Mathematical classification of growth, remodeling, aging, and morphogenesis according to Marcelo Epstein

Epstein defines pure remodeling as a reaccommodation of the same material within the same material neighborhood, in the sense that the time dependence of the constitutive equation for the strain energy at a given point $\mathbf X$ has the form

where $\mathbf P(t)$ is an automorphism from $T_{\mathbf X}\mathscr M$ to itself, i.e. a non-singular matrix. This notion corresponds to material isomorphism in the sense of Noll. A more mundane, though somewhat misleading, way to visualize this fact is to imagine that the constitutive properties can be boiled down to a linear spring with stiffness k and rest length $L_0$. In a process of pure remodeling, the stiffness coefficient k remains constant, while the rest length $L_0$ changes with time. Though incomplete and elementary, this visualization is sometimes heuristically useful.

Any material evolution process of the form that is not expressible in the form $\eqref{pippo2}$ is, by definition, an aging process. In this context, the word aging does not imply any consideration about beneficial or pernicious effects of the passage of time, and may be equally refferred to as rejuvenation.

Aging may affect or not the symmetry group of the constitutive response. According to Epstein,

• morphogenesis is a special type of aging process by which the material evolution process entails also a sudden change in the material symmetry type, as when the material changes from fully isotropic to orthotropic.

Accordingly,

• a material point with constitutive equation $W=W(\mathbf F,t)$ is said to undergo a process of aging without morphogenesis in a time interval T if the symmetry groups are conjugate subgroups of $GL(3;\mathbb R)$ for all t∈T.

In the language of Section 2.2, the symmetry groups at different times in a morphogenetic process are no longer necessarily conjugate. This symmetry breaking is indeed considered as essential for pattern formation in plants and animals, as already envisioned in the epoch-making article by Turing (1952).

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