transient elastic and viscoelastic thermal stresses during
TRANSCRIPT
Transient Elastic and Viscoelastic Thermal Stresses During LaserDrilling of Ceramics
Michael F. ModestDepartment of Mechanical Engineering
The Pennsylvania State UniversityUniversity Park, PA 16802
Abstract
Lasers appear to be particularly well suited to drill and shape hard and brittle ceramics, which are almostimpossible 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 localizedthermal stresses within the narrow heat-affected zone, which can lead to micro-cracks, significant decrease in bendingstrength, and even catastrophic failure. In order to assess the where, when and what stresses occur during laserdrilling, that are responsible for cracks and decrease in strength, elastic and viscoelastic stress models have beenincorporated into our two-dimensional drilling code. The code is able to predict temporal temperature fields as wellas the receding solid surface during CW or pulsed laser drilling. Using the resulting drill geometry and temperaturefield, elastic stresses as well as viscoelastic stresses are calculated as they develop and decay during the drillingprocess. The viscosity of the ceramic is treated as temperature-dependent, limiting viscoelastic effects to a thin layernear 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 remainthat are poorly, or not at all, understood. These problems include thermal stress, redeposition of evaporatedor liquified material, poor surface finish, undesirable hole and groove tapers, etc. It is well known thatlaser 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 (Copleyet 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 notethat ceramic materials will exhibit creep by diffusional processes at levels of temperature at which vacancyconcentrations 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 highertemperature levels are reached; however, experimental evidence suggests that such rapid stress relaxationdoes occur before the material melts or decomposes. Extrapolating data for alumina given by Hasselman [5]to a temperature just below the melting point (T � 2300K) gives a thermal stress relaxation time of only30�s. While creep may reduce the probability of failure by thermal stresses during the heating-up of theceramics, the resulting stress relaxation would be expected to lead to very strong tensile stresses duringcool-down, which in turn could cause the generation of microcracks, overall weakening of the material’smechanical strength, or catastrophic failure.
Data on creep behavior have been obtained by a number of investigators for a number of ceramicmaterials, although no creep properties appear to have been measured for temperatures approaching themelting/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 tothe melting/decomposition point indicate that plastic deformation accompanied by thermal stress relaxationis to be expected during laser shaping of ceramics. This was confirmed by Gross et al. [8], who investigatedcrack formation during laser (CO2 and Nd:YAG) drilling of thin silicon wafers. The existence of a plasticallydeformed 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 ofcracks confirming, at least qualitatively, the nature of the expected stresses in the presence of creep prior tomelting.
Most analytical investigations have been limited to thermoelastic bodies, using a one-dimensional analysisor commercial finite-element programs. Hasselman et al. [9�11] investigated analytically the transientthermal stress field in a one-dimensional slab subjected to external radiation, with internal absorption ofthis 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 describethe 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 problemin 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 andCao [14] predicted residual stresses during welding of thin plates using a 2-D elasto-plastic finite elementmodel. Ferrari and Harding [15] modeled the residual thermal stress field for a one-dimensional sphere withseveral plasma-sprayed ceramic coatings, finding moderate compressive stresses in the radial direction, butlarge 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 predictthermal stresses in the wafer. The predictions indicate that, during cooldown, residual circumferential stressesare tensile in the deformed zone and compressive outside. Radial thermal stresses are tensile everywherewith 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 typesof cracks confirming, at least qualitatively, the nature of the predicted stresses. Bahr et al. [16] presenteda one-dimensional, transient model to predict thermal stresses inside an opaque solid irradiated by a shortlaser 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 periodof 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 mspulse, but only a few �m for a 1 ms pulse. Interestingly, for shorter pulses the tensile stresses are not onlylimited 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 viscoelasticstress model, to predict thermal stresses as they develop and decay during CW and pulsed laser drilling ofceramics.
Theoretical Background
To make analyses for the thermal and the stress problems tractable, a number of limiting and simplifyingassumptions need to be made. Assumptions for the heat transfer problem are identical to those in previouspapers 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 notpenetrate appreciably into the solid.
2. Change of phase from solid to vapor (or decomposition products) occurs in a single step with a rategoverned 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 externalgas 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 toshallow 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 simpleorder-of-magnitude analysis).
7. Inertia effects are negligible during stress development (always true for opaque ceramics, but maybecome questionable for semitransparent ceramics subject to ns laser pulses).
Heat TransferThe 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 � (krT ) =
1�r
�
��r
�k�r�T
��r
�+�
��z
�k�T
��z
� (1)
subject to the boundary conditions
�r = 0 :�T
��r= 0 (2a)
�r�� : T � T� (2b)
�z = 0 : �F � n = �n � (krT ) + vn��hre (2c)
�z = D :�T
��z= 0 (2d)
and an appropriate initial condition, such as�t = 0 : T (�r �z �t = 0) = T� (2e)
s(�r�t = 0) = �s0(�r) (2f)
r
z
2w0
z0
D
Figure 1: Laser drilling setup and coordi-nate system.
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 isthe local depth of a hole (i.e., the �z-coordinate of the top suface) and n is a unit vector normal to the surface(pointing into solid); vn is the total surface recession velocity during drilling, and it is assumed that thesolid is originally at a uniform temperature T�; F is energy intensity distribution, for a focussed Gaussianlaser beam with a waist w0 at the focal plane �z0 (some quantities have been barred to distinguish the presentdimensional 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 ablationand/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�� = �vn = �C1e�E��RT (3)
where E is the decomposition energy, �R is the universal gas constant, and C1 is a preexponential factor thatdepends on the nature of the ablation process.
The governing equations and boundary conditions are non-dimensionalized using the 86%-beam radiusat the focal point, w0:
r = �r�w0; z = �z�w0; t =kre�t
�crew20
; s = �s�w0; =T � T�Tre � T�
(4a)
leading to two basic non-dimensional parameters governing the laser/material interaction:
Nk =kre(Tre � T�)
F0w0; Ste =
�hre
cre(Tre � T�)� (5)
Boundary-fitted coordinates are employed in the numerical solution, i.e., the physical domain (r z), istransformed to a uniformly spaced rectangular coordinate region (� �). Detailed discussions of the heattransfer analysis and the numerical implementation are given by Modest [17], (general development for amoving laser) and [18] (details on through-cutting and drilling).
Thermal StressesThe 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 strongthermal stresses. While under most conditions ceramics may be considered elastic, during (thermal) laserdrilling there will be a thin zone near the receding interface (and at extremely high temperature), overwhich significant creep may occur. To assess the importance of this nonelastic zone on the overall thermalstress development, the present analysis includes a simple linear viscoelastic model (Maxwell body). Thedeviatoric stress-strain relation for a Maxwell body is [23].
��e =1
2���s +
12
�s (6)
where �e and �s are the dimensional deviatoric strain and stress tensors, respectively, � is one of Lame’sconstants (= G, the shear modulus), and is the viscosity of the viscoelastic solid, which–unlike all otherproperties–is assumed to be temperature-dependent, since for ceramics its value changes by many orders ofmagnitude between room temperature and ablation/decomposition temperature. The viscosity-temperaturedependence of �e is known from creep studies to follow an Arrhenius relationship, i.e.,
1
= Ae��Q��RT =
1 re
e��Q��RTre(1�Tre�T ) (7)
where �Q is the activation energy, A is a preexponential factor, and re is the viscosity at the material’sremoval temperature. The complete stress-strain relation for a Maxwell body may then be stated [23] innon-dimensional form as
�� + �()� = �� + ��
3�1� 2�
�� �1 + �
1� 2�� +
1 + �1� 2�
�()(� � )�
(8)
� =12
�ru +ruT
�; � =
13
trace (�) (9)
where
� =��
2��v(Tre � T�) � =
��
�v(Tre � T�) u =
�u
w0�v(Tre � T�) (10a)
�() =�crew
20�
kre (T )= �ree
�Q(1�Tre�T ) Q =�Q
�RTre(10b)
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 ��
and �� are stress and strain tensors, respectively, � is the identity tensor, �u is the displacement vector, �v isthe coefficient of thermal expansion, � is Poisson’s ratio, and the dot upon a symbol denotes differentiationwith respect to time. Equation (8) requires initial conditions for � and �; we will here assume that the solidis 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 conditionr � � = 0 (since stress cannot be eliminated). Since a numerical solution willbe attempted, the time derivatives will first be eliminated through a simple implicite finite difference [as isdone in the solution of equation (1)], i.e.,
� =(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) (13)
A(n)() =1
1 + �(n)�t; b =
1 + �1� 2�
C(n) = A(n) [� � � + �� � �b(�� )](n�1) � (14)
Note that equation (13) reduces to the thermoelastic case for infinite viscosity (� � 0 : A� 1 C � O).For the two-dimensional, axisymmetric problem at hand equation (13) in long-hand becomes
�rr =�u
�r ��� =
u
r �zz =
�w
�z (15a)
�rz =12
��u
�z+�w
�r
� �r� = ��z = 0 (15b)
�rr = A�u
�r+
13
(b� A)��u
�r+u
r+�w
�z
�� b + Crr (16a)
��� = Au
r+
13
(b� A)��u
�r+u
r+�w
�z
�� b +C�� (16b)
�zz = A�w
�z+
13
(b�A)��u
�r+u
r+�w
�z
�� b + Czz (16c)
�rz =A
2
��u
�z+�w
�r
�+ Crz (16d)
and the equilibrium conditions reduce to
��rr�r
+��rz�z
+1r
(�rr � ���) = 0 (17a)
��rz�r
+��zz�z
+1r�rz = 0� (17b)
For simplicity the superscript (n) has been dropped from these equations. Equation (17) is a set of ellipticequations in the unknown displacements u and w, thus requiring boundary conditions along the boundingsurface of the volume under consideration. Assuming zero traction on top and bottom surfaces (� � n = 0),and zero displacement far away from the laser leads to
r = 0 : u = 0�w
�r= 0 (18a)
r�� : u = w = 0 (18b)
z = 0 : �rrnr + �rznz = 0 �rznr + �zznz = 0 (18c)
z = L : �rz = �zz = 0 (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 intothe medium [actually, in the numerical implementation, here and for boundary condition (18b), we use thefact 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 extremelylong and tedious relations, which will not be reproduced here. Special consideration must be given to thetop and bottom boundaries because of the out-of-plane derivatives. These are taken care of by integratingequations (17) over the half-nodes near the surface, eliminating out-of-plane derivatives through the use ofequations (18c) and (18d). Another trouble spot is the triangular node at the bottomof a hole once it forms (seeFig. 1), which is dealt with by integrating equations (17) over the triangular element (in physical coordinates).Finally, the evaluation ofC requires special attention. Recall that equation (8) was finite-differenced in timebefore transformation to computational coordinates: values at the previous time step must be evaluated atthe same physical coordinates. Therefore, if the computational coordinates move with speed �t �t:
C�(n�1)(r(n) z(n)) = C�(n�1)(r(n�1) z(n�1)) ���t�C�
��+ �t
�C�
��
�(n�1)
(r(n�1) z(n�1))�t (19a)
C� = � � � + ��� �b(� � )� (19b)
The result is a set of two equations for each of the N� �N� nodes making up the overall grid (assuming noburn-through). In nine-point stencil form this may be written as
p �uik + n � ui�k+1 + ne � ui+1�k+1 + e �ui+1�k + se � ui+1�k�1 + s �ui�k�1
+sw � ui�1�k�1 +w � ui�1�k + nw � ui�1�k+1 = f (20)
where each of the p, n, etc. are 2�2 tensors. The set of simultaneous equations (20) may be invertedin a number of ways. Since temperature varies fastest in the �-direction, we ordered equation (20) intoa block-tridiagonal system for constant �, which was solved directly, and iteratively swept over � usingsuccessive overrelaxation. This works reasonably well, but will be improved before implementation in the3D laser machining problem. Note that, for the thermoelastic case, equation (20) needs to be solved onlyat 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, severaldrilling operations on �-SiC were simulated, using silicon carbide physical properties from Ramanathanand Modest [25] (all taken at removal temperatures of Tre = 3000 K, which gives good agreement withvariable property calculations [25]) and Edington et al. [26]: kre = 20 W/mK, �cre = 5 � 106 J/m3K,�hre � 12�1MJ/kg, �v = 10�6�K; E = 400 GPa, � = 0�17; and the viscoelastic properties were curvefitted to data given by Lane et al. [7] as A = 5�30 � 1014�(MPa s) and �Q = 840 kJ/mol. Laser parameterswere typical values for a CO2 laser (such as the one in our laboratory), using a w0 = 175�m and an averageabsorbed power of �P = 500 W. Several CW and pulsed laser drilling events have been simulated, includinga 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 elasticbody during CW drilling (up to a non-dimensional time of t = 0�25, or �t � 2 ms), and cool-down afterthe 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); thethird principal stress was found to be always compressive (with maximum values of ��3 � �1�3 GPa). As
0 1 2r
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Figure 2: Hoop stress development during CW CO2 laser drilling of SiC; thermoelastic body (time � 0.25:heating/drilling; time � 0.25: cooling).
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0
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time = 0.600000
Figure 3: Principal stresses (approx. normal to top surface) duringCW CO2 laser drilling of SiC; thermoelastic body (time� 0.25:heating/drilling; time � 0.25: cooling).
Figure 4: SEM cross-section of CW CO2
laser scribed of �-SiC (power=600W,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-loadboundary conditions. Interestingly, substantial tensile stresses in, both, the �1 and �2 directions developparallel to the hole surface inside the material (up to values of about +0.1 or 80 MPa). While these tensilestresses are barely sufficient to cause substantial damage at room temperature, the effects may be morepronounced at elevated temperatures. On a qualitative level the results explain beautifully the damage wehave routinely observed when scribing �-SiC with our CW CO2 laser (see, e.g., Fig. 4, which shows �-SiCscribed at 1 cm/s and CW power of 600 W). Similar observations are made when the laser operates in pulsedmode, here assumed to be running at 500 Hz (2 ms pulse time, or t = 0�25) with a 25% duty cycle (500 �son-time). The frames in Fig. 5 show the hoop stresses just before (t = 0�06 � 0�00625) the end of several
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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 thehole surface, with the maximum tensile stresses below the rim of the hole (at �r � 1w0).
Finally, the effects of viscoelasticity are shown in Fig. 6, which shows the viscoelastic case equivalentto the frames in Fig. 3. Not surprisingly, the compressive stresses (hoop and radial) near the surface aresubstantially reduced (from a maximum of � �1�3 GPa to approximately�400 MPa). This is accompaniedby a strong buildup of compressive normal stresses just below the surface, as seen from Fig. 6. Also, thebelow-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 valuesof � = 0�8 or �� � 600 MPa), in particular near the hole’s rim. Viscoelasticity during pulsed drilling, at leastfor the conditions and extrapolated properties employed here, always only affects the immediate vicinity ofthe surface. The effects do not seem to propagate into the material; therefore, one may assume that this layerwill simply spall off during drilling. However, some very preliminary acoustic emission experiments in ourlaboratory for alumina indicate that crack formation primarily occurs immediately after the laser is turned offand during cooldown. Perhaps it is the combination of strong tensile stresses at the surface and the internallayer 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 temperaturedrops off so rapidly that after 3 �m (center) to 10 �m (rim) the viscoelasticity has decreased by three ordersof 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, elasticand viscoelastic stress models have been incorporated into our two-dimensional drilling code. Simulationshave been performed to predict temporal temperature fields, the receding solid surface during CW or pulsedlaser drilling of thin ceramic wafers, and – based on these results – elastic stresses as well as viscoelasticstresses as they develop and decay during the drilling process. It was observed that during drilling substantial
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0 5 10 15 20 25distance into material [μm]
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Figure 6: Effects of viscoelasticity on hoop stresses during CWCO2 laser drilling of SiC (time � 0.25: heating/drilling; time �0.25: cooling).
Figure 7: Decay of temperature and vis-coelasticity into substrate during pulsedlaser drilling of SiC.
hoop and normal tensile stresses develop over a thick layer below and parallel to the surface, which may bethe cause for experimentally observed subsurface cracks. It was also found that viscoelastic effects (treatingthe viscosity of the ceramic as temperature-dependent) were mostly limited to an extremely thin layer nearthe ablation front, where the ceramic has softened, relaxing compressive stresses during heating, followedby 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 performedby 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 byMetzbower, E. A.), ASME, Metals Park, Ohio.
2. Yamamoto, J., and Y. Yamamoto (1987), Laser Machining of Silicon Nitride, In International Conference on Laser AdvancedMaterials Processing – Science and Applications, 297–302. High Temperature Society of Japan, Japan Laser ProcessingSociety, Osaka, Japan.
3. DeBastiani, D., M. F. Modest, and V. S. Stubican (1990), Mechanisms of Reactions During CO2-Laser Processing of SiliconCarbide, 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.
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 AbsorbingBrittle 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 AbsorbingCeramic 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 PartiallyAbsorbing 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 theSurface 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 TemperatureMoire 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 EnergyResources Technology 114, 105–109.
16. Bahr, H.-A., B. Schultrich, H.-J. Weim, I. Pflugbeil, E. Rudiger, K. Wetzig, and S. Menzel (February 1993), Ther-moschockrißbildung durch Laserinduzierte Hochtemperaturrelaxation, In Proceedings of Vortragsveranstaltung des DVM-Arbeitskreises Bruchvorgange, 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. HeatTransfer 108, 602–607.
20. Bang, S. Y., S. Roy, and M. F. Modest (1993), CW Laser Machining of Hard Ceramics � Part II: Effects of MultipleReflections, Int. J. Heat Mass Transfer 36(14), 3529–3540.
21. Bang, S. Y., and M. F. Modest (1991), Multiple Reflection Effects on Evaporative Cutting with a Moving CW Laser, J. HeatTransfer 113(3), 663–669.
22. Bang, S. Y., and M. F. Modest (1992), Evaporative Scribing with a Moving CW Laser�Effects of Multiple Reflections andBeam 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.
24. Roy, S., and M. F. Modest (1993), CW Laser Machining of Hard Ceramics� Part I: Effects of Three-Dimensional Conductionand 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 CWLaser, 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 CarbidePart 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 Engineeringfrom 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 ispresently a professor in the Mechanical Engineering Department at the Pennsylvania State University. Hisresearch interests cover two major areas in experiment as well as in theory: radiative heat transfer, and heattransfer during laser machining of ceramics.
Additional Color Pictures not in Proceedings Article
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Figure 8: Principal stresses (approx. normal to top surface) during CW CO2 laser drilling of SiC; thermoelasticbody (time = 0.05: shortly after laser turn-on, heating/drilling).
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Figure 9: Principal stresses (approx. normal to top surface) during CW CO2 laser drilling of SiC; thermoelasticbody (time = 0.25: at end of laser turn-on time, heating/drilling).
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Figure 10: Principal stresses (approx. normal to top surface) during CW CO2 laser drilling of SiC; thermoe-lastic body (time = 0.30: shortly after laser turn-off, cooling).
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Figure 11: (Principal) hoop stresses during CW CO2 laser drilling of SiC; thermoelastic body (time = 0.05:shortly after laser turn-on, heating/drilling).
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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).
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Figure 13: (Principal) hoop stresses during CW CO2 laser drilling of SiC; thermoelastic body (time = 0.30:shortly after laser turn-off, cooling).
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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).
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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).
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Figure 16: Effects of viscoelasticity on principal stresses (approx. normal to top surface) during CW CO2
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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).
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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).
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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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Figure 20: Principal stresses (approx. normal to top surface) during pulsed CO2 laser drilling of SiC; ther-moelastic body (time = 0.06: just before end of first laser pulse).
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Figure 21: Principal stresses (approx. normal to top surface) during pulsed CO2 laser drilling of SiC; ther-moelastic body (time = 0.31: just before end of second laser pulse).
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Figure 22: Principal stresses (approx. normal to top surface) during pulsed CO2 laser drilling of SiC; ther-moelastic body (time = 1.06: just before end of fifth laser pulse).
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Figure 23: (Principal) hoop stresses during pulsed CO2 laser drilling of SiC; thermoelastic body (time = 0.06:just before end of first laser pulse).
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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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0.15
0.150.10.050
-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9-1
time = 1.06000
Figure 25: (Principal) hoop stresses during pulsed CO2 laser drilling of SiC; thermoelastic body (time = 1.06:just before end of fifth laser pulse).