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Michell Solution
The Michell solution is a general solution to the elasticity equations in polar coordinates ( r, \theta \,) developed by J. H. Michell. The solution is such that the stress components are in the form of a Fourier series in \theta \, . Michell showed that the general solution can be expressed in terms of an Airy stress function of the form : \begin \varphi(r,\theta) &= A_0~r^2 + B_0~r^2~\ln(r) + C_0~\ln(r) \\ & + \left(I_0~r^2 + I_1~r^2~\ln(r) + I_2~\ln(r) + I_3~\right) \theta \\ & + \left(A_1~r + B_1~r^ + B_1'~r~\theta + C_1~r^3 + D_1~r~\ln(r)\right) \cos\theta \\ & + \left(E_1~r + F_1~r^ + F_1'~r~\theta + G_1~r^3 + H_1~r~\ln(r)\right) \sin\theta \\ & + \sum_^ \left(A_n~r^n + B_n~r^ + C_n~r^ + D_n~r^\right)\cos(n\theta) \\ & + \sum_^ \left(E_n~r^n + F_n~r^ + G_n~r^ + H_n~r^\right)\sin(n\theta) \end The terms A_1~r~\cos\theta\, and E_1~r~\sin\theta\, define a trivial null state of stress and are ignored. Stress components T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These 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 infinitesimal strain- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the ''pole'', and the ray from the pole in the reference direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Henry Michell
John Henry Michell, FRS (26 October 1863 – 3 February 1940) was an Australian mathematician, Professor of Mathematics at the University of Melbourne. Early life Michell was the son of John Michell (pronounced Mitchell), a miner, and his wife Grace, ''née'' Rowse and was born at Maldon, Victoria. His parents had migrated from Devonshire in 1854. Educated at first at Maldon, he went to Wesley College, Melbourne, in 1877, where he won the Draper and Walter Powell scholarships. In 1881 he began the arts course at the University of Melbourne, and qualified for the B.A. degree at the end of 1883. He had an outstanding course, heading the list with first-class honours each year, and winning the final honour scholarship in mathematics and physics. Michell then went to the University of Cambridge, obtained a major scholarship at Trinity College, and was bracketed senior wrangler with three others in the first part of the mathematical tripos in 1887. In the second part of the tripos i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Airy Stress Function
In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation: :\sigma_=0\, where \sigma is the stress tensor, and the Beltrami-Michell compatibility equations: :\sigma_+\frac\sigma_=0 A general solution of these equations may be expressed in terms of the Beltrami stress tensor. Stress functions are derived as special cases of this Beltrami stress tensor which, although less general, sometimes will yield a more tractable method of solution for the elastic equations. Beltrami stress functions It can be shown that a complete solution to the equilibrium equations may be written as :\sigma=\nabla \times \Phi \times \nabla Using index notation: :\sigma_=\varepsilon_\varepsilon_\Phi_ : where \Phi_ is an arbitrary second-rank tensor field that is at least twice differentiable, and is known as the ''Beltram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress (physics)
In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elongation which is also known as deformation, like the stretching of an elastic band, it is called tensile stress. But, when the forces result in the compression of an object, it is called compressive stress. It results when forces like Tension (physics), tension or Compression (physics), compression act on a body. The greater this force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Therefore, stress is measured in newton per square meter (N/m2) or pascal (Pa). Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while deformation (mechanics)#Strain, strain is the measure of the deformation of the material. For example, when a solid vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylindrical Coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference direction ''(axis A)'', and the distance from a chosen reference plane perpendicular to the axis ''(plane containing the purple section)''. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The ''origin'' of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is variously called the ''cylindrical'' or ''longitudinal'' axis, to differentiate it from the ''polar axis'', which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. Other directions perpendicular to the longitudinal axis are called ''radial lines''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Displacement Field (mechanics)
A displacement field is an assignment of displacement vectors for all points in a region or body that is displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position. For example, a displacement field may be used to describe the effects of deformation on a solid body. Before considering displacement, the state before deformation must be defined. It is a state in which the coordinates of all points are known and described by the function: \vec_0: \Omega \to P where *\vec_0 is a placement vector *\Omega are all the points of the body *P are all the points in the space in which the body is present Most often it is a state of the body in which no forces are applied. Then given any other state of this body in which coordinates of all its points are described as \vec_1 the displacement field is the difference between two body states: \vec = \vec_1 - \vec_0 where \vec is a displacement fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These 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 infinitesimal strain- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson's Ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. For small values of these changes, \nu is the amount of transversal elongation divided by the amount of axial compression. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2–0.3. The ratio is named after the French mathematician and physicist Siméon Poisson. Origin Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shear Modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stackrel\ \frac = \frac = \frac where :\tau_ = F/A \, = shear stress :F is the force which acts :A is the area on which the force acts :\gamma_ = shear strain. In engineering :=\Delta x/l = \tan \theta , elsewhere := \theta :\Delta x is the transverse displacement :l is the initial length of the area. The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousand pounds per square inch (ksi). Its dimensional form is M1L−1T−2, replacing ''force'' by ''mass'' times ''acceleration''. Explanation The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law: * Young's modulus ''E'' describes the mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigid Body Displacement
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body is usually considered as a continuous distribution of mass. In the study of special relativity, a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near the speed of light. In quantum mechanics, a rigid body is usually thought of as a collection of point masses. For instance, molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors). Kinematics Linear and angular position The position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
