Thick composite plates cutting process in cryogenic environment Ovidiu Nemes*, Horatiu Iancau**, Teodor Potra** *
LGM Toulouse, 50, chemin des Maraîchers, 31077 Toulouse Cedex, France, tel:05.62.25.87.37, E-Mail:
[email protected] ** UT Cluj-Napoca, B-dul Muncii, Nr. 103-105, RO 3400 Cluj-Napoca, Romania
ABSTRACT. The paper contains a finite element numerical model based on the homogenisation method. This numerical model applies to the thermal transfer through composite materials. A concrete application is illustrated in the case of a splitting disk. KEY WORDS: Nitrogen, cryogenic liquid, finite element
1. Introduction. In almost all problems of much importance, which appear in the research of some technical phenomenon’s governed by elliptical equations with variable coefficients on put the homogenisation problem of equation:
∑ ( − 1)
Au ≡
α
(
)
Dα aα β ( x )Dβ u = f , x ∈ V ⊂ R n
[1]
α , β =1
We’ll determine the constant coefficients a α β ∈ R from variable coefficients
aα β ( x ) such that (1) to be transform into an equation with constant coefficients, that is, in homogenisation equation. In [1] we’ll suppose that A is a strong elliptical operator, aα β ( x ) ∈ C ( V ) , f 1
is sufficiently smooth in V. We’ll consider the domain V of the form:
V =
N
UV ,
[2]
k
k=1
Vk are congruent parallelepipeds from V. We’ll suppose that a α β ( x ) is Vk - periodic functions, together with their derivatives until 1 order. The calculation of constants coefficients supposes [BEN78] the usage of an
asymptotic development of the unknown function u ∈ H ( V ), and the integration of a homogeneous elliptical equation on a parallelepiped Vk, in which the boundary conditions constitute the Vk - periodicity condition of the solution. H1(V) is the Sobolov space. 2
The determination of the solution for the homogeneous problem will be realised, using a finite element type proceeding, for which we’ll study the error of finiteelement approximation and the quadratic error. The notation for
D α and α , β are those usual.
This type of the problems have a large applicability in the study of composite materials with periodic structures, [IAN95]. The transfer of the heat through composite materials has some peculiarities in connection with the structure and the properties of the materials. A composite material has a basic matrix with particular mechanical characteristics in which are introduced armouring materials in form of independent fibres or particles, which have superior mechanical characteristics in comparison of each component of the composite. An important characteristic of the composite materials is that they have a particular periodicity in space relative to the physical-mechanical characteristics. The periodicity allows the use of the homogenisation method during the transformation of the Fourier-Kirchoff equation with variable coefficients, corresponding to the an isotropic medium, into an equation with constant coefficients, which correspond to an homogeneous and isotropic medium equivalent with the composite materials, from the point of view of the thermal transfer. In obtaining the constants coefficients (homogenised) it is utilised an orthogonal finite element method. The solution of the equation with constant coefficients approximates the solution of the equation corresponding to the composite material.
2. The microscopic and macroscopic equations. Suppose that the coefficients a α β ( y ) , y ∈ V , are Vk - periodic functions and
ε > 0 is a small parameter.
aεα β ( x ) = aα β ( y ) ,
[3]
with
y = 1 x = 1 ( x1, x2,..., x n ) ε ε
[4]
It comes out that a α β ( x ) is ε Vk - periodic. ε
Ω ⊂ R n be the transformed domain by the change 1 x and ε 1 f ∈ C ( Ω ). Let
We consider the following boundary value problem: We determine u ∈ H ( Ω ), such that, ε
2
∑ ( − 1)
Aε u ε ≡
α
(
)
Dα aεα β Dβ uε( x ) = f
[5]
α , β =1
in Ω , and
uε ∂Ω = 0
[6]
For ε>0 fixed, the problem [5], [6] has a single solution in H2(Ω ), because Aε is strong elliptical and f∈ C1(Ω ). We suppose that uε (x) admits an asymptotic development, when ε→ 0 in asymptotic power series, of the form:
u (x) = ψ
(x) +
ε
0
εψ
(x) +
1
εψ 2
(x) +
2
...
where (ψ n) is an asymptotic base for uε (x), and y =
[7]
1 x. ε
We assume that the base (ψ n) is Vk - periodic in y variable. If we note
vα =
∑
β ε
aα β D u =
n
∑
i= 1
β =1
ε aα β1 ∂ u , ∂ xi
β = ( β1, β 2,..., β n )
[8]
[9]
β = 1 , it holds:
∑
β ε
Du =
β =1
∑
i =1
=
n
∑
i= 1
and
n
∂ uε + 1 ∂ xi ε
n
∑
i =1
∂ uε = ∂ yi
n ∂ψ 1 ∂ψ 2 ∂ψ 0 ∂ψ 1 + + ε + + ... ∑ ∂ yi ∂ xi ∂ yi ∂ xi i= 1
[10]
vα ( x, y ) = v α ( x, y ) + ε v α ( x, y ) + ... + ε v α ( x, y ) + ... [11] 0
1
n
where
v α ( x, y ) = n
n
∑
i= 1
∂ ψ n ∂ ψ n +1 , n = 0, 1, 2, ... [12] a α β i( x ) + ∂ yi ∂ xi
If we replace [11] in [5] and we identify afterε we get n
∑
i= 1
∂vαo i = 0 , α = ( α 1, α 2,..., α n ), ∂y i α = α 1 + α 2 + ... + α n = 1
[13]
respectively
−
n
∑
i= 1
∂vαo ( x, y ) ∂v1α ( x, y ) i i = f ( x ) in Ω + ∂xi ∂y i
[14]
Theorem 1. If u2∈ H2(V) is the solution of the problem [5], [6], ε>0 fixed and Lk:H2(Vk)→ ⊕ is the meaning operator
Lk[w ] =
1 Vk
∫ w( y ) dy
[15]
Vk n
, where Vk is one from the parallelepipeds in which is decomposed V⊂ ⊕ , then hold:
−
n
∑
i =1
[ ]
∂ L vo = f ( x ) ∂xi k α i
in Ω , |Vk| is the measure of Vk.
[16]
The proof is in [POT89]. The equation [16] is called the macroscopic equation or the homogenised equation. Applying theLk operator of equation [13] we get the microscopic or local equation. If v is Vk - periodic and g has the property Lk [g]=0, then, if aα β ∈ H1(Vk) the equation
∑ ( − 1)
α
(
)
Dα aα β ( y ) Dβ v = g( y ) , ∀ y ∈ Vk
[17]
α , β =1
admit a single solution v∈ H2 (Vk), modulo an additive constant. Also, [13] is equivalent with the following boundary valuevariational problem: Find ψ 1∈ H1(Vk) such that
∫ Vk
∑
α, β
aα β ( y ) Dα ψ Dβ w dy = =1
=
∑
Dα ψ
α , β =1
o
( x ) ∫ Dβa α β ( y ) w( y )dy
[18]
Vk
∀ w∈ H1p(R), where H1p(Rn) denotes the Sobolev space H 1(Rn) or the Vk – periodic functions. Note That [18] frames in theBensoussan-Lions theorem. If in [18] we suppose that ψ o(x) is known, the determination of ψ following boundary-value variational problem: Find hj∈ H1p(Rn) that satisfies Lk[hβ j]=0, (j=1÷ n), such that
∫
Vk
∑
α,β
aα β (y)D hβ j D w dy = ∫ Vk =1 α
β
1
reduces the
∑ D aα β j w dy, α =1 α
[19]
∀ w∈ H1p(Rn). The equation [19] constitute the object of this paper, because, knowing its solutions hβ j, (j=1÷ n) we are leaded [POT89] to the homogenised coefficients a α β , given by
[ ]
aα iβ j = Lk aα iβ j +
n
∑
l= 1
∂ hβ j Lk aα iβ l ∂y l
[20]
In this way we come to the homogenised problem:
∑ ( − 1)
α
(
)
Dα aα β Dβ u = f
[21]
α , β =1
in Ω , with
= 0
u
[22]
∂Ω
The equation [19] is called the macroscopic or local equation. We have used the notation β =(0,...,1j,...,0):=(β 1,...,β j,...,β n):=β j putting aα β j in the place of aα (0,0,...,1j,0,...,0).
3. Finite element for the microscopic equation Consider the equation [19] in that aα β (y) are Vk-periodic, search hβ j∈ H1p(Vk) such that [19] holds for any w∈ H1p(Vk). The condition from Bensoussan-Lions existence’s theorem is satisfied, because
∫
Vk
∑ D aα β j( y ) dy = 0, α =1 α
[23]
since aα β j(y) are Vk-periodic. Therefore [19] is a correctly formulated boundary-value problem. ( e)
Let the ∆ M-discretisation in M∈ N simplexes of Vk, we note with K one of the (e )
(r
r
r
)
r
simplexes of ∆ M, K = conv e1, e2,..., en + 1 , where ek, ( k = 1 ÷ n + 1 ) are
( e)
ˆ the Standardthe position-vectors of the K - simplex vertexes. Denote with K ( e)
simplex in which is transformed each ofK therefore
eˆk = ( 0,0,...,0,1,0,...,0 ) ,
(k
= 1 ÷ n)
[24]
and
eˆk = ( 0,0,...,0)
[25] (e)
The transformation T : K → (1)
xk = xk+
ˆ achieves by K
( j + 1 ) ( 1 ) ˆ x k − x k x k,
n
∑
j= 1
(k
= 1 ÷ n)
[26]
where ( j) r ( j ) ( j) e j = x1, x 2,..., x n ,
(j
= 1 ÷ n + 1)
[27]
Using [26] it is sufficient to know the finite-element local base on the standard
{
}nk += 11 , given by:
ˆ , that is: ϕˆ ( x ) element K k
ϕˆk( xˆ) = xˆk, ( k = 1 ÷ n ) , ϕˆn + 1( xˆ) = 1 −
n
∑
xˆk
[28]
k=1
The geometry of Vk allows such a simple choice. *
Theorem 2. Let [19] be the boundary value variational problem; if h β j is its
~
exact solutions and h β j its finite-element solution, generated by the base (21), then there
is
a
~ h*β j − hβ j of Vk,
(j
constant
H ( Vk ) 1
c>0
independent
of
h
and
h β* j
such
that
< c.h2 , where h denotes the finite-element discretisation norm
= 1 ÷ n ).
The proof of this theorem founds in [POT88]. Generally, the calculation of the coefficients from the finite-element local system obtain approximating the integrals ,which appear, by the quadratic formula. We’ll use the C.Kalik formula [KAL84], which frame this theory.
4. Thermal transfer in the splitting disc By this theory we can obtain with a certain approximation the temperatures resulted in the cutting process of the thick composites plates, with or without cryogenic cooling using the Fourier generalized equation:
∂u − ∂t
∑ ( − 1)
α , β =1
α
D α (aα β ( x )D β u) = f , x ∈ V ⊂ R 3
[29]
These temperatures appear as known values in the calorimetrical equation of the cutting speed. In conclusion this theory leads us to the optimisation of the technological parameters of the cutting process, parameters corresponding with the cutting method under cryogenic conditions. This modern method of machining, proposed by the authors, consists of eliminating the thermal energy from the process by cooling with liquid nitrogen of the splitting disc, using for this purpose a special installation. The liquid nitrogen has not only the task to eliminate the thermal energy from the cutting zone, but also the task to decrease its appearance through minimising the frictions between the disc and the part out of composite material. The mathematical model describes before was tested for the cut of plates in STRATITEX S672 where has used a disc with diamonds teeth having a diameter of 350 mm to 3000 tours/minute, assembled on a woodworking machinery. The experiment established that the optimal position of the cooling module is in the proximity of the output of the disc of the hardware (figure 1, position III). To be able to optimise the technological parameters (speed of machining, flow) one used the calorimetric equation [30].
qvρ ( θ − θ 'asc ) c = Qρ ' ∆ θ c' 1000 asc
[30]
where: v - speed of machining, m/min, q - surface of the section of the chip, mm2, ? - density of material to be machined, kg/dm3, Q asc - temperature of the chip during dry machining, K, Q' asc - temperature of the chip during machining with cooling, K, c - specific heat of material to be machined, Kcal/kg .grd, Q - flow of the cryogenic liquid, l/min, ?' - density of the cryogenic liquid, kg/dm3, ∆ θ - increase in the temperature of the cryogenic liquid, K, c’ - specific heat of the cryogenic liquid, Kcal/kg .grd.
Figure 1: Position of the cooling module
In the figure 2 is presented the variation of machining speed according to the flow of liquid nitrogen. By increasing the flow of nitrogen we can increase the machining speed up to a value which corresponds to one imposed economic lifespan.
350 300
v (m/min)
250 200 150 100 50
1, 89 15
1, 76 1
1, 63 06
1, 50 01
0, 19 56 7 0, 32 61 1 0, 45 65 6 0, 58 70 0 0, 71 74 5 0, 84 78 9 0, 97 83 4 1, 10 87 8 1, 23 92 3 1, 36 96 7
0
Q (l/min) Figure 2. Machining speed according to the flow of the cryogenic liquid According to the experiments carried out, we can notice the following aspects: •
the plates are not separated,
•
the heating of the disc decreased,
• during work, on the faces of the disc settles a liquid film of nitrogen, which has a role of lubricant, • the productivity of the cut increases because we can increase the advance of the part, • the forces of machining decrease because we decrease the friction between the disc and the part. In conclusion, this method is very good for the cut of material plates difficult to machine, for the composite materials for which the disc is very requested from the thermal point of view.
5. References [BEN78] Bensoussan, A., et al., Asymptotic analysis for periodic structures, NorthHolland, Amsterdam, 1978 [IAN97] Iancau, H., et al., Finite elements in the homogenisation problem with applications in composites materials, Journal of Plastic Deformation, col. 3, nr. 1-2, ISSN 1222-605X, p. 76-79, 1997. [IAN95] Ianc>u, H., et al., Calculul cu element finit al transferului termic în materiale compozite, MteM’95, p. 161-164, 1995. [KAL84] Kalik, C., Formules d’intégration approchée sur les domaines polyédriques, Université “Babe•-Bolyai”, Preprint nr.7, 1984. [NEM98] Nemes, O., et al., T., Research regarding the thermal transfer through composites materials parts”, Le Symposium “Stiinta, Inginerie, Eficienta” dans le cadre des Journées Académiques de Cluj, 8-9 juin 1998, UT Press, p. 331-336. [POT88] Potra, T., Regarding the convergence in the finite-element approximation, “Babes-Bolyai” University, Preprint nr.7, p.179-186,1988. [POT89] Potra, T., Finite elements in the homogenisation problem,Babes-Bolyai” University, Preprint nr.7, p.119-128,1989. [TAP88] Tapalaga, I., Achimas, Gh., Iancau, H., Cryogenic technique in machines building, EditionDacia, 1988
