Reduced-Order Network Models for Biological Scaffolding - Multiscale Modeling in Biomechanics and Mechanobiology - page 108

In the present analysis, any surrounding soft tissue contribution to the response is
considered negligible relative to that of the relatively stiff load carrying fibers. 1
By employing enough of these simple structural elements, one can build an entire
macroscale sheet of fibrous tissue, as shown in Fig. 6.1 .

Remark 1 The use of pin-joints, i.e., not allowing moments at each node, greatly
speeds up the computation in this approach. Bending can be taken into account, and
adds a degree of complexity that may be warranted in certain applications. Probably,
there are situations where the adopted simplification could be adequate, and others
where it is not. Also, as mentioned previously, we ignore buckling phenomena and
consider cases where compressive stresses are of somewhat less importance than
tensile states. The objective of this work is simply to illustrate the main modeling
and solution techniques, without overly complicating issues. In other words, this
model and solution techniques serve as a starting point for more in-depth analyses.

6.3 Fiber-Segment Network Representation

For the mechanical portion of the modeling of network structures, we assume that: (1)
the fiber segments are quite thin, experiencing a uniaxial-stress condition, whereby
the forces only act along the length of the fiber segments, (2) the fiber segments
remain (macroscopic) straight, undergoing a homogeneous (axial) stress state, (3)
the compressive response of a fiber segment is insignificant (relative to tensile states),
and (4) fiber-segment buckling phenomena are ignored. We write one-dimensional
constitutive laws in terms of a one-dimensional scalar Piola-Kirchhoff-like stresses
(mimicking 3-D approaches), defined by

force on referential area
referential area

P = , (6.1)

and then transform the result to a scalar second Piola-Kirchhoff-like stress via P =

L

US , where U

L o is the stretch ratio, L is the deformed length of the fiber segment,
L o is its original length and where we note that for a relaxed model, when U

=

≤ 1

(compression), we enforce P is

then employed, with the primary objective being to extract the force carried in the
fiber segment (

= 0. A standard constitutive relation S = F( U )

fiber ), which is needed later for the dynamics of the lumped masses.

ʨ

Specifically,

1 At the end of this chapter, we shall return to this issue, and indicate how soft tissue-fiber interaction
can be computed.