Volume 13, Number 2, Summer 2005
A MOVING BOUNDARY VALUE PROBLEM IN
SOFT TISSUE MECHANICS
MAREK STASTNA
Abstract. Mechanical theories of soft matter such as biological
tissues often tread a fine line between a theory that
can represent the maximal range of experimentally observed
behaviour and one that can yield solutions in a reasonably efficient
manner. While the advent of cheap, high speed computation
has extended the range of theories that can be used
for numerical analysis of a given situation, questions of software
validation, interpretation of experiment and the determination
of mechanical parameters still require a set of simple to
interpret, analytically tractable theories. In the following we
consider the linear theory of consolidation in an idealized con
guration with a moving boundary. Employing an invertible
coordinate transformation we derive a nonlinear problem on a
fixed domain. In the steady limit we derive analytical solutions
on the fixed domain, and by inverting the transformation, for
the steady state free boundary value problem. We employ standard
numerical methods to show that the timedependent nonlinear
problem tends to the steady state limit and discuss the
rate at which the evolution to the steady state depends on the
physical parameters. The steady state results are subsequently
utilized to construct an algorithm for solving the free boundary
value problem for steady state consolidation in a tethered tube
geometry. Finally, the algorithm is implemented to discuss several
case studies which outline the portions of parameter space
in which the effect of the free boundary is significant.
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