DELAYED FRACTURE OF CERAMICS CAUSED BY STRESS-DEPENDENT SURFACE REACTIONS H. H. YU and Z. SUO{ Mechanical and Aerospace Engineering Department, and Princeton Materials Institute, Princeton University, Princeton, NJ 08544, U.S.A. (Received 21 July 1998; accepted in revised form 23 September 1998) AbstractÐConsider a ceramic in an environment, corroding gradually by a surface reaction. When in ad- dition subject to a mechanical load, the ceramic loses mass preferentially at grain-boundary grooves where stress concentrates, so that atomistically sharp cracks may nucleate. Before becoming a crack, a groove maintains local equilibrium at its root; after, it loses local equilibrium. The crack further propagates by breaking atomic bonds, often assisted by environmental molecules. This paper models the groove-to-crack evolution. The groove changes shape to reduce the free energy due to the combined eects of surface ten- sion, grain-boundary tension, elasticity, and chemical potential dierence between the solid and the en- vironment. At any point on the surface, the reaction rate is taken to be proportional to the free energy reduction per unit volume of mass loss. The ceramic body is modeled by a half plane bounded by a curve, whose shape is described by a conformal mapping of many terms, allowing the elastic ®eld in the body to be solved analytically. A variational method leads to a set of ordinary dierential equations to evolve the shape. The model predicts threshold loads, and the times required, for crack nucleation. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION A brittle solid may withstand a static load for a long time and then, without warning, break sud- denly. The smaller the load, the longer the delay time. The phenomenon is known as delayed frac- ture or static fatigue. A practically important ques- tion is whether a threshold stress existsÐthat is, whether there is a stress limit below which the solid can sustain the load inde®nitely. Another question is how long the solid can sustain a stress above the threshold. Orowan [1] attributed the phenomenon to an en- vironmental eect on surface tension. According to Grith, the strength S of a solid scales with its sur- face tension g as SAg 1/2 . Certain molecules in air, adsorbing on the surfaces, reduce the surface ten- sion. To aect fracture, the molecules must diuse to a crack front, and assist in breaking atomic bonds. Both steps take time. Wiederhorn [2] reviewed evidence of glass weakened by water mol- ecules. Within the framework of fracture mechanics, this theory has led to a procedure to predict life- times of engineering components [2, 3]. Cracks are assumed to pre-exist on the surface of a given en- gineering component, and grow slowly under a sta- tic stress. The time-to-fracture is the time required for one of the cracks to grow to a critical size. An upper bound of the initial crack size on the com- ponent surface is estimated by a proof test. To obtain a crack growth law, one cuts a sample of the same material with a long crack, loads it in a con- trolled environment, and measures the crack vel- ocity as a function of stress intensity factor. The two pieces of informationÐthe initial crack size and the crack growth lawÐare then used to predict a lower bound of the lifetime of the component. This procedure has been successfully applied to components with relatively large initial crack sizes and short lifetimes. How to apply the procedure to components like microelectronic chips is uncertain. The feature size on a chip is about 1 mm. The initial ¯aw is ill-de®ned, and the total crack growth should be less than the feature size. Suppose that 10 years of lifetime is required; the allowed crack velocity should be lower than 10 14 m/s. Direct measure- ment of such low crack velocities is extremely di- cult. One is then forced to do accelerated tests, or ®nd a way to determine the threshold condition. To extrapolate the data to predict lifetime, one must understand what happens at low crack velocities or near the threshold condition. That atomistically sharp cracks pre-exist in ma- terials has always been a vexing assumption. Hillig and Charles [4] and others [5±15] have considered a dierent theory of delayed fracture. Initial ¯aws can be blunt. A solid loses mass to its environment by a surface reaction; the rate of the reaction depends on local stress. Because the stress along a ¯aw surface is non-uniform, the reaction may proceed faster at the ¯aw root than elsewhere, gradually changing the ¯aw into a sharp crack. Alternatively, at elev- Acta mater. Vol. 47, No. 1, pp. 77±88, 1999 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/99 $19.00 + 0.00 PII: S1359-6454(98)00344-9 {To whom all correspondence should be addressed. 77