Metal Machining - Theory and Applications Episode 2 Part 9 doc

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 To It may be check

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 ∂2T ∂2T ∂2T

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 To

It may be checked by substitution that

z2

rC 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

∞

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 – T0)/H for a material with k = 10 mm2/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/ 4kt < 1 This result is obtained by integrating equation (A2.6) from z = 0 to  4kt Values of the error function erf p,

erf p = —— ∫ e–u2

p 0

that results are tabulated in Carslaw and Jaeger (1959) Physically, one can visualize the temperature front as travelling a distance ≈ 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 To

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)

d(T – T0) = —— ————— 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

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 To

In this case of three-dimensional heat flow, the equivalent to equation (A2.6) is

x2+y2+z2

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 To

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+z2

qdx ′dy′dt′ – —————

d(T – T0) = ————————— 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),

d(T – T0) = —— ∫ ∫————————————— dx′ (A2.13)

2pK –a –b ((x – x′)2+ (y – y′)2+ z2)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

2qa b b a

(T – T0)max= —— pK (sinh–1— + — sinha a –1—b)

}

(T – T0)av= (T – T0)max– ——3pK[(— + —b a)(1 + ——a2 )– —— – —a2 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

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

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 To

In the steady state, the form of equation (A2.4) (with q* = 0) to be satisfied is

∂2T ∂T

∂z2 ∂z

The temperature distribution

(T – T0) = ——— , z≥ 0; (T – T0) = ——— e——, z≤ 0 (A2.16)

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˘xsteady heating rate q per

unit area over the rectangle –a < x < a, –b < y < b, z = 0 (Figure A2.2(c)); ambient temperature To

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 a2/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 Pe

When the ratio is large (Pe << 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,

— = 1:(T – To)max= 1.12 —— ; (T – T0)av= 0.94 ——

— = 5:(T – To)max= 2.10 —— ; (T – To)av= 1.82 ——

(A2.17a)

At the other extreme (Pe>> 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),

u˘x a/(2k) >> 1: (T – T0)max= 1.13 —— K (——u x a) ; (T – T0)av= 0.75 —— K (——u x a )

(A2.17b)

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 Pe<< 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 = Tson surfaces STof specified temperature,

K ∂T/∂n = 0 on thermally insulated surfaces Sqo,

K ∂T/∂n = –h(T–To) on surfaces Shwith 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

I(T) = ∫V[— 2 {(—— ∂x )+ (—— ∂y )+ (——∂z ) }

– {q* – rC(u˘x —— + u˘ y —— + u˘ z——)}T]dV

h

+ ∫S q qTdS + ∫S h — (T2– 2T0T)dS

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

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 ewith respect to changes in nodal temperatures can also be evaluated and set to zero, to produce an element thermal stiffness equation of the form

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] edepend mainly on the conduction and

convec-tion terms of I e (T) Assembly of all the element equations to create a global equation

[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 Te

anywhere in the element is related to the nodal temperatures {T} = {T i T j T k T l}Tby

Te= [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)

6Ve

where

a i=

|x k y k z k

|, b i= –

|1 y k z k

|

Fig A2.3 A tetrahedral finite element

x k 1 z j x j y j 1

c i= –

| x k 1 z k

|, d i= –

|x k y k 1

|

and

1

1 x i y i z i

V c= —

| 1 x j y j z j

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, Tetakes the nodal values N j , N kand

N l are similarly obtained by cyclic permutation of the subscripts in the order i, j, k, l Veis the volume of the tetrahedron

In the same way, temperature Tsover the surface ikj may be expressed as a linear

func-tion of the surface’s nodal temperatures:

Ts= [N i ′N j ′N k ′]{T} = [N′]{T} (A2.23) where

1

N i ′ = ——— (a i ′ + b i ′x′ + c i ′y′)

2Dik j 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′

Dik j= —

|1 x k ′ y k′|

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 ikjis 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

Dik j= — 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 Ie(T) of equation A2.19 The variation of Ie(T) with respect to T i , Tj, T k and T l

is established by differentiation and set equal to zero [H]eand {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

]

36Ve 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

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 ck+ 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

hDik j 2 1 1 0 + ——[1 2 1 0

]

and

{F}e= ——— { }– ——— { }– ——— { } (A2.27)

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 qsand qfon 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, qsand the average value of qfcan be determined as follows:

where

FCcos f – FTsin f FCsin a + FTcos a

ts= ————————— sin f; tf= —————————

}

Vs= ———— Uwork; Vc= ———— Uwork

(A2.29)

In general, qsis assumed to be uniform over the primary shear plane, but qfmay 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

[C]e{——}+ [H]e{T} = {Fe} (A2.30)

∂t

with

[C]e= ———

| 1 2 1 1

|

([C] is given here for a four-node tetrahedron).

Fig A2.4 Thermal boundary conditions for a shear plane model of machining

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

{——} = (1 – q){——} + q{——} (A2.31a)

or

∂T T n+1 – Tn

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

(—— + q[K]){T} n+1=(—— – (1 – q)[K]){T} n + {F} (A2.32)

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.