MATHEMATICS:

This part provides the conceptual tools needed for an introduction to the mechanics of linearly elastic solids.

Tensor algebra. Vectors and tensors. Dot, cross and triple products. Components and dyadic products. Symmetry and skew-symmetry. Principal invariants: trace and determinant. Eigenvalues and eigenvectors. Characteristic equation. Diagonalisation of a symmetric tensor, spectral decomposition theorem.

Multivariable calculus and kinematics of deformations. Two-variable functions. Limits, continuity and partial derivatives. Tangent plane, unit normal to a surface. Vector fields. Gradient and divergence of a vector field. Infinitesimal deformations and strain tensor. Normal strain, shear strain and dilatation. Geometrical interpretation of the strain tensor. Principal strains.

Double integrals and moments of inertia. Double integrals, simple regions, iterated integrals. Applications of double integrals: centre of mass, moment of inertia. Inertia tensor for a plane region. Huygens-Steiner theorem. Principal moments and principal axes of inertia.

Surfaces. Parametric surfaces, coordinate lines and coordinate vectors. Surface integrals and divergence theorem.

MECHANICS:

The course aims to give the students the deeper knowledge about the mechanical behavior of structures with respect to the basis acquired in the bachelor degree.

Program of the lectures

The course is organized in lessons ex-cathedra and exercises. The covered topics are:

• Review of kinematics and statics and internal forces of a rigid plane beam subjected to concentrated and distributed forces.

• The transmission of forces. The traction vector and the stress tensor. The Mohr circle

• The linear elastic constitutive theory

• The cases of de Saint Venant: axial force, bending moment, shear and torsion

• The strength criteria for fragile and ductile materials

• The principle of virtual work for studying isostatic and hyperstatic structures

There are separate exams for Mathematics and Mechanics. Each examination comprises a written tests, with exercises and theoretical questions, and an oral exam. The oral examinations are usually optional, but you may be asked to sit a compulsory oral exam in some cases.

Notes are posted on Beep, and are available for download.