
	Where results make sense

Topic: Constitutive equations

Related Topics

Linear elasticity

Continuum mechanics

In the News (Tue 17 Nov 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	BME 456: Constitutive Equations: Elasticity
		Therefore, whenever we develop a constitutive equation to model a tissue, we need to balance the need to accurately model tissue behavior under the range of loading with the need to have a constitutive equation that is simple enough to in a numerical model and to experimentally measure all the constants in the constitutive equation.
		Therefore, constitutive equations consist of two major components: constants that must be fit to experimental data and measures of deformation, which may include small or finite deformation as well as the rate of deformation.
		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.
	www.engin.umich.edu /class/bme456/ch5consteqelasticity/bme332consteqelasticity.htm (6846 words)

	SFB298 - Unit 6
		For the description of history dependent material behavior of metals in addition to the partial differential equations derived from the balance laws of moment and mass constitutive equations expressing the dependence of the stress from the deformation history are used.
		In the case of metals, the formulation of these constitutive equations is based on the use of internal variables, and the constitutive equations consist of a system of nonlinear ordinary differential equations, called evolution equations, for these internal variables.
		It was also shown that for many constitutive equations, which are not of monotone type, the interior variables can be transformed so that the transformed variables satisfy equations of monotone type, which means that the boundary value problem for the transformed constitutive equations can be solved.
	www.mathematik.tu-darmstadt.de:8080 /ags/ag6/Projekte/SFB298/A1_en.html (403 words)

	Constitutive Equations (Site not responding. Last check: 2007-11-07)
		The equations for a continuum flow consist of up to five conservation laws in up to three space dimensions and time.
		The dependent variables are the mass density r, the fluid velocity v, the Cauchy stress tensor s, the body force g, the specific internal energy e, and the heat flux q.
		For example in an inviscid gas flow, the heat and entropy fluxes are zero, the body force is independent of the state variables (perhaps constant), and the Cauchy stress is a scalar tensor sij = -Pdij, where dij = 1 for I = j, 0 otherwise.
	www.ams.sunysb.edu /~shock/FTnotes/frontier/lecture01/sld013.htm (190 words)

	[No title]
		These field equations, variously known as the equations of motion, the equations of change, or simply the conservation equations, are nonlinear, partial differential equations that can be solved, in principle, when combined with the appropriate constitutive information1 and boundary conditions.
		The inherent nonlinearity of the conservation equations, which is due to convective transport of momentum, energy, and chemical species, is responsible for certain fluid mechanical phenomena, such as turbulence, that have no electrodynamic analog and that complicate solution of the conservation equations.
		When the equations can be rendered linear (e.g., when transport of the conserved quantities of interest is dominated by diffusion rather than convection) analytical solutions are often possible, provided the geometry of the domain and the boundary conditions are not too complicated.
	www.nae.edu /nae/bridgecom.nsf/BridgePrintView/MKEZ-5HUM3J?OpenDocument (2583 words)

	RMP Lecture Notes
		These may be supplemented by one or more constitutive equations that further define terms in the balance equations.
		Constitutive equations may be needed to define system properties such as density in terms of composition, temperature, pressure, etc.
		Equations of state are typically used to express vapor densities in terms of system temperature and pressure.
	www.cbu.edu /~rprice/lectures/modeleqn.html (750 words)

	I.1.2 Constitutive equations
		The conservation equations presented in section I.1.1 are not sufficient to determine the unknowns corresponding to the flow.
		Constitutive equations are introduced to relate the history of a material point to its extra-stress tensor
		The second normal stress difference may be related to the presence of a lower convected derivative in the constitutive equations.
	users.skynet.be /keyFE2/manual/I_1_2_Constitutive_equation.html (1595 words)

	Constitutive equation - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.tamu.edu) (Site not responding. Last check: 2007-11-07)
		More generally, in physics, a constitutive equation is a relation between two physical quantities (often tensors) that is specific to a material or substance, and does not follow directly from physical law.
		It is combined with other equations that do represent physical laws to solve some physical problem, like the flow of a fluid in a pipe, or the response of a crystal to an electric field.
		Some constitutive equations are simply phenomenological; others are derived from first principles.
	en.wikipedia.org.cob-web.org:8888 /wiki/Constitutive_equation (164 words)

	eFunda: Constitutive Law of Piezo Materials
		For mechanical problems, a constitutive equation describes how a material strains when it is stressed, or vice-versa.
		Constitutive equations exist also for electrical problems; they describe how charge moves in a (dielectric) material when it is subjected to a voltage, or vice-versa.
		Engineers are already familiar with the most common mechanical constitutive equation that applies for everyday metals and plastics.
	www.efunda.com /materials/piezo/piezo_math/background.cfm (221 words)

	Navier-Stokes Equations: Constitutive Relations (Site not responding. Last check: 2007-11-07)
		These relations are normally derived in courses on continuum mechanics from the fundamental principles and the constitutive assumptions described at the highlighted link.
		In the standard approach to the analysis of constitutive relations carried out in courses on continuum mechanics, (21)-(23) are a direct result of requiring that all solutions to the Navier-Stokes equations satisfy the Clausius-Duhem entropy inequality.
		The final form of the Navier-Stokes equations can be determined by substituting (10)-(11) in the local form of our balance equations (6)-(8).
	www.navier-stokes.net /nscr.htm (737 words)

	Thermoviscoelastic Constitutive Equations for Polycrystalline Ice (Site not responding. Last check: 2007-11-07)
		Linear and nonlinear viscoelastic constitutive equations for freshwater and sea ice are addressed.
		It is observed that a broad time spectrum representation is needed for an adequate characterization of the delayed elastic strain in the linear and nonlinear ranges of behavior.
		A specific nonlinear constitutive equation for uniaxial loading is proposed and successfully applied to the strain response of S2 saline ice subjected to multiple cycles of in-plane tensile loading and unloading.
	www.pubs.asce.org /WWWdisplay.cgi?9702191 (113 words)

	Navier-Stokes Equations: Constitutive Assumptions (Site not responding. Last check: 2007-11-07)
		All materials are expected to satisfy the fundamental conservation principles of physics, i.e., the balance equations seen on the "previous" page.
		As pointed out in the discussion of balance equations, the mathematical expression of the constitutive relations is the statement of the dependence of the stress tensor T, the heat flux q, the energy e, and the entropy s on the fields r(x,t), T(x,t), and v(x,t).
		When these constitutive assumptions are combined with certain universal physical principles, the familiar form of the Navier-Stokes equations is easily derived.
	www.navier-stokes.net /nsca.htm (672 words)

	G
		A constitutive equation is and equation that relates the response of a material to the strength of a perturbation, e.g.
		The "dog-bark" constitutive equation reflects only the observed behavior and does not imply a mechanism, which is quite complex in this case, between the perturbation and response.
		For some constitutive equations there is a mechanistic understanding of the relationship between the perturbation and the response.
	www.eng.uc.edu /~gbeaucag/Classes/Physics/DynChapter1html/Chapter1.html (4019 words)

	The Society of Rheology: 69th Annual Meeting (Oct 1997) Abstract of Paper
		During the development of these constitutive equations, the nonequilibrium Helmholtz free energy was expanded in a Frechet series in terms of the temperature and deformation histories, where the deformation was given in terms of right Cauchy-Green deformation tensor C for solids and the relative reight Cauchy-Green deformation tensor Ct for fluids.
		Objectivity and material symmetry only require that the constitutive functionals depend upon C or Ct; however, the constitutive functionals could just as well depend upon any functon of C or Ct. We have introduced generalized deformation tensors, which are arbitrary functions of C or Ct respectively, and then rederived the relevant constitutive equations.
		The predictions of the new constitutive equations have been compared with nonlinear viscoelastic data for an epoxy resin system, whre predictions using the generalized deformations are better able to describe the experimental data.
	www.rheology.org /sor97a/abstract.asp?PaperID=203 (313 words)

	eFunda: Mathematics Used in Piezo Materials
		For more on constitutive equations, visit the Constitutive Background page.
		It is possible to transform piezo constitutive data in one form to another form.
		To view the 4 piezoelectric constitutive equations and their mutual transformations, visit the Constitutive Transform page.
	www.efunda.com /materials/piezo/piezo_math/math_index.cfm (137 words)

	Constitutive Equations for Concrete in Failure State (Site not responding. Last check: 2007-11-07)
		For concrete, two simple three-dimensional failure conditions depending on the first invariant of the stress tensor as well as on the second and third invariant of the stress deviator tensor with two and four material parameters and a nonassociated flow rule are proposed in this paper.
		First, the basic equations like stress deviator tensor and invariants of the stress tensor and a stress deviator tensor are introduced.
		An example illustrates the usefulness of the proposed constitutive equations in the failure state.
	www.pubs.asce.org /WWWdisplay.cgi?8903479 (137 words)

	INTEGRATION ALGORITHMS (Site not responding. Last check: 2007-11-07)
		In this section, attention is focused on step 2 which may be regarded as the central problem of computational plasticity since it is the main role played by the constitutive equations in the computations.
		The purpose of these subroutines is the integration of the elastic-plastic constitutive equations.
		The returning mapping is achieved by integrating the nonlinear plastic evolution equations, and there are several ways this can be implemented (see e.g., Nguyen (1977); Simo and Ortiz (1985); Simo and Taylor (1986); Ortiz and Simo (1986); Simo and Hughes (1987)).
	www.cee.princeton.edu /~radu/papers/const/node8.html (789 words)

	[No title]
		The constitutive equations used to describe polymers and polymer flow are relatively complex.
		The appropriateness of such an equation depends on many factors such as the equation's accuracy in predicting data, its simplicity, the soundness of its mathematical basis, and the range of phenomena one wishes to address.
		The weighting of these factors is very subjective, resulting in a wide variety of constitutive equations over the years, many of which are still in use.
	www.win.tue.nl /oowi/08/8a/index_8a4.html (649 words)

	Constitutive equations for ionic transport in porous shales
		Constitutive equations for ionic transport in porous shales
		The constitutive coupled equations describing ionic transport in a porous shale are obtained at the scale of a representative elementary volume by volume averaging the local Nernst-Planck and Stokes equations.
		After upscaling the local equations the material properties entering the macroscopic constitutive equations are explicitly related to the porosity of the shale, its cation exchange capacity, and some textural properties such as the electrical cementation exponent entering Archie's law.
	www.agu.org /pubs/crossref/2004/2003JB002755.shtml (283 words)

	[No title]
		The wave-equation (4) is obtained from Maxwell’s equations in sourceless domains (2),including the constitutive equations, e.g., (3), by repeated substitution reduction, or equivalently, by equating to zero the symbolic determinant of the system.
		Inasmuch as in (16)-(18) no constitutive relations are incorporated, inverse formulas are obtained by interchanging primed and unprimed fields and coordinates, and replacing EMBED Equation.3 by EMBED Equation.DSMT4 , yielding EMBED Equation.DSMT4 (19) It seems interesting that in (18) the electric field is derived from the magnetic one, and vice-versa.
		In view of the identical structure of (52), and the corresponding (34) for the case of the cylinder, the same conclusions apply: Thus in the far field and for a limited-broadband spectrum, the pulse is characterized by the equation of motion (41) for the present case as well.
	www.ee.bgu.ac.il /~censor/diff.doc (4633 words)

	Navier-Stokes Equations: Constitutive Relations
		In the standard approach to the analysis of constitutive relations carried out in courses on continuum mechanics, (16)-(18) are a direct result of requiring that all solutions to the Navier-Stokes equations satisfy the Clausius-Duhem entropy inequality.
		If conditions (16) and (18) are combined with a simplified form of (5) and (6), it can be shown that they ensure that fluid friction always tends to oppose shear gradients in the flow and that heat flows from hot to cold.
		The final form of the Navier-Stokes equations can be determined by substituting (5)-(6) in our balance equations (1)-(3).
	www.eng.vt.edu /fluids/msc/ns/nscr.htm (702 words)

	Constitutive equations (Site not responding. Last check: 2007-11-07)
		A conservation law on its own does not usually give us a partial differential equation.
		Consider as an example the case of an incompressible fluid.
		Given that this is true, we then have
	www.soton.ac.uk /~jhr/MA361/node26.html (86 words)

	Constitutive Equations for Frictional Granular Flows (Part. 4)
		Typically in a computer model of granular flow we must solve a momentum equation in which one term will account for the momentum contribution from all the stresses within the flow.
		The last equation is again the famous Levy-von Mises flow rule, which is a direct consequence of the compressibility of the material in applying the Plastic Potential Theory.
		Now, as we have done in the previous paragraph, we want to use those results for solving the momentum equations in a computer model for instance.
	www.granular-volcano-group.org /frictional_constitutive_equations.html (1895 words)

	Geomaterials: Constitutive Equations and Modelling - F. Darve - Microsoft Reader eBook
		Geomaterials: Constitutive Equations and Modelling is a revised, English language version of the successful French book Manuel de Rheologie des Geomateriaux published in 1987.
		Geomaterials: Constitutive Equations and Modelling reviews four main subject areas: the mechanical behaviour of geomaterials; a study of the various constitutive equations; an illustration of the relationship linking macroscale behaviour of a sample to the microstructure of its material; computational aspects of both static and dynamic loading especially related to problems of soil-structure interaction.
		Geomaterials: Constitutive Equations and Modelling will enable computer code users to judge the capabilities and limitations of the constitutive relations incorporated in their codes, and will be of wide interest to students, researchers and engineers.
	www.ebookmall.com /ebook/80972-ebook.htm (823 words)

	Navier-Stokes Equations: Introduction
		The Navier-Stokes equations are the foundation of fluid mechanics and, strangely enough, are rarely recorded in their entirety.
		The main idea of these notes is to write an easily accessible, i.e., web-based, summary of these equations which are both correct and complete.
		Again, the idea here is to focus on the governing equations: the rest will follow easily once you have important stuff.
	www.eng.vt.edu /fluids/msc/ns/nsintro.htm (502 words)

	Graduate Courses in Aero/Mech Engineering (Site not responding. Last check: 2007-11-07)
		A first-year graduate course that introduces the subject of continuum mechanics, including derivation of fundamental equations and the development of constitutive equations characterizing the behavior of idealized materials.
		To introduce students to the topic of continuum mechanics, with analysis of the kinematic and mechanical behavior of materials modeled on the continuum assumption (where size scales far exceed molecular distances).
		This includes the derivation of fundamental equations, based on the classical laws of physics, and the development of constitutive equations characterizing the behavior of idealized materials, such as the elastic solid and the viscous fluid.
	www.nd.edu /~ame/graduate/ame.657.html (182 words)

	2 Constitutive Equations (Site not responding. Last check: 2007-11-07)
		Time dependent conservation law PDEs with flux ``down gradient'' are usually parabolic while steady state conservation laws with such flux are generally elliptic.
		These parabolic and elliptic conservation law equations will be the vehicles by which a number of important computational issues surrounding PDE problems will be introduced.
		7 reduces to the constant coefficient diffusion equation
	www.phy.ornl.gov /csep/CSEP/PDE2/NODE2A.html (96 words)

	A constitutive law for dense granular flows : Nature
		A continuum description of granular flows would be of considerable help in predicting natural geophysical hazards or in designing industrial processes.
		However, the constitutive equations for dry granular flows, which govern how the material moves under shear, are still a matter of debate
		For the two extreme regimes, constitutive equations have been proposed based on kinetic theory for collisional rapid flows
	www.nature.com /nature/journal/v441/n7094/abs/nature04801.html (341 words)

Try your search on: Qwika (all wikis)