Euler-Lagrange Elasticity: Differential Equations for Elasticity without Stress or Strain

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1 Journal of Appled Mathematcs and Physcs,,, 6- Publshed Onlne December ( Euler-Lagrange Elastcty: Dfferental Equatons for Elastcty wthout Stress or Stran H H Hardy Math and Physcs Department, Pedmont College, Demorest, USA Emal: hhardy@pedmontedu Receved November 5, ; revsed December 5, ; accepted December, Copyrght H H Hardy hs s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted ABSRAC Dfferental equatons to descrbe elastcty are derved wthout the use of stress or stran he ponts wthn the body are the ndependent parameters nstead of stran and surface forces replace stress tensors hese dfferental equatons are a contnuous analytcal model that can then be solved usng any of the standard technques of dfferental equatons Although the equatons do not requre the nton stress or stran, these quanttes can be calculated as dependent parameters hs approach to elastcty s smple, whch avods the need for multple ntons of stress and stran, and provdes a smple expermental procedure to fnd scalar representatons of materal propertes n terms of the energy of ormaton he derved dfferental equatons descrbe both nfntesmal and fnte ormatons Keywords: Elastcty; Stress; Stran; nfntesmal Deformatons; nte Deformatons; Dscrete Regon Model ntroducton Euler, Lagrange, and Posson all descrbed the ormaton of materals n terms of postons of ponts and forces wthn the body Cauchy ntroduced the concepts of stress and stran, whch are now the standard for elastcty equatons (An accessble early hstory s found n Ref [] wth ponters to the source documents) n ths paper, wll return to the earler use of ponts and forces to descrbe elastcty Surprsngly, only a few basc assumptons are requred he Dfferental Equatons Usng the notaton of Spencer [], the ponts wthn the body before ormaton are descrbed as = (,, ), wth,, correspondng to the ntal x, y, and z coordnates of each pont, respectvely he correspondng ponts wthn the body after ormaton are descrbed as x x, x, x he poston of each pont after ormaton s a functon of the orgnal locaton of that pont, e x f,,, x = f (,, ), and x f,, When an elastc materal s ormed, work s done on the materal and energy s stored n the materal As the materal s returned to ts orgnal shape, the energy returns to ts orgnal value hus the energy depends only upon the fnal poston of the ponts wthn the body Experts n elastcty theory call ths hyperelastcty Hyperelastcty wll be assumed for the remander of ths paper he dfferental equatons are derved by assumng the energy per unt volume of the materal s a functon of the fnal pont locatons, x, and the relatve dsplace- x ments of near-by ponts, he total energy of the body s then x tot E x, d V () t s also assumed that when the body s moved or ormed, the nternal ponts wthn the body move so as to mnmze the total energy hat s x tot E x, ddd () for =,, and =,, and the ntegral s taken over the entre body Mnmzng ths energy functon Equaton () results n the followng three Euler-Lagrange equatons, E d E for,, () x d x

2 H H HARDY 7 hese are the dfferental equatons of elastcty All that remans s to approprately descrbe the energy functon E and the boundary condtons Energy o descrbe the energy per unt volume E, dvde the energy nto two parts Ebody whch nes the energy assocated wth body forces and E whch nes the energy assocated wth the ormaton of the body Ebody s typcally only a functon of the postons of the ponts wthn the body (eg, the energy per unt volume assocated wth a gravtatonal force would be ust gx, wth the x axs vertcal) E s typcally a functon only of the relatve postons of the body, e a functon x only of he total energy assocated wth the body s then the sum of the contrbutons from these dfferent energes, x body E E x E he energy of ormaton, E, must be nvarant to coordnate translatons and rotatons A common way of accomplshng ths [] s to ne energy n terms of the x nvarants of the ormaton gradent tensor, Hardy and Shmdheser [4] noted that the nvarants for sotropc bodes can be descrbe as sngular value decompostons (,, ) of the matrx or dfferent algebrac combnatons of these nvarants A partcularly useful set of these nvarants s (5) whch can also be wrtten drectly n terms of the matrx elements of as aabbcc ababacacbcbc (6) a bc a, b, and c are the column vectors of he nvarants for ansotropc bodes can be the sx values produced by a Gram-Schmdt QRDecomposton of hs QRDecomposton results n an upper trangular matrx, (7) (4) he elements of ths upper trangular matrx can also be wrtten n terms of the column vectors of as follows: aa aa ab aa ac aa abab aa abac aaabab abc aaabab (8) Any algebrac combnaton of these 6 values can also be used as nvarants for ansotropc bodes Boundary Condtons Boundary condtons for materal ormaton problems usually consst of ether specfyng the fnal postons of boundary ponts, or the forces on the surfaces he Drchlet boundary condtons consst of smply nng x at the desred surfaces he Neumann boundary condtons requre convertng the surface forces nto dervatves of x hs can be accomplshed by frst notng that once x,, are known for a gven ormaton, E and tot are known hus the change n energy due to the work done on the body s a functon of only the ntal and fnal postons of the ponts wthn the body hs mples that the ormaton forces are conservatve As a result the nternal forces wthn the body can be wrtten as the negatve gradent of the energy, x tot, whch s equvalent to E ddd (9) x Care must be taken here, however, because the gradent of E s wth respect to x, as the ntegral s over the orgnal Usng Equaton (), d E ddd d x () Applyng the n-dmensonal dvergence theorem gves E d A () x s the th component of the force at the sur-

3 8 H H HARDY face ned by the vector da wth da dd, da dd, da dd hese nternal forces must be balanced by an equvalent surface force n the opposte drecton Snce ths must be true for all surfaces, the surface forces d surface are d surface E d A x () Once the energy per unt volume, E, s expressed n terms of the nvarants of, whch are n turn func- x tons of, Equaton () provde the constrant equatons for the Neumann boundary condtons Some Applcatons he dfferental equatons of elastcty have now been completely descrbed No reference has been made to the magntude of the ormaton As a result, Equaton () apply to both nfntesmal and fnte elastc ormatons Also they apply to both sotropc and ansotropc materals Although the equatons are general, for smplcty wll lmt the rest of ths paper to sotropc materals nfntesmal Elastcty n order to make the connecton between the dfferental equatons of elastcty gven here and the standard dfferental equatons of nfntesmal elastcty, wll show that a aylor expanson of E for an sotropc body yelds the nfntesmal free energy as descrbed by Landau [5] (Equaton 4, p 9) hs nfntesmal free energy when substtuted nto Equaton () results n the dfferental equatons for nfntesmal ormatons that Landau derved assumng stran to be the ndependent parameter x he aylor expanson of E yelds E x E kl k l k E x x x E x x x x x x k l l () the subscrpt corresponds to no ormaton (e when x m x and m m ) he frst term n the expanson s a constant and s physcally rrelevant o evaluate the second term, wll choose to ne the ntal state, when no ormaton has occurred, as correspondng to the state of the body when all nternal forces are zero Snce ths must be true for all ponts wthn the body, Equaton () gves E x (4) whch s the coeffcent of the second term n the aylor expanson, Equaton () As a result, the thrd term, s the leadng term n the aylor expanson Before evaluatng ths however, let s see what constrant Equaton (4) provdes o do ths, express E n terms of the three nvarants for an sotropc body gven n Equaton (6) hen E E r x r x (5) Drect substtuton of the r nvarants n terms of x nto ths equaton yelds E E E 4 (6) t now remans to expand and evaluate the thrd term n the aylor expanson Agan wrtng E n terms of r allows us to proceed he thrd term n the aylor expanson can be expanded quckly usng algebrac computaton software lke Mathematca t s a bt more tedous by hand, but the result of ether s the free energy E as ned by Landau [5], E E u u (7) k u uk wth uk k u x and and u x, so that E E E E E E E 6 8 (8)

4 H H HARDY 9 E E (9) f ths aylor expanson of E s substtuted nto Equaton () and a gravtatonal body force g s added, we get Landau s equaton [5] (Equaton 7, p 6) of equlbrum for sotropc bodes of an nfntesmal ormaton: Y u u Y l g l k Young s modulus son s rato Y () and Pos- he boundary condtons for classcal nfntesmal elastcty consst of settng stress and stran on boundares o complete the comparson, stress and stran need to be ned here are many dfferent ntons of stress and stran n the lterature As an example, the La- k k grangan stran tensor s, and stress as ned by Landau [5] can be nferred from Equaton () to be E () x Note that ths stress corresponds to force dvded by orgnal area We can now conclude that all of the solutons to problems n classcal nfntesmal elastcty descrbed by Equaton () are also solutons of Equaton () One other observaton should be made here As ned by Landau, ned n Equaton () s the stress exerted by the surroundngs on the materal volume Notng ths and consderng the case the materal s n the gravtatonal feld of the earth, we see that Equaton () are ust the equatons of equlbrum n terms of stress, e g () g,, g nte Elastcty he real power of Equaton () s not that they can reproduce standard nfntesmal elastcty equatons, but that they apply equally well to fnte elastcty Consder for example a fnte ormaton wth no body forces x Snce E s a functon only of, the dervatve of E n Equaton () results n equatons wth every term con- tanng a second dervatve of x wth respect to As a result, any homogeneous ormaton wthout body forces automatcally solves Equaton (), dependng only upon the boundary condtons Consder two such homogeneous ormatons he frst homogeneous ormaton smulates the expermental results of Rvln [6] hese expermental results can be used to ne an energy functon E he second homogeneous ormaton uses ths energy functon to calculate the surface forces necessary to produce a smple shear of 5, whch s much larger than would be possble wth nfntesmal elastcty theory Matchng Expermental Data Any of the many models of elastcty expressed n terms of nvarants of (eg Ogden []) can be used for E n Equaton () as long as they span the range of ormaton condtons to be encountered n the smulaton As a specfc example, consder E E gven as 4 E a b cc c c () Note that the coeffcent of has been chosen so that Equaton (6) s satsfed (e so that the forces wth no ormaton are zero) Consderng solutons of Equaton () of the form x a, the ormaton gradent matrx s a x a (4) a and the sngular value decomposton of gves a,,, and were measured by Rvln No force was appled n the z drecton, so able n Rvln s paper [6] recorded computed stress, t h f t w = orgnal wdth = 8 cm, h = w orgnal thckness = 7 mm of the ormed rubber sample hs nformaton allows the measured and to be reconstructed Substtutng these values nto Equaton () provdes three constran equatons for fndng a, b, c, and c Unfortunately Rvln assumed the rubber he ormed to be ncompressble, so = and b can not be evaluated usng hs data However, f a, c, c are computed assumng ncompressblty, an estmate of b can be obtaned usng compressblty 6 55 kg cm from Wood [7],, Equaton (8), and Equaton (9) he result of ths ft s a 4698, b 998, c = 9, c = 799 all n Rvln s unts of kg cm 98 4 J m Posson s rato s and Young s modulus s 46 kg cm hese results are

5 H H HARDY at least consstent wth Rvln s data he process followed here to derve E s not deal What s really needed s expermental data for the entre energy cube as descrbe by Hardy and Shmdheser [4] Lackng that expermental data, wll use the energy functon Equaton () to demonstrate a case of fnte smple shear Smple Shear A smple shear, correspondng to a 5 ormaton n the y drecton, s x tan 5, x, and x he forces requred for ths ormaton on a cm cube can be found usng Equaton () and the energy functon that fts Rvln s data, Equaton () he result s f x 8775, 849,, f y, 957,, and f z,, 47 all n Newtons, wth f x, f y, and f z beng the force aganst what was the orgnal yz, xz, and xy face, respectvely Of course not all ormatons are homogeneous, but Equaton () are approprate for fnte element or any other technque for solvng dfferental equatons for more complcated problems when homogeneous ormatons do not satsfy the boundary condtons of a partcular problem 4 Concluson Elastcty theory has been domnated by Cauchy s stress and stran Cauchy s approach works well for nfntesmal elastcty as used n most engneerng applcatons or fnte ormatons, however, Cauchy s approach becomes unduly complcated, spawnng a number of new ntons of stress and stran he Euler-Lagrange approach presented here avods these complcatons n addton, materal propertes n terms of the energy of ormaton are easly nput nto the elastcty equatons, Equaton () t s my hope that engneers and physcsts who requre computer models of fnte ormatons wll consder the Euler-Lagrange approach also hope that t wll encourage expermentalsts to map out the energy of ormaton for more materals as Rvln has done wth rubber 5 Acknowledgements would lke to thank Jens eder, orsten Jossang, Rchard Beer, Paul Meakn, and Jonathan Gaston for ther support and encouragement of ths work REERENCES [] odhunter, A Hstory of the heory of Elastcty and of the Strength of Materals from Galle to the Present me, Cambrdge Unversty Press, Cambrdge, 886 [] A J M Spencer, Contnuum Mechancs, Dover, New York, 98 [] R W Ogden, Non-Lnear Elastc Deformatons, Dover Publcatons, Mneola, New York, 984 [4] H H Hardy and H Shmdheser, A Dscrete Regon Model of sotropc Elastcty, Mathematcs and Mechancs of Solds, Vol 6, No,, pp 7- [5] L D Landau and E M Lfshtz, heory of Elastcty, Course of heoretcal Physcs, Vol 7, Elsever, London, 5 [6] R S Rvln and D W Saunders, Large Elastc Deformatons of sotropc Materals V Experments on the Deformaton of Rubber, Phlosophcal ransactons of the Royal Socety of London Seres A, Mathematcal and Physcal Scences, Vol 4, No 865, 95, pp [7] L A Wood and G M Martn, Compressblty of Natural Rubber at Pressures Below 5 kg/cm, Journal of Research of the Natonal Bureau of Standards, Vol 68A, No, 964, p 59