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On a 2D hydro-mechanical lattice approach for modelling hydraulic fracture. (English) Zbl 1349.74033
Summary: A 2D lattice approach to describe hydraulic fracturing is presented. The interaction of fluid pressure and mechanical response is described by Biot’s theory. The lattice model is applied to the analysis of a thick-walled cylinder, for which an analytical solution for the elastic response is derived. The numerical results obtained with the lattice model agree well with the analytical solution. Furthermore, the coupled lattice approach is applied to the fracture analysis of the thick-walled cylinder. It is shown that the proposed lattice approach provides results that are independent of the mesh size. Moreover, a strong geometrical size effect on nominal strength is observed which lies between analytically derived lower and upper bounds. This size effect decreases with increasing Biot’s coefficient.

##### MSC:
 74A45 Theories of fracture and damage 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
##### Keywords:
hydraulic fracture; lattice; hydro-mechanical; damage
OOFEM
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##### References:
 [1] Asahina, D.; Houseworth, J. E.; Birkholzer, J. T.; Rutqvist, J.; Bolander, J. E., Hydro-mechanical model for wetting/drying and fracture development in geomaterials, Comput. Geosci., 65, 13-23, (2014) [2] Aurenhammer, F., Voronoi diagrams—a survey of a fundamental geometric data structure, ACM Comput. Surv., 23, 345-405, (1991) [3] Bažant, Z. P., Scaling of structural strength, (2002), Hermes-Penton London · Zbl 1110.74321 [4] Bažant, Z. P.; Salviato, M.; Chau, V. T.; Viswanathan, H.; Zubelewicz, A., Why fracking works, J. Appl. Mech., 81, (2014), 101010-1-101010-10 [5] Bažant, Z. P.; Tabbara, M. R.; Kazemi, M. T.; Pijaudier-Cabot, G., Random particle model for fracture of aggregate or fiber composites, J. Eng. Mech. ASCE, 116, 1686-1705, (1990) [6] Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164, (1941) · JFM 67.0837.01 [7] Bishop, R. F.; Hill, R.; Mott, N. F., The theory of indentation and hardness tests, Proc. Phys. Soc., 57, 147, (1945) [8] Bolander, J. E.; Berton, S., Simulation of shrinkage induced cracking in cement composite overlays, Cement Concr. Compos., 26, 861-871, (2004) [9] Bolander, J. E.; Saito, S., Fracture analysis using spring networks with random geometry, Eng. Fract. Mech., 61, 569-591, (1998) [10] Coussy, O., Mechanics and physics of porous solids, (2010), John Wiley & Sons Chichester [11] Delaplace, A.; Pijaudier-Cabot, G.; Roux, S., Progressive damage in discrete models and consequences on continuum modelling, J. Mech. Phys. Solids, 44, 99-136, (1996) · Zbl 1054.74507 [12] Goulty, N. R., Emplacement mechanism of the great whin and midland valley dolerite sills, J. Geol. Soc., 162, 1047-1056, (2005) [13] Grassl, P., A lattice approach to model flow in cracked concrete, Cement Concr. Compos., 31, 454-460, (2009) [14] Grassl, P.; Grégoire, D.; Solano, L. R.; Pijaudier-Cabot, G., Meso-scale modelling of the size effect on the fracture process zone of concrete, Int. J. Solids Struct., 49, 1818-1827, (2012) [15] Grassl, P.; Jirásek, M., Meso-scale approach to modelling the fracture process zone of concrete subjected to uniaxial tension, Int. J. Solids Struct., 47, 957-968, (2010) · Zbl 1193.74119 [16] Griffiths, D. V.; Mustoe, G. G.W., Modelling of elastic continua using a grillage of structural elements based on discrete element concepts, Int. J. Numer. Methods Eng., 50, 1759-1775, (2001) · Zbl 1006.74086 [17] Jirásek, M.; Bažant, Z. P., Particle model for quasibrittle fracture and application to sea ice, J. Eng. Mech. ASCE, 121, 1016-1025, (1995) [18] van der Meer, J. J.M.; Kjær, K. H.; Krüger, J.; Rabassa, J.; Kilfeather, A. A., Under pressureclastic dykes in glacial settings, Quat. Sci. Rev., 28, 708-720, (2009) [19] Patzák, B., OOFEM—an object-oriented simulation tool for advanced modeling of materials and structures, Acta Polytech., 52, 59-66, (2012) [20] Rice, J. R.; Cleary, M. P., Some basic stress diffusion solutions for fluid-saturated elastic porous-media with compressible constituents, Rev. Geophys., 14, 227-241, (1976) [21] Shawki, G. S.A.; Elwahi, S. H., Strength of thick-walled permeable cylinders, Int. J. Mech. Sci., 12, 535-551, (1970) [22] Slowik, V.; Saouma, E. V., Water pressure in propagating concrete cracks, J. Struct. Eng., 126, 235-242, (2000) [23] Terzaghi, K., Erdbaumechanik auf bodenphysikalischer grundlage, (1925), Franz Deuticke Leipzig and Vienna · JFM 51.0655.07 [24] Timoshenko, S. P.; Goodier, J. N., Theory of elasticity, (1987), McGraw-Hill New York · Zbl 0266.73008 [25] Šavija, B.; Pacheco, J.; Schlangen, E., Lattice modeling of chloride diffusion in sound and cracked concrete, Cement Concr. Compos., 42, 30-40, (2013) [26] Wang, L.; Ueda, T., Mesoscale modelling of the chloride diffusion in cracks and cracked concrete, J. Adv. Concr. Technol., 9, 241-249, (2011) [27] Yip, M.; Mohle, J.; Bolander, J. E., Automated modeling of three-dimensional structural components using irregular lattices, Comput.-Aided Civil Infrastruct. Eng., 20, 393-407, (2005) [28] Yu, H. S., Cavity expansion methods in geomechanics, (2000), Springer Dordrecht · Zbl 1026.74001 [29] Yu, H. S.; Houlsby, G. T., Finite cavity expansion in dilatant soilsloading analysis, Geotechnique, 41, 173-183, (1991) [30] Zubelewicz, A.; Bažant, Z. P., Interface modeling of fracture in aggregate composites, J. Eng. Mech. ASCE, 113, 1619-1630, (1987)
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