continuum mechanics

A crystallite is a domain of solid-state matter that has the same structure as a single crystal. Solid objects that are large enough to see and handle are rarely composed of a single crystal, except for a few cases (gems, silicon single crystals for the electronics industry, certain types of fiber, and single crystals of a nickel-based superalloy for turbojet engines). Most materials are polycrystalline; they are made of a large number of single crystals—crystallites—held together by thin layers of amorphous solid. The crystallite size can ...

Including:

• Crystallite - Grain boundaries

Read more here: » Crystallite: Encyclopedia - Crystallite

Biomechanics is the research and analysis of the mechanics of living organisms. Aristoteles might be considered the first biomechanicist. He wrote the first book called "De Motu Animalium" - On the Movement of Animals. He not only saw animals' bodies as mechanical systems, but pursued such questions as the physiological difference between imagining performing an action and actually doing it. (Read more about the history of Biomechanics in A Genealogy of Biomechanics.) The research and analysis can be carried forth on multiple levels, ...

Including:

• Biomechanics - Applications
• Biomechanics - Continuum Mechanics
• Biomechanics - Biomechanics of Circulation
• Biomechanics - Biomechanics of the bones
• Biomechanics - Biomechanics of the Muscle
• Biomechanics - Biomechanics of Soft Tissues
• Biomechanics - Viscoelasticity
• Biomechanics - Nonlinear Theories

Read more here: » Biomechanics: Encyclopedia - Biomechanics

The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of this encyclopedia, set out in a systematic way. The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — m ...

Including:

• Areas of mathematics - Foundations / general
• Areas of mathematics - Algebra
• Areas of mathematics - Analysis
• Areas of mathematics - Geometry
• Areas of mathematics - Applied mathematics
• Areas of mathematics - Probability and statistics
• Areas of mathematics - Computational sciences
• Areas of mathematics - Physical sciences
• Areas of mathematics - Non-physical sciences

Read more here: » Areas of mathematics: Encyclopedia - Areas of mathematics

Emergence is the process of complex pattern formation from simpler rules. This can be a dynamic process (occurring over time), such as the evolution of the human brain over thousands of successive generations; or emergence can happen over disparate size scales, such as the interactions between a great number of neurons producing a human brain capable of thought (even though the constituent neurons are not individually capable of thought). The original term wa ...

Including:

• Emergence - Emergent properties
• Emergence - Emergence in games
• Emergence - Emergent structures in nature
• Emergence - Emergence in culture and engineering
• Emergence - Emergence in physics
• Emergence - Bibliography

Read more here: » Emergence: Encyclopedia - Emergence

A contravariant tensor of order 1(Ti) is defined as: A covariant tensor of order 1(Ti) is defined as: ...

See also:

Classical treatment of tensors, Classical treatment of tensors - Contravariant and covariant tensors, Classical treatment of tensors - General tensors

Read more here: » Classical treatment of tensors: Encyclopedia II - Classical treatment of tensors - Contravariant and covariant tensors

A tensor may be expressed as the sequence of values represented by a function with a vector valued domain and a scalar valued range. These vectors in the domain are vectors of counting numbers, and these numbers are called indexes. For example, a rank 3 tensor might have dimensions 2, 5, and 7. Here, the vectors range from <1, 1, 1> through <2, 5, 7>. Here, the tensor would have one value at <1, 1, 1>, another at <1, 1, 2>, and so on for a total of 70 values. (Likewise, vectors may be expressed as a sequence of values ...

See also:

Tensor, Tensor - Brief overview, Tensor - Importance and usage, Tensor - History, Tensor - The choice of approach, Tensor - Examples, Tensor - Approaches in detail, Tensor - Tensor densities, Tensor - Notation, Tensor - Foundational, Tensor - Applications, Tensor - Reference books, Tensor - Tensor software

Read more here: » Tensor: Encyclopedia II - Tensor - Examples

A more sophisticated equation of state was derived by F. D. Murnaghan of Johns Hopkins University in 1944[1]. To begin with, we consider the pressure and the bulk modulus Experimentally, the bulk modulus pressure derivative is found to change little with pressure. If we take B' = B'0 to be a constant, then See also:

Birch-Murnaghan equation of state, Birch-Murnaghan equation of state - Murnaghan equation of state, Birch-Murnaghan equation of state - Birch-Murnaghan equation of state

Read more here: » Birch-Murnaghan equation of state: Encyclopedia II - Birch-Murnaghan equation of state - Murnaghan equation of state

In practice, rheology is principally concerned with extending the "classical" disciplines of elasticity and (Newtonian) fluid mechanics to materials whose mechanical behaviour cannot be described with the classical theories. It is also concerned with establishing predictions for mechanical behaviour (on the continuum mechanical scale) based on the micro- or nanostructure of the material, e.g. the molecular size and architecture of polymers in solution or the particle size d ...

See also:

Rheology, Rheology - Terminology, Rheology - Scope, Rheology - Applications, Rheology - Elasticity viscosity solid- and liquid-like behaviour and plasticity

Read more here: » Rheology: Encyclopedia II - Rheology - Scope

The basic mathematics underlying spacetime physics is affine differential geometry, in which we endow an n dimensional differentiable manifold M with a law of parallel translation of vectors along paths in M. (At each point of a differentiable manifold, we have a linear space of tangent vectors, but we have no way to transport vectors to another point, or to compare vectors at two points in M.) The parallel translation preserves linear relationships between vectors; that is, if two vectors u and v at the same point of M parallel translate along a curve to vectors u' and v ...

See also:

Einstein-Cartan theory, Einstein-Cartan theory - Introduction, Einstein-Cartan theory - Derivation of field equations of Einstein-Cartan theory, Einstein-Cartan theory - Geometric insights from Einstein-Cartan theory, Einstein-Cartan theory - First geometric insight, Einstein-Cartan theory - Second geometric insight, Einstein-Cartan theory - Third geometric insight, Einstein-Cartan theory - Fourth geometric insight, Einstein-Cartan theory - General relativity plus matter with spin implies Einstein-Cartan theory

Read more here: » Einstein-Cartan theory: Encyclopedia II - Einstein-Cartan theory - Introduction

Deformation gradient tensor F is defined as: or where are the coordinates of a point in deformed state and are coordinate of a point in undeformed state. By doing so we assume that can be expressed as a differentiable function of and time t: this will not be the case if a crack develops in the deformed body. If we have a small vector in the undeformed body, then the correspon ...

See also:

Finite deformation tensors, Finite deformation tensors - Deformation gradient tensor, Finite deformation tensors - Finger tensor, Finite deformation tensors - Cauchy-Green tensor, Finite deformation tensors - Examples, Finite deformation tensors - Uniaxial extension of an incompressible material, Finite deformation tensors - Simple shear, Finite deformation tensors - Solid body rotation, Finite deformation tensors - Source

Read more here: » Finite deformation tensors: Encyclopedia II - Finite deformation tensors - Deformation gradient tensor

Note that some practitioners often consider rheology a sub-field of materials science, because it can cover any material that flows. However, modern rheology typically deals with non-Newtonian fluid dynamics, so it is often consider a sub-field of continuum mechanics. See also granular material. Surface science --- interactions and structures between solid-gas solid-liquid or solid-solid interfaces. Ceramics, which can be subdivided into: electronic materials such as complex oxides, structural ceramics ...

See also:

Materials science, Materials science - Classes of materials by bond types, Materials science - Sub-fields of materials science, Materials science - Topics that form the basis of materials science, Materials science - A non-exhaustive list of some materials science research institutions and facilities

Read more here: » Materials science: Encyclopedia II - Materials science - Sub-fields of materials science

For the case of uniaxial elongation, true stress can be calculated as: and engineering stress can be calculated as: The Mooney-Rivlin solid model usually fits experimental data better than Neo-Hookean solid does, but requires an additional empirical constant. ...

See also:

Mooney-Rivlin solid, Mooney-Rivlin solid - Uniaxial elongation, Mooney-Rivlin solid - Source

Read more here: » Mooney-Rivlin solid: Encyclopedia II - Mooney-Rivlin solid - Uniaxial elongation

It is often appropriate to model living tissues as continuous media. For example, at the tissue level, the arterial wall can be modeled as a continuum. This assumption breaks down when the length scales of interest approach the order of the microstructural details of the material. The basic postulates of continuum mechanics are conservation of linear and angular momentum, conservation of mass, conservation of energy, and the entropy inequality. Solids are usually modeled using "reference" or "Lagrangian" coordinates, whereas fluids are often ...

See also:

Biomechanics, Biomechanics - Applications, Biomechanics - Continuum Mechanics, Biomechanics - Biomechanics of Circulation, Biomechanics - Biomechanics of the bones, Biomechanics - Biomechanics of the Muscle, Biomechanics - Biomechanics of Soft Tissues, Biomechanics - Viscoelasticity, Biomechanics - Nonlinear Theories

Read more here: » Biomechanics: Encyclopedia II - Biomechanics - Continuum Mechanics

Although the term "crystallite" is more precise, the boundary between two crystallites is traditionally known as a grain boundary. The term "crystallite boundary" is rarely used, and the fact that powder grains are not attached to one another, and so do not form boundaries, helps to remove ambiguity in this case. Grain boundaries disrupt the motion of dislocations through a material; reducing crystallite size is therefore a common way to improve strength, often without any sacrifice in toughness. The high interfacial energy and ...

See also:

Crystallite, Crystallite - Grain boundaries

Read more here: » Crystallite: Encyclopedia II - Crystallite - Grain boundaries

Although the term "crystallite" is more precise, the boundary between two crystallites is traditionally known as a grain boundary. The term "crystallite boundary" is rarely used, and the fact that powder grains are not attached to one another and so do not form boundaries helps to remove ambiguity in this case. Grain boundaries disrupt the motion of dislocations through a material; reducing crystallite size is therefore a common way to improve strength, often without any sacrifice in toughness. The high interfacial energy and r ...

See also:

Crystallite, Crystallite - Grain boundaries

Read more here: » Crystallite: Encyclopedia II - Crystallite - Grain boundaries

An emergent behaviour or emergent property can appear when a number of simple entities (agents) operate in an environment, forming more complex behaviours as a collective. If emergence happens over disparate size scales, then the reason is usually a causal relation across different scales. In other words there is often a form of top-down feedback in systems with emergent properties. These are two of the major reasons why emergent behaviour occurs: intricated causal relations across different scales and feedback. The property it ...

See also:

Emergence, Emergence - Emergent properties, Emergence - Emergence in games, Emergence - Emergent structures in nature, Emergence - Emergence in culture and engineering, Emergence - Emergence in physics, Emergence - Bibliography

Read more here: » Emergence: Encyclopedia II - Emergence - Emergent properties

Note that some practitioners often consider rheology a sub-field of materials science, because it can cover any material that flows. However, modern rheology typically deals with non-Newtonian fluid dynamics, so it is often consider a sub-field of continuum mechanics. See also granular material. Surface science --- interactions and structures between solid-gas solid-liquid or solid-solid interfaces. Ceramics, which can be subdivided into: electronic materials such as complex oxides, structural ceramics ...

See also:

Materials science, Materials science - Classes of materials by bond types, Materials science - Sub-fields of materials science, Materials science - Topics that form the basis of materials science, Materials science - A short list of non-academic materials facilities

Read more here: » Materials science: Encyclopedia II - Materials science - Sub-fields of materials science

The study of structure starting with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about ruler-and-compass constructions were finally settled by Galois ...

See also:

Areas of mathematics, Areas of mathematics - Foundations / general, Areas of mathematics - Algebra, Areas of mathematics - Analysis, Areas of mathematics - Geometry, Areas of mathematics - Applied mathematics, Areas of mathematics - Probability and statistics, Areas of mathematics - Computational sciences, Areas of mathematics - Physical sciences, Areas of mathematics - Non-physical sciences

Read more here: » Areas of mathematics: Encyclopedia II - Areas of mathematics - Algebra

A multi-order (general) tensor is simply the tensor product of single order tensors: such that: This is sometimes termed the tensor transformation law. ...

See also:

Classical treatment of tensors, Classical treatment of tensors - Contravariant and covariant tensors, Classical treatment of tensors - General tensors

Read more here: » Classical treatment of tensors: Encyclopedia II - Classical treatment of tensors - General tensors

Einstein-Cartan theory - First geometric insight. Spin (intrinsic angular momentum) consists of dislocations in the fabric of spacetime. For ordinary fermions (particles with spin such as protons, neutrons and electrons), these are screw dislocations (parking garage ramps) with timelike direction of the screw. That is, for a particle with spin in the +z direction, traversing a space-like loop in the x-y plane around the particle parallel translates you into the past or the future by a smal ...

See also:

Einstein-Cartan theory, Einstein-Cartan theory - Introduction, Einstein-Cartan theory - Derivation of field equations of Einstein-Cartan theory, Einstein-Cartan theory - Geometric insights from Einstein-Cartan theory, Einstein-Cartan theory - First geometric insight, Einstein-Cartan theory - Second geometric insight, Einstein-Cartan theory - Third geometric insight, Einstein-Cartan theory - Fourth geometric insight, Einstein-Cartan theory - General relativity plus matter with spin implies Einstein-Cartan theory

Read more here: » Einstein-Cartan theory: Encyclopedia II - Einstein-Cartan theory - Geometric insights from Einstein-Cartan theory