3.3 Internal Stress. Cauchy s Concept of Stress

Transcription

1 INTERNL TRE 3.3 Inernal ress The idea of sress considered in 3.1 is no difficul o concepualise since objecs ineracing wih oher objecs are encounered all around us. more difficul concep is he idea of forces and sresses acing inside a maerial, wihin he inerior where neiher ee nor eperimen can reach as Euler pu i. I ook man grea minds working for cenuries on his quesion o arrive a he concep of sress we use oda, an idea finall brough o us b ugusin Cauch, who presened a paper on he subjec o he cadem of ciences in Paris, in ugusin Cauch Cauch s Concep of ress Uniform Inernal ress Consider firs a long slender block of maerial subjec o equilibraing forces a is ends, ig a. If he complee block is in equilibrium, hen an sub-division of he block mus be in equilibrium also. B imagining he block o be cu in wo, and considering free-bod diagrams of each half, as in ig b, one can see ha forces mus be acing wihin he block so ha each half is in equilibrium. Thus eernal loads creae inernal forces; inernal forces represen he acion of one par of a maerial on anoher par of he same maerial across an inernal surface. If he maerial ou of which he block is made is uniform over his cu, one can ake i ha a uniform sress / acs over his inerior surface, ig b. imaginar cu N (a ) (c) ( b) igure 3.3.1: a slender block of maerial; (a) under he acion of eernal forces, (b) inernal normal sress σ, (c) inernal normal and shear sress 42

2 Noe ha, if he inernal forces were no acing over he inernal surfaces, he wo halfblocks of ig b would fl apar; one can hus regard he inernal forces as hose required o mainain maerial in an un-cu sae. If he inernal surface is a an incline, as in ig c, hen he inernal force required for equilibrium will no ac normal o he surface. There will be componens of he force normal and angenial o he surface, and hus boh normal ( N ) and shear ( ) sresses mus arise. Thus, even hough he maerial is subjeced o a purel normal load, inernal shear sresses develop. rom ig a, he normal and shear sresses arising on an inerior surface inclined a angle o he horional are { Problem 1} N cos 2, sin cos (3.3.1) N inernal poin inernal surface igure 3.3.2: sress on inclined surface; (a) decomposing he force ino normal and shear forces, (b) sress a an inernal poin lhough sress is associaed wih surfaces, one can speak of he sress a a poin. or eample, consider some poin inerior o he block, ig 3.3.2b. The sress here evidenl depends on which surface hrough ha poin is under consideraion. rom Eqn a, he normal sress a he poin is a maimum / when 0 and a minimum of ero o when 90. The maimum normal sress arising a a poin wihin a maerial is of special significance, for eample i is his sress value which ofen deermines wheher a maerial will fail ( break ) here. I has a special name: he maimum principal sress. rom Eqn b, he maimum shear sress a he poin is / 2 and arises on o surfaces inclined a 45. Non-Uniform Inernal ress ( a) (b) Consider a more comple geomer under a more comple loading, as in ig gain, using equilibrium argumens, here will be some sress disribuion acing over an given inernal surface. To evaluae hese sresses is no an eas maer, and much of Par 43

3 II is devoed o doing jus ha. uffice o sa here ha he will invariabl be nonuniform over a surface, ha is, he sress a some paricle will differ from he sress a a neighbouring paricle N 1 igure 3.3.3: a componen subjeced o a comple loading, giving rise o a nonuniform sress disribuion over an inernal surface Tracion and he Phsical Meaning of Inernal ress ll maerials have a comple molecular microsrucure and each molecule eers a force on each of is neighbours. The comple ineracion of counless molecular forces mainains a bod in equilibrium in is unsressed sae. When he bod is disurbed and deformed ino a new equilibrium posiion, ne forces ac, ig a. n imaginar plane can be drawn hrough he maerial, ig b. Unlike some of his predecessors, who aemped he eremel difficul ask of accouning for all he molecular forces, Cauch discouned he molecular srucure of maer and simpl replaced he molecular forces acing on he plane b a single force, ig 3.3.4c. This is he force eered b he molecules above he plane on he maerial below he plane and can be aracive or repulsive. Differen planes can be aken hrough he same porion of maerial and, in general, a differen force will ac on he plane, ig 3.3.4d. ( a) ( b) ( c) (d) igure 3.3.4: a muliude of molecular forces represened b a single force; (a) molecular forces, a plane drawn hrough he maerial, replacing he molecular forces wih an equivalen force, a differen equivalen force acs on a differen plane hrough he same maerial The definiion of sress will now be made more precise. irs, define he racion a some paricular poin in a maerial as follows: ake a plane of surface area hrough he poin, on which acs a force. Ne shrink he plane as i shrinks in sie boh and ge smaller, and he direcion in which he force acs ma change, bu evenuall he raio / will remain consan and he force will ac in a paricular direcion, ig The 44

4 limiing value of his raio of force over surface area is defined as he racion vecor (or sress vecor) : lim (3.3.2) 0 a plane passing hrough some poin in he maerial igure 3.3.5: he racion vecor - he limiing value of force over area, as he surface area of he elemen on which he force acs is shrunk n infinie number of racion vecors ac a an single poin, since an infinie number of differen planes pass hrough a poin. Thus he noaion lim 0 / is ambiguous. or his reason he plane on which he racion vecor acs mus be specified; his can be done b specifing he normal n o he surface on which he racion acs, ig The racion is hus a special vecor associaed wih i is no onl he direcion in which i acs bu also a second direcion, he normal o he plane upon which i acs. n1 2 same poin wih differen planes passing hrough i (defined b differen normals) n n 1 differen forces ac on differen planes hrough he same poin n 2 lim ( n 1) 0 lim ( n 2 ) 0 igure 3.3.6: wo differen racion vecors acing a he same poin 45

5 ress Componens The racion vecor can be decomposed ino componens which ac normal and parallel o he surface upon which i acs. These componens are called he sress componens, or simpl sresses, and are denoed b he smbol ; subscrips are added o signif he surface on which he sresses ac and he direcions in which he sresses ac. Consider a paricular racion vecor acing on a surface elemen. Inroduce a Caresian coordinae ssem wih base vecors i, j, k so ha one of he base vecors is a normal o he surface, and he origin of he coordinae ssem is posiioned a he poin a which he racion acs. or eample, in ig , he k direcion is aken o be normal o he ( plane, and k ) i j k. (nˆ ) (k) i k j igure 3.3.7: he componens of he racion vecor Each of hese componens i is represened b ij where he firs subscrip denoes he direcion of he normal o he plane and he second denoes he direcion of he ( componen. Thus, re-drawing ig as ig : k ) i j k. The firs wo sresses, he componens acing angenial o he surface, are shear sresses, whereas, acing normal o he plane, is a normal sress 1. (k) i k j igure 3.3.8: sress componens he componens of he racion vecor The racion vecor shown in igs , 3.3.8, represens he force (per uni area) eered b he maerial above he surface on he maerial below he surface. B Newon s hird 1 his convenion for he subscrips is no universall followed. Man auhors, paricularl in he mahemaical communi, use he eac opposie convenion, he firs subscrip o denoe he direcion and he second o denoe he normal. I urns ou ha boh convenions are equivalen, since, as will be shown laer, he sress is smmeric, i.e. ij ji 46

6 law, an equal and opposie racion mus be eered b he maerial below he surface on (k) he maerial above he surface, as shown in ig (hick doed line). If has sress ( k) componens,,, hen so should ( k) ( k) : ( i) ( j) ( k). (k) k i j ( k) igure 3.3.9: equal and opposie racion vecors each wih he same sress componens ign Convenion for ress Componens The following convenion is used: The sress is posiive when he direcion of he normal and he direcion of he sress componen are boh posiive or boh negaive The sress is negaive when one of he direcions is posiive and he oher is negaive ccording o his convenion, he hree sresses in igs are all posiive. Looking a he wo-dimensional case for ease of visualisaion, he (posiive and negaive) normal sresses and shear sresses on eiher side of a surface are as shown in ig Normal sresses which pull (ension) are posiive; normal sresses which push (compression) are negaive. Noe ha he shear sresses alwas go in opposie direcions. ( a) (b) igure : sresses acing on eiher side of a maerial surface: (a) posiive sresses, (b) negaive sresses Eamples of negaive sresses are shown in ig { Problem 4}. 47

7 e 2 k j i ( j) k j i ( j) ( j) ( i j k j ) i j k ( a) (b) igure : eamples of negaive sress componens Real Problems and ain-venan s Principle ome eamples have been given earlier of eernal forces acing on maerials. In reali, an eernal force will be applied o a real maerial componen in a comple wa. or eample, suppose ha a block of maerial, welded o a large objec a one end, is pulled a is oher end b a rope aached o a meal hoop, which is iself aached o he block b a number of bols, ig a. The block can be idealised as in ig b; here, he precise deails of he region in which he eernal force is applied are negleced. (a) (e) (b) (c) / ( d) (f ) sress he same (g) sress differs here / 2 / 2 igure : a block subjeced o an eernal force: (a) real case, (b) ideal model, (c) sress in ideal model, (d) sress in acual maerial, (e) he sress in he real maerial, awa from he righ hand end, is modelled well b eiher (f) or (g) ccording o he earlier discussion, he sress in he ideal model is as in ig c. One will find ha, in he real maerial, he sress is indeed (approimael) as prediced, bu 48

8 onl a an appreciable disance from he righ hand end. Near where he rope is aached, he force will differ considerabl, as skeched in ig d. Thus he ideal models of he pe discussed in his secion, and in much of his book, are useful onl in predicing he sress field in real componens in regions awa from poins of applicaion of loads. This does no presen oo much of a problem, since he sresses inernal o a srucure in such regions are ofen of mos ineres. If one wans o know wha happens near he boled connecion, hen one will have o creae a comple model incorporaing all he deails and he problem will be more difficul o solve. I is an eperimenal fac ha if wo differen force ssems are applied o a maerial, bu he are equivalen force ssems, as in ig (f,g), hen he sress fields in regions awa from where he loads are applied will be he same. This is known as ain- Venan s Principle. Tpicall, one needs o move a disance awa from where he loads are applied roughl equal o he disance over which he loads are applied Problems 1. Derive Eqns The four sides of a square block are subjeced o equal forces, as illusraed. The lengh of each side is l and he block has uni deph (ino he page). Wha normal and shear sresses ac along he (doed) diagonal? [Hin: draw a free bod diagram of he upper lef hand riangle.] 3. shaf is concreed firml ino he ground. hick seel rope is looped around he shaf and a force is applied normal o he shaf, as shown. The shaf is in saic equilibrium. Draw a free bod diagram of he shaf (from he op down o ground level) showing he forces/momens acing on he shaf (including he reacion forces a he ground-level; ignore he weigh of he shaf). Draw a free bod diagram of he secion of shaf from he op down o he cross secion a. Draw a free bod diagram of he secion of shaf from he op o he cross secion a B. Roughl skech he sresses acing over he (horional) inernal surfaces of he shaf a and B. B ground 49

9 4. In ig , which of he sress componens is/are negaive? 5. Label he following sress componen acing on an inernal maerial surface. Is i a posiive or negaive sress? acing parallel o surface 6. Label he following shear sresses. re he posiive or negaive? 7. Label he following normal sresses. re he posiive or negaive? 8. B he definiion of he racion vecor which acs on he plane, ( j) i j k. kech hese hree sress componens on he figure below. 50

10 ource: hp://homepages.engineering.auckland.ac.n/~pkel015/olidmechanicsbooks/par_i/ BookM_Par_I/03_ress/03_ress_03_Inernal_ress.pdf 51