Advances in Applied Mechanics (eBook)

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2013 | 1. Auflage
144 Seiten
Elsevier Science (Verlag)
978-0-12-396466-3 (ISBN)

Infos

Advances in Applied Mechanics draws together recent significant advances in various topics in applied mechanics. Published since 1948, Advances in Applied Mechanics aims to provide authoritative review articles on topics in the mechanical sciences, primarily of interest to scientists and engineers working in the various branches of mechanics, but also of interest to the many who use the results of investigations in mechanics in various application areas, such as aerospace, chemical, civil, environmental, mechanical and nuclear engineering. - Covers all fields of the mechanical sciences - Highlights classical and modern areas of mechanics that are ready for review - Provides comprehensive coverage of the field in question

Inhalt

Front Cover 1
Advances in Applied Mechanics 4
Copyright 5
Contents 6
Contributors 8
Preface 10
References 12
Chapter One: Continuum Theory for the Edge of an Open Lipid Bilayer 14
1. Introduction 19
2. Mathematical Preliminaries 24
2.1. Superficial fields 24
2.2. Differential geometry of the surface 26
2.3. Useful differential and integral identities 26
2.4. Differential geometry of the edge 27
2.5. Notational conventions 30
3. Variations of Geometric Quantities, Integrals over Surfaces and Curves, and Volume 31
3.1. Areal quantities 33
3.1.1. Unit normal 33
3.1.2. Projector 34
3.1.3. Curvature tensor 34
3.1.4. Mean curvature 35
3.1.5. Gaussian curvature 37
3.1.6. Virtual areal Jacobian 39
3.1.7. Surface integral of a spatial field 40
3.2. Lineal quantities 40
3.2.1. Virtual lineal stretch 41
3.2.2. Unit tangent 42
3.2.3. Tangent-normal vector 42
3.2.4. Arclength derivative of a generic quantity 43
3.2.5. Curvature vector 43
3.2.6. Normal curvature 44
3.2.7. Geodesic curvature 44
3.2.8. Geodesic torsion 45
3.2.9. Curve integral of a spatial field 46
3.3. Volume 46
4. Variational Derivation of the Equilibrium Equations of a Lipid Vesicle 47
4.1. Variation of the net free-energy 49
4.2. Virtual volumetric work 50
4.3. Virtual work of the areal loads 51
4.4. Combined results 51
5. Variational Derivation of the Equilibrium Equations of an Open Lipid Bilayer with Edge Energy 52
5.1. Constant edge-energy density 53
5.1.1. Variation of the net free-energy of the surface 53
5.1.2. Virtual work of the areal loads 54
5.1.3. Variation of the net free-energy of the edge 55
5.1.4. Virtual work of the lineal loads 55
5.1.5. The equilibrium equations 56
5.2. Geometry-dependent edge-energy density 57
6. Force and Bending Moment Exerted by an Open Lipid Bilayer on Its Edge 60
6.1. Force and bending moment expressions 62
7. Alternative Treatment of the Edge 63
7.1. Edge kinematics 64
7.1.1. Geometry of deformation 64
7.1.2. Commutator and transport identities 66
7.2. Balance laws 67
7.3. Constitutive equations and thermodynamic restrictions 68
7.3.1. Free-energy imbalance 68
7.3.2. Constitutive assumptions 70
7.3.3. Local form of the free-energy imbalance 71
7.3.4. Thermodynamic restrictions on the elastic contributions to the internal force and internal moment 71
7.3.5. Reduced dissipation inequality. Restrictions on the viscous contributions to the internal force and internal moment 72
7.4. Governing equations 73
7.5. Retrieving the Euler-Lagrange equations at the edge 75
8. Summary 77
Acknowledgments 78
References 79
Chapter Two: A Variational Approach to Modeling Coupled Thermo-Mechanical Nonlinear Dissipative Behaviors 82
1. Introduction 83
2. General Modeling Framework 87
2.1. Local thermodynamic model 87
2.2. Balance equations 89
3. Variational Formulation of Coupled Thermo-Mechanical Boundary-Value Problems 91
3.1. Variational updates 91
3.1.1. Local evolution problem 91
3.1.2. Local time-discrete constitutive problem 93
3.2. Variational boundary-value problem 96
3.2.1. Rate problem 97
3.2.2. Incremental boundary-value problem 98
3.2.3. Mixed thermal boundary conditions 99
3.3. Dynamics 101
3.4. Linearization 102
4. Thermo-Visco-Elasticity 103
4.1. Linearized kinematics 103
4.1.1. Kelvin-Voigt model 103
4.1.2. Generalized Maxwell model 105
4.2. Finite thermo-visco-elasticity 106
4.2.1. Kelvin-Voigt model 106
4.2.2. Generalized Maxwell model 108
4.2.3. Viscous fluids 110
5. Thermo-Elasto-Visco-Plasticity 111
5.1. Crystal plasticity 111
5.1.1. Constitutive modeling 111
5.1.2. Incremental update 114
5.2. Macroscopic plasticity 115
5.2.1. Linear kinematics 115
5.2.1.1. Rate problem of visco-plasticity 116
5.2.1.2. Constitutive updates 117
5.2.2. Finite strains 119
5.2.2.1. Rate problem of finite visco-plasticity 120
5.2.2.2. Exponential mapping 121
5.2.2.3. Hencky hyperelasticity 121
5.2.2.4. General isotropic hyperelasticity 122
5.2.3. Alternative flow rules 124
5.2.4. Heat generated by visco-plastic dissipation 125
6. Numerical Approximation Methods 126
6.1. Variational finite element approximations 126
6.1.1. Standard galerkin formulation 126
6.1.2. Mixed formulations 129
6.2. Alternative variational Ritz-Galerkin approximations 132
7. Examples of Applications 133
7.1. Variational multiscale models 133
7.2. Variational adaptive methods 134
8. Conclusions 134
Acknowledgements 135
References 135
Index 140

Textauszug

Chapter Two

A Variational Approach to Modeling Coupled Thermo-Mechanical Nonlinear Dissipative Behaviors

Laurent Stainier,    Research Institute in Civil and Mechanical Engineering (GeM), Ecole Centrale Nantes, 1 rue de la Noë, F-44321 Nantes, France

Abstract

This chapter provides a general and self-contained overview of the variational approach to nonlinear dissipative thermo-mechanical problems initially proposed in Ortiz and Stainier (1999) and Yang, Stainier, and Ortiz (2006). This approach allows to reformulate boundary-value problems of coupled thermo-mechanics as an optimization problem of an energy-like functional. The formulation includes heat transfer and general dissipative behaviors described in the thermodynamic framework of Generalized Standard Materials. A particular focus is taken on thermo-visco-elasticity and thermo-visco-plasticity. Various families of models are considered (Kelvin–Voigt, Maxwell, crystal plasticity, von Mises plasticity), both in small and large strains. Time-continuous and time-discrete (incremental) formulations are derived. A particular attention is dedicated to numerical algorithms which can be constructed from the variational formulation: for a broad class of isotropic material models, efficient predictor–corrector schemes can be derived, in the spirit of the radial return algorithm of computational plasticity. Variational approximation methods based on Ritz–Galerkin approach (including standard finite elements) are also described for the solution of the coupled boundary-value problem. Some pointers toward typical applications for which the variational formulation proved advantageous and useful are finally given.

Keywords

Thermo-mechanics; Variational principles; Nonlinear dissipative behavior; Thermo-visco-elasticity; Thermo-visco-plasticity; Constitutive update algorithms; Ritz–Galerkin approaches

1 Introduction

Variational principles have played an important role in mechanics for several decades, if not more than a century (see for example Lanczos, 1986 or Lippmann, 1978). They have been mostly developed, and widely used, for conservative systems: the most eminent examples are Hamilton’s principle in dynamics and the principle of minimum potential energy in statics. Some variational principles with application to dissipative systems have been around for a long time as well, such as principles of maximum dissipation for limit analysis (notably in plasticity).

Variational approaches present many attractive features, especially regarding the possibilities that they offer for mathematical analysis, but also for numerical approaches. They open an easier way to unicity, convergence, and stability analysis of mathematical formulations and associated numerical methods. This has motivated a very large quantity of published work and an exhaustive review is thus out of the scope of this chapter. To directly focus on the category of variational approaches envisioned here, let us then simply say that, following the pioneering work of Biot (1956, 1958), the variational form of the coupled thermo-elastic and thermo-visco-elastic problems has been extensively investigated (see for example Batra, 1989; Ben-Amoz, 1965; Herrmann, 1963; Molinari & Ortiz, 1987; Oden & Reddy, 1976). On the other hand, several authors have proposed variational principles for the equilibrium problem of general dissipative solids in the isothermal setting: see for example Carini (1996), Comi, Corigliano, & Maier (1991), Hackl (1997), Han, Jensen, & Reddy (1997), Martin, Kaunda, & Isted (1996), Mialon (1986), Ortiz & Stainier (1999), in elasto-visco-plasticity, and also Balzani & Ortiz (2012), Bourdin, Francfort, & Marigo (2008), Francfort & Marigo (1998), Kintzel & Mosler (2010, 2011), in brittle and ductile damage. By contrast, the case of thermo-mechanical coupling (i.e. with conduction) in these latter classes of dissipative materials has received comparatively less attention (cf. Armero & Simo, 1992, 1993; Simo & Miehe, 1992, for notable exceptions).

This chapter is intended to provide an overview of recent and less recent work by the author and colleagues on a specific variational approach (initially described in Yang et al., 2006) to coupled thermo-mechanical problems involving nonlinear dissipative behaviors, such as thermo-visco-elasticity and thermo-elasto-visco-plasticity. It will also be the occasion to fill a few gaps between previously published material, in particular by providing a more detailed account of thermal coupling aspects for a variety of constitutive models written under variational form. Links toward closely related work by other researchers are also provided.

We start in Section 2 by setting the general thermodynamic modeling framework serving as a foundation for the proposed variational formulation of coupled thermo-mechanical boundary-value problems. This framework follows closely that of Generalized Standard Materials (GSM) (Halphen & Nguyen, 1975), with a local state description based on internal variables. We will also recall some elements of finite transformations kinematics, as well as balance equations in Lagrangian and Eulerian formulations (although we will mostly work in a Lagrangian setting). This part is quite standard, with a few departures from the mainstream approach (e.g. the use of a Biot conduction potential). We then proceed (Section 3) to reset the constitutive and balance equations defining thermo-mechanical boundary-value problems under a variational form. By variational, we here understand that the problem is formulated as an optimization (or at least a stationarity) problem, with respect to fields of state variables. Since we work with a local state description based on internal variables, we will split the presentation in two parts: first, the local constitutive problem (determination of internal variables or their rate), and, second, the boundary-value problem (determination of fields of external variables). Each part is itself structured in two subparts: we first present the time-continuous evolution problem and its variational formulation, followed by the time-discrete (or incremental) variational formulation. This structure of presentation, which somewhat differs from that adopted in Yang et al. (2006), allows to show that the variational boundary-value problem is formally identical to a thermo-elastic problem, internal variables being handled locally through a nested constitutive variational problem. This is probably the most interesting result presented in that specific part of the paper. After presenting the variational formulation for the incremental boundary-value problem, we show how to add more complex thermo-mechanical boundary conditions, such as mixed thermal conditions (e.g. convective exchange). The variational formulation of coupled thermo-mechanical boundary-value problems is initially presented within a quasi-stationary context (yet including combined heat capacity and conduction effects), but we show in subsection 3.3 how it can be extended to account for inertia effects in the time-discrete framework. We conclude this part by describing linearization procedure in the case of infinitesimal (small) displacements and temperature variations. Note that nonlinearities can remain within this “linearized” context, due to the presence of thermo-mechanical coupling terms.

In Sections 4 and 5, we look in more details at the variational formulation of (continuous and incremental) constitutive equations for some specific models in thermo-visco-elasticity and thermo-elasto-visco-plasticity. We start with the simplest thermo-visco-elastic model possible: linearized kinematics Kelvin–Voigt model with linear elasticity and viscosity. Given its relative simplicity of formulation, the variational update for this constitutive model is treated in details, including all possible temperature dependence effects (thermo-elasticity, thermal softening of elastic and viscous moduli). Note that this is the only model for which a complete treatment is provided here, some simplifying hypotheses being taken for later, more complex, models, for the sake of clarity in the presentation. We then move on to generalized Maxwell models, which introduce internal variables (viscous strains). The analysis of Kelvin and Maxwell models is then repeated for finite strains kinematics. Maybe the most interesting point in that part is the fact that, for isotropic materials at least, constitutive updates can be reduced to solving a reduced number of scalar equations by adopting a spectral approach [as given in Fancello, Ponthot, and Stainier (2006)], independently of the complexity of elastic and viscous potentials adopted in the models. Section 5 deals with thermo-elasto-visco-plasticity. We start with crystal plasticity, which describes fine scale behavior of crystalline materials (mostly metals for our purpose, but also some organic materials and rocks). For this class of models, the variational formulation offers the power of optimization algorithms to solve the complex problem of determining (incremental) plastic slip activity, a problem which can become quite acute when considering complex latent hardening models. The section continues by considering plasticity models at the macroscopic scale, chiefly von Mises (J2) plasticity. We look at both the time-continuous and time-discrete variational formulations of linearized...

Produkt-Details

• Erscheint lt. Verlag: 13.11.2013
• Mitarbeit: Herausgeber (Serie): Stephane P.A. Bordas
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-12-396466-0 / 0123964660
• ISBN-13: 978-0-12-396466-3 / 9780123964663