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Bahador Bahmani*, Katherine A. Acton**, Justin M. Garrard*, Connor Sherod**, Philip L. Clarke*, Reza Abedi* *University of Tennessee Space Institute / Knoxville (UTK), ** University of St. Thomas (UST) The use of statistical volume elements for homogenization and fracture analysis of quasi - brittle materials Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332 and two supplementary ROA and RUI awards (UTK / UST) Motivation ➢ Response of brittle materials is highly sensitive to their microstructure due to the lack of energy dissipative mechanisms Al-Ostaz et. al. (1997) Kozicki and J Tejchman (2007) Nonlinear response Different fracture patterns Variations in strength and toughness Approach: Statistical Volume Elements (SVEs) for brittle fracture Statistical Volume Element (SVE) 1 2 Random field statistics 3 1. By using SVEs material inhomogeneities & sample to sample variations (randomness) are preserved. 2. Still no need to resolve all microscale details! A very efficient and accurate model for fracture modeling Macroscopic domain Random field realization 4 Macroscopic fracture simulation 5 Ostoja-Starzewski 1998 RVE limit: is very large PDF of a material property, e.g. stiffness SVEs appropriate for fracture modeling (still reduce problem size by homogenization) • RVEs are good for many elastic regime problems • For fracture modeling (particularly quasi-brittle material): • Need to preserve spatial inhomogeneity • Randomness (sample to sample variations) Spatial inhomogeneity: more realistic fracture patterns 1x1 SVEs 2x2 SVEs 4x4 SVEs Uniform strength Homogenized by: Corresponding fracture patterns under horizontal tensile loading Smaller SVEs are used for homogenization: More realistic dynamic fracture patterns are observed Voronoi tessellation vs. square SVEs: Effect of shape / size of SVEs Original Volume Element Square SVEs: Concern: Intersection of inclusions Voronoi Tessellation based SVEs: - SVE is generated by Voronoi cells around inclusions. - Better homogenized properties are expected. No inclusion intersection problem Sample SVEs at 2, 5, 10, 25% of VE size Square SVEs: - Large variations especially for small SVE sizes. Voronoi SVEs: - Much smaller variation of homogenized properties. - Fracture strength does not decrease significantly as SVE size increases. Large variation and degradation of strength as SVE size increases PDE of bulk modulus Size effect plot for bulk modulus Size effect plot for normal strength Anisotropy of homogenized properties Interesting question: How does fracture strength for a given angle (for example 10 degrees) behave as a random field? Mean strength ෩ Angle-dependent strength () Measures of anisotropy: - Span / mean - COV = s.div / mean span Objectives: - Define measures of anisotropy for fracture and elasticity properties. - How does the shape of SVE affect anisotropy? - By what rate anisotropy decreases as SVE size increases? Small SVE: d = 3.125 Large SVE: d = 12.5 Square SVEs: normal strength - High strength - Low strength Small SVE: d = 3.125 Large SVE: d = 12.5 - High strength - Low strength Square SVEs: shear strength Small SVE: d = 3.125 Large SVE: d = 12.5 Voronoi SVEs: normal strength Isotropic! =2 = 10 Small SVE: d = 3.125 Large SVE: d = 12.5 Voronoi SVEs: shear strength - High strength - Low strength Isotropic! =2 = 10 - Shape of SVE result in artificially weaker and stronger angles. - Voronoi-based SVEs result in an isotropic response (for a macroscopically isotropic material) when SVE size is small. Effect of SVE size on measures of anisotropy - X axis: SVE size - Y axis: Measure of anisotropy (coefficient of variance) for different elastic and fracture properties. As SVE size increases -> homogenized properties become more isotropic. Voronoi-based SVEs (blue) result in more isotropic properties than square SVEs (black) Fracture properties (red) are more anisotropic than elastic properties (black) Another application: Microcracked rock + circular VEs PDFs of microcrack length: - Weibull model (m = 1 2 , 1, 2, 4) - vs. Power law - Same mean length for all m = 1 2 m = 1 m = 2 m = 4 - Yuen-Long marbles all with crack density 24.3% and same mean microcrack length. - Weibull model with m = 1 2 has a distribution closest to power law. It has a wider range of crack lengths. 2a (mm) PDF Strength (MPa) PDF PDFs of strengths obtained by size 1 SVEs for different distributions m = 1 2 m = 1 m = 2 m = 4 m = 4 results in higher strength and more uniform fracture strength field. Macroscopic fracture simulation under dynamic tension in vertical direction Strain Stress Macroscopic strain-stress: m = 1 2 : weakest m = 4: strongest Concluding remarks: - Homogenization by SVEs maintains sample-to-sample variation and material inhomogeneity which are very important in brittle fracture – Shape and size of SVE affect the inhomogeneity and anisotropy of homogenized properties – The distribution of defects affect macroscopic response. Fracture strength fields
