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Abstract:
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The mechanical analysis of soft tissues in biomechanics has undergone an increasing
progress during the last decade. Part of this success is due to the development and
application of some techniques of continuum mechanics, in particular, the decomposition
of the deformation gradient, and the introduction of mass, density or volume
changes in the reference configuration. Resorting to the common terminology employed
in the literature, the changes in biomechanical processes may be classified as
growth (change of mass), remodelling (change of density or other material properties
such as fibre orientation) or morphogenesis (change of shape). Although the use of
those concepts in bone and cardiovascular analysis is well extended, their use in morphogenesis during embryo development has been far less studied. The reasons of this
fact may be found in the large shape changes encountered during this process, or the
complexity of the material changes involved.
In this chapter we develop a general framework for the modelling of morphogenesis
by introducing a growth process in the structural elements of the cell, which in
turn depends on the stress state of the tissue. Some experimental observations suggest this feedback mechanism during embryo development, and only very recently this behaviour has started to be simulated.
We here derive the necessary equilibrium equations of a stress controlled growth
mechanism in the context of continuum mechanics. In these derivations we assume
a free energy source which is responsible of the active forces during the elongation
process, and a passive hyperelastic response of the material. In addition, we write
the necessary conditions that the active elongation law must satisfy in order to be
thermodynamically consistent. We particularise these equations and conditions for the relevant elements of the cytoskeleton, namely, microfilaments and microtubules. We apply themodel to simulate the shape changes observed during embryomorphogenesis in truss element. As a salient result, themodel reveals that by imposing boundary stress conditions, unbounded elongation would be obtained. Therefore, either prescribed displacements or cross-links between fibres are necessary to reach a homeostatic state. |
