1 Linear Shapes-The 'Elastic Line' 2 Linear Displacement (Plane Sections) 3 Linear Stress Strain Behavior (Hooke's Law) 4 Geometric Linearity 5 Linear Tangent Transformation

*Introduction*

Solid mechanics deals with the
calculation of the displacements of a deform-able body subjected to the action
of forces in equilibrium for the purpose of designing structures better.
Throughout the history of engineering and sci-ence from Archimedes to Einstein,
this endeavor has occupied many great minds, and the evolution of solid
mechanics reflects the revolution of applied science for which no end is in
sight.

**Types of Linearity **

The development of
various concepts of linearity is one central theme in solid mechanics. A brief
review of five distinct meanings of 'linear analysis' can, therefore, serve to
introduce the subject from a historical perspective* setting the stage for the
presentation in this text of the field theory of deformable solids for
engineering applications. Admittedly, any scheme to introduce such an
incredibly rich subject in a few pages with one approach is ridicu-lously
simplistic. However, discussing types of linearity can serve as a useful
heuristic fiction.

**1 ****Linear
Shapes-The 'Elastic Line' **

One fundamental idealization of
structures is that, for long slender members, the geometric properties and
therefore the stiffness to resist axial torsion and bending deformation, are
functions of only the one variable along the length of the rod. This is the
so-called elastic line used by Euler in his famous solu-tion for buckling.

If the internal stress
resultants, moments, torque, shears, and axial force are only dependent on the
position, *s*, along the member, then, too, must be the displacements and
stresses.* This, then, is the tacit idealization made in classic structural
analysis when we draw line diagrams of the structure itself and plot line
diagrams for stress resultants or changes in geometry. Structural analysis for
internal forces and moments and, then deformations is, therefore, essentially
one-dimensional analysis having disposed of the other two dimensions in
geometric properties of the cross-section.**

**2 ****Linear
Displacement (Plane Sections) **

The basic problem that
preoccupied structural mechanics in the 17th century from Galileo in 1638
onward was the behavior and resistance to failure of beams in bending. The
hypothesis by Bernoulli*** that the cross-section of a bent beam remains plane
led directly to the result that the resistance to bend-ing is a couple
proportional to the curvature. This result, coupled with the concept of linear
shape, allowed Euler**** to develop and study the deforma-tion of his 'elastic
line' under a variety of loadings. Doing this, he was able to derive the
fundamental equations of flexure with great generality includ-ing initial
curvature and large deflections as well as for axial forces causing buckling
with or without transverse load. Bernoulli and Euler assumed 'elas-tic'
material implicitly lumping the modulus in with their geometric stiffness
'constant.'

Plane sections is, of
course, *the* fundamental idealization of 'Strength of Materials'
('simple' solid mechanics) which, for pure bending, is a special case of the more
general elasticity theory in two and three dimensions.* 'Strength of Materials'
is, in turn, divided into 'simple' and 'advanced' solu-tions: simple being when
the bar is straight or, if curved, thin enough so all the fibers have
approximately the same length. For cases where the axial fibers have the same
base length, then linear axial displacements (Bernouilli's Hypothesis) implies
linear strains, and therefore linear stresses in the axial direction.

Relatively simple
strength of materials solutions are, to the engineer, the most important of
solid mechanics. They:

a. may
be 'exact' (e.g., pure bending, axial loading, or torsion of cir-cular bars);

b. or
so close to correct it makes no difference; and

c. are
generally a reasonable approximation for preliminary design and useful as a
benchmark for more exact analysis.

One important purpose
in studying more advanced solid mechanics is, in fact, to appreciate the great
power of the plane-section idealization while rec-ognizing its limitations as,
for example, in areas of high shear or when the shape is clearly not
one-dimensional and therefore an elastic line idealization is dubious or
impossible.

**3 ****Linear
Stress Strain Behavior (Hooke's Law) **

Of the many fundamental
discoveries by Hooke,** linear material behavior is the only one named for him.
In his experiments, he loaded a great variety of materials in tension and found that the
elongation was proportional to the load. He did not, however, express the concept
of strain as proportional to stress, which required a gestation period of more
than a century.

Although often called
'elastic behavior' or 'elasticity,' these terms are misnomers in the sense that
'elastic' denotes a material which, when unloaded, returns to its original
shape but not necessarily along a linear path. However, elastic, as shorthand
for linear elastic, has become so pervasive that linearity is always assumed
unless it is specifically stated otherwise.

As already discussed, Hooke's Law
combined with the previous idealiza-tion of plane sections, leads to the
elastic line and the fundamental solutions of 'Strength of Materials' and
'Structural Mechanics.' As we will see, Hooke's Law-that deformation is
proportional to load-can be broadly interpreted to include time effects
(viscoelasticity) and temperature (ther-moelasticity). Moreover, elastic
behavior directly implies superposition of any number of elastic effects as
long as they add up to less than the propor-tional limit. Adding the effects of
individual loads applied separately is a powerful strategy in engineering
analysis.

**4 ****Geometric
Linearity **

The basic assumption
that changes in base lengths and areas can be disre-garded in reducing
displacements and forces to strains and stresses is really a first-order or
linear approximation. Related to it is the assumption that the overall
deformation of the structure is not large enough to significantly affect the
equilibrium equations written in terms of the original geometry.*

**5 ****Linear
Tangent Transformation **

The fundamental concept
of calculus is that, at the limit of an arbitrary small baseline, the change as
a nonlinear function can be represented by the slope or tangent. When applied
to functions with two or more variables, this basic idea gives us the
definition of a total derivative, which when applied to dis-placements will, as
presented in next pages, define strains and rotations.

A profound physical assumption is
involved when calculus is used to describe a continuum since, as the limiting
process approaches the size of a molecule, we enter the realm of atomic physics
where Bohr and Einstein argued about the fundamental nature of the universe.
The question: At what point does the limit process of calculus break down? is
also significant in engi-neering. Even for steel, theoretical calculations of
strength or stiffness from solid-state physics are not close to measured
values. For concrete, or better yet soil, the idea of a differential base
length being arbitrarily small is locally dubi-ous. Yet calculus, even for such
discrete materials as sand, works on the average. Advanced analysis, based on,
for example, 'statistical mechanics' or the 'theory of dislocations,' is
unnecessary for most engineering applications.

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