HAL (Le Centre pour la Communication Scientifique Directe), 2007
For the stress analysis in a plastic body Ω, we prove that there exists a maximal positive number... more For the stress analysis in a plastic body Ω, we prove that there exists a maximal positive number C, the load capacity ratio, such that the body will not collapse under any external traction field t bounded by Y_0C, where Y_0 is the elastic limit. The load capacity ratio depends only on the geometry of the body and is given by 1/C=_w∈ LD(Ω)_D∫_∂Ω|w|dA/∫_Ω|ϵ(w)|dV=γ_D. Here, LD(Ω)_D is the space of isochoric vector fields w for which the corresponding stretchings ϵ(w) are assumed to be integrable and γ_D is the trace mapping assigning the boundary value γ_D(w) to any w∈ LD(Ω)_D.
As a sequel to a recent work we consider the generalized stress concentration factor, a purely ge... more As a sequel to a recent work we consider the generalized stress concentration factor, a purely geometric property of a body that for the various loadings, indicates the ratio between the maximum of the optimal stress and maximum of the loading fields. The optimal stress concentration factor pertains to a stress field that satisfies the principle of virtual work and for which the stress concentration factor is minimal. Unlike the previous work, we require that the external loading be equilibrated and that the stress field be a symmetric tensor field.
Here, $LD(\Omega)_D$ is the space of isochoric vector fields $w$ for which the corresponding stre... more Here, $LD(\Omega)_D$ is the space of isochoric vector fields $w$ for which the corresponding stretchings $\epsilon(w)$ are assumed to be integrable and $\gamma_D$ is the trace mapping assigning the boundary value $\gamma_D(w)$ to any $w\in LD(\Omega)_D$.
Continuum Mechanics and Thermodynamics, Jun 13, 2015
The flux of a certain extensive physical quantity across a surface is often represented by the in... more The flux of a certain extensive physical quantity across a surface is often represented by the integral over the surface of the component of a pseudo-vector normal to the surface. A pseudo-vector is in fact a possible representation of a second-order differential form, i.e., a skew-symmetric second-order covariant tensor, which follows the regular transformation laws of tensors. However, because of the skew-symmetry of differential forms, the associated pseudo-vector follows a transformation law that is different from that of proper vectors, and is named after the Italian mathematical physicist Gabrio Piola (1794-1850). In this work, we employ the methods of Differential Geometry and the representation in terms of differential forms to demonstrate how the flux of an extensive quantity transforms from the spatial to the material point of view. After an introduction to the theory of differential forms, their transformation laws, and their role in Integration Theory, we apply them to the case of first-order transport laws such as Darcy's law and Ohm's law.
We present a simple control scheme for attitude regulation of a rigid spacecraft actuated by gas-... more We present a simple control scheme for attitude regulation of a rigid spacecraft actuated by gas-jet systems. It is shown that a simple output-based controller stabilizes the highly-nonlinear system under consideration. The controller is based on measurements of rotation angles only, and hence noisy signals associated with measurements of velocity variables are not involved in the control process. Provided the
A continuum mechanical framework for the description of the geometry and kinematics of defects in... more A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism structure. In addition, both continuous distributions of defects as well as singular distributions are encompassed by the theory. In the general case, the material structure is specified by a de Rham current T and the associated defects are given by its boundary ∂T. For a motion of defects associated with a family of diffeomorphisms of a material body, it is shown that the rate of change of the distribution of defects is given by the dual of the Lie derivative operator.
A. A geometric analysis of high order stresses in continuum mechanics is presented. Virtual veloc... more A. A geometric analysis of high order stresses in continuum mechanics is presented. Virtual velocity fields take their values in a vector bundle W over the n-dimensional space manifold. A stress field of order k is represented mathematically by an n-form valued in the dual of the vector bundle of k-jets of W. While only limited analysis can be performed on high order stresses as such, they may be represented by non-holonomic hyper-stresses, n-forms valued in the duals of iterated jet bundles. For non-holonomic hyperstresses, the analysis that applies to first order stresses may be iterated. In order to determine a unique value for the tangent surface stress field on the boundary of a body and the corresponding edge interactions, additional geometric structure should be specified, that of a vector field transversal to the boundary.
Philosophical Transactions of the Royal Society A, Apr 28, 2016
One contribution of 11 to a theme issue 'Trends and challenges in the mechanics of complex materi... more One contribution of 11 to a theme issue 'Trends and challenges in the mechanics of complex materials' .
Journal of The Franklin Institute-engineering and Applied Mathematics, Sep 1, 1986
A geometrical setting for the Newtonian mechanics of mechanical manipulators is presented. The co... more A geometrical setting for the Newtonian mechanics of mechanical manipulators is presented. The configuration space of the mechanical system is modelled by a differentiable manifold. The kinematics of the system is formulated on the tangent and double tangent bundles of the corifiguration space, and forces are defined as elements of the cotangent bundle. The dynamical properties of the system are introduced by specifying a Riemannian metric on the configuration space. The metric is used in order to generate the generalized momenta and the kinetic energy from the generalized velocities, and the connection it induces makes it possible to formulate a generalization of Newton's second law relating generalized forces and generalized accelerations.
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
We use a metric invariant stress theory of continuum mechanics to formulate a simple generalizati... more We use a metric invariant stress theory of continuum mechanics to formulate a simple generalization of the the basic variables of electrodynamics and Maxwell's equations to general differentiable manifolds of any dimension, thus viewing generalized electrodynamics as a special case of continuum mechanics. The basic variable is the potential, or a variation thereof, which is represented as an r-form in a ddimensional spacetime. The stress for the case of generalized electrodynamics, is assumed to be represented by a (d − r − 1)-form, a generalization of the Maxwell 2-form.
Continuum Mechanics and Thermodynamics, Aug 1, 2018
Although the final result presented in Equation (34) of our work [1] is correct, the proof of Equ... more Although the final result presented in Equation (34) of our work [1] is correct, the proof of Equation (34) contains an error. The text starting immediately after Equation (33) with "When a metric tensor g..." and ending immediately before Equation (36) with "...in the alternative notation" should be replaced with the text below. Correction of the Proof of Equation (34) Let us assume that the space S is equipped with a metric tensor g, i.e., a symmetric and positive-definite tensor field valued in [T S] 0 2 , defining the scalar product of two vectors u and v as u.v = g(u, v). The metric g induces the musical isomorphisms : T S → T S : v → (v) ≡ v , which maps a vector v with components v c to a covector v with components g ac v c , and its inverse : T S → T S : α → (α) ≡ α , which maps a covector α with components α c to a vector α with components g ac α c , where g ac are the components of the inverse of the matrix [[g ab ]] of g. The isomorphism and the metric tensor g induce the scalar product of covectors α.β = g(α , β) = α(β). The (n − 1)-dimensional tangent bundle T s of the hypersurface s determines a 1-dimensional sub-bundle of T S containing the annihilators of T s, i.e., the covectors ν such that ν u ≡ ν(u) = 0, for every u in T s. Moreover, using the scalar product of covectors, we can define the unit normal covector n to the hypersurface s as the annihilating covector such that n 2 = n.n = 1. The integral (33) of an (n − 1)-form ω on the hypersurface s can be expressed in terms of the axial vector field w of ω with respect to the volume form μ, i.e., w is such that ι w μ = ω. If we introduce the axial projector a = n ⊗ n (in components, a a b = n a n b
Mathematical Methods in The Applied Sciences, Aug 13, 2012
A metric independent geometric framework for some fundamental objects of continuum mechanics is p... more A metric independent geometric framework for some fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, balance principles for extensive properties are formulated and Cauchy's theorem for fluxes is proved. Fluxes in an n-dimensional space are represented as differential .n 1/-forms. In an analogous formulation of stress theory, a distinction is made between the traction stress, enabling the evaluation of the traction on the boundaries of the various regions, and the variational stress, which acts on the derivative of a virtual velocity field to produce the virtual power density. The relation between the two stress fields is examined as well as the resulting differential balance law. As an application, metric-invariant aspects of electromagnetic theory are presented within the framework of the foregoing flux and stress theory.
HAL (Le Centre pour la Communication Scientifique Directe), 2007
For the stress analysis in a plastic body Ω, we prove that there exists a maximal positive number... more For the stress analysis in a plastic body Ω, we prove that there exists a maximal positive number C, the load capacity ratio, such that the body will not collapse under any external traction field t bounded by Y_0C, where Y_0 is the elastic limit. The load capacity ratio depends only on the geometry of the body and is given by 1/C=_w∈ LD(Ω)_D∫_∂Ω|w|dA/∫_Ω|ϵ(w)|dV=γ_D. Here, LD(Ω)_D is the space of isochoric vector fields w for which the corresponding stretchings ϵ(w) are assumed to be integrable and γ_D is the trace mapping assigning the boundary value γ_D(w) to any w∈ LD(Ω)_D.
As a sequel to a recent work we consider the generalized stress concentration factor, a purely ge... more As a sequel to a recent work we consider the generalized stress concentration factor, a purely geometric property of a body that for the various loadings, indicates the ratio between the maximum of the optimal stress and maximum of the loading fields. The optimal stress concentration factor pertains to a stress field that satisfies the principle of virtual work and for which the stress concentration factor is minimal. Unlike the previous work, we require that the external loading be equilibrated and that the stress field be a symmetric tensor field.
Here, $LD(\Omega)_D$ is the space of isochoric vector fields $w$ for which the corresponding stre... more Here, $LD(\Omega)_D$ is the space of isochoric vector fields $w$ for which the corresponding stretchings $\epsilon(w)$ are assumed to be integrable and $\gamma_D$ is the trace mapping assigning the boundary value $\gamma_D(w)$ to any $w\in LD(\Omega)_D$.
Continuum Mechanics and Thermodynamics, Jun 13, 2015
The flux of a certain extensive physical quantity across a surface is often represented by the in... more The flux of a certain extensive physical quantity across a surface is often represented by the integral over the surface of the component of a pseudo-vector normal to the surface. A pseudo-vector is in fact a possible representation of a second-order differential form, i.e., a skew-symmetric second-order covariant tensor, which follows the regular transformation laws of tensors. However, because of the skew-symmetry of differential forms, the associated pseudo-vector follows a transformation law that is different from that of proper vectors, and is named after the Italian mathematical physicist Gabrio Piola (1794-1850). In this work, we employ the methods of Differential Geometry and the representation in terms of differential forms to demonstrate how the flux of an extensive quantity transforms from the spatial to the material point of view. After an introduction to the theory of differential forms, their transformation laws, and their role in Integration Theory, we apply them to the case of first-order transport laws such as Darcy's law and Ohm's law.
We present a simple control scheme for attitude regulation of a rigid spacecraft actuated by gas-... more We present a simple control scheme for attitude regulation of a rigid spacecraft actuated by gas-jet systems. It is shown that a simple output-based controller stabilizes the highly-nonlinear system under consideration. The controller is based on measurements of rotation angles only, and hence noisy signals associated with measurements of velocity variables are not involved in the control process. Provided the
A continuum mechanical framework for the description of the geometry and kinematics of defects in... more A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism structure. In addition, both continuous distributions of defects as well as singular distributions are encompassed by the theory. In the general case, the material structure is specified by a de Rham current T and the associated defects are given by its boundary ∂T. For a motion of defects associated with a family of diffeomorphisms of a material body, it is shown that the rate of change of the distribution of defects is given by the dual of the Lie derivative operator.
A. A geometric analysis of high order stresses in continuum mechanics is presented. Virtual veloc... more A. A geometric analysis of high order stresses in continuum mechanics is presented. Virtual velocity fields take their values in a vector bundle W over the n-dimensional space manifold. A stress field of order k is represented mathematically by an n-form valued in the dual of the vector bundle of k-jets of W. While only limited analysis can be performed on high order stresses as such, they may be represented by non-holonomic hyper-stresses, n-forms valued in the duals of iterated jet bundles. For non-holonomic hyperstresses, the analysis that applies to first order stresses may be iterated. In order to determine a unique value for the tangent surface stress field on the boundary of a body and the corresponding edge interactions, additional geometric structure should be specified, that of a vector field transversal to the boundary.
Philosophical Transactions of the Royal Society A, Apr 28, 2016
One contribution of 11 to a theme issue 'Trends and challenges in the mechanics of complex materi... more One contribution of 11 to a theme issue 'Trends and challenges in the mechanics of complex materials' .
Journal of The Franklin Institute-engineering and Applied Mathematics, Sep 1, 1986
A geometrical setting for the Newtonian mechanics of mechanical manipulators is presented. The co... more A geometrical setting for the Newtonian mechanics of mechanical manipulators is presented. The configuration space of the mechanical system is modelled by a differentiable manifold. The kinematics of the system is formulated on the tangent and double tangent bundles of the corifiguration space, and forces are defined as elements of the cotangent bundle. The dynamical properties of the system are introduced by specifying a Riemannian metric on the configuration space. The metric is used in order to generate the generalized momenta and the kinetic energy from the generalized velocities, and the connection it induces makes it possible to formulate a generalization of Newton's second law relating generalized forces and generalized accelerations.
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
We use a metric invariant stress theory of continuum mechanics to formulate a simple generalizati... more We use a metric invariant stress theory of continuum mechanics to formulate a simple generalization of the the basic variables of electrodynamics and Maxwell's equations to general differentiable manifolds of any dimension, thus viewing generalized electrodynamics as a special case of continuum mechanics. The basic variable is the potential, or a variation thereof, which is represented as an r-form in a ddimensional spacetime. The stress for the case of generalized electrodynamics, is assumed to be represented by a (d − r − 1)-form, a generalization of the Maxwell 2-form.
Continuum Mechanics and Thermodynamics, Aug 1, 2018
Although the final result presented in Equation (34) of our work [1] is correct, the proof of Equ... more Although the final result presented in Equation (34) of our work [1] is correct, the proof of Equation (34) contains an error. The text starting immediately after Equation (33) with "When a metric tensor g..." and ending immediately before Equation (36) with "...in the alternative notation" should be replaced with the text below. Correction of the Proof of Equation (34) Let us assume that the space S is equipped with a metric tensor g, i.e., a symmetric and positive-definite tensor field valued in [T S] 0 2 , defining the scalar product of two vectors u and v as u.v = g(u, v). The metric g induces the musical isomorphisms : T S → T S : v → (v) ≡ v , which maps a vector v with components v c to a covector v with components g ac v c , and its inverse : T S → T S : α → (α) ≡ α , which maps a covector α with components α c to a vector α with components g ac α c , where g ac are the components of the inverse of the matrix [[g ab ]] of g. The isomorphism and the metric tensor g induce the scalar product of covectors α.β = g(α , β) = α(β). The (n − 1)-dimensional tangent bundle T s of the hypersurface s determines a 1-dimensional sub-bundle of T S containing the annihilators of T s, i.e., the covectors ν such that ν u ≡ ν(u) = 0, for every u in T s. Moreover, using the scalar product of covectors, we can define the unit normal covector n to the hypersurface s as the annihilating covector such that n 2 = n.n = 1. The integral (33) of an (n − 1)-form ω on the hypersurface s can be expressed in terms of the axial vector field w of ω with respect to the volume form μ, i.e., w is such that ι w μ = ω. If we introduce the axial projector a = n ⊗ n (in components, a a b = n a n b
Mathematical Methods in The Applied Sciences, Aug 13, 2012
A metric independent geometric framework for some fundamental objects of continuum mechanics is p... more A metric independent geometric framework for some fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, balance principles for extensive properties are formulated and Cauchy's theorem for fluxes is proved. Fluxes in an n-dimensional space are represented as differential .n 1/-forms. In an analogous formulation of stress theory, a distinction is made between the traction stress, enabling the evaluation of the traction on the boundaries of the various regions, and the variational stress, which acts on the derivative of a virtual velocity field to produce the virtual power density. The relation between the two stress fields is examined as well as the resulting differential balance law. As an application, metric-invariant aspects of electromagnetic theory are presented within the framework of the foregoing flux and stress theory.
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