Milton, Graeme Walter 1956
Overview
Works:  19 works in 54 publications in 2 languages and 1,304 library holdings 

Genres:  Conference papers and proceedings Academic theses Filmed lectures Nonfiction films 
Roles:  Author, Editor, Other 
Publication Timeline
.
Most widely held works by
Graeme Walter Milton
The theory of composites by
Graeme Walter Milton(
)
23 editions published between 2001 and 2004 in English and Spanish and held by 1,096 WorldCat member libraries worldwide
Annotation The theory of composite materials is the study of partial differential equations with rapid oscillations in their coefficients. Although extensively studied for more than a hundred years, an explosion of ideas in the past four decades has dramatically increased our understanding of the relationship among the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective moduli that govern the macroscopic behavior. This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification. This book surveys these exciting developments at the frontier of mathematics and presents many new results
23 editions published between 2001 and 2004 in English and Spanish and held by 1,096 WorldCat member libraries worldwide
Annotation The theory of composite materials is the study of partial differential equations with rapid oscillations in their coefficients. Although extensively studied for more than a hundred years, an explosion of ideas in the past four decades has dramatically increased our understanding of the relationship among the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective moduli that govern the macroscopic behavior. This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification. This book surveys these exciting developments at the frontier of mathematics and presents many new results
Random media and composites by SIAM Workshop on Random Media and Composites(
Book
)
5 editions published in 1989 in English and held by 86 WorldCat member libraries worldwide
5 editions published in 1989 in English and held by 86 WorldCat member libraries worldwide
Mathematics of multiscale materials by
Kenneth M Golden(
)
3 editions published in 1998 in English and held by 53 WorldCat member libraries worldwide
Polycrystalline metals, porous rocks, colloidal suspensions, epitaxial thin films, gels, foams, granular aggregates, sea ice, shapememory metals, magnetic materials, and electrorheological fluids are all examples of materials where an understanding of the mathematics on the different length scales is a key to interpreting their physical behavior. In their analysis of these media, scientists coming from a number of disciplines have encountered similar mathematical problems, yet it is rare for researchers in the various fields to meet. The 19951996 program at the Institute for Mathematics and its Applications was devoted to Mathematical Methods in Material Science, and was attended by materials scientists, physicists, geologists, chemists engineers, and mathematicians. The present volume contains chapters which have emerged from four of the workshops held during the year, focusing on the following areas: Disordered Materials; Interfaces and Thin Films; Mechanical Response of Materials from Angstroms to Meters; and Phase Transformation, Composite Materials and Microstructure. The scales treated in these workshops ranged from the atomic to the microstructural to the macroscopic, the microstructures from ordered to random, and the treatments from "purely" theoretical to the highly applied. Taken together, these works form a compelling and broad account of many aspects of the science of multiscale materials, and will hopefully inspire research across the selfimposed barriers of twentieth century science
3 editions published in 1998 in English and held by 53 WorldCat member libraries worldwide
Polycrystalline metals, porous rocks, colloidal suspensions, epitaxial thin films, gels, foams, granular aggregates, sea ice, shapememory metals, magnetic materials, and electrorheological fluids are all examples of materials where an understanding of the mathematics on the different length scales is a key to interpreting their physical behavior. In their analysis of these media, scientists coming from a number of disciplines have encountered similar mathematical problems, yet it is rare for researchers in the various fields to meet. The 19951996 program at the Institute for Mathematics and its Applications was devoted to Mathematical Methods in Material Science, and was attended by materials scientists, physicists, geologists, chemists engineers, and mathematicians. The present volume contains chapters which have emerged from four of the workshops held during the year, focusing on the following areas: Disordered Materials; Interfaces and Thin Films; Mechanical Response of Materials from Angstroms to Meters; and Phase Transformation, Composite Materials and Microstructure. The scales treated in these workshops ranged from the atomic to the microstructural to the macroscopic, the microstructures from ordered to random, and the treatments from "purely" theoretical to the highly applied. Taken together, these works form a compelling and broad account of many aspects of the science of multiscale materials, and will hopefully inspire research across the selfimposed barriers of twentieth century science
#x98;The#x9C; theory of composites by
Graeme Walter Milton(
)
2 editions published between 2001 and 2002 in English and held by 26 WorldCat member libraries worldwide
2 editions published between 2001 and 2002 in English and held by 26 WorldCat member libraries worldwide
ETOPIM 6 : proceedings of the Sixth International Conference on Electrical Transport and Optical Properties of Inhomogeneous
Media ; held in Snowbird, UT, USA 1519 July 2002 by
2002, Snowbird, Utah) ETOPIM (6(
Book
)
1 edition published in 2003 in English and held by 14 WorldCat member libraries worldwide
1 edition published in 2003 in English and held by 14 WorldCat member libraries worldwide
Extending the theory of composites to other areas of science(
Book
)
2 editions published in 2016 in English and held by 7 WorldCat member libraries worldwide
2 editions published in 2016 in English and held by 7 WorldCat member libraries worldwide
Proceedings of the Sixth International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, ETOPIM6,
held in Snowbird, UT, USA, 1519 July 2002 by International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media(
Book
)
3 editions published in 2003 in English and held by 5 WorldCat member libraries worldwide
3 editions published in 2003 in English and held by 5 WorldCat member libraries worldwide
Planning a merger of FE colleges by
Allyson Cass(
Book
)
2 editions published in 1997 in English and held by 5 WorldCat member libraries worldwide
2 editions published in 1997 in English and held by 5 WorldCat member libraries worldwide
Some exotic models in statistical physics by
Graeme Walter Milton(
)
2 editions published in 1985 in English and held by 3 WorldCat member libraries worldwide
2 editions published in 1985 in English and held by 3 WorldCat member libraries worldwide
Mathematical problems in micromechanics and composite materials by
Robert V Kohn(
Book
)
2 editions published in 1992 in English and held by 2 WorldCat member libraries worldwide
This is the final technical report on ARO Contract DAAL0389K0039, which began February 1, 1989 and terminated January 31, 1992. The scientific focus of this work is at the frontier where mathematics meets the materials sciences. Physically, we are concerned with the effective moduli of composites, the formation of fine scale structure in coherent phase transitions, and with oscillatory solutions of nonlinear partial differential equations from continuum. mechanics. Mathematically, we bring to bear a variety of tools including homogenization, the calculus of variations, and the theory of stochastic Processes. Our accomplishments include: (1) new bounds on the effective moduli of twocomponent and polycrystalline composites; (2) a new understanding of the role of surface energy in coherent phase transitions; and (3) rigorous results on the effective diffusivity due to a turbulent velocity field. The training of young scientists, both postdocs and students, has been a major part of our activity. composite materials; effective moduli; phase transitions; turbulent transport
2 editions published in 1992 in English and held by 2 WorldCat member libraries worldwide
This is the final technical report on ARO Contract DAAL0389K0039, which began February 1, 1989 and terminated January 31, 1992. The scientific focus of this work is at the frontier where mathematics meets the materials sciences. Physically, we are concerned with the effective moduli of composites, the formation of fine scale structure in coherent phase transitions, and with oscillatory solutions of nonlinear partial differential equations from continuum. mechanics. Mathematically, we bring to bear a variety of tools including homogenization, the calculus of variations, and the theory of stochastic Processes. Our accomplishments include: (1) new bounds on the effective moduli of twocomponent and polycrystalline composites; (2) a new understanding of the role of surface energy in coherent phase transitions; and (3) rigorous results on the effective diffusivity due to a turbulent velocity field. The training of young scientists, both postdocs and students, has been a major part of our activity. composite materials; effective moduli; phase transitions; turbulent transport
The set of forces that ideal trusses, or wire webs, under tension can support(
)
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
Abstract: The problem of determining those multiplets of forces, or sets of force multiplets, acting at a set of points, such that there exists a truss structure, or wire web, that can support these force multiplets with all the elements of the truss or wire web being under tension, is considered. The twodimensional problem where the points are at the vertices of a convex polygon is essentially solved: each multiplet of forces must be such that the net anticlockwise torque around any vertex of the forces summed over any number of consecutive points clockwise past the vertex must be nonnegative; and one can find a truss structure that supports under tension, and only supports, those force multiplets in a convex polyhedron of force multiplets that is generated by a finite number of force multiplets each satisfying the torque condition. Progress is also made on the problem where only a subset of the points are at the vertices of a convex polygon, and the other points are inside. In particular, in the case where only one point is inside, an explicit procedure is described for constructing a suitable truss, if one exists. An alternative recipe to that provided by Guevara Vasquez etal. (2011), based on earlier work of CamarEddine and Seppecher (2003), is given for constructing a truss structure, with elements under either compression or tension, that supports an arbitrary collection of balanced forces at the vertices of a convex polygon. Finally some constraints are given on the forces that a threedimension truss, or wire web, under tension must satisfy
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
Abstract: The problem of determining those multiplets of forces, or sets of force multiplets, acting at a set of points, such that there exists a truss structure, or wire web, that can support these force multiplets with all the elements of the truss or wire web being under tension, is considered. The twodimensional problem where the points are at the vertices of a convex polygon is essentially solved: each multiplet of forces must be such that the net anticlockwise torque around any vertex of the forces summed over any number of consecutive points clockwise past the vertex must be nonnegative; and one can find a truss structure that supports under tension, and only supports, those force multiplets in a convex polyhedron of force multiplets that is generated by a finite number of force multiplets each satisfying the torque condition. Progress is also made on the problem where only a subset of the points are at the vertices of a convex polygon, and the other points are inside. In particular, in the case where only one point is inside, an explicit procedure is described for constructing a suitable truss, if one exists. An alternative recipe to that provided by Guevara Vasquez etal. (2011), based on earlier work of CamarEddine and Seppecher (2003), is given for constructing a truss structure, with elements under either compression or tension, that supports an arbitrary collection of balanced forces at the vertices of a convex polygon. Finally some constraints are given on the forces that a threedimension truss, or wire web, under tension must satisfy
On Characterizing the Set of Possible Effective Tensors of Composites: The Variational Method and the Translation Method(
Book
)
1 edition published in 1990 in English and held by 1 WorldCat member library worldwide
A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarizationproblem simplifies the derivation of the HashinShtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant nonlocal projection operator, gamma. An important class of bounds, namely trace bounds on the effective tensors of twocomponent media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the twocomponents are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. Keywords: Reprints; Mathematical models; Composite materials; Elastic properties; Conduction. (aw)
1 edition published in 1990 in English and held by 1 WorldCat member library worldwide
A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarizationproblem simplifies the derivation of the HashinShtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant nonlocal projection operator, gamma. An important class of bounds, namely trace bounds on the effective tensors of twocomponent media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the twocomponents are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. Keywords: Reprints; Mathematical models; Composite materials; Elastic properties; Conduction. (aw)
Near optimal pentamodes as a tool for guiding stress while minimizing compliance in 3dprinted materials: A complete solution
to the weak Gclosure problem for 3dprinted materials(
)
1 edition published in 2018 in English and held by 1 WorldCat member library worldwide
1 edition published in 2018 in English and held by 1 WorldCat member library worldwide
Characterizing the possible conductivity functions of composite materials by Karen Elizabeth Clark(
Book
)
1 edition published in 1992 in English and held by 1 WorldCat member library worldwide
The thesis is in three separate parts. In the first part a continued fraction representation for the effective conductivity tensor is derived for a polycrystalline material in two dimensions. A basis for the Hilbert space of fields in the composite is chosen so that the linear operators which are relevant to determining the effective conductivity take simple forms as infinite matrices. These infinite matrices are then used in the variational definition of the effective conductivity to formulate the continued fraction
1 edition published in 1992 in English and held by 1 WorldCat member library worldwide
The thesis is in three separate parts. In the first part a continued fraction representation for the effective conductivity tensor is derived for a polycrystalline material in two dimensions. A basis for the Hilbert space of fields in the composite is chosen so that the linear operators which are relevant to determining the effective conductivity take simple forms as infinite matrices. These infinite matrices are then used in the variational definition of the effective conductivity to formulate the continued fraction
Bounds on the macrosopic properties of composite materials by
Graeme Walter Milton(
Book
)
1 edition published in 1982 in English and held by 1 WorldCat member library worldwide
1 edition published in 1982 in English and held by 1 WorldCat member library worldwide
Discrete transformation elastodynamics and active exterior acoustic cloaking by
Graeme Walter Milton(
Visual
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
We recall the transformation of elastodynamic equations under coordinate changes. The idea is to use coordinate transformations to manipulate waves propagating in an elastic material. Then we study the effect of transformations on a massspring network model. The transformed networks can be realized with "torque springs", which are introduced here and are springs with a force proportional to the displacement in a direction other than the direction of the spring terminals. Possible homogenizations of the transformed networks are presented, with potential applications to cloaking. Then we present cloaking methods that are based on cancelling an incident field using active devices which are exterior to the cloaked region and that do not generate significant fields far away from the devices. The exterior cloaking problem for the Laplace equation is reformulated as the problem of polynomial approximation of a function which takes the value one inside a disk and zero inside a disjoint disk. We also consider the active exterior cloaking problem for the Helmholtz equation in 2D and 3D. Our method uses Green's formula and an addition theorem for outgoing waves to design devices that mimic the effect of the single and double layer potentials in Green's formula. The advantage of the method is that it is broadband and leaves "throats" connecting the cloaked region with the outside
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
We recall the transformation of elastodynamic equations under coordinate changes. The idea is to use coordinate transformations to manipulate waves propagating in an elastic material. Then we study the effect of transformations on a massspring network model. The transformed networks can be realized with "torque springs", which are introduced here and are springs with a force proportional to the displacement in a direction other than the direction of the spring terminals. Possible homogenizations of the transformed networks are presented, with potential applications to cloaking. Then we present cloaking methods that are based on cancelling an incident field using active devices which are exterior to the cloaked region and that do not generate significant fields far away from the devices. The exterior cloaking problem for the Laplace equation is reformulated as the problem of polynomial approximation of a function which takes the value one inside a disk and zero inside a disjoint disk. We also consider the active exterior cloaking problem for the Helmholtz equation in 2D and 3D. Our method uses Green's formula and an addition theorem for outgoing waves to design devices that mimic the effect of the single and double layer potentials in Green's formula. The advantage of the method is that it is broadband and leaves "throats" connecting the cloaked region with the outside
ETOPIM 6(
Book
)
1 edition published in 2003 in English and held by 1 WorldCat member library worldwide
1 edition published in 2003 in English and held by 1 WorldCat member library worldwide
Big chief elizabeth by
Graeme Walter Milton(
Book
)
1 edition published in 2003 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 2003 in English and held by 0 WorldCat member libraries worldwide
On bounding the effective conducitvity of anisotropic composites by
Robert V Kohn(
Book
)
1 edition published in 1986 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 1986 in English and held by 0 WorldCat member libraries worldwide
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Related Identities
 Kohn, Robert V. Other Author Editor
 Society for Industrial and Applied Mathematics Other
 United States Air Force Office of Scientific Research Other
 James, Richard D.
 Grimmett, Geoffrey R.
 Sen, P. N. (Pabitra N.)
 Golden, Kenneth M. Author
 ETOPIM (6, 2002, Snowbird, Utah)
 MyiLibrary
 Cass, Alan Author
Useful Links
Associated Subjects
ChemistryMathematics Composite materials Composite materialsElectric properties Composite materialsMathematical models Composite materialsOptical properties Continuum mechanics Differential equations, Partial Dynamics Education Elasticity Electromagnetic wavesTransmissionMathematical models Engineering Homogenization (Differential equations) Inhomogeneous materialsElectric properties Inhomogeneous materialsOptical properties Length measurement Management MaterialsMathematical models Materials scienceMathematics Micromechanics Phase transformations (Statistical physics) Phase transformations (Statistical physics)Mathematical models Statistics Transport theory