Models for the Mechanics of Joining Dissimilar Materials - Structural Interfaces and Attachments in Biology - page 43

4. Newton's second law, written for a continuous system that is not accelerating
and for which gravitational forces are negligible:

s ij;j ¼ 0 (3.5)

Traction boundary conditions enter the problem by noting that T i ¼ s ji n j , where
n is the outward normal vector of the body's outer boundary at a point at which a
traction vector is applied (e.g., the portion S T of the body in Fig. 3.1 ). Finding the
set of internal vector and tensor fields that uniquely satisfy the above four sets of
equations is often not straightforward and is often difficult to do by inspection. Most
of the solutions presented in this chapter were derived using planar assumptions
(either plane stress, in which out-of-plane stress tensor components are taken as
zero or constant, or plane strain, in which out-of-plane strain tensor components are
taken as zero or constant), and using a mathematical trick known as a potential
function. Although the solution procedures are well beyond the scope of this
chapter, we present here a simple example with the hope of preparing a student
interested in the subject to follow a derivation in one of the original articles. The
example presented here is the stress function of Airy, who defined a function

' such

that s xx ¼ @

'
@y 2

2

; s yy ¼ @

@x 2 , and s xy ¼ @

2

' '

@x@y

2

@'
@r

1
r

or, in cylindrical coordinates, s rr ¼

. Substituting this function into Newton's
second law results in Newton's second law being identically satisfied within the
plane. Substituting these definitions into the constitutive equations, and then
substituting the resulting expressions for strain into the compatibility relations
results in a single equation for

2 2

r 2 @

'

, s yy ¼ @

' @'

@y

1

@r 2 , and s ry ¼ @r

1
r

þ

@

2

y

4

' ¼ 0. The identification of a solution then
involves finding which of the functions

' : r

that satisfy this biharmonic equation
meets the boundary conditions for the problem of interest.
One solution presented in this chapter makes use of an energy argument. We
note that the average strain energy U per unit volume over a portion of a body, say
V T as in Fig. 3.1 , can be written:

'

ð

1
2

U ¼

V T s ij e ij d V

(3.6)

and the total strain energy in that part of the body is the product of the volume and
the strain energy density, V T U .

3.3 An Isotropic Solid Attached to a Rigid Substrate

The simplest possible idealizations of an engineering attachment yield insight into
how the morphology of the attachment governs the stress field, and we begin by
discussing two such idealizations. We will relate these to the attachment of tendon