∂s xx ∂s xy ∂s xz ∂s yx ∂s yy ∂s yz ∂s zx ∂s zy ∂s zz —— + —— + —— = —— + —— + —— = —— + —— + —— = 0 ∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z } ∂u˘ ∂v˘ ∂w˘ —— + —— + —— = 0 ∂x ∂y ∂z (A1.15) where u˘, v˘ and w˘ are the x, y and z components of the material’s velocity. The general three-dimensional situation is complicated. However, in plane strain conditions, and if the work hardening of the material is negligible, the integration of the equilibrium and compatibility equations, under the constraint of the constitutive equations, is simplified by describing the stresses and velocities not in a Cartesian coordinate system but in a curvi- linear system that is everywhere tangential to the maximum shear stress directions. The net of curvilinear maximum shear stress lines is known as the slip-line field. Determining the shape of the net for any application and then the stresses and velocities in the field is achieved through slip-line field theory. This theory is now outlined. A1.2.1 Constitutive laws for a non-hardening material in plane strain When the strain in one direction, say the z-direction, is zero, from the flow rules (equation (A1.13)) the deviatoric stresses in that direction are also zero. Then s zz = s m = (1/2)(s xx + s yy ). The yield criterion, equation (A1.12), and flow rules, equation (A1.13), become (s xx – s yy ) 2 +4s 2 xy =4k 2 de xx –de yy de xy —————— = —————— = —— } (A1.16) 1/2(s xx – s yy ) 1/2(s xx – s yy ) s xy When the material is non-hardening, the shear yield stress k is independent of strain. If, in a plastic region, the x, y directions are chosen locally to coincide with the maximum shear stress directions, s xx becomes equal to s yy (and equal to s m ), so (s xx – s yy ) = 0. Equation (A1.16) becomes a statement that (i) the maximum shear stress is constant throughout the plastic region and (ii) there is no extension along maximum shear stress directions. The consequences of these statements for stress and velocity variations throughout a plastic region are developed in the next two subsections. A1.2.2 Stress relations in a slip-line field Figure A1.4(a) shows a network of slip-lines in a plastic field. The pressure p (= –s m ) and the shear stress k is shown at a general point O in the field. The variation of pressure throughout the field may be found by integrating the equilibrium equations along the slip- lines. How this is done, and some consequences for the shape of the field, are now described. First, the two families of lines, orthogonal to each other, are labelled a and b. Which is labelled a and which is b is chosen, by convention, so that, if a and b are regarded as a right-handed coordinate system, the direction of the largest principal stress lies in the first quadrant. (This means that the shear stresses k are positive as shown in the figure.) The Perfectly plastic material in plane strain 333 Childs Part 3 31:3:2000 10:41 am Page 333 direction of an a line at any point is described by its anticlockwise rotation from a datum direction, for example f from the +X direction. By stress analysis (Figure A1.4(b)), the stresses p and k at O have (x,y) components (s xx , s yy , s xy ) s xx = –p – k sin2f; s yy = –p + k sin2f; s xy = k cos2f (A1.17) Substituting these into the equilibrium equations ∂s xx ∂s xy ∂s yy ∂s xy —— + —— = 0; —— + —— = 0 (A1.18a) ∂x ∂y ∂y ∂x gives, after noting that k is a constant, ∂p ∂f ∂f –——– 2k cos 2f —— – 2k sin2f —— = 0 ∂x ∂x ∂y } (A1.18b) ∂p ∂f ∂f –——+ 2k cos 2f —— – 2k sin2f —— = 0 ∂y ∂y ∂x If the direction of X is chosen so that f = 0, that is so that the a slip-line is tangential to X, sin 2f = 0 and cos 2f = 1 and ∂∂ —— (p + 2kf) = 0; —— (p – 2kf) = 0 (A1.18c) ∂x ∂y or p + 2kf = constant along an a-line; p – 2kf = constant along a b-line. (A1.18d) If the geometry of the slip-line field and the pressure at any one point is known, the pressure at any other point can be calculated. Equation (A1.19) relates, for the example of Figure A1.4, pressures along the a-lines AB and DC, and along the b-lines AD and BC 334 Appendix 1 Fig. A1.4 (a) A slip-line net and (b) free body equilibrium diagrams around O Childs Part 3 31:3:2000 10:41 am Page 334 p A + 2kf A = p B + 2kf B; p C + 2kf C = p D + 2kf D (A1.19) p A + 2kf A = p D – 2kf D; p B – 2kf B = p C + 2kf C } Geometry of the field The inclinations f A , f B , f C and f D are not independent. The pressure p C at C may be calcu- lated from that at A in two ways from equations (A1.19), either along the path ABC or ADC. For p C to be single valued f B – f A = f C – f D ; f D – f A = f C – f B (A1.20) Figure A1.5(a) gives some common examples of curvilinear nets that satisfy this condi- tion: a grid of straight lines in which the pressure is constant, a centred fan and a net constructed on two circular arcs. Systematic methods for constructing more complicated fields are described by Johnson et al. (1982). Force boundary conditions Figure A1.5(b) shows a and b slip-lines meeting a tool surface on which there is a friction stress t f . Equilibrium of forces on the triangle ABC, in the direction of t f ,gives t f = k cos 2z (A1.21) Thus, the magnitude of the friction stress relative to k determines the angle z at which the a-line intersects the tool face. Similarly, a and b slip-lines meet a free surface at 45˚ (t f /k = 0). Because there is no normal stress on a free surface, p = ± k there, depending on the direction of k. Perfectly plastic material in plane strain 335 Fig. A1.5 (a) Nets satisfying internal force equilibrium and (b) slip-lines meeting a friction boundary Childs Part 3 31:3:2000 10:41 am Page 335 A1.2.3 Velocity relations in a slip-line field Analogous equations to equations (A1.18d) exist for the variation of velocity along the slip lines. However, the statement that there is no extension along a slip line (Section A1.2.1) may directly be used to develop velocity relations and further rules for the geometry of a slip-line field. Figure A1.6(a) repeats the net of Figure A1.4(a). Figure A1.6(b) represents, in a velocity diagram, possible variations of velocity in the field. Because there is no exten- sion along a slip-line, every element of the velocity net is perpendicular to its correspond- ing element in the physical plane of Figure A1.6(a). Thus, equations (A1.20) also apply in the velocity diagram. Velocity boundary conditions Other constraints on slip-line fields may be derived from velocity diagrams (in addition to the obvious boundary condition that the velocity of work material at an interface with a tool must be parallel to the tool surface). Figure A1.7(a) shows proposed boundaries AB and CDE between a plastic region and a rigid region in a metal forming process. Because this is a book on metal machining, the example is of continuous chip formation, but any example could have been chosen in which part of the work is plastically deformed and part is not. First, the boundary between a plastic and a rigid region must be a slip-line. Secondly, the boundary between a plastic region and a rotating rigid region (for example CDE in Figure A1.7(a) must have the same shape in the physical plane as in the velocity diagram. Both these can be shown by considering the second case. Suppose that any boundary such as CD is not a slip-line. Then any point such as H inside the plastic region can be joined to the boundary in two places by two slip-lines, for example to F and G by HF and HG. Figure A1.7(b) is the velocity diagram. The velocities v F and v G of points F and G are determined from the rigid body rotation of the chip to be wOF and wOG, where w is the angular velocity of the chip. The velocity v H relative to v F is perpendicular to HF and that of v H relative to v G is perpendicular to HG. By comparing the positions of v F , v G and v H relative to v O , the origin of the velocity diagram, with the positions of F, G and H relative to the centre of rotation O in the physical diagram, it is 336 Appendix 1 Fig. A1.6 (a) The physical net of Figure A1.4(a) and (b) a possible associated velocity diagram Childs Part 3 31:3:2000 10:41 am Page 336 seen that the velocity of H is part of the rigid-body rotation: if the boundary CD is not a slip-line, it cannot accommodate velocity changes that must occur in a plastic field. If the boundary is a slip-line, a point H can only be joined to the boundary in two places by three slip-lines: thus, the argument above can no longer be made. For continuity of flow between a plastic and a rigid region, the boundary between the two must be a slip-line. Figure A1.7(b) also shows the whole boundary v C v D v E . It is visually obvious that only if it has the same shape relative to the origin of velocity that CDE has relative to O, can it be consistent with a rigid body rotation. Velocity discontinuities The usual procedure in slip-line field analysis is to construct fields that satisfy the geom- etry and force requirements of a problem and then to check that the velocity requirements are met. In this last part, one more feature of the theory must be introduced: the possibil- ity of velocity jumps (discontinuities) occurring. Figure A1.7(c) returns to the considera- tion of the velocity of a point H in the plastic field. H is connected to the boundary by slip-lines, both directly to G and indirectly to F through H′. It is possible for there to be a finite velocity difference between H and G, however short is the length HG, i.e. a discon- tinuity. If there is a discontinuity, then the rules for constructing the velocity net require that there be a discontinuity of equal size between H′ and F. A velocity discontinuity can exist across a slip line, but only if it is of constant size along the line. It is not implied that there is a discontinuity in the condition of the example described here: examples of actual machining slip-line fields are given in Section A1.2.5. A1.2.4 Further considerations Slip-line fields must satisfy more than the force and velocity conditions considered in Sections A1.2.2 and A1.2.3. First, they must (as must every plastic flow) satisfy a work criterion, that everywhere the work rate on the flow is positive. This means that the direc- tion of the shear stresses in the physical diagram must be the same as the direction of the shear strain rates deduced from the velocity diagram. Perfectly plastic material in plane strain 337 Fig. A1.7 (a) A possible machining process with (b) a partial velocity diagram and (c) an illustration of a velocity discontinuity across a slip-line Childs Part 3 31:3:2000 10:41 am Page 337 It must also be checked that it is possible that regions in the work material that are assumed to be rigid can in fact be rigid. For example, in Figure A1.7(a), in the rigid regions KBA and LCD, the loads change from values on BA and CD determined by the plastic flow to zero on the free surfaces KB and LC. It must be checked that such load changes can be accommodated without the yield stress being exceeded in the rigid regions in the neighbourhood of the vertices B and C. Checking for overstressing is introduced in another context in Appendix A5. The overstressing limits developed in Appendix 5 (Hill, 1954) apply here too. A1.2.5. Machining slip-line fields Figure A1.8 collects a range of slip-line fields, and their velocity diagrams (due to Lee and Shaffer, 1951, Kudo, 1965, and Dewhurst, 1978), which describe steady state chip forma- tion by a flat-faced cutting tool. The first is Lee and Shaffer’s field. It describes formation of a straight chip. The work velocity U work is transformed to the chip velocity U chip by a discontinuous change U OA tangential to the slip-line OA. The angle at which OA meets the free surface is not set by a free surface boundary condition. A is a singularity where the surface direction is not defined. Instead, the direction of OA is determined by its being perpendicular to BD. The inclination of BD to the rake face is given by equation (A1.21). Because all the slip-lines are straight, the hydrostatic pressure is constant along them (equation (A1.19)). The chip region above ADB is free, i.e. there are no forces acting on it. This determines that p = k and AD = DB. The second is due to Kudo. It may be thought of as a modification of Lee and Shaffer’s field in which the primary shear plane OA is replaced by a fan-shaped zone of angular extent d, still with a singularity at the free surface A. It still describes a straight chip. The slip-lines intersecting the rake face do so at a constant angle z: the field therefore contin- ues to describe a condition of constant friction stress along the rake face. The free-chip boundary condition still requires p = k on AD and DB and AD = DB. However d can take a range of values, from zero up to a maximum at which the region below AE becomes overstressed. For the same friction condition, tool rake angle and feed, f, as in the Lee and Shaffer field, the Kudo field describes chip formation with a larger shear plane angle and a shorter contact length. Two further fields suggested by Kudo are the third and fourth examples in Figure A1.8. These describe rotating chips. The boundaries ADB in the physical plane between the fields and the chips can be seen to transform into their own shapes in their velocity diagrams. The third field may be thought of as a distortion of the Lee and Shaffer field and the fourth as a distortion of Kudo’s first field. The slip-lines in the secondary shear zone intersect the rake face at angles z which vary from O to B: these fields describe conditions of friction stress reducing from O to B. Because the slip-lines are curved, the hydrostatic stress now varies throughout the field. Again the allowable fields are limited by the requirement that material assumed rigid outside the flow zone around A must be able to be rigid. However, the possibility arises that it is the chip material downstream of A that becomes overstressed. The last example shows another way in which a rotating chip may be formed. A fan region OED is centred on the cutting edge O and the remainder DA of the primary shear region is a single plane. With this field, the slip-lines intersect the rake face at a constant angle, so that it describes constant friction stress conditions. The fan angle y can take a 338 Appendix 1 Childs Part 3 31:3:2000 10:41 am Page 338 Perfectly plastic material in plane strain 339 Fig. A1.8 Metal machining slip-line fields (left) and their velocity diagrams (right), due to (1) Lee and Shaffer (1951), (2–4) Kudo (1965) and (5) Dewhurst (1978) Childs Part 3 31:3:2000 10:41 am Page 339 range of values, limited only by its effect on overstressing material around A. For the same friction condition, tool rake angle and feed, f, as in the Lee and Shaffer field, this last field describes chip formation with a lower shear plane angle and a longer contact length. A1.3 Yielding and flow in a triaxial stress state: advanced analysis A1.3.1 Yielding and flow rules referred to non-principal axes – analysis of stress The yield criterion is stated in equation (A1.7) in principal stress terms. It is extended to non-principal stresses in equation (A1.12): this has been justified in the two special cases when it represents principal stress and maximum shear stress descriptions of stress. It is now justified more generally, by showing that the function of stress which is the left-hand side of equation (A1.12) has a magnitude that is independent of the orientation of the (x,y,z) coordinate system. If it is valid in one case (as it is when the axes are the principal axes), it is valid for all cases. Tensor analysis is chosen as the tool for proving this, in part to introduce it for later use. Tensor notation and the summation convention Figure A1.9 shows two Cartesian coordinate systems (x,y,z) and (x*,y*,z*) rotated arbitrar- ily with respect to each other. In the (x,y,z) system the stresses are s ij with i and j denoting 340 Appendix 1 Fig. A1.9 ( x , y , z ) and ( x *, y *, z *) co-ordinate systems Childs Part 3 31:3:2000 10:42 am Page 340 x, y or z as appropriate. In the (x*,y*,z*) system the stresses are s* kl , with k,l denoting x*, y* or z*. The figure also shows a tetrahedron OABC, the faces of which are normal to the x, y, z and x* directions. Writing the direction cosines of x* with x, y and z as a x*x , a x*y and a x*z , with similar quantities a y*x , a y*y , a y*z and a z*x , a z*y , a z*z for the direction cosines of y* and z* with x, y and z, first, by geometry, the ratios of the areas OAC, OAB and OBC to ABC are respec- tively a x*x , a x*y and a x*z . Then, from force equilibrium on the tetrahedron, for example s* x*x* = a x*x a x*x s xx + a x*x a x*y s yx + a x*x a x*z s zx + a x*y a x*y s yy + a x*y a x*x s xy + a x*y a x*z s zy (A1.22a) + a x*z a x*z s zz + a x*z a x*y s yz + a x*z a x*x s xz In general and more compactly, any of the stresses s* kl may be written 33 s* kl = ∑∑ a ki a lj s ij (A1.22b) j=1 i=1 Quantities which transform like this are called tensors, and the study of the properties of the transformation is tensor analysis. By the summation convention, the summation signs are omitted, but are implied by the repetition of the suffixes i and j among the coefficients a ki and a lj . Thus equation (A1.22b) becomes s* kl = a ki a lj s ij (A1.22c) Furthermore, the repetition of k and l, between the left and right-hand sides of the equa- tion, implies that it represents all nine equations for the components of s*. The meaning of the equation is unchanged by substituting another pair of letter suffixes, say m and n, for i and j: suffixes such as i and j, repeated on the same side of an equation, are called dummy suffixes and are said to be interchangeable. Suffixes such as k and l are called free suffixes. In the special case when k = l, the summation convention extends to include s* kk = s* x*x* + s* y*y* + s* z*z* (A1.22d) Properties of the direction cosines Because the angle between a direction i and another direction k is the same as the angle between the direction k and the direction i, a ik = a ki . Because the scalar product of two unit vectors is unity if they are parallel and zero if they are perpendicular to each other, the same is true of the sum of the scalar products of their components in any other coordinate system. In repeated suffix notation, a ik a jk = 1 if i = j and 0 if i ≠ j. This can be written a ik a jk = d ij (A1.23) where d ij is defined as 1 or 0 depending respectively on whether or not i = j. Transformations of stress Now consider the summation of the direct stresses s* kk = a ki a kj s ij = d ij s ij = s ii (A1.24) This demonstrates that the sum of direct stresses s* kk in the (x*,y*,z*) system equals the sum s ii in the (x,y,z) system. One of the systems could be the principal stress Yielding and flow in a triaxial stress state 341 Childs Part 3 31:3:2000 10:42 am Page 341 system: thus, the hydrostatic stress s m is a stress invariant (it is known as the first stress invariant). Consider now the product of stresses s* kl s* lk , with the transformations of equation (A1.22c), the interchangeability of dummy suffixes and equation (A1.23): s* kl s* lk =(a ki a lj s ij )(a lm a kn s mn )=(a ki a kn s ij )(a lm a lj s mn ) (A1.25) =(d in s ij )(d mj s mn )=s nj s jn = s ij s ji In principal stress space, s ij s ji = s r 2 . So s r is also a stress invariant (it is known as the second stress invariant). From equation (A1.25) s r 2 = s 2 xx + s 2 yy + s 2 zz +2(s 2 xy + s 2 yz + s 2 zx ) (A1.26a) As s m and s r are stress invariants, so is s′ r . From Figure A1.2(a) s′ r 2 = s r 2 –3s m 2 . From this, equation (A1.4) and similar manipulations as in equations (A1.6) to (A1.7), the yield criterion becomes 2s – 2 ≡ 3(s r 2 – 3s m 2 ) ≡ 3(s xx ′ 2 + s yy ′ 2 + s zz ′ 2 ) + 6(s 2 xy + s 2 yz + s 2 zy ) ≡ (s xx – s yy ) 2 + (s yy – s zz ) 2 + (s zz – s xx ) 2 + 6(s 2 xy + s 2 yz + s 2 zy ) = 6k 2 or 2Y 2 (A1.26b) which is the same as equation (A1.12) of Section A1.1. Strain transformations The strain increments also transform as a tensor: de* ij = a ik a jl de kl (A1.27) It follows, as for stress, that the resultant strain increment and the equivalent strain incre- ment are invariants of the strain. The extension of the definition of resultant strain to a general strain state is de r 2 = de 2 xx + de 2 yy + de 2 zz + 2(de 2 xy +de 2 yz +de 2 zx ) (A1.28) where, as in equation (A1.13), de xy = de yx = (1/2)(∂u/∂y + ∂v/∂x) and similarly for de yz and de zx . Equivalent strain increments are √(2/3) times resultant strain increments. 1.3.2 Further developments The repeated suffix notation may be used to write the plastic flow rules (equation (A1.13)) more compactly and to express various relations between changes in equivalent stress and the deviatoric stress components that will be of use in Section A1.4. First, from equation (A1.13), 3de – 3 s′ ij de ij = — s′ ij —— or — —— ds – (A1.29) 2 s – 2 H′s – The dependence of ds – on its components s′ kl is ∂s – ds – = —— ds′ kl (A1.30) ∂s′ kl 342 Appendix 1 Childs Part 3 31:3:2000 10:42 am Page 342 [...]... q*dxdydzdt (A2.2c) Equating the sum of the terms (equations (A2.2a) to (A2.2c)) to the product of temperature rise and heat capacity of the volume: ∂T ∂ 2T dK ∂T 2 ∂T rC —— = K —— + —— —— – u˘z rC —— + q* 2 ∂t ∂z dT ∂z ∂z ( ) (A2.3a) The extension to three dimensions is straightforward: ∂T ∂ 2T ∂ 2T ∂ 2T dK rC —— = K —— + —— + —— + —— 2 2 2 ∂t ∂x ∂y ∂z dT ( ( ∂T 2 ∂T 2 ∂T —— + —— + —— ∂x ∂y ∂z 2 ) (( )... 27 8 28 5 Kobayashi, S., Oh, S-I and Altan, T (1 989 ) Metal Forming and the Finite Element Method New York: Oxford University Press Kudo, H (1965) Some new slip-line solutions for two-dimentional steady state machining Int J Mech Sci 7, 43–55 Johnson, W and Mellor, P B (1973) Engineering Plasticity London: van Nostrand Johnson, W., Sowerby, R and Venter, R D (19 82 ) Plane-Strain Slip-Line Fields for MetalDeformation... the shear modulus G = 0.5E/(1 + n) Childs Part 3 31:3 :20 00 10: 42 am Page 3 48 3 48 Appendix 1 9G 2 [De–p] = [De] – ————— s 2( H′ + 3G) [ s xx s′ ′ xx s yys xx ′ ′ s zzs′ ′ xx s xx s′ ′ yy s yys yy ′ ′ s zzs′ ′ yy s xx s′ ′ zz s yys zz ′ ′ s zzs zz ′ ′ 2s′ s′ 2s′ s yz xx xy xx ′ 2s yys xy 2s yys yz ′ ′ ′ ′ 2s zzs′ 2s′ s yz ′ xy zz ′ 2s′ s zx xx ′ 2s yys zx ′ ′ 2s′ s zx zz ′ s xys xx ′ ′ s yzs xx ′ ′ s... )} dxdydt (A2.2a) 2 ( )) dxdydzdt The heat accumulating due to convection, Hconv, is Hconv = { ∂T u˘z rCT – u˘z rC T + —— dz ∂z ( ∂T dxdydt ≅ –u˘z rC —— dxdydzt ∂z )} (A2.2b) Childs Part 3 31:3 :20 00 10: 42 am Page 3 52 3 52 Appendix 2 Fig A2.1 (a) A control volume for temperature analysis and (b) dependence of temperature on position and time for the example of Section A2 .2. 1 (κ = 10 mm2/s) Internal heat... Pergamon Press Lee, E H and Shaffer, B W (1951) The theory of plasticity applied to a problem of machining Trans ASME J Appl Mech 18, 405–413 Osakada, K., Nakano, N and Mori, K (19 82 ) Finite element method for rigid plastic analysis of metal forming Int J Mech Sci 24 , 459–4 68 Prager, W and Hodge, P G (1951) The Theory of Perfectly Plastic Solids New York: Wiley Thompsen, E G., Yang, C T and Kobayashi, S (1965)... of Plastic Deformation in Metal Processing New York: MacMillan Zienkiewicz, O C (1 989 ) The Finite Element Method, 4th edn London: McGraw-Hill Zienkiewicz, O C and Godbole, P N (1975) A penalty function approach to problems of plastic flow of metals with large surface deformation J Strain Analysis 10, 180 – 183 Childs Part 3 31:3 :20 00 10: 42 am Page 351 Appendix 2 Conduction and convection of heat in... s′ s zz yz ′ s′ s zz zx ′ 2s xys xy 2s xys yz ′ ′ ′ ′ 2s yzs xy 2s yzs yz ′ ′ ′ ′ 2s zxs xy 2s zxs yz ′ ′ ′ ′ 2s xys yz ′ ′ 2s yzs zx ′ ′ 2s zxs zx ′ ′ ] (A1.49b) If gij instead of eij is used for the shear strains the factors 2 are replaced by 1 Rigid–plastic conditions The basic relation between stress and strain increment, equation (A1.39), for a base set of stresses so and strain increments deo... and penalty functions (for example Kobayashi et al., 1 989 and Zienciewicz and Godbole, 1975) give the same results References Dewhurst, P (19 78) On the non-uniqueness of the machining process Proc Roy Soc Lond A360, 587 –610 Hill, R (1950) Plasticity Oxford: Clarendon Press Hill, R (1954) On the limits set by plastic yielding to the intensities of singularities of stress J Mech Phys Solids, 2, 27 8 28 5... total strain increment and deviatoric strain increment components) 3 s 2 = — (s′ s′ + gs 2 ); ij ji m 2 2 2 dev – de 2 = — de′ de′ + —— ij ji 3 g ( ) (A1.38b) then by the same procedure that led to equation (A1.11), c remains equal to (3 /2) (de–/s–) Then, noting that sij = s′ + dij sm and deij ≡ de′ + dijdev /3, ij ij 2s– 1 1 2s– dev sij = s′ + dij sm = —— de′ + dij —— ≡ —— deij + dij — – — dev ij... z-direction Consider the heat flow into and out of the control volume through the two surfaces of area dxdy, at z = z and z + dz The heat accumulating in time dt due to conduction, Hcond, is (equation (A2.1) with n as z, allowing K and ∂T/∂z to vary with z): Hcond = ∂T dK ∂T ∂T ∂ 2T –K —— + K + —— —— dz —— + ——— dz ∂z dT ∂z ∂z ∂z 2 { ( ( )( ( ∂ 2T dK ∂T ≅ K ——— + —— —— 2 ∂z dT ∂z ) )} dxdydt (A2.2a) . 3(s xx ′ 2 + s yy ′ 2 + s zz ′ 2 ) + 6(s 2 xy + s 2 yz + s 2 zy ) ≡ (s xx – s yy ) 2 + (s yy – s zz ) 2 + (s zz – s xx ) 2 + 6(s 2 xy + s 2 yz + s 2 zy ) = 6k 2 or 2Y 2 (A1 .26 b) which is the same. slip-line is tangential to X, sin 2f = 0 and cos 2f = 1 and ∂∂ —— (p + 2kf) = 0; —— (p – 2kf) = 0 (A1.18c) ∂x ∂y or p + 2kf = constant along an a-line; p – 2kf = constant along a b-line. (A1.18d) If. s r 2 . So s r is also a stress invariant (it is known as the second stress invariant). From equation (A1 .25 ) s r 2 = s 2 xx + s 2 yy + s 2 zz +2( s 2 xy + s 2 yz + s 2 zx ) (A1 .26 a) As s m and