• 097723 - MATHEMATICAL AND PHYSICAL MODELING IN ENGINEERING [1]

The main objective of the overall course is to acquire knowledge of mathematical and physical modeling of continuum material bodies and to develop critical skills for their application in solving engineering problems.

The main objective of part [1] of the course is to provide the student with an overall understanding of the basics of advanced Continuum Mechanics and Continuum Thermodynamics. The student is expected to become knowledgable about all the main concepts and ideas, so that he might be ready to use them in more application-oriented contexts, which are part of the program in Mathematical Engineering, in particular for the "scientific computation track".

The student is expected:

- to master the mathematical develoment of all the mathematical and physical methods exposed during the course;

- to develop a good capability of handling and working with tensorial quantities, of writing balance equations and boundary conditions for simple problems in elasticity theory;

- to understand the structure and the construction of basic theories of Continuum Mechanics;

- to become articulate and fluent in expressing and communicating all the key ideas and proofs of the basic theorems and concepts.

Cauchy stress tensor and mechanical balance equations (reminder).

Theorem of kinetic energy and power of internal stresses. Piola transformation for vector and tensor fields. Piola stress tensor: definition and properties.

Elastic solids. Constitutive relation. Local balance equations in a material form. Material frame indifference. Symmetry group. Isotropicity. Representation formula. Strain energy in finite elasticity. Strain energy for isotropic materials. Examples: Blatz-Ko, Mooney-Rivlin, Ogden, Neo-Hookean. Internal constraints and reactive stresses. Examples: incompressibility, inextensibility in a given direction.

Linearization of constitutive relation for elasticity. Infinitesimal strain tensor. Elasticity tensor. Linear constitutive relation. Symmetry group. Isotropic linear elasticity. Lame' constants. Navier equations. Wave propagation in linear isotropic elasticity: transverse and longitudinal wave popagation. Simple shear, uniform extensions and uniform tractions. Poisson's ratio and Young's modulus.

Energy balance for continuum thermodynamics. Entropy inequality. Axioms of Continuum Thermodynamics. Constitutive relations for thermoelasticity. Free energy and entropy inequality. Coleman and Noll method and consequences. Linear thermoelasticity. Field equations for linear thermoelasticity. Fourier's law. Heat equation. Symmetry group for elastic fluids.

Constitutive relations for materials with stress tensor depending on F and its time derivative. Viscoelastic materials. Fluids. Viscous fluids. Newtonian and non-Newtonian fluids.

Tensor algebra and general coordinates in continuum mechanics. Natural basis and dual basis. The metric tensor. Covariant and controvariant components. Covariant derivative and Christoffel symbols.

Basic ideas of calculus. Theory of ordinary differential equations and basic notions about partial differential equations. Elementary knowledege of classical Rational Mechanics and Continuum Mechanics (forces, balance equations, Cauchy stress tensor, kinematics of continua).

The exams wil be organized in agreement with the Academic Calendar and the Rules of the School of Engineering.

The verification of knowledge and capabilities acquired by the student is based, for each part, on a written test of about 2h, followed by an oral part.

The two written tests can be taken together during the same exam session, or separately in different exam sessions.

The final grade is based on an overall evaluation of the two parts of the exam.

The verification of knowledge and capabilities acquired by the student in part [1] of the course is based on a written test of about 2h, followed by an oral part.

The written test consists in an exercise on Finite Elasticity theory, which requires the student to be able to work with tensorial constitutive relations and writing and solving simple equilibrium problems. This will provide a basis on which to evaluate the capability of the student to apply correctly knowledge and comprehension acquired during the Course.

In the oral part of the exam the student will show his knowledge and his communication skills through appropriate exposition of all the main concepts and correct presentation of the proofs of all the main results and theorems.

In the oral part the student is also expected to answer theoretical questions on course topics, and to verify student's skills at making connections between course topics and previous courses.

Material prepared by the Teachers will be placed in the Beep area of the Course.