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Flamant Solution
The Flamant solution provides expressions for the stresses and displacements in a linear elastic wedge loaded by point forces at its sharp end. This solution was developed by A. Flamant in 1892 by modifying the three-dimensional solution of Boussinesq. The stresses predicted by the Flamant solution are (in polar coordinates) : \begin \sigma_ & = \frac + \frac \\ \sigma_ & = 0 \\ \sigma_ & = 0 \end where C_1, C_3 are constants that are determined from the boundary conditions and the geometry of the wedge (i.e., the angles \alpha,\beta) and satisfy : \begin F_1 & + 2\int_^ (C_1\cos\theta + C_3\sin\theta)\,\cos\theta\, d\theta = 0 \\ F_2 & + 2\int_^ (C_1\cos\theta + C_3\sin\theta)\,\sin\theta\, d\theta = 0 \end where F_1,F_2 are the applied forces. The wedge problem is ''self-similar'' and has no inherent length scale. Also, all quantities can be expressed in the separated-variable form \sigma = f(r)g(\theta). The stresses vary a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flamant Solution Wedge
Flamant is the French for flamingo. It may refer to: * Flamant (company), a European interior decoration brand *The Dassault MD 315 Flamant, an aircraft * Flamant class patrol vessel, a type of ship *Flamant solution, the solution to a problem in 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 mec ... provided by A. Flamant in 1892 See also * Flamand (other) {{disambig ... [...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 ... [...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] |
Flamant Bounded Wedge
Flamant is the French for flamingo. It may refer to: * Flamant (company), a European interior decoration brand *The Dassault MD 315 Flamant, an aircraft * Flamant class patrol vessel, a type of ship *Flamant solution, the solution to a problem in 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 mec ... provided by A. Flamant in 1892 See also * Flamand (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress (mechanics)
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 or 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 strain is the measure of the deformation of the material. For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential e ... [...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] |
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 ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...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 or 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 strain is the measure of the deformation of the material. For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flamant Solution Half Plane
Flamant is the French for flamingo. It may refer to: * Flamant (company), a European interior decoration brand *The Dassault MD 315 Flamant, an aircraft * Flamant class patrol vessel, a type of ship *Flamant solution, the solution to a problem in 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 mec ... provided by A. Flamant in 1892 See also * Flamand (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-similar
__NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed. A time developing phenomenon is said to exhibit self-similarity if the numeri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
