EN2210: Continuum Mechanics. Homework 4: Balance laws, work and energy, virtual work Due 12:00 noon Friday February 4th

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1 EN: Contnuum Mechncs Homewok 4: Blnce lws, wok nd enegy, vtul wok Due : noon Fdy Feuy 4th chool of Engneeng Bown Unvesty. how tht the locl mss lnce equton t cn e e-wtten n sptl fom s xconst v y v t yconst y Note tht v t xconst t yconst y nd susttute nto the fst equton. Expndng out the devtve of the poduct n the second equton shows tht the two expessons e equvlent.. The stess feld 3Pk yk y y j j R y 5 k yk 4 R epesents the stess n n nfnte, ncompessle lne elstc sold tht s sujected to pont foce wth components P k ctng t the ogn (you cn vsulze pont foce s vey lge ody foce whch s concentted n vey smll egon ound the ogn). () Vefy tht the stess feld s n sttc equlum (except t the ogn) j 3Pk k y y j yk y j yk yj yk y y j y 5 y R R R R R () Consde sphecl egon of mtel centeed t the ogn. Ths egon s sujected to () the ody foce ctng t the ogn; nd () foce exeted y the stess feld on the oute sufce of the sphee. Clculte the esultnt foce exeted on the oute sufce of the sphee y the stess, nd show tht t s equl n mgntude nd opposte n decton to the ody foce. The tcton ctng on the exteo sufce s n n y / R. The esultnt foce s thus j 3 ykyj Fj Pk da 4 3 R The ntegl clely vnshes fo k j y symmety. Choosng k=j=3 wthout loss of genelty we cn evlute the emnng ntegl n sphecl-pol coodntes s

2 3 Rcos F3 P3 R sndd P R 3. The fgue shows test desgned to mesue the vscosty of flud. The smple s hollow cylnde wth ntenl dus nd extenl dus. The nsde dmete s onded to fxed gd cylnde. The extenl dmete s onded nsde gd tue, whch s otted wth ngul velocty () t. Assume tht ll mtel ptcles n the specmen (geen) move ccumfeentlly, wth velocty feld (n sptl coodntes) v v(, t) e. () Clculte the sptl velocty gdent L n the ss { e, e, ez} nd hence deduce the stetch te tenso D. e e ez Defomed The gdent opeto fo cylndcl-pol coodntes s (n D) e e. Ths gves v v The velocty gdent follows s L e e e e. v v The stetch te s the symmetc pt of L,.e. D e e e e () Clculte the cceleton feld The cceleton s v v v L v e e t yconst t (c) uppose tht the specmen s homogeneous, hs mss densty, nd my e delzed s vscous flud, n whch the Kchhoff stess s elted to stetch te y τ D p(, t) I whee p s hydosttc pessue (to e detemned) nd s the vscosty. Use ths to wte down n expesson fo the Cuchy stess tenso n tems of p, expessng you nswe s components n { e, e, e z} usttutng fo the stetch te, we see tht v v τ p e e e e e e e e (d) Assume stedy defomton. Expess the equtons of equlum n tems of v (, t) The equlum equton s τ, whch educes to.

3 p v v v v e e e (e) olve the equlum equton, togethe wth ppopte oundy condtons, to clculte v (, t) nd p(). (The pessue cn only e detemned to wthn n ty constnt). The tngentl equlum equton cn e e-wtten s v ntegted to see tht,. Ths equton cn esly e B v A. The oundy condtons e v v These two equtons cn e solved fo A,B, wth the esult v cn lso e esly done n mple o mthemtc ). (The clculton The pessue feld cn e computed fom p v p log p Hee, p s the pessue t. (f) Fnd n expesson fo the toque (pe unt out of plne dstnce) necessy to otte the extenl cylnde The toque s v v 4 ( ) ( ) (g) Clculte the te of extenl wok done y the toque ctng on the ottng exteo cylnde The te of wok s Q 4 ( ) (h) Clculte the te of ntenl dsspton n the sold s functon of. The dsspton te s v : : v τ D D D () how tht the totl ntenl dsspton s equl to the te of wok done y the extenl moment.

4 Clcultng D: D d gves the sme esult s (g) 4. A sold wth volume V s sujected to dstuton of tcton t on ts sufce. Assume tht the sold s n sttc equlum (ths eques tht t exets no esultnt foce o moment on the oundy). By consdeng vtul velocty of the fom v Aj y j, whee A j s constnt tenso, use the pncple of vtul wok to show tht the vege stess n sold cn e computed fom the shpe of the sold nd the tctons ctng on ts sufce usng the expesson jdv t y j t j y da V V The vtul wok pncple gves Note lso tht fo moment equlum V V A dv t A y dv j j j j y t da y t da ( y t y t ) da jk j k pq jk j k p q q p Thus t Aj y jdv Aj ( t y j t j y ) Aj ( t y j t j y ) dv Aj ( t y j t j y ) da Hence, engng nd notng tht A j s constnt A j jdv Aj ( t y j t j y ) da V Ths must hold fo ll A j whch shows the equed esult. 5. The shell shown n the fgue s sujected to dl ody foce ( R) e R, nd dl pessue p, p ctng on the sufces t R nd R. The lodng nduces spheclly symmetc R e e stte of stess n the shell, whch cn e expessed n tems of ts components n sphecl-pol coodnte system s e RReR er e e e e. By consdeng vtul e velocty of the fom v wr ( ) e R, show tht the stess stte s n sttc equlum f dw w RR 4 R dr ( R) w( R)4 R dr 4 pw( ) 4 pw( ) dr R fo ll w(r). Hence, show tht the stess stte must stsfy d RR RR RR p R RR p R dr R e 3 e R

5 w w Fo spheclly symmetc stte the stetch te s R R The vtul wok pncple theefoe educes to * σ : DdV vdv t vda R e e R e e e e V V dw w RR 4 R dr ( R) w( R)4 R dr 4 pw( ) 4 pw( ) dr R Integtng the fst tem y pts gves dr ( ) RR ( R) w4 R dr R dr R 4 ( RR ( ) p ) w( ) 4 ( RR ( ) p ) w( ) Ths must vnsh fo ll w, whch gves the equed soluton. 6. An del gs wth mss densty, pessue p nd tempetue hs specfc ntenl enegy, specfc Helmholtz fee enegy, nd stess gven y p cv cv cv log Rlog s j pj R j ( ) whee R s the gs constnt, c v s the specfc het cpcty ( postve constnt), nd s s n ty constnt. Idel gses e lso chctezed y the specfc het t constnt pessue cp cv R nd the to c / c. In ddton, het conducton though n del gs s often modeled usng Foue s lw p v q y whee s the theml conductvty ( postve constnt). how tht ths consttutve model oeys the fee enegy mlnce jdj q s y t t Consde the fst tem v jdj R R fom mss consevton y t Consde the second tem, q, whch s non-negtve. y y y Fnlly ecll tht s s cv log Rlog s, nd note tht c v log Rlog R s t t t t t

6 Comnng, s R t t t. Hence jdj q s y t t y y