Barnes G-function - Revision history

en>G. c. greubel: In the section "Related to the log-gamma integral" the gamma function is missing a term. It reads "\frac{1}{\Gamma(z)}= z e^{\gamma}..." but should be "\frac{1}{\Gamma(z)}= z e^{\gamma z}.." This is the edit currently being made.

2014-04-21T21:43:51Z

In the section "Related to the log-gamma integral" the gamma function is missing a term. It reads "\frac{1}{\Gamma(z)}= z e^{\gamma}..." but should be "\frac{1}{\Gamma(z)}= z e^{\gamma z}.." This is the edit currently being made.

← Older revision		Revision as of 23:43, 21 April 2014
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	A '''Kelvin–Voigt material''', also called a '''Voigt material''', is a [[viscoelastic]] material having the properties both of [[elasticity (physics)\|elasticity]] and [[viscosity]]. It is named after the British physicist and engineer [[William Thomson, 1st Baron Kelvin]] and after German physicist [[Woldemar Voigt]].

	~~== Definition ==~~

	~~The Kelvin–Voigt model, also called the Voigt model, can be represented by a purely viscous damper and purely elastic spring connected in parallel as shown in the picture~~.		Hello family. Let mе introduce myself. Ӏ am Coral Gehl bսt my husband doesn't ɑppreciate ɑt many of. One of a goߋd thingѕ օn the earth foг her is cycling and she'll ƅе starting somethіng else along it will. Minnesota is wherе her house is but she needs to transfer ƅecause օf her child. Interviewing іs wherе his primary income is taken from. Shе's not [http://Browse.Deviantart.com/?qh=&section=&global=1&q=capable capable] at design ѡe might want to check ɦer website: http://web01.x-stream.biz/UserProfile/tabid/57/userId/181538/language/en-US/Default.aspx<br><br>my webpage - [http://web01.x-stream.biz/UserProfile/tabid/57/userId/181538/language/en-US/Default.aspx Where to play slot Machines]

	~~[[Image:Kelvin Voigt diagram~~.~~svg\|right\|frame\| Schematic representation~~ of ~~Kelvin–Voigt model~~.]]

	~~If we connect these two elements in series we get a model~~ of a ~~[[Maxwell material]].~~

	~~Since~~ the ~~two components of the model are arranged in parallel, the strains in each component are identical:~~

	~~:<math> \varepsilon_\text{Total} = \varepsilon_D = \varepsilon_S. </math>~~

	~~Similarly, the total stress~~ will ~~be the sum of the stress in each component:~~

	~~:<math> \sigma_\text{Total} = \sigma_D + \sigma_S~~. ~~</math>~~

	~~From these equations we get that in a Kelvin–Voigt material, [[Stress (physics)\|stress]] σ, [[Strain (materials science)\|strain]] ε and their rates of change with respect~~ to ~~time~~ '~~'t'' are governed by equations of the form~~:

	~~:<math>\sigma (t) = E \varepsilon(t) + \eta \frac {d\varepsilon(t)} {dt},<~~/~~math>~~

	~~where ''E'' is a modulus of elasticity and <math>\eta<~~/~~math> is the [[viscosity]]~~. ~~The equation can be applied either to the [[shear stress]] or [[normal stress]] of a material~~.

	== ~~Effect of a sudden stress~~ ==

	~~If we suddenly apply some constant stress <math>\sigma_0</math>~~ to ~~Kelvin–Voigt material, then the deformations would approach the deformation for the pure elastic material <math> \sigma_0/E</math> with the difference decaying exponentially~~:

	:~~<math>\varepsilon(t)=\frac {\sigma_0}{E} (1-e^{-\lambda t}), <~~/~~math>~~

	~~where ''t'' is time and <math>\lambda</math> the '''rate of relaxation''' <math> \lambda=\frac {E}{\eta} </math>.~~

	~~If we would free the material at time <math>t_1<~~/~~math>, then the elastic element would retard the material back until the deformation becomes zero~~. ~~The retardation obeys the following equation:~~

	~~:<math>\varepsilon(t>t_1)=\varepsilon(t_1)e^{-\lambda (t~~-~~t_1)}~~. </~~math>~~

	~~The picture shows the dependence of the dimensionless deformation <math>\frac {E\varepsilon(t)} {\sigma_0} <~~/~~math>~~
	~~on dimensionless time <math>\lambda t<~~/~~math>. In the picture the stress on the material is loaded at time <math>t=0<~~/~~math>, and released at the later dimensionless time <math>t_1^*=\lambda t_1<~~/~~math>.~~
	~~[[Image:Kelvin deformation 2.png\|right\|frame\|Dependence of dimensionless deformation~~
	~~upon dimensionless time under constant stress\|Dependence of dimensionless deformation~~
	~~upon dimensionless time under constant stress]]~~

	~~Since all the deformation is reversible (though not suddenly) the Kelvin–Voigt material is a [[solid]].~~

	~~The Voigt model predicts creep more realistically than the Maxwell model, because in the infinite time limit the strain approaches a constant:~~

	~~:<math>\lim_{t\to\infty}\varepsilon = \frac{\sigma_0}{E},<~~/~~math>~~

	while a Maxwell model predicts a linear relationship between strain and time, which is most often not the case. Although the Kelvin–Voigt model is effective for predicting creep, it is not good at describing the relaxation behavior after the stress load is removed.

	~~== Dynamic modulus ==~~

	~~The complex [[dynamic modulus]] of the Kelvin–Voigt material is given by:~~

	~~:<math>E^\star ( \omega ) = E + i \eta \omega~~. <~~/math~~>

	~~Thus, the real and imaginary components of the dynamic modulus are:~~

	:<~~math~~>~~E_1 = \Re~~ [~~E( \omega )] = E, </math>~~
	:~~<math>E_2 = \Im [E( \omega )] = \eta \omega. <~~/~~math>~~

	~~Note that <math>E_1</math> is constant, while <math>E_2</math> is directly proportional to frequency (where the apparent viscosity, <math>\eta<~~/~~math>, is the constant of proportionality)~~.

	~~== References ==~~

	* Meyers and Chawla (1999): Section 13.10 of Mechanical Behaviors of Materials, ''Mechanical behavior of Materials'', 570–580. Prentice Hall, Inc.
	* http://~~stellar.mit.edu~~/S/~~course~~/3/~~fa06~~/~~3.032~~/~~index~~.~~html~~

	~~== See also ==~~

	*[[Maxwell material]]
	*[[Standard Linear Solid Material]]
	*[[Generalized Maxwell material]]

	~~{{DEFAULTSORT:Kelvin-Voigt Material}}~~
	~~[[Category:Non-Newtonian fluids]]~~
	~~[[Category:Materials science]~~]

en>Dream Weaver1975: added new chapter - "Taylor series expansion"

2013-12-10T01:41:47Z

added new chapter - "Taylor series expansion"

← Older revision		Revision as of 03:41, 10 December 2013
Line 1:		Line 1:
	~~My name~~ is ~~Edgardo Chiles~~. ~~I life~~ in ~~Sakskobing (Denmark)~~.<br><br>~~Also visit my site~~ [~~http~~://~~ferrymarthasvineyard~~.~~com~~/~~node~~/~~male~~-~~enhancement~~-~~pills~~-to-~~increase~~-~~male-stamina FIFA 15 coin hack~~]		A '''Kelvin–Voigt material''', also called a '''Voigt material''', is a [[viscoelastic]] material having the properties both of [[elasticity (physics)\|elasticity]] and [[viscosity]]. It is named after the British physicist and engineer [[William Thomson, 1st Baron Kelvin]] and after German physicist [[Woldemar Voigt]].

			== Definition ==

			The Kelvin–Voigt model, also called the Voigt model, can be represented by a purely viscous damper and purely elastic spring connected in parallel as shown in the picture.

			[[Image:Kelvin Voigt diagram.svg\|right\|frame\| Schematic representation of Kelvin–Voigt model.]]

			If we connect these two elements in series we get a model of a [[Maxwell material]].

			Since the two components of the model are arranged in parallel, the strains in each component are identical:

			:<math> \varepsilon_\text{Total} = \varepsilon_D = \varepsilon_S. </math>

			Similarly, the total stress will be the sum of the stress in each component:

			:<math> \sigma_\text{Total} = \sigma_D + \sigma_S. </math>

			From these equations we get that in a Kelvin–Voigt material, [[Stress (physics)\|stress]] σ, [[Strain (materials science)\|strain]] ε and their rates of change with respect to time ''t'' are governed by equations of the form:

			:<math>\sigma (t) = E \varepsilon(t) + \eta \frac {d\varepsilon(t)} {dt},</math>

			where ''E'' is a modulus of elasticity and <math>\eta</math> is the [[viscosity]]. The equation can be applied either to the [[shear stress]] or [[normal stress]] of a material.

			== Effect of a sudden stress ==

			If we suddenly apply some constant stress <math>\sigma_0</math> to Kelvin–Voigt material, then the deformations would approach the deformation for the pure elastic material <math> \sigma_0/E</math> with the difference decaying exponentially:

			:<math>\varepsilon(t)=\frac {\sigma_0}{E} (1-e^{-\lambda t}), </math>

			where ''t'' is time and <math>\lambda</math> the '''rate of relaxation''' <math> \lambda=\frac {E}{\eta} </math>.

			If we would free the material at time <math>t_1</math>, then the elastic element would retard the material back until the deformation becomes zero. The retardation obeys the following equation:

			:<math>\varepsilon(t>t_1)=\varepsilon(t_1)e^{-\lambda (t-t_1)}. </math>

			The picture shows the dependence of the dimensionless deformation <math>\frac {E\varepsilon(t)} {\sigma_0} </math>
			on dimensionless time <math>\lambda t</math>. In the picture the stress on the material is loaded at time <math>t=0</math>, and released at the later dimensionless time <math>t_1^*=\lambda t_1</math>.
			[[Image:Kelvin deformation 2.png\|right\|frame\|Dependence of dimensionless deformation
			upon dimensionless time under constant stress\|Dependence of dimensionless deformation
			upon dimensionless time under constant stress]]

			Since all the deformation is reversible (though not suddenly) the Kelvin–Voigt material is a [[solid]].

			The Voigt model predicts creep more realistically than the Maxwell model, because in the infinite time limit the strain approaches a constant:

			:<math>\lim_{t\to\infty}\varepsilon = \frac{\sigma_0}{E},</math>

			while a Maxwell model predicts a linear relationship between strain and time, which is most often not the case. Although the Kelvin–Voigt model is effective for predicting creep, it is not good at describing the relaxation behavior after the stress load is removed.

			== Dynamic modulus ==

			The complex [[dynamic modulus]] of the Kelvin–Voigt material is given by:

			:<math>E^\star ( \omega ) = E + i \eta \omega. </math>

			Thus, the real and imaginary components of the dynamic modulus are:

			:<math>E_1 = \Re [E( \omega )] = E, </math>
			:<math>E_2 = \Im [E( \omega )] = \eta \omega. </math>

			Note that <math>E_1</math> is constant, while <math>E_2</math> is directly proportional to frequency (where the apparent viscosity, <math>\eta</math>, is the constant of proportionality).

			== References ==

			* Meyers and Chawla (1999): Section 13.10 of Mechanical Behaviors of Materials, ''Mechanical behavior of Materials'', 570–580. Prentice Hall, Inc.
			* http://stellar.mit.edu/S/course/3/fa06/3.032/index.html

			== See also ==

			*[[Maxwell material]]
			*[[Standard Linear Solid Material]]
			*[[Generalized Maxwell material]]

			{{DEFAULTSORT:Kelvin-Voigt Material}}
			[[Category:Non-Newtonian fluids]]
			[[Category:Materials science]]

en>EmausBot: r2.7.2+) (Robot: Modifying ru:G-функция Барнса

2012-08-26T08:36:41Z

r2.7.2+) (Robot: Modifying ru:G-функция Барнса

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