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1. Fracture Diodes: Directional Asymmetry of Fracture Toughness

   https://doi.org/10.1103/PhysRevLett.126.025503  

   Brodnik, N. R.; Brach, S.; Long, C. M.; Ravichandran, G.; Bourdin, B.; Faber, K. T.; Bhattacharya, K. (January 2021, Physical Review Letters)
2. Fracture toughness of thermoelectric materials

   https://doi.org/10.1016/j.mser.2021.100607  

   Li, Guodong; An, Qi; Duan, Bo; Borgsmiller, Leah; Malki, Muath Al; Agne, Matthias; Aydemir, Umut; Zhai, Pengcheng; Zhang, Qingjie; Morozov, Sergey I.; et al (April 2021, Materials Science and Engineering: R: Reports)

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3. Nine circles of elastic brittle fracture: A series of challenge problems to assess fracture models

   https://doi.org/10.1016/j.cma.2025.118449  

   Kamarei, Farhad; Zeng, Bo; Dolbow, John E; Lopez-Pamies, Oscar (January 2026, Computer Methods in Applied Mechanics and Engineering)
4. Variable-order fracture mechanics and its application to dynamic fracture

   https://doi.org/10.1038/s41524-021-00492-x  

   Patnaik, Sansit; Semperlotti, Fabio (February 2021, npj Computational Materials)

   Abstract This study presents the formulation, the numerical solution, and the validation of a theoretical framework based on the concept of variable-order mechanics and capable of modeling dynamic fracture in brittle and quasi-brittle solids. More specifically, the reformulation of the elastodynamic problem via variable and fractional-order operators enables a unique and extremely powerful approach to model nucleation and propagation of cracks in solids under dynamic loading. The resulting dynamic fracture formulation is fully evolutionary, hence enabling the analysis of complex crack patterns without requiring any a priori assumption on the damage location and the growth path, and without using any algorithm to numerically track the evolving crack surface. The evolutionary nature of the variable-order formalism also prevents the need for additional partial differential equations to predict the evolution of the damage field, hence suggesting a conspicuous reduction in complexity and computational cost. Remarkably, the variable-order formulation is naturally capable of capturing extremely detailed features characteristic of dynamic crack propagation such as crack surface roughening as well as single and multiple branching. The accuracy and robustness of the proposed variable-order formulation are validated by comparing the results of direct numerical simulations with experimental data of typical benchmark problems available in the literature. 

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5. Random field realization and fracture simulation of rocks with angular bias for fracture strength

   Garrard, J. M.; Abedi, Reza; Clarke, P. L. (January 2018, 52nd U.S. Rock Mechanics/Geomechanics Symposium)

   Realistic fracture simulations in rock as a heterogeneous brittle material with significant inherent randomness require the use of models that incorporate its inhomogeneities and statistical variability. The high dependence of their fracture progress on microstructural defects results in wide scatter in their ultimate strength and the so-called size effect. This paper proposes an approach based on statistical volume elements (SVEs) to characterize rock fracture strength at the mesoscale. The use of SVEs ensures that the material randomness is maintained upon averaging of microscale features. Because the fracture strength varies not just spatially, but also by the angle of loading, this work includes angular variability to properly model a heterogeneous rock domain. Two different microcrack distributions, one angularly uniform and one angularly biased towards a specific angle, are used to show that implementing angle into the random field provides the most realistic fracture simulation. An adaptive asynchronous spacetime discontinuous Galerkin (aSDG) finite element method is used to perform the dynamic fracture simulations. 

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