Equations of Motion. Figure 1.1.1: a differential element under the action of surface and body forces
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1 Equtions of Motion In Prt I, lnce of forces nd moments cting on n component ws enforced in order to ensure tht the component ws in equilirium. Here, llownce is mde for stresses which vr continuousl throughout mteril, nd force equilirium of n portion of mteril is enforced. One-Dimensionl Eqution Consider one-dimensionl differentil element of length nd cross sectionl re A, Fig Let the verge od force per unit volume cting on the element e nd the verge ccelertion nd densit of the element e nd. Stresses ct on the element. () A, ( ) Figure 1.1.1: differentil element under the ction of surfce nd od forces The net surfce force cting is ( ) A ( ) A. If the element is smll, then the od force nd velocit cn e ssumed to vr linerl over the element nd the verge will ct t the centre of the element. Then the od force cting on the element is A nd the inertil force is A. Appling Newton s second lw leds to ( ) A ( ) A A A ( ) ( ) (1.1.1) so tht, the definition of the derivtive, in the limit s 0, d 1-d Eqution of Motion (1.1.2) d which is the one-dimensionl eqution of motion. Note tht this eqution ws derived on the sis of phsicl lw nd must therefore e stisfied for ll mterils, whtever the e composed of. The derivtive d / d is the stress grdient phsicll, it is mesure of how rpidl the stresses re chnging. Emple Consider r of length l which hngs from ceiling, s shown in Fig Solid Mechnics Prt II 3 Kell
2 l Figure 1.1.2: hnging r The grvittionl force is F mg downwrd nd the od force per unit volume is thus g. There re no ccelerting mteril prticles. Tking the is positive down, n integrtion of the eqution of motion gives d g 0 g c (1.1.3) d where c is n ritrr constnt. The lower end of the r is free nd so the stress there is ero, nd so Two-Dimensionl Equtions ( l ) g (1.1.4) Consider now two dimensionl infinitesiml element of width nd height nd unit depth (into the pge). nd Looking t the norml stress components cting in the direction, nd llowing for vritions in stress over the element surfces, the stresses re s shown in Fig (, ) (, ) (, ) (, ) Figure 1.1.3: vring stresses cting on differentil element Using (two dimensionl) Tlor series nd dropping higher order terms then leds to the nd the prtil linerl vring stresses illustrted in Fig (where (, ) derivtives re evluted t ( ) element is smll., ), which is resonle pproimtion when the Solid Mechnics Prt II 4 Kell
3 Figure 1.1.4: linerl vring stresses cting on differentil element The effect (resultnt force) of this liner vrition of stress on the plne cn e replicted constnt stress cting over the whole plne, the sie of which is the verge stress. For the left nd right sides, one hs, respectivel, 1 2, 1 2 (1.1.5) One cn tke w the stress ( 1/ 2) / from oth sides without ffecting the net force cting on the element so one finll hs the representtion shown in Fig (, ) Figure 1.1.5: net stresses cting on differentil element Crring out the sme procedure for the sher stresses contriuting to force in the direction leds to the stresses shown in Fig (, ) 2 1, v 1 1 (, ) Figure 1.1.6: norml nd sher stresses cting on differentil element Tke, to e the verge ccelertion nd od force, nd to e the verge densit. Newton s lw then ields Solid Mechnics Prt II 5 Kell
4 Solid Mechnics Prt II Kell 6 (1.1.6) which, dividing through nd tking the limit, gives (1.1.7) A similr nlsis for force components in the direction ields nother eqution nd one then hs the two-dimensionl equtions of motion: 2-D Equtions of Motion (1.1.8) Three-Dimensionl Equtions Similrl, one cn consider three-dimensionl element, nd one finds tht 3-D Equtions of Motion (1.1.9) These three equtions epress force-lnce in, respectivel, the,, directions.
5 Figure 1.1.7: from Cuch s Eercices de Mthemtiques (1829) The Equtions of Equlirium If the mteril is not moving (or is moving t constnt velocit) nd is in sttic equilirium, then the equtions of motion reduce to the equtions of equilirium, D Equtions of Equilirium (1.1.10) These equtions epress the force lnce etween surfce forces nd od forces in mteril. The equtions of equilirium m lso e used s good pproimtion in the nlsis of mterils which hve reltivel smll ccelertions.
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