Metal Machining - Theory and Applications 2008 Part 13 pot

A2.2 Selected problems, with no convection When u˘ x = u˘ y = u˘ z = 0, and q* = 0 too, equation (A2.4) simplifies further, to 1 ∂T ∂ 2 T ∂ 2 T ∂ 2 T — —— = ( —— + —— + —— ) (A2.5) k ∂t ∂x 2 ∂y 2 ∂z 2 where the diffusivity k equals K/rC. In this section, some solutions of equation (A2.5) are presented that give physical insight into conditions relevant to machining. A2.2.1 The semi-infinite solid z > 0: temperature due to an instantaneous quantity of heat H per unit area into it over the plane z = 0, at t = 0; ambient temperature T o It may be checked by substitution that z 2 H 1 – —— T – T 0 = —— ——— e 4kt (A2.6) rC ǰ˭˭˭ pkt is a solution of equation (A2.5). It has the property that, at t = 0, it is zero for all z > 0 and is infinite at z = 0. For t > 0, ∂T/∂z = 0 at z = 0 and ∞ ∫ rC(T – T 0 )dz = H (A2.7) 0 Equation (A2.6) thus describes the temperature rise caused by releasing a quantity of heat H per unit area, at z = 0, instantaneously at t = 0; and thereafter preventing flow of heat across (insulating) the surface z = 0. Figure A2.1(b) shows for different times the dimen- sionless temperature rC(T – T 0 )/H for a material with k = 10 mm 2 /s, typical of metals. The increasing extent of the heated region with time is clearly seen. At every time, the temperature distribution has the property that 84.3% of the associ- ated heat is contained within the region z/ ǰ˭˭˭ 4kt < 1. This result is obtained by integrating equation (A2.6) from z = 0 to ǰ˭˭˭ 4kt. Values of the error function erf p, 2 p erf p = —— ∫ e –u 2 du (A2.8) Ȉȉ p 0 that results are tabulated in Carslaw and Jaeger (1959). Physically, one can visualize the temperature front as travelling a distance ≈ ǰ˭˭˭ 4kt in time t. This is used in considering temperature distributions due to moving heat sources (Section A2.3.2). A2.2.2 The semi-infinite solid z > 0: temperature due to supply of heat at a constant rate q per unit area over the plane z = 0, for t > 0; ambient temperature T o Heat dH = qdt′ is released at z = 0 in the time interval t′ to t ′ + dt′. The temperature rise that this causes at z at a later time t is, from equation (A2.6) Selected problems, with no convection 353 Childs Part 3 31:3:2000 10:42 am Page 353 z 2 qdt′ 1 – —— d(T – T 0 ) = —— ————— e 4k(t–t′) (A2.9) rC (pk(t – t ′)) ½ The total temperature is obtained by integrating with respect to t′ from 0 to t. The temper- ature at z = 0 will be found to be of interest. When q is independent of time 2 q (T – T 0 ) = —— — Ȉȉ kt (A2.10) Ȉȉ p K The average temperature at z = 0, over the time interval 0 to t, is 2/3rds of this. A2.2.3 The semi-infinite solid z > 0: temperature due to an instantaneous quantity of heat H released into it at the point x = y = z =0 , at t = 0; ambient temperature T o In this case of three-dimensional heat flow, the equivalent to equation (A2.6) is x 2 +y 2 +z 2 H 1 – ——— T – T 0 = —— ——— e 4kt (A2.11) 4rC (pkt) 3/2 Equation (A2.11) is a building block for determining the temperature caused by heating over a finite area of an otherwise insulated surface, which is considered next. A2.2.4 The semi-infinite solid z > 0: uniform heating rate q per unit area for t > 0, over the rectangle – a < x < a ,– b < y < b at z = 0; ambient temperature T o Heat flows into the solid over the surface area shown in Figure (A2.2a). In the time inter- val t′ to t′ + dt′, the quantity of heat dH that enters through the area dA = dx′dy′ at (x′, y′) is qdAdt′. From equation (A2.11) the contribution of this to the temperature at any point (x, y, z) in the solid at time t is (x–x′) 2 +(y–y′) 2 +z 2 qdx′dy′dt′ – ————— d(T – T 0 ) = ————————— e 4k(t–t′) (A2.12) 4rC(pk) 3/2 (t – t ′ ) 3/2 Integrating over time first, in the limit as t and t ′ approach infinity (the steady state), q +a +b dy′ d(T – T 0 ) = —— ∫∫ ————————————— dx′ (A2.13) 2pK –a –b ((x – x′) 2 + (y – y′) 2 + z 2 ) ½ Details of the integration over area are given by Loewen and Shaw (1954). At the surface z = 0, the maximum temperature (at x = y = 0) and average temperature over the heat source are respectively 354 Appendix 2 Childs Part 3 31:3:2000 10:42 am Page 354 2qa b b a (T – T 0 ) max = —— ( sinh –1 — + — sinh –1 — ) pKaab } 2qa a b b 2 ½ b 2 a (T – T 0 ) av = (T – T 0 ) max –—— [( — + — )( 1 + —— ) – —— – — ] 3pKb a a 2 a 2 b (A2.14) A2.3 Selected problems, with convection Figures A2.2(b) and (c) show two classes of moving heat source problem. In Figure A2.2(b) heating occurs over the plane z = 0, and the solid moves with velocity u˘ z through the source. In Figure A2.2(c), heating also occurs over the plane z = 0, but the solid moves tangentially past the source, in this case with a velocity u˘ x in the x- direction. Selected problems, with convection 355 Fig. A2.2 Some problems relevant to machining: (a) surface heating of a stationary semi-infinite solid; (b) an infinite solid moving perpendicular to a plane heat source; (c) a semi-infinite solid moving tangentially to the plane of a surface heat source Childs Part 3 31:3:2000 10:42 am Page 355 A2.3.1 The infinite solid with velocity u ˘ z : steady heating at rate q per unit area over the plane z = 0 (Figure A2.2b); ambient temperature T o In the steady state, the form of equation (A2.4) (with q* = 0) to be satisfied is ∂ 2 T ∂T k —— = u˘ z —— (A2.15) ∂z 2 ∂z The temperature distribution qq u˘ z z (T – T 0 ) = ——— , z ≥ 0; (T – T 0 ) = ——— e —— , z ≤ 0 (A2.16) rCu˘ ˘z rCu˘ z k satisfies this. For z > 0, the temperature gradient is zero: all heat transfer is by convection. For z = – 0, ∂T/∂z = q/K: from equation (A2.1), all the heating rate q is conducted towards –z. It is eventually swept back by convection towards + z. A2.3.2 Semi-infinite solid z > 0, velocity: u ˘ x steady heating rate q per unit area over the rectangle –a < x < a , – b < y < b , z = 0 (Figure A2.2(c)); ambient temperature T o Two extremes exist, depending on the ratio of the time 2a/u˘ x , for an element of the solid to pass the heat source of width 2a to the time a 2 /k for heat to conduct the distance 2a (Section A2.2.1). This ratio, equal to 2k/(u˘ x a), is the inverse of the more widely known Peclet number P e . When the ratio is large (P e << 1), the temperature field in the solid is dominated by conduction and is no different from that in a stationary solid, see Section A2.2.4. Equations (A2.14) give maximum and average temperatures at the surface within the area of the heat source. When b/a = 1 and 5, for example, bqaqa — = 1:(T – T o ) max = 1.12 —— ; (T – T 0 ) av = 0.94 —— u x ˘a/(2k) << 1: aKK } bqaqa — = 5:(T – T o ) max = 2.10 —— ; (T – T o ) av = 1.82 —— aKK (A2.17a) At the other extreme (P e >> 1), convection dominates the temperature field. Beneath the heat source, ∂T/∂z >> ∂T/∂x or ∂T/∂y; heat conduction occurs mainly in the z-direction and temperatures may be found from Section A2.2.2. At z = 0, the temperature variation from x = – a to x = + a is given by equation (A2.10), with the heating time t from 0 to 2a/u˘ x . Maximum and average temperatures are, after rearrangement to introduce the dimension- less group (qa/K), qa 2k ½ qa 2k ½ u˘ x a/(2k) >> 1: (T – T 0 ) max = 1.13 —— ( —— ) ;(T – T 0 ) av = 0.75 —— ( —— ) Ku x aKu x a (A2.17b) 356 Appendix 2 Childs Part 3 31:3:2000 10:42 am Page 356 Because these results are derived from a linear heat flow approximation, they depend only on the dimension a and not on the ratio b/a, in contrast to P e << 1 conditions. A more detailed analysis (Carslaw and Jaeger, 1959) shows equations (A2.16) and (A2.17) to be reasonable approximations as long as u˘ x a/(2k) < 0.3 or > 3 respectively. Applying them at u˘ x a/(2k) = 1 leads to an error of ≈20%. A2.4 Numerical (finite element) methods Steady state (∂T/∂t = 0) solutions of equation (A2.4), with boundary conditions T = T s on surfaces S T of specified temperature, K∂T /∂n = 0 on thermally insulated surfaces S qo , K∂T/∂n = –h(T–T o ) on surfaces S h with heat transfer (heat transfer coefficient h), K∂T/∂n = –q on surfaces S q with heat generation q per unit area. may be found throughout a volume V by a variational method (Hiraoka and Tanaka, 1968). A temperature distribution satisfying these conditions minimizes the functional K ∂T 2 ∂T 2 ∂T 2 I(T) = ∫ V [ — {( —— ) + ( —— ) + ( —— )} 2 ∂x ∂y ∂z ∂T – ∂T – ∂T – – { q* – rC ( u˘ x —— + u˘ y —— + u˘ z —— )} T ] dV ∂x ∂y ∂z h + ∫ S q qTdS + ∫ S h — (T 2 – 2T 0 T)dS 2 (A2.18) where the temperature gradients ∂T – /∂x, ∂T – /∂y, ∂T – /∂z, are not varied in the minimization process. The functional does not take into account possible variations of thermal proper- ties with temperature, nor radiative heat loss conditions. Equation (A2.18) is the basis of a finite element temperature calculation method if its volume and surface integrations, which extend over the whole analytical region, are regarded as the sum of integrations over finite elements: m I(T) = ∑ I e (T) (A2.19) e=1 where I e (T) means equation (A2.18) applied to an element and m is the total number of elements. If an element’s internal and surface temperature variations with position can be written in terms of its nodal temperatures and coordinates, I e (T) can be evaluated. Its vari- ation dI e with respect to changes in nodal temperatures can also be evaluated and set to zero, to produce an element thermal stiffness equation of the form [H] e {T} = {F} e (A2.20a) where the elements of the nodal F-vector depend on the heat generation and loss quanti- ties q*, q and h, and the elements of [H] e depend mainly on the conduction and convec- tion terms of I e (T). Assembly of all the element equations to create a global equation Numerical (finite element) methods 357 Childs Part 3 31:3:2000 10:42 am Page 357 [H]{T} = {F} (A2.20b) and its solution, completes the finite element calculation. The procedure is particularly simple if four-node tetrahedra are chosen for the elements, as then temperature variations are linear within an element and temperature gradients are constant. Thermal properties varying with temperature can also be considered, by allowing each tetrahedron to have different thermal properties. In two-dimensional problems, an equally simple procedure may be developed for three-node triangular elements (Tay et al., 1974; Childs et al., 1988). A2.4.1 Temperature variations within four-node tetrahedra Figure A2.3 shows a tetrahedron with its four nodes i, j, k, l, ordered according to a right- hand rule whereby the first three nodes are listed in an anticlockwise manner when viewed from the fourth one. Node i is at (x i , y i , z i ) and so on for the other nodes. Temperature T e anywhere in the element is related to the nodal temperatures {T} = {T i T j T k T l } T by T e = [N i N j N k N l ]{T} = [N]{T} (A2.21) where [N] is known as the element’s shape function. 1 N i = —— (a i + b i x + c i y + d i z) 6V e where x j y j z j 1 y j z j a i = | x k y k z k | , b i =– | 1 y k z k | x l y l z l 1 y l z l 358 Appendix 2 Fig. A2.3 A tetrahedral finite element Childs Part 3 31:3:2000 10:42 am Page 358 x k 1 z j x j y j 1 c i =– | x k 1 z k | , d i =– | x k y k 1 | x l 1 z l x l y l 1 and 1 1 x i y i z i V c = — | 1 x j y j z j | (A2.22) 6 1 x k y k z k 1 x l y l z l This may be checked by showing that, at the nodes, T e takes the nodal values. N j , N k and N l are similarly obtained by cyclic permutation of the subscripts in the order i, j, k, l. V e is the volume of the tetrahedron. In the same way, temperature T s over the surface ikj may be expressed as a linear func- tion of the surface’s nodal temperatures: T s = [N i ′N j ′N k ′]{T} = [N′]{T} (A2.23) where 1 N i ′ = ——— (a i ′ + b i ′x′ + c i ′y′) 2D ikj and a i ′ = x k ′y j ′ – x j ′y k ′; b i ′ = y k ′ – y j ′; c i ′ = x j ′ – x k ′ (A2.24) 1 1 x i ′ y i ′ D ikj = — | 1 x k ′ y k ′ | 2 1 x j ′ y j ′ The other coefficients are obtained by cyclic interchange of the subscripts in the order i, k, j. x′, y′ are local coordinates defined on the plane ikj. D ikj is the area of the element’s trian- gular face: it may also be written in global coordinates as 1 y k – y i y j – y j 2 z k – z i z j – z i 2 x k – x i x j – x i 2 ½ D ikj = — ( || + || + || ) 2 z k – z i z j – z i x k – x i x j – x i y k – y i y j – y i (A2.25) A2.4.2 Tetrahedral element thermal stiffness equation Equation (A2.21), after differentiation with respect to x, y and z, and equation (A2.23) are substituted into I e (T) of equation A2.19. The variation of I e (T) with respect to T i , T j , T k and T l is established by differentiation and set equal to zero. [H] e and {F} e (equation (A2.20a)) are [H] e = K b i b i + c i c i + d i d i b j b i + c j c i + d j d i b k b i + c k c i + d k d i b l b i + c l c i + d l d i —— [ b i b j + c i c j + d i d j b j b j + c j c j + d j d j b k b j + c k c j + d k d j b l b j + c l c j + d l d j ] 36V e b i b k + c i c k + d i d k b j b k + c j c k + d j d k b k b k + c k c k + d k d k b l b k + c l c k + d l d k b i b l + c i c l + d i d l b j b l + c j c l + d j d l b k b l + c k c l + d k d l b l b l + c l c l + d l d l Numerical (finite element) methods 359 Childs Part 3 31:3:2000 10:42 am Page 359 rC u˘ x b i + u˘ y c i + u˘ z d i u˘ ˘x b j + u˘ y c j + u˘ z d j u˘ ˘x b k + u˘ y c k + u˘ ˘z d k u˘ x b l + u˘ y c l + u˘ z d l + —— [ u˘ x b i + u˘ y c i + u˘ z d i u˘ x b j + u˘ y c j + u˘ ˘z d j u˘ x b k + u˘ y c k + u˘ ˘z d k u˘ x b l + u˘ y c l + u˘ z d l ] 24 u˘ x b i + u˘ y c i + u˘ z d i u˘ x b j + u˘ y c j + u˘ z d j u˘ x b k + u˘ ˘y c k + u˘ z d k u˘ x b l + u˘ ˘y c l + u˘ ˘z d l u˘ ˘x b i + u˘ y c i + u˘ ˘z d i u˘ x b j + u˘ ˘y c j + u˘ ˘z d j u˘ x b k + u˘ ˘y c k + u˘ z d k u˘ ˘x b l + u˘ y c l + u˘ z d l hD ikj 2110 + —— [ 1210 ] 12 1120 0000 (A2.26) and 11 1 q*V e 1 qD ikj 1 hT 0 D ikj 1 {F} e = ——— {} – ——— {} – ——— {} (A2.27) 4 1 3 1 3 1 10 0 Global assembly of equations (A2.20a), with coefficients equations (A2.26) and (A2.27), to form equation (A2.20b), or similarly in two-dimensions, forms the thermal part of closely coupled steady state thermal–plastic finite element calculations. A2.4.3 Approximate finite element analysis Finite element calculations can be applied to the shear-plane cutting model shown in Figure A2.4. There are no internal volume heat sources, q*, in this approximation, but internal surface sources q s and q f on the primary shear plane and at the chip/tool inter- face. If experimental measurements of cutting forces, shear plane angle and chip/tool contact length have been carried out, q s and the average value of q f can be determined as follows: q s = t s V s (A2.28a) q f = t f V c (A2.28b) where F C cos f – F T sin f F C sin a + F T cos a t s = ————————— sin f; t f = ————————— fd l c d } cos a sin f V s = ———— U work ; V c = ———— U work cos(f – a) cos(f – a) (A2.29) In general, q s is assumed to be uniform over the primary shear plane, but q f may take on a range of distributions, for example triangular as shown in Figure A2.4. A2.4.4 Extension to transient conditions The functional, equation (A2.18), supports transient temperature calculation if the q* term is replaced by (q* – rC∂T – /∂t). Then the finite element equation (A2.20a) becomes 360 Appendix 2 Childs Part 3 31:3:2000 10:42 am Page 360 ∂T [C] e { —— } + [H] e {T} = {F e } (A2.30) ∂t with rCV e 2111 [C] e = ——— | 1211 | 20 1121 1112 ([C] is given here for a four-node tetrahedron). Numerical (finite element) methods 361 Fig. A2.4 Thermal boundary conditions for a shear plane model of machining Childs Part 3 31:3:2000 10:42 am Page 361 Over a time interval Dt, separating two instants t n and t n+1 , the average values of nodal rates of change of temperature can be written in two ways ∂T ∂T ∂T { —— } = (1 – q) { —— } + q { —— } (A2.31a) ∂t av ∂t n ∂t n+1 or ∂TT n+1 – T n { —— } = { ———— } (A2.31b) ∂t av Dt where q is a fraction varying between 0 and 1 which allows the weight given to the initial and final values of the rates of change of temperature to be varied. After multiplying equa- tions (A2.31) by [C], substituting [C]{∂T/∂t}terms in equation (A2.31a) for ({F}–[H]{T}) terms from equation (A2.30), equating equations (A2.31a) and (A2.31b), and rearranging, an equation is created for temperatures at time t n+1 in terms of temperatures at time t n :in global assembled form [C][C] ( —— + q[K] ) {T} n+1 = ( —— – (1 – q)[K] ) {T} n + {F} (A2.32) Dt Dt This is a standard result in finite element texts (for example Huebner and Thornton, 1982). Time stepping calculations are stable for q ≥ 0.5. Giving equal weight to the start and end rates of change of temperature (q = 0.5) is known as the Crank–Nicolson method (after its originators) and gives good results in metal cutting transient heating calculations. References Carslaw, H. S. and Jaeger, J. C. (1959) Conduction of Heat in Solids, 2nd edn. Oxford: Clarendon Press. Childs, T. H. C., Maekawa, K. and Maulik, P. (1988) Effects of coolant on temperature distribution in metal machining. Mat. Sci. and Technol. 4, 1006–1019. Hiraoka, M. and Tanaka, K. (1968) A variational principle for transport phenomena. Memoirs of the Faculty of Engineering, Kyoto University 30, 235–263. Huebner, K. H. and Thornton, E. A. (1982) The Finite Element Method for Engineers, 2nd edn. New York: Wiley. Loewen, E. G. and Shaw, M. C. (1954) On the analysis of cutting tool temperatures. Trans. ASME 76, 217–231. Tay, A. O., Stevenson, M. G. and de Vahl Davis, G. (1974) Using the finite element method to deter- mine temperature distributions in orthogonal machining. Proc. Inst. Mech. Eng. Lond. 188, 627–638. 362 Appendix 2 Childs Part 3 31:3:2000 10:42 am Page 362 [...]... contact stresses sn and t, and on the hydrostatic stress in the bulk field, will now be developed, still for conditions of plane strain and a non-hardening plastic material to which slipline field theory can be applied Figure A3.9(a) is adapted from Sutcliffe (1988) It shows a combined asperity and bulk slip-line field The bulk field is described by the hydrostatic pressure pE and slip-lines inclined at... section suggest pr/k < 1 How pr/k – and hence (with sn/k) Ar/An – depends on pE and zbulk can be determined from the slip-line field and its limits of validity However, it is simpler to consider the overall force balances between the bulk field, the nominal contact stresses sn and t, and the real contact stresses pr and s Force equilibrium between pE, k, zbulk and sn and t creates the relations: t = k... (Figure A4.1) The copper and copper alloys (left-hand panel) are all initially in the annealed state They show the low initial yield and large amount of strain hardening typical of these face centred cubic metals The aluminium and aluminium alloys (right-hand panel) show a similar behaviour, but generally at a lower level of stress Some aluminium alloys can be hardened Childs Part 3 31:3:2000 10:43 am... appropriate for an austenitic 18%Mn-5%Cr steel, with negligible strain path dependence (Maekawa et al 1993): s— = 3.02e ˘— 0.00714[45400/(273 + T) + 58.4 + a(860 – T)e— b] where, for e— ≤ 0.5 a + 0.87, b = 0.8; e— ≥ 0.5 a = 0.57, b = 0.2 Other forms have been given for a Ti-6Al-4V alloy (Usui et al 1984) and a Ti-6Al-6V-2Sn alloy (Maekawa et al 1994b) For the Ti-6Al-4V alloy: { [ ∫strain path e–aT/N(e˘—/1000)–m/Nde—]N}... for the Ti-6Al-6V-2Sn alloy were fitted to equation (A4.2a) with 2 2 2 A = 2160e–0.0013T + 29e–0.00 013( T–80) + 7.5e–0.00014(T–300) + 47e–0.0001(T–700) M = 0.026 + 0.0000T a = 0.00009 2 N = 0.18e–0.0016T + 0.015e–0.00001(T–700) m = 0.0055 References ASM (1990) Metals Handbook, 10th edn Ohio: ASM Ashby, M F and Jones, D R H (1986) Engineering Materials, Vol 2 Oxford: Pergamon Press Childs, T H C and Maekawa,... K and Kitagawa, T (1994b) The effects of cutting speed and feed on chip flow and tool wear in the machining of a titanium alloy In: Proc 3rd Int Conf on Behaviour of Materials in Machining, Warwick, 15–17 November pp 152–167 Childs Part 3 31:3:2000 10:43 am Page 382 382 Appendix 4 Maekawa, K., Ohhata, T., Kitagawa, T and Childs, T H C (1996) Simulation analysis of machinability of leaded Cr-Mo and. .. 3.5–4.2 12–14 8.7 3.8–4.7 12–14 8.9 4.2–5.3 Titanium pure titanium* α, α–β, β alloys Ti-6Al-4V 9.5 2.2–5.0 2.2–3.0 7.6 2.7–5.5 2.7–3.5 6.8 3.2–6.0 3.2–3.8 – 3.7–6.4 3.8–4.2 – 3.7–6.6 3.8–4.7 *: high and commercial purity; **: including cobalt- and ferrous-base superalloys source of information has been the ASM (1990) Metals Handbook but it has been necessary also to gather information from a range of other... the left-hand part of Figure A3.2, the asperity is shown flattened by a depth d1, and the flat by a depth d2, in accommodating the total overlap d and creating a contact width 2a From the geometry of overlap, supposing 2a to be a fixed fraction of the chordal length 2ac, and when ac . the elements of the nodal F-vector depend on the heat generation and loss quanti- ties q*, q and h, and the elements of [H] e depend mainly on the conduction and convec- tion terms of I e (T) stresses s n and t, and on the hydrostatic stress in the bulk field, will now be devel- oped, still for conditions of plane strain and a non-hardening plastic material to which slip- line field theory. p r varies with the contact width 2a, or with d; and with R or b; and with Young’s modulus E 1 and E 2 and Poisson’s ratio n 1 and n 2 of the asperity and counterface respectively, is developed here. The