8.1 Introduction to Plasticity

Transcription

1 Section Introduction to Platicity 8.. Introduction The theory of linear elaticity i ueful for modelling material which undergo mall deformation and which return to their original configuration uon removal of load. Almot all real material will undergo ome ermanent deformation, which remain after removal of load. With metal, ignificant ermanent deformation will uually occur when the tre reache ome critical value, called the yield tre, a material roerty. latic deformation are termed reverible; the energy exended in deformation i tored a elatic train energy and i comletely recovered uon load removal. Permanent deformation involve the diiation of energy; uch rocee are termed irreverible, in the ene that the original tate can be achieved only by the exenditure of more energy. The claical theory of laticity grew out of the tudy of metal in the late nineteenth century. It i concerned with material which initially deform elatically, but which deform latically uon reaching a yield tre. In metal and other crytalline material the occurrence of latic deformation at the micro-cale level i due to the motion of dilocation and the migration of grain boundarie on the micro-level. In and and other granular material latic flow i due both to the irreverible rearrangement of individual article and to the irreverible cruhing of individual article. Similarly, comreion of bone to high tre level will lead to article cruhing. The deformation of microvoid and the develoment of micro-crack i alo an imortant caue of latic deformation in material uch a rock. A good art of the dicuion in what follow i concerned with the laticity of metal; thi i the imlet tye of laticity and it erve a a good background and introduction to the modelling of laticity in other material-tye. There are two broad grou of metal laticity roblem which are of interet to the engineer and analyt. The firt involve relatively mall latic train, often of the ame order a the elatic train which occur. Analyi of roblem involving mall latic train allow one to deign tructure otimally, o that they will not fail when in ervice, but at the ame time are not tronger than they really need to be. In thi ene, laticity i een a a material failure. The econd tye of roblem involve very large train and deformation, o large that the elatic train can be diregarded. Thee roblem occur in the analyi of metal manufacturing and forming rocee, which can involve extruion, drawing, forging, rolling and o on. In thee latter-tye roblem, a imlified model known a erfect laticity i uually emloyed (ee below), and ue i made of ecial limit theorem which hold for uch model. Platic deformation are normally rate indeendent, that i, the tree induced are indeendent of the rate of deformation (or rate of loading). Thi i in marked two other tye of failure, brittle fracture, due to dynamic crack growth, and the buckling of ome tructural comonent, can be modelled reaonably accurately uing elaticity theory (ee, for examle, Part I, 6., Part II, 5.) 4

2 Section 8. contrat to claical Newtonian fluid for examle, where the tre level are governed by the rate of deformation through the vicoity of the fluid. Material commonly known a latic are not latic in the ene decribed here. They, like other olymeric material, exhibit vicoelatic behaviour where, a the name ugget, the material reone ha both elatic and vicou comonent. Due to their vicoity, their reone i, unlike the latic material, rate-deendent. Further, although the vicoelatic material can uffer irrecoverable deformation, they do not have any critical yield or threhold tre, which i the characteritic roerty of latic behaviour. When a material undergoe latic deformation, i.e. irrecoverable and at a critical yield tre, and thee effect are rate deendent, the material i referred to a being vicolatic. Platicity theory began with Treca in 864, when he undertook an exerimental rogram into the extruion of metal and ublihed hi famou yield criterion dicued later on. Further advance with yield criteria and latic flow rule were made in the year which followed by Saint-Venant, Levy, Von Mie, Hencky and Prandtl. The 94 aw the advent of the claical theory; Prager, Hill, Drucker and Koiter amongt other brought together many fundamental aect of the theory into a ingle framework. The arrival of owerful comuter in the 98 and 99 rovided the imetu to develo the theory further, giving it a more rigorou foundation baed on thermodynamic rincile, and brought with it the need to conider many numerical and comutational aect to the laticity roblem. 8.. Obervation from Standard Tet In thi ection, a number of henomena oberved in the material teting of metal will be noted. Some of thee henomena are imlified or ignored in ome of the tandard laticity model dicued later on. At iue here i the fact that any model of a comonent with comlex geometry, loaded in a comlex way and undergoing latic deformation, mut involve material arameter which can be obtained in a traight forward manner from imle laboratory tet, uch a the tenion tet decribed next. The Tenion Tet Conider the following key exeriment, the tenile tet, in which a mall, uually cylindrical, ecimen i gried and tretched, uually at ome given rate of tretching (ee Part I, 5..). The force required to hold the ecimen at a given tretch i recorded, Fig If the material i a metal, the deformation remain elatic u to a certain force level, the yield oint of the material. Beyond thi oint, ermanent latic deformation are induced. On unloading only the elatic deformation i recovered and the ecimen will have undergone a ermanent elongation (and conequent lateral contraction). In the elatic range the force-dilacement behaviour for mot engineering material (metal, rock, latic, but not oil) i linear. After aing the elatic limit (oint A in Fig. 8..), the material give and i aid to undergo latic flow. Further increae in load are uually required to maintain the latic flow and an increae in dilacement; thi 4

3 Section 8. henomenon i known a work-hardening or train-hardening. In ome cae, after an initial latic flow and hardening, the force-dilacement curve decreae, a in ome oil; the material i aid to be oftening. If the ecimen i unloaded from a latic tate (B) it will return along the ath BC hown, arallel to the original elatic line. Thi i elatic recovery. The train which remain uon unloading i the ermanent latic deformation. If the material i now loaded again, the force-dilacement curve will retrace the unloading ath CB until it again reache the latic tate. Further increae in tre will caue the curve to follow BD. Two imortant obervation concerning the above tenion tet (on mot metal) are the following: () after the onet of latic deformation, the material will be een to undergo negligible volume change, that i, it i incomreible. () the force-dilacement curve i more or le the ame regardle of the rate at which the ecimen i tretched (at leat at moderate temerature). force B hardening D elatic loading ield oint A unload load latic deformation C elatic deformation dilacement Figure 8..: force/dilacement curve for the tenion tet Nominal and True Stre and Strain There are two different way of decribing the force F which act in a tenion tet. Firt, normaliing with reect to the original cro ectional area of the tenion tet ecimen A, one ha the nominal tre or engineering tre, F (8..) n A Alternatively, one can normalie with reect to the current cro-ectional area A, leading to the true tre, F (8..) A 4

4 Section 8. in which F and A are both changing with time. For very mall elongation, within the elatic range ay, the cro-ectional area of the material undergoe negligible change and both definition of tre are more or le equivalent. Similarly, one can decribe the deformation in two alternative way. Denoting the original ecimen length by l and the current length by l, one ha the engineering train l l (8..) l Alternatively, the true train i baed on the fact that the original length i continually changing; a mall change in length dl lead to a train increment d dl / l and the total train i defined a the accumulation of thee increment: l dl l ln t (8..4) l l l The true train i alo called the logarithmic train or Hencky train. Again, at mall deformation, the difference between thee two train meaure i negligible. The true train and engineering train are related through t ln (8..5) Uing the aumtion of contant volume for latic deformation and ignoring the very mall elatic volume change, one ha alo { Problem } l. (8..6) n l The tre-train diagram for a tenion tet can now be decribed uing the true tre/train or nominal tre/train definition, a in Fig The hae of the nominal tre/train diagram, Fig. 8..a, i of coure the ame a the grah of force veru dilacement (change in length) in Fig A here denote the oint at which the maximum force the ecimen can withtand ha been reached. The nominal tre at A i called the Ultimate Tenile Strength (UTS) of the material. After thi oint, the ecimen neck, with a very raid reduction in cro-ectional area omewhere about the centre of the ecimen until the ecimen ruture, a indicated by the aterik. Note that, during loading into the latic region, the yield tre increae. For examle, if one unload and re-load (a in Fig. 8..), the material tay elatic u until a tre higher than the original yield tre. In thi reect, the tre-train curve can be regarded a a yield tre veru train curve. 44

5 Section 8. n A A ( a) (b) t Figure 8..: tyical tre/train curve; (a) engineering tre and train, (b) true tre and train Comreion Tet A comreion tet will lead to imilar reult a the tenile tre. The yield tre in comreion will be aroximately the ame a (the negative of) the yield tre in tenion. If one lot the true tre veru true train curve for both tenion and comreion (abolute value for the comreion), the two curve will more or le coincide. Thi would indicate that the behaviour of the material under comreion i broadly imilar to that under tenion. If one were to ue the nominal tre and train, then the two curve would not coincide; thi i one of a number of good reaon for uing the true definition. The Bauchinger ffect If one take a virgin amle and load it in tenion into the latic range, and then unload it and continue on into comreion, one find that the yield tre in comreion i not the ame a the yield trength in tenion, a it would have been if the ecimen had not firt been loaded in tenion. In fact the yield oint in thi cae will be ignificantly le than the correonding yield tre in tenion. Thi reduction in yield tre i known a the Bauchinger effect. The effect i illutrated in Fig The olid line deict the reone of a real material. The dotted line are two extreme cae which are ued in laticity model; the firt i the iotroic hardening model, in which the yield tre in tenion and comreion are maintained equal, the econd being kinematic hardening, in which the total elatic range i maintained contant throughout the deformation. 45

6 Section 8. t kinematic hardening iotroic hardening Figure 8..: The Bauchinger effect The reence of the Bauchinger effect comlicate any laticity theory. However, it i not an iue rovided there are no reveral of tre in the roblem under tudy. Hydrotatic Preure Careful exeriment how that, for metal, the yield behaviour i indeendent of hydrotatic reure. That i, a tre tate ha negligible effect on the yield tre of a material, right u to very high reure. Note however that thi i not true for oil or rock. 8.. Aumtion of Platicity Theory Regarding the above tet reult then, in formulating a baic laticity theory with which to begin, the following aumtion are uually made: () the reone i indeendent of rate effect () the material i incomreible in the latic range () there i no Bauchinger effect (4) the yield tre i indeendent of hydrotatic reure (5) the material i iotroic The firt two of thee will uually be very good aroximation, the other three may or may not be, deending on the material and circumtance. For examle, mot metal can be regarded a iotroic. After large latic deformation however, for examle in rolling, the material will have become aniotroic: there will be ditinct material direction and aymmetrie. Together with thee, aumtion can be made on the tye of hardening and on whether elatic deformation are ignificant. For examle, conider the hierarchy of model illutrated in Fig below, commonly ued in theoretical analye. In (a) both the elatic and latic curve are aumed linear. In (b) work-hardening i neglected and the 46

7 Section 8. yield tre i contant after initial yield. Such erfectly-latic model are articularly aroriate for tudying rocee where the metal i worked at a high temerature uch a hot rolling where work hardening i mall. In many area of alication the train involved are large, e.g. in metal working rocee uch a extruion, rolling or drawing, where u to 5% reduction ratio are common. In uch cae the elatic train can be neglected altogether a in the two model (c) and (d). The rigid/erfectly-latic model (d) i the crudet of all and hence in many way the mot ueful. It i widely ued in analying metal forming rocee, in the deign of teel and concrete tructure and in the analyi of oil and rock tability. (a) Linear latic-platic (b) latic/perfectly-platic (c) Rigid/Linear Hardening (d) Rigid-Perfectly-Platic Figure 8..4: Simle model of elatic and latic deformation 8..4 The Tangent and Platic Modulu Stre and train are related through in the elatic region, being the oung modulu, Fig The tangent modulu K i the loe of the tre-train curve in the latic region and will in general change during a deformation. At any intant of train, the increment in tre d i related to the increment in train d through d Kd (8..7) the ymbol here rereent the true train (the ubcrit t ha been droed for clarity); a mentioned, when the train are mall, it i not neceary to ecify which train i in ue ince all train meaure are then equivalent 47

8 Section 8. K d d e d Figure 8..5: The tangent modulu After yield, the train increment conit of both elatic, d d d e, and latic, d, train: e (8..8) The tre and latic train increment are related by the latic modulu H: d H d (8..9) and it follow that { Problem 4} K (8..) H 8..5 Friction Block Model Some additional inight into the way latic material reond can be obtained from friction block model. The rigid erfectly latic model can be imulated by a Coulomb friction block, Fig No train occur until reache the yield tre. Then there i movement although the amount of movement or latic train cannot be determined without more information being available. The tre cannot exceed the yield tre in thi model: (8..) If unloaded, the block to moving and the tre return to zero, leaving a ermanent train, Fig. 8..6b. 48

9 Section 8. unload (a) ermanent deformation (b) Figure 8..6: (a) Friction block model for the rigid erfectly latic material, (b) reone of the rigid-erfectly latic model The linear elatic erfectly latic model incororate a free ring with modulu in erie with a friction block, Fig The ring tretche when loaded and the block alo begin to move when the tre reache, at which time the ring to tretching, the maximum oible tre again being. Uon unloading, the block to moving and the ring contract. Figure 8..7: Friction block model for the elatic erfectly latic material The linear elatic latic model with linear train hardening incororate a econd, hardening, ring with tiffne H, in arallel with the friction block, Fig Once the yield tre i reached, an ever increaing tre need to be alied in order to kee the block moving and elatic train continue to occur due to further elongation of the free ring. The tre i then lit into the yield tre, which i carried by the moving block, and an overtre carried by the hardening ring. Uon unloading, the block lock the tre in the hardening ring remain contant whilt the free ring contract. At zero tre, there i a negative tre taken u by the friction block, equal and ooite to the tre in the hardening ring. The loe of the elatic loading line i. For the latic hardening line, e H K d d H H (8..) It can be een that H i the latic modulu. 49

10 Section 8. (a) H (b) e (c) e (d) e Figure 8..8: Friction block model for a linear elatic-latic material with linear train hardening; (a) tre-free, (b) elatic train, (c) elatic and latic train, (d) unloading 8..6 Problem. Give two difference between latic and vicoelatic material.. A tet ecimen of initial length. m i extended to length. m. What i the ercentage difference between the engineering and true train (relative to the engineering train)? What i thi difference when the ecimen i extended to length.5m?. Derive the relation 8..6, / n l / l. 4. Derive qn Which i larger, H or K? In the cae of a erfectly-latic material? 6. The Ramberg-Ogood model of laticity i given by e b where i the oung modulu and b and n are model contant (material arameter) obtained from a curve-fitting of the uniaxial tre-train curve. n 5

11 Section 8. (i) (ii) Find the tangent and latic moduli in term of latic train (and the material contant). A material with model 5.5 x 8 7GPa arameter n 4, 5 7GPa and 4.5 b 8MPa 4 b 8MPa i trained n 4.5 n in tenion to. and i ubequently.5 n unloaded and ut into comreion. Find the.5 tre at the initiation of n comreive yield.5 auming iotroic hardening [Note that the yield tre i actually zero in thi model, although the latic train at relatively low tre level i mall for larger value of n.] 7. Conider the laticity model hown below. (i) What i the elatic modulu? (ii) What i the yield tre? (iii) What are the tangent and latic moduli? Draw a tyical loading and unloading curve. 8. Draw the tre-train diagram for a cycle of loading and unloading to the rigid - latic model hown here. Take the maximum load reached to be max 4 and. What i the ermanent deformation after comlete removal of the load? [Hint: lit the cycle into the following region: (a), (b), (c) 4, then unload, (d) 4, (e), (f).] 5

12 Section Stre Analyi for Platicity Thi ection follow on from the analyi of three dimenional tre carried out in 7.. The latic behaviour of material i often indeendent of a hydrotatic tre and thi feature neceitate the tudy of the deviatoric tre. 8.. Deviatoric Stre Any tate of tre can be decomoed into a hydrotatic (or mean) tre mi and a deviatoric tre, according to m = m + m (8..) where + + m = (8..) and = ( ) ( ) ( ) (8..) In index notation, = δ + (8..4) m In a comletely analogou manner to the derivation of the rincial tree and the rincial calar invariant of the tre matrix, 7..4, one can determine the rincial tree and rincial calar invariant of the deviatoric tre matrix. The former are denoted,, and the latter are denoted by J, J, J. The characteritic equation analogou to qn. 7.. i J J J = (8..5) and the deviatoric invariant are (comare with 7..4, 7..6) unfortunately, there i a convention (adhered to by mot author) to write the characteritic equation for tre with a + I term and that for deviatoric tre with a J term; thi mean that the formulae for J in qn are the negative of thoe for I in qn

13 Section 8. J J J = + + = + + = = = = ( + + ) ( + + ) + (8..6) Since the hydrotatic tre remain unchanged with a change of coordinate ytem, the rincial direction of tre coincide with the rincial direction of the deviatoric tre, and the decomoition can be exreed with reect to the rincial direction a m = m + m (8..7) Note that, from the definition qn. 8.., the firt invariant of the deviatoric tre, the um of the normal tree, i zero: J = ( 8..8) The econd invariant can alo be exreed in the ueful form { Problem } ( + ) J = +, (8..9) and, in term of the rincial tree, { Problem 4} [( ) + ( ) + ( ) ] J =. (8..) 6 Further, the deviatoric invariant are related to the tre tenor invariant through { Problem 5} A State of Pure Shear ( I I ), J = ( I 9I I I ) J = + 7 (8..) The tre tate at a oint i one of ure hear if for any one coordinate axe through the oint one ha only hear tre acting, i.e. the tre matrix i of the form 7 [ ] = (8..) 5

14 Section 8. Alying the tre tranformation rule 7..6 to thi tre matrix and uing the fact that the tranformation matrix Q i orthogonal, i.e. QQ T = Q T Q = I, one find that the firt invariant i zero, + +. Hence the deviatoric tre i one of ure hear. = 8.. The Octahedral Stree xamine now a material element ubjected to rincial tree,, a hown in Fig By definition, no hear tree act on the lane hown. Figure 8..: tree acting on a material element Conider next the octahedral lane; thi i the lane hown haded in Fig. 8.., whoe normal n a make equal angle with the rincial direction. It i o-called becaue it cut a cubic material element (with face erendicular to the rincial direction) into a triangular lane and eight of thee triangle around the origin form an octahedron. n a Figure 8..: the octahedral lane Next, a new Carteian coordinate ytem i contructed with axe arallel and erendicular to the octahedral lane, Fig One axi run along the unit normal n a ; 54

15 thi normal ha comonent ( /,/,/ ) Section 8. with reect to the rincial axe. The angle θ the normal direction make with the direction can be obtained from e = e,, i a unit vector in the direction, Fig To n a, where ( ) coθ comlete the new coordinate ytem, any two erendicular unit vector which lie in (arallel to) the octahedral lane can be choen. Chooe one which i along the rojection of the axi down onto the octahedral lane. The comonent of thi vector = are { Problem 6} = ( /, / 6, / 6) n c. The final unit vector b that it form a right hand Carteian coordinate ytem with In ummary, n a and n c, i.e. n i choen o n n = n. n = =, = a, n b n c (8..) 6 a b c b a θ n b e n a θ n c θ e c e Figure 8..: a new Carteian coordinate ytem To exre the tre tate in term of comonent in the tre tranformation matrix: a, b, c direction, contruct the e / / 6 n a e nb e n c Q = e = / / / 6 n a e n b e n c (8..4) e n a e nb e n c / / / 6 and the new tre comonent are aa ba ca ab bb cb ac bc = cc = Q T ( ) ( ) ( ) ( ) ( + ) ( ) 6 ( ) ( ) ( ) Q (8..5) 55

16 Section 8. Now conider the tre comonent acting on the octahedral lane, aa, ab, ac, Fig Recall from Cauchy law, qn. 7..9, that thee are the comonent of ) the traction vector ( n t a acting on the octahedral lane, with reect to the (a,b,c) axe: ( a t n ) = n + n + n aa a ab b ac c (8..6) ( n t a ) ab aa = oct ac τ oct Figure 8..4: the tre vector and it comonent The magnitude of the normal and hear tree acting on the octahedral lane are called the octahedral normal tre oct and the octahedral hear tre τ oct. Referring to Fig. 8..4, thee can be exreed a { Problem 7} τ oct oct = aa = = ab + = ( + + ) ac ( ) + ( ) + ( ) = I = J (8..7) The octahedral normal and hear tree on all 8 octahedral lane around the origin are the ame. Note that the octahedral normal tre i imly the hydrotatic tre. Thi imlie that the deviatoric tre ha no normal comonent in the direction n a and only contribute to hearing on the octahedral lane. Indeed, from qn. 8..5, aa ba ca ab bb cb ac bc cc = ( ) ( ) 6 ( ) ( ) ( ) 6 6 ( ) ( ) ( ) 6 (8..8) 56

17 Section 8. The on the right here can be relaced with ince =. i j i j 8.. Problem. What are the hydrotatic and deviatoric tree for the uniaxial tre =? What are the hydrotatic and deviatoric tree for the tate of ure hear = τ In both cae, verify that the firt invariant of the deviatoric tre i zero: J =.?. For the tre tate (a) the hydrotatic tre (b) the deviatoric tree (c) the deviatoric invariant = 4 4, calculate. The econd invariant of the deviatoric tre i given by qn. 8..6, J = ( + + ) By quaring the relation J = + +, derive qn. 8..9, ( + ) J + = = 4. Ue qn (and your work from Problem ) and the fact that =, etc. to derive 8.., J = + + [( ) ( ) ( ) ] 6 5. Ue the fact that J = + + = to how that I = m I = ( + + ) + m I = + m ( + + ) + m Hence derive qn. 8.., J = I I, J = I 9II + 7I ( ) ( ) 6. Show that a unit normal n c in the octahedral lane in the direction of the rojection of the axi down onto the octahedral lane ha coordinate ( ),,, Fig To do thi, note the geometry hown below and the fact that when the axi i rojected down, it remain at equal angle to the and axe. 7 co θ = n a θ x roject down n c Octahedral lane 57

18 Section Ue qn to derive qn For the tre tate of roblem, calculate the octahedral normal tre and the octahedral hear tre 58

19 Section ield Criteria in Three Dimenional Platicity The quetion now arie: a material yield at a tre level in a uniaxial tenion tet, but when doe it yield when ubjected to a comlex three-dimenional tre tate? Let u begin with a very general cae: an aniotroic material with different yield trength in different direction. For examle, conider the material hown in Fig Thi i a comoite material with long fibre along the x direction, giving it extra trength in that direction it will yield at a higher tenion when ulled in the x direction than when ulled in other direction. x x x x fibre binder bundle ( a ) ( b ) x Figure 8..: an aniotroic material; (a) microtructural detail, (b) continuum model We can aume that yield will occur at a article when ome combination of the tre comonent reache ome critical value, ay when F,,,,, ) k. (8..) ( Here, F i ome function of the 6 indeendent comonent of the tre tenor and k i ome material roerty which can be determined exerimentally. Alternatively, it i very convenient to exre yield criteria in term of rincial tree. Let u uoe that we know the rincial tree everywhere, (,, ), Fig ield mut deend omehow on the microtructure on the orientation of the axe x, x, x, but thi information i not contained in the three number (,, ). Thu we exre the yield criterion in term of rincial tree in the form F,,, i ) k (8..) ( n F will no doubt alo contain other arameter which need to be determined exerimentally 59

20 Section 8. where n i rereent the rincial direction thee give the orientation of the rincial tree relative to the material direction x, x, x. If the material i iotroic, the reone i indeendent of any material direction indeendent of any direction the tre act in, and o the yield criterion can be exreed in the imle form F,, ) k (8..) ( Further, ince it hould not matter which direction i labelled, which and which, F mut be a ymmetric function of the three rincial tree. Alternatively, ince the three rincial invariant of tre are indeendent of material orientation, one can write or, more uually, F( I, I, I ) k (8..4) F( I, J, J ) k (8..5) where J, J are the non-zero rincial invariant of the deviatoric tre. With the further retriction that the yield tre i indeendent of the hydrotatic tre, one ha F( J, J ) k (8..6) 8.. The Treca and Von Mie ield Condition The two mot commonly ued and ucceful yield criteria for iotroic metallic material are the Treca and Von Mie criteria. The Treca ield Condition The Treca yield criterion tate that a material will yield if the maximum hear tre reache ome critical value, that i, qn. 8.. take the form max,, k (8..7) The value of k can be obtained from a imle exeriment. For examle, in a tenion tet,,, and failure occur when reache, the yield tre in tenion. It follow that k. (8..8) 6

21 Section 8. In a hear tet,,,, and failure occur when reache, the yield tre of a material in ure hear, o that k. The Von Mie ield Condition The Von Mie criterion tate that yield occur when the rincial tree atify the relation 6 k (8..9) Again, from a uniaxial tenion tet, one find that the k in qn i k. (8..) Writing the Von Mie condition in term of, one ha (8..) The quantity on the left i called the Von Mie Stre, ometime denoted by VM. When it reache the yield tre in ure tenion, the material begin to deform latically. In the hear tet, one again find that k, the yield tre in ure hear. Sometime it i referable to work with arbitrary tre comonent; for thi uroe, the Von Mie condition can be exreed a { Problem } 6 k (8..) 6 The iecewie linear nature of the Treca yield condition i ometime a theoretical advantage over the quadratic Mie condition. However, the fact that in many roblem one often doe not know which rincial tre i the maximum and which i the minimum caue difficultie when working with the Treca criterion. The Treca and Von Mie ield Criteria in term of Invariant From qn. 8.. and 8..9, the Von Mie criterion can be exreed a f ( J ) J k (8..) Note the relationhi between J and the octahedral hear tre, qn. 8..7; the Von Mie criterion can be interreted a redicting yield when the octahedral hear tre reache a critical value. With, the Treca condition can be exreed a 6

22 Section f ( J, J ) 4J 7J 6k J 96k J 64k (8..4) but thi exreion i too cumberome to be of much ue. xeriment of Taylor and Quinney In order to tet whether the Von Mie or Treca criteria bet modelled the real behaviour of metal, G I Taylor & Quinney (9), in a erie of claic exeriment, ubjected a number of thin-walled cylinder made of coer and teel to combined tenion and torion, Fig Figure 8..: combined tenion and torion of a thin-walled tube The cylinder wall i in a tate of lane tre, with and all other tre comonent zero. The rincial tree correonding to uch a tre-tate are (zero and) { Problem } 4 (8..5) and o Treca' condition reduce to 4 4k or / (8..6) The Mie condition reduce to { Problem 4} k or / (8..7) Thu both model redict an ellitical yield locu in, tre ace, but with different ratio of rincial axe, Fig The origin in Fig. 8.. correond to an untreed tate. The horizontal axe refer to uniaxial tenion in the abence of hear, wherea the vertical axi refer to ure torion in the abence of tenion. When there i a combination of and, one i off-axe. If the combination remain inide the yield locu, the material remain elatic; if the combination i uch that one reache anywhere along the locu, then laticity enue. 6

23 Section 8. / Mie / Treca Figure 8..: the yield locu for a thin-walled tube in combined tenion and torion Taylor and Quinney, by varying the amount of tenion and torion, found that their meaurement were cloer to the Mie ellie than the Treca locu, a reult which ha been reeatedly confirmed by other worker. D Princial Stre Sace Fig. 8.. give a geometric interretation of the Treca and Von Mie yield criteria in, ace. It i more uual to interret yield criteria geometrically in a rincial tre ace. The Taylor and Quinney tet are an examle of lane tre, where one rincial tre i zero. Following the convention for lane tre, label now the two non-zero rincial tree and, o that (even if it i not the minimum rincial tre). The criteria can then be dilayed in, D rincial tre ace. With, one ha Treca: max, Von Mie:, (8..8) Thee are lotted in Fig The Treca criterion i a hexagon. The Von Mie criterion i an ellie with axe inclined at 45 to the rincial axe, which can be een by exreing qn. 8..8b in the canonical form for an ellie: the maximum difference between the redicted tree from the two criteria i about 5%. The two criteria can therefore be made to agree to within 7.5% by chooing k to be half-way between / and / 6

24 Section 8. / / / / / / / / / / / / / where, are coordinate along the new axe; the major axi i thu and the minor axi i /. Some tre tate are hown in the tre ace: oint A correond to a uniaxial tenion, B to a equi-biaxial tenion and C to a ure hear. B C D A Figure 8..4: yield loci in D rincial tre ace Again, oint inide thee loci rereent an elatic tre tate. Any combination of rincial tree which uh the oint out to the yield loci reult in latic deformation. 8.. Three Dimenional Princial Stre Sace The D rincial tre ace ha limited ue. For examle, a tre tate that might tart out two dimenional can develo into a fully three dimenional tre tate a deformation roceed. 64

25 Section 8. In three dimenional rincial tre ace, one ha a yield urface f,,, Fig In thi cae, one can draw a line at equal angle to all three rincial tre axe, the ace diagonal. Along the ace diagonal and o oint on it are in a tate of hydrotatic tre. Aume now, for the moment, that hydrotatic tre doe not affect yield and conider ome arbitrary oint A,,, a, b, c, on the yield urface, Fig A ure hydrotatic tre can be uerimoed on thi tre tate without affecting yield, o h,, any other oint a, b, c h h h will alo be on the yield urface. xamle of uch oint are hown at B, C and D, which are obtained from A by moving along a line arallel to the ace diagonal. The yield behaviour of the material i therefore ecified by a yield locu on a lane erendicular to the ace diagonal, and the yield urface i generated by liding thi locu u and down the ace diagonal. hydrotatic tre the π - lane yield locu D ρ C B A a, b, c deviatoric tre Figure 8..5: ield locu/urface in three dimenional tre-ace The -lane Any urface in tre ace can be decribed by an equation of the form, cont f (8..9), and a normal to thi urface i the gradient vector f e f e f e (8..) where e, e, e are unit vector along the tre ace axe. In articular, any lane erendicular to the ace diagonal i decribed by the equation a mentioned, one ha a ix dimenional tre ace for an aniotroic material and thi cannot be viualied 65

26 Section 8. cont (8..) Without lo of generality, one can chooe a a rereentative lane the π lane, which i defined by. For examle, the oint,,,, i on the π lane and, with yielding indeendent of hydrotatic tre, i equivalent to oint in rincial tre ace which differ by a hydrotatic tre, e.g. the oint,,,,,, etc., can be regarded a the um of the tre tate at the correonding oint on the π lane, D, rereented, together with a hydrotatic tre rereented by the vector The tre tate at any oint A rereented by the vector, by the vector, ρ,, : m m m,,,,,, m m m (8..) The comonent of the firt term/vector on the right here um to zero ince it lie on the π lane, and thi i the deviatoric tre, whilt the hydrotatic tre i / m. Projected view of the -lane Fig. 8..6a how rincial tre ace and Fig. 8..6b how the π lane. The heavy line,, in Fig. 8..6b rereent the rojection of the rincial axe down onto the lane (o one i looking down the ace diagonal). Some oint, A, B, C in tre ace and their rojection onto the lane are alo hown. Alo hown i ome oint D on the lane. It hould be ket in mind that the deviatoric tre vector in the rojected view of Fig. 8..6b i in reality a three dimenional vector (ee the correonding vector in Fig. 8..6a). m m m D A B C ace diagonal B C D A ( a) (b) Figure 8..6: Stre ace; (a) rincial tre ace, (b) the π lane 66

27 Section 8. Conider the more detailed Fig below. Point A here rereent the tre tate,,, a indicated by the arrow in the figure. It can alo be reached in different way, for examle it rereent,, and,,. Thee three tre tate of coure differ by a hydrotatic tre. The actual lane value for A i the one for which 5 4, i.e.,,,,,,. Point B and C alo rereent multile tre tate { Problem 7}. B A C D biector Figure 8..7: the -lane The biector of the rincial lane rojection, uch a the dotted line in Fig. 8..7, rereent tate of ure hear. For examle, the lane value for oint D i,,, correonding to a ure hear in the lane. The dahed line in Fig are helful in that they allow u to lot and viualie tre tate eaily. The ditance between each dahed line along the direction of the rojected axe rereent one unit of rincial tre. Note, however, that thee unit are not conitent with the actual magnitude of the deviatoric vector in the lane. To create a more comlete icture, note firt that a unit vector along the ace diagonal i n,,, Fig The comonent of thi normal are the direction coine; for examle, a unit normal along the rincial axi i,, between the axi and the ace diagonal i given by n e and o the angle e co. From Fig. 8..8, the angle between the axi and the lane i given by co, and o a length of unit get rojected down to a length.,, which i on the lane. The length of the vector out to in Fig i unit. To For examle, oint in Fig rereent a ure hear,,, 67

28 Section 8. convert to actual magnitude, multily by to get, which agree with. n ace diagonal n -lane Figure 8..8: rincial tre rojected onto the -lane Tyical -lane ield Loci Conider next an arbitrary oint ( a, b, c) on the lane yield locu. If the material i iotroic, the oint ( a, c, b), ( b, a, c), ( b, c, a), ( c, a, b) and ( c, b, a) are alo on the yield locu. If one aume the ame yield behaviour in tenion a in comreion, e.g. neglecting the Bauchinger effect, then o alo are the oint ( a, b, c), ( a, c, b), etc. Thu oint become and one need only conider the yield locu in one o ector of the lane, the ret of the locu being generated through ymmetry. One uch ector i hown in Fig. 8..9, the axe of ymmetry being the three rojected rincial axe and their (ure hear) biector. yield locu Figure 8..9: A tyical ector of the yield locu The Treca and Von Mie ield Loci in the -lane The Treca criterion, qn. 8..7, i a regular hexagon in the lane a illutrated in Fig Which of the ix ide of the locu i relevant deend on which of,, i the maximum and which i the minimum, and whether they are tenile or comreive. 68

29 Section 8. For examle, yield at the ure hear /,, / i indicated by oint A in the figure. Point B rereent yield under uniaxial tenion, magnitude of the hexagon, i therefore,,., i. The ditance ob, the ; the correonding oint on the lane A criticim of the Treca criterion i that there i a udden change in the lane uon which failure occur uon a mall change in tre at the har corner of the hexagon. o B A Figure 8..: The Treca criterion in the -lane Conider now the Von Mie criterion. From qn. 8.., 8.., the criterion i J /. From qn. 8..9, thi can be re-written a (8..) Thu, the magnitude of the deviatoric tre vector i contant and one ha a circular yield locu with radiu k, which trancribe the Treca hexagon, a illutrated in Fig

30 Section 8. Treca Von Mie Figure 8..: The Von Mie criterion in the -lane The yield urface i a circular cylinder with axi along the ace diagonal, Fig The Treca urface i a imilar hexagonal cylinder. Treca yield urface Von Mie yield urface - lane yield locu lane tre yield locu ( ) ( ) Figure 8..: The Von Mie and Treca yield urface 8.. Haigh-Wetergaard Stre Sace Thu far, yield criteria have been decribed in term of rincial tree (,, ). It i often convenient to work with,, coordinate, Fig. 8..; thee cylindrical coordinate are called Haigh-Wetergaard coordinate. They are articularly ueful for decribing and viualiing geometrically reure-deendent yield-criteria. 7

31 Section 8. The coordinate, are imly the magnitude of, reectively, the hydrotatic tre vector ρ m, m, m and the deviatoric tre vector,,. Thee are given by ( can be obtained from qn. 8..9) ρ I J (8..4) m /, e ρ ( a) (b) Figure 8..: A oint in tre ace i meaured from the ( ) axi in the lane. To exre in term of invariant, conider a unit vector e in the lane in the direction of the axi; thi i the ame vector n c conidered in Fig in connection with the octahedral hear tre, and it ha coordinate,,,, 6 now be obtained from e co { Problem 9}: 6, Fig The angle can co (8..5) J Further maniulation lead to the relation { Problem } J co (8..6) J / Since J and J are invariant, it follow that through co i alo. Note that J enter co, and doe not aear in or ; it i J which make the yield locu in the -lane non-circular. From qn and Fig. 8..b, the deviatoric tree can be exreed in term of the Haigh-Wetergaard coordinate through 7

32 Section 8. J co co / co / (8..7) The rincial tree and the Haigh-Wetergaard coordinate can then be related through { Problem } co co / (8..8) co / In term of the Haigh-Wetergaard coordinate, the yield criteria are Von Mie: f ( ) k Treca f (, ) in (8..9) 8..4 Preure Deendent ield Criteria The Treca and Von Mie criteria are indeendent of hydrotatic reure and are uitable for the modelling of laticity in metal. For material uch a rock, oil and concrete, however, there i a trong deendence on the hydrotatic reure. The Drucker-Prager Criteria The Drucker-Prager criterion i a imle modification of the Von Mie criterion, whereby the hydrotatic-deendent firt invariant I i introduced to the Von Mie qn. 8..: f I, J ) I J k (8..) ( with i a new material arameter. On the lane, I, and o the yield locu there i a for the Von Mie criterion, a circle of radiu k, Fig. 8..4a. Off the lane, the yield locu remain circular but the radiu change. When there i a tate of ure hydrotatic tre, the magnitude of the hydrotatic tre vector i { Problem } ρ k /, with. For large reure,, the I term in qn. 8.. allow for large deviatoric tree. Thi effect i hown in the meridian lane in Fig. 8..4b, that i, the (, ) lane which include the axi. 7

33 Section 8. - lane meridian lane k k (a) (b) Figure 8..4: The Drucker-Prager criterion; (a) the -lane, (b) the Meridian Plane The Drucker-Prager urface i a right-circular cone with aex at k /, Fig Note that the lane tre locu, where the cone interect the lane, i an ellie, but whoe centre i off-axi, at ome, ). ( ρ Figure 8..5: The Drucker-Prager yield urface In term of the Haigh-Wetergaard coordinate, the yield criterion i f (, ) 6 k (8..) The Mohr Coulomb Criteria The Mohr-Coulomb criterion i baed on Coulomb 77 friction equation, which can be exreed in the form c tan (8..) n 7

34 Section 8. where c, are material contant; c i called the coheion 4 and i called the angle of internal friction. and n are the hear and normal tree acting on the lane where failure occur (through a hearing effect), Fig. 8..6, with tan laying the role of a coefficient of friction. The criterion tate that the larger the reure n, the more hear the material can utain. Note that the Mohr-Coulomb criterion can be conidered to be a generalied verion of the Treca criterion, ince it reduce to Treca when with c k. n Figure 8..6: Coulomb friction over a lane Thi criterion not only include a hydrotatic reure effect, but alo allow for different yield behaviour in tenion and in comreion. Maintaining iotroy, there will now be three line of ymmetry in any deviatoric lane, and a tyical ector of the yield locu i a hown in Fig (comare with Fig. 8..9) yield locu Figure 8..7: A tyical ector of the yield locu for an iotroic material with different yield behaviour in tenion and comreion Given value of c and, one can draw the failure locu (line) of the Mohr-Coulomb criterion in ( n, ) tre ace, with intercet c and loe tan, Fig Given ome tre tate, a Mohr tre circle can be drawn alo in ( n, ) ace (ee 7..6). When the tre tate i uch that thi circle reache out and touche the failure line, yield occur. 4 c correond to a coheionle material uch a and or gravel, which ha no trength in tenion 74

35 Section 8. failure line c c n Figure 8..8: Mohr-Coulomb failure criterion From Fig. 8..8, and noting that the large Mohr circle ha centre radiu, one ha, and co n in (8..) Thu the Mohr-Coulomb criterion in term of rincial tree i co in c (8..4) The trength of the Mohr-Coulomb material in uniaxial tenion, f t, and in uniaxial comreion, f c, are thu c co c co ft, fc (8..5) in in In term of the Haigh-Wetergaard coordinate, the yield criterion i co in 6c co f (,, ) in in (8..6) The Mohr-Coulomb yield urface in the lane and meridian lane are dilayed in Fig In the lane one ha an irregular hexagon which can be contructed from two length: the magnitude of the deviatoric tre in uniaxial tenion at yield, t, and the correonding (larger) value in comreion, c ; thee are given by: in 6 f in 6 fc c t, c (8..7) in in 75

36 Section 8. In the meridian lane, the failure urface cut the axi at c cot { Problem 4}. - lane t t meridian lane c c ccot (a) (b) Figure 8..9: The Mohr-Coulomb criterion; (a) the -lane, (b) the Meridian Plane The Mohr-Coulomb urface i thu an irregular hexagonal yramid, Fig Figure 8..: The Mohr-Coulomb yield urface By adjuting the material arameter, k, c,, the Drucker-Prager cone can be made to match the Mohr-Coulomb hexagon, either incribing it at the minor vertice, or circumcribing it at the major vertice, Fig

37 Section 8. Figure 8..: The Mohr-Coulomb and Drucker-Prager criteria matched in the - lane Caed ield Surface The Mohr-Coulomb and Drucker-Prager urface are oen in that a ure hydrotatic reure can be alied without affecting yield. For many geomaterial, however, for examle oil, a large enough hydrotatic reure will induce ermanent deformation. In thee cae, a cloed (caed) yield urface i more aroriate, for examle the one illutrated in Fig Figure 8..: a caed yield urface An examle i the modified Cam-Clay criterion: J I IM c or M, (8..8) c with M and read c material contant. In term of the tandard geomechanic notation, it q M c (8..9) where 77

38 Section 8. I, q J (8..4) The modified Cam-Clay locu in the meridian lane i hown in Fig Since i contant for any given, the locu in lane arallel to the - lane are circle. The material arameter c i called the critical tate reure, and i the reure which carrie the maximum deviatoric tre. M i the loe of the dotted line hown in Fig. 8.., known a the critical tate line. - lane q Critical tate line c M c Figure 8..: The modified Cam-Clay criterion in the Meridian Plane c 8..5 Aniotroy Many material will dilay aniotroy. For examle metal which have been roceed by rolling will have characteritic material direction, the tenile yield tre in the direction of rolling being tyically 5% greater than that in the tranvere direction. The form of aniotroy exhibited by rolled heet i uch that the material roertie are ymmetric about three mutually orthogonal lane. The line of interection of thee lane form an orthogonal et of axe known a the rincial axe of aniotroy. The axe are (a) in the rolling direction, (b) normal to the heet, (c) in the lane of the heet but normal to rolling direction. Thi form of aniotroy i called orthotroy (ee Part I, 6..). Hill (948) rooed a yield condition for uch a material which i a natural generaliation of the Mie condition: f( ) F G H L M N (8..4) where F, G, H, L, M, N are material contant. One need to carry out 6 tet: uniaxial tet in the three coordinate direction to find the uniaxial yield trength ( ) x,( ) y,( ) z, and hear tet to find the hear trength ( ) xy, ( ) yz, ( ) zx. For a uniaxial tet in the x direction, qn reduce to GH /( ) x. By conidering the other imle uniaxial and hear tet, one can olve for the material arameter: 78

39 Section 8. F L y z x yz G M z x y zx H N x y z xy (8..4) The criterion reduce to the Mie condition 8.. when F L M N G H (8..4) 6k The,, axe of reference in qn are the rincial axe of aniotroy. The form aroriate for a general choice of axe can be derived by uing the uual tre tranformation formulae. It i comlicated and involve cro-term uch a, etc Problem. A material i to be loaded to a tre tate 5 9 MPa What hould be the minimum uniaxial yield tre of the material o that it doe not fail, according to the (a) Treca criterian (b) Von Mie criterion What do the theorie redict when the yield tre of the material i 8MPa?. Ue qn. 8..6, J to derive qn. 8.., 6 6 k Mie criterion., for the Von. Ue the lane tre rincial tre formula to derive qn for the Taylor-Quinney tet. 4. Derive qn for the Taylor-Quinney tet., 5. Decribe the tate of tre rereented by the oint D and in Fig (The comlete tre tate can be viualied with the hel of Mohr circle of tre, Fig ) 79

40 Section Suoe that, in the Taylor and Quinney tenion-torion tet, one ha / and / 4. Plot thi tre tate in the D rincial tre tate, Fig (Ue qn to evaluate the rincial tree.) Keeing now the normal tre at /, what value can the hear tre be increaed to before the material yield, according to the von Mie criterion? 7. What are the lane rincial tre value for the oint B and C in Fig. 8..7? 8. Sketch on the lane Fig a line correonding to and alo a region correonding to 9. Uing the relation e co and, derive qn. 8..5, co.. Uing the trigonometric relation co 4co co and qn. 8..5, co, how that co J. Then uing the relation 8..6, / J J J, with J, derive qn. 8..6, J co / J. Conider the following tre tate. For each one, evaluate the ace coordinate (,, ) and lot in the lane (ee Fig. 8..b): (a) triaxial tenion: T T (b) triaxial comreion: (thi i an imortant tet for geomaterial, which are deendent on the hydrotatic reure) (c) a ure hear xy :,, (d) a ure hear xy in the reence of hydrotatic reure :,,, i.e.. Ue relation 8..4, I /, J and qn to derive qn Show that the magnitude of the hydrotatic tre vector i ρ k / for the Drucker-Prager yield criterion when the deviatoric tre i zero 4. Show that the magnitude of the hydrotatic tre vector i c cot for the Mohr-Coulomb yield criterion when the deviatoric tre i zero 5. Show that, for a Mohr-Coulomb material, in ( r ) /( r ), where r f c / ft i the comreive to tenile trength ratio 6. A amle of concrete i ubjected to a tre, A where the contant A. Uing the Mohr-Coulomb criterion and the reult of Problem 5, how that the material will not fail rovided A fc / r J 8

41 Section latic Perfectly Platic Material Once yield occur, a material will deform latically. Predicting and modelling thi latic deformation i the toic of thi ection. For the mot art, in thi ection, the material will be aumed to be erfectly latic, that i, there i no work hardening Platic Strain Increment When examining the train in a latic material, it hould be emhaied that one work with increment in train rather than a total accumulated train. One reaon for thi i that when a material i ubjected to a certain tre tate, the correonding train tate could be one of many. Similarly, the train tate could correond to many different tre tate. xamle of thi tate of affair are hown in Fig ε ε Figure 8.4.: tre-train curve; (a) different train at a certain tre, (b) different tre at a certain train One cannot therefore make ue of tre-train relation in latic region (excet in ome ecial cae), ince there i no unique relationhi between the current tre and the current train. However, one can relate the current tre to the current increment in train, and thee are the tre-train law which are ued in laticity theory. The total train can be obtained by umming u, or integrating, the train increment The Prandtl-Reu quation e An increment in train d ε can be decomoed into an elatic art and a latic art. If the material i iotroic, it i reaonable to uoe that the rincial latic train increment are roortional to the rincial deviatoric tree i : i = = = dλ (8.4.) Thi relation only give the ratio of the latic train increment to the deviatoric tree. To determine the recie relationhi, one mut ecify the oitive calar d λ (ee later). Note that the latic volume contancy i inherent in thi relation: + +. = 8