**Objectives**

To provide the elements of theory of elastic structures, linear and nonlinear solid mechanics, in view of the applications to biomechanics.

The second part of the course is aimed at providing the student with the basic knowledge of fluid dynamics with particular emphasis on biofluids.

**Prerequisites**

Calculus. Linear algebra. Elementary physics.

**Contents**

1.Introduction

-Motivation: Masonry & the microstructure of nacre. Truss structures, the vulture’s wing & the vertebrates. The brunelleschi dome & the way natural shells are broken. Impact: egg & skull. The cytoskeleton & tensegrity. Nonlinear solid mechanics & soft tissues. Brain mechanics.

2.Elementary structures

-Motivation: how living organisms transmit loading

-bending moment distribution in tetrapods & in a limb of a bird

-insects and arachnids are indetermined structures

3.Linear Solid Mechanics

-Motivation: stress in the arteries

-Solutions:(1)arteries;(2)beams;(3)failure of bone; (4)wave propagation in solids

-Curiosities: Isotropy and pebbles; transversal shrinking under compression

4.Complex structures

-The shark tooth

-The human femur and hip complex

-The human forearm complex

-Truss structures (vulture’s wing, sand dollars and the vertebrate skeleton)

-Tensegrity & the cell cytoskeleton

5.Buckling

-The size of bones (allometry)

-Curiosities: Hedgehog needles, Alan Turing, gastrulation & growth of sunflower; Brain convolutions; Flutter & snake locomotion; The coating of seawolf nuclear submarines; Did someone really chop the last tree down on Easter Island? Should towers necessarily lean? Coke cans & dislocations in solids.

6.Fracture Mechanics

-Motivation: bones & chalk, rodent teeth and the shape of the hop stem cross section

-Stiffness & strength are not the only concepts

-Stress intensity factor & criticality

-Curiosities: Pizarro, the emeralds and the Dominican monk who knew the difference between strength and toughness; the mother-of-pearl toughness.

7.Nonlinear Solid Mechanics

-Motivation: soft tissues (aneurysms in arteries)

-Artery subject to internal pressure

Curiosities: Elastic energy & insect jumping. Are bones polar materials?

8.Two-phases tissues

-Motivation: the hydrated nature of brain parenchyma, cartilage & soft tissues

-Exercise in consolidation theory and comparison with experiments on human brain tissue

-Curiosities: Silly putty, continental drift & cranial artificial deformations.

Motivation for the biofluid mechanic study. Basic assumptions.

Lagrangian kinematics: definition of position, velocity and acceleration in Cartesian and intrinsic coordinates; definition of Lagrangian system and time derivatives.

Conservation equations in Lagrangian dynamics: conservation of mass, linear momentum and energy.

Eulerian kinematics: limits of the Lagrangian approach for fluids; the velocity field and its visualization; field variables; Lagrangian derivatives in terms of partial Eulerian derivatives. The Reynolds theorem and the concept of control volume.

Eulerian expression of integral balance laws for control volume. Mass conservation for fixed and elastic volumes. Linear momentum and energy.

The motion in the neighborhood of a point: translation, deformation, rotation. Relevant tensors.

Stresses in fluids: the state of stress in a point and the Cauchy theorem

Constitutive relations between stress and rate of deformation. Newtonian fluids. Stresses in hydrostatic conditions. Non Newtonian behaviour: pseudoplatic, dilatant, thixotropic, rheopectic. Rheological properties of blood.

Balance laws in differential form: mass and linear momentum. Navier-Stokes equations for incompressible Newtonian fluids. Euler equations for inviscid fluids: Cartesian and intrinsic expression. Bernoulli theorems.

Some Exact solutions of viscous: steady flow between parallel flat planes and in circular tubes. Unsteady periodic flow in circular tubes.

Hints of Turbulence.

Integral Balance Laws to quasi-unidimensional flows: rigid conduits. Energy conservation: application to pipe flows.

Unsteady flows in elastic tubes.

**Teaching Methods**

Chalk and blackboard.

Systematic use of models.

The course will be equipped with some seminars held by dr. Bonmassari, director of the Cardiologic Department of the Hospital of Trento, regarding some physiological, medical and pathological aspect of the human hemodynamic.

**Verification of learning**

Oral test.

**Texts**

- C.R. Ethier and C.A. Simmons ‘Introductory biomechanics’ Cambridge University Press, 2007

- J.Z. Young ‘The life of Mammals’ Oxford University Press, 1957

- D. Bigoni ‘Nonlinear Solid Mechanics’ Cambridge University Press, 2012

- The Feynman Lectures in Physics, Vol II, Sections 38 and 39.

- J.R. Rice Solid Mechanics article on Encyclopædia Britannica.

-JD Humphrey, S.L. Delange. An introduction to Biomechanics. Springer

-C.G. Caro, T.J. Pedley,R.C. Schroter, W.A. Seed. The mechanics of the circulation. Cambridge university press.

-J.R. Levick. An introduction to Cardiovascular physiology. Hodder Arnold. (only cap. 8)

**More Information**

Further readings:

- H.J. Cooke, K.F.P. Burkitt & W.B. Barker ‘Biology’ Longmans, 1949

- D'Arcy W. Thompson ‘On growth and form’ (1919) Dover reprint 1992

-J.R. Levick. An introduction to Cardiovascular physiology. Hodder Arnold.

-G. K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press.

For a detailed list of contents see here