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Constitutive Equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance or field, and approximates its response to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations. Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. Howe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Relationship
In science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ..., an empirical relationship or phenomenological relationship is a relationship or correlation that is supported by experiment or observation but not necessarily supported by theory. Analytical solutions without a theory An empirical relationship is supported by confirmatory data irrespective of theoretical basis such as first principles. Sometimes theoretical explanations for what were initially empirical relationships are found, in which case the relationships are no longer considered empirical. An example was the Rydberg formula to predict the wavelengths of hydrogen spectral lines. Proposed in 1876, it perfectly predicted the wavelengths of the Lyman series, but lacked a theoretical basis until Niels Bohr produce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green–Kubo Relations
The Green–Kubo relations ( Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for a transport coefficient \gamma in terms of the integral of the equilibrium time correlation function of the time derivative of a corresponding microscopic variable A (sometimes termed a "gross variable", as in ): \gamma = \int_0^\infty \left\langle \dot(t) \dot(0) \right\rangle \;t. One intuitive way to understand this relation is that relaxations resulting from random fluctuations in equilibrium are indistinguishable from those due to an external perturbation in linear response. Green-Kubo relations are important because they relate a macroscopic transport coefficient to the correlation function of a microscopic variable. In addition, they allow one to measure the transport coefficient without perturbing the system out of equilibrium, which has found much use in molecular dynamics simulations. Thermal and mechanical transport processes Thermodynamic systems m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condensed Matter Physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and electrons. More generally, the subject deals with condensed phases of matter: systems of many constituents with strong interactions among them. More exotic condensed phases include the superconductivity, superconducting phase exhibited by certain materials at extremely low cryogenic temperatures, the ferromagnetic and antiferromagnetic phases of Spin (physics), spins on crystal lattices of atoms, the Bose–Einstein condensates found in ultracold atomic systems, and liquid crystals. Condensed matter physicists seek to understand the behavior of these phases by experiments to measure various material properties, and by applying the physical laws of quantum mechanics, electromagnetism, statistical mechanics, and other theoretical physics, physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Truesdell
Clifford Ambrose Truesdell III (February 18, 1919 – January 14, 2000) was an American mathematician, natural philosopher, and historian of science. Life Truesdell was born in Los Angeles, California. After high school, he spent two years in Europe learning French, German, and Italian, and improving his Latin and Greek. His linguistic skills stood him in good stead in his later historical investigations. At Caltech he was deeply influenced by the teaching of Harry Bateman. In particular, a course in partial differential equations "taught me the difference between an ordinary good teacher and a great mathematician, and after that I never cared what grade I got in anything." He obtained a B.Sc. in mathematics and physics in 1941, and an MSc. in mathematics in 1942. In 1943, he completed a Ph.D. in mathematics at Princeton University. For the rest of the decade, the U.S. Navy employed him to do mechanics research. Truesdell taught at Indiana University 1950–61, where his students ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Noll
Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics, thermodynamics, and continuum mechanics. Biography Born in Berlin, Weimar Germany, Noll had his school education in a suburb of Berlin. In 1954, Noll earned a Ph.D. in Applied Mathematics from Indiana University Bloomington in 1954 under Clifford Truesdell. His thesis "On the Continuity of the Solid and Fluid States" was published both in '' Journal of Rational Mechanics and Analysis'' and in one of Truesdell's books. Noll thanks Jerald Ericksen for his critical input to the thesis. Noll has served as a visiting professor at the Johns Hopkins University, the University of Karlsruhe, the Israel Institute of Technology, the Institut National Polytechnique de Lorraine in Nancy, the University of Pisa, the University of Pavia, and the University of Oxford. In 2012 he became a fellow of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Elastic Material
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics. The fundamental assumptions of linear elasticity are infinitesimal strains — meaning, "small" deformations — and linear relationships between the components of stress and strain — hence the "linear" in its name. Linear elasticity is valid only for stress states that do not produce yielding. Its assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis. Mathematical formulation Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six infin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hooke's Law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. Hooke's equation holds (to some extent) in many other situations where an elastic body is deformed, such as wind blowing on a tall building, and a musician plucking a string of a guitar. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean. Hooke's law is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Hooke
Robert Hooke (; 18 July 16353 March 1703) was an English polymath who was active as a physicist ("natural philosopher"), astronomer, geologist, meteorologist, and architect. He is credited as one of the first scientists to investigate living things at microscopic scale in 1665, using a compound microscope that he designed. Hooke was an impoverished scientific inquirer in young adulthood who went on to become one of the most important scientists of his time. After the Great Fire of London in 1666, Hooke (as a surveyor and architect) attained wealth and esteem by performing more than half of the Boundary (real estate), property line surveys and assisting with the city's rapid reconstruction. Often vilified by writers in the centuries after his death, his reputation was restored at the end of the twentieth century and he has been called "England's Leonardo da Vinci, Leonardo [da Vinci]". Hooke was a Fellow of the Royal Society and from 1662, he was its first Curator of Experimen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Response Function
A linear response function describes the input-output relationship of a signal transducer, such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as susceptibility, impulse response or impedance; see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related. Mathematical definition Denote the input of a system by h(t) (e.g. a force), and the response of the system by x(t) (e.g. a position). Generally, the value of x(t) will depend not only on the present value of h(t), but also on past values. Approximately x(t) is a weighted sum of the previous values of h(t'), with the weights given by the linear response function \chi(t-t'): x(t) = \int_^t dt'\, \chi(t-t') h(t') + \cdots\,. The explicit te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-linear
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
