Models for the Mechanics of Joining Dissimilar Materials - Structural Interfaces and Attachments in Biology - page 54

Fig. 3.9 Tailoring of the insertion angle can radically reduce the region in Dundurs parameter
space over which a free edge singularity occurs. However, even in this example that precludes a
free edge singularity for nearly all engineering material attachments, a singular stress field is
possible for attachment of materials within the range of material properties reported for tendon and
bone. Data from Wang and Xu [ 26 ]
3.5 Free Edge Singularities Between Orthotropic Solids
Section 3.4 focused on attachment of two isotropic materials. Since a focus of this
text is biologic attachment and biomaterials are often quite anisotropic, we extend
the discussion in this section to the problem of free edge singularities in the
attachment of orthotropic materials.
The problem of attachment of general anisotropic materials is much studied,
with specialized numerical methodologies well established (e.g., [ 29 - 32 ]), and
many published analytical treatments [ 33 - 37 ]. A central challenge is that the
number of parameters needed to describe an anisotropic material can be much
greater than that needed to describe an isotropic material. Although anisotropic
generalizations of Dundurs parameters exist, they have not yet been explored in the
context of threshold values for edge singularities. We focus here on a specific case,
namely the attachment of an isotropic bone to an orthotropic tendon.
The mechanical properties of a generally anisotropic linear elastic material can
include up to 21 constants, but this number can be reduced to five based upon
symmetry arguments for a material such as tendon that can be approximated as
transversely isotropic. This means that we will consider the case studied in the
introductory chapter, in which the idealized tendon has the same stiffness for
stressing in any direction perpendicular to that of the dominant direction of the
fibers that comprise them. The general anisotropic constitutive law, e ij ¼ S ijkl s kl ,
involves a fourth order tensor S called the compliance tensor. This tensor is