While creep may reduce the probability of failure by thermal stresses during the .... While under most conditions ceramics may be considered elastic, during (thermal) laser ... the coefficient of thermal expansion, is Poisson's ratio, and the dot upon a .... -0.7. -0.8. -0.9. -1 time = 0.600000. Figure 2: Hoop stress development ...
Transient Elastic and Viscoelastic Thermal Stresses During Laser Drilling of Ceramics Michael F. Modest Department of Mechanical Engineering The Pennsylvania State University University Park, PA 16802 Abstract Lasers appear to be particularly well suited to drill and shape hard and brittle ceramics, which are almost impossible to netshape to tight tolerances, and can presently only be machined by diamond grinding. Unfortunately, the large, focussed heat flux rates that allow the ready melting and ablation of material, also result in large localized thermal stresses within the narrow heat-affected zone, which can lead to micro-cracks, significant decrease in bending strength, and even catastrophic failure. In order to assess the where, when and what stresses occur during laser drilling, that are responsible for cracks and decrease in strength, elastic and viscoelastic stress models have been incorporated into our two-dimensional drilling code. The code is able to predict temporal temperature fields as well as the receding solid surface during CW or pulsed laser drilling. Using the resulting drill geometry and temperature field, elastic stresses as well as viscoelastic stresses are calculated as they develop and decay during the drilling process. The viscosity of the ceramic is treated as temperature-dependent, limiting viscoelastic effects to a thin layer near the ablation front where the ceramic has softened.
Introduction Nearly all ceramics can be efficiently drilled, scribed or cut with a laser, although massive problems remain that are poorly, or not at all, understood. These problems include thermal stress, redeposition of evaporated or liquified material, poor surface finish, undesirable hole and groove tapers, etc. It is well known that laser irradiation causes damage in ceramics due to thermal stresses, resulting in micro-cracks and, often, catastrophic failure; in all cases laser processing severly reduces the bending strength of the ceramic (Copley et al. [1], Yamamoto and Yamamoto [2], deBastiani, Modest and Stubican [3]). Criteria for stress failure of ceramics have been discussed in detail by Hasselman and Singh [4]. They note that ceramic materials will exhibit creep by diffusional processes at levels of temperature at which vacancy concentrations and mobility become appreciable. These temperatures correspond to about 0.5 to 0:7 Tmelt of the material. At the fast heating rates during laser machining, severe compressive stresses develop, and creep rates fast enough to effect appreciable stress relaxation may not occur until somewhat higher temperature levels are reached; however, experimental evidence suggests that such rapid stress relaxation does occur before the material melts or decomposes. Extrapolating data for alumina given by Hasselman [5] to a temperature just below the melting point (T 2300 K ) gives a thermal stress relaxation time of only 30 s. While creep may reduce the probability of failure by thermal stresses during the heating-up of the ceramics, the resulting stress relaxation would be expected to lead to very strong tensile stresses during cool-down, which in turn could cause the generation of microcracks, overall weakening of the material’s mechanical strength, or catastrophic failure. Data on creep behavior have been obtained by a number of investigators for a number of ceramic materials, although no creep properties appear to have been measured for temperatures approaching the melting/decomposition point, which are needed to fully understand the laser shaping process. For example, creep rates for alumina at 1500 C have been determined by Folweiler [6]; Lane et al. [7] measured creep
rates for sintered -SiC, finding rates of 10;7 =s at 1750 C and 200 MPa. Extrapolation of these data to the melting/decomposition point indicate that plastic deformation accompanied by thermal stress relaxation is to be expected during laser shaping of ceramics. This was confirmed by Gross et al. [8], who investigated crack formation during laser (CO2 and Nd:YAG) drilling of thin silicon wafers. The existence of a plastically deformed zone was shown by etch pit studies. Radial cracks, terminating at the deformed zone boundary, were observed as well as circumferential cracks following the boundary of the deformed zone; both types of cracks confirming, at least qualitatively, the nature of the expected stresses in the presence of creep prior to melting. Most analytical investigations have been limited to thermoelastic bodies, using a one-dimensional analysis or commercial finite-element programs. Hasselman et al. [9{11] investigated analytically the transient thermal stress field in a one-dimensional slab subjected to external radiation, with internal absorption of this irradiation. They found that greatest tensile stresses occurred in slabs of medium optical thickness (L 3 ; 5). A similar analysis for an opaque slab was made by Bradley [12], and a procedure to describe the total strain energy at fracture due to thermal stresses is given. Sumi et al. [13] give an analytical/numerical solution for transient stresses for a simplified 3-D problem in which a local square surface heat source moves in the x-direction across an infinite flat x-y plane plate. The resulting stresses turn out to be mostly compressive stresses with some small tensile stresses. However, the tensile stresses occur near the (unrealistically abrupt) edge of the heat source and are, thus, exaggerated. Very few theoretical investigation have addressed thermal stresses accompanied by creep. Guan and Cao [14] predicted residual stresses during welding of thin plates using a 2-D elasto-plastic finite element model. Ferrari and Harding [15] modeled the residual thermal stress field for a one-dimensional sphere with several plasma-sprayed ceramic coatings, finding moderate compressive stresses in the radial direction, but large tensile stresses in the transverse direction. Gross et al. [8] investigated crack formation during laser (CO2 and Nd:YAG) drilling of thin silicon wafers. They developed a simple one-dimensional model incorporating compressive plastic deformation to predict thermal stresses in the wafer. The predictions indicate that, during cooldown, residual circumferential stresses are tensile in the deformed zone and compressive outside. Radial thermal stresses are tensile everywhere with a maximum at the deformed zone boundary. Radial cracks, terminating at the deformed zone boundary, were observed as well as circumferential cracks following the boundary of the deformed zone; both types of cracks confirming, at least qualitatively, the nature of the predicted stresses. Bahr et al. [16] presented a one-dimensional, transient model to predict thermal stresses inside an opaque solid irradiated by a short laser pulse. At high temperatures the material was allowed to deform viscoelastically as a Maxwell body. Results show that compressive stresses build up until—over a small range of temperature and a short period of time—stress relaxation takes place due to creep. During cooldown the solid behaves elastically again, resulting in tensile stresses throughout the heat-affected zone, which extends a few hundred m for a 100 ms pulse, but only a few m for a 1 ms pulse. Interestingly, for shorter pulses the tensile stresses are not only limited to a shallower depth, but they are also of much smaller magnitude. In the present paper our two-dimensional drilling code is being augmented by an elastic and viscoelastic stress model, to predict thermal stresses as they develop and decay during CW and pulsed laser drilling of ceramics.
Theoretical Background To make analyses for the thermal and the stress problems tractable, a number of limiting and simplifying assumptions need to be made. Assumptions for the heat transfer problem are identical to those in previous papers of the author [17, 18], and are very briefly given here: 1. The solid is isotropic, has constant density, and the material is opaque, i.e., the laser beam does not penetrate appreciably into the solid.
2. Change of phase from solid to vapor (or decomposition products) occurs in a single step with a rate governed by a simple Arrhenius relation, modeled through a “heat of removal”, hre , [17]. 3. The evaporated material does not interfere with the incoming laser beam (or is removed by an external gas jet). 4. Heat losses by convection and radiation (on top surface and sidewalls) are negligible, [18, 19]. 5. Multiple reflections of laser radiation within the groove are neglected, restricting the present model to shallow holes, holes with steep sidewalls or materials with high absorptivities. [20{22] 6. Heat transfer is unaffected by thermal expansion (always true for ceramics as shown by a simple order-of-magnitude analysis). 7. Inertia effects are negligible during stress development (always true for opaque ceramics, but may become questionable for semitransparent ceramics subject to ns laser pulses). Heat Transfer The transient heat conduction equation for a solid plate of thickness D, irradiated by a Gaussian laser beam may be expressed in terms of temperature T as (see Fig. 1)
c @T @t
=
r (kr
T ) = 1r @@r
@ @T @T kr @ r + @ z k @ z
(1) 2w0
subject to the boundary conditions
r = 0 : r ! 1 : z = 0 : z = D :
z0
@T = 0 @ r T ! T1 F nˆ = ;nˆ (krT ) + vn hre @T = 0 @ z
and an appropriate initial condition, such as t = 0 : T (r z t = 0) = T1 s(r t = 0) = s0 (r)
r
(2a) (2b) (2c)
z
D
(2d) (2e) Figure 1: Laser drilling setup and coordinate system. (2f)
where c k and are density, specific heat, thermal conductivity, and laser absorptance, respectively. Also, r is radial distance measured from the center of the laser beam, z is axial distance through the plate and s is the local depth of a hole (i.e., the z-coordinate of the top suface) and nˆ is a unit vector normal to the surface (pointing into solid); v n is the total surface recession velocity during drilling, and it is assumed that the solid is originally at a uniform temperature T1 ; F is energy intensity distribution, for a focussed Gaussian laser beam with a waist w0 at the focal plane z0 (some quantities have been barred to distinguish the present dimensional quantities from the nondimensional ones introduced below). Boundary conditions (2) are sufficient to solve equation (1) for the temperature if the shape of the hole, s is already established (vn = 0) or if vn is otherwise known. We will assume in this paper that the ablation and/or decomposition of the solid material is governed by a simple reaction equation of the Arrhenius type, i.e., the rate of mass loss per unit area is described by
m_ 00 = vn = C1e;E=RT
(3)
is the universal gas constant, and C1 is a preexponential factor that where E is the decomposition energy, R depends on the nature of the ablation process.
The governing equations and boundary conditions are non-dimensionalized using the 86%-beam radius at the focal point, w0:
r = r=w0; z = z=w0; t = ckrewt 2 ; s = s=w0; = TT ;;TT1 re 1 re 0
(4a)
leading to two basic non-dimensional parameters governing the laser/material interaction:
Nk = kre (TFre w; T1 ) ; 0
Ste =
0
hre : cre (Tre ; T1)
(5)
Boundary-fitted coordinates are employed in the numerical solution, i.e., the physical domain (r z ), is transformed to a uniformly spaced rectangular coordinate region ( ). Detailed discussions of the heat transfer analysis and the numerical implementation are given by Modest [17], (general development for a moving laser) and [18] (details on through-cutting and drilling). Thermal Stresses The extreme temperature gradients that occur during laser machining (in space and in time) result in extreme nonuniformities in the local thermal expansion of material (strain) which, in turn, cause strong thermal stresses. While under most conditions ceramics may be considered elastic, during (thermal) laser drilling there will be a thin zone near the receding interface (and at extremely high temperature), over which significant creep may occur. To assess the importance of this nonelastic zone on the overall thermal stress development, the present analysis includes a simple linear viscoelastic model (Maxwell body). The deviatoric stress-strain relation for a Maxwell body is [23].
e_ =
1 1 s_ + s 2 2
(6)
where e and s are the dimensional deviatoric strain and stress tensors, respectively, is one of Lam´e’s constants (= G, the shear modulus), and is the viscosity of the viscoelastic solid, which–unlike all other properties–is assumed to be temperature-dependent, since for ceramics its value changes by many orders of magnitude between room temperature and ablation/decomposition temperature. The viscosity-temperature is known from creep studies to follow an Arrhenius relationship, i.e., dependence of e ;Q=RT (1;T =T ) (7)
re e is the activation energy, A is a preexponential factor, and re is the viscosity at the material’s where Q
1
= Ae;Q=RT =
1
re
re
removal temperature. The complete stress-strain relation for a Maxwell body may then be stated [23] in non-dimensional form as
_ + ( ) = _ + = 1 2
1+ 3 1+ _ + _;
( )( ; ) 1 ; 2 1 ; 2 1 ; 2
(8)
ru + ruT
(9)
;
= 13 trace ()
where
=
2v (Tre ; T1 )
u v (Tre ; T1 ) u = w0v (Tre ; T1 ) 2 Q re w0 = re e;Q(1;T =T ) Q = ( ) = c k (T ) RT
=
re
re
(10a) (10b)
re
with varying along lines essentially parallel to top and bottom surfaces and perpendicular to it (see Fig. 1); non-dimensional time, coordinates and temperature have already been defined in equation (4). Here
is the displacement vector, v is and are stress and strain tensors, respectively, is the identity tensor, u the coefficient of thermal expansion, is Poisson’s ratio, and the dot upon a symbol denotes differentiation with respect to time. Equation (8) requires initial conditions for and ; we will here assume that the solid is initially unstressed and unstrained: t = 0 : = 0; u = 0: (11)
Because of the temperature dependence of the viscosity, or , it is inconvenient to substitute equation (8) into the equilibrium condition = 0 (since stress cannot be eliminated). Since a numerical solution will be attempted, the time derivatives will first be eliminated through a simple implicite finite difference [as is done in the solution of equation (1)], i.e.,
r
(n) (n;1)
_ = ;t etc.
(12)
where the superscript (n) denotes the n-th time step. Equation (8) may then be rewritten as
(n) = A(n) ((n) ; (n) ) + b((n) ; (n) ) + C (n) A(n)( ) = 1 + 1(n) t ; b = 11;+ 2 C (n) = A(n) [ ; + ; b( ; )](n;1) :
Note that equation (13) reduces to the thermoelastic case for infinite viscosity ( ! 0 : A ! 1 C For the two-dimensional, axisymmetric problem at hand equation (13) in long-hand becomes
rr
=
=
zz
=
rz
=
rr
=
rz
=
@u = u = @w @r r zz @z 1 @u @w + r = z = 0 2 @z @r
1 @u @u u @w A @r + 3 (b ; A) @r + r + @z ; b + Crr u @u u @w 1 A r + 3 (b ; A) @r + r + @z ; b + C 1 @w u @w @u A @z + 3 (b ; A) @r + r + @z ; b + Czz A @u + @w + C rz 2 @z @r
(13) (14)
! O). (15a) (15b)
(16a) (16b) (16c) (16d)
and the equilibrium conditions reduce to
@rr + @rz + 1 ( ; ) = 0 @r @z r rr @rz + @zz + 1 = 0: @r @z r rz
(17a) (17b)
For simplicity the superscript (n) has been dropped from these equations. Equation (17) is a set of elliptic equations in the unknown displacements u and w, thus requiring boundary conditions along the bounding surface of the volume under consideration. Assuming zero traction on top and bottom surfaces ( nˆ = 0), and zero displacement far away from the laser leads to
r=0: r!1: z=0: z=L:
u = 0 @w @r = 0 u=w=0 rr nr + rz nz = 0 rz nr + zz nz = 0 rz = zz = 0
(18a) (18b) (18c) (18d)
In equation (18d) use has been made of the fact that the bottom surface is always perpendicular to the z -axis. For very thick specimens equation (18d) may be replaced by a no-displacement condition far enough into the medium [actually, in the numerical implementation, here and for boundary condition (18b), we use the fact that displacement decays as 1=r2 far away from a point source]. After substituting equations (16) into (17), the equations are transformed from physical coordinates (r z t) to computational coordinates ( ), followed by finite differencing [24]. This results in extremely long and tedious relations, which will not be reproduced here. Special consideration must be given to the top and bottom boundaries because of the out-of-plane derivatives. These are taken care of by integrating equations (17) over the half-nodes near the surface, eliminating out-of-plane derivatives through the use of equations (18c) and (18d). Another trouble spot is the triangular node at the bottom of a hole once it forms (see Fig. 1), which is dealt with by integrating equations (17) over the triangular element (in physical coordinates). Finally, the evaluation of C requires special attention. Recall that equation (8) was finite-differenced in time before transformation to computational coordinates: values at the previous time step must be evaluated at the same physical coordinates. Therefore, if the computational coordinates move with speed t t:
C (n;1) (r(n) z (n)) C
= =
@ C @ C (n;1) C (n;1) (r(n;1) z (n;1) ) ; t @ + t @ (r(n;1) z (n;1) )t ; + ; b( ; ):
(19a) (19b)
The result is a set of two equations for each of the N N nodes making up the overall grid (assuming no burn-through). In nine-point stencil form this may be written as
p uik + n uik+1 + ne ui+1k+1 + e ui+1k + se ui+1k;1 + s uik;1 +sw ui;1k;1 + w ui;1k + nw ui;1k+1 = f
(20)
where each of the p, n, etc. are 2 2 tensors. The set of simultaneous equations (20) may be inverted in a number of ways. Since temperature varies fastest in the -direction, we ordered equation (20) into a block-tridiagonal system for constant , which was solved directly, and iteratively swept over using successive overrelaxation. This works reasonably well, but will be improved before implementation in the 3D laser machining problem. Note that, for the thermoelastic case, equation (20) needs to be solved only at times of interest while, for the viscoelastic case, an inversion must be carried out after every time step. However, using the previous time step as an initial guess causes very rapid convergence.
Results and Discussion In order to assess the development of thermal stresses, and the importance of viscoelastic effects, several drilling operations on -SiC were simulated, using silicon carbide physical properties from Ramanathan and Modest [25] (all taken at removal temperatures of Tre = 3000 K, which gives good agreement with variable property calculations [25]) and Edington et al. [26]: kre = 20 W/mK, cre = 5 106 J/m3K, hre ' 12:1 MJ/kg, v = 10;6 =K; E = 400 GPa, = 0:17; and the viscoelastic properties were curve = 840 kJ/mol. Laser parameters fitted to data given by Lane et al. [7] as A = 5:30 1014 =(MPa s) and Q were typical values for a CO2 laser (such as the one in our laboratory), using a w0 = 175m and an average absorbed power of P = 500 W. Several CW and pulsed laser drilling events have been simulated, including a cool-down period after the laser has been turned off, all on large wafers with a thickness of 0.7 mm (= 4w0 ). Figures 2 and 3 are sequences of frames showing the development of principal stresses for an elastic body during CW drilling (up to a non-dimensional time of t = 0:25, or t ' 2 ms), and cool-down after the laser is turned off at t = 0:25. Figure 2 shows hoop stresses (1 = ), while Fig. 3 depicts the stress 2 perpendicular to the principal plane more or less parallel to the top surface (i.e., zz at large r); the third principal stress was found to be always compressive (with maximum values of 3 ' ;1:3 GPa). As
time = 0.200000
time = 0.100000
0.05
2
2.5
0
-0.1 0
-0.1
0
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
5
.8 .6 -0 -0 .4 .3 0.1 1.5-0 -0 0 5 0.0
2
0.0
5
0.05
2.5
-0 .2
-0 -1 .7
0.0
1
05 0.
0
1 -0.
1.5
0
z 1.5
.2 -0
z
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
5 0.0
.1 -0
-0 .9 -0 .5
1
z
0.05
5
05 0.
1
0.5
.2
0.5
5
0.0
0.0
time = 0.600000 0
-0 .5
-1 .7 -0. 9
0
-0 -0.60.8 . -04 .3
9 -0. .5 -0.3 -1 -0.8--00.6 -0.4 .2 -0 .1 -0 0
-0
.4 -0 .2 -0 1 -0.
.7 -0
0.5
-0.4
0
.9 -0 .6 -1-0.8 -0-0.5 0.3 -
-0
0
2
2.5
0
0
0
0
1
2
0
1
2
0
1
2
Figure 2: Hoop stress development during CW CO2 laser drilling of SiC; thermoelastic body (time < 0.25: heating/drilling; time > 0.25: cooling). r
r
r
time = 0.200000 0
time = 0.600000
0
0
0
0 0
0.5
5
0.05
0.0
0.5 -0.1
2 -0 .
2
1 -0 .
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
0.05 0. 05
2.5
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
0
0 05 0.
z
.1
1.5
2 0
1.5
-0
5
.3 .4 -0-0
1
0.0
z
1
2.5
0
1
2
r
0
1
2
r
Figure 3: Principal stresses (approx. normal to top surface) during Figure 4: SEM cross-section of CW CO2 CW CO2 laser drilling of SiC; thermoelastic body (time < 0.25: laser scribed of -SiC (power=600W, heating/drilling; time > 0.25: cooling). scan velocity=1cm/s). expected, strong compressive hoop stresses develop near the surface of the hole (up to 1 ' ;1:3 GPa), however, not strong enough to cause serious damage; 2 is close to zero at the surface due to the no-load boundary conditions. Interestingly, substantial tensile stresses in, both, the 1 and 2 directions develop parallel to the hole surface inside the material (up to values of about +0.1 or 80 MPa). While these tensile stresses are barely sufficient to cause substantial damage at room temperature, the effects may be more pronounced at elevated temperatures. On a qualitative level the results explain beautifully the damage we have routinely observed when scribing -SiC with our CW CO2 laser (see, e.g., Fig. 4, which shows -SiC scribed at 1 cm/s and CW power of 600 W). Similar observations are made when the laser operates in pulsed mode, here assumed to be running at 500 Hz (2 ms pulse time, or t = 0:25) with a 25% duty cycle (500 s on-time). The frames in Fig. 5 show the hoop stresses just before (t = 0:06 < 0:00625) the end of several
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
time = 1.06000 -0.2
0
1
2
r
3
4
0
0
0.05
0 .0
0.05
3
-0.5
3.5
1
2
r
3
4
.2 -0
0
0.0
-0.2 -0.6 -0.3 -0.1 0 0.05
-0.9-1 -0.7
-0.4
2.5
1 0.
4
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
-0.5
2
-1-0. -0.7 9 -0.8 -0.6 -0.4 -0.3 -0.2 -0.1
5 0.0
z
5
1.5
5
0
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
5
-0.8
-0.8 -0.4 0
-0.7
-0.1
0.0
-0.6 -1 -0.2
-0.5 -0.3
-0 -0.4.8 -0.7 -0 0 .1
-1
-0.6
3.5
5
3
0.1
0.1
5
1
-0.9
z
2.5
-0.3 -0.1
0
-0.2
-0.9
3.5
0.0
0.0
0
2
-0.2
0
1.5
-0.30.5
2.5
0.5
-0 -0.3 .5
5 0.0
.5 -0
-0-.9 1
-0-0-0. .3 .1 2
-0.8 -0 .70.6
-0.4
z
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 0-0.6 -0.7 -0.8 -0.9 -1
5 0. 0
2 0.05
3
0.5
5
05 0.
1.5 .9 -1 .3 -0.1 .2 -0 -0
5
1
0.0
1
0.0
0
-0.5 -0.4
-0.6
-0 .1
.4 -0
0.5
-1 -0.9 -0.7 -0.6
-1
-0
.7
0.05
0
0.00 5
time = 0.060000
0
0
1
2
r
Figure 5: Hoop stress development during pulsed CO2 laser drilling of SiC; thermoelastic body (time=0.06: near end of first pulse; time=0.31: near end of second pulse; time=1.06: near end of fifth pulse). laser pulses. As in the CW case substantial tensile stresses are seen to form inside the medium parallel to the r ' 1w0). hole surface, with the maximum tensile stresses below the rim of the hole (at Finally, the effects of viscoelasticity are shown in Fig. 6, which shows the viscoelastic case equivalent to the frames in Fig. 3. Not surprisingly, the compressive stresses (hoop and radial) near the surface are substantially reduced (from a maximum of ' ;1:3 GPa to approximately ;400 MPa). This is accompanied by a strong buildup of compressive normal stresses just below the surface, as seen from Fig. 6. Also, the below-surface tensile stresses are increased substantially in the viscoelastic material (by approximately 50%), making the failure depicted in Fig. 4 much more likely. During cooling the viscoelastic material contracts, producing very strong tensile stresses in all three principal directions very close to the surface (up to values of = 0:8 or ' 600 MPa), in particular near the hole’s rim. Viscoelasticity during pulsed drilling, at least for the conditions and extrapolated properties employed here, always only affects the immediate vicinity of the surface. The effects do not seem to propagate into the material; therefore, one may assume that this layer will simply spall off during drilling. However, some very preliminary acoustic emission experiments in our laboratory for alumina indicate that crack formation primarily occurs immediately after the laser is turned off and during cooldown. Perhaps it is the combination of strong tensile stresses at the surface and the internal layer of tensile stresses that cause cracks to occur (although one also needs to keep in mind that alumina, unlike SiC, melts and resolidifies, thus generating a much thicker creep zone). The reasons for the thin tensile stress surface layer are obvious from Fig. 7, which shows equation (7) and temperature vs. depth at two locations, i.e., near the center and the rim of a typical hole: the temperature drops off so rapidly that after 3 m (center) to 10 m (rim) the viscoelasticity has decreased by three orders of magnitude and, for a typical time of 1 ms, has become negligible, i.e., 1= t= [see equation (6)].
Conclusions To assess the where, when and what stresses occur during CW and pulsed laser drilling of ceramics, elastic and viscoelastic stress models have been incorporated into our two-dimensional drilling code. Simulations have been performed to predict temporal temperature fields, the receding solid surface during CW or pulsed laser drilling of thin ceramic wafers, and – based on these results – elastic stresses as well as viscoelastic stresses as they develop and decay during the drilling process. It was observed that during drilling substantial
3
time = 0.200000 0
time = 0.600000 0
0 0
0
3000
10
0
0
-0
0.1
.4--00 ..32
1 -0.
0.05
-0
.5
T at center 1/η at center T at rim 1/η at rim
5 0.0
0.05
1 0. 5 0.0
2.5
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
1.5
2
2.5
1/η [1/MPa s]
0
1 0.
2
0 0.1
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
5
0
0.0
10-1
T [K]
0
.6 -0
0. 05
.5
1
z
z 1.5
-0 .2 -0 -0.1
2500
0
1
0.5
0.05
-0 .4-0 0.32 -0 .5 .7 -0 . -0 1.4 0.6 -0 .3 -0 .1 0.2
0.5
2000
10-2
1500
0
0
0
1
2
r
0
1
2
r
5
10
15
20
distance into material [μm]
Figure 6: Effects of viscoelasticity on hoop stresses during CW Figure 7: Decay of temperature and visCO2 laser drilling of SiC (time < 0.25: heating/drilling; time > coelasticity into substrate during pulsed laser drilling of SiC. 0.25: cooling). hoop and normal tensile stresses develop over a thick layer below and parallel to the surface, which may be the cause for experimentally observed subsurface cracks. It was also found that viscoelastic effects (treating the viscosity of the ceramic as temperature-dependent) were mostly limited to an extremely thin layer near the ablation front, where the ceramic has softened, relaxing compressive stresses during heating, followed by strong tensile stresses during cooling.
Acknowledgments Support by the National Science Foundation through Grant no. CMS-9634744 is gratefully acknowledged. Code verification tests on a commercial FEM stress code, as well as property data evaluations were performed by Mr. T. Mallison. References 1. Copley, S. W., R. J. Wallace, and M. Bass (1983), Laser Shaping of Materials, In Lasers in Materials Processing (Edited by Metzbower, E. A.), ASME, Metals Park, Ohio. 2. Yamamoto, J., and Y. Yamamoto (1987), Laser Machining of Silicon Nitride, In International Conference on Laser Advanced Materials Processing – Science and Applications, 297–302. High Temperature Society of Japan, Japan Laser Processing Society, Osaka, Japan. 3. DeBastiani, D., M. F. Modest, and V. S. Stubican (1990), Mechanisms of Reactions During CO2 -Laser Processing of Silicon Carbide, J. Amer. Cer. Soc. 73(7), 1947–1952. 4. Hasselman, D. P. H., and J. P. Singh, author and editor In Thermal Stresses I (Edited by Hetnarski, R. B.), Criteria for the Thermal Stress Failure of Brittle Structural Ceramics, Ch. 4. North-Holland, New York (1986). 5. Hasselman, D. P. H. (1967), Approximate Theory of Thermal Stress Resistance of Brittle Ceramics Involving Creep, J. Amer. Cer. Soc. 50, 454–457. 6. Folweiler, R. C. (1961), Creep Behavior of Pore-Free Polycrystalline Alluminum Oxide, J. Appl. Phys. 32(5), 773–778. 7. Lane, J. E., C. H. Carter, and R. F. Davis (1988), Kinetics and Mechanisms of High-Temperature Creep in Silicon Carbide: III, Sintered -Silicon Carbide, J. Amer. Cer. Soc. 71(4), 281–295.
10-3 25
8. Gross, T. S., S. D. Hening, and D. W. Watt (1991), Crack Formation during Laser Cutting of Silicon, J. Appl. Phys. 69(2), 983–989. 9. Hasselman, D. P. H., J. R. Thomas, M. P. Kamat, and K. Satyamurthy (1980), Thermal Stress Analysis of Partially Absorbing Brittle Ceramics Subjected to Symmetric Radiation Heating, J. Amer. Cer. Soc. 63(1-2), 21–25. 10. Thomas, J. R., J. P. Singh, and D. P. H. Hasselman (1981), Analysis of Thermal Stress Resistance of Partially Absorbing Ceramic Plate Subjected to Asymmetric Radiation, I: Convective Cooling at Rear Surface, J. Amer. Cer. Soc. 64(3), 163–173. 11. Singh, J. P., N. Sumi, J. R. Thomas, and D. P. H. Hasselman (1981), Analysis of Thermal Stress Resistance of Partially Absorbing Ceramic Plate Subjected to Asymmetric Radiation, II: Convective Cooling at Front Surface, J. Amer. Cer. Soc. 64, 169–173. 12. Bradley, F. (May 1988), Thermoelastic Analysis of Radiation-Heating Thermal Shock, High Temperature Technology 6(2), 63–72. 13. Sumi, N., R. B. Hetnarski, and N. Noda (1987), Transient Thermal Stresses due to a Local Source of Heat Moving over the Surface of an Infinite Elastic Slab, Journal of Thermal Stresses 10, 83–96. 14. Guan, Q., and Y. Cao (1993), Verification of FE Programs for Welding Thermal Strain– Stress Analysis Using High Temperature Moir´e Measurement, Journal of the International Institute of Welding 31(1), 344–347. 15. Ferrari, M., and J. H. Harding (1992), Thermal Stress Field in Plasma-Sprayed Ceramic Coatings, Journal of Energy Resources Technology 114, 105–109. 16. Bahr, H.-A., B. Schultrich, H.-J. Weim, I. Pflugbeil, E. R¨udiger, K. Wetzig, and S. Menzel (February 1993), Thermoschockrißbildung durch Laserinduzierte Hochtemperaturrelaxation, In Proceedings of Vortragsveranstaltung des DVMArbeitskreises Bruchvorg¨ange, 149–157, Karlsruhe, Germany. 17. Modest, M. F. (1996), Three-Dimensional, Transient Model for Laser Machining of Ablating/Decomposing Materials, Int. J. Heat Mass Transfer 39(2), 221–234. 18. Modest, M. F. (June 1997), Laser Through-Cutting and Drilling Models for Ablating/ Decomposing Materials, J. Laser Appl. 9(3), 137–146. 19. Modest, M. F., and H. Abakians (1986), Evaporative Cutting of a Semi-Infinite Body With a Moving CW Laser, J. Heat Transfer 108, 602–607. 20. Bang, S. Y., S. Roy, and M. F. Modest (1993), CW Laser Machining of Hard Ceramics Reflections, Int. J. Heat Mass Transfer 36(14), 3529–3540.
| Part II: Effects of Multiple
21. Bang, S. Y., and M. F. Modest (1991), Multiple Reflection Effects on Evaporative Cutting with a Moving CW Laser, J. Heat Transfer 113(3), 663–669.
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22. Bang, S. Y., and M. F. Modest (1992), Evaporative Scribing with a Moving CW Laser Effects of Multiple Reflections and Beam Polarization, In Proceedings of ICALEO ’91, Laser Materials Processing, Vol. 74, 288–304, San Jose, CA. 23. Boley, B. A., and J. H. Weiner (1960), Theory of Thermal Stresses, Wiley, New York.
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24. Roy, S., and M. F. Modest (1993), CW Laser Machining of Hard Ceramics Part I: Effects of Three-Dimensional Conduction and Variable Properties and Various Laser Parameters, Int. J. Heat Mass Transfer 36(14), 3515–3528. 25. Ramanathan, S., and M. F. Modest (1990), Effects of Variable Thermal Properties on Evaporative Cutting with a Moving CW Laser, In Heat Transfer in Space Systems, Vol. HTD–135, 101–108, ASME. 26. Edington, J. W., D. J. Rowcliffe, and J. L. Henshall (1975), The Mechanical Properties of Silicon Nitride and Silicon Carbide Part I: Materials and Strength, Powder Metallurgy International 7(2), 82–96.
Meet the Author Michael F. Modest was born in Germany and received his Dipl.-Ing. degree in Mechanical Engineering from the Technical University in Munich in 1968. After moving to the U.S. he obtained his M.S. and Ph.D. degrees, also in Mechanical Engineering, from the University of California at Berkeley in 1972. He is presently a professor in the Mechanical Engineering Department at the Pennsylvania State University. His research interests cover two major areas in experiment as well as in theory: radiative heat transfer, and heat transfer during laser machining of ceramics. Additional Color Pictures not in Proceedings Article
time = 0.050000 0
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1.5
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r Figure 8: Principal stresses (approx. normal to top surface) during CW CO2 laser drilling of SiC; thermoelastic body (time = 0.05: shortly after laser turn-on, heating/drilling).
time = 0.250000 0
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r Figure 9: Principal stresses (approx. normal to top surface) during CW CO2 laser drilling of SiC; thermoelastic body (time = 0.25: at end of laser turn-on time, heating/drilling).
time = 0.300000 0
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0.5
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r Figure 10: Principal stresses (approx. normal to top surface) during CW CO2 laser drilling of SiC; thermoelastic body (time = 0.30: shortly after laser turn-off, cooling).
.4 -0
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0.5
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r Figure 11: (Principal) hoop stresses during CW CO2 laser drilling of SiC; thermoelastic body (time = 0.05: shortly after laser turn-on, heating/drilling).
time = 0.250000
0.05
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05 0.
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r Figure 12: (Principal) hoop stresses during CW CO2 laser drilling of SiC; thermoelastic body (time = 0.25: at end of laser turn-on time, heating/drilling).
time = 0.300000 0 -0.2
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r Figure 13: (Principal) hoop stresses during CW CO2 laser drilling of SiC; thermoelastic body (time = 0.30: shortly after laser turn-off, cooling).
time = 0.200000
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0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
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1.1
1.2
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r Figure 14: Effects of viscoelasticity on principal stresses (approx. normal to top surface) during CW CO2 laser drilling of SiC (time = 0.20: well after laser turn-on, heating/drilling).
time = 0.250000 0
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r Figure 15: Effects of viscoelasticity on principal stresses (approx. normal to top surface) during CW CO2 laser drilling of SiC (time = 0.25: at end of laser turn-on time, heating/drilling).
time = 0.300000 0
0.05 0
0.1 0.05 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
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1.1
1.2
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r Figure 16: Effects of viscoelasticity on principal stresses (approx. normal to top surface) during CW CO2 laser drilling of SiC (time = 0.30: shortly after laser turn-off, cooling).
time = 0.200000 -0.7
-0 .1
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r Figure 17: Effects of viscoelasticity on (principal) hoop stresses during CW CO2 laser drilling of SiC (time = 0.20: well after laser turn-on, heating/drilling).
time = 0.250000
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r Figure 18: Effects of viscoelasticity on (principal) hoop stresses during CW CO2 laser drilling of SiC (time = 0.25: at end of laser turn-on time, heating/drilling).
time = 0.300000 0
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r Figure 19: Effects of viscoelasticity on (principal) hoop stresses during CW CO2 laser drilling of SiC (time = 0.30: shortly after laser turn-off, cooling).
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r Figure 20: Principal stresses (approx. normal to top surface) during pulsed CO2 laser drilling of SiC; thermoelastic body (time = 0.06: just before end of first laser pulse).
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r Figure 21: Principal stresses (approx. normal to top surface) during pulsed CO2 laser drilling of SiC; thermoelastic body (time = 0.31: just before end of second laser pulse).
time = 1.06000
0
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r Figure 22: Principal stresses (approx. normal to top surface) during pulsed CO2 laser drilling of SiC; thermoelastic body (time = 1.06: just before end of fifth laser pulse).
time = 0.060000 0
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r Figure 23: (Principal) hoop stresses during pulsed CO2 laser drilling of SiC; thermoelastic body (time = 0.06: just before end of first laser pulse).
time = 0.310000
05 0.
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r Figure 24: (Principal) hoop stresses during pulsed CO2 laser drilling of SiC; thermoelastic body (time = 0.31: just before end of second laser pulse).
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r Figure 25: (Principal) hoop stresses during pulsed CO2 laser drilling of SiC; thermoelastic body (time = 1.06: just before end of fifth laser pulse).
