Rheological Considerations in the Physical Properties of Bone 



99 



Bone has long been known to be an anisotropic material. As such, the laws and 

 formulae that are utilized in determining p.p. should be applied with caution. Our 

 use of the standard formulae has been for the sole purpose of comparing the effects of 

 methods of handling, or differences in individuals. An example of this would be the 

 use of the straightest portion of the stress-strain curve in determining E, when, in 

 fact, only a rare curve in our series possessed a truly linear component. To account 

 for this phenomenon, and to reconcile it and other findings with statements that bone 

 is an elastic material, visco-elastic material, two or three phase material, we were 

 lead into the field of rheology, which in the broadest sense is the study of the deforma- 

 tion of all materials (Reiner, 1958). 



There are four fundamental properties of materials, all others being reducible to 

 definitions in terms of these. They are elasticity, plasticity, viscosity, and strength 

 (Reiner, 1958). We will consider the first three of these at this time, the fourth being 

 beyond the limits set for discussion. To better understand what is meant by the terms, 

 one can use idealized models, as are presented in Fig. 1. Thus, we can speak of the 



[y^^" D- 



(a) 



(b) 



(e) 



(?) 



(H) 



Fig. 1. Several models to depict the ideal behavior of certain materials. a = stress. £ = strain. t = time. Os = yield 

 point, a) The perfectly elastic body. This is represented by a spring. A stress-strain diagram for this body (e) 

 is a straight line, the slope of which is determined by the modulus of elasticity, b) The perfectly viscous body. 

 Symbolized by a dash-pot. it is typified by a linear relationship between strain and time (o = k), and between 

 stress and the rate of strain (f). c) The rigid plastic. This material resists deformation below Og, and if o 

 exceeds this point, it flows indeterminately (g). d) The perfectly plastic or St. Venant body. Here a spring is 

 linked in series with a plastic. The material is able to deform in a linear fashion until a,, is reached, after 

 which the internal solid friction of the plastic is overcome, and the material flows indeterminately (h) 



perfectly elastic body, the perfectly viscous body, the rigid plastic body, and the 

 perfectly plastic body (Jaeger, 1956). All materials can be characterized by various 

 combinations of these basic elements. In the perfectly elastic body, o is proportional 

 to strain (e) and Is related by the constant E. If this material be loaded. It Is Immedi- 

 ately strained, and If unloaded, the strain Is Instantaneously recovered, so that no 

 deformation remains when o = 0. In the viscous body, the rate of strain (f) is pro- 

 portional to o, and Is related by a constant //. These are statements of Hooke's and 

 Newton's laws. In the rigid plastic body, no deformation whatsoever takes place 

 below the yield point (og), and If a = Os, the material flows indeterminately. In the 

 perfectly plastic body, o = E f for o < o^ , and when o = Os the material flows indeter- 

 minately. Our stress-strain curves for all axes of testing show that bone Is a more com- 

 plex material than can be typified by the Ideal bodies cited. Other types of testing. In 

 addition to the ones cited have lead us to formulate a model for the behavior of bone 

 under moderate load (Fig. 2). This is a model of a Hooke body linked In series to a 



