James CASEY - Academia.edu (original) (raw)
Papers by James CASEY
American Journal of Physics, 1994
A concise but general derivation of Lagrange's equations is given for a system of fi... more A concise but general derivation of Lagrange's equations is given for a system of finitely many particles subject to holonomic and nonholonomic constraints. Based directly on Newton's second law, it takes advantage of an inertia-based metric to obtain a geometrically transparent ...
Journal of Elasticity, 2011
A general approach to continuum thermodynamics that was advocated by R.S. Rivlin is carried out f... more A general approach to continuum thermodynamics that was advocated by R.S. Rivlin is carried out for thermoelastic materials which can also depend on strain rate. An entropy function is constructed (rather than assumed to exist). A method for treating thermomechanical internal constraints for such materials is also presented. In this method, the properties of a constrained material are inherited from those of a related equivalence class of unconstrained materials.
Journal of Applied Mechanics, 1987
is incorrect in his conclusion that the presence of microstructural anisotropy in crystalline med... more is incorrect in his conclusion that the presence of microstructural anisotropy in crystalline media conflicts with the demand that intermediate stress-free configurations be subject to invariance requirements under arbitrary superposed rotations. One should distinguish between two types of theories of plasticity: Type A, those in which no explicit account is taken of microstructure; and Type B, those in which microstructural concepts play an essential role. In both types of theories, a decomposition F = F e ¥ p of the deformation gradient F into an elastic part F e and a plastic part F p can be effected through the use of intermediate stress-free local configurations. But, in theories of Type B the continuum is augmented with additional fields intended to represent microstructural properties. The full invariance requirements advocated by Naghdi and coworkers (see Dashner's Ref. 2 and its list of references) are based on the transformations
The Mathematics Teacher
Hardcover book rolls very well on pencils of circular cross section. If pencils of hexagonal cros... more Hardcover book rolls very well on pencils of circular cross section. If pencils of hexagonal cross section are used instead, the book can still be rolled, but not so nicely. Can we quantify such differences? In other words, can we put our intuitions about the physical situation into mathematical form? The purpose of the present, article is to propose a mathematical way of thinking about how good a roller is and to describe classroom activities that were carried out to measure and evaluaLe model of roller. The subject is a beautiful blend of geometry and analysis, encourages students to think quantitatively, and afford them an opportunity for bridging the gap between mathematics and physical reality.
Mathematics and Mechanics of Solids
It is argued that the concept of body, as it is usually employed in continuum mechanics, is somew... more It is argued that the concept of body, as it is usually employed in continuum mechanics, is somewhat too general. Specifically, it is standard practice to associate a set of dynamical processes with an abstract body. However, many of these processes cannot be experienced by any physical body. Yet, Euler’s laws for the balance of linear momentum and angular momentum are asserted to hold for such abstract bodies. It is suggested here that Euler’s laws should be postulated only for abstract bodies that are composed of ideal materials belonging to some large collection.
International Journal of Solids and Structures
Abstract Within the framework of finite plasticity, it is shown how a number of kinematical appro... more Abstract Within the framework of finite plasticity, it is shown how a number of kinematical approximations can be systematically derived from the multiplicative decomposition of the deformation gradient. Particular attention is devoted to two types of approximate theories: (a) those in which elastic deformations are of a different order of magnitude than plastic deformations; and (b) those in which rotations are of a different order of magnitude than strains.
Zeitschrift Fuer Angewandte Mathematik Und Physik, 1995
Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids, 1995
Journal of Elasticity, 2014
The concept of parallelism along a surface curve, which was introduced by Levi-Civita in the cont... more The concept of parallelism along a surface curve, which was introduced by Levi-Civita in the context of n-dimensional Riemannian manifolds, is re-examined from a kinematical viewpoint. A special type of frame, whose angular velocity is determined by the rate at which the tangent plane turns as one moves along a surface curve, is defined and is called a Levi-Civita frame. The surface may be orientable or not. Vectors and tensors fixed on Levi-Civita frames are parallel transported. Covariant differentiation of vectors and tensors along a surface curve can be expressed in terms of the corresponding corotational rates measured on Levi-Civita frames. Relevant results on ruled surfaces are also included.
Exploring Curvature, 1996
The most deeply rooted geometrical properties of a surface are its topological ones: these are pr... more The most deeply rooted geometrical properties of a surface are its topological ones: these are preserved under all homeomorphisms of the surface. Hence also, they are preserved under all deformations. (Some examples of deformations of surfaces were studied at the end of Chapter 11.) There is a broader class of properties that are intimately bound up with the geometry of the surface and that are preserved under a large subclass of homeomorphisms. These are ones that Gauss discovered, and which we will now explore.
Exploring Curvature, 1996
In this chapter, we describe a particular way of moving a vector along a given curve on a surface... more In this chapter, we describe a particular way of moving a vector along a given curve on a surface. It provides an especially revealing means of exploring the non-Euclideanness of the surface.
Exploring Curvature, 1996
Like Mozart’s, Bernhard Riemann’s life was short but marvelously creative. He solved several of t... more Like Mozart’s, Bernhard Riemann’s life was short but marvelously creative. He solved several of the most difficult problems in pure and applied mathematics, introduced entirely new concepts and techniques, and profoundly changed the way in which mathematicians, physicists, and philosophers view space.1
Exploring Curvature, 1996
As we saw in the preceding chapter, on a curved surface, geodesics replace the straight lines of ... more As we saw in the preceding chapter, on a curved surface, geodesics replace the straight lines of the Euclidean plane. It also became clear that, in general, when figures are formed from geodesics, some of the most cherished results of Euclidean geometry - such as the theorem on angle sum of a triangle - must be given up. But, Euclid’s theorems are derived by logical arguments from a set of postulates (or axioms). Physical intuition is allowed to enter the theory only through the postulates; after these are stated, only rules of logic can be appealed to in the proofs (although, of course, the inspiration for a theorem can come from intuition). It must therefore be the case that some of the properties ascribed to lines in Euclid’s postulates cannot be true in general for geodesics. For, if all of Euclid’s postulates held for geodesics, then so would his theorems. Let us examine some of the Euclidean postulates to see what goes wrong when they are interpreted to hold for geodesics.
Exploring Curvature, 1996
Mathematics, like music or literature or art, is an activity for which we human beings possess im... more Mathematics, like music or literature or art, is an activity for which we human beings possess immeasurable stores of talent and passion. It is a highly intellectual activity, but it should not be regarded as an elitist one. Even those of us who have never created a song, or a story, or a piece of mathematics, can still experience much pleasure from playing or listening to music, or from reading a book or attending a play, or from doing a calculation or studying a proof. Furthermore, after an initial period of practice, many of us become quite accomplished at an activity, and continue to derive pleasure from it throughout their lives. Some others among us have sufficient talent to become professional musicians, writers, or mathematicians. And, scattered throughout history, there are those rare individuals whose genius leaves us in awe. Thus, in music, Bach (1685–1750), Mozart (1756–1791), and Beethoven (1770–1827) seem to possess almost superhuman powers. In literature, we have Shakespeare (1564–1616), Milton (1608–1674), Goethe (1749–1832), and several others. In mathematics, Archimedes (287–212 B.C.), Newton (1642–1727), and Gauss are ranked at the top, but magnificent contributions were also made by a large number of others.
Foreword.- Paul M. Naghdi (1924-1994).- List of Publication of P. M. Naghdi.- I. Nonlinear and Li... more Foreword.- Paul M. Naghdi (1924-1994).- List of Publication of P. M. Naghdi.- I. Nonlinear and Linear Elasticity.- Interfacial and surface waves in pre-strained isotropic elastic media.- Deformations of an elastic, internally constrained material. Part 3: Small superimposed deformations and waves.- On axisymmetric solutions for compressible nonlinearly elastic solids.- On obtaining closed form solutions for compressible nonlinearly elastic materials.- Stress and the moment-twist relation in the torsion of a cylinder with a nonconvex stored energy function.- Conditions on the elastic strain-energy function.- On the stability of a biaxially stressed elastic material with a free surface under variations in surface shape.- Expressions for the gradients of the principal stresses and their application to interior stress concentration.- On the number of distinct elastic constants associated with certain anisotropic elastic symmetries.- Stress and deformation in moderately anisotropic inhomogeneous elastic plates.- II. General Continuum Mechanics.- Remarks concerning forces on line defects.- On entropy and incomplete information in irreversible heat flow.- A system of hyperbolic conservation laws with frictional damping.- Numerical solution of two- and three-dimensional thermomechanical problems using the theory of a Cosserat point.- The relaxation of a decompressed inclusion.- The common conjugate directions of plane sections of two concentric ellipsoids.- III. Plasticity.- Stable response in the plastic range with local instability.- A work-hardening elastic-plastic wedge.- Finite elastic-plastic deformations of an ideal fibre-reinforced beam bent around a cylinder.- The elasto-plastic plate with a hole: Analytical solutions derived by singular perturbations.- Exact stress states and velocity fields in bicrystals at the yield point in channel die compression.- IV. Biological and New Technological Materials.- Stress, strain, growth, and remodeling of living organisms.- Experimental determination of tribological properties of ultra-thin solid films.- The hierarchy of microstructures for low density materials.- Swelling and shrinking of polyelectrolytic gels.- V. Fluid Mechanics.- On gravity waves in channels.- Bragg scattering of water waves by Green-Naghdi theory.- Water waves over a sloping beach in a rotating frame.- An approximate theory for velocity profiles in the near wake of a flat plate.- On the motion of a non-rigid sphere in a perfect fluid.- An adaptive hp-finite element method for incompressible free surface flows of generalized Newtonian fluids.- Viscoelastic effects in film casting.- Experimental evidence for intense vortical structures in grid turbulence.- Evaluation of Reynolds stress turbulence closures in compressible homogeneous shear flow.- A structural theory of anisotropic turbulence.- VI. Dynamics.- The effect of damping on the stability of gyroscopic pendulums.- Dynamical systems considered as ordering machines.- On the advantages of a geometrical viewpoint in the derivation of Lagrange's equations for a rigid continuum.- Author Index.
American Journal of Physics - AMER J PHYS, 1989
The Physics Teacher, 1993
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2004
Abstract Pseudo–rigid bodies are regarded here as globally constrained three–dimensional homogene... more Abstract Pseudo–rigid bodies are regarded here as globally constrained three–dimensional homogeneous continua. The constraint reaction stresses play a fundamental role in maintaining the homogeneity of the deformation field in pseudo–rigid bodies, and the ...
Meccanica, 2011
For motion of a material point along a space curve, a kinematical decomposition, discovered by Si... more For motion of a material point along a space curve, a kinematical decomposition, discovered by Siacci, expresses the acceleration vector as the sum of two special oblique components in the osculating plane to the curve. A new proof of Siacci's theorem is presented.
American Journal of Physics, 1994
A concise but general derivation of Lagrange's equations is given for a system of fi... more A concise but general derivation of Lagrange's equations is given for a system of finitely many particles subject to holonomic and nonholonomic constraints. Based directly on Newton's second law, it takes advantage of an inertia-based metric to obtain a geometrically transparent ...
Journal of Elasticity, 2011
A general approach to continuum thermodynamics that was advocated by R.S. Rivlin is carried out f... more A general approach to continuum thermodynamics that was advocated by R.S. Rivlin is carried out for thermoelastic materials which can also depend on strain rate. An entropy function is constructed (rather than assumed to exist). A method for treating thermomechanical internal constraints for such materials is also presented. In this method, the properties of a constrained material are inherited from those of a related equivalence class of unconstrained materials.
Journal of Applied Mechanics, 1987
is incorrect in his conclusion that the presence of microstructural anisotropy in crystalline med... more is incorrect in his conclusion that the presence of microstructural anisotropy in crystalline media conflicts with the demand that intermediate stress-free configurations be subject to invariance requirements under arbitrary superposed rotations. One should distinguish between two types of theories of plasticity: Type A, those in which no explicit account is taken of microstructure; and Type B, those in which microstructural concepts play an essential role. In both types of theories, a decomposition F = F e ¥ p of the deformation gradient F into an elastic part F e and a plastic part F p can be effected through the use of intermediate stress-free local configurations. But, in theories of Type B the continuum is augmented with additional fields intended to represent microstructural properties. The full invariance requirements advocated by Naghdi and coworkers (see Dashner's Ref. 2 and its list of references) are based on the transformations
The Mathematics Teacher
Hardcover book rolls very well on pencils of circular cross section. If pencils of hexagonal cros... more Hardcover book rolls very well on pencils of circular cross section. If pencils of hexagonal cross section are used instead, the book can still be rolled, but not so nicely. Can we quantify such differences? In other words, can we put our intuitions about the physical situation into mathematical form? The purpose of the present, article is to propose a mathematical way of thinking about how good a roller is and to describe classroom activities that were carried out to measure and evaluaLe model of roller. The subject is a beautiful blend of geometry and analysis, encourages students to think quantitatively, and afford them an opportunity for bridging the gap between mathematics and physical reality.
Mathematics and Mechanics of Solids
It is argued that the concept of body, as it is usually employed in continuum mechanics, is somew... more It is argued that the concept of body, as it is usually employed in continuum mechanics, is somewhat too general. Specifically, it is standard practice to associate a set of dynamical processes with an abstract body. However, many of these processes cannot be experienced by any physical body. Yet, Euler’s laws for the balance of linear momentum and angular momentum are asserted to hold for such abstract bodies. It is suggested here that Euler’s laws should be postulated only for abstract bodies that are composed of ideal materials belonging to some large collection.
International Journal of Solids and Structures
Abstract Within the framework of finite plasticity, it is shown how a number of kinematical appro... more Abstract Within the framework of finite plasticity, it is shown how a number of kinematical approximations can be systematically derived from the multiplicative decomposition of the deformation gradient. Particular attention is devoted to two types of approximate theories: (a) those in which elastic deformations are of a different order of magnitude than plastic deformations; and (b) those in which rotations are of a different order of magnitude than strains.
Zeitschrift Fuer Angewandte Mathematik Und Physik, 1995
Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids, 1995
Journal of Elasticity, 2014
The concept of parallelism along a surface curve, which was introduced by Levi-Civita in the cont... more The concept of parallelism along a surface curve, which was introduced by Levi-Civita in the context of n-dimensional Riemannian manifolds, is re-examined from a kinematical viewpoint. A special type of frame, whose angular velocity is determined by the rate at which the tangent plane turns as one moves along a surface curve, is defined and is called a Levi-Civita frame. The surface may be orientable or not. Vectors and tensors fixed on Levi-Civita frames are parallel transported. Covariant differentiation of vectors and tensors along a surface curve can be expressed in terms of the corresponding corotational rates measured on Levi-Civita frames. Relevant results on ruled surfaces are also included.
Exploring Curvature, 1996
The most deeply rooted geometrical properties of a surface are its topological ones: these are pr... more The most deeply rooted geometrical properties of a surface are its topological ones: these are preserved under all homeomorphisms of the surface. Hence also, they are preserved under all deformations. (Some examples of deformations of surfaces were studied at the end of Chapter 11.) There is a broader class of properties that are intimately bound up with the geometry of the surface and that are preserved under a large subclass of homeomorphisms. These are ones that Gauss discovered, and which we will now explore.
Exploring Curvature, 1996
In this chapter, we describe a particular way of moving a vector along a given curve on a surface... more In this chapter, we describe a particular way of moving a vector along a given curve on a surface. It provides an especially revealing means of exploring the non-Euclideanness of the surface.
Exploring Curvature, 1996
Like Mozart’s, Bernhard Riemann’s life was short but marvelously creative. He solved several of t... more Like Mozart’s, Bernhard Riemann’s life was short but marvelously creative. He solved several of the most difficult problems in pure and applied mathematics, introduced entirely new concepts and techniques, and profoundly changed the way in which mathematicians, physicists, and philosophers view space.1
Exploring Curvature, 1996
As we saw in the preceding chapter, on a curved surface, geodesics replace the straight lines of ... more As we saw in the preceding chapter, on a curved surface, geodesics replace the straight lines of the Euclidean plane. It also became clear that, in general, when figures are formed from geodesics, some of the most cherished results of Euclidean geometry - such as the theorem on angle sum of a triangle - must be given up. But, Euclid’s theorems are derived by logical arguments from a set of postulates (or axioms). Physical intuition is allowed to enter the theory only through the postulates; after these are stated, only rules of logic can be appealed to in the proofs (although, of course, the inspiration for a theorem can come from intuition). It must therefore be the case that some of the properties ascribed to lines in Euclid’s postulates cannot be true in general for geodesics. For, if all of Euclid’s postulates held for geodesics, then so would his theorems. Let us examine some of the Euclidean postulates to see what goes wrong when they are interpreted to hold for geodesics.
Exploring Curvature, 1996
Mathematics, like music or literature or art, is an activity for which we human beings possess im... more Mathematics, like music or literature or art, is an activity for which we human beings possess immeasurable stores of talent and passion. It is a highly intellectual activity, but it should not be regarded as an elitist one. Even those of us who have never created a song, or a story, or a piece of mathematics, can still experience much pleasure from playing or listening to music, or from reading a book or attending a play, or from doing a calculation or studying a proof. Furthermore, after an initial period of practice, many of us become quite accomplished at an activity, and continue to derive pleasure from it throughout their lives. Some others among us have sufficient talent to become professional musicians, writers, or mathematicians. And, scattered throughout history, there are those rare individuals whose genius leaves us in awe. Thus, in music, Bach (1685–1750), Mozart (1756–1791), and Beethoven (1770–1827) seem to possess almost superhuman powers. In literature, we have Shakespeare (1564–1616), Milton (1608–1674), Goethe (1749–1832), and several others. In mathematics, Archimedes (287–212 B.C.), Newton (1642–1727), and Gauss are ranked at the top, but magnificent contributions were also made by a large number of others.
Foreword.- Paul M. Naghdi (1924-1994).- List of Publication of P. M. Naghdi.- I. Nonlinear and Li... more Foreword.- Paul M. Naghdi (1924-1994).- List of Publication of P. M. Naghdi.- I. Nonlinear and Linear Elasticity.- Interfacial and surface waves in pre-strained isotropic elastic media.- Deformations of an elastic, internally constrained material. Part 3: Small superimposed deformations and waves.- On axisymmetric solutions for compressible nonlinearly elastic solids.- On obtaining closed form solutions for compressible nonlinearly elastic materials.- Stress and the moment-twist relation in the torsion of a cylinder with a nonconvex stored energy function.- Conditions on the elastic strain-energy function.- On the stability of a biaxially stressed elastic material with a free surface under variations in surface shape.- Expressions for the gradients of the principal stresses and their application to interior stress concentration.- On the number of distinct elastic constants associated with certain anisotropic elastic symmetries.- Stress and deformation in moderately anisotropic inhomogeneous elastic plates.- II. General Continuum Mechanics.- Remarks concerning forces on line defects.- On entropy and incomplete information in irreversible heat flow.- A system of hyperbolic conservation laws with frictional damping.- Numerical solution of two- and three-dimensional thermomechanical problems using the theory of a Cosserat point.- The relaxation of a decompressed inclusion.- The common conjugate directions of plane sections of two concentric ellipsoids.- III. Plasticity.- Stable response in the plastic range with local instability.- A work-hardening elastic-plastic wedge.- Finite elastic-plastic deformations of an ideal fibre-reinforced beam bent around a cylinder.- The elasto-plastic plate with a hole: Analytical solutions derived by singular perturbations.- Exact stress states and velocity fields in bicrystals at the yield point in channel die compression.- IV. Biological and New Technological Materials.- Stress, strain, growth, and remodeling of living organisms.- Experimental determination of tribological properties of ultra-thin solid films.- The hierarchy of microstructures for low density materials.- Swelling and shrinking of polyelectrolytic gels.- V. Fluid Mechanics.- On gravity waves in channels.- Bragg scattering of water waves by Green-Naghdi theory.- Water waves over a sloping beach in a rotating frame.- An approximate theory for velocity profiles in the near wake of a flat plate.- On the motion of a non-rigid sphere in a perfect fluid.- An adaptive hp-finite element method for incompressible free surface flows of generalized Newtonian fluids.- Viscoelastic effects in film casting.- Experimental evidence for intense vortical structures in grid turbulence.- Evaluation of Reynolds stress turbulence closures in compressible homogeneous shear flow.- A structural theory of anisotropic turbulence.- VI. Dynamics.- The effect of damping on the stability of gyroscopic pendulums.- Dynamical systems considered as ordering machines.- On the advantages of a geometrical viewpoint in the derivation of Lagrange's equations for a rigid continuum.- Author Index.
American Journal of Physics - AMER J PHYS, 1989
The Physics Teacher, 1993
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2004
Abstract Pseudo–rigid bodies are regarded here as globally constrained three–dimensional homogene... more Abstract Pseudo–rigid bodies are regarded here as globally constrained three–dimensional homogeneous continua. The constraint reaction stresses play a fundamental role in maintaining the homogeneity of the deformation field in pseudo–rigid bodies, and the ...
Meccanica, 2011
For motion of a material point along a space curve, a kinematical decomposition, discovered by Si... more For motion of a material point along a space curve, a kinematical decomposition, discovered by Siacci, expresses the acceleration vector as the sum of two special oblique components in the osculating plane to the curve. A new proof of Siacci's theorem is presented.