The Essentials of Biomechanics echanics is the study of the action of forces in promoting movement or equilibrium. It is important to many engineering disciplines and has a long and interesting history dating back to the time of the Egyptians. Although it flowered first as a science in Greek and Roman times, the mechanics of solids is generally dated from the work of Galileo in the early part of the seventeenth century. Prior to his investigations into the behavior of solid bodies under loads, builders followed precedent and empirical rules. Galileo was the first to attempt to explain the behavior of some of the building members under load on a rational basis. He studied members in tension and compression, and notably beams used in the construction of hulls of ships for the Italian navy. Of course much progress has been made since that time, but in passing it must be noted that much is owed in the development of this subject to the French investigators, among whom a group of outstanding men such as Coulomb, Poisson, Navier, St. Venant, and Cauchy. These gentlemen worked at the break of the nineteenth century, and have left an indelible impression on the subject. Mathematical Conventions in Biomechanics We shall define and describe some mathematical terms and usage; terms like: scalar, vector, tensor, and dolor. The last is real but unquantifiable and therefore will not be discussed in detail here. [1] A scalar is simply a real number and is a zero–rank tensor. A vector is a quantity having both magnitude and direction; it is a first–rank tensor. A tensor per se is a quantity expressing the ratio in which the length of a vector is increased. The mathematical language used in defining stresses and strains can become quite complex, especially when three–dimensional, curved objects are studied. Albeit, these all reduce or decompose into smaller, more discrete terms and expressions much like bricks in a wall. Figure 7. a. All force components are resolved around a right- hand system of Cartesian coordinates. b. The positive sense of the force components viewed toward the origin coincide with the positive direction of the coordinate axes. In this treatise, all force components will be resolved along a right–hand system of Cartesian coordinates as shown in (Figure 7a). Usually the x and y axes will be taken in the plane of the paper with the z axis pointing toward the reader. The positive sense of the force components viewed toward the origin coincides with the positive directions of the coordinate axes, (Figure 7b). The equilibrating forces on the far side of the element (not shown in the figure) act in the direction opposite to those of the positive coordinate axes. Hence, the force vector P of the facing surface can be expressed as:        where , ,  are the components of the force P, and i, j, k are the unit vectors along the x, y, and z–axes, respectively (Figure 8). Figure 8. Unit force vectors. The behavior of a beam or other solid subjected to forces depends not only on the fundamental laws of Newtonian mechanics governing the equilibrium of the forces but also on the mechanical characteristics of the materials of which the member is fabricated. The necessary information regarding the latter comes from the laboratory where materials are subjected to the action of accurately known forces and the behavior of test specimens is observed with particular regard to such phenomena as the occurrence of deformations, etc. The mechanics of solids is hence a blended science of experiment and Newtonian postulates of analytical mechanics; from the latter is borrowed the branch of that science called statics. Biomechanics is a subdiscipline of mechanics which concerns itself with the medical application of mechanical concepts. The eye is a structure that moves or changes shape in response to forces such as the IOP, extraocular muscles and surgical intervention and hence can be studied through applied mechanics. Table 1. Corneal variables for modeling. The first step in establishing a mechanical view of the cornea must be the determination of variables capable of describing the system (Table 1). Quantities such as stress, strain, Poisson’s ratio, and displacement may not be familiar to many ophthalmologists and yet provide the framework for this new description of the cornea. Although many of the measurements involve changes of lengths in microns or less as well as equally tiny applied forces, instruments and techniques have been developed to measure these quantities with precision — both in biologic systems and in mechanical engineering. Since advances in our understanding of this relatively new discipline will only come with a close cooperation between biomechanicians and ophthalmic physicians, we must perforce make the effort to digest this material. It is not essential to be a engineer in order to make use of the biomechanical properties of the eye (one can, after all, operate an automobile without understanding physics), however, some facility with mathematics is paramount for a complete understanding of these properties. Just as a lay person cannot be expected to grasp all the nuances of medicine — especially eye surgery — so too it cannot be expected that the eye surgeon will in turn be facile in structural engineering. Be of good cheer, however. Some physicists have difficulty with this topic as well. Still an intelligent non–physician is capable of acquiring sufficient understanding of a particular topic to hold his own with a physician — just talk to a young diabetic some time. In any event, our goal in this chapter is to give basic grounding in terms and concepts — not conduct a course in biomechanics, hence we will merely skim the high points. To accomplish this goal, it will be necessary to reduce FEA (finite element analysis) to its elements — no pun intended. As in any profession — there is a vocabulary to be learned and some different usage of terms to be made aware of. It does no good to rail against such things, however. The concept: “When in Rome ...”, must prevail. Thus certain expressions will be used that will require defining if we are to understand any part of this fascinating aspect of engineering. Such terms as: stress strain shear continuum homogeneity isotropy/anisotropy/orthotropy displacement Then too there are: Constitutive and Hooke’s Law, La Place’s formula, and Poisson’s ratio, to name but a few. [1]   Dolor occurs frequently during the study of the first three terms. In its mildest form it occurs between C1 and C3 and in its most severe form in the area of the gluteus maximus. © Leo D. Bores, MD - 2002