PORTALE DELLA DIDATTICA

### Introduction to Geometric Continuum Mechanics (didattica di eccellenza)

01DTRRT

A.A. 2021/22

Course Language

Inglese

Course degree

Doctorate Research in Matematica Pura E Applicata - Torino

Course structure
Teaching Hours
Lezioni 20
Teachers
Teacher Status SSD h.Les h.Ex h.Lab h.Tut Years teaching
Grillo Alfio   Professore Associato MAT/07 2 0 0 0 1
Teaching assistant
Context
SSD CFU Activities Area context
*** N/A ***
The basic notions of continuum mechanics are formulated in the geometric framework of general differentiable manifolds and stress fields may be as irregular as Radon measures. Metric-free electrodynamics and generalization thereof are presented as special cases.
The basic notions of continuum mechanics are formulated in the geometric framework of general differentiable manifolds and stress fields may be as irregular as Radon measures. Metric-free electrodynamics and generalization thereof are presented as special cases.
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Prof. Reuven Segev - University of Negev The theoretical foundations of modern continuum mechanics have been under continuous study since the middle of the 20th century. Of particular interest is the notion of stress that plays important roles in solid mechanics, fluid mechanics, electromagnetism, and relativity theory. Stresses cannot be measured experimentally, and their existence is based on a mathematical theorem, the Cauchy theorem, that relies on a number of assumptions of physical and mathematical nature. In particular, standard stress theory is formulated in the geometric framework of a three-dimensional Euclidean space. Geometric continuum mechanics generalizes the geometric and analytic framework in order to uncover the essential features of the theory. In particular, the theory is formulated on general manifolds free of any Riemannian structure. The course starts with consideration of the algebraic properties of uniform fluxes as operators on the collection of simplices in an affine space, It is shown that natural assumptions lead to the representation of fluxes by completely antisymmetric tensors — algebraic Cauchy theorem. Next, the algebraic Cauchy theorem is used in the formulation of general flux and stress theories. We show that at this level of generality, metric free electromagnetism and its generalizations to higher dimensional spaces may be viewed as particular cases of stress theory. The essence of stress theory is revealed through a global approach to continuum mechanics. The basic mathematical object is the configuration space — the Banach manifold of k-times continuously differentiable sections C)fa fiber bundle over a compact base manifold — the material body. The choice of topology is natural so that the set of embeddings of the body manifold in a space manifold is open in the manifold of all mappings. Forces are defined to be elements of the cotangent bundle of the configuration space and their action on virtual velocities — elements of the tangent bundle — is interpreted physically as virtual power. Stresses and hyper-stresses emerge naturally from a representation theorem as measures, valued in the dual bunclles of some jet bundles, which represent forces.
Prof. Reuven Segev - University of Negev The theoretical foundations of modern continuum mechanics have been under continuous study since the middle of the 20th century. Of particular interest is the notion of stress that plays important roles in solid mechanics, fluid mechanics, electromagnetism, and relativity theory. Stresses cannot be measured experimentally, and their existence is based on a mathematical theorem, the Cauchy theorem, that relies on a number of assumptions of physical and mathematical nature. In particular, standard stress theory is formulated in the geometric framework of a three-dimensional Euclidean space. Geometric continuum mechanics generalizes the geometric and analytic framework in order to uncover the essential features of the theory. In particular, the theory is formulated on general manifolds free of any Riemannian structure. The course starts with consideration of the algebraic properties of uniform fluxes as operators on the collection of simplices in an affine space, It is shown that natural assumptions lead to the representation of fluxes by completely antisymmetric tensors — algebraic Cauchy theorem. Next, the algebraic Cauchy theorem is used in the formulation of general flux and stress theories. We show that at this level of generality, metric free electromagnetism and its generalizations to higher dimensional spaces may be viewed as particular cases of stress theory. The essence of stress theory is revealed through a global approach to continuum mechanics. The basic mathematical object is the configuration space — the Banach manifold of k-times continuously differentiable sections C)fa fiber bundle over a compact base manifold — the material body. The choice of topology is natural so that the set of embeddings of the body manifold in a space manifold is open in the manifold of all mappings. Forces are defined to be elements of the cotangent bundle of the configuration space and their action on virtual velocities — elements of the tangent bundle — is interpreted physically as virtual power. Stresses and hyper-stresses emerge naturally from a representation theorem as measures, valued in the dual bunclles of some jet bundles, which represent forces.
In presenza
On site
Presentazione orale
Oral presentation
P.D.2-2 - Maggio
P.D.2-2 - May