Section 8. 8. 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 complex three-dimenional tre tate? Let u begin with a very general cae: an aniotropic material with different yield trength in different direction. For example, conider the material hown in Fig. 8... Thi i a compoite material with long fibre along the x direction, giving it extra trength in that direction it will yield at a higher tenion when pulled in the x direction than when pulled in other direction. x x x x fibre binder bundle ( a ) ( b ) x Figure 8..: an aniotropic material; (a) microtructural detail, (b) continuum model We can aume that yield will occur at a particle when ome combination of the tre component reache ome critical value, ay when F,,,,, ) k. (8..) ( Here, F i ome function of the 6 independent component of the tre tenor and k i ome material property which can be determined experimentally. Alternatively, it i very convenient to expre yield criteria in term of principal tree. Let u uppoe that we know the principal tree everywhere, (,, ), Fig. 8... ield mut depend 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 expre the yield criterion in term of principal tree in the form F,,, i ) k (8..) ( n F will no doubt alo contain other parameter which need to be determined experimentally 59
Section 8. where n i repreent the principal direction thee give the orientation of the principal tree relative to the material direction x, x, x. If the material i iotropic, the repone i independent of any material direction independent of any direction the tre act in, and o the yield criterion can be expreed in the imple 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 principal tree. Alternatively, ince the three principal invariant of tre are independent 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 principal invariant of the deviatoric tre. With the further retriction that the yield tre i independent 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 iotropic 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, Eqn. 8.. take the form max,, k (8..7) The value of k can be obtained from a imple experiment. For example, in a tenion tet, 0, 0, and failure occur when 0 reache, the yield tre in tenion. It follow that k. (8..8) 60
Section 8. In a hear tet,, 0,, and failure occur when reache, the yield tre of a material in pure hear, o that k. The Von Mie ield Condition The Von Mie criterion tate that yield occur when the principal tree atify the relation 6 k (8..9) Again, from a uniaxial tenion tet, one find that the k in Eqn. 8..9 i k. (8..0) 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 pure tenion, the material begin to deform platically. In the hear tet, one again find that k, the yield tre in pure hear. Sometime it i preferable to work with arbitrary tre component; for thi purpoe, the Von Mie condition can be expreed a { Problem } 6 k (8..) 6 The piecewie linear nature of the Treca yield condition i ometime a theoretical advantage over the quadratic Mie condition. However, the fact that in many problem one often doe not know which principal 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 Eqn. 8..0 and 8..9, the Von Mie criterion can be expreed a f ( J ) J k 0 (8..) Note the relationhip between J and the octahedral hear tre, Eqn. 8..7; the Von Mie criterion can be interpreted a predicting yield when the octahedral hear tre reache a critical value. With, the Treca condition can be expreed a 6
Section 8. 4 6 f ( J, J ) 4J 7J 6k J 96k J 64k 0 (8..4) but thi expreion i too cumberome to be of much ue. Experiment 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 experiment, ubjected a number of thin-walled cylinder made of copper and teel to combined tenion and torion, Fig. 8... Figure 8..: combined tenion and torion of a thin-walled tube The cylinder wall i in a tate of plane tre, with and all other tre component zero. The principal tree correponding 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 predict an elliptical yield locu in, tre pace, but with different ratio of principal axe, Fig. 8... The origin in Fig. 8.. correpond to an untreed tate. The horizontal axe refer to uniaxial tenion in the abence of hear, wherea the vertical axi refer to pure 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 platicity enue. 6
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 ellipe than the Treca locu, a reult which ha been repeatedly confirmed by other worker. D Principal Stre Space Fig. 8.. give a geometric interpretation of the Treca and Von Mie yield criteria in, pace. It i more uual to interpret yield criteria geometrically in a principal tre pace. The Taylor and Quinney tet are an example of plane tre, where one principal tre i zero. Following the convention for plane tre, label now the two non-zero principal tree and, o that 0 (even if it i not the minimum principal tre). The criteria can then be diplayed in, D principal tre pace. With 0, one ha Treca: max, Von Mie:, (8..8) Thee are plotted in Fig. 8..4. The Treca criterion i a hexagon. The Von Mie 0 criterion i an ellipe with axe inclined at 45 to the principal axe, which can be een by expreing Eqn. 8..8b in the canonical form for an ellipe: the maximum difference between the predicted 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
Section 8. / / / / / 0 / / / / 0 / / / / 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 pace: point A correpond to a uniaxial tenion, B to a equi-biaxial tenion and C to a pure hear. B E C D A Figure 8..4: yield loci in D principal tre pace Again, point inide thee loci repreent an elatic tre tate. Any combination of principal tree which puh the point out to the yield loci reult in platic deformation. 8.. Three Dimenional Principal Stre Space The D principal tre pace ha limited ue. For example, a tre tate that might tart out two dimenional can develop into a fully three dimenional tre tate a deformation proceed. 64
Section 8. In three dimenional principal tre pace, one ha a yield urface f,, 0, Fig. 8..5. In thi cae, one can draw a line at equal angle to all three principal tre axe, the pace diagonal. Along the pace diagonal and o point 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 point A,,, a, b, c, on the yield urface, Fig. 8..5. A pure hydrotatic tre can be uperimpoed on thi tre tate without affecting yield, o h,, any other point a, b, c h h h will alo be on the yield urface. Example of uch point are hown at B, C and D, which are obtained from A by moving along a line parallel to the pace diagonal. The yield behaviour of the material i therefore pecified by a yield locu on a plane perpendicular to the pace diagonal, and the yield urface i generated by liding thi locu up and down the pace diagonal. hydrotatic tre the π - plane yield locu D ρ C σ B A a, b, c deviatoric tre Figure 8..5: ield locu/urface in three dimenional tre-pace The -plane Any urface in tre pace 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..0) where e, e, e are unit vector along the tre pace axe. In particular, any plane perpendicular to the pace diagonal i decribed by the equation a mentioned, one ha a ix dimenional tre pace for an aniotropic material and thi cannot be viualied 65
Section 8. cont (8..) Without lo of generality, one can chooe a a repreentative plane the π plane, which i defined by 0. For example, the point,,,,0 i on the π plane and, with yielding independent of hydrotatic tre, i equivalent to point in principal tre pace which differ by a hydrotatic tre, e.g. the point,0,, 0,,, etc. σ, can be regarded a the um of the tre tate at the correponding point on the π plane, D, repreented, together with a hydrotatic tre repreented by the vector The tre tate at any point A repreented by the vector, by the vector, ρ,, : m m m,,,,,, m m m (8..) The component of the firt term/vector on the right here um to zero ince it lie on the π plane, and thi i the deviatoric tre, whilt the hydrotatic tre i / m. Projected view of the -plane Fig. 8..6a how principal tre pace and Fig. 8..6b how the π plane. The heavy line,, in Fig. 8..6b repreent the projection of the principal axe down onto the plane (o one i looking down the pace diagonal). Some point, A, B, C in tre pace and their projection onto the plane are alo hown. Alo hown i ome point D on the plane. It hould be kept in mind that the deviatoric tre vector in the projected view of Fig. 8..6b i in reality a three dimenional vector (ee the correponding vector in Fig. 8..6a). m m m D A B C pace diagonal B C D A ( a) (b) Figure 8..6: Stre pace; (a) principal tre pace, (b) the π plane 66
Section 8. Conider the more detailed Fig. 8..7 below. Point A here repreent the tre tate,,0, a indicated by the arrow in the figure. It can alo be reached in different way, for example it repreent,0, and,,. Thee three tre tate of coure differ by a hydrotatic tre. The actual plane value for A i the one for which 5 4 0, i.e.,,,,,,. Point B and C alo repreent multiple tre tate { Problem 7}. B E A C D biector Figure 8..7: the -plane The biector of the principal plane projection, uch a the dotted line in Fig. 8..7, repreent tate of pure hear. For example, the plane value for point D i 0,,, correponding to a pure hear in the plane. The dahed line in Fig. 8..7 are helpful in that they allow u to plot and viualie tre tate eaily. The ditance between each dahed line along the direction of the projected axe repreent one unit of principal tre. Note, however, that thee unit are not conitent with the actual magnitude of the deviatoric vector in the plane. To create a more complete picture, note firt that a unit vector along the pace diagonal i n,,, Fig. 8..8. The component of thi normal are the direction coine; for example, a unit normal along the principal axi i,0,0 between the axi and the pace diagonal i given by n e and o the angle 0 e co 0. From Fig. 8..8, the angle between the axi and the plane i given by co, and o a length of unit get projected down to a length.,, which i on the plane. The length of the vector out to E in Fig. 8..7 i unit. To For example, point E in Fig. 8..7 repreent a pure hear,,,0 67
Section 8. convert to actual magnitude, multiply by to get, which agree with. n pace diagonal n 0 -plane Figure 8..8: principal tre projected onto the -plane Typical -plane ield Loci Conider next an arbitrary point ( a, b, c) on the plane yield locu. If the material i iotropic, the point ( 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 compreion, e.g. neglecting the Bauchinger effect, then o alo are the point ( a, b, c), ( a, c, b), etc. Thu point become and one need only conider the yield locu in one 0 o ector of the plane, 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 projected principal axe and their (pure hear) biector. yield locu Figure 8..9: A typical ector of the yield locu The Treca and Von Mie ield Loci in the -plane The Treca criterion, Eqn. 8..7, i a regular hexagon in the plane a illutrated in Fig. 8..0. Which of the ix ide of the locu i relevant depend on which of,, i the maximum and which i the minimum, and whether they are tenile or compreive. 68
Section 8. For example, yield at the pure hear /, 0, / i indicated by point A in the figure. Point B repreent yield under uniaxial tenion, magnitude of the hexagon, i therefore,,., i. The ditance ob, the ; the correponding point on the plane A criticim of the Treca criterion i that there i a udden change in the plane upon which failure occur upon a mall change in tre at the harp corner of the hexagon. o B A Figure 8..0: The Treca criterion in the -plane Conider now the Von Mie criterion. From Eqn. 8..0, 8.., the criterion i J /. From Eqn. 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. 8... 69
Section 8. Treca Von Mie Figure 8..: The Von Mie criterion in the -plane The yield urface i a circular cylinder with axi along the pace diagonal, Fig. 8... The Treca urface i a imilar hexagonal cylinder. Treca yield urface Von Mie yield urface - plane yield locu plane tre yield locu ( 0) ( 0) Figure 8..: The Von Mie and Treca yield urface 8.. Haigh-Wetergaard Stre Space Thu far, yield criteria have been decribed in term of principal tree (,, ). It i often convenient to work with,, coordinate, Fig. 8..; thee cylindrical coordinate are called Haigh-Wetergaard coordinate. They are particularly ueful for decribing and viualiing geometrically preure-dependent yield-criteria. 70
Section 8. The coordinate, are imply the magnitude of, repectively, the hydrotatic tre vector ρ m, m, m and the deviatoric tre vector,,. Thee are given by ( can be obtained from Eqn. 8..9) ρ I J (8..4) m /, e ρ ( a) (b) Figure 8..: A point in tre pace i meaured from the ( ) axi in the plane. To expre in term of invariant, conider a unit vector e in the plane in the direction of the axi; thi i the ame vector n c conidered in Fig. 8..4 in connection with the octahedral hear tre, and it ha coordinate,,,, 6 now be obtained from e co { Problem 9}: 6, Fig. 8... The angle can co (8..5) J Further manipulation lead to the relation { Problem 0} 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 appear in or ; it i J which make the yield locu in the -plane non-circular. From Eqn. 8..5 and Fig. 8..b, the deviatoric tree can be expreed in term of the Haigh-Wetergaard coordinate through 7
Section 8. J co co / co / (8..7) The principal 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 0 Treca f (, ) in 0 (8..9) 8..4 Preure Dependent ield Criteria The Treca and Von Mie criteria are independent of hydrotatic preure and are uitable for the modelling of platicity in metal. For material uch a rock, oil and concrete, however, there i a trong dependence on the hydrotatic preure. The Drucker-Prager Criteria The Drucker-Prager criterion i a imple modification of the Von Mie criterion, whereby the hydrotatic-dependent firt invariant I i introduced to the Von Mie Eqn. 8..: f I, J ) I J k 0 (8..0) ( with i a new material parameter. On the plane, I 0, and o the yield locu there i a for the Von Mie criterion, a circle of radiu k, Fig. 8..4a. Off the plane, the yield locu remain circular but the radiu change. When there i a tate of pure hydrotatic tre, the magnitude of the hydrotatic tre vector i { Problem } ρ k /, with 0. For large preure, 0, the I term in Eqn. 8..0 allow for large deviatoric tree. Thi effect i hown in the meridian plane in Fig. 8..4b, that i, the (, ) plane which include the axi. 7
Section 8. - plane meridian plane k k (a) (b) Figure 8..4: The Drucker-Prager criterion; (a) the -plane, (b) the Meridian Plane The Drucker-Prager urface i a right-circular cone with apex at k /, Fig. 8..5. Note that the plane tre locu, where the cone interect the 0 plane, i an ellipe, but whoe centre i off-axi, at ome 0, 0). ( ρ Figure 8..5: The Drucker-Prager yield urface In term of the Haigh-Wetergaard coordinate, the yield criterion i f (, ) 6 k 0 (8..) The Mohr Coulomb Criteria The Mohr-Coulomb criterion i baed on Coulomb 77 friction equation, which can be expreed in the form c tan (8..) n 7
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 plane where failure occur (through a hearing effect), Fig. 8..6, with tan playing the role of a coefficient of friction. The criterion tate that the larger the preure 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 0 with c k. n Figure 8..6: Coulomb friction over a plane Thi criterion not only include a hydrotatic preure effect, but alo allow for different yield behaviour in tenion and in compreion. Maintaining iotropy, there will now be three line of ymmetry in any deviatoric plane, and a typical ector of the yield locu i a hown in Fig. 8..7 (compare with Fig. 8..9) yield locu Figure 8..7: A typical ector of the yield locu for an iotropic material with different yield behaviour in tenion and compreion Given value of c and, one can draw the failure locu (line) of the Mohr-Coulomb criterion in ( n, ) tre pace, with intercept c and lope tan, Fig. 8..8. Given ome tre tate, a Mohr tre circle can be drawn alo in ( n, ) pace (ee 7..6). When the tre tate i uch that thi circle reache out and touche the failure line, yield occur. 4 c 0 correpond to a coheionle material uch a and or gravel, which ha no trength in tenion 74
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,0 and co n in (8..) Thu the Mohr-Coulomb criterion in term of principal tree i co in c (8..4) The trength of the Mohr-Coulomb material in uniaxial tenion, f t, and in uniaxial compreion, 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 0 f (,, ) in in (8..6) The Mohr-Coulomb yield urface in the plane and meridian plane are diplayed in Fig. 8..9. In the plane one ha an irregular hexagon which can be contructed from two length: the magnitude of the deviatoric tre in uniaxial tenion at yield, t0, and the correponding (larger) value in compreion, c0 ; thee are given by: in 6 f in 6 fc c t0, c0 (8..7) in in 75
Section 8. In the meridian plane, the failure urface cut the 0 axi at c cot { Problem 4}. - plane t0 t0 meridian plane c0 c0 ccot (a) (b) Figure 8..9: The Mohr-Coulomb criterion; (a) the -plane, (b) the Meridian Plane The Mohr-Coulomb urface i thu an irregular hexagonal pyramid, Fig. 8..0. Figure 8..0: The Mohr-Coulomb yield urface By adjuting the material parameter, 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. 8... 76
Section 8. Figure 8..: The Mohr-Coulomb and Drucker-Prager criteria matched in the - plane Capped ield Surface The Mohr-Coulomb and Drucker-Prager urface are open in that a pure hydrotatic preure can be applied without affecting yield. For many geomaterial, however, for example oil, a large enough hydrotatic preure will induce permanent deformation. In thee cae, a cloed (capped) yield urface i more appropriate, for example the one illutrated in Fig. 8... Figure 8..: a capped yield urface An example i the modified Cam-Clay criterion: J p I IM c or M p, 0 (8..8) c with M and read p c material contant. In term of the tandard geomechanic notation, it q M p p c p (8..9) where 77
Section 8. p I, q J (8..40) The modified Cam-Clay locu in the meridian plane i hown in Fig. 8... Since i contant for any given, the locu in plane parallel to the - plane are circle. The material parameter p c i called the critical tate preure, and i the preure which carrie the maximum deviatoric tre. M i the lope of the dotted line hown in Fig. 8.., known a the critical tate line. - plane q Critical tate line p c M p c p Figure 8..: The modified Cam-Clay criterion in the Meridian Plane p c 8..5 Aniotropy Many material will diplay aniotropy. For example metal which have been proceed by rolling will have characteritic material direction, the tenile yield tre in the direction of rolling being typically 5% greater than that in the tranvere direction. The form of aniotropy exhibited by rolled heet i uch that the material propertie are ymmetric about three mutually orthogonal plane. The line of interection of thee plane form an orthogonal et of axe known a the principal axe of aniotropy. The axe are (a) in the rolling direction, (b) normal to the heet, (c) in the plane of the heet but normal to rolling direction. Thi form of aniotropy i called orthotropy (ee Part I, 6..). Hill (948) propoed a yield condition for uch a material which i a natural generaliation of the Mie condition: f( ) F G H ij L M N 0 (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, Eqn. 8..4 reduce to GH /( ) x. By conidering the other imple uniaxial and hear tet, one can olve for the material parameter: 78
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 Eqn. 8..4 are the principal axe of aniotropy. The form appropriate for a general choice of axe can be derived by uing the uual tre tranformation formulae. It i complicated and involve cro-term uch a, etc. 8..6 Problem. A material i to be loaded to a tre tate 50 0 0 ij 0 90 0 MPa 0 0 0 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 predict when the yield tre of the material i 80MPa?. Ue Eqn. 8..6, J to derive Eqn. 8.., 6 6 k Mie criterion., for the Von. Ue the plane tre principal tre formula to derive Eqn. 8..5 for the Taylor-Quinney tet. 4. Derive Eqn. 8..7 for the Taylor-Quinney tet., 5. Decribe the tate of tre repreented by the point D and E in Fig. 8..4. (The complete tre tate can be viualied with the help of Mohr circle of tre, Fig. 7..7.) 79
Section 8. 6. Suppoe that, in the Taylor and Quinney tenion-torion tet, one ha / and / 4. Plot thi tre tate in the D principal tre tate, Fig. 8..4. (Ue Eqn. 8..5 to evaluate the principal tree.) Keeping 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 plane principal tre value for the point B and C in Fig. 8..7? 8. Sketch on the plane Fig. 8..7 a line correponding to and alo a region correponding to 0 9. Uing the relation e co and, derive Eqn. 8..5, co. 0. Uing the trigonometric relation co 4co co and Eqn. 8..5, co, how that co J. Then uing the relation 8..6, / J J J, with J 0, derive Eqn. 8..6, J co / J. Conider the following tre tate. For each one, evaluate the pace coordinate (,, ) and plot in the plane (ee Fig. 8..b): (a) triaxial tenion: T T 0 (b) triaxial compreion: p p 0 (thi i an important tet for geomaterial, which are dependent on the hydrotatic preure) (c) a pure hear xy :, 0, (d) a pure hear xy in the preence of hydrotatic preure p: p, p, p, i.e.. Ue relation 8..4, I /, J and Eqn. 8..7 to derive Eqn. 8..8.. 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 compreive to tenile trength ratio 6. A ample of concrete i ubjected to a tre p, Ap where the contant A. Uing the Mohr-Coulomb criterion and the reult of Problem 5, how that the material will not fail provided A fc / p r J 80