EN: Continuum Mechnics Homewok 6: Aliction of continuum mechnics to elstic solids Due Decembe th, School of Engineeing Bown Univesity. Exeiments show tht ubbe-like mteils hve secific intenl enegy ( ) nd het ccity c( ) tht e both essentilly indeendent of stin C. () Show tht the het ccity of n elstic mteil (egdless of the fom of ) is elted to the secific entoy by s c Recll lso tht which then gives the nswe stted. s s c s s (b) Hence, show tht the entoy fo ubbe-like mteils must hve the seble fom s( C, ) g( C ) h( ) Integting s c c s d h( ) C (c) As secific exmle, conside n incomessible, isotoic mteil of this kind, fo which g is only function of I Bkk with given by Fom notes we hve tht /3 B B / J Show tht Cuchy stess in such mteil is ij ij dg ij B ij I 3 ij di ij J ji I I I J I JF F /3 5/3 /3 /3 ij ij ij 3 ij ij F 3 ji F ij I F F F J F J F J J 3 ji Futhemoe W ( s), nd W W I W ij Fik Fik Fik F /3 jk IFkj J Fjk J I Fjk I J 3 Noting tht J= fo n incomessible mteil gives the nswe stted.
(d) Conside the simle she defomtion shown in the figue. Show tht the she stess in the solid is elted to the she stin tn by dg di (The genelized she modulus must stisfy ) e 3 e e This follows by noting nd substituting into the constitutive lw [ B] (e) Show tht the unixil Cuchy stess-stin esonse of the mteil is given by ( / ) (the solid is loded in unixil tension llel to e, nd l/ L is the incil stetch llel to the loding xis, with l the defomed nd L the undefomed length of the b, esectively) Fo volume eseving unixil defomtion we hve tht [ B] / / The constitutive lw theefoe gives [ ] / 3 / Since the stess is unixil, it follows tht 3 nd substituting bck gives the esult stted. (f) Suose tht b of this ubbe-like solid is loded in unixil tension t constnt stess. How does the length of the b chnge when its temetue is incesed? The modulus inceses with temetue, so the b gets shote when heted.
(g) Suose tht the b is stetched qusi-stticlly t constnt temetue, with nomlized dl extension te (neglect body foces). Show tht the het flow e unit volume into the L dt b is Q (i.e. when stetched, the b gives off het) d The fist lw of themodynmics gives ( KE) Q W, whee dt W DijijdV V The stetch te tenso is / / 3/ [ D] [ F][ F ] / / 3/ / / Since the intenl enegy is function only of temetue, nd temetue is constnt, it follows tht Q [ D][ ] V which gives the exession stted. (h) Show tht, if loded s descibed in the eceding section unde dibtic conditions (no het flow though the solid) the te of chnge of temetue of the solid is () Unixil tension. Deive the stess-stin eltions fo n incomessible, Neo-Hooken mteil subjected to e 3 e 3 S 3 (b) Equibixil tension (c) Pue she Deive exessions fo the Cuchy stess, L 3 l 3 S S the Nominl stess, nd the Mteil stess e tensos (the solutions fo nominl stess e e listed in the notes). You should use the L L e l l e following ocedue: (i) ssume tht the secimen exeiences the length chnges Undefomed Defomed listed in the tble in the notes ; (ii) use the stess-stetch eltions to comute the Cuchy stess, leving the hydosttic t of the stess s n unknown; (iii) Detemine the hydosttic stess fom the boundy conditions (e.g. fo unixil tensile llel to 33 ; fo equibixil tension o ue she in the e, e lne you know tht 33 ) Fo the incomessible neo-hooken solid, the Cuchy stess-ight stetch eltion cn be witten s
B ij ij ij The defomtion mesues fo the thee cses e listed in the tble below Unixil tension Bixil tension Pue she / L l / L l / L l / L l / L l l / / L l3 / L3 I F / / B S Σ 3 l 3/ L3 4 I l 3/ L3 I 4 4 4 4 5 3 5 3 6 3 6 4
3. In model exeiment intended to dulicte the oulsion mechnism of the lystei bcteium, sheicl bed with dius is coted with n Rigid bed enzyme known s n A/3 ctivto. When susended in solution of ctin, the enzyme cuses the ctin to olymeize t the sufce of the bed. The olymeiztion ection cuses sheicl gel of dense ctin b netwok to fom ound the bed. New gel is continuously fomed t the bed/gel intefce, focing the est of the gel to exnd dilly ound the bed. The ctin gel is long-chin olyme nd consequently cn be idelized s ubbe-like incomessible neo-hooken mteil. Exeiments show tht fte eching citicl dius the ctin gel loses sheicl symmety nd occsionlly will fctue. Stesses in the ctin Actin gel netwok e believed to dive both ocesses. In this oblem you will clculte the stess stte in the gowing, sheicl, ctin gel. () Note tht this is n unusul boundy vlue oblem in solid mechnics, becuse comtible efeence configution cnnot be identified fo the solid. Nevetheless, it is ossible to wite down defomtion gdient field tht chcteizes the chnge in she of infinitesiml volume elements in the gel. To this end: (i) wite down the length of cicumfeentil line t the sufce of the bed; (ii) wite down the length of cicumfeentil line t dius in the gel; (iii) use these esults, togethe with the incomessibility condition, to wite down the defomtion gdient chcteizing the she chnge of mteil element tht hs been dislced fom = to genel osition. Assume tht the bed is igid, nd tht the defomtion is sheiclly symmetic. Length of cicumfeentil line t the sufce of the bed is L Length of cicumfeentil line t dius isl This gives F F /. Fo incomessibility, F / (b) Suose tht new ctin olyme is geneted t volumetic te V. Use the incomessibility condition to wite down the velocity field in the ctin gel in tems of V, nd (think bout the volume of mteil cossing dil line e unit time) The volume of mteil inside sheicl shell with inne dius nd oute dius must emin constnt. This mens tht the volume of mteil flowing coss cicumfeentil line t dius must blnce the volume being geneted t dius. Thus 4 v V v V / (4 ) (c) Clculte the velocity gdient v in the gel (i) by diect diffeentition of (b) nd (ii) by using the esults of (). Show tht the esults e consistent. By diect diffeentition, we hve
v V 3 4 v [ ] V L 3 4 v V 3 4 The ltentive method is v v 3 v [ ] [ ][ ] v L F F v v (d) Clculte the comonents of the left Cuchy-Geen defomtion tenso field nd hence wite down n exession fo the Cuchy stess field in the solid, in tems of n indeteminte hydosttic essue. 4 4 [ B] The stess cn be exessed s Whee J=. Thus ij B 5/3 ij ij J 4 4 [ ]
(e) Use the equilibium equtions nd boundy condition to clculte the full Cuchy stess distibution in the bed. Assume tht the oute sufce of the gel (t =b) is tction fee. The equilibium eqution with sheicl symmety is d d b This leds to d 4 d 4 Integting this exession nd using the boundy condition gives The stesses then follow s 6 4 6 4 6 4 4 6 4 4 ( ) ( b 3 b 3 b ) / ( b ) 6 6 4 4 4 4 ( b )(3 b 3 b ) / ( b ) 6 4 6 4 6 4 4 6 4 4 ( b 3 b 4 b ) / ( b ) 4. A comessible, neo-hooken solid hs stess-ight C-G stin eltion given by ij B 5/3 ij Bkkij K J ij J 3 Suose tht solid consisting of such mteil is fist subjected to defomtion chcteized by Fij, JB ij, inducing stess ij. This defomtion ms mteil ticle t osition X i in the efeence configution to osition y i in the defomed solid. The solid is then subjected to futhe smll defomtion tht induces n dditionl dislcement distibution ui in the mteil. Let u u i j ij yj y i denote the incement of infinitesiml stin ssocited with this dislcement, nd exnd the stess s Tylo seies in stin s ij ij Cijkl kl ( kl ) Show tht the tngent modulus fo this defomtion is (Hint: note, eg, tht the Jcobin fte the incementl defomtion cn be oximted s u J J m ym The defomtion gdient fte the incementl dislcement cn be exessed s
In ddition u i Fij i F j y u i u j Bij Fik Fjk i Fk jq F qk y y q u i uj uj ui i Bq jq Bij Biq B j y y q yq y u 5/3 5/3 5 m u J J J J m ym 3 ym Substituting these eltions into the stess-stin eltion, nd noting tht ij 5/3 Bij Bkkij K J ij J 3 shows tht the stess cn be exessed s u j u i uk u k ij ij B 5/3 iq Bj Bkq Bk ij J y 3 q y yq y 5 uk u B k 5/3 ij Bnnij KJ ij 3 J 3 y k yk Noting tht B is symmetic, this cn be e-witten s 5 u k ij ij 5/3 Bjlik Bik jl Bklij B 5/3 ij Bnnij kl KJ klij J 3 3 3 y J l 5. A solid, sheicl nucle fuel ellet with oute dius is subjected to unifom intenl distibution of het due to nucle ection. The heting induces stedy-stte temetue field T() T T T whee T nd T e the temetues t the cente nd oute sufce of the ellet, esectively. Assume tht the ellet cn be idelized s line elstic solid with Young s modulus E, Poisson s tio nd theml exnsion coefficient. Clculte the distibution of stess in the ellet. The solution should follow the stndd ocess. (i) Clculte the dislcement field by solving the following ODE d u du u d d dt R u b( R) dr R dr R dr R dr dr E The solution contins two bity constnts the fist is detemined by the condition tht the dislcement vnishes t the oigin. (ii) Detemine the stess stte fom the constitutive lw
du RR E dr ET u R The second constnt follows fom the boundy condition RR t =. The lgeb is somewht tedious - A MATLAB solution (using mud) is shown below.