Effect of Flaw Shape on Strength of Ceramics

Abstract

IN m ' r m P R E r A T t o N s of the strength of ceramic materials, the size of the fracture-initiating surface flaw is invariably described by a single dimension-depth.A variable factor, @, is then used to account for the flaw shape, and determination of @ requires knowledge of the ratio a / c , wherea and c are the axes of an ellipse which describes the flaw shape. This complexity is not necessary; a, @, and c can be eliminated, and the flaw area, A , used to describe the flaw size.For a flaw much smaller than component dimensions and stresses well below the yield stress, Irwin' gives the following equation for the stress-intensity factor, K I , for the opening mode in plane stress*:where G is the strain energy releaserate, E is Young's modulus, u is applied stress, the angle 0 is as shown in Fig. 1, and Y is a geometrical parameter.'For surface flaws, Y =2.0, and for subsurface flaws, Y 1.8. The integral @ varies only with a/c as follows:Note that ifaic in Eq. (2a) has the same value ascia in Eq. (2b), the expressions under the integrals (i .e. elliptic integrals) are numerically equal, but the values of @ differ as follows:(3) A flaw will extend from that point on its periphery where K i is maximum, and this occurs where the first derivative of K I , dKl/ d8, is zero and the second derivative, d2Kl/d8', is negative. From Eq. (l), the maxima are found at 8 = n / 2 f o r a < c and at 8=0 for u >c. It follows that an elliptical flaw of any a/c ratio will tend to reduce its ellipticity during growth. When a circular shape is achieved, Eq. (1) shows that K I is constant along the periphery.Maximum values of K I are given by:In the case of materials which fracture well below the yield stress, Eqs. (4a), (4h), and (4c) apply with u andKl(,n,,, replaced, respectively, by g f , the fracture stress, andKlp, the critical stress-intensity factor, and the following equations are obtained:Equations (5u) and (5b) show that the smaller dimension of the flaw controls fracture, and not necessarily the depth, a . Equation (5b) can, however, be modified to include a dependence of C T~ on a: U Fig. 1. (A) Semielliptical flaw in plate subjected to uniform tensile stress. In ( B ) , the flaw depth,a, is less than its half-width, c, and in (C), a is greater than c .Equations (5a) and (5c) can now be combined in a general relation:where Z , known as the flaw shape ~arameter,".~ is as follows:The effect of flaw shape, aic, on Z is plotted in Fig. 2 . ' Values of the flaw area, A ( A _ = n u c / 2 ) , were calculated for several a/c ratios, and Z 2 vs a / d A has been plotted in Fig. 3. For practical purposes, the curve in Fig. 3 can be approximated by a straight line, Z 2 = 2 . 8 2 a l f i , which, when substituted into Eq. (6) gives 1.68 Kic (7) uj=-~ The use of Eq. (7) in place of Eq. (6) introduces a maximum error of 5 % in calculating the fracture stress associated with any realistic elliptical flaw including where a =c. For flaws with 0.2>a/c>3, Eqs. ( 6) and (7) differ significantly. Eq. (7) may, however, be quite useful for n...

Fracture Origin and Strength in Advanced Pressureless‐Sintered Alumina

“…The Y factors required for different geometrical configurations were taken from the work of Bansal. 12 As a result, the observed flaw sizes gave K Ic data with an average of 3.85 MPaиm 1/2 , and none of the individual K Ic values show a significant discrepancy with the previously measured toughness (Table III). The agreement of the observed flaw sizes with measured strength data and the known toughness supports the identification of the flaws as the origins of failure.…”

Analysis of symmetric fracture mirrors in glass bottles

“…Therefore, there is no reason why the consequent fracture front propagation should not be circular. This description is at variance with [5] which assumes a gradual transition between an elliptic flaw and a more circular ellipse at the mirror boundary.…”

“…There were 17 such samples and it is for these that the detailed calculations were pursued ( Table 2). The K1c values were derived from [10,5]:…”

“…We are concerned here with two concepts explaining the mirror boundaries that have been developed. The first [5], implies that the flaw/mirror boundary is the fracture front which bifurcates at a specific time. The ellipticity of this front is lowered from the flaw to the mirror boundary due to a non-uniform stress intensity at the flaw boundaries.…”

“…There is a dispute about its cause [2,5] while its dependence on fracture parameters and surface effects [6] is also of interest.…”