Quasi-static anti-plane shear crack propagation in nonlinear strain-limiting elastic solids using phase-field approach

International Journal of Fracture Mechanics - Tập 227 - Trang 153-172 - 2021
Hyun C. Yoon1, Sanghyun Lee2, S. M. Mallikarjunaiah1
1Department of Mathematics & Statistics, Texas A&M University-Corpus Christi, Corpus Christi, USA
2Department of Mathematics, Florida State University, Tallahassee, USA

Tóm tắt

We study a quasi-static evolution of anti-plane crack with the nonlinear strain-limiting model using the phase-field approach. The nonlinear strain-limiting models, a subclass of the implicit constitutive relations, allow the linearized strain value to remain small even if the stress value tends to infinity. To compute the quasi-static crack, we solve the constrained energy minimization for the nonlinear bulk energy coupled with diffusive crack energy. An iterative staggered scheme (L-scheme) is employed for coupling the nonlinear mechanics and phase-field, and augmented Lagrangian is utilized to accommodate crack irreversibility. Several numerical experiments of the proposed framework substantiate the bounded strain in the neighborhood of the crack-tip for both static and quasi-static cracks. The comparisons of bulk and the crack energies and the discrete crack-tip speed between the LEFM and our model are presented.

Tài liệu tham khảo

Aifantis E (1992) On the role of gradients in the localization of deformation and fracture. Int J Eng Sci 30(10):1279–1299 Ambrosio L, Tortorelli V (1992) On the approximation of free discontinuity problems. Boll Unione Mat Ital B 6:105–123 Ambrosio L, Tortorelli V (1990) Approximation of functionals depending on jumps by elliptic functionals via \(\gamma \)-convergence. Commun Pure Appl Math 43:999–1036 Arndt D, Bangerth W, Clevenger TC, Davydov D, Fehling M, Garcia-Sanchez D, Harper G, Heister T, Heltai L, Kronbichler M, Kynch RM, Maier M, Pelteret J-P, Turcksin B, Wells D (2019) The deal.II library, version 9.1. J Numer Math. https://doi.org/10.1515/jnma-2019-0064 Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129 Borregales M, Radu FA, Kumar K, Nordbotten JM (2018) Robust iterative schemes for non-linear poromechanics. Comput Geosci 22(4):1021–1038 Bourdin B, Francfort G, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826 Bourdin B, Marigo J-J, Maurini C, Sicsic P (2014) Morphogenesis and propagation of complex cracks induced by thermal shocks. Phys Rev Lett 112:014301 Bridges C, Rajagopal K (2015) Implicit constitutive models with a thermodynamic basis: a study of stress concentration. Z angew Math Phys 66(1):191–208 Broberg KB (1999) Cracks and fracture. Elsevier, London Brun MK, Ahmed E, Berre I, Nordbotten JM, Radu FA (2020a) Monolithic and splitting solution schemes for fully coupled quasi-static thermo-poroelasticity with nonlinear convective transport. Comput Math Appl 80(8):1964–1984 Brun MK, Wick T, Berre I, Nordbotten JM, Radu FA (2020b) An iterative staggered scheme for phase field brittle fracture propagation with stabilizing parameters. Comput Methods Appl Mech Eng 361:112752. https://doi.org/10.1016/j.cma.2019.112752 Bulíček M, Málek J, Rajagopal KR, Walton JR (2015) Existence of solutions for the anti-plane stress for a new class of strain-limiting elastic bodies. Calc Var Partial Differ Equ 54(2):2115–2147 Bustamante R, Rajagopal K (2010) A note on plane strain and plane stress problems for a new class of elastic bodies. Math Mech Solids 15(2):229–238 Bustamante R, Rajagopal K (2011) Solutions of some simple boundary value problems within the context of a new class of elastic materials. Int J Non-Linear Mech 46(2):376–386 Bustamante R, Rajagopal K (2014) A note on some new classes of constitutive relations for elastic bodies. IMA J Appl Math 80(5):1287–1299 Bustamante R, Rajagopal K (2015) Implicit constitutive relations for nonlinear magnetoelastic bodies. Proc R Soc A 471(2175):20140959 Bustamante R, Rajagopal K (2018) A nonlinear model for describing the mechanical behaviour of rock. Acta Mech 229(1):251–272 Choo J, Sun W (2018) Cracking and damage from crystallization in pores: coupled chemo-hydro-mechanics and phase-field modeling. Comput Methods Appl Mech Eng 335:347–379 Dal Maso G, Toader R (2002) A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch Ration Mech Anal 162(2):101–135 Dal Maso G, Francfort G, Toader R (2005) Quasistatic crack growth in nonlinear elasticity. Arch Ration Mech Anal 176(2):165–225 Dugdale D (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104 Español M, Kochmann D, Conti S, Ortiz M (2013) A \(\gamma \)-convergence analysis of the quasicontinuum method. Multiscale Model Simul 11(3):766–794 Ferguson LA, Muddamallappa M, Walton JR (2015) Numerical simulation of mode-III fracture incorporating interfacial mechanics. Int J Fract 192(1):47–56 Fortin M, Glowinski R (2000) Augmented Lagrangian methods: applications to the numerical solution of boundary-value problems, vol 15. Elsevier, Amsterdam Francfort G, Marigo J-J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342 Francfort GA, Larsen CJ (2003) Existence and convergence for quasi-static evolution in brittle fracture. Commun Pure Appl Math J Courant Inst Math Sci 56(10):1465–1500 Garwood S (1997) Investigation of the MV Kurdistan casualty. Eng Fail Anal 4(1):3–24 Glowinski R, Le Tallec P (1989) Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, vol 9. SIAM, Philadelphia Gou K, Mallikarjuna M, Rajagopal K, Walton J (2015) Modeling fracture in the context of a strain-limiting theory of elasticity: a single plane-strain crack. Int J Eng Sci 88:73–82 Griffith A (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond 221:163–198 Guo X, Bao Z, Shang F (2013) Mixed-mode mechanical responses of Ni(111)/\(\alpha \)-Al\(_2\)O\(_3\)(0001) interface by first-principle calculations. J Mater Res 28(21):3018–3028 Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323 Irwin G (1957) Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech 24:361–364 Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87:045501. https://doi.org/10.1103/PhysRevLett.87.045501 Kim C, Schiavone P, Ru C-Q (2010) The effects of surface elasticity on an elastic solid with mode-III crack: complete solution. J Appl Mech 77(2):021011 Kulvait V, Málek J, Rajagopal K (2013) Anti-plane stress state of a plate with a V-notch for a new class of elastic solids. Int J Fract 179(1–2):59–73 Kulvait V, Málek J, Rajagopal K (2019) The state of stress and strain adjacent to notches in a new class of nonlinear elastic bodies. J Elast 135(1–2):375–397 Lawn B (1993) Fracture of brittle solids, 2nd edn. Cambridge solid state science series. Cambridge University Press, Cambridge Mai T, Walton JR (2015) On strong ellipticity for implicit and strain-limiting theories of elasticity. Math Mech Solids 20(2):121–139. https://doi.org/10.1177/1081286514544254 Mallikarjunaiah SM, Walton JR (2015) On the direct numerical simulation of plane-strain fracture in a class of strain-limiting anisotropic elastic bodies. Int J Fract 192(2):217–232. https://doi.org/10.1007/s10704-015-0006-5 Miehe C, Hofacker M, Schaenzel L-M, Aldakheel F (2015) Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic-plastic solids. Comput Methods Appl Mech Eng 294:486–522 Mikelić A, Wheeler MF, Wick T (2015) A quasi-static phase-field approach to pressurized fractures. Nonlinearity 28(5):1371–1399 Muddamallappa MS (2015) On two theories for brittle fracture: modeling and direct numerical simulations. PhD Thesis, Texas A&M University Noii N, Wick T (2019) A phase-field description for pressurized and non-isothermal propagating fractures. Comput Methods Appl Mech Eng 351:860–890 Ortiz A, Bustamante R, Rajagopal K (2012) A numerical study of a plate with a hole for a new class of elastic bodies. Acta Mech 223(9):1971–1981 Ortiz-Bernardin A, Bustamante R, Rajagopal K (2014) A numerical study of elastic bodies that are described by constitutive equations that exhibit limited strains. Int J Solids Struct 51(3–4):875–885 Rajagopal KR (2003) On implicit constitutive theories. Appl Math 48(4):279–319 Rajagopal K (2007) The elasticity of elasticity. Z Angew Math Phys 58(2):309–317 Rajagopal K (2011a) Non-linear elastic bodies exhibiting limiting small strain. Math Mech Solids 16(1):122–139 Rajagopal K (2011b) Conspectus of concepts of elasticity. Math Mech Solids 16(5):536–562 Rajagopal K (2014) On the nonlinear elastic response of bodies in the small strain range. Acta Mech 225(6):1545–1553 Rajagopal K, Srinivasa A (2007) On the response of non-dissipative solids. Proc R Soc Lond A 463:357–367 Rajagopal K, Srinivasa A (2009) On a class of non-dissipative materials that are not hyperelastic. Proc R Soc Lond A 465:493–500 Rajagopal K, Walton J (2011) Modeling fracture in the context of a strain-limiting theory of elasticity: a single anti-plane shear crack. Int J Fract 169(1):39–48 Sendova T, Walton JR (2010) A new approach to the modeling and analysis of fracture through extension of continuum mechanics to the nanoscale. Math Mech Solids 15(3):368–413 Lee S, Wheeler MF, Wick T (2016) Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model. Comput Methods Appl Mech Eng 305:111–132 Shiozawa S, Lee S, Wheeler MF (2019) The effect of stress boundary conditions on fluid-driven fracture propagation in porous media using a phase-field modeling approach. Int J Numer Anal Methods Geomech 43(6):1316–1340 Tadmor E, Phillips R, Ortiz M (1996) Mixed atomistic and continuum models of deformation in solids. Langmuir 12(19):4529–4534 Walton JR (2012) A note on fracture models incorporating surface elasticity. J Elast 109(1):95–102 Walton JR (2014) Plane-strain fracture with curvature-dependent surface tension: mixed-mode loading. J Elast 114(1):127–142 Walton JR, Muddamallappa M (2016) Plane strain fracture with surface mechanics: non-local boundary regularization. In: International congress of theoretical and applied mechanics, vol XXIV Wheeler MF, Wick T, Wollner W (2014) An augmented-Lagrangian method for the phase-field approach for pressurized fractures. Comput Methods Appl Mech Eng 271:69–85 Wheeler MF, Wick T, Lee S (2020) IPACS: integrated phase-field advanced crack propagation simulator. an adaptive, parallel, physics-based-discretization phase-field framework for fracture propagation in porous media. Comput Methods Appl Mech Eng 367:113124. https://doi.org/10.1016/j.cma.2020.113124 Xu X, Needleman A (1993) Void nucleation by inclusion debonding in a crystal matrix. Model Simul Mater Sci Eng 1(2):111–132 Yoffe E (1951) LXXV. The moving Griffith crack. Lond Edinb Dublin Philos Mag J Sci 42(330):739–750 Yoshioka K, Parisio F, Naumov D, Lu R, Kolditz O, Nagel T (2019) Comparative verification of discrete and smeared numerical approaches for the simulation of hydraulic fracturing. GEM Int J Geomath 10(1):13 Zemlyanova AY (2013) The effect of a curvature-dependent surface tension on the singularities at the tips of a straight interface crack. Q J Mech Appl Math 66(2):199–219 Zemlyanova AY (2016) Curvilinear mode-I/mode-II interface fracture with a curvature-dependent surface tension on the boundary. IMA J Appl Math 81(6):1112–1136 Zemlyanova AY, Walton JR (2012) Modeling of a curvilinear planar crack with a curvature-dependent surface tension. SIAM J Appl Math 72(5):1474–1492