Energy Engineering

PRELIMINARY CONCEPTS Vectors, dot product, cross product, right orthonormal basis. Second-order tensors, transpose, tensor product, representative matrices, trace, determinant, orthogonal tensors, absolute and index notation. KINEMATICS OF CONTINUA Definition of a body. Placement, deformation, deformation gradient. Displacement, displacement gradient. Homogeneous deformations. Exact geometric variations: length variation, volume variation, area variation (Nanson's formula), angle variation. Cauchy-Green strain tensor. Measures of strain in exact theory: Green-Saint-Venant tensor. Elementary deformations: uniform dilation, extension, shear. Infinitesimal theory: infinitesimal strain tensor, infinitesimal rotation. Geometric variations in infinitesimal theory: length variation, volume variation, area variation, angle variation. Interpretation of components of the infinitesimal strain tensor. STATIC OF CONTINUA Volume forces and contact forces. Cauchy's postulate, stress. Euler's axioms. Hamel-Noll theorem. Cauchy's lemma. Cauchy's theorem. Stress tensor. Interpretation of components of the stress tensor. Pointwise equilibrium equations: Cauchy's relation for interior points and boundary points, stress tensor symmetry. Normal and tangential stress. Principal stresses and directions. PRINCIPLES OF CONTINUUM THERMODYNAMICS Energy balance (First Law of Thermodynamics). Mechanical sources, power of external forces and stress power. Thermal sources, heat flux and internal heat source. Local energy balance equation. Entropy imbalance (Second Law of Thermodynamics). Internal entropy production, entropy flux and internal entropy source. Local entropy imbalance equation. Relationship between entropy flux and heat flux. Helmholtz free energy. Reduced dissipation inequality. Coleman-Noll procedure. Constitutive theory. ELASTIC MATERIALS Linear elastic materials, elasticity tensor, major and minor symmetries. Material symmetry, material symmetry group. Representation theorem for isotropic materials. Interpretation of elastic constants. The elastic problem, displacement formulation (Navier's equation). Formulation of the elastic problem in curvilinear coordinates. RIGID CONDUCTIVE MATERIALS Consequences of the Coleman-Noll procedure. Gibbs relations, specific heat. Constitutive equation for heat flux, Fourier's law. Heat equation. Solution of the heat equation in the one-dimensional case, eigenvalues and eigenfunctions, boundary conditions, equilibrium temperature. THERMOELASTIC MATERIALS Thermoelasticity. Consequences of the Coleman-Noll procedure. Gibbs relations. Constitutive equations for stress, entropy, heat flux. Force and energy balance equations. Weakly coupled theory. Solution of one-dimensional and three-dimensional problems.

Basic notions acquired in courses of mathematical analysis, geometry, and physics.

Notes prepared by the professor. Further references: R. Paroni, Scienza delle Costruzioni - Elementi di teoria dell'elasticità lineare, Springer, 2022. M.E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1982. M.E. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2010. P. Podio Guidugli, Continuum Thermodynamics, Springer, 2019.

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