Multiscale nonlinear continuum mechanics of solids undergoing disarrangements

Lecturers:  Prof. Luca Deseri (UNITN/DICAM)

Timetable: this class can be taken only after having participating in the course of prof. Pandolfi. The course will be held after February 2024.

The detailed timetable and schedule will be posted as soon as available.

Programme:

The course, open to all the interested PhD students, will be delivered online by Luca Deseri after February 2024. Seminars from scientists in the field may also be included.

Prerequisites:

The graduate course “An Introduction to Nonlinear Solid Mechanics” (this year taught by Prof. Anna Pandolfi, Politecnico di Milano) is a pre-requisite for this class.
Overview. This is an introduction to multiscale to nonlinear solid mechanics of continua undergoing (i) diffuse submacroscopic rearrangements of their microstructure, such as the occurrence of damage, voids, slips, microfractures, distributed instabilities, submacroscopic reconfigurations, phase transitions, etc., and (ii) fractures, etc. Multiscale elasticity with disarrangements and dissipation will be introduced and analyzed through constitutive models within a thermodynamic framework. Paradigmatic examples will also be analyzed.

Specific topics:
1 - Multiscale elasticity allowing disarrangements and dissipation (13 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 in local measure of the deformation without disarrangements plus deformation due to them;
multiplicative decompositions generated by the two-scale geometry holding throughout the whole configuration of
the body: relationship of such decomposition with finite plasticity, growth, etc.;
• Stresses with and without disarrangements; a universal decomposition of continuum fluxes for first order
structured deformations.
• Examples and seminars.

2 - Gradient (second order) disarrangements (7 hours)
• Examples: twinning in solids; submacroscopic rotational disarrangements, etc.
• Additive decompositon of the gradient of the local measure of the deformation without disarrangements and
multiscale identification relation for its curl;
• Structured configurations and continuum fluxes revisited; power and balance laws; additive decomposition of the
stress-power; constitutive relations; frame-indifference;
• Field relations for elasticity allowing gradient and first order disarrangements;
• Coherent, submacroscopically affine motions and strain-gradient elasticity; defect-dominant gradient energetics
and strain-gradient plasticity;
• Example: diffused tensile and compressive instability; discrete to continuum multiscale instabilities; further
examples
• Preliminary applications to biological systems. Application to the multiphysics governing ligand-binding of
transmembrane proteins in the cell membrane.
• Hints about elasticity of hierarchical bodies experiencing disarrangements at the various length scales.

Selected references
1. S. Palumbo, L. Deseri, D.R. Owen, M. Fraldi (2018). Disarrangements and instabilities in augmented 1D hyperelasticity, PROCEEDINGS OF THE
ROYAL SOCIETY-A, doi.org/10.1098/rspa.2018.0312
2. L. Deseri, D. R. Owen (2019). Elasticity with Hierarchical Disarrangements: A Field Theory That Admits Slips and Separations at Multiple
Submacroscopic Levels, JOURNAL OF ELASTICITY 135 (1-2), 149-182
3. L. DESERI, D. R. Owen (2016). Submacroscopic Disarrangements Induce a Unique, Additive and Universal Decomposition of Continuum
Fluxes, JOURNAL OF ELASTICITY 2016, 122, (2), pp 223-230
4. D. R. Owen, Elasticity with Gradient-Disarrangements: A Multiscale Perspective for Strain-Gradient Theories of Elasticity and of Plasticity,
JOURNAL OF ELASTICITY (2017) 127:115–150
5. L. Deseri, D. R. Owen (2015). Stable Disarrangement Phases Arising from Expansion/Contraction or from Simple Shearing of a Model
Granular Medium, INT. J. ENGINEERING SCIENCES 96 111-130
7. L. Deseri (2004). Crystalline plasticity and structured deformations. In “Multiscale Modeling in Continuum Mechanics and Structured
Deformations”, 203-230, Edited by G. Del Piero and D. R. Owen, Springer New York, Wien
9. A. R. Carotenuto, N. M. Pugno, L. Deseri, S. Palumbo, A. Cutolo, M. Fraldi (2019). Buckling soft tensegrities: fickle elasticity and
configurational switching in living cells, J. OF THE MECHANICS & PHYSICS OF SOLIDS, doi.org/10.1016/j.jmps.2018.10.017
10. S. Palumbo, A.R. Carotenuto, A. Cutolo, D.R. Owen, L. Deseri, M. Fraldi (2021), Bulky auxeticity, tensile buckling and deck-of-cards
kinematics emerging from structured continua. PROCEEDINGS OF THE ROYAL SOCIETY-A 477:20200729
11. M. Morandotti (2017), Structured deformations and applications, PAMM 17, 711 – 712
12. J. Matias , M. Morandotti , D. R. Owen (2023) Energetic Relaxation to Structured Deformations: A Multiscale Geometrical Basis for
Variational Problems in Continuum Mechanics, Springer
13. Ana Cristina Barroso, José Matias, Marco Morandotti, David R. Owen & Elvira Zappale (2022), The Variational Modeling of Hierarchical
Structured Deformations, https://doi.org/10.1007/s10659-022-09961-w

Time duration: 20 hours (4 credits)

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