Torsion theory. Shear stresses in bending. Skew bending, bending with axial force. Eccentric loading (the section‘s core region, inactive area). Shear flow in thin-walled members (shear center). The principle of energy conservation. Energy methods (principle of virtual work, Castigliano’s theorems, reciprocal theorems of Betti and Maxwell Mohr ). Buckling. Elastoplastic behaviour of solids (yield criteria of Tresca, Mises and Mohr Coulomb).
Computational Mechanics I
Introduction. General description of the finite element method. The displacements method. The principle of virtual work and the transformation of coordinates. Creation of the final stiffness matrix. Choose of the displacements’ function. Convergence criteria. Conformal and non-conformal approximation. Practical remarks for the application of the finite element method. Truss and beam elements. Plane elements Three-dimensional stress analysis. Axisymmetric elements. General families of elements. Isoparametric elements.
Computational Mechanics II
A general description of the finite element method. Formation of the global stiffness matrix with the application of the virtual work principle in the whole body. Calculation of the reduced global stiffness and loading matrices. Elasto-Dynamic field problems (static and dynamic behavior). Thin plate and shell elements in bending. Convergence criteria and the patch test in the case of plate bending. Discretization of large constructions with the finite element method: the case of infinite bodies. Pre-processing and post-processing of data and other techniques in the FEM. A generalization of the finite element method. The method of weighted residuals, variational methods and the Rayleigh – Ritz Method. Applications in the case of field and fluid mechanics problems.
Mechanics of Coupled Fields
Basic relations and constitutive equations of linear theory of elasticity and thermoelasticity. Energy theorems and complex potentials. Basic relations and constitutive equations of linear theory of electroelasticity. Elements of crystallography and crystallophysics. Interaction of physical fields in piezoelectric mediums. Waves in piezoelectric mediums. Fracture mechanics of piezoelectric materials. Basic relations and constitutive equations of magneto-thermo-elasticity.
Special Courses of Computational Mechanics
Introduction to the fracture mechanics. Fatigue of metallic materials. Crack propagation laws. Fatigue of composite materials. Use of finite element programs in crack problems. Basic principles of damage tolerance. Analysis of the mechanism of crack propagation based on damage tolerance. Methodology of prediction of the remaining life of a structure. Example of computational study of a structure based on damage tolerance. Computational programs of crack propagation (AFGROW-NASGRO-RAPID). Smart materials and Structures. Introduction in the inspection of the structural integrity (structural health monitoring).
Laboratory: Exhibition of the fatigue experiment in the test machines INSTRON
Numerical applications with the finite element metho
Analysis of Surface Mechanical Systems
Elements of the differential geometry of three dimensional surfaces in oblique and orthogonal systems. The general bending theory of elastic shells (Applications). The bending theory of thin elastic shells (Applications).Methods of decoupling of partial linear differential systems of high order. The membrane theory of elastic shells (Applications).Analysis of cylindrical shells and shells of revolution under bending and membrane loading.
Advanced Computational Methods I (Program Computational Mechanics and Applied Mechanics)
Generalized Finite Element Method: Weighted Residual Methods (Budnov-Galerkin Method, Petrov-Galerkin Method, Μέθοδος (Rayleigh)-Ritz Method, Constraint Extrema, Lagrange Multipliers, Penalty functions).
Shape functions in finite elements: Creation of Shape functions. Shape functions in one dimensional domain. Shape functions in two dimensional domain. General families of elements. Isoparametric mapping. Numerical Integration.
Analysis of one dimensional problems. Dicretization of the domain. Calculation of the stiffness and loading matrices. Composition of the global stiffness matrix. Calculation of the displacements and the stresses. Applications in the case of elasticity, in the heat transfer and in fluid mechanics problems.
Analysis of two dimensional problems. Dicretization of the domain. Calculation of the stiffness and loading matrices. Composition of the global stiffness matrix. Calculation of the displacements and the stresses. Applications in the case of elasticity, in the heat transfer and in fluid mechanics problems.
Three dimensional stress analysis. The displacement functions of the three dimensional elasticity. Tetrahedral elements. Eight nodes elements. Plate and Shell elements. Bending of thin Plates and Shells Elastodynamics field problems.
Convergence criteria in the FEM. The error in the FEM.
Laboratory: Programming of new elements, solving problems with the Finite Element Program Ansys.
Mechanics of Coupled Fields (Program Applied Mechanics)
Basic relations and constitutive equations of linear theory of elasticity and thermoelasticity. Energy theorems and complex potentials.Basic relations and constitutive equations of linear theory of electroelasticity. Elements of crystallography and crystallophysics. Interaction of physical fields in piezoelectric mediums. Waves in piezoelectric mediums. Fracture mechanics of piezoelectric materials. Basic relations and constitutive equations of magneto-thermo-elasticity.
Atomic structure of crystals. Operators of groups and symmetries. Crystalline ties. Potentials of interaction. Introduction in statistical mechanics. Critical phenomena and computational methods. Ising model. Calculation of free energy. Theory of phases changing. Theory of middle field. Systems in non equilibrium. Brownian dynamics and the equation of Langevin. The Fokker – Plack equation. Equations of movement and calculation of forces of interacting coupled fields. Algorithms of solution and stability. Simulations of Dynamics of coupled fields in different statistical models. Introduction in Monte Carlo method. Metropolis algorithm. Monte Carlo simulations in different statistical models.
Theory of Plates and Shells (Program Applied Mechanics)
Curves lines in space. Parametric representation of a surface/ Fundamental quantities of 1st and 2nd order. Equations of Godazzi – GaussPrincipal Curvatures. Special cases of parametric nets. Special cases of surfaces. The deformation of a shell. Fundamental quantities of 1st and 2nd order. Surface parallel of the middle surface. Equations of compatibility of deformations. Stress state of a shell- Stresses and strains. Resultants Shear forces-Torsion Strains. Equations of Equilibrium. Strain Energy. Constitutive System. Membrane Theory of Elastic Shells. Membrane Theory of Elastic Shells of Revolution. Applications.
Computational Fracture Mechanics (Program Computational Mechanics and Applied Mechanics)
Mathematical Theory of Non-Linear Fracture Mechanics.
Elastoplastic Fracture Mechanics. Fracture Mechanics Mechanisms in Metals, Ceramics, Polymeric and Composite Materials. Crack propagation due to Fatigue.Special Finite Elements and Special Methods in Fracture Mechanics. Examples of Structure Calculation with Damage Tolerance Philosophy. Programs of Calculation of Crack Propagation (AFGROW – NASGRO – RAPID). Smart Materials and Structures. Introduction in the Observation of the Structural Integrity.
Laboratory: Showing of Fatigue Test in Laboratory Machine INSTRON
Boundary Element Method (Program Computational Mechanics and Applied Mechanics)
Some fundamentals of singular integrals. The definite integral of unbounded functions and the definition in case: as generalized, as main value and finite part. Introduction to the types of integral equations: Fredholm και Voltera integral equations of first and second kind. Numerical solution of no singular integral equations: i) the Nystrom method, and II) the technique of boundary method method "BEM". The Green function, the fundamental solution and the reduction of one-dimensional problems to integral equations. The potential problem, the fundamental solution of the Laplacian and the development of complete formulation of the potential problem of the isotropic or the non-isotropic medium in two and three dimensions. The numerical solution of the potential problem in two dimensions with the method of boundary elements (BEM) for stable, linear and secondary data. Applications of the BEM method: the Torsion problem, the problem of heat convection. The development of complete formulation of the linear elastostatik problem in two and three dimensions: The second Betti theorem (reciprocity theorem). The fundamental solution of the problem of linear elasticity of the Navier equation (the Kelvin solution). The Somigliana equation. The mathematical formulation of elastostatik problem with integral equation. The stress problem in interior points. The numerical solution of two-dimensional elastostatik problem with the method of boundary elements (BEM) for stable, linear and secondary data. Applications: Stress concentrationw for plane problems in elasticity.