Variational Continuum Multiphase Poroelasticity (eBook)

Theory and Applications

Roberto Serpieri, Francesco Travascio (Autoren)

eBook Download: PDF

2017 | 1st ed. 2017
XIII, 198 Seiten
Springer Singapore (Verlag)
978-981-10-3452-7 (ISBN)

Infos

This book collects the theoretical derivation of a recently presented general variational macroscopic continuum theory of multiphase poroelasticity (VMTPM), together with its applications to consolidation and stress partitioning problems of interest in several applicative engineering contexts, such as in geomechanics and biomechanics.

The theory is derived based on a purely-variational deduction, rooted in the least-Action principle, by considering a minimal set of kinematic descriptors. The treatment herein considered keeps a specific focus on the derivation of most general medium-independent governing equations.

It is shown that VMTPM recovers paradigms of consolidated use in multiphase poroelasticity such as Terzaghi's stress partitioning principle and Biot's equations for wave propagation. In particular, the variational treatment permits the derivation of a general medium-independent stress partitioning law, and the proposed variational theory predicts that the external stress, the fluid pressure, and the stress tensor work-associated with the macroscopic strain of the solid phase are partitioned according to a relation which, from a formal point of view, turns out to be strictly compliant with Terzaghi's law, irrespective of the microstructural and constitutive features of a given medium. Moreover, it is shown that some experimental observations on saturated sandstones, generally considered as proof of deviations from Terzaghi's law, are ordinarily predicted by VMTPM.

As a peculiar prediction of VMTPM, the book shows that the phenomenon of compression-induced liquefaction experimentally observed in cohesionless mixtures can be obtained as a natural implication of this theory by a purely rational deduction. A characterization of the phenomenon of crack closure in fractured media is also inferred in terms of macroscopic strain and stress paths.

Altogether the results reported in this monograph exemplify the capability of VMTPM to describe and predict a large class of linear and nonlinear mechanical behaviors observed in two-phase saturated materials.

 

Inhalt

Foreword 7
Preface 9
Contents 11
1 Variational Multi-phase Continuum Theories of Poroelasticity: A Short Retrospective 14
1.1 Introduction 14
1.2 Variational Theories from the 70s to the 80s 18
1.2.1 Cowin's Theory 19
1.2.2 Mindlin's Variational Single-Phase Theory 19
1.2.3 The Variational Theory of Immiscible and Structured Mixtures by Bedford and Drumheller 20
1.3 Most Recent Theories 22
1.3.1 Variational Theories by Lopatnikov and Co-workers 22
1.3.2 Variational Higher Gradient Theories by dell'Isola and Co-workers 22
1.4 Conclusions 24
References 25
2 Variational Macroscopic Two-Phase Poroelasticity. Derivation of General Medium-Independent Equations and Stress Partitioning Laws 29
2.1 Introduction 29
2.2 Variational Formulation 33
2.2.1 Basic Configuration Descriptors 34
2.2.2 Variational Formulation 45
2.2.3 Integral Equations 55
2.2.4 Strong Form Equations 59
2.2.5 Additional Solid-Fluid Interaction 66
2.2.6 The Kinematically-Linear Medium-Independent Problem 69
2.2.7 Equations for Static and Quasi-static Problems 72
2.3 Discussion and Conclusions 79
References 82
3 The Linear Isotropic Variational Theory and the Recovery of Biot's Equations 86
3.1 Introduction 86
3.2 Two-Phase Medium-Independent Variational Equations ƒ 88
3.3 Linear Elastic Isotropic Constitutive Theory ƒ 92
3.4 Governing PDEs for the Isotropic Linear Problem 96
3.4.1 baru(s)-baru(f) Hyperbolic PDEs with Inertial Terms 96
3.4.2 Analysis of Wave Propagation 97
3.4.3 PDE for Static and Quasi-static Interaction 102
3.5 Bounds and Estimates of Elastic Moduli 105
3.5.1 Basic Application of CSA 107
3.5.2 Application of CSA to the Extrinsic/Intrinsic Description 110
3.6 The Limit of Vanishing Porosity 116
3.7 Comparison with Biot's Theory and Concluding Remarks 121
References 124
4 Stress Partitioning in Two-Phase Media: Experiments and Remarks on Terzaghi's Principle 126
4.1 Introduction 127
4.2 Boundary Conditions with Unilateral Contact 134
4.3 Kinematic and Static Characterization of Undrained Flow Conditions 136
4.3.1 Static Characterization of Undrained Flow 137
4.4 Stress Partitioning in Ideal Compression Tests 144
4.4.1 Ideal Jacketed Drained Test 149
4.4.2 Ideal Unjacketed Test 151
4.4.3 Ideal Jacketed Undrained Test 155
4.4.4 Creep Test with Controlled Pressure 159
4.5 Analysis of Nur and Byerlee Experiments 162
4.5.1 Determination of bare(s) 165
4.5.2 Estimates of (s) 165
4.6 Domain of Validity of Terzaghi's Principle According to VMTPM 167
4.6.1 Recovery of Terzaghi's Law for Cohesionless Frictional Granular Materials 168
4.6.2 Extensibility of Terzaghi's Effective Stress and Terzaghi's Principle Beyond Cohesionless Granular Materials 170
4.7 Discussions and Conclusions 172
References 175
5 Analysis of the Quasi-static Consolidation Problem of a Compressible Porous Medium 179
5.1 Introduction 179
5.2 Theoretical Background 180
5.2.1 Dimensionless Analysis 180
5.2.2 Semi-analytical Solution of the Stress-Relaxation Problem 182
5.2.3 Numerical Solutions 185
5.3 Discussion and Conclusions 186
References 190
Appendix A Notation and Identities for Differential Operations 192
Appendix B Variation of Individual Terms in Lagrange Function 202

Produkt-Details

• Erscheint lt. Verlag: 19.1.2017
• Reihe/Serie: Advanced Structured Materials
• Reihe/Serie: Advanced Structured Materials
• Zusatzinfo: XIII, 198 p. 20 illus., 16 illus. in color.
• Verlagsort: Singapore
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Biot's Equations • Compression-induced Liquefaction • Least-action Principle • Multiphase Porous Media • Saturated Sandstones • Terzaghi's Stress Partitioning Principle • Two-phase Saturated Materials • VMTPM
• ISBN-10: 981-10-3452-4 / 9811034524
• ISBN-13: 978-981-10-3452-7 / 9789811034527