Pergamon J. Mech. Phys. Solids, Vol. 42, No. 11, pp. 1653-1677, 1994 Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-5096/94 $7.00 + 0.00 0022-5096(94)00052-2 EMERGENCE OF CRACKS BY MASS TRANSPORT IN ELASTIC CRYSTALS STRESSED AT HIGH TEMPERATURES B. SUN, Z. SUO and A. G. EVANS Mechanical and Environmental Engineering Department, Materials Department, University of California, Santa Barbara, CA 93106, U.S.A. (Received 10 June 1994; in revised form 11 July 1994) ABSTRACT Single crystals are used under high temperatures and high stresses in hostile environments (usually gases). A void produced in the fabrication process can change shape and volume, as atoms migrate under various thermodynamic forces. A small void under low stress remains rounded in shape, but a large void under high stress evolves to a crack. The material fractures catastrophically when the crack becomes sufficiently large. In this article three kinetic processes are analyzed : diffusion along the void surface, diffusion in a low melting point second phase inside the void, and surface reaction with the gases. An approximate evolution path is simulated, with the void evolving as a sequence of spheroids, from a sphere to a penny- shaped crack. The free energy is calculated as a functional of void shape, from which the instability conditions are determined. The evolution rate is calculated by using variational principles derived from the balance of the reduction in the free energy and the dissipation in the kinetic processes. Crystalline anisotropy and surface heterogeneity can be readily incorporated in this energetic framework. Comparisons are made with experimental strength data for sapphire fibers measured at various strain rates. 1. INTRODUCTION Inorganic solids stressed at elevated temperatures often fail by creep deformation and rupture. These processes involve diffusion along grain boundaries and dislocation cell boundaries, motivated by applied stresses. They have been comprehensively studied (Chan and Page, 1993). The usual approach to enhancing creep resistance is to fabricate materials with large grains. The limiting case involves single crystals, exem- plified by superalloys used for turbine blades and oxides used for fiber reinforcements. Diffusional phenomena also limit the performance of these materials. However, the appropriate mechanisms have not been established. The intent of this article is to provide an analysis of time-dependent rupture mechanisms that occur in single crys- tals. The rationale for the choice of mechanisms is provided by measurements and observations of high temperature rupture of single crystal aluminum oxide (sapphire) fibers. Newcomb and Tressler (1993) demonstrated that, when stressed at high tem- peratures, cracks originate from internal pores, grow slowly at first and, upon attaining a critical size, cause catastrophic fracture. We suggest a mechanism sketched in Fig. 16531654 B. SUN, Z. SUO and A. G. EVANS a) Initial Void J$ <3== .t"~'~'~',, ~ Void ".~ Z." j~~..~. Growth b) Surface Diffusion Mechanism t t t Solid c) Diffusion in a Uquid Phase Fig. 1. (a) The cross-section of a spheroidal void in an infinite solid. (b) A void changes shape by surface diffusion. (c) A void filled with a liquid that provides a fast diffusion path. 1. Motivated by the applied stress, atoms diffuse on the void surface, changing the void shape from a sphere to a crack [Fig. 1 (b)]. Such crack growth is accelerated if a fluid (either liquid or gas) exists within the crack. This mechanism has two features : (i) it provides a path for rapid mass relocation [Fig. 1 (c)] ; (ii) the solid and fluid may chemically react, leading to dissolution and reprecipitation along the void surface. Each of these kinetic processes is analyzed in this article. A comparison with exper- imental data is then made. It appears that stress-induced mass transport in elastic solids was first studied by Asaro and Tiller (1972) and by Stevens and Dutton (1971). Asaro and Tiller studied the stability of a flat surface of a stressed solid against surface diffusion. Their linearEmergence of cracks in stressed crystals 1655 stability analysis showed that a slight undulation of the surface decays if the surface energy dominates, but amplifies itself if the elastic energy dominates. They suggested that the positive feedback in the latter case would lead to surface cracks. The same instability has been rediscovered by Grinfel'd (1986, 1993), Srolovitz (1989) and Gao (1991), and found particular relevance in thin epitaxial films (Spencer et al., 1991 ; Freund and Jonsdottir, 1993). Nonlinear analyses of Chiu and Gao (1993) and Yang and Srolovitz (1993) have confirmed that cracks will emerge from the surface undulation. The same concepts apply to the instability of cylindrical voids in elastic solids (Stevens and Dutton, 1971 ; Gao, 1992, 1993 ; Suo and Wang, 1994). Building upon these previous investigations, this article will study the evolution of a three dimensional void, with the kinetic processes significant to single crystals used as structural components. The free energy is computed as a functional of the void shape. The free energy landscape determines the instability conditions. The method of Suo and Wang (1994) will be adopted to study the kinetics. In this method, evolution is traced by using variational principles that govern the rates. The func- tionals to be minimized involve the rate of reduction in the free energy and the rate of dissipation associated with the kinetic processes. The energetics of these problems are examined first, followed by analysis of the kinetics for the three mechanisms. Both the energetics and kinetics differ for the diffusion and the reaction problems. They will be analyzed separately. 2. ENERGETICS OF VOID INSTABILITY 2.1. Diffusion problem Sketched in Fig. 1 (a) is a void in an infinite elastic solid having the cross-section in the (X, Z) plane. An axially symmetric problem will be analyzed in which the void evolves as a sequence of oblate spheroids. The three semi-axes satisfy a = b >t c and the solid is under a remote triaxial stress state Ohl = 0"22 ~ 0"33. For the diffusion problem, the void changes shape but volume is conserved. The volume-conserving spheroids are described