Hyperelastic material - Revision history

83.233.54.174: Undid revision 639065895 by 119.159.190.117 (talk)

2014-12-21T19:58:39Z

Undid revision 639065895 by 119.159.190.117 (talk)

← Older revision		Revision as of 21:58, 21 December 2014
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	~~{{Lowercase\|title=d'Alembert's Solution of the wave equation}}~~		Let me [http://citymama.com.ua/profile_info.php?ID=38588 navigate to these guys] first begin by [http://Www.news-Medical.net/health/What-is-an-STD-(Sexually-Transmitted-Disease).aspx introducing] myself. My title is Boyd Butts although it is not the name on my beginning certification. One of the issues she enjoys most is to do aerobics and now she is trying to make money with it. South Dakota is her at home std test beginning place but she needs to move because of her family. [http://Phpfoxdev.azurewebsites.net/index.php?do=/profile-44231/info/ http://Phpfoxdev.azurewebsites.net/index.php?do=/profile-44231/info] Hiring is her day occupation now but over the counter std test she's always needed her own company.<br><br>Also visit my blog post - [http://www.animecontent.com/user/D2456 http://www.animecontent.com/user/D2456]
	~~{{Refimprove\|date=September 2009}}~~
	~~{{Textbook\|date=June 2010}}~~
	In [~~[mathematics]], and specifically [[partial differential equations]], '''d´Alembert's formula''' is the general solution to the one-dimensional [[wave equation]]:~~
	:~~<math>u_{tt}-c^2u_{xx}=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x),<~~/~~math>~~
	~~for <math>-\infty < x<\infty,\,\, t>0<~~/~~math>~~. ~~It is named after the mathematician [[Jean le Rond d'Alembert]]~~.~~<ref>D'Alembert (1747) [http:/~~/~~books~~.~~google.com/books~~?id=~~lJQDAAAAMAAJ&pg=PA214#v=onepage&q&f=false "Recherches sur la courbe que forme une corde tenduë mise en vibration"] (Researches on the curve that a tense cord forms [when~~] ~~set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 214-219. See also: D'Alembert (1747)~~ [http://~~books~~.~~google~~.~~com~~/~~books?id=lJQDAAAAMAAJ&pg=PA220#v=onepage&q&f=false "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration"]~~ (~~Further researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 220~~-~~249. See also: D'Alembert (1750~~) ~~[http://books.google~~.~~com/books?id=m5UDAAAAMAAJ&pg=PA355#v=onepage&q&f=false "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration,"~~] ~~''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol~~. ~~6, pages 355-360~~.~~</ref>~~

	~~The [[method of characteristics\|characteristics]]~~ of the ~~PDE are <math>x\pm ct=\mathrm{const}\,</math>, so use the change of variables <math>\mu=x+ct, \eta=x-ct\,</math>~~ to ~~transform the PDE~~ to ~~<math>u_{\mu\eta}=0\,</math>~~. ~~The general solution~~ of ~~this PDE is <math>u(\mu,\eta) = F(\mu) + G(\eta)\,</math> where <math>F\,</math> and <math>G\,</math> are <math>C^1\,</math> functions~~. ~~Back in <math>x,t\,</math> coordinates,~~

	~~:<math>u(x,t)=F(x+ct)+G(x-ct)\,</math>~~
	:~~<math>u\,<~~/~~math> is <math>C^2\,<~~/~~math> if <math>F\,</math> and <math>G\,</math> are <math>C^2\,</math>~~.

	~~This solution <math>u\,<~~/~~math> can be interpreted as two waves with constant velocity <math>c\,</math> moving in opposite directions along the x-axis~~.

	~~Now consider this solution with the [[Cauchy data]] <math>u(x,0)~~=~~g(x), u_t(x,0)=h(x)\,<~~/~~math>.~~

	~~Using <math>u(x,0)=g(x)\,<~~/~~math> we get <math>F(x)+G(x)=g(x)\,<~~/~~math>.~~

	~~Using <math>u_t(x,0)=h(x)\,<~~/~~math> we get <math>cF'(x)-cG'(x)=h(x)\,<~~/~~math>~~.

	~~Integrate the last equation to get~~

	~~:<math>cF(x)-cG(x)=\int_{-\infty}^x h(\xi) \, d\xi + c_1~~.~~\,<~~/~~math>~~

	~~Now solve this system of equations to get~~

	~~:<math>F(x)~~ = ~~\frac{-1}{2c}\left(-cg(x)-\left(\int_{-\infty}^x h(\xi) \, d\xi +c_1 \right)\right)\,<~~/~~math>~~

	~~:<math>G(x) = \frac{-1}{2c}\left(-cg(x)+\left(\int_{~~-~~\infty}^x h(\xi) d\xi +c_1 \right)\right).\,<~~/~~math>~~

	~~Now, using~~

	~~:<math>u(x,t) = F(x+ct)+G(x-ct)\,</math>~~

	~~d´Alembert~~'s ~~formula becomes:~~

	:<~~math~~>~~u(x,t) = \frac{1}{2}\left[g(x-ct) + g(x+ct)\right] + \frac{1}{2c} \int_{x-ct}^{x+ct} h(\xi) \, d\xi.~~<~~/math~~>

	~~==See also==~~
	*[[D'Alembert operator]]

	~~== Notes ==~~

	~~{{reflist}}~~

	~~==External links==~~
	*[http://www.~~exampleproblems~~.com/~~wiki~~/~~index.php~~/~~PDE27 An example] of solving a nonhomogeneous wave equation from~~ www.~~exampleproblems~~.com

	~~[[Category:Partial differential equations]~~]

130.88.123.93: /* Consistency conditions for isotropic hyperelastic models */

2013-10-23T10:16:26Z

Consistency conditions for isotropic hyperelastic models

← Older revision		Revision as of 12:16, 23 October 2013
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	~~The name~~ of the ~~writer is Numbers but it~~'s ~~not~~ the ~~most masucline title out there~~. ~~To play baseball~~ is the ~~hobby he will by no means quit performing~~. ~~South Dakota~~ is where me and ~~my spouse live~~. ~~Supervising~~ is ~~my profession~~.<br><br>~~My page;~~ [~~http:~~//~~Test~~.~~Ithink~~-Now.~~org~~/~~content~~/~~think~~-~~f83d96d08fe71e25bf80a6cb59048a18 over the counter std test~~]		{{Lowercase\|title=d'Alembert's Solution of the wave equation}}
			{{Refimprove\|date=September 2009}}
			{{Textbook\|date=June 2010}}
			In [[mathematics]], and specifically [[partial differential equations]], '''d´Alembert's formula''' is the general solution to the one-dimensional [[wave equation]]:
			:<math>u_{tt}-c^2u_{xx}=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x),</math>
			for <math>-\infty < x<\infty,\,\, t>0</math>. It is named after the mathematician [[Jean le Rond d'Alembert]].<ref>D'Alembert (1747) [http://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA214#v=onepage&q&f=false "Recherches sur la courbe que forme une corde tenduë mise en vibration"] (Researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 214-219. See also: D'Alembert (1747) [http://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA220#v=onepage&q&f=false "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration"] (Further researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 220-249. See also: D'Alembert (1750) [http://books.google.com/books?id=m5UDAAAAMAAJ&pg=PA355#v=onepage&q&f=false "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration,"] ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 6, pages 355-360.</ref>

			The [[method of characteristics\|characteristics]] of the PDE are <math>x\pm ct=\mathrm{const}\,</math>, so use the change of variables <math>\mu=x+ct, \eta=x-ct\,</math> to transform the PDE to <math>u_{\mu\eta}=0\,</math>. The general solution of this PDE is <math>u(\mu,\eta) = F(\mu) + G(\eta)\,</math> where <math>F\,</math> and <math>G\,</math> are <math>C^1\,</math> functions. Back in <math>x,t\,</math> coordinates,

			:<math>u(x,t)=F(x+ct)+G(x-ct)\,</math>
			:<math>u\,</math> is <math>C^2\,</math> if <math>F\,</math> and <math>G\,</math> are <math>C^2\,</math>.

			This solution <math>u\,</math> can be interpreted as two waves with constant velocity <math>c\,</math> moving in opposite directions along the x-axis.

			Now consider this solution with the [[Cauchy data]] <math>u(x,0)=g(x), u_t(x,0)=h(x)\,</math>.

			Using <math>u(x,0)=g(x)\,</math> we get <math>F(x)+G(x)=g(x)\,</math>.

			Using <math>u_t(x,0)=h(x)\,</math> we get <math>cF'(x)-cG'(x)=h(x)\,</math>.

			Integrate the last equation to get

			:<math>cF(x)-cG(x)=\int_{-\infty}^x h(\xi) \, d\xi + c_1.\,</math>

			Now solve this system of equations to get

			:<math>F(x) = \frac{-1}{2c}\left(-cg(x)-\left(\int_{-\infty}^x h(\xi) \, d\xi +c_1 \right)\right)\,</math>

			:<math>G(x) = \frac{-1}{2c}\left(-cg(x)+\left(\int_{-\infty}^x h(\xi) d\xi +c_1 \right)\right).\,</math>

			Now, using

			:<math>u(x,t) = F(x+ct)+G(x-ct)\,</math>

			d´Alembert's formula becomes:

			:<math>u(x,t) = \frac{1}{2}\left[g(x-ct) + g(x+ct)\right] + \frac{1}{2c} \int_{x-ct}^{x+ct} h(\xi) \, d\xi.</math>

			==See also==
			*[[D'Alembert operator]]

			== Notes ==

			{{reflist}}

			==External links==
			*[http://www.exampleproblems.com/wiki/index.php/PDE27 An example] of solving a nonhomogeneous wave equation from www.exampleproblems.com

			[[Category:Partial differential equations]]

en>Bbanerje: /* Cauchy stress */

2012-02-11T04:25:02Z

Cauchy stress

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