Unit sphere - Revision history

en>Cydebot: Robot - Moving category One to :Category:1 (number) per CFD at Wikipedia:Categories for discussion/Log/2014 November 19.

2014-11-28T22:02:03Z

Robot - Moving category One to Category:1 (number) per CFD at Wikipedia:Categories for discussion/Log/2014 November 19.

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	'~~''Lame's stress ellipsoid''' (Figure coming)~~ is ~~an alternative to~~ [~~[Mohr's circle]~~] for ~~the graphical representation~~ of ~~the [[Stress (mechanics)#Stress tensor\|stress state at~~ a ~~point]]~~. ~~The surface~~ of ~~the~~ [~~[ellipsoid~~]~~] represents~~ the ~~locus~~ of the ~~endpoints~~ of ~~all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints~~ of ~~all stress vectors at a given point in~~ the ~~continuum body lie on~~ the ~~stress ellipsoid surface, i~~.e.~~, the radius-vector from the center of~~ the ~~ellipsoid~~, ~~located~~ at the ~~material point in consideration~~, ~~to a point on~~ the ~~surface of~~ the ~~ellipsoid is equal to the stress vector on some plane passing through the point~~. ~~In two dimensions,~~ the ~~surface is represented by an~~ [~~[ellipse~~]~~] (Figure coming).~~		On the other hand, you could also place these sturdy shop displays in a entire 'nother distinct kind of retailer a single a bit far more rugged. This is because you can use [http://asgaard.Pf-control.de/index.php?site=profile&id=41 shop displays] for looking things like guns, knives, pocket knives, bullets, or scopes and optical products. Truly, it will not make a difference what sort of a store you have as long as you want to spot objects in these shop shows to generate a far more skilled look, you can do it with these displays. www.sturdystoredisplays.com are companies of [http://www.anjomanma.Cloudsite.ir/F6672 showcases] and retail shows, and the residence of the ever popular Streamline Showcases We current total regular traces of<br>I was reminded of the explanation we came listed here and concluded my beer as the waitress brought out our meals. It looked questionable just as I figured. We all tentatively but hungrily ate the grub, sitting on wobbly stools at the bar. Right after we concluded up and paid the silent waitress, we quietly exited the noisy bar. The sentry the doorway grimaced as we walked out. We all received into the auto and [http://Www.harrietoneilllaw.com/node/590503 Dorothy retraced] the road we arrived on till we noticed an odd, 50 percent useless, oversize oak tree. Subsequent these eleven ideas can give you a leg up on your competitiveness and lead you to fantasy baseball glory.<br>When you have your jacket prepared, decide on the studs you desire to adorn your jacket with. The quantity, designs, measurements, and also the positioning of the studs will all depend on your personal creative imagination and type. Just make [http://act.jinbo.net/forum2006/index.php/Who_Makes_The_Best_Tactical_Knives positive] you do not discover the jacket uncomfortable to use, specifically if you are generating one for your personal little one. A very good minor ways" is considerably farther than "a little ways". I really like my mom-in-legislation even even though I at times want a moment to translate what she's declaring. She by no means could give instructions, bless her heart. Never tell her I stated that or she'll go out back and cut a hickory!<br> Asha Tysons possess rise from abuse and homelessness to prominence deems her an expert on individual good results. Born the only kid to a one mom who was dying. At the age of seventeen, Asha lived deserted and forgotten. With no connections, no mentors and only $eighty dollars in her pocket, Asha went to university, financed her personal [http://www.thebestpocketknifereviews.com/best-combat-knife-fightingtactical-pocket-knives-reviews/ Best combat knife In the world] schooling, and attained both Bachelor and Grasp levels in document time. By means of persistence and perseverance, this very first era university graduate quickly introduced a professional job then retired at the tender age of 26!<br> To prepare a cooked chilly fish you fundamentally follow the measures outlined for cooked rooster. Put together the pill, get a little bit of cooked fish and cut or split it up into small pieces. Make it moist, drain off the excessive h2o and zap it in the microwave for a number of seconds - do not make the meals scorching! Incorporate the crushed pill to the heat soaked fish and provide it up. Utilising several of the attributes that you would discover on a bigger Smartphone, the HTC Hd Mini manages to pack a large sum of engineering into a really tiny handset.<br> The 1st stage to finding the best knife is to know precisely what the knife will be used for. Types needs to know the objective for the knife whether it be a searching knife , a pocket knife , a survival knife or even a kitchen area knife No query, pocket knives are one particular of the most adaptable instruments out there, not only for tenting but also for each day tasks. So, if you are preparing to buy 1 of these truly useful tools, below are three suggestions that can aid you to decide on the proper 1 for you.

	~~Once~~ the ~~equations of the ellipsoid is known~~, ~~the magnitude of the stress vector~~ can ~~then be obtained for any plane passing through that point~~.

	~~To determine the equation of the stress ellipsoid we consider the coordinate axes~~ <~~math~~>~~x_1~~, ~~x_2~~, ~~x_3\~~,~~\!</math> taken in~~ the ~~directions~~ of the ~~principal axes, i~~.e.~~, in a principal stress space~~. ~~Thus,~~ the ~~coordinates of the stress vector <math>\mathbf T^{(\mathbf n)}\~~,~~\!</math> on~~ a ~~plane with normal unit vector <math>\mathbf n\,\!</math> passing through~~ a ~~given point <math>P\,\!</math> is represented~~ by

	~~:<math>T_1^{(\mathbf n)}=\sigma_1n_1~~, ~~\qquad T_2^{(\mathbf n)}=\sigma_2n_2, \qquad T_3^{(\mathbf n)}=\sigma_3n_3\,\!</math>~~

	~~And knowing~~ that ~~<math>\mathbf n\,\!</math> is~~ a ~~unit vector we have~~

	~~:<math>n_1^2+n_2^2+n_3^2=\frac{T_1}{{\sigma_1}^2}^2+\frac{T_2}{{\sigma_2}^2}^2+\frac{T_3}{{\sigma_3}^2}^2=1\,\~~!<~~/math~~>

	~~which is the equation of~~ an ~~ellipsoid centered at the origin of the coordinate system, with the lengths of the semiaxes of~~ the ~~ellipsoid equal~~ to the ~~magnitudes~~ of ~~the principal stresses~~, ~~i.e~~. ~~the intercepts of the ellipsoid with the principal axes are <math>\pm\sigma_1~~, ~~\pm\sigma_2~~, ~~\pm\sigma_3\~~,~~\!<~~/~~math>~~.

	* The first stress invariant <math>I_1\,\!</~~math> is directly proportional to~~ the ~~sum~~ of the ~~principal radii~~ of ~~the ellipsoid.~~
	* The second stress invariant <math>I_2\,\!<~~/math~~> ~~is directly proportional to~~ the ~~sum of the three principal areas of the ellipsoid~~. ~~The tree principal areas are~~ the ~~ellipses on each principal plane.~~
	* The third stress invariant <math>I_3\,~~\!</math> is directly proportional to the volume~~ of ~~the ellipsoid~~.
	* If two of the ~~three principal stresses are numerically equal the stress ellipsoid becomes an [[spheroid\|ellipsoid of revolution]].<ref name="timoshenko2">Timoshenko</ref> Thus, two principal areas are ellipses~~ and the ~~third is~~ a ~~[[circle]].~~
	* If all of the ~~principal stresses are equal~~ and of the ~~same sign~~, the ~~stress ellipsoid becomes~~ a ~~[[sphere]] and any three perpendicular directions can be taken as principal axes~~.<~~ref name="timoshenko2"/~~>

	The ~~stress ellipsoid by itself, however, does not indicate~~ the ~~plane on which~~ the ~~given traction vector acts~~. ~~Only for the case where the stress vector lies along one of the principal directions it is possible~~ to know the ~~direction of~~ the ~~plane~~, as the principal stresses act perpendicular to their planes. To find the orientation of any other plane we used the ''stress-director surface''<ref name="timoshenko2"/> or ''stress director quadric''<ref name=timoshenko2>Timoshenko</ref> represented by the equation

	~~:<math>n_1^2+n_2^2+n_3^2=\frac{T_1^2}{{\sigma_1}^2}+\frac{T_2^2}{{\sigma_2}^2}+\frac{T_3^2}{{\sigma_3}^2}=1\~~,~~\!</math>~~

	~~The stress represented by~~ a ~~radius-vector of the stress ellipsoid acts on~~ a ~~plane oriented parallel to the tangent plane to the stress-director surface at the point~~ of ~~its intersection with~~ the ~~radius-vector~~.~~<ref name="timoshenko2"/>~~

	~~==References==~~
	~~{{Reflist\|3}}~~

	~~==Bibliography==~~
	*{{cite book
	~~\| last = [[Stephen Timoshenko\|Timoshenko]]~~
	~~\| first = Stephen P.~~
	~~\| coauthors = James Norman Goodier~~
	~~\| title = Theory~~ of ~~Elasticity~~
	~~\| publisher = McGraw-Hill International Editions~~
	~~\| year = 1970~~
	~~\| edition = Third~~
	~~\| isbn = 0-07-085805-5}}~~
	*{{cite book
	~~\| last = [[Stephen Timoshenko\|Timoshenko]]~~
	~~\| first = Stephen P.~~
	~~\| series= Dover Books~~ on ~~Physics~~
	~~\| title = History of strength of materials: with a brief account of~~ the ~~history of theory of elasticity and theory of structures~~
	~~\| publisher = Dover Publications~~
	~~\| year = 1983~~
	~~\| isbn = 0-486-61187-6}}~~

	~~{{DEFAULTSORT:Lame's Stress Ellipsoid}}~~
	~~[[Category:Classical mechanics]]~~
	~~[[Category:Materials science]]~~
	~~[[Category:Elasticity (physics)]]~~
	~~[[Category:Solid mechanics]]~~
	~~[[Category:Mechanics]]~~

en>Kxx: /* Unit balls in normed vector spaces */ use svg image

2014-01-26T00:40:56Z

Unit balls in normed vector spaces: use svg image

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+'''Lame's stress ellipsoid''' (Figure coming) is an alternative to [[Mohr's circle]] for the graphical representation of the [[Stress (mechanics)#Stress tensor|stress state at a point]]. The surface of the [[ellipsoid]] represents the locus of the endpoints of all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints of all stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i.e., the radius-vector from the center of the ellipsoid, located at the material point in consideration, to a point on the surface of the ellipsoid is equal to the stress vector on some plane passing through the point. In two dimensions, the surface is represented by an [[ellipse]] (Figure coming).
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+Once the equations of the ellipsoid is known, the magnitude of the stress vector can then be obtained for any plane passing through that point.
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+To determine the equation of the stress ellipsoid we consider the coordinate axes <math>x_1, x_2, x_3\,\!</math> taken in the directions of the principal axes, i.e., in a principal stress space. Thus, the coordinates of the stress vector <math>\mathbf T^{(\mathbf n)}\,\!</math> on a plane with normal unit vector <math>\mathbf n\,\!</math> passing through a given point <math>P\,\!</math> is represented by
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+:<math>T_1^{(\mathbf n)}=\sigma_1n_1, \qquad T_2^{(\mathbf n)}=\sigma_2n_2, \qquad T_3^{(\mathbf n)}=\sigma_3n_3\,\!</math>
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+And knowing that <math>\mathbf n\,\!</math> is a unit vector we have
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+:<math>n_1^2+n_2^2+n_3^2=\frac{T_1}{{\sigma_1}^2}^2+\frac{T_2}{{\sigma_2}^2}^2+\frac{T_3}{{\sigma_3}^2}^2=1\,\!</math>

en>CarrieVS: /* Unit balls in normed vector spaces */ Repairing links to disambiguation pages - You can help!

2012-06-09T19:45:01Z

Unit balls in normed vector spaces: Repairing links to disambiguation pages - You can help!

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