Models for the Mechanics of Joining Dissimilar Materials - Structural Interfaces and Attachments in Biology - page 41

notation that will be used in this chapter, or to provide the initiate with a broad sense
of what underlies the boundary value problems presented in this chapter. For a more
meaningful discourse on the subject, we suggest Allan Bower's online text [ 6 ].
3.2 A Whirlwind Tour of Linear Elasticity
This section presents the basic elements of a boundary value problem in linear
elasticity. The goal in each of the model problems discussed in this chapter is to
understand how the details of the way two materials are attached and loaded elevate
internal forces (stresses) at the attachment site. In each case studied, the boundary
value problem involves an idealization of an attachment into a linear elastic
“tendon” and a linear elastic “bone,” with a sharp interface between them
(Fig. 3.1 ). The inputs to the problem are:
1. The initial shapes of tendon, bone, and their interface, which will often look
somewhat less like a potato than Fig. 3.1 but nevertheless be highly idealized.
2. A set of constants that describe the mechanical responses of the tissues, which, in
the simplest model problems, consist of two constants for tendon and two
constants for bone, which are an elastic modulus E and a Poisson ratio n .
3. A set of boundary conditions that describe either the displacement or the
mechanical tractions applied in each of three orthogonal directions at each
point on the outer boundaries of the model.
The outputs are:
1. The internal stress field.
2. The internal displacement field.
Fig. 3.1 A potato, with some notation useful to the theory of elasticity