LETTER Mode-II shielding-curve of Al 2 O 3 from measurement of cone crack angles Theo Fett Æ Gabriele Rizzi Æ Michael J. Hoffmann Æ Rainer Oberacker Æ Susanne Wagner Received: 31 October 2007 / Accepted: 26 December 2007 / Published online: 20 January 2008 Ó Springer Science+Business Media, LLC 2008 Many ceramic materials exhibit the effect of an increasing crack growth resistance during crack extension. Espe- cially in the case of coarse-grained materials, cracks generally follow the grain boundaries, leave the original crack plane and produce a crack-face roughness. In the theoretical analysis forces transmitted by local links between the two crack faces are smoothed and replaced by so-called continuous bridging stresses acting against the crack opening. As the consequence of such bridging stresses, there exists a shielding stress intensity factor term that shields the crack tip partially from the applied loads. This effect as a reason for the occurrence of R-curves is well documented in literature for cracks under pure mode-I loading conditions [1]. In addition it has to be expected that crack-face inter- actions will also affect crack extension under pure or superimposed mode-II loading as for instance outlined in literature for frictional crack-face interactions [2–4]. In [5] it was outlined that the shear tractions generated under small mode-II load contributions may cause a disappearing effective crack-tip stress intensity factor K II,tip . Also the externally applied mode-II stress intensity factor K II,appl can be reduced by the mode-II shielding so that a disap- pearing K II,tip can occur. The distribution of the intensity of crack-face inter- locking has the same consequences on the bridging tractions normal to the crack face r br as for the tangentially transferred tractions s br . At the same distance from the crack tip where the bridging interactions disappear, both the shear and the normal stresses must vanish. Under the assumption of the same traction versus displacement characteristics, the two stresses are proportional sðxÞ¼ kr br ðxÞ ð1Þ with the crack coordinate x as shown in Fig. 1a. The mode-II shielding stress intensity factor can be computed from the distribution of the shear tractions over the crack. It holds by use of the mode-II weight function h II K II;sh ¼ Z a a 0 sðxÞ h II ðxÞ dx ð2Þ For small scale considerations with the crack-face interactions concentrated near the crack tip, exclusively the asymptotic singular part of the weight function is of interest. This part is identical for mode-I and mode-II (see e.g., [6]) h II ¼ h I ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pða  xÞ s ð3Þ From (1) to (3) it simply results K II;sh ¼ k K I;sh ð4Þ Let us now apply this general relation to two possible bridging relations. A simple bridging relation that fulfills the required properties reads [7] r br ¼ r 0 expðd y =d y0 Þ ð5Þ with a characteristic stress value r 0 \ 0 and a characteristic normal displacement d y0 . It is plotted in Fig. 2a as the solid T. Fett  G. Rizzi Institut fu ¨r Materialforschung II, Forschungszentrum Karlsruhe, Karlsruhe, Germany T. Fett (&)  M. J. Hoffmann  R. Oberacker  S. Wagner Institut fu ¨r Keramik im Maschinenbau, Universita ¨t Karlsruhe, Karlsruhe, Germany e-mail: theo.fett@ikm.uni-karlsruhe.de 123 J Mater Sci (2008) 43:2077–2081 DOI 10.1007/s10853-007-2432-x