Random polycrystals of grains containing cracks: Model ofquasistatic elastic behavior for fractured systems Page: 3 of 38
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It is known [1, 2] that the quasistatic behavior of cracked or fractured systems cannot be
successfully modeled using traditional effective medium theory methods: (a) in part because
most of those methods are based on energy stored in the volume of the inclusions, whereas
the effects of cracks are known to be nearly independent of their volume fraction, and (b)
in part because addition of more cracks results in shielding effects, thereby reducing the
softening influence of all the cracks on each other and on the overall system. This reduced
softening effect for multiple cracks is usually missed entirely by the traditional volume-
fraction-based effective medium theories, typically based on Eshelby's analysis of ellipsoidal
To circumvent the volume fraction issue, we consider herein a model based on grains
containing cracks. The grains are assembled into (for example) an isotropic polycrystal of
cracked grains. (The assumed overall isotropy is not a requirement of the approach, but
it does greatly simplify presentation of the modeling results.) The analysis of polycrystal
behavior then proceeds using ensemble averaging and, therefore, is not limited by the lack
of crack-volume sensitivity of such cracked systems. This model of polycrystals of cracked
grains also contains within it an effect similar to the shielding effect observed in high crack-
density systems. In particular, it is not difficult to show that the natural definition of the
bulk modulus of an anisotropic grain is always given precisely by the Reuss average of the
bulk modulus. (Imagine immersing a grain in a water bath, and then measuring total grain
strain as a function of fluid pressure.) But since the same Reuss average is also the rigorous
lower bound of the bulk modulus of a polycrystal composed of like grains, it is certain that
a polycrystal of grains will be hydrostatically stiffer than the grains themselves . We can
attribute this effect to grain-to-grain bridging of the strongest components (i.e., the large
volume of solid that is not cracked in the present study). The effect just described will always
be present in true polycrystals, and may be contributing part of the observed "shielding" in
cracked systems. But, we do not expect that this is the only type of crack-crack shielding
present in real systems. In particular, the assumed granular structure of polycrystals also
prevents various long-range connections among cracks from occurring, and thus limits the
range of behaviors that can be present in the model; by assumption, cracks never intersect
grain boundaries in these models, so these systems are thereby inherently constrained never
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Berryman, James G. & Grechka, Vladimir. Random polycrystals of grains containing cracks: Model ofquasistatic elastic behavior for fractured systems, article, July 8, 2006; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc897902/m1/3/: accessed November 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.