Field Theory

Field Theory

At the turn of the century, although few realized it, the ingredients were in place for a flowering of the natural sciences with the development of field theory. This was certainly the case in the study of the mechanics of fluids and solids, which led the way for the new physics of electricity, magnetism, and the propagation of light.

An uncharitable observer of the solid-mechanics scene in 1800 might, with the benefit of hindsight, characterize the state of knowledge then as a jumble of incorrect solutions for collapse loads, an incomplete theory of bending, an unclear definition for Young's modulus, a strange discussion of the frictional strength of brittle materials, a semigraphical solution for arches, a theory for the longitudinal vibration of bars that was erroneous when extended to plates or shells, and the wrong equation for torsion. However, this assess-ment would be wrong. While no general theory was developed, 120 years of research from Galileo to Coulomb had developed the basic mental tools of the scientific method (hypothesis, deduction, and verification) and compiled the necessary ingredients to formulate the modern field theories for strain, stress, and displacement.

The differentiation between shear and normal displacement and the gener-alization of equilibrium at a point to the cross-section of a beam were both major steps in the logic of solid mechanics as, of course, was Young's insight in relating strain and stress linearly in tension or compression. Newton first proposed bodies made up of small points or 'molecules' held together by self-equilibriating forces and the generalization of calculus to two and three dimensions allowed the mathematics of finite linear transformation to be reduced to an arbitrarily small size to describe deformation at a differential scale. Thus the stage was set. The physical concepts and the mathematical tools were available to produce a general field theory of elasticity. Historical events conspired to produce it in France.

The French Revolution destroyed the old order and replaced it with republican chaos. The great number of persons separated at the neck by Dr. Guillotine's invention is symbolic of the beheading of the Royal Society as the leader of an elite class of intellectuals supported by the King's treasury and beholden to imperial dictate.

The new school, L'Ecole Polytechnique founded in 1794, was unlike any seen before. Based on equilitarian principles, entrance was by competitive examination so that boys without privileged birth could be admitted. More-over, the curriculum was entirely different. Perhaps because there were so may unemployed scientists and mathematicians available, Gospareed Monge (1746-1818), who organized the new school, was able to select a truly remarkable faculty including among others, Lagrange, Fourier, and Poisson. Together they agreed on a new concept of engineering education.They would, for the first two years, concentrate on instruction in the basic sciences of mechanics, physics, and chemistry, all presented with the fundamental language of mathematics as the unifying theme. Only in the third year, once the fundamentals that apply to all branches of engineering were mastered, would the specific training in applications be covered.*

Thus the modern 'institute of technology' was born and the consequences were immediate and profound. The basic field theory of mathematical elas-ticity would appear within 25 years, developed by Navier and Cauchy not only as an intellectual construction but for application to the fundamental problems left by their predecessors.* The first generation of graduates of the Ecole Polytechique such as Navier and Cauchy, became professors and edu-cated many great engineers who would come to dominate structural design in the later half of the 19th century.**

The French idea of amalgamating the fundamental concepts of mathe-matics and mechanics as expressed by field theory for engineering applica-tions, is the theme of this text. Today, two centuries of history have proven this concept not only as an educational approach, but as a unifying princi-ple in thinking about solid mechanics.*** In bygone days, the term 'Ratio-nal Mechanics' was popular to differentiate this perspective of visualizing fields graphically with mathematics and experiments so as to understand how structures work rather than just solving specific boundary value prob-lems. The phrase 'Rational Mechanics' is now old-fashioned, but historically correct for the attitude adopted in succeeding chapters of combining elastic and plastic behavior as a continuous visual progression to yield and then collapse.