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Kinematics of surface growth. (English) Zbl 0883.92005
Summary: A general mathematical framework is developed to describe cases of fixed and moving growth surfaces. This formulation has the mathematical structure suggested by R. Skalak [Growth as a finite displacement field. In: Proc. IUTAM Symp. Finite Elasticity. The Hague (1981)], but is extended herein to include discussion of possible singularities, incompatibilities, residual stresses and moving growth surfaces. Further, the general theoretical equations necessary for the computation of the final form of a structure from the distribution of growth velocities on a growth surface are presented and applied in a number of examples.
It is shown that although assuming growth is always in a direction normal to the current growth surface is generally sufficient, growth at an angle to the growth surface may represent the biological reality more fully in some respects. From a theoretical viewpoint, growth at an angle to a growth surface is necessary in some situations to avoid postulating singularities in the growth velocity field. Examples of growth on fixed and moving surfaces are developed to simulate the generation of horns, seashells, antlers, teeth and similar biological structures.

MSC:
92C15 Developmental biology, pattern formation
92C05 Biophysics
92C10 Biomechanics
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