Biomechanics: Concepts and Computation
Using a quantitative biomechanics approach, this book integrates both general and specific topics, theoretical background and biomedical engineering applications, as well as analytical and numerical approaches.
TABLE OF CONTENTS Biomechanics: Concepts and Computation
Chapter 8: Stress in Three-Dimensional Continuous Media
In this chapter the concepts introduced in Chapter 6 for a one-dimensional continuous system are generalized to two-dimensional configurations. Extension to three-dimensional problems is briefly discussed. First the equilibrium conditions in a two- or three-dimensional body are derived from force equilibrium of an infinitesimally small volume element. Thereafter, the concept of a stress tensor, as a sum of dyads, is introduced to compute the stress vector acting on an arbitrary surface in a material point of the body.
8.1 Stress Vector
Before examining the equilibrium conditions in a two-dimensional body, the concept of a stress vector is introduced. For this purpose we consider an infinitesimally small surface element having area ? A, see Fig. 8.1.
Figure 8.1: Force
on an infinitesimal surface element having area ?A.On this surface an infinitesimally small force vector is applied with components in the x-, y- and z-direction: 
. Following the definition of stress, Eq. (6.8), three stresses may be defined:
that acts in the x-direction, and
that acts in the y-direction, and
that acts in the z-direction. Hence, a stress vector may be defined:
8.2 From Stress to Force
Suppose, that a free body diagram is created by means of an imaginary cutting plane through a body. The cutting plane is chosen in such a way that it coincides with the xy-plane (Fig. 8.2). On the imaginary cutting plane a stress vector 
is given as a function of x
Copyright C, Oomens, M. Brekelmans and F. Baaijens 2009 under license agreement with Books24x7
