An Introduction to Biomechanics pp 47-107 | Cite as

# Stress, Strain, and Constitutive Relations

## Abstract

Consider the two structural members in Fig. 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 **Σ****F****= 0** and **Σ****M****= 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.

### Keywords

Anisotropy Hydrate Europe Steam Rubber### References

- Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P (2002) Molecular biology of the cell. Garland, New York, NYGoogle Scholar
- Askeland DR (1994) The science and engineering of materials, 3rd edn. PWS Kent, Boston, MAGoogle Scholar
- Carter DR, Beaupré GS (2001) Skeletal function and form: mechanobiology of skeletal development, aging, and regeneration. Cambridge University Press, CambridgeGoogle Scholar
- Cowin SC (2001) Bone biomechanics handbook, 2nd edn. CRC Press, Boca Raton, FLGoogle Scholar
- Fung YC (1990) Biomechanics: motion, flow, stress, and growth. Springer, New York, NYCrossRefGoogle Scholar
- Genovese K, Lee YU, Lee AY, Humphrey JD (2013) An improved panoramic digital image correlation method for vascular strain analysis and material characterization. J Mech Behav Biomed Mater 27:132–142PubMedCentralPubMedCrossRefGoogle Scholar
- Humphrey JD (2001) Stress, strain and mechanotransduction in cells. J Biomech Eng 123:638–641PubMedCrossRefGoogle Scholar
- Humphrey JD (2002) Cardiovascular solid mechanics: cells, tissues, and organs. Springer, New York, NYCrossRefGoogle Scholar
- Khan AS, Huang S (1995) Continuum theory of plasticity. John Wiley & Sons, New York, NYGoogle Scholar
- Mow VC, Hayes WC (1991) Basic orthopedic biomechanics. Raven, New York, NYGoogle Scholar
- Nigg BM, Hertzog W (1994) Biomechanics of the musculoskeletal system. John Wiley & Sons, ChichesterGoogle Scholar
- Özkaya N, Nordin M (1999) Fundamentals of biomechanics: equilibrium, motion, and deformation. Springer, New York, NYCrossRefGoogle Scholar
- Ratner BD (2003) Biomaterials science: introduction to materials in medicine. Academic Press, San Diego, CAGoogle Scholar
- Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York, NYGoogle Scholar
- Waldman LK, Fung YC, Covell JW (1985) Transmural myocardial deformation in the canine left ventricle: normal in vivo three-dimensional finite strains. Circ Res 57:152–163PubMedCrossRefGoogle Scholar