INTRODUCTORY CONCEPTS. The model of structural response in bioengineering. The nine fundamental principles of biomechanics. Anatomical and functional concepts. The musculoskeletal system. Mechanical properties of biological tissues.
RIGID BODY KINEMATICS: General formula for infinitesimal rigid displacement. Planar rigid displacements. Kinematic characterization of external and internal constraints. The kinematic problem. Exercises.
STATICS OF RIGID BODY SYSTEMS. Forces and couples. Resultant and resultant moment of a system of applied forces. Equivalent systems of forces. Elementary equivalence operations. Balance of a rigid body or a system of rigid bodies. Cardinal equations of statics. In-plane force systems. Distributed forces over a volume (gravity), over a surface, over a line; concentrated forces and couples. Constraint reactions for smooth, fixed, and bilateral constraints; static characterization of external and internal in-plane constraints. Search for balanced reactive states. Isostatic, hyperstatic, labile, degenerate structures. Characteristics of internal stress in beam systems: normal stress, shear stress, bending moment, torque; indefinite equilibrium equations for the rectilinear beam.
AREA GEOMETRY. Area, static moment, centroid, moment of inertia, radius of gyration, mixed moment of inertia, Huygens theorem. Principal axes of inertia, central ellipse of inertia.
INTRODUCTION TO THE THEORY OF ELASTIC STRUCTURES. Limits of the rigid body model. Elementary deformable model: rod, linear elastic bond. Equilibrium, compatibility, and constitutive equations for the straight rod. The linear elastic problem; solution methods: force method and displacement method for the straight rod.
ELASTIC BEAM AND HYPERSTATIC SYSTEMS OF BEAMS. Differential relations between transverse displacement of the axis line, rotation of the straight section, and bending curvature; curvature due to thermal distortions or bending moment; integration of the elastic line equation. Deformation characteristics (bending and torsional curvature, extension, sliding); elastic relationship between stress and deformation characteristics. Virtual Work Theorem for deformable beams; application of the Theorem of virtual work to determine displacements and rotations in isostatic structures. Resolution of hyperstatic structures through compatibility equations (Muller-Breslau equations).
CAUCHY CONTINUUM. STRESS ANALYSIS. Cauchy's stress. Cauchy's lemma. Decomposition of the Cauchy stress tensor. Cauchy's stress formula. Indefinite equilibrium equations and boundary conditions. Principal stresses and directions around the point. Triaxial, cylindrical, and spherical stress states. Deviatoric stress. Octahedral stress. Main reference. Mohr's circles. State of plane stress, purely tangential, and uniaxial. Mohr circles for stress analysis at a point on the De Saint Venant beam. Isostatic lines.
DEFORMATION ANALYSIS. Act of rigid motion. Decomposition of displacement around the point: deformation tensor and rigid rotation tensor. Mechanical meaning of deformation components: elongations and angular displacements. Decomposition of the deformation process. Cubic dilatation. Cauchy's formula for deformation. Principal directions of deformation. Triaxial, cylindrical, and spherical deformation states. Cubic dilation. Main reference.
HOOKE'S LAW. Hooke's law for uniaxial tension, linearity, and plastic behavior. Tensile test and torsion test. Elastic behavior under triaxial stress: generalized Hooke's law. Elastic constants: Young's modulus and Poisson's ratio, cubic modulus of elasticity. Inelastic deformations.
ELASTIC PROBLEM. Existence and uniqueness of the solution to the problem of elastic equilibrium, Navier and Beltrami equations. Solution of the De Saint Venant problem for pressure bending, tension approach. The Neumann problem for torsion, approach to displacements.
TECHNICAL TREATMENT OF THE BEAM. THE SOLID OF DE SAINT VENANT. Determination of stress and deformation state for a beam of homogeneous linear elastic material starting from stress characteristics (technical treatment): hypothesis of conservation of flat sections; centered normal stress; straight bending; near straight and deviated bending. Torsion in closed thin sections; Bredt's formula. Torsion in elongated rectangular sections; C, L sections or in any case developed in thin rectangle. Torsion in the full circular section. Approximate treatment of Shear (Jourawski). Compound stresses: deflected bending; eccentric normal stress, relation between stress center and neutral axis, central inertial core; cutting and twisting, cutting center.
RESISTANCE CRITERIA. Brief mention of resistance criteria in monoaxial and triaxial stress regimes.
CONSTITUTIVE MODELS IN BIOENGINEERING AND BIOMECHANICS. The biomechanical model of bones. The biomechanical model of muscles. The biomechanical model of fibrous-reinforced soft tissues.
