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Stress intensities at notch singularities
Engineering Fracture Mechanics Vol. 57, No. 4, pp. 417-430, 1997 © 1997ElsevierScienceLtd. All rights reserved Printed in Great Britain PII: S0013-7944(9"/)00019-2 0013-7944/97 $17.00+ 0.00
Pergamon
STRESS
INTENSITIES
AT
NOTCH
SINGULARITIES
MARTIN L. DUNN and WAN SUWlTO Center for Acoustics, Mechanics and Materials, Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309-0427, U.S.A. SHAWN CUNNINGHAM Ford Microelectronics Inc., Colorado Springs, Colorado 80921, U.S.A. Abstract--In the context of linear elasticity, a stress singularity of the type K~r6 (3 < 0) may exist at re-entrant notch comers, with an intensity/~. The magnitude of the intensity, referred to here as the notch stress intensity, characterizes the stress state in the region of the notch tip. We determined/(' for two mode 1-type specimens that are proposed to characterize a critical value of K~. Such a critical Kn may be appropriate as a failure criterion in situations where sharp re-entrant comers exist, and the region around the corner dominated by the singular field is large compared to intrinsic flaw sizes, inelastic zones and process zone sizes. For these specimens we show that the region dominated by the singularity increases as the notch angle increases and is smallest for the limiting case of a crack. At the same time though, the strength of the stress singularity decreases with increasing notch angle. Estimates of the plastic zone size also decrease with increasing notch angles. For notch angles less than about 60°, the stress singularity, stress intensity factor, region dominated by the singularity and the estimated plastic zone size do not differ substantially from those of a crack. © 1997 Elsevier Science Ltd
1. I N T R O D U C T I O N PREDICTION o f failure in solids typically proceeds f r o m one o f two points o f view: stress-strength or stress intensity-fracture toughness. The former entails c o m p u t i n g stresses where failure is anticipated, typically at sites o f stress concentration, and using the c o m p u t e d stresses in a failure criteria based on measured strengths, m o s t c o m m o n l y the yield or ultimate strength in uniaxial tension. The latter entails c o m p u t i n g the stress intensity at the tip o f a sharp crack (of course, singular stresses are predicted by elasticity theory), and using the c o m p u t e d stress intensity in a failure criteria based on a measured fracture toughness, which is presumed to be a material property. A l t h o u g h b o t h approaches apparently proceed f r o m quite different points o f view, a relation between the two can be seen by considering the stress analysis o f sharp notches. Williams[l] showed that in the context o f elasticity theory the asymptotic stress state near a corner or notch depends only on the notch geometry and is often singular. If we consider a polar coordinate system centered at the tip o f a notch o f angle V as shown in Fig. 1, the stresses near the notch tip can be expressed as tr~/~ = Kr ~ - l f ~ ( O ) . The order o f the stress singularity 2 - 1 and the angular function f~a(O) for a given angle 7 can be completely determined by an asymptotic analysis. The stress intensity K is a function o f the geometry o f the solid and the far-field loading. Thus, for a given geometry and loading, K completely characterizes the stress state in a region near the notch tip. The limiting case o f the notch angle ~ = 0 corresponds to a crack. This results in 2 - 1 = - 1 / 2 and on 0 = 0, a00 = K r-1/2, where K is the widely-used stress intensity factor o f linear elastic fracture mechanics. The case o f 7 = n corresponds to a solid with no notch and ~ . - 1 = 0. F o r uniaxial loading o f an edge-notched strip with 7 = n, croo = K on 0 = 0. I f failure is presumed to occur at a critical value o f K in both o f these cases, then in the former the critical K is the fracture toughness o f the material while in the latter it is the uniaxial strength. Based on the above discussion, the two approaches to failure analysis can be reconciled if we consider the analysis o f notched solids and presume, irrespective o f the notch angle, that failure occurs when K reaches a critical value. O f course, only for the limiting case o f a crack is there a rigorous connection between the stress intensity factor and the energy release rate o f fracture mechanics. Thus a critical K failure criterion appears to be suitable only for fracture initiation. Interestingly, very little w o r k has been done in the middle o f these two extremes. There are p r o b a b l y a couple o f reasons for this. In structural design, it is widely appreciated that sharp corners lead to high stresses and g o o d design proceeds by rounding corners. 417
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y
r
0 1'
-
-
X
Fig. 1. Geometry of the c o m e r considered in this study.
This reduces the problem from one of stress intensities to one of stress concentrations, where elasticity theory can be applied. Fracture mechanics plays a role in the case where cracks occur, not by design, but as a result of service. Indeed, in practice, it is common to drill holes at the tips of cracks in a structure. Again, this reduces the problem from one of stress intensities to one of stress concentrations, but during the life of the structure, not the design stage. Thus, it is rare in traditional structural design to be faced with sharp re-entrant corners. We either have intentionally rounded corners or unintentional sharp cracks. However, an exception arises when dealing with microelectromechanical structures that are machined, for example from silicon, by anisotropic chemical etching processes. The resulting structures have sharp corners formed by intersections of crystallographic planes. Appropriate approaches to failure prediction are needed for these structures and a critical K criterion appears reasonable as with these brittle materials stable crack propagation is rarely observed and an initiation failure criterion seems particularly well-suited. There are a handful of related studies that are pertinent to our work. We focus only on those concerned with computing the stress intensity at a sharp corner and do not mention the many studies directed at computing only the order of the stress singularity. Perhaps the earliest studies concerning a generalized fracture mechanics approach for notched materials were aimed at wood structures with sharp reentrant corners, and specifically at scaling effects with different size specimens [2--4]. Sinclair and Kondo [5] discussed the idea of a generalized stress intensity at sharp corners, but did not attempt to correlate fracture with the stress intensity, opting instead to pursue a stress-concentration approach. Sinclair et al. [6] also outlined an approach to obtain the generalized stress intensity at sharp corners based on the evaluation of a contour integral, which is an extension of the work of Stern et al. [7]. Independently, Carpenter [8] also applied the reciprocal work contour integral method of Stern et al. [7] to compute fracture mechanics parameters for a general corner. A series of subsequent papers by Carpenter[9-11] studied the problem further and introduces a collocation approach to compute stress intensities. The application of the approaches of Sinclair et al. [6] and Carpenter [8] to mixed-mode loading problems are further discussed by Sinclair [12]. Neville [13] proposed a statistical approach for failure prediction that makes use of the singularity at a sharp notch. Gross and Mendelson [14] studied stress intensities at notch tips from the viewpoint of addressing crack-notch configurations such as those that occur after fatigue precracking of standard fracture specimens, and understanding how crack-notch configurations influence measured fracture toughness. The stability of notched solids was studied by Knesl[15] who introduced as a length scale a distance ahead of the notch tip over which an average opening stress is computed. Perhaps the most detailed study of stress
Stress intensities at notch singularities
419
intensities at sharp corners is that of Carpinteri [16]. Carpinteri performed three-point flexure tests with notched bars and attempted to experimentally obtain the connection between a critical stress intensity at various angles and structure sizes. Although it may be possible to empirically correlate the critical stress intensity at various angles, it may not be formally appropriate. This is because although for a given notch angle the asymptotic stress field is universal, the universal field differs for different angles. For a given notch angle, regardless of the geometry of the structure containing the notch and regardless of the external loading, the complicated stress state at the notch tip is fully characterized by the magnitude of the generalized stress intensity. The proposition that a critical stress intensity can be used as a failure criterion really means that the stress state at the notch is presumed to cause failure. Thus, for two specimens with different geometry and loading, but with the same notch angle, the stress state at the notch tip is the same at failure. This stress state is different, though, for specimens with different notch angles, thus casting doubt on the possibility of a rigorous connection between the critical stress intensities at different notch angles. Indeed as an extreme case, such a connection would imply a rigorous connection between the fracture toughness and the uniaxial strength of a material. Nevertheless, it may prove valuable in practice to be able to correlate critical stress intensities at different notch angles. Despite the different viewpoints adopted in the above-mentioned studies, they all lend support to the use of a critical generalized stress intensity to correlate fracture of structures with sharp notches. Further support for such an approach is provided by the recent studies of Reedy [17-20] and Reedy and Guess [21-23]. They correlated failure of adhesive-bonded butt tensile joints with a generalized stress intensity at the free edge of an interface corner between elastic and rigid layers. In their studies the order of the elastic singularity is about 2 - 1 = - 1/3, as compared to 2 - 1 = - 1/2 for a crack. The success of these approaches has motivated us to attempt to use a critical stress intensity at re-entrant notch corners to correlate failure of micromachined silicon structures which often posses sharp corners due to etching. Over the last few years applications of micromachined silicon structures have increased considerably. The fabrication of silicon structures proceeds using technology developed in the integrated circuit industry: thin film deposition, photolithography and etching. In many applications the resulting structure contains sharp re-entrant corners. This is a result of the etching process that attacks various crystallographic planes at dramatically different rates. Failure mechanisms for these structures are not well understood and appropriate methodologies for predicting failure are needed. At the device or structure level, designers require knowledge of mechanical properties such as stiffness, strength and fracture toughness to proceed with rational design. Currently designers compute stresses in structures using finite element methods. The use of these results for quantitative failure prediction is problematic because predicted stresses near the sharp corner depend on the finite element idealization, and due to the existence of the elastic singularity, convergence in the stresses is not obtained with mesh refinement. In addition, both the necessary material properties and appropriate failure criteria are lacking. Only a few studies have even considered fracture of micromachined silicon [24, 25]. These have focused on measuring fracture toughness and assessing the influence of environment on crack growth. Of course, a pre-existing crack is necessary. Our studies of micromachined silicon structures seem to indicate that in many cases neither the traditional stress-strength or stress intensity-fracture toughness approaches are appropriate for describing failure. We have found the former problematic for a couple of reasons. First, it is very difficult to even measure a uniaxial tensile strength because failure always occurs at a stress raiser [26]. Even if a true uniaxial strength can be obtained, it is not clear how it would be of use in micromachined structures where failure often occurs at sharp corners. Since the epitaxial silicon of our studies is very brittle, it might be appealing to assume the existence of a very small flaw so that fracture mechanics can be applied. This approach, however, does not appear to be realistic. Scanning and transmission electron microscopy studies have failed to reveal any cracks at the sharp corners of our structures. To overcome these difficulties, we are pursuing an approach that takes advantage of the elastic singularity at sharp corners that exist in many micromachined structures[27]. In this paper we lay out our initial efforts in the development and application of a generalized stress intensity approach for correlating failure of micromachined silicon structures. We
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et al.
focus on isotropic elastic solids. The extension to anisotropic solids representative of single crystal silicon, for example, will be presented in a future study. We use the asymptotic analysis of Williams[l] to determine the order of the elastic singularity at a sharp corner. From the asymptotic analysis we obtain the order of the singularity and the angular dependence of the stress and displacement fields. These fully determine the stress state at the notch tip within an arbitrary constant, the notch stress intensity K", which is a function of the geometry of the structure and the loading. We obtain the stress intensity by detailed finite element calculations for two specimens proposed to obtain a critical value of K": a center-notched and a double-edge-notched tensile specimen. The solutions are obtained for a ratio of notch depth to plate width representative of a notch in an infinite plate. The only finite length scale that enters the problem is then the notch width, implying a simple connection between strengths of specimens with varying notch widths.
2. ASYMPTOTIC ANALYSIS OF SINGULAR FIELDS Consider the notched body shown in Fig. 1. It is loaded by tractions on remote boundaries and the surfaces 0 = __+~ are traction free. In the following, we redundantly, but conveniently, refer to 7 as the notch angle where ? is related to ct by ?/2 = n - ~. For simplicity we assume linear isotropic material behavior. The asymptotic, possibly singular, fields near the corner can be obtained using Williams' [1] eigenfunction expansion method for both the plane and antiplane problems. For the plane problem, the appropriate Airy stress function is:
q~ = r~+lf(O),
(1)
f(O) = Ai cos(~. + 1)0 + A2 sin(X + 1)0 + A3 cos(X - 1)0 + A4 sin(Z - 1)0.
(2)
where:
In the standard manner, the stresses can be obtained from ~b as: O'rr =
rk-l[(~, -st- 1)f(0) +f"(0)]
0"00 = ff'k-l~o~ + l)f(O) OrO = - - r g - l ~ , f
t(0),
(3)
where a prime denotes differentiation with respect to 0. Note that 2 > 0 for boundedness of the strain energy and 2 < 1 for singular fields. Imposing traction-free boundary conditions on 0 = + ~ results in a system of four equations for the five unknowns Af and 2. A nontrivial solution requires that the determinant of the coefficient matrix vanishes. Upon imposing this requirement, two equations result, one for symmetric fields with respect to the x-axis (0 = 0), and one for antisymmetric fields[l]: ), sin 2or + sin 2~.a = 0 ~. sin 2or - sin 2~.a = 0
for symmetric fields
(4)
for antisymmetric fields.
(5)
Borrowing from the terminology of fracture mechanics we refer to the symmetric and antisymmetric fields as mode I and mode II fields, respectively. For the antiplane (mode III) problem, the field equations reduce to V2w = 0 and admit an eigenfunction expansion:
Izw = r~f (O),
(6)
f(O) = Al cos ~.0 + A2 sin X0.
(7)
where # is the shear modulus and:
The shear stresses are: Orz .~
r~-I~.f(O)
Stress intensities at notch singularities
421
0.0 -
-
In-plane(synmaetric)
..'
. . . . . . . . In-plane (antisymmetric) Antiplane
-0.1
,
/
"
"~/ )'"
.-,
,-. /
-0.2
/
,y'"
''
,
I
"Y
/
/
-0.3
-0.4
"'"
/"
.,- / . - ' /
-0.5
'' 0
'' . 20
. 40
. 60
. . 80 100 T (degrees)
120
~ ...... 140 160
180
Fig. 2. Order of the stress singularity 2 - 1 as a function of the notch angle 7 for modes I, II and Ill.
(8)
aoz = ? - ~ f ' ( o ) .
Imposing traction-free boundary conditions on the notch faces yields two equations for the three unknowns Ai and 2. These result in the following equation for 2: cos 2or = 0
for antiplane fields.
(9)
The order of the stress singularity 2 - 1 is plotted in Fig. 2 for the three modes as a function of 7. Once 2 is obtained from eq. (4), eq. (5) or eq. (9), the coefficients Ai can not be completely determined. They can, however, be expressed in terms of only one Ai. We normalize the system so that a00(0 = 0) = /~ir ~ - 1, arO(0 = 0) = K'i'nr~ - 1 and azo(O = 0) = K~nir~ - 1 under mode I, II and III loading, respectively. This allows us to write the singular stresses and displacements as: Mode I
tra# = K ~rX-lf le(o ) Ua = K ~r~gla(O)
(10)
Mode II a~e = K ~irX-lf ~(0) II n
II
u~ = K i i r ~ g a (0)
(11)
Mode III a~# = K i n r - 7
Illoa-~
w = K ~nrXgz(O).
(12)
In eqs (10)-(12) ~ and fl take on the values (x,y,z) in cartesian coordinates and (r,O,z) in cylindrical coordinates. 2, f~/~(O) and g=(O) are determined by the asymptotic analysis. 2 and f,/~(O) are functions of ~, and g,(O) are functions of • and the elastic properties of the material. f,#(O) and g,(O) are tabulated in Appendix A for mode I. The stress intensity K" is a function of the specific geometry and loading. Its magnitude fully determines the stress state at a notch tip
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(a)
co
(b)
cyo
Y
J
(yo
co
Fig. 3. Geometry of tensile ~ specimens: (a) center-notched and (b) edge-notched.
for a given c~. Note that for a notch angle of y = 90 °, 2 - 1 = - 0.4555, -0.0915 and -0.3333 for modes I, II, and III, respectively. Thus, the mode I singularity is nearly as strong as that for a crack, while the mode II singularity is quite weak. This suggests that in opening, and perhaps in antiplane situations, a critical value of K" might be used successfully as a failure criterion. If so, we can measure a critical value o f / C ' using a simple test geometry, then use the critical K ~ so obtained to predict failure in practical structures. The analyses of two specimens proposed for extracting a critical K ~ are presented in the following section. 3. M O D E I S P E C I M E N S : / C AND F U L L - F I E L D FINITE E L E M E N T S O L U T I O N S
We focus on two potential specimens designed to determine the mode I stress intensity factor K~I, deferring the consideration of mode II (and mode III) specimens to a later date. We emphasize mode I because the singularity is much stronger than that for mode II for geometries of practical interest with y ,-~ 90 ° and so mode I failure is thought to dominate. The two specimens studied here are shown in Fig. 3; center-notched and edge-notched panels. These are both thin panels with notch geometries defined by the angle Y and the depth a, subjected to far-field tensile loading, cr°. The notched panels are appropriate for brittle materials such as silicon where the notches can be machined using standard etching procedures. Such specimens can be fabricated from thin silicon wafers, or epitaxial silicon on a silicon substrate. Here we will take the notch depth to panel width, a/w, to be small enough so that a becomes the only finite length scale. Since a is the only finite length scale, dimensional considerations lead to the following form for K}'I: K~ "~ ~r°a 1-~.
(13)
Here, 2 - 1 is the order of the mode I stress singularity and the dimensionless factor that multiplies (r°a 1 - ~ in eq. (13) differs for the center-notched and edge-notched specimens. We performed finite element calculations to determine K~ and obtain full-field solutions for the center-notched and double-edge notched specimens. A representative mesh is shown in Fig. 4 for y = 90 °. Due to symmetry only a quarter of the specimen needs to be modeled and the same mesh can be used for both specimens. The boundary conditions for the two specimens are all that differ in the analysis and they are indicated in Fig. 4. Near the corner we use a highlyrefined mesh in order to accurately d e t e r m i n e / ~ . The specimen length to width and a/w ratios are large enough so that a uniform stress state exists in the panel far from the notch. The results
Stress intensities at notch singularities
ttttttttttttt
ttttttttt?ttf
423 Illl 1111 I
i
E
C,qster Notch
Edge N~rb
IIII [IIII IIIII [IIII illll llIll IIIII IIIII IIIII Hill fllll IIIII IIIII Hill IIIII IIIII Hill IIIII ~II!
I1 I1
Hill IIIII IIIII flill IIIII IIIII IIIII IIIII Hill IIIII llIll IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIl I IIIII
Ill III
I I
Ill
i
III III III III I II III III III III III III I II III III III III I II III Ill Ill
I I I I I I I I I I I
III
i
III I II I II I II I li I I I I II III I II I II I II I II III III II I III III I II I II III I II III I II III
I I I I II I I I
Fig. 4. Finite element mesh used to determine K~Ifor the center-notchedand edge-notched specimens, along with boundary conditions for each specimen. are thus fully representative of a notch in an infinite panel. The mesh shown in Fig. 4 contains 1487 8-node quadratic elements and 9316 degrees of freedom. As previously mentioned, there are a handful of approaches that can be used to compute the stress intensities [6, 8-10]. Here we choose to determine K~I by correlating either calculated stresses or displacements with the asymptotic formula of eq. (10) as is done in computational fracture mechanics. In a finite element calculation based on a displacement formulation (the case here), displacements are typically computed more accurately than stresses, and so we use calculated nodal displacements to determine K~I. The most appropriate displacements to use are probably those along the notch flanks, 0 = ___cc For the calculations we performed, the displacement component Uy is greater than uo so we used it to compute K~I. In addition, using Uy is advantageous because it is directly computed and output in the finite element calculation. ~ can then be determined from a least-squares fit (to find C) of: Uy(O = Ol) = Cr )~,
(14)
where again 2 is determined from the asymptotic analysis. K~ is then computed from K~I = C/gy(O = ~) where gy(O = ~) is also determined from the asymptotic analysis and is
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M. L. DUNN et al. i
10.6
10-7 t
I
10 -3
I
I
I
I
I
I
10-2 r (mm)
Fig. 5. Displacement Uy along the notch flank 0 = ct for the edge-notched specimens for various notch angles, ~. The curves, from the bottom to top, are for ), = 0°, 15°, 30°,.... 150°. recorded in Appendix A for both plane stress and plane strain. A plot of uy(cO vs r is shown in Fig. 5 for the various notch angles. For all cases Uy is essentially linear in the region r < 0.01 (on a log-log plot). It is to these curves that eq. (14) was fit to determine K~I. From eq. (13) we define: K ~ = Ytr°a 1-~.
(15)
Computed values of Y are given in Table 1 for both the center-notched and edge-notched specimens.
4. D I S C U S S I O N We begin by looking at the order of the stress singularity 2 - 1 in Fig. 2. The symmetric (mode I) singularity varies from 0 at ~ = 180 ° to -0.5 at 7 = 0°. The singularity is nearly as strong as that for a crack for V up to about 90 °. This suggests that for V less than about 90 °, a critical value of the stress intensity may be successful in correlating failure in mode I loadings, particularly in situations where traditional linear elastic fracture mechanics is successful for cracks. We note that Reedy and Guess [21-23] were able to successfully correlate failure of adhesive butt joints based on a critical free-edge stress intensity factor at an adhesive/adherend interface where an elastic stress singularity of 2 - 1 ~ - 1/3 exists. Their success suggests that the present approach may even be successful for 7 up to about 130 ° where 2 - 1 ~ - 1/3. Over the entire range of 7, the antisymmetric (mode II) stress singularity is much lower than that in mode I. In fact, for ~ > 102.6 ° there is no singularity at all. At 7 = 0° (a crack), 2 - 1 is of course -0.5. At ), = 90 °, 2 - 1 = - 0 . 0 9 1 5 and thus is quite weak. Due to the relatively weak singularity in mode II for all but very small notch angles, the success of a critical value of the mode II stress intensity being able to characterize mode II failure is questionable. On the other hand, the weakness of the mode II singularity as compared to that for mode I suggests that for a given notch angle, mixed mode failure may be predominantly controlled by mode I. If this is the case, mixed-mode failure correlation may turn out to be easier for notches than for cracks. The behavior of the antiplane (mode III) singularity with varying notch angles falls in between that for modes I and II. Like mode I, a singularity exists for the entire range of 0 ° < ~ < 180 °. In Table 1 we present Y (and thus K~I) for both the center-notched and edge-notched geometries. Our results for YEN are within 0.5% of those computed by Theocaris [28] for ~ = 60 °, 90 °, and 120 °. Again, we realize that a direct comparison between Y at different notch angles is
Stress intensities at notch singularities
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Table 1. Order of the corner stress singularity and coefficient of the notch stress intensity factor for infinite centernotched and edge-notched panels. The subscripts CN and EN refer to the center-notched and edge-notched geometries, respectively 7 (degrees) 0 15 30 45 60 75 90 105 120 135 150
2
~,- 1
YCN
YEN
0.5000 0.5002 0.5015 0.5050 0.5122 0.5247 0.5445 0.5739 0.6157 0.6736 0.7520
---0.5000 --0.4998 -0.4985 -0.4950 -0.4878 -0.4753 -0.4555 -0.4261 -0.3843 -0.3264 -0.2480
0.7047 0,7126 0,7271 0.7515 0.7885 0.8350 0.8962 0.9718 1.0625 1.1579 1.2406
0.7914 0.7917 0.8004 0.8145 0.8392 0.8747 0.9252 0,9912 1.0719 1,1617 1.2372
probably not strictly meaningful from a theoretical point of view because the stress state differs for each notch angle not only by the single parameter KTI, but in the angular and radial variation also. Nevertheless, looking at the dependence of K~ on notch angle (realizing that the units of K~I vary with the notch angle) may provide insight into the possibility of correlating failure in cases with different notch angles. More specifically, it is reasonable to ask whether tests performed on specimens with a given non-zero notch angle could be used to estimate the fracture toughness (the critical stress intensity that would be obtained from a test with a notch angle of zero). The question is not only one of academic interest, but also one of practical significance. This is because for reasonably large notch angles, specimen preparation can be accomplished by standard machining techniques resulting in considerable savings in effort as compared to that required to produce specimens with real cracks. From Table 1 we see that for edge-cracked panels, K~ for a notch angle of ~, = 60 ° increases only 6% above that for a crack (7 = 0°). This means that if a test were done with an edge-cracked panel (7 = 0 °) to extract a critical stress intensity factor (fracture toughness), the error that would be introduced in estimating the critical KT~ would only be 6% if the solution for an edge notch of ? = 60 ° were used instead of the solution for an edge crack. This can be looked at in the other direction also. It is possible that if a test were done with a notch of 7 = 60°, the fracture toughness could be estimated. Again, this possible correlation is not rigorously justified because of the different stress states that exist for each notch angle, however, it may prove useful. Indeed, it is probably this line of thinking that is behind the ideas for the ASTM sharp-notch tensile testing standard (ASTM E338, Standard Method of Sharp-Notch Tension Testing of High-Strength Sheet Materials). In this test, panels with sharp edge notches are tested in tension, but no effort is taken to interpret the failures in the context of a critical stress intensity at an elastic singularity. The test, however, is recommended for quantitative ranking of materials based on their resistance to fracture, implying that the results of such tests should be at least qualitatively similar to results for actual fracture tests with cracked panels. As ~ increases beyond 60 °, the difference between YEN(~) and YEN(0°) increases rapidly. The behavior of the stress intensity for the center-notched panel as a function of 7 is similar to that for the edge-notched panel. Note that at 1' = 0°, YEN/YCN = 1.12 which is a well-known result available in most texts on fracture mechanics. As ~ increases YEN and YCN approach each other. In order to be useful as a failure criterion, the region dominated by the asymptotic stress state must be sufficiently large relative to the size of intrinsic flaws, any plastic zone and the fracture process zone. In Fig. 6 we plot the stress aoo(O = 0 °) ahead of the notch tip for the edge-notched panel for three notch angles: ~, = 0 °, 90 ° and 150 °. The plotted stresses are normalized by the nominal stress tr°. Both the asymptotic (solid line) and full-field (open circles) solutions are shown. The stresses from the full-field finite element solutions were obtained by extrapolating calculated stresses in each element from the integration points (where they are most accurate) to the nodes, and then averaging the nodal quantity for each of the connecting elements. As a result these stresses are not as accurate as the computed displacements. Nevertheless, the accuracy is sufficient to infer a qualitative understanding of the behavior of the stress field. We immediately note that although the strength of the singularity (the slope of the asymptotic solution on the log-log plot) decreases as y increases, the distance ahead of the
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101
y=150
°
10o
o
101
OO
II D
10o
. . . . . . . . . . . . . . .
101
10o ............... lO-S 10-2
, ,'%..°,o,
o
, .~°.°..o . . . . . . . . . . . . 10-1 10o 101 r (mm)
Fig. 6. Stress component tr0o on 0 = 0 ahead of the notch tip for the edge-notched specimens. The solid line is the asymptotic solution a00 = /~Ir~ - 1 and the open circles are the full-field finite element solutions.
notch tip that is adequately described by the one-term asymptotic solution increases. This is made more apparent if we chose, albeit somewhat arbitrarily, a criterion to describe when the asymptotic estimate of the stress field is valid. For example, in Fig. 7 we plot the normalized distance ahead of the notch tip r*/a where the asymptotic solution underestimates the full-field solution by at most 10%, vs the notch angle 7. We see that r*/a increases monotonically as y increases, but the increase is small over the range 0 ° < 7 < 90°. As ? increases beyond about 90 ° though, the region dominated by the singular field increases rapidly. The fact that the region dominated by the singular stress field for a notch exceeds that for a crack suggests that for a material and geometry where linear elastic fracture mechanics (LEFM) works, failure correlation based on a critical K~I may also be successful. This argument is particularly appealing for notch angles less than about 90 ° where the strength of the stress singularity 2 - 1 differs little from that for a crack.
427
Stress intensitiesat notch singularities 0.4
0.3
--~
0.2
0.1 J
0.0 0
30
60
90 T (degrees)
120
150
180
Fig. 7. Normalized region r*/a dominated by the singular stress field ahead of the notch tip vs the notch angle for the edge-notchedspecimens. The distance r*/a is defined as the distance within which the asymptoticsolutionagrees with the full-fieldsolution to within 10%.
We can estimate the size of any plastic zone that may exist based on the elastic asymptotic solution in the same way crack-tip plastic zones have been estimated [29]. Assuming no hardening, the size of the plastic zone rp can be estimated as: rp=\K~,]
or
rp=~
,
(16)
where try is the uniaxial yield strength of the material. The first of eq. (16) is obtained by simply setting the elastic asymptotic solution of eq. (10) at 0 = 0 equal to try and solving for r. In reality, the elastic stresses must redistribute so that equilibrium is satisfied. The forces carried by the region between 0 < r < rp must be carried by an increased plastic zone. Applying a simple force balance then leads to the second estimate of eq. (16). A slightly more detailed analysis based on the von Mises effective stress yields the shape of the plastic zone
rp(0) =
( ~-~ ) 1/(h-1)feI(0)1/(1 -~').
(17)
Equation (17) is obtained by setting the von Mises effective stress equal to Cry and solving for r(O). The von Mises effective stress, based on the asymptotic elastic solution, is expressed as ae = I~ilra- lf~(O) and is computed from the stress components in eq. (10) in the standard manner. Note that fXe(O) depends on fIrr(O ), f~o0(O) and flro(O), and differs depending on whether plane stress or plane strain conditions are assumed. Notch-tip plastic zone shapes estimated from the asymptotic elastic solutions are shown in Fig. 8 for both plane strain and plane stress conditions for notch angles of y = 0 °, 45 °, 90 ° and 135 °. Except for the reduced area, the plastic zone shapes for the notch angles are similar to those for the crack, particularly for ~ = 45 ° and 90 °. Ahead of the notch on 0 = 0 °, the plastic zone reaches its greatest extent for the crack, and its length decreases as the notch angle increases. The key point to be made here is that with K~I in hand, an estimate for the plastic zone size can be made. Note that for cracks in hardening materials, the RTi-based estimate for the plastic zone size is conservative for high hardening rates [30]. Again, to be useful as a failure criterion, the region dominated by the singularity must be large relative to the size of inherent flaws, the plastic zone and the fracture process zone. For
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1.5
(a) 1.0 0.5 0.0 0.5
J
1.0 1.5 1.5
(b) 1.0 0.5 0.0 v
0.5 1.0 1.5 Fig. 8. Estimated plastic zone shapes for edge-notched specimens with notch angles of V = 0 °, 45 °, 90 ° and 135 ° for: (a) plane stress and (b) plane strain. The notch tip as defined in Fig. 1 is located at the origin of the polar plot.
micromachined silicon, our micro-tensile tests have shown brittle fracture with no plasticity [26, 27]. Furthermore, scanning and transmission electron microscopy study of the micromachined silicon specimens show no apparent flaws such as cracks. Thus, it appears that for micromachined silicon the inherent flaw size and plastic zone will fall well within the region dominated by the singular stress field. We conclude our discussion by pointing out a consequence of eq. (15) for K~I for the centernotched and edge-notched specimens. If tests are performed with two notched plates with the same notch angle but different notch depths a, and failure occurs when K~i reaches a critical value, then the failure stresses of the two specimens can be related by: O'~cr
= ( a 2 ) x-1 ,
(18)
where the subscripts "1" and "2" refer to the two specimens. The simple relation of eq. (18) results from the fact that the notch depth a is the only finite length scale of the specimens.
Stress intensities at notch singularities
429
5. C O N C L U S I O N S W e a n a l y z e d two c a n d i d a t e specimens for m o d e I fracture testing o f m a t e r i a l s with s h a r p r e - e n t r a n t corners such as m i c r o m a c h i n e d silicon. I n all c a l c u l a t i o n s the w i d t h o f the centern o t c h e d a n d e d g e - n o t c h e d p a n e l s t h a t have been a n a l y z e d is such t h a t the solutions are fully representative o f n o t c h e d infinite panels. A s a result, the o n l y finite length scale that enters the p r o b l e m is the n o t c h d e p t h a a n d so d i m e n s i o n a l c o n s i d e r a t i o n s i m m e d i a t e l y yield the simple f o r m for the stress intensity factor: K ~ = Ytr°a l-x, where the n o n - d i m e n s i o n a l factor Y varies with n o t c h angle a n d specimen geometry. A critical value o f K~ m a y be a p p r o p r i a t e as a failure criterion in situations where s h a r p r e - e n t r a n t corners exist, a n d the region a r o u n d the c o r n e r d o m i n a t e d b y the singular field is large c o m p a r e d to intrinsic flaw sizes, inelastic zones a n d process zone sizes. A c o m p r e h e n s i v e e x p e r i m e n t a l p r o g r a m is required to verify this a n d we are currently p r o c e e d i n g with such efforts. W e showed t h a t the region d o m i n a t e d by the singularity increases as the n o t c h angle increases a n d is smallest for a crack. A t the same time t h o u g h , the strength o f the stress singularity decreases with increasing n o t c h angle. Estimates o f the plastic zone size also decrease with increasing n o t c h angles, t a k i n g o n a m i n i m u m value for a crack. F o r n o t c h angles less t h a n a b o u t 60 °, the stress singularity, the stress intensity factor, the region d o m i n a t e d by the singularity, a n d the e s t i m a t e d plastic zone size differ little f r o m those o f a crack. Acknowledgements--The support of Ford Microelectronics Inc. and the National Science Foundation (CMS-9523073) is
gratefully acknowledged.
REFERENCES 1. Williams, M. L., Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. appl. Mech., 1952, 74, 526-528. 2. Leicester, R. H., Effect of size on the strength of structures. CSIRO Forest Products Laboratory, Division of Building Research, Paper No. 71, 1973. 3. Walsh, P. F., Linear fracture mechanics solutions for zero and right angle notches. CSIRO Forest Products Laboratory, Division of Building Research, Paper No. 2, 1974. 4. Walsh, P. F., Crack initiation in plain concrete. Mag. Concrete Res., 1976, 25, 37~,1. 5. Sinclair, G. B. and Kondo, M., On the stress concentration at sharp re-entrant comers in plates. Int. J. Mech. Sci., 1984, 26, 477-487. 6. Sinclair, G. B., Okajima, M, and Griffin, J. H.. Int. J. numer. Meth. Engng, 1984, 20, 999-1008. 7. Stern, M., Becket, E. B. and Dunham, R-S., A contour integral computation of mixed-mode stress intensity factors. Int. J. Fracture, 1976, 12, 359-368. 8. Carpenter, W. C., Calculation of fracture parameters for a general corner. Int. J. Fracture, 1984, 24, 45-58. 9. Carpenter, W. C., A collocation procedure for determining fracture mechanics parameters at a corner. Int. J. Fracture, 1984, 24, 255-266. 10. Carpenter, W. C., Mode I and Mode II stress intensities for plates with cracks of finite opening. Int. J. Fracture, 1984, 26, 201-214. I I. Carpenter, W. C., The eigenvector Solution for a general comer or finite opening crack with further studies on the collocation procedure. Int. J. Fracture, 1985, 27, 63-74. 12. Sinclair, G. B., A remark on the determination of Mode I and II stress intensity factors for sharp re-entrant corners. Int. J. Fracture, 1985, 27, R81-R85. 13. Neville, D. J., The statistics of fracture at sharp notches. Int. J. Fracture, 1988, 36, 233-239. 14. Gross, B. and Mendelson, A., Plane elastostatics of V-notched plates. Int. J. Fracture Mech., 1972, 8, 267-276. 15. Knesl, Z., A criterion of V-notch stability. Int. J. Fracture, 1991, 48, R79-R83. 16. Carpinteri, A., Stress-singularity and generalized fracture toughness at the vertex of re-entrant corners. Engng Fracture Mech., 1987, 26, 143-155. 17. Reedy, J. E. Jr, Intensity of the stress singularity at the interface corner between a bonded elastic and rigid layer. Engng Fracture Mech., 1990, 38, 575-583. 18. Reedy, J. E. Jr, Intensity of the stress singularity at the interface corner of a bonded elastic layer subjected to shear. Engng Fracture Mech., 1991, 38, 273-281. 19. Reedy, J. E. Jr, Asymptotic interface corner solutions for butt tensile joints. Int. J. Solids Structures, 1993, 30, 767777. 20. Reedy, J. E. Jr, Free-edge stress intensity factor for a bonded ductile layer subjected to shear. J. appl. Mech., 1993, 60, 715-720. 21. Reedy, J. E. Jr and Guess, T. R., Comparison of butt tensile strength data with interface corner stress intensity factor prediction. Int. J. Solids Structures, 1993, 30, 2929-2936. 22. Reedy, J. E. Jr and Guess, T. R., Butt tensile joint strength: interface corner stress intensity factor prediction. J. Adhesion Sci. Technol., 1995, 9, 237-251.
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23. Reedy, J. E., Jr and Guess, T. R., Butt joint strength: effects of residual stress and stress relaxation. J. Adhesion Sci. TechnoL, 1996 (in press). 24. Brown, S., Povirk, G. and Connally, J., Measurement of slow crack growth in silicon and nickel micromechanical devices. In MEMS 93, IEEE, 1993, pp. 99-104. 25. Connally, J. and Brown, S., Micromechanical fatigue testing. Exp, Mech., 1993, 33, 81-90. 26. Cunningham, S. J., Suwito, W. and Read, D. T., Tech. Digests of Transducers, 1995, 2, 56-59. 27. Suwito, W., Dunn, M. L. and Cunningham, S. J., To appear in Transducers 97, Proceedings of The 9th International Conference on Solid-State Sensors and Activators, Chicago, IL, U.S.A., 16-19 June, 1997. 28. Theocaris, P. S. and Ioakimidis, N. I., Mode I stress intensity factors at comer points in plane elastic media. Engng Fracture Mech., 1980, 13, 699-708. 29. Irwin, G. R., Fracture. In Handbuch der Physik VI, ed. Flugge, Springer, 1958, pp. 551-590. 30. Anderson, T. L., Fracture Mechanics, Fundamentals and Applications, 2nd edn. CRC Press, 1995.
APPENDIX A Explicit expressions for the mode I angular-dependent terms in Section 2 are:
flrr(O) ~_ -[CIC4 cos C20 -1- (el q- 2)(;'5 cos ClO] C9
f~(O)
Cz C4 cos C20 - C2C5 cos CIO C9 Cj (Ca sin C20 - Cs sin C10)
flr°(O) =
C9
For plane strain:
~3C2+C2-3C~C6-(~C~+C~)
f~(O)=
C9
giy(O) = (1 + v)[CtC4 sin ~.0 + ~C5 sin(2 - ~.)0 - (3 - 4v)C5 sin ~.0] E~.C9 where E and v are Young's modulus and Poisson's ratio, respectively, and: Cl = 1 - 2 C4 = sin C : t C7 = cos 2C2ct
C 2 = 1 + 2 C5 = sin C : t Ca = cos 2a - cos 2).ct
C3 = 1 - 2 v (;'6 = cos 2 C : t C9 = CIC4 - C2C5
For plane stress, replace Ca with 1 in f~(0), and replace E and v with (1 + 2u)/(l + v)2E and v/(1 + v) in g~,(0), respectively.
(Received 15 October 1996, in final form 10 January 1997, accepted 11 January 1997)
Pergamon
STRESS
INTENSITIES
AT
NOTCH
SINGULARITIES
MARTIN L. DUNN and WAN SUWlTO Center for Acoustics, Mechanics and Materials, Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309-0427, U.S.A. SHAWN CUNNINGHAM Ford Microelectronics Inc., Colorado Springs, Colorado 80921, U.S.A. Abstract--In the context of linear elasticity, a stress singularity of the type K~r6 (3 < 0) may exist at re-entrant notch comers, with an intensity/~. The magnitude of the intensity, referred to here as the notch stress intensity, characterizes the stress state in the region of the notch tip. We determined/(' for two mode 1-type specimens that are proposed to characterize a critical value of K~. Such a critical Kn may be appropriate as a failure criterion in situations where sharp re-entrant comers exist, and the region around the corner dominated by the singular field is large compared to intrinsic flaw sizes, inelastic zones and process zone sizes. For these specimens we show that the region dominated by the singularity increases as the notch angle increases and is smallest for the limiting case of a crack. At the same time though, the strength of the stress singularity decreases with increasing notch angle. Estimates of the plastic zone size also decrease with increasing notch angles. For notch angles less than about 60°, the stress singularity, stress intensity factor, region dominated by the singularity and the estimated plastic zone size do not differ substantially from those of a crack. © 1997 Elsevier Science Ltd
1. I N T R O D U C T I O N PREDICTION o f failure in solids typically proceeds f r o m one o f two points o f view: stress-strength or stress intensity-fracture toughness. The former entails c o m p u t i n g stresses where failure is anticipated, typically at sites o f stress concentration, and using the c o m p u t e d stresses in a failure criteria based on measured strengths, m o s t c o m m o n l y the yield or ultimate strength in uniaxial tension. The latter entails c o m p u t i n g the stress intensity at the tip o f a sharp crack (of course, singular stresses are predicted by elasticity theory), and using the c o m p u t e d stress intensity in a failure criteria based on a measured fracture toughness, which is presumed to be a material property. A l t h o u g h b o t h approaches apparently proceed f r o m quite different points o f view, a relation between the two can be seen by considering the stress analysis o f sharp notches. Williams[l] showed that in the context o f elasticity theory the asymptotic stress state near a corner or notch depends only on the notch geometry and is often singular. If we consider a polar coordinate system centered at the tip o f a notch o f angle V as shown in Fig. 1, the stresses near the notch tip can be expressed as tr~/~ = Kr ~ - l f ~ ( O ) . The order o f the stress singularity 2 - 1 and the angular function f~a(O) for a given angle 7 can be completely determined by an asymptotic analysis. The stress intensity K is a function o f the geometry o f the solid and the far-field loading. Thus, for a given geometry and loading, K completely characterizes the stress state in a region near the notch tip. The limiting case o f the notch angle ~ = 0 corresponds to a crack. This results in 2 - 1 = - 1 / 2 and on 0 = 0, a00 = K r-1/2, where K is the widely-used stress intensity factor o f linear elastic fracture mechanics. The case o f 7 = n corresponds to a solid with no notch and ~ . - 1 = 0. F o r uniaxial loading o f an edge-notched strip with 7 = n, croo = K on 0 = 0. I f failure is presumed to occur at a critical value o f K in both o f these cases, then in the former the critical K is the fracture toughness o f the material while in the latter it is the uniaxial strength. Based on the above discussion, the two approaches to failure analysis can be reconciled if we consider the analysis o f notched solids and presume, irrespective o f the notch angle, that failure occurs when K reaches a critical value. O f course, only for the limiting case o f a crack is there a rigorous connection between the stress intensity factor and the energy release rate o f fracture mechanics. Thus a critical K failure criterion appears to be suitable only for fracture initiation. Interestingly, very little w o r k has been done in the middle o f these two extremes. There are p r o b a b l y a couple o f reasons for this. In structural design, it is widely appreciated that sharp corners lead to high stresses and g o o d design proceeds by rounding corners. 417
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y
r
0 1'
-
-
X
Fig. 1. Geometry of the c o m e r considered in this study.
This reduces the problem from one of stress intensities to one of stress concentrations, where elasticity theory can be applied. Fracture mechanics plays a role in the case where cracks occur, not by design, but as a result of service. Indeed, in practice, it is common to drill holes at the tips of cracks in a structure. Again, this reduces the problem from one of stress intensities to one of stress concentrations, but during the life of the structure, not the design stage. Thus, it is rare in traditional structural design to be faced with sharp re-entrant corners. We either have intentionally rounded corners or unintentional sharp cracks. However, an exception arises when dealing with microelectromechanical structures that are machined, for example from silicon, by anisotropic chemical etching processes. The resulting structures have sharp corners formed by intersections of crystallographic planes. Appropriate approaches to failure prediction are needed for these structures and a critical K criterion appears reasonable as with these brittle materials stable crack propagation is rarely observed and an initiation failure criterion seems particularly well-suited. There are a handful of related studies that are pertinent to our work. We focus only on those concerned with computing the stress intensity at a sharp corner and do not mention the many studies directed at computing only the order of the stress singularity. Perhaps the earliest studies concerning a generalized fracture mechanics approach for notched materials were aimed at wood structures with sharp reentrant corners, and specifically at scaling effects with different size specimens [2--4]. Sinclair and Kondo [5] discussed the idea of a generalized stress intensity at sharp corners, but did not attempt to correlate fracture with the stress intensity, opting instead to pursue a stress-concentration approach. Sinclair et al. [6] also outlined an approach to obtain the generalized stress intensity at sharp corners based on the evaluation of a contour integral, which is an extension of the work of Stern et al. [7]. Independently, Carpenter [8] also applied the reciprocal work contour integral method of Stern et al. [7] to compute fracture mechanics parameters for a general corner. A series of subsequent papers by Carpenter[9-11] studied the problem further and introduces a collocation approach to compute stress intensities. The application of the approaches of Sinclair et al. [6] and Carpenter [8] to mixed-mode loading problems are further discussed by Sinclair [12]. Neville [13] proposed a statistical approach for failure prediction that makes use of the singularity at a sharp notch. Gross and Mendelson [14] studied stress intensities at notch tips from the viewpoint of addressing crack-notch configurations such as those that occur after fatigue precracking of standard fracture specimens, and understanding how crack-notch configurations influence measured fracture toughness. The stability of notched solids was studied by Knesl[15] who introduced as a length scale a distance ahead of the notch tip over which an average opening stress is computed. Perhaps the most detailed study of stress
Stress intensities at notch singularities
419
intensities at sharp corners is that of Carpinteri [16]. Carpinteri performed three-point flexure tests with notched bars and attempted to experimentally obtain the connection between a critical stress intensity at various angles and structure sizes. Although it may be possible to empirically correlate the critical stress intensity at various angles, it may not be formally appropriate. This is because although for a given notch angle the asymptotic stress field is universal, the universal field differs for different angles. For a given notch angle, regardless of the geometry of the structure containing the notch and regardless of the external loading, the complicated stress state at the notch tip is fully characterized by the magnitude of the generalized stress intensity. The proposition that a critical stress intensity can be used as a failure criterion really means that the stress state at the notch is presumed to cause failure. Thus, for two specimens with different geometry and loading, but with the same notch angle, the stress state at the notch tip is the same at failure. This stress state is different, though, for specimens with different notch angles, thus casting doubt on the possibility of a rigorous connection between the critical stress intensities at different notch angles. Indeed as an extreme case, such a connection would imply a rigorous connection between the fracture toughness and the uniaxial strength of a material. Nevertheless, it may prove valuable in practice to be able to correlate critical stress intensities at different notch angles. Despite the different viewpoints adopted in the above-mentioned studies, they all lend support to the use of a critical generalized stress intensity to correlate fracture of structures with sharp notches. Further support for such an approach is provided by the recent studies of Reedy [17-20] and Reedy and Guess [21-23]. They correlated failure of adhesive-bonded butt tensile joints with a generalized stress intensity at the free edge of an interface corner between elastic and rigid layers. In their studies the order of the elastic singularity is about 2 - 1 = - 1/3, as compared to 2 - 1 = - 1/2 for a crack. The success of these approaches has motivated us to attempt to use a critical stress intensity at re-entrant notch corners to correlate failure of micromachined silicon structures which often posses sharp corners due to etching. Over the last few years applications of micromachined silicon structures have increased considerably. The fabrication of silicon structures proceeds using technology developed in the integrated circuit industry: thin film deposition, photolithography and etching. In many applications the resulting structure contains sharp re-entrant corners. This is a result of the etching process that attacks various crystallographic planes at dramatically different rates. Failure mechanisms for these structures are not well understood and appropriate methodologies for predicting failure are needed. At the device or structure level, designers require knowledge of mechanical properties such as stiffness, strength and fracture toughness to proceed with rational design. Currently designers compute stresses in structures using finite element methods. The use of these results for quantitative failure prediction is problematic because predicted stresses near the sharp corner depend on the finite element idealization, and due to the existence of the elastic singularity, convergence in the stresses is not obtained with mesh refinement. In addition, both the necessary material properties and appropriate failure criteria are lacking. Only a few studies have even considered fracture of micromachined silicon [24, 25]. These have focused on measuring fracture toughness and assessing the influence of environment on crack growth. Of course, a pre-existing crack is necessary. Our studies of micromachined silicon structures seem to indicate that in many cases neither the traditional stress-strength or stress intensity-fracture toughness approaches are appropriate for describing failure. We have found the former problematic for a couple of reasons. First, it is very difficult to even measure a uniaxial tensile strength because failure always occurs at a stress raiser [26]. Even if a true uniaxial strength can be obtained, it is not clear how it would be of use in micromachined structures where failure often occurs at sharp corners. Since the epitaxial silicon of our studies is very brittle, it might be appealing to assume the existence of a very small flaw so that fracture mechanics can be applied. This approach, however, does not appear to be realistic. Scanning and transmission electron microscopy studies have failed to reveal any cracks at the sharp corners of our structures. To overcome these difficulties, we are pursuing an approach that takes advantage of the elastic singularity at sharp corners that exist in many micromachined structures[27]. In this paper we lay out our initial efforts in the development and application of a generalized stress intensity approach for correlating failure of micromachined silicon structures. We
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et al.
focus on isotropic elastic solids. The extension to anisotropic solids representative of single crystal silicon, for example, will be presented in a future study. We use the asymptotic analysis of Williams[l] to determine the order of the elastic singularity at a sharp corner. From the asymptotic analysis we obtain the order of the singularity and the angular dependence of the stress and displacement fields. These fully determine the stress state at the notch tip within an arbitrary constant, the notch stress intensity K", which is a function of the geometry of the structure and the loading. We obtain the stress intensity by detailed finite element calculations for two specimens proposed to obtain a critical value of K": a center-notched and a double-edge-notched tensile specimen. The solutions are obtained for a ratio of notch depth to plate width representative of a notch in an infinite plate. The only finite length scale that enters the problem is then the notch width, implying a simple connection between strengths of specimens with varying notch widths.
2. ASYMPTOTIC ANALYSIS OF SINGULAR FIELDS Consider the notched body shown in Fig. 1. It is loaded by tractions on remote boundaries and the surfaces 0 = __+~ are traction free. In the following, we redundantly, but conveniently, refer to 7 as the notch angle where ? is related to ct by ?/2 = n - ~. For simplicity we assume linear isotropic material behavior. The asymptotic, possibly singular, fields near the corner can be obtained using Williams' [1] eigenfunction expansion method for both the plane and antiplane problems. For the plane problem, the appropriate Airy stress function is:
q~ = r~+lf(O),
(1)
f(O) = Ai cos(~. + 1)0 + A2 sin(X + 1)0 + A3 cos(X - 1)0 + A4 sin(Z - 1)0.
(2)
where:
In the standard manner, the stresses can be obtained from ~b as: O'rr =
rk-l[(~, -st- 1)f(0) +f"(0)]
0"00 = ff'k-l~o~ + l)f(O) OrO = - - r g - l ~ , f
t(0),
(3)
where a prime denotes differentiation with respect to 0. Note that 2 > 0 for boundedness of the strain energy and 2 < 1 for singular fields. Imposing traction-free boundary conditions on 0 = + ~ results in a system of four equations for the five unknowns Af and 2. A nontrivial solution requires that the determinant of the coefficient matrix vanishes. Upon imposing this requirement, two equations result, one for symmetric fields with respect to the x-axis (0 = 0), and one for antisymmetric fields[l]: ), sin 2or + sin 2~.a = 0 ~. sin 2or - sin 2~.a = 0
for symmetric fields
(4)
for antisymmetric fields.
(5)
Borrowing from the terminology of fracture mechanics we refer to the symmetric and antisymmetric fields as mode I and mode II fields, respectively. For the antiplane (mode III) problem, the field equations reduce to V2w = 0 and admit an eigenfunction expansion:
Izw = r~f (O),
(6)
f(O) = Al cos ~.0 + A2 sin X0.
(7)
where # is the shear modulus and:
The shear stresses are: Orz .~
r~-I~.f(O)
Stress intensities at notch singularities
421
0.0 -
-
In-plane(synmaetric)
..'
. . . . . . . . In-plane (antisymmetric) Antiplane
-0.1
,
/
"
"~/ )'"
.-,
,-. /
-0.2
/
,y'"
''
,
I
"Y
/
/
-0.3
-0.4
"'"
/"
.,- / . - ' /
-0.5
'' 0
'' . 20
. 40
. 60
. . 80 100 T (degrees)
120
~ ...... 140 160
180
Fig. 2. Order of the stress singularity 2 - 1 as a function of the notch angle 7 for modes I, II and Ill.
(8)
aoz = ? - ~ f ' ( o ) .
Imposing traction-free boundary conditions on the notch faces yields two equations for the three unknowns Ai and 2. These result in the following equation for 2: cos 2or = 0
for antiplane fields.
(9)
The order of the stress singularity 2 - 1 is plotted in Fig. 2 for the three modes as a function of 7. Once 2 is obtained from eq. (4), eq. (5) or eq. (9), the coefficients Ai can not be completely determined. They can, however, be expressed in terms of only one Ai. We normalize the system so that a00(0 = 0) = /~ir ~ - 1, arO(0 = 0) = K'i'nr~ - 1 and azo(O = 0) = K~nir~ - 1 under mode I, II and III loading, respectively. This allows us to write the singular stresses and displacements as: Mode I
tra# = K ~rX-lf le(o ) Ua = K ~r~gla(O)
(10)
Mode II a~e = K ~irX-lf ~(0) II n
II
u~ = K i i r ~ g a (0)
(11)
Mode III a~# = K i n r - 7
Illoa-~
w = K ~nrXgz(O).
(12)
In eqs (10)-(12) ~ and fl take on the values (x,y,z) in cartesian coordinates and (r,O,z) in cylindrical coordinates. 2, f~/~(O) and g=(O) are determined by the asymptotic analysis. 2 and f,/~(O) are functions of ~, and g,(O) are functions of • and the elastic properties of the material. f,#(O) and g,(O) are tabulated in Appendix A for mode I. The stress intensity K" is a function of the specific geometry and loading. Its magnitude fully determines the stress state at a notch tip
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M. L. DUNN et al.
(a)
co
(b)
cyo
Y
J
(yo
co
Fig. 3. Geometry of tensile ~ specimens: (a) center-notched and (b) edge-notched.
for a given c~. Note that for a notch angle of y = 90 °, 2 - 1 = - 0.4555, -0.0915 and -0.3333 for modes I, II, and III, respectively. Thus, the mode I singularity is nearly as strong as that for a crack, while the mode II singularity is quite weak. This suggests that in opening, and perhaps in antiplane situations, a critical value of K" might be used successfully as a failure criterion. If so, we can measure a critical value o f / C ' using a simple test geometry, then use the critical K ~ so obtained to predict failure in practical structures. The analyses of two specimens proposed for extracting a critical K ~ are presented in the following section. 3. M O D E I S P E C I M E N S : / C AND F U L L - F I E L D FINITE E L E M E N T S O L U T I O N S
We focus on two potential specimens designed to determine the mode I stress intensity factor K~I, deferring the consideration of mode II (and mode III) specimens to a later date. We emphasize mode I because the singularity is much stronger than that for mode II for geometries of practical interest with y ,-~ 90 ° and so mode I failure is thought to dominate. The two specimens studied here are shown in Fig. 3; center-notched and edge-notched panels. These are both thin panels with notch geometries defined by the angle Y and the depth a, subjected to far-field tensile loading, cr°. The notched panels are appropriate for brittle materials such as silicon where the notches can be machined using standard etching procedures. Such specimens can be fabricated from thin silicon wafers, or epitaxial silicon on a silicon substrate. Here we will take the notch depth to panel width, a/w, to be small enough so that a becomes the only finite length scale. Since a is the only finite length scale, dimensional considerations lead to the following form for K}'I: K~ "~ ~r°a 1-~.
(13)
Here, 2 - 1 is the order of the mode I stress singularity and the dimensionless factor that multiplies (r°a 1 - ~ in eq. (13) differs for the center-notched and edge-notched specimens. We performed finite element calculations to determine K~ and obtain full-field solutions for the center-notched and double-edge notched specimens. A representative mesh is shown in Fig. 4 for y = 90 °. Due to symmetry only a quarter of the specimen needs to be modeled and the same mesh can be used for both specimens. The boundary conditions for the two specimens are all that differ in the analysis and they are indicated in Fig. 4. Near the corner we use a highlyrefined mesh in order to accurately d e t e r m i n e / ~ . The specimen length to width and a/w ratios are large enough so that a uniform stress state exists in the panel far from the notch. The results
Stress intensities at notch singularities
ttttttttttttt
ttttttttt?ttf
423 Illl 1111 I
i
E
C,qster Notch
Edge N~rb
IIII [IIII IIIII [IIII illll llIll IIIII IIIII IIIII Hill fllll IIIII IIIII Hill IIIII IIIII Hill IIIII ~II!
I1 I1
Hill IIIII IIIII flill IIIII IIIII IIIII IIIII Hill IIIII llIll IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIl I IIIII
Ill III
I I
Ill
i
III III III III I II III III III III III III I II III III III III I II III Ill Ill
I I I I I I I I I I I
III
i
III I II I II I II I li I I I I II III I II I II I II I II III III II I III III I II I II III I II III I II III
I I I I II I I I
Fig. 4. Finite element mesh used to determine K~Ifor the center-notchedand edge-notched specimens, along with boundary conditions for each specimen. are thus fully representative of a notch in an infinite panel. The mesh shown in Fig. 4 contains 1487 8-node quadratic elements and 9316 degrees of freedom. As previously mentioned, there are a handful of approaches that can be used to compute the stress intensities [6, 8-10]. Here we choose to determine K~I by correlating either calculated stresses or displacements with the asymptotic formula of eq. (10) as is done in computational fracture mechanics. In a finite element calculation based on a displacement formulation (the case here), displacements are typically computed more accurately than stresses, and so we use calculated nodal displacements to determine K~I. The most appropriate displacements to use are probably those along the notch flanks, 0 = ___cc For the calculations we performed, the displacement component Uy is greater than uo so we used it to compute K~I. In addition, using Uy is advantageous because it is directly computed and output in the finite element calculation. ~ can then be determined from a least-squares fit (to find C) of: Uy(O = Ol) = Cr )~,
(14)
where again 2 is determined from the asymptotic analysis. K~ is then computed from K~I = C/gy(O = ~) where gy(O = ~) is also determined from the asymptotic analysis and is
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10.6
10-7 t
I
10 -3
I
I
I
I
I
I
10-2 r (mm)
Fig. 5. Displacement Uy along the notch flank 0 = ct for the edge-notched specimens for various notch angles, ~. The curves, from the bottom to top, are for ), = 0°, 15°, 30°,.... 150°. recorded in Appendix A for both plane stress and plane strain. A plot of uy(cO vs r is shown in Fig. 5 for the various notch angles. For all cases Uy is essentially linear in the region r < 0.01 (on a log-log plot). It is to these curves that eq. (14) was fit to determine K~I. From eq. (13) we define: K ~ = Ytr°a 1-~.
(15)
Computed values of Y are given in Table 1 for both the center-notched and edge-notched specimens.
4. D I S C U S S I O N We begin by looking at the order of the stress singularity 2 - 1 in Fig. 2. The symmetric (mode I) singularity varies from 0 at ~ = 180 ° to -0.5 at 7 = 0°. The singularity is nearly as strong as that for a crack for V up to about 90 °. This suggests that for V less than about 90 °, a critical value of the stress intensity may be successful in correlating failure in mode I loadings, particularly in situations where traditional linear elastic fracture mechanics is successful for cracks. We note that Reedy and Guess [21-23] were able to successfully correlate failure of adhesive butt joints based on a critical free-edge stress intensity factor at an adhesive/adherend interface where an elastic stress singularity of 2 - 1 ~ - 1/3 exists. Their success suggests that the present approach may even be successful for 7 up to about 130 ° where 2 - 1 ~ - 1/3. Over the entire range of 7, the antisymmetric (mode II) stress singularity is much lower than that in mode I. In fact, for ~ > 102.6 ° there is no singularity at all. At 7 = 0° (a crack), 2 - 1 is of course -0.5. At ), = 90 °, 2 - 1 = - 0 . 0 9 1 5 and thus is quite weak. Due to the relatively weak singularity in mode II for all but very small notch angles, the success of a critical value of the mode II stress intensity being able to characterize mode II failure is questionable. On the other hand, the weakness of the mode II singularity as compared to that for mode I suggests that for a given notch angle, mixed mode failure may be predominantly controlled by mode I. If this is the case, mixed-mode failure correlation may turn out to be easier for notches than for cracks. The behavior of the antiplane (mode III) singularity with varying notch angles falls in between that for modes I and II. Like mode I, a singularity exists for the entire range of 0 ° < ~ < 180 °. In Table 1 we present Y (and thus K~I) for both the center-notched and edge-notched geometries. Our results for YEN are within 0.5% of those computed by Theocaris [28] for ~ = 60 °, 90 °, and 120 °. Again, we realize that a direct comparison between Y at different notch angles is
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Table 1. Order of the corner stress singularity and coefficient of the notch stress intensity factor for infinite centernotched and edge-notched panels. The subscripts CN and EN refer to the center-notched and edge-notched geometries, respectively 7 (degrees) 0 15 30 45 60 75 90 105 120 135 150
2
~,- 1
YCN
YEN
0.5000 0.5002 0.5015 0.5050 0.5122 0.5247 0.5445 0.5739 0.6157 0.6736 0.7520
---0.5000 --0.4998 -0.4985 -0.4950 -0.4878 -0.4753 -0.4555 -0.4261 -0.3843 -0.3264 -0.2480
0.7047 0,7126 0,7271 0.7515 0.7885 0.8350 0.8962 0.9718 1.0625 1.1579 1.2406
0.7914 0.7917 0.8004 0.8145 0.8392 0.8747 0.9252 0,9912 1.0719 1,1617 1.2372
probably not strictly meaningful from a theoretical point of view because the stress state differs for each notch angle not only by the single parameter KTI, but in the angular and radial variation also. Nevertheless, looking at the dependence of K~ on notch angle (realizing that the units of K~I vary with the notch angle) may provide insight into the possibility of correlating failure in cases with different notch angles. More specifically, it is reasonable to ask whether tests performed on specimens with a given non-zero notch angle could be used to estimate the fracture toughness (the critical stress intensity that would be obtained from a test with a notch angle of zero). The question is not only one of academic interest, but also one of practical significance. This is because for reasonably large notch angles, specimen preparation can be accomplished by standard machining techniques resulting in considerable savings in effort as compared to that required to produce specimens with real cracks. From Table 1 we see that for edge-cracked panels, K~ for a notch angle of ~, = 60 ° increases only 6% above that for a crack (7 = 0°). This means that if a test were done with an edge-cracked panel (7 = 0 °) to extract a critical stress intensity factor (fracture toughness), the error that would be introduced in estimating the critical KT~ would only be 6% if the solution for an edge notch of ? = 60 ° were used instead of the solution for an edge crack. This can be looked at in the other direction also. It is possible that if a test were done with a notch of 7 = 60°, the fracture toughness could be estimated. Again, this possible correlation is not rigorously justified because of the different stress states that exist for each notch angle, however, it may prove useful. Indeed, it is probably this line of thinking that is behind the ideas for the ASTM sharp-notch tensile testing standard (ASTM E338, Standard Method of Sharp-Notch Tension Testing of High-Strength Sheet Materials). In this test, panels with sharp edge notches are tested in tension, but no effort is taken to interpret the failures in the context of a critical stress intensity at an elastic singularity. The test, however, is recommended for quantitative ranking of materials based on their resistance to fracture, implying that the results of such tests should be at least qualitatively similar to results for actual fracture tests with cracked panels. As ~ increases beyond 60 °, the difference between YEN(~) and YEN(0°) increases rapidly. The behavior of the stress intensity for the center-notched panel as a function of 7 is similar to that for the edge-notched panel. Note that at 1' = 0°, YEN/YCN = 1.12 which is a well-known result available in most texts on fracture mechanics. As ~ increases YEN and YCN approach each other. In order to be useful as a failure criterion, the region dominated by the asymptotic stress state must be sufficiently large relative to the size of intrinsic flaws, any plastic zone and the fracture process zone. In Fig. 6 we plot the stress aoo(O = 0 °) ahead of the notch tip for the edge-notched panel for three notch angles: ~, = 0 °, 90 ° and 150 °. The plotted stresses are normalized by the nominal stress tr°. Both the asymptotic (solid line) and full-field (open circles) solutions are shown. The stresses from the full-field finite element solutions were obtained by extrapolating calculated stresses in each element from the integration points (where they are most accurate) to the nodes, and then averaging the nodal quantity for each of the connecting elements. As a result these stresses are not as accurate as the computed displacements. Nevertheless, the accuracy is sufficient to infer a qualitative understanding of the behavior of the stress field. We immediately note that although the strength of the singularity (the slope of the asymptotic solution on the log-log plot) decreases as y increases, the distance ahead of the
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101
y=150
°
10o
o
101
OO
II D
10o
. . . . . . . . . . . . . . .
101
10o ............... lO-S 10-2
, ,'%..°,o,
o
, .~°.°..o . . . . . . . . . . . . 10-1 10o 101 r (mm)
Fig. 6. Stress component tr0o on 0 = 0 ahead of the notch tip for the edge-notched specimens. The solid line is the asymptotic solution a00 = /~Ir~ - 1 and the open circles are the full-field finite element solutions.
notch tip that is adequately described by the one-term asymptotic solution increases. This is made more apparent if we chose, albeit somewhat arbitrarily, a criterion to describe when the asymptotic estimate of the stress field is valid. For example, in Fig. 7 we plot the normalized distance ahead of the notch tip r*/a where the asymptotic solution underestimates the full-field solution by at most 10%, vs the notch angle 7. We see that r*/a increases monotonically as y increases, but the increase is small over the range 0 ° < 7 < 90°. As ? increases beyond about 90 ° though, the region dominated by the singular field increases rapidly. The fact that the region dominated by the singular stress field for a notch exceeds that for a crack suggests that for a material and geometry where linear elastic fracture mechanics (LEFM) works, failure correlation based on a critical K~I may also be successful. This argument is particularly appealing for notch angles less than about 90 ° where the strength of the stress singularity 2 - 1 differs little from that for a crack.
427
Stress intensitiesat notch singularities 0.4
0.3
--~
0.2
0.1 J
0.0 0
30
60
90 T (degrees)
120
150
180
Fig. 7. Normalized region r*/a dominated by the singular stress field ahead of the notch tip vs the notch angle for the edge-notchedspecimens. The distance r*/a is defined as the distance within which the asymptoticsolutionagrees with the full-fieldsolution to within 10%.
We can estimate the size of any plastic zone that may exist based on the elastic asymptotic solution in the same way crack-tip plastic zones have been estimated [29]. Assuming no hardening, the size of the plastic zone rp can be estimated as: rp=\K~,]
or
rp=~
,
(16)
where try is the uniaxial yield strength of the material. The first of eq. (16) is obtained by simply setting the elastic asymptotic solution of eq. (10) at 0 = 0 equal to try and solving for r. In reality, the elastic stresses must redistribute so that equilibrium is satisfied. The forces carried by the region between 0 < r < rp must be carried by an increased plastic zone. Applying a simple force balance then leads to the second estimate of eq. (16). A slightly more detailed analysis based on the von Mises effective stress yields the shape of the plastic zone
rp(0) =
( ~-~ ) 1/(h-1)feI(0)1/(1 -~').
(17)
Equation (17) is obtained by setting the von Mises effective stress equal to Cry and solving for r(O). The von Mises effective stress, based on the asymptotic elastic solution, is expressed as ae = I~ilra- lf~(O) and is computed from the stress components in eq. (10) in the standard manner. Note that fXe(O) depends on fIrr(O ), f~o0(O) and flro(O), and differs depending on whether plane stress or plane strain conditions are assumed. Notch-tip plastic zone shapes estimated from the asymptotic elastic solutions are shown in Fig. 8 for both plane strain and plane stress conditions for notch angles of y = 0 °, 45 °, 90 ° and 135 °. Except for the reduced area, the plastic zone shapes for the notch angles are similar to those for the crack, particularly for ~ = 45 ° and 90 °. Ahead of the notch on 0 = 0 °, the plastic zone reaches its greatest extent for the crack, and its length decreases as the notch angle increases. The key point to be made here is that with K~I in hand, an estimate for the plastic zone size can be made. Note that for cracks in hardening materials, the RTi-based estimate for the plastic zone size is conservative for high hardening rates [30]. Again, to be useful as a failure criterion, the region dominated by the singularity must be large relative to the size of inherent flaws, the plastic zone and the fracture process zone. For
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1.5
(a) 1.0 0.5 0.0 0.5
J
1.0 1.5 1.5
(b) 1.0 0.5 0.0 v
0.5 1.0 1.5 Fig. 8. Estimated plastic zone shapes for edge-notched specimens with notch angles of V = 0 °, 45 °, 90 ° and 135 ° for: (a) plane stress and (b) plane strain. The notch tip as defined in Fig. 1 is located at the origin of the polar plot.
micromachined silicon, our micro-tensile tests have shown brittle fracture with no plasticity [26, 27]. Furthermore, scanning and transmission electron microscopy study of the micromachined silicon specimens show no apparent flaws such as cracks. Thus, it appears that for micromachined silicon the inherent flaw size and plastic zone will fall well within the region dominated by the singular stress field. We conclude our discussion by pointing out a consequence of eq. (15) for K~I for the centernotched and edge-notched specimens. If tests are performed with two notched plates with the same notch angle but different notch depths a, and failure occurs when K~i reaches a critical value, then the failure stresses of the two specimens can be related by: O'~cr
= ( a 2 ) x-1 ,
(18)
where the subscripts "1" and "2" refer to the two specimens. The simple relation of eq. (18) results from the fact that the notch depth a is the only finite length scale of the specimens.
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5. C O N C L U S I O N S W e a n a l y z e d two c a n d i d a t e specimens for m o d e I fracture testing o f m a t e r i a l s with s h a r p r e - e n t r a n t corners such as m i c r o m a c h i n e d silicon. I n all c a l c u l a t i o n s the w i d t h o f the centern o t c h e d a n d e d g e - n o t c h e d p a n e l s t h a t have been a n a l y z e d is such t h a t the solutions are fully representative o f n o t c h e d infinite panels. A s a result, the o n l y finite length scale that enters the p r o b l e m is the n o t c h d e p t h a a n d so d i m e n s i o n a l c o n s i d e r a t i o n s i m m e d i a t e l y yield the simple f o r m for the stress intensity factor: K ~ = Ytr°a l-x, where the n o n - d i m e n s i o n a l factor Y varies with n o t c h angle a n d specimen geometry. A critical value o f K~ m a y be a p p r o p r i a t e as a failure criterion in situations where s h a r p r e - e n t r a n t corners exist, a n d the region a r o u n d the c o r n e r d o m i n a t e d b y the singular field is large c o m p a r e d to intrinsic flaw sizes, inelastic zones a n d process zone sizes. A c o m p r e h e n s i v e e x p e r i m e n t a l p r o g r a m is required to verify this a n d we are currently p r o c e e d i n g with such efforts. W e showed t h a t the region d o m i n a t e d by the singularity increases as the n o t c h angle increases a n d is smallest for a crack. A t the same time t h o u g h , the strength o f the stress singularity decreases with increasing n o t c h angle. Estimates o f the plastic zone size also decrease with increasing n o t c h angles, t a k i n g o n a m i n i m u m value for a crack. F o r n o t c h angles less t h a n a b o u t 60 °, the stress singularity, the stress intensity factor, the region d o m i n a t e d by the singularity, a n d the e s t i m a t e d plastic zone size differ little f r o m those o f a crack. Acknowledgements--The support of Ford Microelectronics Inc. and the National Science Foundation (CMS-9523073) is
gratefully acknowledged.
REFERENCES 1. Williams, M. L., Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. appl. Mech., 1952, 74, 526-528. 2. Leicester, R. H., Effect of size on the strength of structures. CSIRO Forest Products Laboratory, Division of Building Research, Paper No. 71, 1973. 3. Walsh, P. F., Linear fracture mechanics solutions for zero and right angle notches. CSIRO Forest Products Laboratory, Division of Building Research, Paper No. 2, 1974. 4. Walsh, P. F., Crack initiation in plain concrete. Mag. Concrete Res., 1976, 25, 37~,1. 5. Sinclair, G. B. and Kondo, M., On the stress concentration at sharp re-entrant comers in plates. Int. J. Mech. Sci., 1984, 26, 477-487. 6. Sinclair, G. B., Okajima, M, and Griffin, J. H.. Int. J. numer. Meth. Engng, 1984, 20, 999-1008. 7. Stern, M., Becket, E. B. and Dunham, R-S., A contour integral computation of mixed-mode stress intensity factors. Int. J. Fracture, 1976, 12, 359-368. 8. Carpenter, W. C., Calculation of fracture parameters for a general corner. Int. J. Fracture, 1984, 24, 45-58. 9. Carpenter, W. C., A collocation procedure for determining fracture mechanics parameters at a corner. Int. J. Fracture, 1984, 24, 255-266. 10. Carpenter, W. C., Mode I and Mode II stress intensities for plates with cracks of finite opening. Int. J. Fracture, 1984, 26, 201-214. I I. Carpenter, W. C., The eigenvector Solution for a general comer or finite opening crack with further studies on the collocation procedure. Int. J. Fracture, 1985, 27, 63-74. 12. Sinclair, G. B., A remark on the determination of Mode I and II stress intensity factors for sharp re-entrant corners. Int. J. Fracture, 1985, 27, R81-R85. 13. Neville, D. J., The statistics of fracture at sharp notches. Int. J. Fracture, 1988, 36, 233-239. 14. Gross, B. and Mendelson, A., Plane elastostatics of V-notched plates. Int. J. Fracture Mech., 1972, 8, 267-276. 15. Knesl, Z., A criterion of V-notch stability. Int. J. Fracture, 1991, 48, R79-R83. 16. Carpinteri, A., Stress-singularity and generalized fracture toughness at the vertex of re-entrant corners. Engng Fracture Mech., 1987, 26, 143-155. 17. Reedy, J. E. Jr, Intensity of the stress singularity at the interface corner between a bonded elastic and rigid layer. Engng Fracture Mech., 1990, 38, 575-583. 18. Reedy, J. E. Jr, Intensity of the stress singularity at the interface corner of a bonded elastic layer subjected to shear. Engng Fracture Mech., 1991, 38, 273-281. 19. Reedy, J. E. Jr, Asymptotic interface corner solutions for butt tensile joints. Int. J. Solids Structures, 1993, 30, 767777. 20. Reedy, J. E. Jr, Free-edge stress intensity factor for a bonded ductile layer subjected to shear. J. appl. Mech., 1993, 60, 715-720. 21. Reedy, J. E. Jr and Guess, T. R., Comparison of butt tensile strength data with interface corner stress intensity factor prediction. Int. J. Solids Structures, 1993, 30, 2929-2936. 22. Reedy, J. E. Jr and Guess, T. R., Butt tensile joint strength: interface corner stress intensity factor prediction. J. Adhesion Sci. Technol., 1995, 9, 237-251.
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23. Reedy, J. E., Jr and Guess, T. R., Butt joint strength: effects of residual stress and stress relaxation. J. Adhesion Sci. TechnoL, 1996 (in press). 24. Brown, S., Povirk, G. and Connally, J., Measurement of slow crack growth in silicon and nickel micromechanical devices. In MEMS 93, IEEE, 1993, pp. 99-104. 25. Connally, J. and Brown, S., Micromechanical fatigue testing. Exp, Mech., 1993, 33, 81-90. 26. Cunningham, S. J., Suwito, W. and Read, D. T., Tech. Digests of Transducers, 1995, 2, 56-59. 27. Suwito, W., Dunn, M. L. and Cunningham, S. J., To appear in Transducers 97, Proceedings of The 9th International Conference on Solid-State Sensors and Activators, Chicago, IL, U.S.A., 16-19 June, 1997. 28. Theocaris, P. S. and Ioakimidis, N. I., Mode I stress intensity factors at comer points in plane elastic media. Engng Fracture Mech., 1980, 13, 699-708. 29. Irwin, G. R., Fracture. In Handbuch der Physik VI, ed. Flugge, Springer, 1958, pp. 551-590. 30. Anderson, T. L., Fracture Mechanics, Fundamentals and Applications, 2nd edn. CRC Press, 1995.
APPENDIX A Explicit expressions for the mode I angular-dependent terms in Section 2 are:
flrr(O) ~_ -[CIC4 cos C20 -1- (el q- 2)(;'5 cos ClO] C9
f~(O)
Cz C4 cos C20 - C2C5 cos CIO C9 Cj (Ca sin C20 - Cs sin C10)
flr°(O) =
C9
For plane strain:
~3C2+C2-3C~C6-(~C~+C~)
f~(O)=
C9
giy(O) = (1 + v)[CtC4 sin ~.0 + ~C5 sin(2 - ~.)0 - (3 - 4v)C5 sin ~.0] E~.C9 where E and v are Young's modulus and Poisson's ratio, respectively, and: Cl = 1 - 2 C4 = sin C : t C7 = cos 2C2ct
C 2 = 1 + 2 C5 = sin C : t Ca = cos 2a - cos 2).ct
C3 = 1 - 2 v (;'6 = cos 2 C : t C9 = CIC4 - C2C5
For plane stress, replace Ca with 1 in f~(0), and replace E and v with (1 + 2u)/(l + v)2E and v/(1 + v) in g~,(0), respectively.
(Received 15 October 1996, in final form 10 January 1997, accepted 11 January 1997)