INTRODUCTION

In memoriam
Professor Marin
DOROBANÞU
INTRODUCTION
That book presents some theoretical problems related to the Mechanics of
a Continuum Solid Body, of particular importance to Applied Geomechanics,
Geological Engineering and Structural Geology. In most cases, only static
aspects are discussed, but some dynamic cases are also presented. As a rule, the modern tensorial approach is used. The linear elasticity
and the homogeneity of the continuum solid body are almost thoroughly
assumed to be valid, but some elements of Rheology are also presented. In most cases, the semi-inverse method is used to solve the problems.
According to it, the solution is supposed to be of a particular form, as a
consequence of the simplified hypothesis previously assumed. It is verified
that solution checks both the corresponding equations and the boundary
conditions. Based on the Uniqueness Theorem of the Linear Elasticity, it
follows the assumed particular solution is just the general solution of the
problem. In all the cases discussed here, the assumed simplified hypotheses
allow one to obtain simple, analytical solutions. At a first glance, the
importance of such solutions is minor with respect to the real cases, where
mainly the non-homogeneity of the medium plays a great role. However, the
analytical solutions are the basis for deriving finite element algorithms,
allowing one to model satisfactory the complex real cases. Such examples
are also presented. The lessons are mainly designed to be used as a part of the course of
Mechanics followed by the students in Geophysics at the Geology and
Geophysics Faculty, University of Bucharest. The author is gratefully to his colleagues and to the referees. CONTENTS
INTRODUCTION
page Chapter A: BASIC ELEMENTS.
..............................................................................
............................ 4 A.1) The displacement vector. Lagrangean (material) and Eulerian (spatial)
co-ordinates. .........................................4
A.2) Invariants of a tensor. Tensor deviator.
..............................................................................
..................4
A.3) Strain tensor. Stress tensor. Equation of motion / equilibrium.
....................................................................5
A.4) HOOKE's law.
..............................................................................
...............................................7
Chapter B: DEFORMATION OF A CYLINDRICAL BODY IN THE PRESENCE OF GRAVITY
......................8
B.1) The model...
..............................................................................
.................................................8
B.2) Equations of equilibrium. Boundary conditions. Simplifying hypothesis.
.........................................................8
B.3) The final shape of the body.
..............................................................................
..............................11
Chapter C: LÉVY's PROBLEM - the triangular
dam...........................................................................
.... 13 C.1) The SAINT-VENANT's equations.
..............................................................................
......................13
C.2) The model. Simplifying hypothesis. The planar deformation state.
...............................................................13
C.3) Equations of equilibrium. AIRY's potential.
..............................................................................
...........14
C.4) Boundary conditions. The final shape of the
dam..........................................................................
..........15
Chapter D: KIRSCH's PROBLEM - the circular bore hole /
tunnel..............................................................18 D.1) The model. .........
..............................................................................
.........................................18
D.2) The planar state of deformation in cylindrical co-ordinate system.
...............................................................19
D.3) The circle of MOHR.
..............................................................................
......................................20
D.4) AIRY's potential in cylindrical co-ordinates. The bi-harmonic
equation........................................................ 20
D.5) The divergence of a tensor in cylindrical co-ordinates.
............................................................................
.21
D.6) The gradient of a vector and the strain tensor in cylindrical co-
ordinates. ......................................................22
D.7) The bi-harmonic equation cylindrical co-ordinates.
............................................................................
....23
D.8) The stress elements. Conditions at infinity for the stress elements.
...............................................................24
D.9) Strain and displacement vector. Conditions at infinity.
............................................................................
26
D.10) Boundary conditions for the stress elements on the wall of the
circular cavity. ...............................................27
D.11) The final shape of the wall. .....................
............................................................................
.........30
Chapter E: BOUSSINESQ's PROBLEM - concentrated load acting on an elastic
semi-space...............................32 E.1) The equations of BELTRAMI and MITCHELL.
..............................................................................
......32
E.2) The model.
..............................................................................
...................................................34
E.3) The equations of equilibrium and strain tensor in spherical co-
ordinates. .......................................................34
E.4) LAPLACE operator in spherical co-ordinates. LEGENDRE's polynomials.
....................................................35
E.5) The displacement field.
..............................................................................
....................................36
E.6) Boundary conditions for the stress elements. The final solution.
..................................................................38 Chapter F: ELEMENTS OF THIN PLATE
THEORY........................................................................
.......40 F.1) The model of a thin elastic plane
plate........................................................................
.........................40
F.2) The planar state of a plate. The bending
state........................................................................
.................40
F.3) Loads acting on the
plate........................................................................
.........................................41
F.4.) Odd and even functions for the planar state and for the bending
state. ..........................................................41
F.5) Mean value of a function. Equilibrium equations for thin plates.
.................................................................42
F.6) Thin plate in the bending state.
..............................................................................
...........................44
F.7) BERNOULLI's hypothesis.
............................................................................
..................................44
F.8) HOOKE's law for a thin plate.
............................................................................
..............................45
F.9) The infinite, 1-dimensional (1-D) plate. The flexure of the
lithosphere. .........................................................46
F.10) Exterior forces on the lateral surface of the plate. Buckling.
......................................................................47
F.11) The buckling of a simply leaning thin plate.
..............................................................................
..........49
F.12) The infinite extended 1-D plate.
..............................................................................
.........................50
F.13) FOURIER transforms. Properties.
..............................................................................
.......................50
F.14) Solution of the flexure equation by using FOURIER transforms.
.................................................................51
F.15) Finite plates.
....................