Continuum Modeling in Mechanobiology by Larry A. Taber

Author:Larry A. Taber
Language: eng
Format: epub
ISBN: 9783030432096
Publisher: Springer International Publishing

(6.57)

where ΩR and S R are the volume and surface area, respectively, of state B R (see Fig. 6.10). The other equations remain the same.

6.6.4 Growth Equations in Principal Cylindrical Coordinates

Numerous problems involving biological growth involve cylindrical or spherical structures. Thus, it is useful to list the basic equations in cylindrical and spherical coordinates. The expressions given below for the Lagrange multiplier p are obtained by substituting the constitutive relations for the Cauchy stresses σ i into the (radial) equilibrium equation. Auxiliary equations (e.g., growth laws), as well as boundary and initial conditions, are given later for specific problems.

Here, the equations are restricted to cases where the coordinates correspond to principal directions of stress, strain, and growth. For cylindrical polar coordinates, we set

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