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CONTENTS
INTRODUCTION
Chapter A: BASIC ELEMENTS
A.1) The displacement vector. Lagrangean (material) and Eulerian (spatial) co-ordinates
A.2) Invariants of a tensor. Tensor deviator
A.3) Strain tensor. Stress tensor. Equation of motion / equilibrium
A.4) HOOKE’s law.
Chapter B: DEFORMATION OF A CYLINDRICAL BODY IN THE PRESENCE OF GRAVITY
B.1) The model
B.2) Equations of equilibrium. Boundary conditions. Simplifying hypothesis
B.3) The final shape of the body
Chapter C: LEVY’s PROBLEM – the triangular dam
C.1) The SAINT-VENANT’s equations
C.2) The model. Simplifying hypothesis. The planar deformation state
C.3) Equations of equilibrium. AIRY’s potential
C.4) Boundary conditions. The final shape of the dam
Chapter D: KIRSCH’s PROBLEM – the circular bore hole / tunnel
D.1) The model
D.2) The planar state of deformation in cylindrical co-ordinate system
D.3) The circle of MOHR
D.4) AIRY’ s potential in cylindrical co-ordinates. The bi-harmonic equation
D.5) The divergence of a tensor in cylindrical co-ordinates
D.6) The gradient of a vector and the strain tensor in cylindrical co-ordinates
D.7) The bi-harmonic equation cylindrical co-ordinates
D.8) The stress elements. Conditions at infinity for the stress elements
D.9) Strain and displacement vector. Conditions at infinity
D.10) Boundary conditions for the stress elements on the wall of the circular cavity
D.11) The final shape of the wall
Chapter E: BOUSSINESQ’s PROBLEM – concentrated load acting on an elastic semi-space
E.1) The equations of BELTRAMI and MITCHELL
E.2) The model
E.3) The equations of equilibrium and strain tensor in spherical co-ordinates
E.4) LAPLACE operator in spherical co-ordinates. LEGENDRE’ s polynomials
E.5) The displacement field
E.6) Boundary conditions for the stress elements. The final solution
Chapter F: ELEMENTS OF THIN PLATE THEORY
F.1) The model of a thin elastic plane plate
F.2) The planar state of a plate. The bending state
F.3) Loads acting on the plate
F.4.) Odd and even functions for the planar state and for the bending state
F.5) Mean value of a function. Equilibrium equations for thin plates
F.6) Thin plate in the bending state
F.7) BERNOULLI’ s hypothesis
F.8) HOOKE’ s law for a thin plate
F.9) The infinite, 1-dimensional (1-D) plate. The flexure of the lithosphere
F.10) Exterior forces on the lateral surface of the plate. Buckling
F.11) The buckling of a simply leaning thin plate
F.12) The infinite extended 1-D plate
F.13) FOURIER transforms. Properties
F.14) Solution of the flexure equation by using FOURIER transforms
F.15) Finite plates
a) Significance of Sij and Mij for the bending state
b) The rectangular plate. Boundary conditions. LEVY’ s solution
F.16) Vibrations of a plate laying on a viscous substratum
a) The differential equation
b) The rectangular plate with 3 embedded sides
Chapter G: THE SPHERICAL SHELL
G.1) The model. BERNOULLI’ s hypothesis. Displacement vector and strain tensor
G.2) Quasi-mean values. Equation of motion
G.3) Integrals of the stress elements. Quasi-moments
G.4) Integrals of displacement vector
G.5) Equation of motion in quasi-mean values
G.6) Quasi-mean value of the shell density. The differential equation
G.7) The buckling of a spherical shell
G.8) Load on the upper face. Stress on the lower surface of the shell
G.9) The differential equation of time dependent flexure
G.10) Spherical effects with respect to the plane plate
Chapter H: ELEMENTS OF RHEOLOGY
H.1) Introduction
H.2) Linear models
a) KELVIN-VOIGT (strong viscous) model
b) MAXWELL (viscous-elastic) model
c) BURGERS (general linear) model
d) Remarks on the linear models
H.3) Non-linear models
H.4) Brittle. Creep. Empirical criteria
H.5) Empirical criteria for shear-faulting. TRESCA criterion. COULOMB-NAVIER criterion
H.6) Von MISES-HENCKY criterion for ductile flow (plasticity)
H.7) Rheological models
a) SAINT VENANT body (elastic-plastic material)
b) BINGHAM body (visco-plastic material)
Chapter I: THE ACCRETION WEDGE
I.1) The model
I.2) Equations of equilibrium. Yield condition. Stress field
I.3) Boundary conditions. Final results
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