The course re-elaborates some theoretical aspects of the mechanics of growth and structural reorganisation (remodelling) of biological tissues by proposing an interpretation suggested by Analytical Mechanics and Variational Principles. For this purpose, the contents of the course will be framed in the context of Field Theories, and studied by following a Continuum Mechanics approach, which will be suitably generalised to include chemo-mechanical phenomena. Since growth and remodelling are the results of phenomena of various nature, their investigation necessitates transversal competencies. This leads to an interdisciplinary environment that, although being very fruitful from the scientific point of view, requires to reconcile well-acknowledged physical principles, such as Balance Laws, with phenomenological laws that are introduced ad hoc and that, in spite of being compatible with experimental data, may sometimes lack of a clear theoretical justification. The main goal of the course is to provide the Participants with the methodological instruments necessary for a systematic formulation of the models of growth and remodelling, thereby proposing a theoretical interpretation also of the phenomenological laws. In particular, for a given tissue and for a given problem, the course proposes: (1) To define the problem’s kinematics by extending the concept of Lagrangian parameter to the descriptors of the internal structure of the tissue; (2) To identify and classify the constraints to which the tissue is subjected, in order to interpret possible phenomenological laws that relate the variations of the kinematic parameters with the other physical quantities characterising the tissue itself; (3) To identify the generalised forces acting on the tissue and, if possible, to characterise them constitutively; (4) To determine the dynamic equations of the systems under study. Once the constraints are given, and the dynamic equations are determined, the course aims at supplying criteria that, through the employment of appropriate invariance (or covariance) principles establish the admissibility of the relationships connecting the variables of a model. Finally, it will be discussed how the structural transformations induced in a tissue by growth and remodelling influence the material symmetries of the tissue itself and, for this purpose, the proposed topics will be discussed under the light shed by Noether’s Theorem.

Basic courses of Mathematical Analysis I and II; Rational Mechanics (optional).

1. Brief review on Continuum Mechancs (5 hours) 1.1. Definition of the principal tensor quantities employed in Continuum Mechanics. 1.2. Bilby-Kröner-Lee decomposition. 2. Brief review on Analytical Mechanics (5 hours) 2.1. Kinamtics, Definition and classification of kinematic constraints. 2.2. Generalised forces. Determination of the dynamic equations for a constrained system. Lagrange’s multipliers. 3. Mechanics of growth and remodelling (10 hours) 3.1. Dynamic equations for a tissue subjected to growth and remodelling. 3.2. Invariance principles for the admissibility of evolution laws. 3.3. Material symmetries and Noether’s Theorem.

Mixed mode

Oral presentation

P.D.1-1 - November

VARIATIONS OF THE TEXT Although the contents of the course remained unchanged (indeed, the present offer proposes again the course already held in 2019), I changed the old Italian text "ai Partecipanti" into "ai/alle Partecipanti", in order to adhere to the inclusive policy of the Politecnico di Torino. OTHER POTENTIALLY INTERESTED PhD PROGRAMS The course is potentially of interest also for the other following PhD Programs: "Bioingegneria e Scienze Medico-Chirurgiche", "Fisica", "Ing. Aerospaziale", "Ing. Chimica" e "Ing. Meccanica". INTERDISCIPLINARITY OF THE TOPICS Although the leading topic of the course is addressed to theoretical Biomechanics, the expertise that will be transferred to the Participants can be employed also in other contexts of mathematical physics and engineering, such as the characterization of industrial materials. DRAFT OF THE SCHEDULE OF THE LECTURES First week - Monday 08/11/2021, from 10:00 to 12:00 (2 h); - Tuesday 09/11/2021, from 10:00 to 13:00 (3 h); - Thursday 11/11/2021, from 15:00 to 18:00 (3 h); - Friday 12/11/2021, from 15:00 to 17:00 (2 h). Second week - Monday 15/11/2021, from 10:00 to 12:00 (2 h); - Tuesday 16/11/2021, from 10:00 to 13:00 (3 h); - Thursday 18/11/2021, from 15:00 to 18:00 (3 h); - Friday 19/11/2021, from 15:00 to 17:00 (2 h).