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Introductory nonlinear mechanics

Lecturers:  Prof. Luca Deseri (UNITN/DICAM)

Timetable and schedule will be posted as soon as available.

Programme:
This course is directed at honour program, doctoral, postdoctoral and other researchers interested in advancing their knowledge of continuum mechanics beyond classical incompressible, nonlinear elastic solids. The course will begin with a review of fundamental principles of continuum mechanics and move on to modeling the behavior of materials with evolving internal structure, with particular regards to soft biological tissues. Topics such as the multiphysics of biomembranes at a small length scale, growth and remodeling in biological tissues will be covered in class.
Outline.
1-Summary of macroscopic nonlinear continuum mechanics (11 hours)
• Kinematics of large deformations;
• Stress measures in the context of solids experiencing large deformations;
• Balance equations for statics and dynamics;
• Constitutive laws: general principles;
• Nonlinear elasticity: relationships among the various measures of stress and the corresponding strain measures; incompressible and compressible solids;
• Application: the mechanics of the cell membrane. The ideal case of lipid bilayers.
Reading in class: 2 hours.
2-Multiscale elasticity of disarrangements and dissipation: application to biomechanics (9 hours)
• Material bodies and their multiscale structured configurations: motivations and examples of first and second order disarrangements (slips, voids, microfractures, micro-shears, rotational changes, solid-solid phase transitions, etc.);
• Two-scales geometrical changes due to the rearrangements of the microstructure; additive decomposition of the deformation gradient; multiplicative decompositions generated by the two-scale geometry and their relationship of such with finite inelasticity, such as growth, etc.
• Stresses with and without disarrangements; a universal decomposition of continuum fluxes for first order structured deformations.
• Submacroscopic rotational disarrangements; example: active proteins experiencing conformational changes.
Reading in class: 2 hours.
3-Mechanics of soft biological tissues (10 hours)
• Multiple configurations in presence of growth. Inelastic deformations and multiplicative decomposition of the deformation gradient. General incompatibility of grown configuration and elastic stretches. Constitutive relations revisited and concept of residual stress in growing biological structures. Material models for soft biological tissues.
• Growth functions and mass balance, relationship between mass change and deformation. Coupling between mechanics and evolutionary equations. Phenomenological growth models, remarks on interspecific nonlinear models. Growth in multi-component/multi-phase systems and remarks to nutrient transport in poro-elastic heterogeneous solids. Mechanical feedback on growth, mechano-sensing and stress-mediated growth in soft tissues.
• Applications to the growth and remodeling of biological structures.
References
A. Goriely (2017). Morphoelasticity: The Mathematics and Mechanics of Biological Growth. Springer-Verlag
S. C. Cowin, S. B. Doty (2007) Tissue Mechanics. Springer
G. Del Piero, D. R. Owen (2004) Multiscale Modeling in Continuum Mechanics and Structured Deformations - CISM Lecture Notes No. 447. Springer, doi 10.1007/978-3-7091-2770-4_7
L. Deseri, D. R. Owen (2019) Elasticity of hierarchical bodies predicted with multilevels Structured Deformations, 135 (1-2), 149-182
S. Palumbo, A.R. Carotenuto, A. Cutolo, D.R. Owen, L. Deseri & M. Fraldi (2021), Bulky auxeticity, tensile buckling and deck-ofcards kinematics emerging from structured continua. Proc. Royal Soc.-A 477:20200729 https://doi.org/10.1098/rspa.2020.0729
A. R. Carotenuto, L. Lunghi, V. Piccolo, M. Babaei, K. Dayal, N. Pugno, L. Deseri & M. Fraldi (2020). Mechanobiology predicts raft formations triggered by ligand-receptor activity across the cell membrane. J. of the Mechanics and Physics of Solids, 141, 103974.

Time duration: 24 hours (3 credits)

Registration: in order to access the course, please send an e-mail to dicamphd [at] unitn.it