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An a posteriori approach to the mechanics of volumetric growth

Alfio Grillo and Salvatore Di Stefano

Vol. 11 (2023), No. 1, 57–86
DOI: 10.2140/memocs.2023.11.57
Abstract

This work focuses on the study of volumetric growth, i.e., the branch of biomechanics that investigates the variation of mass of biological systems, modeled as continuous media. Our purpose is to analyze in detail a perspective of growth, taken from the literature, that follows the paradigm of analytical mechanics. The analysis unfolds through (i) the introduction of suitable structural kinematic descriptors associated with growth; (ii) the identification of generalized forces dual to the virtual variations of such descriptors; (iii) the formulation of the principle of virtual work; and (iv) the establishment of a constitutive framework capable of capturing the phenomenology of interest. Within this setting, we determine the growth law of the problem at hand as a consequence of the dynamics of its structural descriptors. Accordingly, we term the growth law obtained this way the “a posteriori growth law”, and we call the resulting overall approach the “a posteriori approach”.

Keywords
tumor growth, growth phenomenology, principle of virtual work, configurational mechanics
Mathematical Subject Classification
Primary: 74B20, 74C99, 74E05, 74F99, 92-10
Milestones
Received: 29 September 2022
Revised: 16 March 2023
Accepted: 12 April 2023
Published: 23 October 2023

Communicated by Emilio Barchiesi
Authors
Alfio Grillo
Dipartimento di Scienze Matematiche “G. L. Lagrange”
Politecnico di Torino
Torino
Italy
Salvatore Di Stefano
Dipartimento di Ingegneria Civile, Ambientale, Edile, del Territorio e di Chimica
Politecnico di Bari
Bari
Italy