Stress, Strain, and Constitutive Relations

Abstract

Consider the two structural members in Figure 2.1, each acted upon by an applied weight W that is much larger than the individual weights mg, which we therefore neglect. From statics, we know that if these two members are in equilibrium, then ΣF = 0 and ΣM = 0. Free-body diagrams of the whole structure and the individual parts reveal that the reaction and internal forces are the same: R _y = f _y = W; that is, from the perspective of statics alone, these two problems are equivalent. Nevertheless, intuition tells us that the behavior of member A need not be the same as that of member B. One may fail before the other. An important question to be answered by mechanics, therefore, may be the following: Which member will likely fail first given increasing weights W? At first glance, we may be inclined to say that A will fail before B, for A is “thinner,” and indeed this may well be. Yet, our information is incomplete: We have not specified what A and B are made of; A could be made of a much stronger material than B. Thinking back to statics, we realize that we never specified the properties of the materials or structures that we studied, we simply assumed that they were always rigid (i.e., infinitely stiff). In this book, however, we will see that the individual properties of materials are central in biomechanics. For example, we often seek to match the properties of man-made or tissue-engineered replacements to those of the native tissue or organ. Indeed, one of the continuing challenges in biomechanics is accurate characterization, or quantification, of the material behavior of both living tissues and biomaterials.