Postgraduate courses

Syllabus:
Mechanical properties of materials, failure criteria, concept of stress concentration, plastic analysis, linear elastic fracture mechanics, elasto-plastic fracture mechanics, characterisation of properties in fracture mechanics. Standards.

Programme:
Failure modes: functional failure, physical failure, overload failure, failure with damage accumulation. Mechanical behaviour: tensile test, stress-strain curve and models, residual stress models, impact test.
Stress concentration: Definition, elliptical hole problem, limit cases, application to static cases, Neuber’s theory, direct problem, inverse problem, applications.
Plastic analysis: yield initiation load, plastic collapse load, application to tension, application to bending, shape factor, residual stresses in bending, residual stresses with stress concentration.
Linear elastic fracture mechanics: Griffith’s theory, FIT concept, Ki, stress field, geometric factor, plasticisation radius.
Elasto-plastic fracture mechanics: Competing failure modes, brittle rupture, plastic collapse, example, crack opening displacement, critical failure stress, J integral.
Toughness tests: ASTM E399, KIc; ASTM E1737, JIc; ASTM E1820; R-curves.
Application standards: API 579, API 530.

Syllabus:
Analysing the phenomenon of fatigue and the response of materials to cyclic stresses. Types of tests and testing machines. Material curves, fatigue life regimes. Estimation of fatigue curves. Stress concentration in the elastic and elasto-plastic regime. Effect of average stresses, importance of residual stress for fatigue. Crack propagation, Paris law, propagation life.

Programme:
THE FATIGUE PHENOMENON Micro and macroscopic behaviour, cyclic stress-strain curve, design criteria.
FATIGUE RESISTANCE OF MATERIALS Fatigue tests, experimental results, estimation of the material’s  – N and  – N curves, cyclic stress-strain curve, accumulated damage, random stresses.
FATIGUE RESISTANCE OF COMPONENTS Effects on the  – N and  – N diagrams, stress concentration, Neuber’s rule.
THE EFFECT OF AVERAGE STRESSES Diagrams a – m, stress concentration under average stresses, safety factor, use of the diagram  – N, combined loading, residual stresses.
CRAFT PROPAGATION Correlation å – K, propagation life, damage tolerance, significance and estimation of defects.

Syllabus:
FEM in 1D problems: bar problem. Weighted residuals, PTV, Principle of Minimum Total Potential Energy. Superposition. Processes for applying boundary conditions. 2D/3D FEM in scalar fields: heat conduction problem. Strong and weak formulation. Global and elementary basis. Inhomogeneous Dirichlet boundary conditions. Convergence curves. FEM for linear elasticity. Strong and weak formulation. FEM for plane strain, stress and axisymmetric states. Volumetric elements. Isoparametric elements. Mapping. Jacobian. Numerical integration. Applications and convergence evaluations. Basic mathematical properties of FEM. 1D linear and bilinear forms. Equivalence between variational and minimum problems. Uniqueness of the solution. A-priori error estimates in 1D FEM. FEM of homogeneous plates. Mindlin-Reissner kinematic model. Pathologies: bending locking.

Programme:
1 FEM in 1D problems: bar problem. Weighted residuals, PTV,
2 Principle of Minimum Total Potential Energy. Superposition. Application of boundary conditions. Lagrangean interpolation functions.
3 2D/3D FEM in scalar field: heat conduction problem.
4 Strong and weak formulation. Galerkin approximation. Global and elementary basis. Inhomogeneous Dirichlet boundary conditions. Convergence curves.
5 FEM for linear elasticity. Strong and weak formulation. Galerkin approximation.
6 FEM for plane strain, stress and axisymmetric states.
7 Volumetric elements. Isoparametric elements. Mapping. Jacobian.
8 Shape functions in intrinsic coordinates. High-order elements.
9 Numerical integration. Applications and convergence evaluations.
10 Basic mathematical properties of FEM. 1D linear and bilinear forms. Equivalence between variational and minimum problems. Uniqueness of the solution.
13 Linear spaces, norms. Symmetry, distributivity and positivity of the bilinear form in elasticity and of the stiffness matrix. Uniqueness. Is the FEM solution the best solution?
14 A-priori error estimates in 1D FEM. Error in the derivative, linear polynomials, quadratic polynomials in the member problem.
15 FEM of homogeneous plates. Mindlin-Reissner kinematic model. Rotation of the element matrix and modelling of shells by plate elements. Pathologies: bending locking. 16 FEA for homogeneous plates.

Programme:

Vibrations in systems with 1 degree of freedom. Types of problems. 1D equation of motion.  Free vibration solutions without and with damping.

Harmonic loading. Non-periodic loading via the Duhamel integral.

FEM in dynamics. Eqs. of 3D motion via PTV and D’Alembert’s Principle.

Lagrange’s equations of motion. FEM in bar, beam and solid problems.

Modal analysis. Undamped free vibrations. Properties of the system’s eigenvalues and eigenvectors. Initial excitation in undamped systems.

General modal analysis. Determination of damping.

Guyan reduction. Harmonic response analysis. Via modal reduction, via Guyan reduction.

Direct integration methods: central difference, diagonalisation of the mass matrix.

Implicit methods. Stability and accuracy.

Integration in non-linear systems. Newmark methods.

FEM of homogeneous plates. Kirchhoff and Mindlin-Reissner kinematic models. Resultant forces, constitutive relations.

PTV in plates. FEM in plates. Rotation of the element matrix and modelling of shells using plate elements. Pathologies: bending locking.

Variational calculation. Overview of weighted residue methods: placement, discontinuities…

Programme:
Introduction. Types and properties of fibres, matrices and composites. Manufacturing processes. Advantages and disadvantages.
Micromechanics of a layer..
Macromechanics of a layer. Mechanical properties of a layer.. Stress-strain relationship. Failure criteria.
Laminate analysis – Classical laminating theory. Determination of stresses in laminates. Initial failure analysis.
Formulation of the 1st order laminated plate problem. Kinematic equations of motion. Principle of virtual work, deformation energies. Particularities for thin plates.
Interlaminar stresses. Analytical solutions for cylindrical bending using TCL, linear elastography and 1st order theory. Shear k-factor for homogeneous and orthotropic laminated plates.
Design of beams with equivalent modulus of elasticity.
Sandwich panels – constructive and design aspects, failure modes.

Programme:
Review of the formulation of anisotropic laminates.
Vibrations of laminated plates.
Stability and critical load. Adjacent equilibrium method. Analytical solution cases.
Analysis of laminated composites using 1st order finite elements.
Linear static formulation, vibrations.
Initial instability problem.
Formulation of laminated degenerate shells.
Higher order theories (equivalent layer theories).
Zig-zag theories. Models by di Sciuva, Ambartsumian. Higher-order global-local models.
Piezoelasticity and intelligent structures. Governing equations. Weak form, FEM, analytical solutions with Mindlin’s model.

Syllabus:
(i) Introduction – phenomenological aspects of damage; (ii) Thermodynamics of irreversible processes; (iii) Thermal dissipation; (iv) Elastoplasticity with isotropic damage; (v) Discretization: incremental Finite Element Analysis; (vi) Viscoplasticity of materials.
Programme:
(i) Introduction – phenomenological aspects of damage: Phenomenological aspects of damage; physical nature of the solid state and damage; irreversible deformations, slip planes, mechanical representation of damage; concept of effective stress; Lemaitre’s principle of equivalent deformation. Low-cycle fatigue, one-dimensional models; numerical integration and Newton’s method.
(ii) Thermodynamics of irreversible processes: Introduction; infinitesimal deformation; Motion, concepts of deformation; infinitesimal deformation; conservation of mass, conservation of linear momentum; conservation of energy; second principle of thermodynamics – Clausius-Duhem inequality; Local variables method; Internal and observable variables; thermodynamic potentials; dissipative potentials; Osanger’s relation.
(iii) Thermal dissipation. Energy equation.
(iv) Elastoplasticity with isotropic damage Elastoplastic constitutive models; additive decomposition; yield criteria; hardening laws; plastic potentials; plastic multiplier; J2 plasticity; isotropic damage; damage evolution laws; low-cycle fatigue; elastic decomposition, operator decomposition, plastic corrector
(v) Discretization: Incremental finite element analysis Formulation of the incremental discretised problem; finite element discretisation, numerical integration; operator B (bar) – projection (vi) Viscoplasticity of materials One-dimensional viscoplasticity models; multiaxial viscoplastic models;

Syllabus:
Definition of reliability, concept of failure modes, failure rate. Main probabilistic
distributions and their applications in reliability.
Order statistics and probability graphs. Numerical methods.

Programme:
Definition of failure. Functional failure, physical failure.
Reliability concepts, definition, failure rate, failure rate curve.
Concepts of probability. Event, sample space, ME events and EI events.
Distributions for discrete sample space: Bernoulli, binomial, Poisson. Stochastic processes. Statistics on the set, statistics on the domain.
Stationary processes, ergodic processes, PSD function. Moments of the PSD function. Bandwidth. Exponential distribution as a particular case of the Poisson distribution.
Distribution of limit cases.
Extreme value distributions.
Weibull distribution.
Analysis of experiments, order statistics, probability graphs.
Numerical methods: Monte Carlo, response surface, finite elements.

Syllabus:
First part of the study of the fundamentals of continuum mechanics theory, with an emphasis on solid mechanics. Development of an appropriate philosophical stance in the study and analysis of mathematical models in their relation to associated physical models and phenomena. Rigorous detailing of concepts, models and theorems concerning: the various types of stress measurements, deformations and rates, general principles of continuum mechanics, elements of constitutive relations and thermodynamics of solids.

Programme:
1. Review and consistent conceptualisation of vectors, tensors, vector bases, tensor operations, vector and tensor functions and operators, indicial notation.
2. Cauchy tensor. Properties, definition, Cartesian transformation, principal values, invariants.
3. Linear kinematics. Concepts and properties of strain tensors and linearised strain rates. Additive decomposition.
4. Finite deformation. Concepts and properties of strain tensors and finite strain rates. Eulerian and Lagrangean formulations. Polar decomposition theorem. Rotation and elongation tensors.

Syllabus:

Second part of the study of the fundamentals of continuum mechanics theory, with an emphasis on solid mechanics. Development of an appropriate philosophical stance in the study and analysis of mathematical models in their relation to associated physical models and phenomena. Rigorous detailing of the concepts, models and theorems concerning: the general principles of continuum mechanics, the elements of constitutive relations and the thermodynamics of solids.

 

Programme:

1. General principles of continuum mechanics. Revision of transformation theorems for area, surface and volume integrals. Conservation of mass, quantity of movement and energy. Formulations deduced and interpreted in both Eulerian and Lagrangean descriptions.
2. Method of weighted residuals and Principle of Virtual Work. Piola-Kirchhoff tensors and PTV in Lagrangean description.
3. Entropy. Second law of thermodynamics in solids, concept of internal variables. Clausius inequality.
4. Constitutive relations. Overview of relations. Thermodynamic constraints.
5.  Breakdown of the linear elastic constitutive relation. Elastic energy density function. Symmetry of the elastic tensor. Material symmetry planes. Anisotropic, allotropic, orthotropic and isotropic materials. Linear theory of elasticity: hypotheses, three-dimensional Navier equations and particularizations for plane states.

Syllabus:

The course aims to study mathematical concepts and numerical optimisation techniques for mechanical design, with an emphasis on improving structural performance. Although this is the focus of application, the conceptual tools are general enough to be understood and applied in other areas of engineering. The course is divided into two parts. The first part aims to present the minimum theoretical concepts needed to solve practical cases and provide a comprehensive overview of the optimisation problem. The second part completes this first vision with new theoretical concepts, algorithms and classic cases of application.

 

Programme:

1. Introduction. Optimal design problem. Design variables, performance function, objective, constraint. Classification: Linear, non-linear problems. With constraints, without constraints. Graphical resolution.
2. General format. Local/global minimum. Weierstrass theorem. Differentiability. Gradient, hessian. Taylor series. Quadratic functions. Unconstrained problems. Optimality conditions based on curves. Examples.
3. Mathematical programming. Basic concepts of algorithms. Algorithms for minimisation without constraints. Direction of descent. One-dimensional search: biserial, secant, Newton, quadratic approximation, GoldenSearch.
4. Direction search methods. Newton. Almost Newton. Conjugate Gradient.
5. Incorporating constraints. Penalisation. Barriers, External penalisation.
6. Formulation of optimal problems in solid mechanics. Sensitivity analysis (gradient calculation) in linear problems. Discrete formulation. Direct and adjoint methods. Examples
7. Sensitivity analysis in non-linear problems.
8. Optimality conditions with equality and inequality constraints. K-K-T conditions.
9. Convexity. Duality. Examples.
10. Linear programming. Simplex.
11. Minimisation methods with constraints. Augmented Lagrangeans. Sequential linear programming. Sequential Quadratic Programming.
12. Topological structural optimisation.
13. Genetic Algorithms.

Syllabus:

1. Structure and deformation of materials – types of atomic bonds in metals and polymers, internal structure, elastic deformation, inelastic deformation;
2. Stress-strain relationships and mechanical behaviour;
3.  Experimental characterisation techniques – uniaxial compression, uniaxial traction, EPD traction, simple shear, impact, dynamic mechanical analysis (DMA), hardness and indentation, bending, torsion, creep and fatigue;
4. 4. Surface characterisation techniques – optical microscopy, scanning electron microscopy, atomic force microscopy;
5.   Measuring mechanical deformation – electrical resistance strain gauges (straingages), digital image correlation (DIC);
6.  Mechanical tests to validate material models (small punch, V-notchshear, cyclic stress-strain curve);
7.  Determining material parameters from experimental data – identification processes, characteristic tests, schematic representation of real behaviour, error functions, algorithms for extracting parameters;
8.  Statistical analysis of experimental data – statistical distributions, confidence intervals, comparison of means, regression, propagation of errors;
9.  Design of experiments – analysis of variance (ANOVA), single factor experiments, 2k factorial;
10.  Applications.