Analytical techniques and solutions for
linear elastic solids
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.
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
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.
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.
this chapter, we present a very brief survey of the field of linear
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
Summary of the governing equations of linear elasticity
already listed the governing equations of
linear elasticity in our discussion of simple axisymmetric problems. They are
repeated here for convenience.
The shape of the
solid in its unloaded condition
stress field in the solid (we will take this to be zero)
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
displacements, strains and stresses satisfying the governing equations of
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
5.1.2 Alternative form of the governing
equations the Navier equation
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
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.
Superposition and linearity of solutions
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
2. If you can find two sets of displacements, strains and
stresses that satisfy the governing equations, you can add them to create more
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
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.
Uniqueness and existence of solutions to the linear elasticity equations
The following results are
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
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
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
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.
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.
version of SVP can be stated as follows.
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 .
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
A uniform pressure
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.
stresses induced by the two different pressures are clearly indistinguishable
for . This example helps to quantify what we mean
by a `large’ distance.
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.
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.