Fundamentals of Continuum Mechanics. Seoul National University Graphics & Media Lab

Transcription

1 Fndmntls of Contnm Mchncs Sol Ntonl Unvrsty Grphcs & Md Lb

2 Th Rodmp of Contnm Mchncs Strss Trnsformton Strn Trnsformton Strss Tnsor Strn T + T ++ T Strss-Strn Rltonshp Strn Enrgy FEM Formlton

3 Lt s Stdy Strss

4 Introdcton to Strss Tnsor,,,,,, Dffrnt nottons r n s. j y z Ech row of strss tnsor s clld th strss vctor, whch cn b ntrprtd s th strss ctng on S th rght fc, S th top fc, S th front fc. Dgonls = norml strsss, Off-dgonls = shr strsss y yy zy z yz zz

5 Mnng of Ech Componnt,,,,,, j : Th prssr ctng on th rght fc to strtch th volm. If ths vl s ngtv t comprsss th volm. : Th prssr ctng on th rght fc to shr th volm to th postv drcton. : Th prssr ctng on th rght fc to shr th volm to th postv drcton.

6 Sgn Convntons,,,, Postv fcs,, rght, top, front fcs Ngtv fcs lft, bottom, bck fcs A strss componnt.. ch j ctng on postv fc s postv f t cts n th postv drcton of th corrspondng coord. s, nd ngtv f t cts n th ngtv drcton of th s. A strss componnt.. ch j ctng on ngtv fc s postv f t cts n th ngtv drcton of th coord. s, nd ngtv f t cts n th postv drcton of n s.

7 Postv Strsss y yy y yz zz z z y zy zy z y z zz yz y z yy

8 Is Strss Tnsor 9 DOF Qntty? No. =, =, = y c j b s ctng on th rght fc. Thn th lft fc gts th sm strss dottd rrow. Th nt ffct s momnt of mgntd bc n th CCW drcton. Smlrly, crts momnt of mgntd bc n th CW drcton. In th qlbrm, thy shold cncl ch othr. If not, ch mtrl pont wll b spnnng. z Strss tnsor s symmtrc. Thrfor = Strss tnsor hs 6 DOFs.

9 Lt s Stdy Strn Tnsor

10 Mthmtcl Dscrpton of Dformton Dformton fncton Invrs fncton,,,, Lgrngn Dscrpton Elrn Dscrpton,,,,

11 Drwng Convnton,, Intrprt th followng s dformton, not s trnslton,,,, B crfl not to b confsd.

12 Comprbl Msrs for Dformton ds ds d d d W r ntrstd n th dffrnc of th two ds ds Tht qntty s zro n pr trnslton or rotton d d d? q p q,, p,,

13 Frst-Ordr Appromton of ds ds????????? d d d d d d ds ds HW: Fnd ths fctorzng mtr HINT: d d d d d d ds ds,, d d d d

14 Wht s th Strn Tnsor? Grn Σ W cll ths Grn s strn tnsor d d d d d d ds ds

15 Bck to th Orgnl Drwng To vod confsons

16 By th dformton, s movd to. By th sm dformton, whr s +d movd to? +d How to clclt d? s sfl qntty for rprsntng th dformton Dffrntl Anlyss of Dformton,,,,,,,,,,,, d d d d d d

17 Grn s Strn Tnsor n trms of Jcobns I T T? Grn

18 th Dsplcmnt Fld Dffrntl Anlyss of th Dformton wth Comprd to, contns pr dsplcmnts. By th sm dformton, whr +d s movd to? +d + +d. It ws +d. How to clclt d? d d s vry sfl qntty for rprsntng strns It s clld dformton grdnt d d d d d

19 Grn s Strn Tnsor n Trms of Grn T T T T j I I I I k j k k j j j Homwork: Vrfy ths

20 Cchy s Infntsml Strn Tnsor Grn T T T T j I I I I k j k k j j j Homwork: Vrfy ths Cchy s nfntsml strn tnsor Cchy s nfntsml strn tnsor mks mor sns whn th dsplcmnts r vry smll. k j k k j j j C

21 Gnrlzd Rltonshp btwn Strss nd Strn

22 Gnrlzd Hook s Lw yy zz zz zz yy yy zz yy v E v E v E Assmpton: Isotropc Mtrls. For mtrl spcfc constnts E, G, nd n, z z yz yz y y G G G yy zz zz zz yy yy zz yy E E E n n n n n n n n n n n n z z yz yz y y G G G HW: Drv th nvrs rltonshp

23 Strn Enrgy

24 Wht s Strn Enrgy? Strn Enrgy U Th nrgy stord n body d to dformton U U V dv Strn Enrgy Dnsty U Th strn nrgy pr nt volm Th r ndrnth th strss-strn crv p to th pont of dformton U d

25 Strn Enrgy n Lnr Elstc Mtrls Strn nrgy dnsty s ddtv yy zz y yz z U U U U U U U Clclton of norml strn nrgy dnsty In lnr lstc mtrls U d Clclton of shr strn nrgy dnsty In lnr lstc mtrls U y So, th totl strn nrgy dnsty s U y y U y y U d d y y y y d yy yy zz zz y y yz yz z z y Homwork: Eprss th strn nrgy dnsty n trms of strn componnts only.

26 For Ftr Us