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Chapter 5
Analytical
techniques and solutions for linear elastic solids
In
the preceding chapter, we solved some simple linear elastic boundary value
problems. The problems were trivial,
however, in that the stress, strain and displacement fields were axially or
spherically symmetric. Most problems
of practical interest involve fully 3D stress and displamement fields.
It
is extremely difficult to solve general boundary value problems. However, some of the best mathematicians
over the past 200 years have turned their attention to this matter, and have
developed several very elegant techniques.
None of these are completely general, but solutions derived using
these techniques have provided invaluable insight into the behavior of
deformable solids.
Sadly,
these days any fool with a PC and a finite element package can solve
virtually any linear elastic boundary value problem, so you will not be able
to make a living calculating exact elasticity solutions. Nevertheless, some exact solutions are of
fundamental practical importance. Examples include contact problems,
solutions for cracks, stress concentrations, thermal stress problems, and
problems involving defects such as dislocations in solids. It is worth seeing how such solutions were
derived.
In
addition, there are some very powerful theorems in linear elasticity (such as
the principle of minimum potential energy, and the reciprocal theorem), which
can be used to directly calculate quantities of interest without necessarily
having to solve all the governing equations directly.
In
this chapter, we present a very brief survey of the field of linear
elasticity. Specifically,
1. We will outline some important general features of
solutions to boundary and initial value problems;
2. We will discuss some solution techniques and present
solutions to selected boundary value problems of interest;
3. We will discuss energy methods for solving problems
involving linear elastic solids, including the principle of minimum potential
energy, the reciprocal theorem, and the Rayleigh-Ritz method for estimating
natural frequencies of vibrating elastic solids.
5.1 General Principles
5.1.1
Summary of the governing equations of linear elasticity
Static problems. We already listed the governing
equations of linear elasticity in our discussion of simple axisymmetric
problems. They are repeated here for convenience.
Given:
1. The shape of the solid in its unloaded condition 
2. The initial stress field in the solid (we will take
this to be zero)
3. The elastic constants for the solid and its mass density
4. The thermal expansion coefficients for the solid,
and temperature change from the initial configuration
5. A body force distribution (per unit mass) acting on the solid
6. Boundary conditions, specifying displacements on a portion or tractions on a portion of the boundary of R
Calculate
displacements, strains and stresses satisfying the governing equations of
linear elastostatics
Dynamic problems Dynamic
problems are essentially identical, except that the boundary conditions must
be specified as functions of time, and the initial displacement and velocity
field must be specified. In this case
the governing equations are
5.1.2 Alternative form of the
governing equations the Navier equation
The
governing equations can be simplified by eliminating stress and strain from
the governing equations, and solving directly for the displacements. In this case the linear momentum balance
equation (in terms of displacement) reduces to
For
the special case of an isotropic solid with shear modulus and Poisson ratio and uniform temperature this equation reduces to
These are known as the
Navier (or Cauchy-Navier) equations of elasticity.
The boundary conditions
remain as given in the preceding section.
5.1.3
Superposition and linearity of solutions
The
governing equations of elasticity are linear. This has two important consequences:
1. The stresses, strains and displacements in a solid
are directly proportional to the loads (or displacements) applied to the
solid.
2. If you can find two sets of displacements, strains
and stresses that satisfy the governing equations, you can add them to create
more solutions.
These
principles can be illustrated clearly using some of the simple solutions
derived in Section 4.1. For example,
examine the solution to the pressurized sphere (Sect 4.1.4). As an example, the radial stress induced
by pressure on the interior, and zero pressure on the
exterior surface is
The radial stress induced by pressure on the exterior surface, with zero pressure
on the interior surface is
Note
that in both cases the stress is directly proportional to the pressure. In addition, to find the radial stress by
combined pressures on the interior and on the exterior surface, you can just add
these two solutions.
Further examples of superposition and linearity will be
given in subsequent sections.
5.1.4
Uniqueness and existence of solutions to the linear elasticity equations
The following results are
useful:
1. If only displacements are prescribed on the boundary
of the solid, the governing equations of linear elasticity always have a
solution, and the solution is unique.
2. If mixed boundary conditions are specified, a static
solution exists and is unique if the displacements constrain rigid motions. A
dynamic solution always exists and is unique, provided the velocity field and
displacement field at time t=0
are known.
3. If only tractions are prescribed on the boundary, a
static solution exists only if the tractions are in equilibrium. In this case, the stresses and strains are
unique, but the displacements are not.
A dynamic solution always exists and is unique, again, providing
initial conditions are known.
5.1.5 Saint-Venant’s principle
Saint-Venant’s
principle is often invoked to justify approximate solutions to boundary value
problems in linear elasticity. For
example, when we solve problems involving bending or axial deformation of slender
beams and rods in elementary strength of materials courses, we only specify
the resultant forces acting on the ends of a rod, or the magnitudes of point
forces acting on a beam, we don’t specify the distribution of traction in
detail. We rely on Saint Venant’s
principle to justify this approach. In this context, the principle states the
following.
The
stresses, strains and displacements far from the ends of a rod or beam
subjected to end loading depend only on the resultant forces and moments acting
on its ends, and do not depend on how the tractions themselves are
distributed.
Although
SVP is widely used, it turns out to be remarkably difficult to prove
mathematically. The difficulty is
partly that it is not easy to state the principle itself precisely enough to
apply any mathematical machinery to it.
A rigorous statement is given by Sternberg (Q. J. Appl. Mech 11 p. 393 1954), among several other
versions. Here, we will just
illustrate the most common applications of the principle through specific
examples.
One
version of SVP can be stated as follows.
Suppose
that we calculate the stress, strain and displacement induced in a solid by
two different traction distributions and that act on some small region of a solid
with characteristic size a. If the
tractions exert the same resultant force and moment, then the stresses,
strains and displacements induced by the two traction distributions at a
distance r from the loaded region are identical for large .
In practice `large’ usually means .
This
principle can be illustrated using a simple example.
Consider a large solid with a flat surface, as shown
in the picture. It is possible to
calculate formulas for the stresses and displacements induced by various
pressure distributions acting on the flat surface the procedure to do this will be outlined
later. For now, we will compare the
stresses induced by
1. A uniform pressure
2. A parabolic pressure
You can verify for yourself that both pressure
distributions exert a resultant force P
acting in the vertical direction on the surface, and exert zero moment about
the origin. The variation of stress
down the axis of symmetry ( ), expressed in cylindrical-polar
coordinates, can be derived as
Case 1: Uniform pressure
Case 2: Parabolic pressure
Now, to demonstrate SVP, we
want to show that the stresses are equal for large z/a. We can do this
graphically the figures below compare the variation of
vertical and radial stress down the axis of symmetry with z/a.
 
The
stresses induced by the two different pressures are clearly indistinguishable
for . This example helps to quantify what we mean
by a `large’ distance.
The
second commonly used application of SVP is a rather vague statement that
A localized geometrical feature with
characteristic size R in a large solid only influences the stress in a region
with size approximately 3R surrounding the feature.
This
is more a rule of thumb than a precise mathematical statement. It can be illustrated by looking at
specific solutions. For example, the
figure below shows the Mises stress contours surrounding a circular hole in a
thin rectangular plate that is subjected to extensional loading (calculated
using the finite element method). Far from the hole, the stress is
uniform. The contours deviate from the
uniform solution in a region that is about three times the hole radius.
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