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Topic: Linear elasticity

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Constitutive equations

Price elasticity of demand

Indifference curve

	EN224: LINEAR ELASTICITY
		We proceed to prove several useful theorems which follow as a consequence of the structure of the field equations of linear elasticity, and which are useful in interpreting or constructing solutions to boundary value problems.
		The proof is trivial: it follows from the linearity and homogeneity of the field equations.
		Boussinesq: An equilibrated system of external forces applied to an elastic body, all of the points of application lying within a given sphere, produces deformations of negligible magnitude at distances from the sphere which are sufficiently large compared to its radius.
	www.engin.brown.edu /courses/en224/superpos/superpos.html (1885 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		Linear elasticity relies upon the continuum hypothesis and is applicable at macroscopic (and sometimes microscopic) length scales.
		Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics.
		Linear elasticity is therefore used extensively in structural analysis and engineering design, often through the aid of finite element analysis.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Linear_elasticity (404 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		Elasticity is a branch of physics which studies the properties of elastic materials.
		The elastic regime is characterized by a linear relationship between stress and strain, denoted linear elasticity.
		Above a certain stress known as the elastic limit or the yield strength of an elastic material, the relationship between stress and strain becomes nonlinear.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=elasticity_(physics) (390 words)

	BME 456: Constitutive Equations: Elasticity
		The use of linear elastic constitutive equations is typically restricted for use with bone tissue, since bone tissue is the only tissue that consistently operates in the small strain regime and exhibits a linear relationship between stress and strain.
		However, in the linear elastic case, we have a linear proportion between the Cauchy stress tensor and the small deformation strain tensor.
		For small deformation linear elasticity the tangent elastic constants are equivalent to the overall elastic properties since the stress-strain curve is linear and stiffness therefore does not change as deformation changes.
	www.engin.umich.edu /class/bme456/ch5consteqelasticity/bme332consteqelasticity.htm (6846 words)

	Elasticity - LoveToKnow 1911 (Site not responding. Last check: )
		The behaviour of an elastic solid body, strained within the limits of its elasticity, is entirely determined by the constants E and a if the body is isotropic, that is to say, if it has the same quality in all dir.
		The limits of perfect elasticity as regards change of shape, on the other hand, are very low, if they exist at all, for glasses and other hard, brittle solids; but a class of metals including copper, brass, steel, platinum are very perfectly elastic as regards distortion, provided that the distortion is not too great.
		Whatever view may ultimately be adopted as to the relation between the conditions of safety of a structure and the state of stress or strain in it, the calculation of this state by means of the theory or by experimental means (as in § 18) cannot be dispensed with.
	www.1911encyclopedia.org /Elasticity (14477 words)

	NationMaster - Encyclopedia: Modulus of elasticity (Site not responding. Last check: )
		In solid mechanics, Young's modulus (also known as the modulus of elasticity or elastic modulus) is a measure of the Stiffness of a given material.
		The SI unit of modulus of elasticity is the Pascal.
		Elastic modulus is proportional to the number of crosslinks, although not in a strictly linear fashion.
	www.nationmaster.com /encyclopedia/Modulus-of-elasticity (641 words)

	Project Description
		For many years one of the most difficult open problems of non-linear elasticity theory has been the use of global continuation methods (via degree theory) to study the governing system of partial differential equations of three-dimensional models, c.f.
		For the three dimensional mixed problem of nonlinear elasticity the required spectral estimates were obtained in [11] and together with the estimates of [1] for elliptic systems, Healey and Simpson were able to apply the generalized degree to get the existence of a global branch of solutions of this problem.
		These linear problems correspond to elliptic systems of partial differential equations on a domain determined by the geometry of the physical problem.
	mate.uprh.edu /~pnm/regconf/prodescp.htm (844 words)

	AEM 5503 Information
		Introduction to the theory of elasticity, with emphasis on linear elasticity.
		Linear and nonlinear strain measures, the boundary value problem for linear elasticity, plane problems in linear elasticity, and three dimensional problems in linear elasticity.
		Linear elasticity has also been found to be able to describe atomic motions of dislocations in ductile metals.
	www.aem.umn.edu /cgi-bin/courses/noauth/class-summary?QFCHK=0&class=32&return=1 (279 words)

	Evidence of Non-Linear Elasticity
		This observation suggests that the elastic moduli of the crust in tension and in compression are different due to the presence of cracks in the crust at shallow depth.
		Linear models of dislocation in an elastic half space (1) are widely used to represent earthquake faults and generally provide a satisfactory fit to near and far field displacement data obtained with conventional geodetic techniques (2).
		Given the symmetry of the fault geometry and slip distribution of the model, the along-strike distribution of the fault-parallel, horizontal component of the displacement vector is symmetric with respect to the auxiliary plane and the distribution of the vertical component is anti-symmetric.
	www-radar.jpl.nasa.gov /sect323/InSar4crust/manyi/science_paper.html (3956 words)

	Linear Visco-Elasticity and the Recruitment Index
		This linear visco-elastic behaviour requires a highly specialised sequence of failure and recruitment of spring-enet mechanical units which are used to incorporate the micro-failure concept into the model (figure (1)).
		Therefore, it seems that the case of linear visco-elasticity for the micro-failure model is a highly specialised condition which sits between these infinitesimally elastic and infinitely viscous states.
		The precise characteristics of this solid is a 50 N/mm elastic spring in parallel with a series arrangement of a second 50 N/mm spring and a linear dashpot with a relaxation constant of 10 seconds (1).
	dspace.dial.pipex.com /jegan/mechmods/qspaper.shtml (1629 words)

	The Concept of Elasticity
		So, even though the formula says that the price elasticity of demand is negative, we would say the elasticity of demand is 1.5 in the first example and 0.67 in the second.
		Although the exact formula for calculating an elasticity is useful for theory, in practice economists usually calculate an approximation called the symmetric midpoint formula elasticity.
		Elasticity measures the magnitude of an economic effect in percentages.
	instruct1.cit.cornell.edu /courses/econ101-dl/lecture-elasticity.html (810 words)

	P&A UBC -- Biophysics Seminars:Smith (Site not responding. Last check: )
		Assuming that short-range cholesterol interactions simply “squeeze” lipid acyl chains, we have used the elastic stretch of pure phosphatidyl-choline (PC) vesicles measured at high precision to establish a transducer relation for quantifying the free energy of lipid:cholesterol interactions in mixed PC:cholesterol vesicles.
		Rapidly stressing vesicles by micropipette aspiration to large tensions, we have exposed the non-linear softening in elasticity expected from an inverse dependence of lipid surface pressure on area.
		Based on the extensional elasticity of polymer chains, we have developed an anharmonic model for lipid surface pressure that agrees with the hyper-stretch response of PC vesicles and correlates well with published monolayer surface pressure-area isotherms.
	www.physics.ubc.ca /events/biogroup/Smith2006-10-27.html (234 words)

	Linear elasticity - Term Explanation on IndexSuche.Com (Site not responding. Last check: )
		The linear theory_of_elasticity models the macroscopic mechanical properties of solids assuming "small" deformations.
		Linear elastodynamics is based on three tensor equations:
		In isotropic media, the elasticity tensor has the form : C_{ijkl} = \kappa \, \delta_{ij}\, \delta_{kl} +\mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\, \delta_{ij}\,\delta_{kl}) where \kappa is ''incompressibility'', and \mu is ''rigidity''.
	www.indexsuche.com /Linear_elasticity.html (256 words)

	MA7011 Applied Nonlinear Elasticity
		The main aim of the theory of elasticity is to provide a description of deformed bodies in terms of certain preferential configurations.
		The primary objective of this course is to illustrates the theory of elasticity with a pervading emphasis on nonlinear aspects and the interplay between mathematics and engineering sciences.
		Principles of linear and angular momentum, the stress tensor, Piola-Kirchhoff tensors, Cauchy stress, principal stresses, stress invariants.
	www.mcs.le.ac.uk /Modules/MA/MA7011.html (605 words)

	Non-linear spring (US5062619)
		The elastic portions are disposed at different pitches along the length so that some of them have different elasticity from the others.
		The elasticity between some portions and the others may be varied by way of changing an inner diameter of a through hole formed in the spring.
		a plurality of elastic portions respectively defined between adjacent slits, wherein the thicknesses of the elastic portions in a first set is different from the thicknesses of the elastic portions in a second set to provide the spring with a non-linear elasticity characteristic.
	www.delphion.com /details?pn=US05062619__ (271 words)

	Elasticity, Total Revenue, and Linear Demand
		Elasticity is a measure of the responsiveness of one variable to changes in some other variable.
		We are interested in the price elasticity of demand, which measures the response of the quantity demanded to a change in price.
		Formally, the price elasticity of demand, which we will call E, is the ratio of the percent change in quantity over the percent change in price.
	www.econtools.com /jevons/java/elastic/Elasticity.html (786 words)

	eFunda: Linear Elastic Fracture Mechanics (LEFM)
		Linear Elastic Fracture Mechanics (LEFM) first assumes that the material is isotropic and linear elastic.
		In Linear Elastic Fracture Mechanics, most formulas are derived for either plane stresses or plane straines, associated with the three basic modes of loadings on a cracked body: opening, sliding, and tearing.
		Based on linear elasticity theories, the stress field near a crack tip is a function of the location, the loading conditions, and the geometry of the specimen or object.
	www.efunda.com /formulae/solid_mechanics/fracture_mechanics/fm_lefm.cfm (320 words)

	Brian P. Tighe (Site not responding. Last check: )
		For a system as simple as a 2D network of linear springs, there will be a geometric nonlinearity if the springs have nonzero equilibrium length.
		Linear elasticity is the result of throwing out all but the quadratic terms.
		But in nonlinear elasticity theory, when you've included some of the higher order terms in the free energy, the differences become apparent.
	www.phy.duke.edu /~btighe/elasticity2.html (196 words)

	Elasticity
		This is a first year graduate textbook in Linear Elasticity.
		Most of the text should be readily intelligible to a reader with an undergraduate background of one or two courses in elementary Mechanics of Materials and a rudimentary knowledge of partial differentiation.
		Emphasis is placed on engineering applications of elasticity and examples are generally worked through to final expressions for the stress and displacement fields in order to explore the engineering consequences of the results.
	www.netcomposites.com /netcommerce_features.asp?731 (270 words)

	Gamasutra - Features "2D Surface Deformation"
		In the relaxed state elastic forces between vertices are equal to zero.
		The resulted elastic forces counter the displaced vertex motion and try to return the vertex to its original location.
		Greater elasticity corresponds to faster deformation and stiffer body, while low elasticity corresponds to softer bodies.
	www.gamasutra.com /features/20000216/deformation_01.htm (597 words)

	Rigid Body Collisions - Physics Simulation
		If elasticity is less than 1, and gravity is greater than zero, then the objects eventually settle onto the floor.
		We now use a standard formula for the velocity of an arbitrary point on a rotating and translating rigid body to get the pre-collision velocities of the points of collision (which is the point P on each body).
		Let e be the elasticity of the collision, having a value between 0 (inelastic) and 1 (perfectly elastic).
	www.myphysicslab.com /collision.html (2985 words)

	NLAbstracts3
		The equivalent linear method has been developed in spherical coordinates to simulate wave propagation between the cavity radius and the elastic radius.
		Damage mechanics are used to describe a type of threshold nonlinear effect, the growth of microcracks caused by a passing elastic wave.
		These tests indicate that the measured pore compressibilities for various deviatoric states of stress is not a simple function of the mean stress, as might be expected if the rock was an isotropic, linear elastic matrix permeated with voids of various sizes and shapes.
	www.science.doe.gov /bes/geo/Publications/Conferences/Nonlinear/body_nlabstracts3.html (2665 words)

	Solid mechanics in TutorGig Encyclopedia
		This region of deformation is known as the linearly elastic region.
		A spring obeying Hooke's law is a one-dimensional linear version of a general elastic body.
		a material that is elastic, but also has damping: on loading, as well as on unloading, some work has to be made against the damping effects.
	www.tutorgig.com /ed/Solid_mechanics (440 words)

	5.2.1 Elasticity Theory, Energy and Forces
		The theory of elasticity is quite difficult just for simple homogeneous media (no crystal), and even more difficult for crystals with dislocations - because the dislocation core cannot be treated with the linear approximations always used when the math gets tough.
		The first element of elasticity theory is to define the displacement field u(x,y,z), where u is a vector that defines the displacement of atoms or, since we essentially consider a continuum, the displacement of any point P in a strained body from its original (unstrained) position to the position P' in the strained state.
		A material under strain contains elastic energy - it is just the sum of the energy it takes to move atoms off their equilibrium position at the bottom of the potential well from the binding potential.
	www.tf.uni-kiel.de /matwis/amat/def_en/kap_5/backbone/r5_2_1.html (1682 words)

	A Hybrid Elastic Model allowing Real-Time Cutting, Deformations and Force-Feedback for Surgery Training and Simulation ...
		Abstract: We propose three different physical models based on linear elasticity theory and finite elements modeling that are well-suited for surgery simulation.
		Furthermore, it can be extended to anisotropic linear elasticity [15] which allows to model ber reinforced materials, very common within...
		We use the Finite Element Method with linear tetrahedral elements and mass lumping in a Newtonian di erential equation with an explicit...
	citeseer.ist.psu.edu /397582.html (766 words)

	Alibris: Elasticity
		A history of the theory of elasticity and of the strength of materials, from Galilei to Lord Kelvin.
		The book is the first to present the topic of linear elasticity in mathematical terms that will be familiar to anyone with a grounding in fluid mechanics.
		This highly regarded hardcover engineering manual is mainly concerned with three important aspects of elasticity theory: finite elastic deformations, complex variable methods for two-dimensional problems for both isotropic and aeolotropic bodies, and shell theory.
	www.alibris.com /search/books/subject/Elasticity (767 words)

	Apel, Thomas; Sändig, Anna-Margarete; Solov'ev, Sergey I. : Computation of 3D vertex singularities for linear ...
		This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes.
		The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains.
		Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions.
	www.mathematik.tu-chemnitz.de /preprint/2001/SFB393_33.html (354 words)

	AHPCRC Workshop on Recent Advances and State-of-the-art in Discontinuous Galerkin Methods in Computational Structural ... (Site not responding. Last check: )
		As an example, a DG method for linear elasticity is proposed and studied.
		In the present work, we obtain a discretization of the equations for linear elasticity by formulating a discrete variational principle.
		We begin by analyzing the linear elasticity problem, with an eye toward a formulation for nonlinear elasticplastic problems and cohesive elements.
	www.ahpcrc.org /conferences/DG-UM/abstracts.html (2259 words)

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