Arch Appl Mech (2006) 76: 667–679 DOI 10.1007/s00419-006-0055-3
O R I G I NA L
T. Fett · D. Munz
Influence of narrow starter notches on the initial crack growth resistance curve of ceramics
Received: 3 February 2006 / Accepted: 20 June 2006 / Published online: 1 August 2006 © Springer-Verlag 2006
Abstract In experimental R-curve investigations crack development usually starts from notches. The validity of R-curves depends on the size of the notch root radius. This influence is completely ignored in most cases. In this theoretical study it is shown how the notch radius affects the formally computed crack resistance curve. First, the influence of the notch radius on the starting point of the R-curve, the so-called crack-tip toughness K I0 , will be addressed. Then, the effect of the notch on the shielding stress intensity factor will be discussed, and, finally, the influence on T-stress and the consequences on local path stability will be shown. Keywords Bridging stresses · Slender notches · Stability · Stress intensity factor · T-stress
1 Introduction Fracture in most ceramic starts from flaws introduced during the processing of these materials. These flaws are conveniently described as cracks, and, hence, linear elastic fracture mechanics can be applied to describe the crack growth behaviour. A characteristic feature of most ceramics is the existence of a rising crack growth resistance curve (R-curve), when the stress intensity factor during crack propagation increases from an initial value K I0 until a critical value is reached. The rising crack growth resistance for coarse-grained ceramics is caused by crack border interactions in the wake of the advancing crack. The stress intensity factor at the crack tip is the sum of K appl from the external load and K sh from the crack border interaction stresses: K tip = K appl + K sh .
(1)
During crack propagation, K tip is constant. K sh is negative and increases with increasing crack extension. Therefore, K appl has to increase to maintain a constant K tip . The following considerations are focused on 2D problems under mode I loading conditions. Most investigations deal with cracks starting from narrow notches. In earlier investigations these were introduced as a saw cut, leading to notch radii of about R = 30 to 100 μm. In recent years, thin saw cuts were produced with a razor blade as proposed by Nishida et al. [1] and successfully applied by Kübler [2]. This notching procedure yields notch roots with a radius of about 5 to 20 μm. During notching, a damage zone T. Fett (B) Institut für Materialforschung II, Forschungszentrum Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany E-mail:
[email protected] Present Address: Institut für Keramik im Maschinenbau, Universität Karlsruhe, Karlsruhe, Germany D. Munz Institut für Zuverlässigkeit von Bauteilen und Systemen, Universität Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany E-mail:
[email protected]
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b)
1
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/R Fig. 1 Crack ahead of a slender notch. a Geometric data. b True stress intensity factor
ahead of the notch may occur, which can be described by a crack of length 0 in front of the notch of depth a0 (Fig. 1a). The stress intensity factor commonly is formally computed for a crack of total length a = a0 + as a K ∗ = σ π(a0 + )F , (2) W where F is the geometric function for an edge crack of length a in a specimen of width W . The geometric function is available from fracture mechanics handbooks. The stress intensity factor K ∗ given by Eq. (2), of course, is correct only if the crack length is clearly larger than the radius of the notch. In the first crack extension phase, where the crack length is comparable to R, the quantity K ∗ deviates strongly from the correct stress intensity factor K. This fact is ignored in many evaluations of experimental results. In the special case of an edge crack ahead of a slender notch, where R is small compared to the notch depth a0 and other specimen dimensions, the true stress intensity factor K is given by [3] K ∼ . (3) = tanh 2.243 ∗ K R This relation was applied for the evaluation of fracture toughness and the R-curve [4–7] and is shown in Fig. 1b. From this plot it is clearly visible that the true stress intensity factor is significantly lower than the formally computed K ∗ . On the other hand, it can be concluded from Fig. 1b that notch effects are of importance if < 1.5R only. To demonstrate the effect of the notch and to discuss the principal influence of notches, two examples of coarse-grained alumina are presented. Figure 2 compares two different methods for the calculation of the critical stress intensity factor K I0 obtained by Kounga et al. [8] for alumina with different grain sizes. The circles were obtained from fracture tests of notched bending bars with notch radii of 15 to 20 μm, where the onset of crack extension was obtained from strain gauge measurements. K I0 was calculated neglecting the notch radius effect. The squares represent results obtained from measurements of the crack opening displacement u (COD) near the tip of a crack after some extension. From the relation between the COD and the distance from the crack tip 8 K I0 √ u= a−x (4) π E the value K I0 is determined. In both cases, a decrease of K I0 with increasing grain size is apparent. The ratio between K I0 from the ∗ from the fracture tests is 0.62. Insertion of this value into Eq. (3) leads to COD measurements and K I0 0 = 1.6–2.1 μm. A second example is shown in Fig. 3, where two R-curves for different initial crack lengths a0 are plotted for an alumina with a grain size of 16 μm [9]. The stress intensity factor again was calculated
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K I0 (MPa√m ) 2
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KIR
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√ with the notch √ effect being neglected. From Fig. 2 K I0 = 2.2 MPa m is expected, whereas from Fig. 3 ∗ = 3.6 MPa m is obtained, leading to a ratio of 0.61. In this experiment the notch root radii were between K I0 30 and 100 μm. Equation (3) then leads to 0 = 3–10 μm. With these data, the R-curves of Fig. 3 were re-evaluated providing the results shown in Fig. 4. In both cases, only the first 3 to 4 data points were affected. Comparing the results of both examples, it can be concluded that the damage length 0 increases with increasing notch radius. 2 Effect of notch radius on R-curve 2.1 R-curve from load-controlled fracture test The R-curve behaviour may be modelled by
√ K sh = −K sh,max 1 − exp(−λ a) .
This relation interpolates the limit cases √ K sh ∝ a for a → 0,
K sh → −K sh,max for a → ∞ .
(5)
(6)
In the following evaluation, the results √ of Steinbrech et al. [10] for an alumina are used. From them λ = 0.35/mm1/2 and K sh,max = 6.85 MPa m are obtained for an initial crack length of a0 /W = 0.4. The crack
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Fig. 4 a True R-curve with stress intensity factor computation via Eq. (3) for a notch root radius of R = 30 μm and b for R = 100 μm obtained from the data of Fig. 3 (different symbols for different initial notch depths) 2.8
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a (mm) Fig. 5 Total stress intensity factor (K total = K tip ) obtained by superposition of applied and shielding stress intensity factors according to Eq. (1); σc = stress at onset of crack extension, σmax ∼ = 1.27σc = maximum stress for stable crack extension
growth resistance curve depends on the specimen geometry and, hence, on the notch radius. This effect is ignored in this section. In Sect. 3 the effect of this assumption will be shown. In Fig. 5 K total = K appl + K sh is plotted vs. crack length for different applied stresses at a zero notch root radius, i.e. for a crack growing from a bridging-free initial crack of length a0 . Crack extension starts at a √ critical stress of 37.7 MPa and a corresponding K total = K I0 = 2.2 MPa m. Due to the large change of K sh compared to K appl , K total first decreases. Consequently, stable crack growth occurs under increasing applied stress. At an applied stress, at which the minimum of K total reaches K I0 , unstable crack extension occurs. The influence of the notch root radius on the total stress intensity factor is illustrated in Fig. 6 for an initial crack size of 0 = 2 μm and two specimen widths. Since an intersection between K total and K I0 is obtained for R < 12.2 μm (W = 7 mm) and R < 8.8 μm (W = 4 mm) only, these radii, denoted as Rc , are limits for the occurrence of stable crack extension. If R > Rc , then unstable crack extension must occur when K total = K I0 is reached for the first time. The different crack extension phases for W = 4 mm are illustrated in Fig. 7 for a notch radius of R = 6μm < Rc and an initial crack size of 0 = 2 μm. Figure 7a shows the total stress intensity factor during crack extension, and Fig. 7b represents the applied stress intensity factor together with the true R-curve K IR . A crack of total length a1 = a0 + 0 starts to propagate if the condition K total = K I0 is fulfilled for the first time (Fig. 7a).
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Fig. 7 Crack extension under increasing stresses applied. a Development of total stress intensity factor. b Applied stress intensity factor and R-curve
At a second crack length a2 , the condition K total = K I0 is fulfilled again. Between the two crack lengths a1 and a2 , spontaneous crack extension occurs, followed by stable crack extension from 1 to 2 under increased load. At crack length a3 , the curves K I0 = constant and K total (a) have the same horizontal tangent. At a > a3 the specimen fails by spontaneous crack growth. Figure 8 shows the apparent R-curve behaviour for very small crack extensions. The starting point of the ∗ , is given as “apparent R-curve”, the “apparent crack tip toughness” K I0 ∗ K I0 =
K I0 √ tanh(2.243 0 /R)
(7)
∗ ∼ 2.55 MPa√m (open square in Fig. 8), which is clearly larger than the true crack tip toughness resulting in K I0 = √ K I0 = 2.2 MPa m (solid square).
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a (mm) Fig. 8 Apparent R-curve and crack tip toughness under load-controlled test conditions
When plotting the formally computed stress intensity factor K ∗ , interpreted as the measured R-curve value, the solid curve in Fig. 8 is obtained. At the beginning of unstable crack extension, the related stress intensity factors differ strongly. The apparent R-curve in terms of K ∗ starts at a significantly higher value than the correct one and, consequently, is flatter. If the starting point of the K ∗ (a)-curve is defined as K I0 , this value becomes too high. Unfortunately, the true crack tip toughness is never visible in such curves. 2.2 R-curve from displacement-controlled fracture test Under constant load conditions, only small regions of stable crack extension are visible from Fig. 7. To allow for a more extended part of the R-curve to be determined, special test procedures were developed. One possibility, for instance, is to use a very rigid loading system where the load drops when compliance increases due to crack propagation. A fast piezoelectric loading system also allows to reduce the applied load within a very short response time. In this case, quasistable crack extension is ensured. In the following considerations the influence of compliance will be studied for displacement-controlled tests. Under displacement-controlled loading conditions, the stress applied at a crack length a > a0 is given by σappl = σappl (a0 )
C(a0 ) , C(a)
(8)
where C(a) is the compliance for a crack depth a and C(a0 ) for the initial notch length. These compliances consist of a part caused by the stiffness of the test machine, the compliance of the uncracked bending bar (depending on the supporting length and Young’s modulus), and the contribution of the notch or crack. The strongest drop of the load applied during spontaneous crack extension, for which the displacement is kept constant, is obtained for a very stiff machine. In the further computations the maximum possible stress reduction will be considered by neglecting the contributions of the test machine and of the uncracked specimens. The contribution of a crack to the compliance is given by Tada et al. [11] as C∝
α2 a 5.93 − 19.69 α + 37.14 α 2 − 35.84α 3 + 13.12 α 4 , α = . 2 (1 − α) W
(9)
Figure 9a represents the total stress intensity factor under increasing bending displacements δ. Apart from a small region of instability at the beginning of the test, further crack extension is completely stable. The unstable phase is plotted in more detail in Fig. 9b, together with the result of the load-controlled test. As is obvious from
Influence of narrow starter notches on the initial crack growth resistance curve of ceramics
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Fig. 9 Crack extension under displacement-controlled loading conditions. a Development of total stress intensity factor. b Comparison with load-controlled tests
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Fig. 10 Applied stress intensity factor in displacement-controlled fracture tests and R-curve
this figure, the region of instability is smaller for a displacement-controlled than for a load-controlled test. In Fig. 10, the applied stress intensity factors are represented together with the R-curve K IR . The initial part of ∗ and K must result as for Fig. 10a is plotted again in more detail in Fig. 10b. The same difference of K I0 I0 the load-controlled test. The apparent R-curves must differ due to the larger unstable crack extension phase in the two test modifications. In the load-controlled test a moderately increasing (unstable) part of the apparent R-curve occurs first. In the case of the displacement-controlled test, a decrease of the R-curve is visible first (Fig. 11). The different crack extensions in load- and displacement-controlled tests yield different types of load vs. displacement records for the experiments. This is illustrated schematically in Fig. 12. Whereas for a sharp precrack a continuous curve results (Fig. 12a), discontinuous records must occur (Fig. 12b,c) for notched specimens with their unstable crack extension phases.
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load controlled 2.7
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displacement controlled 0
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Fig. 11 Comparison of apparent R-curves K ∗ (a) for a load-controlled (dashed curve) and a displacement-controlled test (solid curve)
P b)
P a)
P c)
load controlled
without notch
δ
δ
displacement controlled
δ
Fig. 12 Effect of unstable and stable crack extension phases on measured load-displacement curves a for a sharp preexisting crack and for a slender notch under b load-controlled and c displacement-controlled loading
3 Calculations applying bridging relation In Sect. 2 it was assumed that the R-curve is independent of the notch radius. This is not correct. The basic physical relation describing crack bridging is the relation between the bridging stress and the crack opening displacement δ [12]. Different relations have been proposed. Here, the relation
δ σbr = σ0 exp − δ0
(10)
shall be used with the parameters σ0 = −60.3 MPa and δ0 = 0.53 μm as determined in [13] for a coarsegrained Al2 O3 with 16 μm mean grain size. In Eq. (10), σbr represents the bridging stresses as a continuously distributed stress averaging the local crack face interactions, and δ is the crack opening displacement, i.e. half of the total crack opening. The general treatment of bridging stress effects is outlined in detail in [9].
Influence of narrow starter notches on the initial crack growth resistance curve of ceramics
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Fig. 13 Some geometric data necessary for using weight function for a crack emanating from a notch root
The shielding stress intensity factor K sh caused by these bridging stresses is given by a K sh =
σbr h(x, a)dx
(11)
0
with the weight function h. The crack opening displacements δ are given by the integral equation 1 δtotal (x) = E
a
K total (a )hda
(12)
x
(E plane strain modulus) and with Eqs. (5) and (11) ⎤ ⎡ a a
⎥ 1 ⎢ δtotal (x) = h(x, a )⎣ h(x , a ) σappl + σbr dx ⎦da . E x
(13)
0
Equation (13) can be solved by successive approximation. First only σappl (x) is introduced in the integral, resulting in a first approximation of δ. From this δ a first estimate of σbr is obtained via Eq. (10). Insertion into the integral of Eq. (13) gives the second approximation of δ and, consequently, of σbr etc. The correct solution σbr (x) is found when the iterative procedure converges to a fixed distribution of δ(x) or σbr (x). The applied stresses ahead of a slender notch of radius R can be concluded from the analysis of Creager and Paris [14] √ σappl = 2σbend F(a0 ) a0
R+ξ , (R + 2ξ )3/2
(14)
where F(a0 ) is the geometric function used in Eq. (1) for a crack of the same length a0 (for the geometric data see Fig. 13). The weight function for the edge crack reads ⎤ ⎡ 1 Aνμ α μ
a 2 ⎣ 1 ν+ 2 ⎦ h edge = + , (15) 1 − ax , α= 3/2 πa (1 − α) W 1− x a
(ν,μ)
with the coefficients Aνμ compiled in [3]. Now, the abbreviations of h (1) = h edge
ξ + a0 a , , = a − a0 a W
(16a)
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as the “long-crack weight function” for a crack of total length a and the coordinate ξ in the x-direction with the origin at the notch root and
ξ (2) h = h edge , (16b) W − a0 as the weight function of an edge crack in a body of reduced width W −a0 are introduced (see [3], pp. 199–203). Then, the weight function for a small crack of length emanating from the notch root reads h notch = λh
(2)
+ (1 − λ)h
(1)
, λ=
R R+
7/2 .
(17)
This weight function is plotted in Fig. 14a for the case of a short crack ahead of a notch of R = 10 μm with /R = 0.25, together with the near-tip solution for an edge crack. It should be mentioned here that the edge crack solution is also valid for a crack ahead a notch if /R 1. Figure 14b shows the crack profile at K = K I0 as the total displacement vs. the coordinate ξ for a0 /W = 0.4, R = 10 μm, and /R = 0.25. The different weight functions for a long crack and the crack ahead of the notch result in different crack profiles and different bridging stress distributions (Fig. 14c). The influence of the different weight functions results in different shielding stress intensity factors and, consequently, in different R-curves (Fig. 14d). For R, the R-curve for the crack emanating from the notch root is slightly larger than that for a long crack. For R, the differences in R-curves disappear. 4 T-stress for cracks at notches and their influence on path stability In Sect. 2 we discussed the stability and instability behaviour of crack extension to find out under which loading condition a controlled increase in crack length is possible. A different type of stability is path stability, which indicates whether a crack will remain on the plane prescribed by the symmetry axis of the narrow notch. In order to answer this question, we have to look for T-stress at the tip of slender notches. Taking into consideration the singular stress term and the first regular term, the near-tip stress field of a crack can be described by KI σi j = √ f i j ( ) + σi j,0 , 2πr
(18)
where K I is the mode-I stress intensity factor and f i j are the well-known angular functions (r, = polar coordinates with the origin at the crack tip). In Cartesian coordinates, σi j,0 reads
σx x,0 σx y,0 T 0 , (19) σi j,0 = = 0 0 σ yx,0 σ yy,0 where T is the so-called “T-stress”. To determine the first deviation from the initial straight crack plane, Cotterell and Rice [15] analysed the local crack path stability in terms of T. The reason for deviations from the crack plane may be small mode-II stress intensity factor parts caused by small misalignments in loading and leading to a small kink angle 0 . For small extensions ξ , the deviation from the initial crack plane, y, is given by √ √
2 8T K2 8ξ T 2ξ T y(ξ ) = 0 2 exp ξ erfc − − 1 − 4 . (20) √ 8T K2 K π K The most important conclusion of [15] is illustrated in Fig. 15b, namely, increasing deviation from the prescribed kink angle for T > 0 and decreasing deviations for T < 0. The maximum normal stress σmax occurring directly at the notch root results from the analysis of Creager and Paris [14] as a a 0 0 σmax = 2σbend F . (21) W R
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Fig. 14 Computation of R-curve from bridging stress relation Eq. (10). a Weight functions for crack ahead of notch and long crack of same total length a = a0 + . b Crack profiles (total displacements) computed with different weight functions. c Bridging stress distributions. d Shielding stress intensity factors computed from bridging relation Eqs. (10) and (13) using the two weight functions of a
For → 0, the T-stress is the same as for an edge crack in a half-space, but now under the high-stress σmax . Since for the limit case /R → 0 of the notch problem for the edge-cracked half-space the T-stress is T = −0.526σappl , a a 0 0 (22) σappl T0 = T/R→0 = −1.052 F W R is obtained [16]. In contrast to this T-stress, the “long-crack solution” T(a) is given by [16] −0.526 + 2.481 α − 3.553 α 2 + 2.6384 α 3 − 0.9276 α 4 a T (a) = , α= . 2 σappl (1 − α) W
(23)
Figure 16a shows the T-stress for cracks ahead of notches as obtained from boundary collocation computations. A rough interpolation relation based on these collocation data is proposed in [16]. It interpolates the
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a) x a a0 ξ
T>0
b) y y(ξ)
T<0 Θ0
ξ Fig. 15 a Geometrical data of a crack growing under mode-I loading (vertical arrows) with a superimposed small mode-II disturbance (horizontal arrows). b General influence of T-stress after crack kinking under mixed-mode loading
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Fig. 16 a T-stress solution for cracks ahead of slender notches in bending bars, Eqs. (23) and (24), as curves compared with boundary collocation results (symbols) from [16]. b T-stress for small cracks
“long-crack solution” and the value T0 by T ≈ T0 + (T (a) − T0 ) tanh
4/3
5 R
3/4 .
(24)
This dependence is shown in Fig. 16b for a four-point bending bar of W = 4 mm as a function of the crack length for different notch root radii and notch depths a0 . From this diagram it can be concluded that the first few micrometres in crack extension from a notch are automatically stable, since T < 0 over about a − a0 = 6–12 μm for notch radii of R = 10 and 20 μm. The crack lengths stable for which path stability occurs at a/W > 0.35 (for a/W < 0.35, the condition T < 0 is trivially fulfilled, since even the “long-crack solution” c´ 6´c6 T (a) is negative in this case, see Fig. 16a) can be roughly expressed as stable = astable − a0 ≈ 0.6R .
(25)
Influence of narrow starter notches on the initial crack growth resistance curve of ceramics
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Due to the disappearing stress intensity factor K I → 0 for → 0 and the very high T-stress T0 under this condition, the ratio T /K tends to −∞. Also, this fact reflects the high path stability in the region described by (25). Summary Measurements of crack-growth-resistance curves are very often performed with specimens containing narrow starter notches. Such notches affect the initial phase of the crack growth behaviour. It is shown by a fracture mechanics analysis in terms of stress intensity factor, weight function and T-stress • How the true stress intensity factor for a small crack emanating from the notch root must be determined, • How the special weight function for this crack/notch problem influences the initial part of the R-curve for a given bridging stress relation and, finally, • How the notch changes the initial path stability. From the computations it becomes evident that influences of the notch root radius on the R-curves and path stability are present as long as the crack size is smaller than the notch root radius. References 1. Nishida, T., Pezzotti, G., Mangialardi, T., Paolini, A.E.: Fracture mechanics evaluation of ceramics by stable crack propagation in bend bar specimens. Fract. Mech. Ceram. 11, 107–114 (1996) 2. Kübler, J.: Fracture toughness using the SEVNB method: preliminary results. Ceram. Eng. Sci. Proc. 18, 155–162 (1997) 3. Fett, T., Munz, D.: Stress Intensity Factors and Weight Functions. Computational Mechanics Publications, Southampton (1997) 4. Fett, T.: Influence of a finite notch root radius on the measured R-curve. J. Mater. Sci. 39, 1061–1063 (2004) 5. Fett, T.: Influence of a finite notch root radius on fracture toughness. J. Eur. Ceram. Soc. 25, 543–547 (2005) 6. Damani R., Gstrein, R., Danzer, R.: Critical notch-root radius effect in SENB-S fracture toughness testing. J. Eur. Ceram. Soc. 16, 695–702 (1996) 7. Damani, R.J., Schuster, C., Danzer, R.: Polished notch modification of SENB-S fracture toughness testing. J. Eur. Ceram. Soc. 17, 1685–1689 (1997) 8. Kounga, A.B., Yousef, S.G., Fett, T., Rödel, J.: Crack tip toughness of coarse-grained alumina. Eng. Fract. Mech. 72, 1011–1019 (2005) 9. Munz, D., Fett, T.: CERAMICS, Failure, Material Selection, Design. Springer, Berlin Heidelberg New York (1999) 10. Steinbrech, R., Reichl, A., Schaarwächter, W.: R-curve behaviour of long cracks in alumina. J. Am. Ceram. Soc. 73, 2009– 2015 (1990) 11. Tada, H., Paris, P.C., Irwin, G.R.: The Stress Analysis of Cracks Handbook. Del Research Corporation, St. Louis (1986) 12. Mai, Y., Lawn, B.R.: Crack-interface grain bridging as a fracture resistance mechanism in ceramics: II. Theoretical fracture mechanics model. J. Am. Ceram. Soc. 70, 289 (1987) 13. Fett, T., Munz, D., Thun, G., Bahr, H.A.: Evaluation of bridging parameters in Al2 O3 from R-curves by use of the fracture mechanical weight function. J. Am. Ceram. Soc. 78, 949–951 (1995) 14. Creager, M., Paris, P.C.: Elastic field equations for blunt cracks with reference to stress corrosion cracking. Int. J. Fract. 3, 247–252 (1967) 15. Cotterell, B., Rice, J.R.: Slightly curved or kinked cracks. Int. J. Fract. 16, 155–169 (1980) 16. Fett, T.: T-Stress and Stress Intensity Factor Solutions for 2-Dimensional Cracks. VDI, Düsseldorf (2002)
