A Micromechanical Inelastic Strain-Damage Constitutive Model Based on Wing- and Secondary- Cracking Mechanisms under Dynamic Loading

Document Type : Research Article


دانشکده مهندسی معدن و متالورژی، دانشگاه صنعتی امیرکبیر، تهران، ایران،


For most rock materials, there exists a coupling between inelastic deformations caused by crack displacements on micro-crack faces and damage evolution due to nucleation and growth of wing- and secondary cracks. While rock material is subjected to dynamic loading, the interaction between micro-cracks plays an important role in materials behavior. The self-consistent homogenization scheme is implemented in this paper to consider micro-cracks interaction and determine the equivalent mechanical properties of micro-cracked rock deteriorated by damage evolution. This article aims to develop a self-consistent based micromechanical damage model by taking into account the wing- and secondary-cracking mechanisms accompanied by inelastic strains caused by crack displacements under dynamic compressive loading. While stress intensity factors in tensile and in-plane shear modes at flaw tips exceed the material fracture toughness in modes I and II, respectively, wing- and secondary cracks are sprouted and damage evolution occurs. For closed cracks, an appropriate criterion for the secondary-crack initiation is proposed in this paper. The developed model algorithm is programmed in the commercial finite difference software environment for numerical simulation of rock material to investigate the relationship between the macroscopic mechanical behavior and the microstructure. The fracture toughness parameters of the rock samples are experimentally determined. The rock microstructure parameters (average initial length and density of flaws) are studied using scanning electron microscopy. To verify the developed model, a series of numerical simulations are carried out to numerically reproduce the Split-Hopkinson pressure bar test results. The simulation results demonstrate that the developed micromechanical model can adequately reproduce many features of the rock behavior such as softening in the post-peak region, damage induced by wing- and secondary cracks, and irreversible deformations caused by crack displacements on micro-cracks.  Furthermore, the softening behavior of rock material in the post-peak region is affected by considering inelasticity and the secondary cracking mechanisms. Therefore, the rock sample simulation with the coupled inelastic-damage model can increase inelastic deformations in the post-peak region as a result of irreversible strains caused by crack displacements on micro-cracks. The simulation by considering the secondary-crack mechanism leads to an increase in the micro-cracking process, damage, and fragmentation in rock material.


Main Subjects

[1] Wong, T.F. Micromechanics of faulting in Westerly granite. Int. J. Rock Mech. Min. Sci. 1982: 19, 49–62.
[2] Fredrich, J.T., Evans, B., Wong, T.F. Micromechanics of the brittle to plastic transition in Carraramarbe. J. Geophys. Res. 1989: 94, 4129–4145.
[3] Horri, H., Nemat-Nasser, S. Brittle failure in compression: splitting, faulting   and brittle-ductile transition. J. Phil. Trans.  R. Soc. A. 1986. 337-374.              
[4] Nemat-Nasser, S., Obata, M. A microcrack model of dilatancy in brittle materials, J. Appl. Mech.1998: 55, 24-35.
[5] Nemat-Nasser, S., Deng, H. Strain-rate effect on brittle failure in compression.ActaMetall.Mater. 1994: 42, 1013-1024.
[6] Zhu Q.Z., Kondo D., Shao J.F. Micromechanical analysis of coupling between anisotropic damage and friction in quasi brittle materials: Role of homogenization scheme. Int. J. Solids. Struct. 2008: 45, 1358-1405.
[7] Zhu Q.Z., Shao J.F., Kondo D. A micromechanics-based thermodynamic formulation of isotropic damage with unilateral and friction effects.Euro. J. Mech. Solids. 2011: 30, 316-325.
[8] Xie N., Zhu Q.Z., Xu L.H., Shao J.F., A micromechanics-based elastoplastic damage model for quasi-brittle rocks. Computers and Geotechnics 2011: 38, 970–977.
[9] Molladavoodi H., Sliding and damage criteria investigation of a micromechanical damage model for closed frictional microcracks. Computers and Geotechnics 2015: 67, 135–141.
[10] Qi, M., Shao, J.F., Giraud, A., Zhu, Q.Z., Colliat, J.B. Damage and plastic friction in initially anisotropic quasi brittle materials. Int. J. Plast; 2016: 82, 260-282.
[11] Paliwal, B., Ramesh, K. An interacting micro-crack damage model for failure of brittle materials under compression. J. Mech. Phys. Solids. 2008: 56, 896-923.
[12] Katcoff, C., Graham-Brady, L. Modeling dynamic brittle behavior of materials with circular flaws or pores. Int. J. Solids. Struct. 2014: 51, 754-766.
[13] Hu, G., Liu, J., Graham-Brady, L., Ramesh, K.T. A 3D mechanistic model for brittle materials containing evolving flaw distributions under dynamic multiaxial loading. J. Mech. Phys. Solids. 2015:78, 269-297.
[14] Liu, J., Graham-Brady, L. Effective anisotropic compliance relationships for wing-cracked brittle materials under compression.Int. J. Solids Struct. 2016, 100-101, 151–168.
[15] Ayyagari, R.S., Daphalapurkar, N.P., Ramesh, K.T. The effective compliance of spatially evolving planar wing-cracks.J.MechPhys Solids. 2018: 111, 503–529.
[16] Part sazeh structural engineering, Structural Health Monitoring – SHM, 2015.
[17] Mohammad Bahmani, Seyed Mehdi Zahrai; “Developing a procedure for simultaneous vibration control and health monitoring of structures using semiactive viscous dampers” Amirkabir Journal of Civil Engineering, 2019, DOi: 10.22060/CEEJ.2019.16737.6324.
[18] Gross, G., Seelig, T. Fracture mechanics with an introduction to micromechanics, 2 Springer Science, 2011. Business Media, New York, NY.
[19] J.Liu, L.Graham-Brady; “Effective anisotropic compliance relationships for wing-cracked brittle materials under compression” Int. J. Solids Struct. 100-101, 151–168. 2016.
[20] B.Budiansky, R-J.O’Connell; “Elastic moduli of a cracked solid” Int. J. Solids Struct. 12, 81–97, 1976.
[21] Daphalapurkar, N., Ramesh, K.T., Graham-Brady, L., Molinari, J-F. Predicting variability in the dynamic failure strength of brittle materials considering pre-existing flaws.J.MechPhys solids. 2011: 59, 297–319.
[22] G.Hu, J.Liu, L.Graham-Brady, K.T.Ramesh; “A 3D mechanistic model for brittle materials containing evolving flaw distributions under dynamic multiaxial loading” J. Mech. Phys. Solids. 78, 269-297, 2015.
[23] Hosseini-Nasab, H., Marji, M.F., 2007. “A semi-infinite higher-order displacement discontinuity method and its application to the quasistatic analysis of radial cracks produced by blasting”. Journal of mechanics of materials and structures, vol. 2, pp. 439-458.
[24] A.Bobet, H.H.Einstein; “Fracture coalescence in rock-type materials under uniaxial and biaxial compression” Int. J. Rock Mech. Min. Sci. 35, 863-888. 1998.
[25] L.B.Freund; “Crack propagation in an elastic solid Subjected to general loading-II. Non-uniform rate of extension” Journal of the Mechanics and Physics of Solids 20, pp. 141-152, 1972.
[26] M.H.ahmadi, H.Molladavoodi; “A micromechanical damage model under uniaxial compreesive with high strain rates for brittle materials”, Journal of Mineral Resources Engineering (JMRE), 2018, Vol. 3, No. 2, pp. 1-16.
[27] A.Sayes, H.Dahy’Gaber, Hassib; “Spectral discrimination between quary blasts and microearthquakes in Southern Eqypt”, Researchjournal of earth sciences 2, 2010.
[28] M.R.Saharan, H.S.Mitri; “Numerical procedure for dynamic simulation of discrete fractures due to blasting”, Rock Mech. Rock Eng, 2008.
[29] Y.Zhou, J.Zhao; “Advances in rock dynamics and applications” first Edition, CRC Press/Balkema, 2011.
[30] L.B. Freund; “Dynamic fracture mechanics” Cambridge University Press. New York, NY, 1990.
[31] K. B.  J.Broberg; “Applied Mechanics” 31, 546, 1964.
[32] R.Burridge, G.Conn, Freund L. B., J. Geophys. Res. 84, 2210, 1979.
[33] A.J.Rosakis, O.Samudrala, D.Coker; “Cracks faster than the shear wave speed” Graduate aeronautical laboratories California institute of technology, Pasadena, ca 91125, 1998.
[34] I.Carol, E.Rizzi, K.Willam; “On the formulation of anisotropic elastic degradation. I. Theory based on a pseudo-logarithmic damage tensor rate” Int. J. Solids Struct. 38, 491-518, 2001.
[35] M.R.M. Aliha, M.R. Ayatollahi; “Rock fracture toughness study using cracked chevron notched Brazilian disc specimen under pure modes I and II loading – A statistical approach” Theoretical and Applied Fracture Mechanics 69, 17–25, 2014.
[36] K.Liu, M.Ostadhassan; “Multi-scale fractal analysis of pores in shale rocks” Journal of Applied Geophysics 140, 1–10, 2017.
[37] X.Q.Zhou, H.Hao; “Mesoscale modelling of compressive behaviour of concrete at high strain rate” In: Proceedings of the Australian Structural Engineering Conference, June 26–27, 2008.
[38] G.W.Ma, A.Dong, J.Li; “Modeling strain rate effect for heterogeneous brittle materials” In First International Conference on Analysis and Design of Structures against Explosive and Impact Loads, September 15–17, 2006, Tianjin, China. Transaction of Tianjin University, 12 (Suppl.), pp. 79–82, 2006.
[39] J.F.Georgin, J.M.Reynouard; “Modeling of structures subjected to impact: concrete behaviour under high strain rate” Cement and Concrete Composites 25, 131–143, 2003.
[40] S.Sharma, V.M. Chavan, R.G. Agrawal, R.J. Patel; “Split-hopkinson pressure bar: An experimental technique for high strain rate tests” Bhabha atomic research centre mumbai, India, 2011.
[41] ] J.Kimberley, K.T.Ramesh, N.P.Daphalapurkar; “A scaling law for the dynamic strength of brittle solids” J. Acta Mater. 61, 3509–3521, 2013.
[42] Z. Zhang; “Rock Fracture and Blasting: Theory and Applications” University Centre in Svalbard, Longyearbyen, Svalbard, NORWAY, 2016.