MAS10 SCHOOL OF MATHEMATICS AND STATISTICS Autumn Semeter 011 1 Continuum Mechanic hour!#"$ % &"('$)*+*,'.-/01"$ 4 % 576 "$ 98:; & <= 8 > ';*?@ BAC>>; 9D E8:; F G7A7AHD.I; B,'( J KL7M N1OPRQBS$TUP@VWPYXYZ%[$\,T][$^Z%OP1_ PY`ba@S$\UZ,PT,L+S$_cX[;[W\,d L+_BSeZ,PTYf e 1, e, fse\,p1\,pyghsez,pdiz,[ Z%O P9QBSeT,PFVWPX Z%[W\UTj[e^kZ%O P9[Wgld#a@S$\UZ,PT,L+S$_mX[;[W\,dLl_BSeZ,PTf e fbq(n f e 1, e, e e i = a ij e `ROPY\,P Se\,PFZ%OP&PglPoP_(Z%T=[$^kZ,OPFZ%\%Se_TU^p[W\UoSZ%L+[$_mo SeZ,\,Lrq LlT=S T,PYX[W_ dut j a ij  T [W\UdP\@Z,P_TU[W\ Z,T1X[Wow![$_P_(Z%T1Ll_Z%O P5[Wg+dxa@Se\UZ%PYT,L+S$_yX[;[W\,d L+_BSeZ,PT1Se\,P{zWLrVWPY_mQ(n Gv \,LlZ,PiZ%OPP quw\uptut,l+[$_}^p[w\fdp Z%P\Uo Ll_L+_ z~z%opcx[$o we[w_py_z,t T ij = T(e i, e j )c T ij [$^@Z,P_TU[W\ T L+_Z%O P_P ` a@s$\uz,pt,l+s$_ XY[u[$\,dLl_BSeZ%PYT O [ƒ` Z,OBSeZ S$_d S$\,P m T ij T \,PYghSeZ,PdxQ(n ij T ij = a ik a jl T kl. O[*` Z,OBSeZ1Z,OL+T1\UPghSZ%L+[$_yL+T1P LrVeS$g+PY_Z1Z,[.Z%OPFo SeZ%\ULrq\,Pg+SeZ%Ll[W_ ˆT = ˆTÂT, `ROPY\,P LlTcZ,OPxZ%\%Se_TU^p[W\UoSZ%L+[$_ˆo SeZ,\,Lrq ^p\,[wo [Wg+dŠZ%[}_PY` X[;[W\,dLl_BSeZ,PT.`RLlZ,O  PglPoP_(Z%T f!s$_d S$_d Se\,P9Z,OPŒo SeZ,\,L+XYPT=[$^GXY[Wo we[w_py_(z%t [e^z%py_t,[w\ a ij ˆT ˆT T [Wgld#Se_dx_PY` XY[u[$\,dLl_BSeZ%PYT1\,PYT,wEPXYZ,LlV$Pgln Ll_ jp ` a@s$\uz,ptulhs$_ X[;[W\,dLl_BSeZ,PTf x 1, x, fs$\,pˆ[wq Z%S$L+_PYd ^p\u[wo f Q(nSy\U[$ZSeZ,L+[W_šS$Q![$ Z5Z%OP x t SquL+T Z,O\,[W zwoœs$_ Se_zWg+P x 1, x, x x Z%O PFZ,\%S$_ TU^p[W\UoSeZ,L+[W_oSeZ,\,L q~llt1zwllv$p_mqn Z%O P [Wgld [W_ PTf O[*` Z%OBSeZ θ  = co θ in θ 0 in θ co θ 0 0 0 1 ŽŸž MAS10 1 Quetion 1 continued on next page
J KXY[W_(Z%Ll_; PdM N1OP&X[WowE[W_P_(Z,T=[e^ Z,P_TU[W\ T L+_xX[;[W\,d L+_BSeZ,PT x 1, x, x S$\UP5zWLlV$P_mQ(n ˆT = a b 0 b c 0 0 0 0 Ll_dyZ%OPFo SeZ%\ULrq ˆT [$^ X[Wow![$_P_(Z%TR[$^ Z%O L+T@Z%PY_T,[$\=Ll_xX[;[W\,dLl_BSeZ,PT /[W S$\UPizWLrVWP_šZ,OBSeZ θ T x 1, x, x Ž ˆ ij = 0 `RO P_ i j fs$_d a = f PYZ,P\,oL+_ P b = c = 1 ŽŸž KL7M 5LlVWP Z%OPydP B_LrZ%L+[$_Š[$^jSšTUZ%\UP:S$og+Ll_P O[*` Z%OSeZ:fGLl_ˆa@S$\UZ,PT,L+S$_ŠX[;[W\UdL+_BSZ%PT f]z%opxtun;t Z%Po [e^ PY ; SeZ%Ll[W_Ti^p[W\.T œ Z%\,PS$og+L+_ PTiXS$_ QEPy`R\,LrZ,Z%PY_ L+_ Z%O P x 1, x, x ^p[w\uo `ROPY\,P dx 1 dλ = v 1(x, t), dx dλ = v (x, t), dx dλ = v (x, t), v = (v 1, v, v ) LlT@Z%OP5VWPYg+[;XLrZ nxs$_d x = (x 1, x, x ) Ž N1OPFVWPYg+[;XLlZ n BPgldx[$^Scwg+S$_BS$\@o[$Z,L+[W_yLlT1zWLlV$P_xQ(n ( v = u x a + ax ) ( 1 x1 e x 1 + x 1 + u a + ax ) e x 1 + x, `ROPY\,P S$\,PœZ%OP}QBS$TUP VWPYXYZ%[$\,Tf9S$_d e 1, e u Se_d a quwghs$ll_.`ron$fu^p[w\z,ol+t]wbse\uz%llx g+s$\kv$pg+[;xlrz n BPYg+d f(z%o PRZ,\%SUPYXYZ,[W\,LlPT/X[WLl_XLldP@`RLlZ,O S$\UP we[wt,lrz%llv$p}xy[w_t ZS$_(Z%T Z%O P&T Z%\UP:S$og+Ll_PT Ž jtup&z,op5ves$\,l+s$qg+pjt, Q TUZ%LrZ% Z,L+[W_ x 1 = r co φ, x = r in φ Z%[.[$Q ZS$Ll_~Z%OP5T n;tuz%pyo [$^ P BSZ%L+[$_T1dP _L+_zcT Z%\,PS$og+L+_ PT Ll_~Z%OP tÿwghse_p Ll_ x 1 x we[wghs$\ X[;[W\,dLl_BSeZ,PT r, φ ^p[w\@z,op5vwpyg+[;xllz n BPgld e u Y Ž} f B_ dcz,op1p BSZ%L+[$_.[$^!S{TUZ%\UP:S$og+Ll_P Z,OBSeZGXY[W_(ZS$Ll_TkZ,OP [W\kZ,OP@VWPYg+[;XLlZ n BPYg+d we[wl+_(zg[$^ez%o P t wg+s$_p `RLrZ%OcwE[WghS$\ X[;[W\,d L+_BSeZ,PT x 1 x (r 0, φ 0 ) L+_.Z,OP@^p[W\,o r = r(φ) Ž} MAS10 Continued
KL7M 5LlVWP Z,OPdP B_ LlZ%Ll[W_Ty[$^9Z%OPw\UL+_XYL+wBSeg{dL+\UPX Z%L+[$_TxS$_d w\,ll_xllwbs$g{t Z%\UPT,TUPT#[$^ Z%O PšTUZ,\,PTUTZ%PY_T,[$\ T f=s$_d `R\,LlZ,P d[*`r_bz,op PY ; SeZ%Ll[W_bdP B_L+_ z Z%OP w\,ll_xllwbs$g dll\,px Z%Ll[W_TRS$_dmTUZ,\,PYT,T,PYT Ž Š PYZ f S$_d QEP.Z%OPcw\UL+_XYL+wBSegdL+\UPX Z%L+[$_T5[$^/Z%OPcTUZ,\,PYT,T9Z,P_TU[W\ e 1 e e T T,wE[W_d L+_zxZ%[mZ%OPw\,Ll_XLlwBS$gT Z%\UPT,TUPT f S$_d O[*` Z%OBSZŒLl_ a@s$\ Z%PYT,LhSe_ X[W\U\,P t T 1 T T x X[;[W\UdL+_BSZ%PT]`RLlZ,O Z,OP QBSeT,L+T {e 1, e, e } Z%OPjoSeZ,\,L q [$^ Z%OP=Z%P_ T,[W\ OS$T Z,OP ˆT T dl+s$zw[w_bseg!^p[w\uo `RLlZ,O f Se_d [W_yZ%OPFo S$L+_ydL+S$zW[W_S$g T 1 T T Ž Š 5LlVWP&Z%OP9P quw\upt,tul+[w_š^p[$\=z,opœtu \U^S$XYP9Z,\%S$X Z%L+[$_ t S$_dxZ%O P& _LlZR_[$\,o S$g!V$PXYZ,[W\ Ll_#Z,P\,oT=[$^GZ%OP9TUZ,\,PTUT Z%PY_T,[$\ n Z%[.Z%OPFTU \U^S$XYP Ž ä [W_TUL+dPY\1Z%OPFT, \U^S$XYPTR`RLlZ%OyZ,OPF_[W\,o S$g! _LlZ1V$PX Z%[W\UT n 1 = 1 (e 1 + e + e ), n = 1 (e 1 + e ) 1 e, n = 1 6 e 1 1 e 1 e, `ROPY\,P f S$_d Se\,P Z,OP _LlZ&VWPYXYZ,[W\,TiLl_œZ,OPw \,L+_ XL+wS$gGdL+\UPX Z%L+[$_T9[e^ e 1 e e PY_[$Z%PyZ,OPyX[W\U\,PYT,wE[W_dLl_zT, \ ^S$XP~Z,\%S$X Z%L+[$_TiQ(n f S$_ d \UPTUw!PYXYZ%LrVWPYgln T t 1 t t /[W S$\,P#zWLlV$P_ Z,OBSeZ LlT wep\uw!py_dllx g+s$\z,[ f L+Tcw!PY\,wEP_d L+X ghs$\cz%[ f t 1 n 1 t n t = 100 eo fs$_d a@s$g+xy ghsez,p f S$_d T 1 > 0 T 1 T T Ž MAS10 Turn Over
M M P KL7M nyxy[w_tul+dpy\,l+_ ziz%op5py ; L+g+LlQ\,Ll o [$^ksctuoseg+gvw[wgl o P5[e^ ` SeZ,P\RL+_~Z,OP5^p[W\,o [$^ks X Q!PefEd P\,LrVWP5Z%O P&P quw\,pyt,t,ll[w_x^p[w\@z%op5` SeZ%PY\jw \,PTUT, \UP p SZjd Pw Z,O h f p = p a + gρh, `ROPY\,P L+T@Z,OP&SeZ%o[WT,w OP\UL+XFw\,PYT,TU \,Pef p a ρ Z%OP5` SeZ%PY\=d P_TULlZ n g Z%OPFzW\,S*V;LrZSeZ,L+[W_BSeg S$XYXPglP\,SeZ%Ll[W_ fbse_d e u Y N1OPY\,PxL+TŒSšT,wOPY\,LlX:S$g@TUOPglg o S$dP~[e^ Se_ŠPghSeTUZ%LlXo SeZ%PY\,L+S$g N1O PyT,OPYg+g LlT Bglg+Pd `RLlZ,OšS$Ll\=SeZ w \,PTUT, \UP eo p 0 = 10 6 N1OPŒ\,S$dL+ Tj[$^kZ%OP9T,OPYg+gHL+T r 0 = 0.1 ZRZ%OPFLl_LlZ,LhS$g!o [WoP_(Z1[e^kZ%Llo PjZ%OPFT,O Pg+gLlT=S$Q![*V$PFZ,OP5`@SZ%P\=T, \U^S$XYP N1OP_ o S ` PL+z$OZj[$^koS$TUT z~s$_dm[$^g_ PzWglL+zWLlQg+P=VW[Wgl o PFLlT SeZUZS$XO Pd#Z,[Z,OP&T,OPYg+gf m = 5 S$_d Z%OP{TUOPglgEZ,[WzWP Z%OPY\ `RLlZ,OZ%O Pj` PL+z$OZ@L+T/L+oo PY\,T,PYdLl_ ` SeZ%PY\ /[W ~Se\,P z$llvwpy_ Z%OSeZFZ%OP.VW[$g+ op.[$^ Z,OPcT,OPYg+gGLlT&w \,[WwE[W\UZ,L+[W_S$g Z,[xZ,OPdL!P\UP_XYPQEPYZ ` PPY_ Z%O P L+_(Z,P\,_S$g w \,PTUT, \UPc[$^GZ%OPŒS$L+\jL+_mZ%OPiTUOPglgS$_ d Z%OPŒP q;z%py\,_bs$gkw\,pyt,t, \,P$fHSe_d Z%O P w\upt,tu \,P9[$^ Z%OP&S$Ll\@L+_TUL+dP{Z,OPFT,OPYg+g LlT1L+_(VWPY\,TUPglnxw \,[WwE[W\UZ,L+[W_S$g!Z,[LrZ%T@V$[Wg+ op K S(M O[*` Z%OSeZ5Z%OPcdPYw Z%O}SeZ5`ROLlXOœZ,OP.T,O Pg+gGLlT5L+oo PY\,TUPdS$_dLrZ%T5\%SedL+ T Se\,P5\,PYghSeZ,PdxQ(nyZ%OPFPY ; SeZ%Ll[W_ ( ) 6 ( ) r r 9 + (1 + 0.1h) 10 = 0. r 0 r 0 K /[W X:S$_}S$TUT, op.z%obsez5z,opco [ez%l+[$_šl+t5 BS$TULrtŸTUZSZ%L+XcSe_d Z%OPcT,O Pg+gGw \,P t TUP\ VWPTmLlZ,T~T,w OP\UL+X:Seg{T,OBSew!P d \,L+_ z Z%OP o[$z%ll[w_ frs$_ d Z%S PšZ,OPš` SeZ%PY\ w \,PTUT, \UPœZ,OPœT%S$oPSeZ#S$_(n we[wl+_(zx[w_ Z%OPTUOPglg /[W S$g+TU[ˆXS$_ ZS g = 10 o et f Z%OPj`@SeZ,P\@d P_TULlZ n z eo ρ = 10 fs$_dz,op5sez,o [WTUwOPY\,L+X w \,PTUT, \UP o p a = 10 5 KQM P\,LrVWPmKpQ Z5d[y_ [$Z5Z%\ n#z,[yt,[wgrvwp*mjz,opipy ; SeZ%Ll[W_š^p[W\ Z,OPidPYw Z%O h(t) `RO L+XÕ Z,OP TUOPglg=`RLlg+gRQ!P SeZZ,OP o[wop_(z [$^{Z%Llo P N1O; TyX:S$glX g+sez%p SZ Z,OPS$XXYPglP\%SZ%L+[$_œ[e^ Z,OPcT,OPYg+g]SeZFZ%O Pd Pw Z,O t h = 10 o K /[$ }XS$_} T,P `RLrZ%O[$ ZŒd P\,LrVeSeZ%Ll[W_ \,XOLlo PYdPT g+s:` ŒZ%OP w\,pyt,tu \,P~^p[W\UXP P qup\ Z%PdŠ[$_ Z,OPFT, \ ^S$XPF[$^kS.Q![;d nl+oo PY\,T,PYdxL+_~` SeZ%PY\jLlT@L+_yZ%O PFV$P\ Z%L+XS$gdLl\,PX Z%Ll[W_ f Se_dLlZ%T os$z$_llz, dpœllt{p BSegZ%[~Z%O Pc` PLlzWO(ZF[$^]`@SZ%P\5d L+T,w ghs$xypdq(n Z%O P QE[;d n /[$ ~S$glT,[&X:S$_~_ PzWglPXYZ/Z%O Pj` PL+z$OZ [$^ Z,OP TUwOPY\,L+XS$gET,OPYg+gf;S$_ d Z%O P ` SeZ,P\R\,PYT,L+T ZS$_ XP5Z%[LlZ,T1o [$Z,L+[W_ Ž} MAS10 4 Continued
P /[$ 9Se\,PGzWLrVWP_5Z,OBSeZ:fƒLl_&glL+_PS$\ PYghS$T Z%LlXLlZ n$f:z%op]o[wop_(z% o P BSeZ,L+[W_5^p[W\!Z%OP]L+_ B_LlZ,PTUL+o S$g d L+T,w ghs$xypop_(z1vwpyxyz%[$\ u OS$T1Z%OP5^p[W\Uo ρ u t = (λ + µ) ( u) + µ u, `RO P\,P ρ = XY[W_TUZ=LlT@Z%OP5dPY_T,LrZ nxs$_d λ S$_ d µ S$\UP{Z%OP HS$o 5X[$_TUZ%S$_(Z%T /[W ms$\up&s$g+tu[ z$llvwpy_ Z%OSeZ u L+TjSwg+S$_BS$\RVWPYXYZ%[$\f fes$_d Z,OBSeZ LlZjdPwEP_d T{[W_š[W_ PŒT,wSeZ%L+S$g u = (w, v, 0) XY[u[$\,dLl_BSeZ%P [W_grnWfL x = x 1 u = u(t, x) KL7M O[*` Z,OBSeZ T%SeZ,L+T PT1Z%OPFPY ; SeZ%Ll[W_m^p[$\@Z%OP{Z,\%S$_T VWPY\,T,P&`@S:V$P$f v ρ v t = µ v x. Ž ˆ O[*` Z%OBSZ{P BSZ%L+[$_ OBS$T5S~TU[Wg+ uz%l+[$_ L+_#Z%OPŒ^p[$\,o v = f(t ± x/c ) fe`rop\up f LlT{S$_ Se\,QLrZ%\%Se\Uny^p _X Z%Ll[W_ f!s$_d B_ ds$_šp quw\uptut,l+[$_š^p[w\ Ll_#Z%PY\,oT [$^ c ρ Se_d µ Ž /[W }S$\,PczWLrVWPY_œZ,OBSeZ f S$_d Ll_Z,OPOBS$gl^htŸT,wBSeXP ρ = ρ λ = λ µ = µ `ROLlg+P f Se_d L+_ Z%O POS$gl^htŸT,wBS$XYP /[$ S$\UPS$g+TU[ ρ = ρ + λ = λ + µ = µ + x > zwlrvwp_xz%osez1z%opftu \U^S$XYP5Z%\%SeXYZ%Ll[W_mSeZ1Z%O P&TU \U^SeXP 0 x = 0 LlT1zWLlV$P_mQ(n t = ( (λ + µ) u ) x, µ v. x x < 0 f TUT, ol+_ z{z%obsez u = (0, v, 0) S$_d v dpwep_ dt][w_.[w_p1tuwbsez%l+s$g XY[u[$\,dLl_BSeZ%P x = x [W_grnWfG TUP~Z%OPX[W_d LlZ%Ll[W_T9[$^=X[W_(Z%Ll_ LlZ n}[$^ 1 u S$_ d t SeZ Z,[ dp\ullv$p~z ` [ x = 0 QE[W _dbse\unyxy[w_dlrz%l+[$_t@^p[w\ v SZ e u Y x = 0 Ž N1OPY\,P/L+THS1wghSe_P]OBS$\,o[W_LlXkZ%\,S$_T VWP\UT,P]`@S:V$P w\u[wwbs$z(sz%l+_ z1l+_5z%op]obsegl^ht TUwBS$XYP 0 L+_9Z,OP we[wtullz%lrvwp x t dll\,pyxyz%ll[w_ N1OLlT ` S*V$P@L+T dp B_Pd.Q(n quwghs$ll_y`rony` PFX:S$_mg+[;[ ^p[w\@z%o P&TU[Wg+ uz%l+[$_~z%[p BSeZ,L+[W_ L+_yZ%OP5^p[W\Uo x < v = a co[ω(t x/c )] v = { a co[ω(t x/c )] + b co[ω(t + x/c )], x < 0, c co[ω(t x/c + )], x > 0. jtupjz,op QE[W _ds$\unx[$_t,lrz%l+[$_t SZ X[;P XYL+P_(Z=[$^ \,P BPX Z%L+[$_ R = b /a x = 0 Z,[ŒXS$g+XY ghsez,p b/a S$_d c/a Se_d B_ dz%op Ž Š End of Quetion Paper MAS10 5