# A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture
## Abbreviations  
The following abbreviations are used in this manuscript:  
* [1]
* [2]
* [3]
* [4]
* [5]
* [6]
* [7]
* [8]
* [9]
* [10]
* [11]
* [12]
* [13]
* [14]
* [15]  
* [PERSON] and [PERSON] (2017) [PERSON]; [PERSON] [PERSON] A peridynamics formulation for quasi-static fracture and contact in rock. _Eng. Geol._**2017**, _225_, 42-48. [CrossRef]
* [PERSON] (1988) [PERSON]; [PERSON] Localization limiters in transient problems. _Int. J. Solids Struct._**1988**, _24_, 581-597. [CrossRef]
* [PERSON] and [PERSON] (1990) [PERSON]; [PERSON] [PERSON] Dynamic strain localization in elasto-(visco-) plastic solids, Part 1. General formulation and one-dimensional examples. _Comput. Methods Appl. Mech. Eng._**1990**, _83_, 247-273. [CrossRef]
* [PERSON] et al. (1998) [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] Gradient-enhanced damage modelling of concrete fracture. _Mech. Coles.-Frict. Mater._**1998**, \(3\), 323-342. [CrossRef]
* [PERSON] (1999) [PERSON] Computational modelling of gradient-enhanced damage in quasi-brittle materials. _Mech. Coles.-Frict. Mater._**1999**, \(4\), 17-36. [CrossRef]
* [PERSON] et al. (2001) [PERSON]; [PERSON]; [PERSON]; [PERSON] A critical comparison of nonlocal and gradient-enhanced softening continua. _Int. J. Solids Struct._**2001**, _38_, 7723-7746. [CrossRef]
* [PERSON] et al. (2017) [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] A new effective rate dependent damage model for dynamic tensile failure of concrete. _Eng. Fract. Mech._**2017**, _176_, 281-299. [CrossRef]
* [PERSON] (1997) [PERSON]; [PERSON] Delayed-damage modelling for fracture prediction of laminated composites under dynamic loading. _Eng. Trans._**1997**, _45_, 29-46. * [PERSON] et al. (2011) [PERSON]; [PERSON]; [PERSON] A non-local visco-elastic damage model and dynamic fracturing. _J. Mech. Phys. Solids_**2011**, _59_, 1752-1776. [CrossRef]
* [PERSON] and [PERSON] (2016) [PERSON] [PERSON]; [PERSON] [PERSON] Modeling of concrete spallation with damaged viscoelasticity and retarded damage. _Int. J. Solids Struct._**2016**, _90_, 153-166. [CrossRef]
* [PERSON] et al. (2017) [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] A relaxation-based approach to damage modeling. _Contin. Mech. Thermodyn._**2017**, _29_, 291-310. [CrossRef]
* [PERSON] and [PERSON] (2016) [PERSON] [PERSON]; [PERSON] Gradient damage vs phase-field approaches for fracture: Similarities and differences. _Comput. Methods Appl. Mech. Eng._**2016**, _312_, 78-94. [CrossRef]
* [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON] Phase field and gradient enhanced damage models for quasi-brittle failure: A numerical comparative study. _Eng. Fract. Mech._**2019**, _207_, 48-67. [CrossRef]
* [PERSON] and [PERSON] (2019) [PERSON]; [PERSON] [PERSON] Asynchronous Spacetime Discontinuous Galerkin Formulation for a Hyperbolic Time-Delay Bulk Damage Model. _J. Eng. Mech._**2019**, accepted. * [PERSON] et al. (2006) [PERSON]; [PERSON]; [PERSON] A spacetime discontinuous Galerkin method for elastodynamics with element-level balance of linear momentum. _Comput. Methods Appl. Mech. Eng._**2006**, _195_, 3247-3273. [CrossRef]
* [PERSON] et al. (2006) [PERSON]; [PERSON]; [PERSON]; [PERSON] An \(h\)-adaptive spacetime-discontinuous Galerkin method for linearized elastodynamics. _Revue Europenne de Mecanique Numerique (Eur. J. Comput. Mech. )_**2006**, _15_, 619-642. * [PERSON] and [PERSON] (1995) [PERSON]; [PERSON] Impact behavior of plain concrete loaded in uniaxial compression. _J. Eng. Mech._**1995**, _121_, 685-693. [CrossRef]
* [PERSON] and [PERSON] (1997) [PERSON]; [PERSON] [PERSON] Crack initiation and propagation in materials with randomly distributed holes. _Eng. Fract. Mech._**1997**, _58_, 395-420. [CrossRef]
* [PERSON] and [PERSON] (2007) [PERSON] [PERSON]; [PERSON] [PERSON] Effect of aggregate structure on fracture process in concrete using 2D lattice model. _Arch. Mech._**2007**, _59_, 365-384. * [PERSON] and [PERSON] (2000) [PERSON]; [PERSON] [PERSON] Probabilistic nonlocal theory for quasibrittle fracture initiation and size effect--I: Theory. _J. Eng.