ES240 Solid Mechanics Z. Suo. Principal stress. . Write in the matrix notion, and we have

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1 ES4 Sold Mehs Z Suo Prpl stress Prpl Stress Imge mterl prtle stte o stress The stte o stress s xed, but we represet the mterl prtle my wys by uttg ubes deret orettos For y gve stte o stress, t s lwys possble to ut ube sutble oretto, suh tht the stress ompoets o ll the ube es re orml to the es, d there re o sher stresses o the ube es These ube es re lled the prpl ples, the orml vetors to these es the prpl dretos, d the stresses o them the prpl stresses Exmples Uxl teso Equl bxl teso Hydrostt pressure Pure sher s the sme stte o stress s the ombto o pullg d pressg 45 Gve stress stte, how to d the prpl stresses? Whe ple s the prpl ple, the trto o the ple s orml to the ple, mely, the trto vetor t must be the sme dreto s the ut orml vetor Let the mgtude o the trto be O the prpl ple, the trto vetor s the dreto o the orml vetor, t Wrte the mtrx oto, d we hve t t t Here both t d re vetors, but s slr represetg the mgtude o the prpl stress Rell tht the trto vetor s the stress mtrx tmes the orml vetor, so tht Ths s egevlue problem Whe the stress stte s kow, e, the stress mtrx s gve, solve the bove egevlue problem to determe the egevlue d the egevetor The egevlue s the prpl stress, d the egevetor s the prpl dreto Ler lgebr o egevlues Beuse the stress tesor s by symmetr mtrx, you lwys d three rel egevlues, e, prpl stresses,, b, We dstgush three ses: () I the three prpl stresses re uequl, the three prpl dretos re orthogol (eg, pure sher stte) () I two prpl stresses re equl, but the thrd s deret, the two equl prpl stresses be y dretos ple, d the thrd prpl dreto s orml to the ple (eg, pure tesle stte) () I ll the three prpl stresses re equl, y dreto s prpl dreto Ths stress stte s lled hydrostt stte Mxmum orml stress Why do we re bout the prpl stress? Chlks re mde o brttle mterl: they brek by tesle stress, ot by sher stress Whe hlk s uder 9/8/ Prpl stress-

2 ES4 Sold Mehs Z Suo bedg, the tesle stress s log the xl dreto o the hlk, so tht the hlk breks o ple orml to the xl dreto Whe hlk s uder torso, the mxmum tesle stress s 45 rom the xl dreto, so tht the hlk breks dreto 45 rom the xl dreto (The rture sure o the hlk uder torso s ot ple, beuse o some D eets) We re bout the prpl stress beuse brttle mterls l by tesle stress, d we wt to d the mxmum tesle stress Let s order the three prpl stresses s b Ths orderg tkes to osderto the sgs: ompressve stress (egtve) s smller th tesle stress (postve) O rbtrry ple, the trto vetor my be deomposed to two ompoets: oe ompoet orml the ple (the orml stress), d the other ompoet prllel to the ple (the sher stress) Obvously, whe you look t ple wth deret orml vetor, you d deret orml d sher stresses You wll be delghted by the ollowg theorem: O ll ples, the prpl ple orrespodg to hs the mxmum orml stress Mxmum sher stress Why do we re bout the mxmum sher stress? Most metls re dutle mterls: they l by plst yeldg Whe mterl s uder omplex stress stte, t s kow emprlly tht yeldg rst ours o ple wth mxmum sher stress To d the mxmum sher stress d the prtulr ple, you re helped by the ollowg theorem: The mxmum sher stress s τ ( )/ mx τ mx ts o ple wth the orml vetor 45 rom the prpl dretos d A proo o the bove theorems s outled below Cosder system o oordtes tht ode wth three orthogol dretos o the prpl stresses,,, The osder rbtrry ple whose ut orml vetor hs ompoets,, ths oordte system The ompoets o the stress tesor ths oordte system s b Thus, o the ple wth ut vetor (,, ), the trto vetor s (, b, ) The orml stress o the ple s + + b + + We eed to mxmze uder the ostrt tht The sher stress o the ple τ s gve by ( ) + ( ) + ( ) ( + ) τ b b We eed to mxmze τ uder the ostrt tht b 9/8/ Prpl stress-

3 ES4 Sold Mehs Z Suo Chge o Coordtes The dreto-ose mtrx reltg two bses I the D spe, let e, e d e be orthoorml bss, mely, e e δ The bse vetors re ordered to ollow the rght-hd rule Let e, e, e be ew orthoorml bss, mely, e e δ Let the gle betwee the two vetors e d e be θ Deote the dreto ose o the two vetors by l e os We ollow the oveto tht the rst dex o l reers to oordte the old bss, d the seod to oordte the ew bss For the two bses, there re totl o 9 dreto oses We lst l s by mtrx By our oveto, the rows reer to the old bss, d the olums to the ew bss Note tht l s the ompoet o the vetor e proeted o the vetor e We express eh ew bse vetor s ler ombto o the three old bse vetors: e l e + l e + l e I you re tred o wrtg sums lke ths, you bbrevte t s e l e, wth the oveto tht repeted dex mples summto over,, Beuse the sum s the sme whtever the repeted dex s med, suh dex s lled dummy dex Smlrly, we express the old bss s ler ombto o the ew bss: e l e + le + le Usg the summto oveto, we wrte more osely s e e l Trsormto o ompoets o vetor due to hge o bss Let be vetor It s ler ombto o the bse vetors: e, where,, re the ompoets o the vetor, d re ommoly wrtte s olum Cosder the vetor potg rom Cmbrdge to Bosto Whe the bss s hged, the vetor betwee Cmbrdge d Bosto rems uhged, but the ompoets o the vetor do hge Let,, be the ompoets o the vetor the ew bss, mely, e Rell the trsormto betwee the two sets o bss, θ e l e, we wrte tht e l e A omprso betwee the two expressos gves tht l Thus, the ompoet olum the ew bss s the trspose o the dreto-ose mtrx tmes the ompoet olum the old bss Smlrly, we show tht 9/8/ Prpl stress-

4 ES4 Sold Mehs Z Suo l The ompoet olum the old bss s the dreto-ose mtrx tmes the ompoet olum the old bss Trsormto o stress ompoets due to hge o bss The stress tesor,, desrbes the stte o stress suered by mterl prtle Represet the mterl prtle by ube The stress ompoets re the ore per ut re o 6 es o the ube The stress tesor s represeted by by symmetr mtrx The stte o stress o mterl prtle s physl obet, d s depedet o your hoe o the bss (e, how you ut ube to represet the prtle) However, the ompoets o the stress tesor do deped o your hoe o the bss How do we trsorm the stress ompoets whe the bss s hged? Cosder the stress stte o mterl prtle, d the trto vetor o gve ple I the old bss e, e d e, deote the ompoets o the stress stte by, the ompoets o the ut vetor orml to the ple by, d the ompoets o the trto vetor o the ple by t Usg the summto oveto, we wrte the trto-stress equtos s t Rell tht we obted ths relto by the ble o ores o tetrhedro I the lguge o ler lgebr, we ll the stress s ler opertor tht mps the ut orml vetor o ple to the trto vetor tg o the ple Smlrly, the ew bsse, e, e, deote the ompoets o the stress stte by, the ompoets o the ut vetor orml to the ple by, d the ompoets o the trto vetor o the ple by t Fore ble requres tht t () We ow exme the reltos betwee the ompoets the old bss d those the ew bss The trto s vetor, so tht ts ompoets trsorm s t l t Isert t to the bove, d we obt tht t l l Cosequetly, we obt tht t l l The ut orml vetor trsorms s (b) Equtos () d (b) re vld or y hoe o the ple Cosequetly, we must requre tht l l Thus, the stress-ompoet mtrx the ew bss s the produt o three mtres: the trspose o the dreto-ose mtrx, the stress-ompoet mtrx the old bss, d the dreto-ose mtrx Slrs, vetors, d tesors Whe the bss s hged, slr (eg, temperture, eergy, d mss) does ot hge, the ompoets o vetor trsorm s l, d the ompoets o tesor trsorm s l l Ths trsormto dees the seod-rk tesor By logy, vetor s rst-rk tesor, d slr s zeroth-rk tesor We lso smlrly dee tesors o hgher rks Multler lgebr The rule o the bove trsormto s bsed o oly oe t: the stress s ler mp rom oe vetor to other vetor You geerte ew tesor rom 9/8/ Prpl stress-4

5 ES4 Sold Mehs Z Suo ler mp rom oe tesor to other tesor You lso geerte tesor by bler orm, eg, bler mp rom two vetors to slr O ourse, multler mp o severl tesors to tesor s yet other tesor Cosequetly, ll tesors ollow smlr rule uder hge o bss We wrte ths rule g or thrd-rk tesor: g g l l l γ k kγ Ivrts o tesor A slr s vrt uder y hge o bss Whe the bss hges, the ompoets o vetor hge, but the legth o the vetor s vrt Let be vetor, d be the ompoets o the vetor or gve bss The legth o the vetor s the squre root o The dex s dummy Thus, ths ombto o the ompoets o vetor s slr, whh s vrt uder y hge o bss For vetor, there s oly oe depedet vrt Ay other vrt o the vetor s uto o the legth o the vetor Ths observto be exteded to hgh-order tesors By deto, vrt o tesor s slr ormed by ombto o the ompoets o the tesor For exmple, or symmetr seod-rk tesor, we orm three depedet vrts:,, k k I eh se, ll des re dummy, resultg slr Ay other vrt o the tesor s uto o the bove three vrts Exerse For osymmetr seod-rk tesor, gve ll the depedet vrts Wrte eh vrt usg the summto oveto, d the expltly ll ts terms Exerse Gve ll the depedet vrts o thrd-rk tesor A spel se: the ew bss d the old bss der by gle θ roud the xs e The sg oveto or θ ollows the rght-hd rule The dreto oses re e osθ, e sθ, e, e e sθ, e osθ, e, e e, e, e e θ Cosequetly, the mtrx o the dreto oses s osθ sθ θ [ ] e l sθ osθ Thus, The ompoets o vetor trsorm s osθ sθ sθ osθ 9/8/ Prpl stress-5

6 ES4 Sold Mehs Z Suo Thus, The ompoets o stress stte trsorm s osθ sθ sθ osθ osθ + sθ + sθ osθ osθ sθ + + osθ + s θ + osθ s θ s θ + osθ osθ + sθ sθ + osθ sθ osθ Stte o ple stress A eve more spel se s tht the stress ompoets out o the ple re bset, mely, + + osθ + s θ + osθ s θ s θ + osθ I Beer s Seto 74, these equtos re represeted by grph (e, Mohr s rle) I ths se, e s oe prpl dreto, d the prpl stress ths dreto s zero To d the other two prpl dretos, we set, so tht the two prpl dretos re t the gle θ p rom the e d e dretos Ths gle s gve by τ t θ p The two prpl stresses re gve by + ± + We eed to order these two prpl stresses d the zero prpl stress the e dreto The mxmum sher stress s the hl o the deree betwee the mxmum prpl stress d the mmum prpl stress The stte o str o mterl prtle s seod-rk tesor I the old bss, the 9/8/ Prpl stress-6

7 ES4 Sold Mehs Z Suo oordtes o mterl prtle re ( x, x, x ) re u ( x, x, x, t) ( x, x, x ), d the ompoets o the dsplemet eld re u ( x, x, x, t) l u, d the ompoets o the dsplemet eld I the ew bss, the oordtes o the sme mterl prtle re u d x l x, so tht u x l u x x x l l u x Rell tht Thus, the grdets o the dsplemet eld, u / x, orm the ompoets o seod-rk tesor Cosequetly, the stte o str, beg the symmetr prt o the dsplemet grdet, s seod-rk tesor Whe the bss s hged, the ompoets o the str stte trsorm s ε ε l l 9/8/ Prpl stress-7