Models for the Mechanics of Joining Dissimilar Materials - Structural Interfaces and Attachments in Biology - page 42

3. The displacement at each point and in each direction for which traction was
prescribed in the boundary conditions (e.g., the portion S u of the boundary of V
in Fig. 3.1 ).
4. The traction at each point and in each direction for which displacement was
prescribed in the boundary conditions (e.g., the portion S t of the boundary of V in
Fig. 3.1 ).

These are obtained by solving the following sets of equations at each point
within the model:

1. Strain-displacement equations that define the strain tensor field « ( x ) at each
point x within the volume V T of the tendon and V B of the bone in terms of the
displacement vector field, u ( x ). Here and throughout the chapter, we write these
equations using index notation, in which subscripts are understood to represent
indices ranging from 1 to 3, commas represent partial differentiation, and
repeated indices within a grouping imply summation over the three values of
that index. The strain displacement relations are then:

1

e ij ¼

2 ðu i;j þ u j;i Þ

(3.1)

in a
particular coordinate system, u i represents the three components of the displace-
ment vector u in that coordinate system, and u i;j ¼ @u i =@x j .
2. Compatibility relations. Since the nominal strain tensor has six independent
components and the displacement vector only three, not every choice of a
spatially varying strain field relates to a displacement field. Strain fields that
do meet this criterion satisfy the compatibility relations:

where e ij represents the 3 3 matrix of components of the strain tensor «

e ij;kl þ e kl;ij e jl;ik e ik;jl ¼ 0 (3.2)

where the additional subscript following the comma represents a second deriva-
tive with respect to the components of the coordinate system, so that e ij;kl ¼ @

2

e ij =@x k @x l .
3. Constitutive relations that predict the strain tensor at a point in terms of the stress
tensor s . For the case of linear, isotropic elasticity, these are:

1
E ½ð 1 þ nÞs ij nd ij s kk

e ij ¼ (3.3)

E
ð 1 þ nÞ e ij þ

E
ð 1 þ nÞð 1 2 nÞ d ij e kk

s ij ¼ (3.4)

where d ij is Kronecker's delta function that equals 1 when i ¼ j and 0 otherwise,
and e kk ¼ e 11 þ e 22 þ e 33 .