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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 time-dependent 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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© 2005, Canadian Applied Mathematics Quarterly (CAMQ)