Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context (eBook)

Based on a Concise Thermodynamic Analysis

Lukas Schneider, Kolumban Hutter (Autoren)

eBook Download: PDF

2009 | 2009
XXIV, 247 Seiten
Springer Berlin (Verlag)
978-3-642-02968-4 (ISBN)

Infos

Mixture concepts are nowadays used in a great number of subjects of the - ological, chemical, engineering, natural and physical sciences (to name these alphabetically) and the theory of mixtures has attained in all these dis- plines a high level of expertise and specialisation. The digression in their development has on occasion led to di?erences in the denotation of special formulations as 'multi-phase systems' or 'non-classical mixtures', 'structured mixtures', etc. , and their representatives or defenders often emphasise the di?erences of these rather than their common properties. Thismonographisanattempttoviewtheoreticalformulationsofprocesses which take place as interactions among various substances that are spatially intermixedandcanbeviewedtocontinuously?llthespacewhichtheyoccupy as mixtures. Moreover, we shall assume that the processes can be regarded to becharacterisedbyvariableswhichobeyacertaindegreeofcontinuityintheir evolution, so that the relevant processes can be described mathematically by balance laws, in global or local form, eventually leading to di?erential and/or integralequations,towhichtheusualtechniquesoftheoreticalandnumerical analysis can be applied. Mixtures are generally called non-classical, if, apart from the physical laws (e. g. balances of mass, momenta, energy and entropy), also further laws are postulated,whicharelessfundamental,butmaydescribesomefeaturesofthe micro-structure on the macroscopic level. In a mixture of ?uids and solids - thesearesometimescalledparticleladensystems-thefractionofthevolume that is occupied by each constituent is a signi?cant characterisation of the micro-structure that exerts some in?uence on the macro-level at which the equations governing the processes are formulated. For solid-?uid mixtures at high solids fraction where particle contact is essential, friction between the particles gives rise to internal stresses, which turn out to be best described by an internal symmetric tensor valued variable.

Inhalt

Preface 6
Acknowledgments 10
Contents 12
Acronyms, Symbols 14
Assumptions 23
Introduction 25
Motivation 26
Mixtures and Debris Flow Models 32
Mixtures with Frictional Components 34
Objectives, Methods and Structure of the Present Work 36
Mathematical Preliminaries and Notations 39
Tensors 39
Results from Exterior Calculus 42
What is integrability? 42
Requirements to be imposed on the `normal fields' 44
On the non-uniqueness of the integrating factors 48
Introduction to Mixture Theory 52
Basic Principles of Mixture Theory 52
Kinematics of Multi-phase Mixtures 56
Balance Equations and Sum Relations 62
Constitutive Assumptions 77
Selection of Balance Equations 78
Constitutive Laws of Single-Material Bodies 80
Constitutive Laws in the Context of Mixture Theory 82
A First Attempt to Incorporate Hypo-plasticity 87
Further Assumptions 90
Remarks on the Principle of Objectivity 93
Entropy Principle, Transformation of the Entropy Inequality 96
Mixture Entropy Principle According to Müller & Liu
Mixture Entropy Inequality 100
Thermodynamic Analysis I 111
Liu Identities and Residual Entropy Inequality 111
Exploiting the Isotropy of the Vector-valued One-form 116
Integrability Conditions 131
Thermodynamic Analysis II 143
Residual Entropy Inequality in Final Form 143
Mixture Thermodynamic Equilibrium 147
`Isotropic' Expansions of Interaction Densities and Extra Entropy Flux 157
Final Representations for i|to.E, q|to.E, |to.E and |to.E 163
Reduced Model 173
Physical Assumptions 174
`Artificial' Assumptions 178
Final Constitutive Laws 202
An Alternative to the Assumption of `Pressure Equilibrium' 203
Discussions and Conclusions 208
A Primer on Exterior Calculus 215
Auxiliary Results 238
Manipulation of the Entropy Inequality 238
Other Auxiliary Results from Section 5.2 242
Deduction of the Liu Identities 244
Isotropic Representation of the Mixture Flux Density 245
Auxiliary results 247
Derivation of Residual Inequality 249
References 254
Index 262

Produkt-Details

• Erscheint lt. Verlag: 25.7.2009
• Reihe/Serie: Advances in Geophysical and Environmental Mechanics and Mathematics
• Reihe/Serie: Advances in Geophysical and Environmental Mechanics and Mathematics
• Zusatzinfo: XXIV, 247 p. 63 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: debris flow • density • Dynamics • Entropy • Geotechnical Engineering • Heat • Hypoplasticity • Phase • Plasticity • quality • second law of thermodynamics • Solid-fluid Mixture • Structure
• ISBN-10: 3-642-02968-X / 364202968X
• ISBN-13: 978-3-642-02968-4 / 9783642029684