Mines Paris PSL, France
2 LMA, Aix-Marseille Université, France
3 EDF Lab Paris-Saclay, EDF R&D/ERMES, Palaiseau, France
For 40 years now, numerical simulation has become essential for studying the behavior of structures, from design to operational analyses, and in many industrial fields (building, transport, energy, manufacturing, etc.). Along with the advances in the physics of materials and computer science, it allows for a more realistic description than ever before. Nonetheless, two phenomena raise numerical difficulties to an extent that they still remain, to this day, the main subject of a large body of research: these are the questions concerning contact and friction, on the one hand, and damage mechanics, on the other hand. This is all the more critical considering that these mechanisms are often involved in the response of a structure to external loading. And due to the lack of robust numerical solutions, the design engineer may be compelled to overlook them – or at least to approach them in a crude manner – at the risk of the simulation incurring potential loss of reliability, which in turn requires that more margins of error be taken, usually with an impact on economic and ecological costs.
Although physically distinct, the phenomena of frictional contact and damage both result in a strong nonlinearity of the mechanical problem. They are actually characterized by a threshold behavior (detachment/contact; elasticity/damage) that restricts the regularity of the problem, now non-differentiable (non-smooth mechanics). In addition, this loss of regularity has the disadvantage of involving the main variables of the problem (displacement, speed and damage), with two direct consequences: on the one hand, the mathematical characteristics of the problem shift away from well-controlled forms and require in particular innovative discretization schemes, and, on the other hand, the low regularity persists at the scale of the degrees of freedom of the discretized problem. This makes it particularly difficult for solving the corresponding large algebraic systems in a robust and efficient manner. Moreover, neither uniqueness nor even the existence of solutions can be established, which is reflected by the presence of bifurcation points, limit loads and structural instabilities, which prove always challenging to overcome at the numerical level. Finally, the coupling of these phenomena with other nonlinearities such as plasticity or large deformations results in further weakening the entire numerical structure.
As far as couplings are concerned, the phenomena of contact-friction and damage also interact with each other. In fact, during the final stage of damage, macroscopic cracks appear. It is then necessary to solve the problem of frictional contact between the crack lips. This is also the case at the microscopic scale, where damage can be interpreted – depending on the materials – as a network of microcracks or decohesion between particles and matrix: the contact and friction conditions then have a direct impact on the damage model, in particular its unilateral response according to whether the microcracks open or close. Finally, some damage models aim to concentrate degradation phenomena on surfaces rather than distributing them in the bulk. These are then referred to as cohesive models whose shape is closely similar to that of frictional contact laws. If on top of that we add the fact that loading and kinematic relations of mechanical parts generally derive from a frictional contact mechanism, it is thus most of the time all of the nonlinearities of the problem that may simultaneously occur, at the cost of increased numerical complexity.
The collection of numerical tools deployed to address these problems includes some of the topics that will be examined in this book and can already be mentioned:
alternatives to the traditional Newton method better suited to regular problems;
methods for solving non-symmetric problems that result from the potentially unassociated nature of damage and friction laws;
continuation methods for passing critical points and adjusting loading changes in the vicinity of the limit loads;
geometric methods for surface detection and changes, namely contact surfaces, cracks resulting from the location of the damage field or the boundary of the damaged area;
spatial discretization schemes based on mixed finite elements for avoiding possible numerical locking problems;
finite element techniques for representing discontinuity surfaces within or between meshes.
The authors of the book have contributed their particular perspectives on these issues. Therefore, Franz Chouly, Patrick Hild and Yves Renard propose Lagrangian, augmented Lagrangian and Nitsche methods, recently applied, for addressing frictional contact problems. They provide a fine mathematical analysis within the framework of small elastic perturbations. They show the relationships between these methods within the framework of small and large deformations. Pierre Alart is interested in solving qualified systems with diffuse non-regularity, that is, for which non-regularity travels through the system through a multitude of potential contacts. This concerns both granular media and structures (tensegrity systems, for example). Domain decomposition-based techniques are analyzed. Vladislav A. Yastrebov is dedicated to numerical methods for addressing contact problems at small scales, namely for surfaces where roughness is taken into account. He shows the peculiarities of numerical processing of such problems. In the second part, Jacques Besson covers ductile damage. He addresses the issue of coupling between large strain plasticity and damage and that of spatial localization of damage, which often requires the introduction of an additional balance equation deriving from the microstructures of the material. This last issue is also considered by Eric Lorentz for brittle damage, more brutal than the previous one, insisting on the role of this new nonlocal equation and its numerical processing, which can alternatively be interpreted as a regularization of a cracking problem, in the spirit of phase field models. It is also an opportunity to examine numerical methods adapted to the treatment of instabilities as a result of the brutal nature of brittle damage. The nonlocal equation leads to a spatial organization of damage that also could be imposed directly: this is the guiding principle behind the thick level set (TLS) method presented by Nicolas Moës. The problem is rewritten in terms of finding a level surface corresponding to the boundary of the damaged area. This modeling of damage also builds a bridge to cracking models in which the major numerical problem remains the representation of a crack without a priori on the mesh, typically via extended finite element (XFEM) elements that encapsulate the description of a displacement discontinuity (the crack) and the surrounding volume. Finally, Sylvia Feld-Payet focuses on the transition from damage to crack. More particularly, she addresses the issues of geometric crack detection from a damage field, remeshing accordingly to explicitly represent the new crack and transferring fields from the old to the new mesh, which is being computed.
October 2024
