Metal Machining Episode 13 pot

A2.3.1 The infinite solid with velocity u˘z: steady heating at rate q perunit area over the plane z = 0 Figure A2.2b; ambient temperature To In the steady state, the form of equation A2.

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 Theincreasing 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,

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)

Selected problems, with no convection 353

A2.2.3 The semi-infinite solid z > 0: temperature due to an

instantaneous quantity of heat H released into it at the point

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

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

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

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

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 byconduction 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,

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

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 ties with temperature, nor radiative heat loss conditions

proper-Equation (A2.18) is the basis of a finite element temperature calculation method if itsvolume and surface integrations, which extend over the whole analytical region, areregarded 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 ation dI ewith respect to changes in nodal temperatures can also be evaluated and set tozero, 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

Numerical (finite element) methods 357

[H]{T} = {F} (A2.20b)and its solution, completes the finite element calculation The procedure is particularlysimple if four-node tetrahedra are chosen for the elements, as then temperature variationsare linear within an element and temperature gradients are constant Thermal propertiesvarying with temperature can also be considered, by allowing each tetrahedron to havedifferent 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

N l are similarly obtained by cyclic permutation of the subscripts in the order i, j, k, l Veisthe 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 jand

a i ′ = x k ′y j ′ – x j ′y k′; b i ′ = y k ′ – y j′; c i ′ = x j ′ – x k′ (A2.24)

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 gular face: it may also be written in global coordinates as

trian-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

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

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 asfollows:

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] 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

Over a time interval Dt, separating two instants t n and t n+1, the average values of nodalrates of change of temperature can be written in two ways

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: inglobal 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.

All engineering components – for example slideways, gears, bearings, and cutting tools– have rough surfaces, characteristic of how they are made When such surfaces are loadedtogether, they touch first at their high spots Figure A3.1 is a schematic view of two rough

surfaces placed in contact under a load W, the top one sliding to the right under the action

of a friction force F.

Figure A3.1(a) shows a contact, the material properties and roughness of which are suchthat the surfaces have deformed to bring the direction of sliding into the planes of the real

areas of contact Ar Resistance to sliding then comes from the surface shear stresses s.

Friction that arises from shear stresses is called adhesive friction If the real areas of

contact on average support a normal contact stress pr, the adhesive coefficient of friction

(trailing) the real contact mean normal n Even in the absence of surface shear stresses, a

resistance to sliding occurs if the normal forces on the leading and trailing portions of thecontacts differ from one another Friction arising from contact normal stresses is called

deformation friction If, on average, the normal stress plon the leading part of a contact of

sub-area Alis inclined at qlto the direction of the load W, and on the trailing part of the contact the equivalent variables are pt, Atand qt, force resolution in the directions of W and

F give the deformation friction coefficient mdas

p1A1sinq1– ptAtsinqt

p1A1cosq1+ ptAtcosqt

Special cases occur If the contact is symmetrical (pl= pt; Al= At; ql= qt), equation (A3.2a)

simplifies to md= 0: this is the case of perfectly elastic deformation At the other extreme,when the indenting surface plastically scratches (abrades) the other, there may be no trail-

ing portion contact: At= 0 Then, equation (A3.2a) becomes

This type of deformation friction (abrasion of metals) is of most relevance to this book.(There is a third situation, of visco-elastic contact, intermediate between perfectly elastic

and totally plastic contact, when mdmay be shown to depend on both tan qland tan d, the

loss factor for the contact deformation cycle.)

Equation (A3.1) shows that adhesive friction depends mainly on material properties s and pralthough, as will become clear, pralso depends on surface contact geometry Bycontrast, equation (A3.2b) shows that abrasive deformation friction depends mainly on

surface geometry, insofar as the angle qlis the same as the slope of the leading part of thecontact, but this could be modified by material properties if, for example, the real pressure

distribution over Alis not uniform

The main focus of this appendix is to review how the friction coefficient varies withmaterial properties and contact geometry, in adhesive and deformation friction conditions,and when both act together

Two further points can usefully be introduced before proceeding with this review The

real contact stress prin equation (A3.1) is the natural quantity to be part of a friction law,but in practice it is the nominal stress, the load divided by the apparent, or nominal, contact

area An, which is set in any given application In Chapter 2, this stress has been written sn

The first point is that, from load force equilibrium, the ratio of snto pris the same as the

ratio of the real to apparent contact area (Ar/An):

The second point is that, in Chapter 2, snis normalized with respect to some shear flow stress

k of the work or chip material The dimensionless ratios pr/k and s/k can be introduced into equation (A3.1) and further pr/k eliminated in favour of sn/k by means of equation (A3.3a):

(s/k) (s/k) Ar

(pr/k) (sn/k) An

In the following sections, a view of how sliding friction depends on material properties,

contact geometry and intensity of loading is developed, by concentrating on how pr/k and

Ar/Anvary in adhesive and deformation friction conditions A more detailed account ofmuch of the contact mechanics is in the standard text by Johnson (1985) Reference will

be made to this work in the abbreviated form (KLJ Ch.x).

A3.2 The normal contact of a single asperity on an elastic foundation

As a first step in building up a view of asperity contact, consider the normal loading of asingle asperity against a flat counterface At the lightest loading, the deformation may beelastic At some heavier load, plastic deformation may set in The purpose of this section

is to establish how transition from an elastic to a plastic state varies with material

proper-ties and asperity shape; and what real contact pressures prare set up

A3.2.1 Elastic contact

Figure A3.2 shows asperities idealized as a sphere or cylinder of radius R, or as a blunt cone or wedge of slope b, pressed on to a flat The dashed lines show the asperity and flat penetrating each other to a depth d, as if the other was not there The solid lines show the deformation required to eliminate the penetration How prvaries with the contact width 2a,

or with d; and with R or b; and with Young’s modulus E1and E2and Poisson’s ratio n1and

n2of the asperity and counterface respectively, is developed here

The contact of an elastic sphere or cylinder on a flat in the absence of interface shear isthe well-known Hertzian contact problem A dimensional approach gives insight into thecontact conditions more simply than does a full Hertzian analysis

In 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<< R,

a2c a2

The surface deformations in the asperity and flat cause sub-surface strains In the

asper-ity, these are in proportion to the dimensionless ratio d1/a and in the flat to d2/a When the

A single asperity on an elastic foundation 365

Fig A3.2 Models of elastic asperity deformation

asperity and flat obey Hooke’s law, the mean contact stress prwill increase in proportion

to the product of Young’s modulus and strain in each:

from the asperity’s point of view, pr∝ E1(d1/a)

(A3.5)from the flat’s point of view, pr∝ E2(d2/a)

Combining equations (A3.4) and (A3.5) gives

and c depends on whether the circular profile of radius R represents a spherically or a

cylindrically capped asperity (Table A3.1)

Similarly, the pressing together of two spherical or two cylindrical asperities with

paral-lel axes, of radii R1and R2, creates a normal contact stress pr:

pr = cE*(a/R*) where 1/R* = 1/R1 + 1/R2 (A3.8)The elastic contact of a wedge or cone on a flat (right-hand part of Figure A3.2(a))

generates a contact pressure pr(KLJ Ch 5):

where c is also given in Table A3.1 The quantities (a/R*) and tan b can be regarded as

representative contact strains Their interpretation as mean contact slopes will be returned

to later As they increase, so does pr

A3.2.2 Fully plastic contact

Figure A3.3 shows a wedge-shaped asperity loaded plastically against a softer (left) and a

harder (right) counterface, so that it indents or is flattened The dependence of pron

asper-ity slope b and shear flow stress k of the softer material is considered here, by means of

slip-line field theory (Appendix 1.2)

In each case, the region ADE is a uniform stress region and the free surface condition

along AE requires that p1= k Region ABC is also uniformly stressed Normal force

equi-librium across AC gives

Table A3.1 Elastic contact parameters, from Johnson (1985, Chs 4 and 5)

Asperity peak shape c, eqns (A3.8) and (3.9) (pr/ τ max ) (pr/ τ max)/c

pr= p2+ k (A3.10)

Slip-line EDBC is an a-line, so

The angle y is chosen to conserve the volume of the flow: material displaced from the

overlap between the flat and the asperity must re-appear in the shoulders of the flow, but

for small values of b, y ≈ p/2 This, with equations (A3.11) and (A3.10), gives

A3.2.3 The transition from elastic to plastic contact

The elastic and plastic views of the previous sub-sections are brought together by

non-dimensionalizing the contact pressures prby k In Figure A3.4(a), the elastic and plastic

model predictions are the dashed lines The solid line is the actual behaviour Departure

from elastic behaviour first occurs in the range 1 < pr/k < 2.6, at values of (E*/k)(a/R* or tanb) from 2 to 6.2 The values depend on the asperity shape: they are the last two columns

in Table A3.1

The fully plastic state is developed for (E*/k)(a/R* or tanb) greater than about 50 pr/k

continues to increase at larger deformations than this due to strain hardening

A single asperity on an elastic foundation 367

Fig A3.3 Plastic indenting by, and flattening of, wedge-shaped asperities

Fig A3.4 (a) Real contact pressure variation with asperity deformation severity; (b) the dependence of degree of

contact on intensity of loading, in the absence of sliding