Ebook: Mathematics of Multiscale Materials

Polycrystalline metals, porous rocks, colloidal suspensions, epitaxial thin films, gels, foams, granular aggregates, sea ice, shape-memory metals, magnetic materials, and electro-rheological 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 1995-1996 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 self-imposed barriers of twentieth century science.

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Polycrystalline metals, porous rocks, colloidal suspensions, epitaxial thin films, gels, foams, granular aggregates, sea ice, shape-memory metals, magnetic materials, and electro-rheological 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 1995-1996 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 self-imposed barriers of twentieth century science.

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Polycrystalline metals, porous rocks, colloidal suspensions, epitaxial thin films, gels, foams, granular aggregates, sea ice, shape-memory metals, magnetic materials, and electro-rheological 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 1995-1996 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 self-imposed barriers of twentieth century science.
Content:
Front Matter....Pages i-x
Scaling Limit for the Incipient Spanning Clusters....Pages 1-24
Bounded and Unbounded Level Lines in Two-Dimensional Random Fields....Pages 25-35
Transversely Isotropic Poroelasticity Arising from Thin Isotropic Layers....Pages 37-50
Bounds on the Effective Elastic Properties of Martensitic Polycrystals....Pages 51-62
Statistical Models for Fracture....Pages 63-80
Anomalous Diffusion in Random Flows....Pages 81-99
Calculating the Mechanical Properties of Materials from Interatomic Forces....Pages 101-108
Granular Media: Some New Results....Pages 109-128
Elastic Freedom in Cellular Solids and Composite Materials....Pages 129-153
Weakly Nonlinear Conductivity and Flicker Noise Near Percolation....Pages 155-178
Fine Properties of Solutions to Conductivity Equations With Applications to Composites....Pages 179-208
Composite Sensors and Actuators....Pages 209-211
Bounding the Effective Yield Behavior of Mixtures....Pages 213-221
Upper Bounds on Electrorheological Properties....Pages 223-230
On Spatiotemporal Patterns in Composite Reactive Media....Pages 231-253
Equilibrium Shapes of Islands in Epitaxially Strained Solid Films....Pages 255-269
Numerical Simulation of the Effective Elastic Properties of a Class of Cell Materials....Pages 271-280
Back Matter....Pages 281-287

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Polycrystalline metals, porous rocks, colloidal suspensions, epitaxial thin films, gels, foams, granular aggregates, sea ice, shape-memory metals, magnetic materials, and electro-rheological 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 1995-1996 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 self-imposed barriers of twentieth century science.
Content:
Front Matter....Pages i-x
Scaling Limit for the Incipient Spanning Clusters....Pages 1-24
Bounded and Unbounded Level Lines in Two-Dimensional Random Fields....Pages 25-35
Transversely Isotropic Poroelasticity Arising from Thin Isotropic Layers....Pages 37-50
Bounds on the Effective Elastic Properties of Martensitic Polycrystals....Pages 51-62
Statistical Models for Fracture....Pages 63-80
Anomalous Diffusion in Random Flows....Pages 81-99
Calculating the Mechanical Properties of Materials from Interatomic Forces....Pages 101-108
Granular Media: Some New Results....Pages 109-128
Elastic Freedom in Cellular Solids and Composite Materials....Pages 129-153
Weakly Nonlinear Conductivity and Flicker Noise Near Percolation....Pages 155-178
Fine Properties of Solutions to Conductivity Equations With Applications to Composites....Pages 179-208
Composite Sensors and Actuators....Pages 209-211
Bounding the Effective Yield Behavior of Mixtures....Pages 213-221
Upper Bounds on Electrorheological Properties....Pages 223-230
On Spatiotemporal Patterns in Composite Reactive Media....Pages 231-253
Equilibrium Shapes of Islands in Epitaxially Strained Solid Films....Pages 255-269
Numerical Simulation of the Effective Elastic Properties of a Class of Cell Materials....Pages 271-280
Back Matter....Pages 281-287
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