In this book, for the first time, a hitherto unknown general solution of the reliably known stress conditions is presented. This general solution forms a reliable and new starting point to get further in stress calculations than before.
In this way, approximately realistic solutions can be found despite a recurring problem: the information deficits that are unavoidable due to the difficulty of exploring glaciers. This issue is demonstrated by the example of stagnating glaciers.
For horizontally isotropic homogeneous tabular iceberg models, even mathematically exact unambiguous solutions of all relevant conditions are presented.
All calculations use only elementary arithmetic operations, differentiations and integrations. The mathematical fundamentals are presented in detail and explained in many application examples. The integral operators specific to calculations of stresses facilitate the mathematical considerations. The stand-alone text allows the reader to understand what is involved even without considering the formulas.
The author
Peter Halfar is a theoretical physicist. He also developed a model of the movement of large ice caps (1983), which is still in use today.
Author(s): Peter Halfar
Publisher: Springer
Year: 2023
Language: English
Pages: 220
City: Berlin
Preface to the German Edition
Preface to the English Edition
Contents
Part I: Introduction and Basics
1: Introduction
1.1 The Calculation of Stresses
1.2 The Physical Mechanisms
1.3 The Weightless Stress Tensor Fields
1.4 Concept and Aim of the Study
1.4.1 General
1.4.2 Special
2: Balance and Boundary Conditions
3: Integral Operators
3.1 Example, General Properties and Minimal Models
3.2 Integral Operators
3.3 Dependency Cones and Products of Integral Operators
3.4 Solutions of Boundary Value Problems of Partial Differential Equations
3.4.1 Boundary Surfaces with Boundary Conditions, Domains of Definition and Minimal Models
3.4.2 Boundary Value Problems in Minimal Models
3.4.3 Solutions of Boundary Value Problems by Integral and Differential Operators
3.5 Integrations of Distributions with Integral Operators
3.5.1 Domain of Definition
3.5.2 Integrations
4: Forces and Torques on Surfaces
4.1 Gauss´s Theorem
4.2 Projection Shadow
4.3 Oriented Volume Integrals
4.4 Projection Masses and Moments
5: Special Solutions of the Balance Conditions
5.1 Disappearing xx, xy and yy Components
5.2 Disappearing Non-diagonal Components
6: Weightless Stress Tensor Fields
6.1 Construction
6.2 Redundancies and Normalizations
6.2.1 Redundancy Functions
6.2.2 Normalizations
Part II: The General Solution
7: Weightless Stress Tensor Fields with Boundary Conditions
7.1 Terms
7.2 Structure
7.3 Construction
7.4 Redundancies and Normalizations
8: The General Solution of the Balance and Boundary Conditions
8.1 Representations with Stress Functions
8.2 Representations with Three Independent Stress Components
8.2.1 Problem Definition and Solution Procedure
8.2.2 Calculation of the Solutions
8.2.3 Surface Shape and Domain of Definition
8.2.4 Solution Dependency Cones
9: Models and Model Selection
9.1 Characterisation of the Models
9.1.1 Models with Stress Functions
9.1.2 Models with Three Selected Independent Stress Components
9.2 Model Selection
9.2.1 Floating Glaciers
9.2.2 Land Glaciers with Multiple Connected Free Surfaces
9.2.3 Land Glaciers with a Simply Connected Free Surface
Part III: Applications and Examples
10: Land Glaciers
10.1 Glaciers with Simply Connected Free Surface: Models with Three Independent Stress Components
10.1.1 Independent Components Sxx, Syy, Sxy
10.1.2 Independent Non-diagonal Components
10.1.3 Independent Deviatoric Components, S′xx, S′yy, Sxy
10.2 Glaciers with Surface Load and with Twofold Connected Free Surface: A Model with Normalized Stress Functions
10.3 Quasi-Stagnant Models
10.3.1 Stagnant Glaciers
10.3.2 Quasi-Stagnant Models
10.3.3 The Quasi-Stagnant Model with Horizontally Acting Gravity Pressure
11: Floating Glaciers
11.1 Glaciers in Local Floating Equilibrium
11.2 Boundary Stresses on Closed Boundaries and the Global Balance Conditions for Icebergs
11.3 Horizontally Isotropic-Homogeneous Table Iceberg Models
11.3.1 Horizontally Isotropic-Homogeneous Stress Tensor Fields
11.3.2 Influence of Lateral Water Pressure
11.3.3 Flow Velocities and Strain Rates
11.3.4 The Unique Solution, Even with Generalized Flow Law and with Generalized Lateral Boundary Conditions
11.3.5 The Flow Law
11.3.6 Calculation of the Solution
Part IV: Mathematical Appendix
12: Vectors and Tensors
13: Tensor Analysis
14: Redundancy Functions and Normalizations
14.1 Redundancy Functions
14.2 Normalizations
14.2.1 xx-yy-zz Normalization
14.2.2 The Normalizations xx-yy-xy, xx-yy-xz, xx-xy-yz, xy-yz-xz
14.3 Normalizations with Boundary Conditions
15: Analysis on Curved Surfaces
15.1 Curvilinear Coordinates
15.2 Differential Operators and Derivatives
15.3 The Boundary Fields
15.4 The Boundary Fields as Functions of Curvilinear Surface Coordinates
16: Calculation of Special Weightless Stress Tensor Fields
16.1 Calculation of T
16.2 Calculation of T
17: The General Solution Expressed by Three Independent Stress Components
17.1 (a) Independent xx-, yy-, zz-Components
17.2 (b) Independent xx-, yy-, xy-Components
17.3 (c) Independent xx-, yy-, xz-Components
17.4 (d) Independent xx-, xy-, yz-Components
17.5 (e) Independent xy-, yz-, xz-Components
17.6 (f) Independent Deviatoric xx-, yy-, xy-Components
17.7 (g) Independent Deviatoric xx-, yy-, xz-Components
17.8 (h) Independent Deviatoric xx, xy, yz Components
18: Transformations
18.1 Spatial Domain of Definition
18.2 Heaviside and Delta Function
18.3 Derivatives
18.4 Integrals
18.5 Transformations in Model Types ``a´´-``e´´
18.6 Transformations in the Model Types ``f´´-``g´´
18.7 Transformations in Model Type ``h´´
19: The Hyperbolic Differential Equation in Three Variables
20: Table Icebergs
20.1 The Functions K1, K2, χ, I1 and I2
20.2 Existence and Uniqueness of the Solution
20.3 Examples
20.4 The Constants C1 and C2
20.4.1 Spatially Non-constant Densities of Ice and Water
20.4.2 Spatially Constant Densities of Ice and Water
Appendix A: Explanation and List of Symbols
References
