Types of Linearity

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.