The main difference of the abrasive-extrusion machining from other kinds of abrasive blasting is the fact that a polymer base capable of viscoelastic deformation ...
ISSN 1068-7998, Russian Aeronautics (Iz.VUZ), 2009, Vol. 52, No. 1, pp. 94–98. © Allerton Press, Inc., 2009. Original Russian Text © V.A. Levko, 2009, published in Izvestiya VUZ. Aviatsionnaya Tekhnika, 2009, No. 1, pp. 59–62.
AIRCRAFT PRODUCTION TECHNOLOGY
Calculation of Surface Roughness in Abrasive-Extrusion Machining on the Basis of Contact–Interaction Model V. A. Levko Siberian State Aerospace University, Krasnoyarsk, Russia Received June 3, 2008
Abstract—A calculation of surface roughness with regard for the abrasive-extrusion machining (AEM) efficiency is considered. The said problem was solved using a model of contact interactions between microasperities of surface being machined and abrasive (diamond) grains being displaced by a viscoelastic medium flux. DOI: 10.3103/S1068799809010164
The main difference of the abrasive-extrusion machining from other kinds of abrasive blasting is the fact that a polymer base capable of viscoelastic deformation is used as an abrasive grain carrier. A flux of the viscoelastic working medium during the AEM process in the multiple shear flow takes the shape of the channel being machined; thus, making it possible to perform finishing of internal surfaces of different complex-shaped parts of aerospace equipment [1]. The flow parameters along with the geometric characteristics of the channel being machined and material properties determine the form of contact interactions between microasperities on the surface layer of the channel being machined and abrasive grains being displaced in the channel by the viscoelastic medium flux [2]. A mechanism of elastic deformation initiation in the working medium flux during AEM process is considered in [3]. It is proposed that a method for determining the contact characteristics in the diamond-abrasive machining be used as a basis for calculations [4]; the method makes it possible to calculate the efficiency and quality of diamond honing. As compared with the contact of two rough surfaces in the diamondabrasive machining, however, the contact of the medium (as an abrasive tool) with a part in AEM is characterized by a number of distinctive features. The rotation of the medium as an abrasive tool is missing from the channel being machined. Bodies different in physico-mechanical properties are in contact, and the working medium is a composition of the hardest abrasive (diamond) grains and the viscoelastic polymer base. The contact occurs in the shear flow of the working medium within a rather narrow velocity range (0.001–0.1 m/s), and the stress-strain state of the viscoelastic base determines an extent of grain mobility in contact. In the experiments we observed a nearwall rotation of abrasive grains in combination with some medium volume. In this case the rotational speed depends on the velocity of the shear flow and the dimensions of the rotating working medium volume. In this connection, in the model considered the influence of the abrasive grain rotation on the conditions of contact with the surface being machined is taken to be insignificant. At the same time, rotation provides a regular change of grain microasperities that is in contact resulting in an effect of constant self-grinding of the medium as an abrasive tool. In the flow of viscoelastic base 1 abrasive grains 2 being in contact with protrusions 3 and recesses 4 of the surface layer microasperities follow its micropattern (Fig. 1). The forces Fσ and Fτ that are due to the stress-strain state of polymer base 1 above microasperity recess 4 extrude a grain out of the base into the recess for a definite amount that depends on grain size 2, protrusion height 3, recess pitch 4 and the distance within which the grain in contact zone 5 approaches the lateral surface of the protrusion (Fig. 2).
94
CALCULATION OF SURFACE ROUGHNESS IN ABRASIVE-EXTRUSION MACHINING
Fig. 1.
95
Fig. 2.
Fig. 1. General scheme of abrasive grain–surface layer microasperity contact. Fig. 2. Scheme of abrasive grain–microasperity recess contact.
At the start of the grain contact with the lateral surface of microasperity at the side of the apex there appears a response to the action. In this case a part of grain microasperity is introduced into the surface material for the amount h, and the grain itself starts to be extruded back into the base performing an oscillatory motion relative to the line of the medium-part contact. The amount h depends on the physicomechanical properties of the bodies being in contact and the stress-strain state of the working medium. The frequency and amplitude of the abrasive grain oscillations are determined by its sizes and the roughness parameters of the part being machined. Based on this, it is possible to assume that the surface layer roughness of the bodies in contact by the height parameters is similar. For further consideration we use a spherical model of microasperity both for the working medium grains and for the part being machined. In the AEM process such an internal contact is realized at which the part being machined and the tool are of the same nominal dimension. The surface microasperity is simulated by spherical segments with the radius ρ p , the microasperity of abrasive tool grains⎯by spheres of the radius ρt , the protruding part of which has a random height. It is assumed that the shear flow velocity is low, and the contact conditions of an abrasive grain along the channel length insignificantly vary. Hence, taking into account that the viscoelastic base follows the shape of the channel being machined, the influence of surface waviness on the contact condition in this model is considered to be slight. We considered two rough surfaces of the part and tool that first are at a distance of Rmax = Rmaxp + Rmaxt and in contact approach each other by the magnitude a. The relative approach of surfaces are determined by the formula
(
)
ε = a Rmax p + Rmax t = a Rmax . In interaction of microasperities of the part being machined and the tool, the contact between them can be elastic, plastic and elastoplastic. In the elastic contact, the calculation can be made by the Hertz formula: 1 /2
4 ⎛ ρ pρt ⎞ Ni = ⎜ ⎟ 3 ⎜⎝ ρ p + ρt ⎟⎠
ai3/2
⎛ 1 − μ 2p 1 − μ 2 ⎞ t ⎟ ⎜ + , ⎜ Ep Et ⎟ ⎝ ⎠
where Ni is the force appearing in the interaction of two microasperities; ai is the approach of two microasperities; E p , Et , µ p , and µt are the moduli of elasticity and Poissson’s ratios of the part and tool, respectively. In the plastic contact without hardening, the equality is known
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Ni = cσ yΔAr ,
(1)
where σ y is the yield strength; c is the constraint coefficient ( c ≈ 3 ); ΔAr is the area of the microroughness contact. For the contact of two spherical microasperities, Eq. (1) can be written as follows ⎛ ρt ρ p ⎞ Ni = 2πα1cσ y ⎜ ⎟a , ⎜ ρt + ρ p ⎟ i ⎝ ⎠ where α1 is the coefficient determining protrusion shrinkage ( 0.5 ≤ ρ1 ≤ 1 ). For the plastic contact with hardening we have
((
Ni = g 8 ρtρ p
( ρt + ρ p ) ) )
1 /2n1
a1i /2n1 ,
where g is the extrapolated yield strength; n1 is the coefficient characterizing the material capability to hardening. The nominal pa , contour pc and real pressure pr in the contact depending on the pressing force Py is determined by the formulas:
pa = Py Aa ;
pc = Py Ac ;
pr = Py Ar ,
where Aa , Ac , Ar are the nominal, contour and real areas of contact. Since in the AEM process the main contact occurs on the lateral surfaces of microasperities, the magnitude of the pressing force Py will be determined from the vector sum of forces Fσ and Fτ that appear in the stressed state of the viscoelastic base. The real area of contact of microasperities with regard for the AEM features can be obtained from the expressions: ⎯elastic contact
Ar = ⎡ 2.35 ( 0.5 Acb) ⎣⎢
1 2v
( I ρ (k 12 c c
12 12 AEM Rmax
)) P ⎤⎦⎥
2v (2v+1)
,
y
(2)
⎯plastic contact
Ar = Py cσ y = Py HB ,
(3)
⎯elastoplastic contact
Ar =
Py cσ y
+
( bAc )1 /v vac ( Py
( v−1 )
( cσ y ) )
v
,
2Rmax
(4)
where Ic is the elastic constant of the contact; ρc is the reduced radius of the bodies in contact; ac is the critical absolute approach corresponding to the change of contact from the elastic to the plastic state; kAEM is the contact factor in AEM (is determined experimentally); vp , vt , bp , bt are the coefficients for calculating the reference curve of the roughness profile; HB is the Brinell hardness:
((
Ic = 1 − µt2
ρc = ρtρ p
( ρt + ρ p ) ;
) E ) + ((1 − µ ) t
2 p
v = vt + vp ; b =
)
Ep ; v kAEM btbp Rmax v
vt p Rmax t + Rmaxp
.
The absolute approach between the surfaces of the part being machined and the medium for the elastic and plastic contacts is the following: RUSSIAN AERONAUTICS
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a=
v 4.7 Icρ1c/2 PyRmax
kAEM Acb
;
( ( A cσ b))
a = Rmax Py
1 /v
c
y
97
(5) .
(6)
In the elastoplastic contact, the general approach is composed of elastic and plastic components. The approach calculated by Eqs. (5) and (6) does not correspond to the depth of abrasive grain introduction into the material being machined, that is, the depth of cutting, since in the contact interaction the grains are displaced in the tool base. The amount and nature of displacement depend on the viscoelastic properties of the base and physico-mechanical properties of the material being machined. A further AEO feature is the fact that there is no clearance between the surface being machined and the tool. The working medium totally fills the channel being machined, and the nominal and contour areas of contact coincide. The products of machining (films, oxides, chip of the main material) are removed from the zone of machining by the working medium flux, gradually accumulating in it. In this connection, for the AEM case, the factor of volume for chip disposal is kV = 1. Then the AEM efficiency in removing the material volume Vm.r can be written as: —elastic contact
Vmr .
⎡ 1 /v ⎛ 4.7 Icρ1c/2 Py ⎛ 1 ⎞ = ⎢⎜ ⎟ − ⎜ 1 /2 ⎢⎝ 2b ⎠ ⎜ kAEM Rmax bAc ⎝ ⎢⎣
⎞ ⎟ ⎟ ⎠
2 (1 + 2v) ⎤
⎥R A ; ⎥ max c ⎥⎦
—plastic contact
Vmr .
1v ⎡ 1 /v ⎛ Py ⎞ ⎤ ⎛ 1 ⎞ ⎢ = ⎜ ⎟ −⎜ ⎟ ⎥R A . ⎜ Ac cσ yb ⎟ ⎥ max c ⎢⎝ 2b ⎠ ⎝ ⎠ ⎦ ⎣
(7)
The roughness of the surface machined can be calculated under condition of specified efficiency by the formulas: —elastic contact −1 ⎡ ⎤ 2 (1 + 2v) ⎞ ⎛ 12 1v ⎛ ⎞ ρ 4.7 I P ⎥ 1 ⎢ ⎛ 1 ⎞ c c y ⎜ ⎟ ⎟ − Rmax t Ac ⎥ ; Rmaxp = ⎢Vmr . ⎜⎜ ⎟ − ⎜⎜ 1 2 ⎟ ⎟ Ac ⎢ ⎜ ⎝ 2b ⎠ ⎟ ⎥ ⎝ kAEM Rmax bAc ⎠ ⎝ ⎠ ⎣⎢ ⎦⎥ —plastic contact
⎡ 1 ⎢ Rmax p = Vmr . Ac ⎢ ⎣⎢
1v ⎤ ⎛ 1v ⎛ ⎞ ⎞ ⎥ ⎜ ⎛ 1 ⎞ − ⎜ Py ⎟ ⎟ − R max t Ac . ⎜ Ac cσ yb ⎟ ⎟ ⎥ ⎜ ⎜⎝ 2b ⎟⎠ ⎝ ⎠ ⎠ ⎝ ⎦⎥
(8)
The initial data for the calculation are determined from the physico-mechanical properties of the part being machined (HB, µ, E), the parameters of its surface layer state Δ = Rmax Rb1 /v and the geometric
(
)
characteristics of the channel being machined, The contour pressure pc and tangential stress τn are established by the results of hydrodynamic and rheological calculations of the working medium flux in the channel being machined using the previously developed techniques [2, 3]. As an example, we present the calcualtion of the AEM parameters for a circular channel with the constant section of the radius r = 0.0125 m and length L = 0.07 m. The material is the 12Kh18N10T steel. According to State Standard 5632-72, this steel has E p = 220 000 MPa, µ p = 0.3. The channel is manufactured by boring and subjected to hardening followed by annealing. The surface hardiness is
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HB = 450. The complex surface indicator is Δ = 6.3× 10-2 , maximal roughness is Rmax p = 4.72 µm,
ρ p = 55 µm, the coefficients are bp = 2 and vp = 1.6. The working medium has the following characteristics: Et = 0.116 MPa, µt = 0.4, the grain effect is Ba = 320 µm, Rmaxt = 40.2 µm, ρt = 160 µm, bt = 1.7, vt = 1.1. On the basis of a number of experiments and calculations [2], we determined the components of the cutting force Pz = 2.28 N, Py = 2.01 N on the contour area of contact at the value of pressure pinl = 6 MPa that shifts the working medium at the inlet of the channel being machined. The force at the contact point
Py = Fσ2 + Fτ2 = 3.04 N is calculated with the use of the stress tensor of the polymer base. Since the motion is steady-state, the force magnitude will be constant over the entire contour area of contact. Using formulas (2)–(4) we will further find the real areas of contact of microasperities. We have for 2 2 the single elastic contact Ar = 1.04 × 10−6 mm , for the single plastic contact Ar = 0.68× 10−8 mm . The total real area of the single elasto-plastic contact of microasperities is Ar = 7 × 10−6 mm . 2
The absolute approach between surfaces is found by Eqs. (5) and (6). The approach to the single microasperity in the elastic contact amounts to 0.6 µm, in the plastic contact it amounts to 37 µm. Such amount of the plastic approach is due to the base flow caused by displacement of the abrasive grain in the contact. The volume of the material layer instantaneously removed in one contact calculated by Eq. (7) 3 amounted to 2.871 × 10−8 mm . The change of microasperity roughness in one contact (Eq. (8)) was 0.000124 µm. If we know the general amount of grains that come into contact with the surface layer microasperity being considered, it is possible to calculate the change of roughness and amount of wear with time. For this purpose it is necessary to introduce corrections for the change of dimensions Δ = 6.3× 10-2 , Rmaxp = 4.72 µm, ρ p = 55 µm, bp = 2, vp = 1.6 into the formulas under consideration. The surface roughness in AEM is formed as a result of mass contact. Since the modulus of elasticity Et is by several orders of magnitude less than that of the material of the part being machined, the part material in the single contact is removed at the nanolevel. It should be taken into account that per one cycle of extrusion the single surface microasperity may come into contact from 500 to 12000 times. Thus, the calculations show that AEM makes it possible to effectively machine any materials starting from nonferrous alloys to difficult-to-machine surfaces (immediately after casting or heat treatment); in this case the total efficiency is highly competitive with that of such processes of finishing as honing or polishing. REFERENCES 1. Sysoev, S.K., Sysoev, A.S., Levko, V.A., Snetkov, P.A., and Lubnin, M.A., Provision of Fuel Component Flowrate Precision Through the Channels of Parts Machined by Extrusion Honing, Tekhnologiya Mashinostroeniya, 2007, no. 6, pp. 48–52. 2. Levko, V.A. and Pshenko, E.B., Influence of the Working Medium Composition on Process Conditions of Abrasive-Extrusion Machining of Complex-Shaped Parts, Vestnik SibGAU, 2006, issue 11, pp. 64–68. 3. Levko, V.A., Special Features of Working Medium Rheology in Abrasive-Extrusion Machining, Vestnik SibGAU, 2005, issue 7, pp. 96–100. 4. Chepovetskii, I.Kh., Osnovy finishnoi almaznoi obrabotki (Fundamentals of Diamond Finishing), Kiev: Naukova Dumka, 1980.
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