journal zyxwv J zyxwv Am Ceram zyx So&, 77[5] zyxw 112338(1994) zy Microstructural Mechanics Model of Anisotropic-Thermal-Expansion-Induced Microcracking Narayanaswamy Sridhar, Wuhua Yang, and David J. Srolovitz Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109 Edwin R. Fuller, Jr. Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 Thermal-expansion-induced microcracking in single-phase ceramics has been simulated using a simple mechanics model based upon a regular lattice of brittle, elastic springs. Microcracks preferentially form at grain boundaries and propagate either into the bulk or along grain boundaries, depending on the toughness of the boundaries relative to the grain interiors. The present results show that aniso- tropic-thermal-expansion-induced microcracking can be more severe for either large or small grain size samples depending on the damage measure employed. At very small misfit strains, the large grain microstructure develops microcracks before the small grain microstructure. How- ever, over most of the misfit strain regime examined, the total length/area of all cracks in a sample is larger when the grain size is small. This is manifested in a larger decrement of the elastic modulus in small grain size samples as com- pared with large grain size samples at the same misfit (AT). However, large grain sizes are more detrimental with regard to fracture properties. This is because the fracture stress scales as inversely with the crack length and large grain samples exhibit larger microcracks than small grain samples. Unlike in the unconstrained samples, when a sam- ple is constrained during a temperature excursion, the stress created by the overall thermal expansion can directly lead to fracture of the entire sample. I. Introduction ANY ceramic materials are known to undergo spontane- M ous cracking when cooled from high processing tempera- tures. The presence of microcracks modifies several physical properties including thermal diffusivity, dielectric constant, acoustic transparency, and elastic moduli.' Microcracking can also lead to an increase in fracture toughness: presumably asso- ciated with the microcracking-induced dilatation and also partly due to the formation of a process zone ahead of a propagating rack.^,^ However, the contribution of microcracking to tough- ening is usually minor.' The tendency to form these cracks is known to increase with increasing temperature excursions and increasing grain size6 and is often attributable to residual stresses that develop from either thermal contraction anisotropy or non-shape-preserving phase transformations.' In multiphase T. Michalske~ontributing editor Manuscript No. 194300. Received August 16, 1993; approved December 20,1993. Support for S.N. was provided by the National Institute of Standards and Tech- nology through Grant No. 70NANBlH1147: support for W.Y. and D.J.S. was pro- vided by the US. Air Force Office of Scientific Research through Grant No. AFOSR-90-0141, materials, thermal-expansion-induced microcracking may also result from the difference in the coefficient of thermal expan- sion of the different phases. In single-phase, polycrystalline materials, however, thermal-expansion-induced microcracking is associated with the crystalline anisotropy of the coefficient of thermal expansion. Since many single-phase, ceramic materials are neither isotropic nor cubic, thermal expansion anisotropy is thought to be the dominant cause of microcracking associated with temperature excursions. This paper focuses on the role of anisotropic thermal expansion in microcracking. Thermal-expansion-anisotropy-induced microcracking has received a great deal of e~perimental~.~~~~' and the~retical'.~~'''~' attention. A fracture mechanics analysis of microcracking"'." reveals an essential dependence of microcracking on micro- structural dimensions. In particular, dimensional considerations dictate that microcracks initiate when grains exceed a critical size. Obtaining an exact solution to this problem is a formidable task, since this involves adding the effects of thermal expansion anisotropy to the tensor Hooke's law and then integrating over the temperature range of interest to yield the thermal-expan- sion-induced stresses. This procedure should be applied to non- uniform microstructures in order to obtain a realistic stress distribution. Therefore, most of the theoretical analyses of anisotropic-thermal-expansion-induced microcracking have made certain simplifying assumptions such as the presence of very idealized microstructures (e.g., Refs. 4, 10, 11, and 12). These include pairs of grains, hexagonal arrays of grains, etc. Real microstructures, on the other hand, exhibit a wide distribu- tion of microstructural geometries." This is particularly important since fracture properties are determined by the extremes in the local (microstructure-dependent) stresses rather than their average values. Thus, an understanding of micro- cracking-related phenomena requires a realistic, microstruc- ture-based description of microcracked microstructures. In this paper, we examine thermal-expansion-anisotropy- induced microcracking based upon a simple mechanical model which is solved n~merically.'~ We refer to this model as the "microstructural mechanics model" because it is capable of describing the stress distribution and fracture behavior of mate- rials with arbitrarily complex microstructure. We employed the microstructural mechanics model to study microcracking in a realistic polycrystalline microstructure as a function of grain size, thermal expansion anisotropy, and relative grain boundary to bulk toughness, and to determine the effect of external stress on microcracking. 11. Microstructural Mechanics Model (1) Basic Model The mechanics model employed in the simulations, described below, is based upon the elastic properties of a net- work of springs. Rather than discretizing the equations of elas- ticity, the elastic continuum has been replaced with a lattice of 1123