# A phase field approach for damage propagation in periodic microstructured materials

- 20 Downloads

## Abstract

In the present work, the evolution of damage in periodic composite materials is investigated through a novel finite element-based multiscale computational approach. The proposed methodology is developed by means of the original combination of asymptotic homogenization with the phase field approach to nonlocal damage. This last is applied at the macroscale level on the equivalent homogeneous continuum, whose constitutive properties are obtained in closed form via a two-scale asymptotic homogenization scheme. The formulation considers different assumptions on the evolution of damage at the microscale (e.g., damage in the matrix and not in the inclusion/fiber), as well as the role played by the microstructural reinforcement, i.e. its volumetric content and shape. Numerical results show that the proposed formulation leads to an apparent tensile strength and a post-peak branch of unnotched and notched specimens dependent not only on the internal length scale of the phase field approach, as for homogeneous materials, but also on microstructural features. Down-scaling relations provide the full reconstruction of the microscopic fields at any point of the macroscopic model, as a simple post-processing operation.

## Keywords

Periodic microstructured material Asymptotic homogenization scheme Phase field approach## Notes

### Acknowledgements

AB would like to acknowledge the financial support by National Group of Mathematical Physics (GNFM-INdAM). MP would like to acknowledge the financial support of the Italian Ministry of Education, University and Research to the Research Project of National Interest (PRIN 2017: “XFAST-SIMS: Extra fast and accurate simulation of complex structural systems” (CUP: D68D19001260001)). The authors would like to thank the IMT School for Advanced Studies Lucca for its support to the stays of FF and JR in the IMT Campus as visiting researchers in 2019, making possible the realization of this joint work.

## References

- Addessi D, De Bellis M, Sacco E (2013) Micromechanical analysis of heterogeneous materials subjected to overall cosserat strains. Mech Res Commun 54:27–34CrossRefGoogle Scholar
- Allaire G (1992) Homogenization and two-scale convergence. SIAM J Math Anal 23:1482–1518CrossRefGoogle Scholar
- Ambati M, Gerasimov T, De Lorenzis L (2015) Phase-field modeling of ductile fracture. Comput Mech 55(5):1017–1040CrossRefGoogle Scholar
- Andrianov I, Bolshakov V, Danishevsḱyy V, Weichert D (2008) Higher order asymptotic homogenization and wave propagation in periodic composite materials. Proc R Soc Lond 464(2093):1181–1201CrossRefGoogle Scholar
- Areias P, Rabczuk T, Msekh M (2016) Phase-field analysis of finite-strain plates and shells including element subdivision. Comput Methods Appl Mech Eng 312:322–350CrossRefGoogle Scholar
- Arteiro A, Catalanotti G, Melro AR, Linde P, Camanho PP (2015) Micro-mechanical analysis of the effect of ply thickness on the transverse compressive strength of polymer composites. Compos Part A 79:127–137CrossRefGoogle Scholar
- Bacca M, Bigoni D, Dal Corso F, Veber D (2013a) Mindlin second-gradient elastic properties from dilute two-phase cauchy-elastic composites. Part i: closed form expression for the effective higher-order constitutive tensor. Int J Solids Struct 50(24):4010–4019CrossRefGoogle Scholar
- Bacca M, Bigoni D, Dal Corso F, Veber D (2013b) Mindlin second-gradient elastic properties from dilute two-phase cauchy-elastic composites part ii: higher-order constitutive properties and application cases. Int J Solids Struct 50(24):4020–4029CrossRefGoogle Scholar
- Bacca M, Dal Corso F, Veber D, Bigoni D (2013c) Anisotropic effective higher-order response of heterogeneous cauchy elastic materials. Mech Res Commun 54:63–71CrossRefGoogle Scholar
- Bacigalupo A (2014) Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits. Meccanica 49(6):1407–1425CrossRefGoogle Scholar
- Bacigalupo A, Gambarotta L (2010) Micro-polar and second order homogenization of periodic masonry. Mater Sci Forum Trans Tech Publ 638:2561–2566CrossRefGoogle Scholar
- Bacigalupo A, Gambarotta L (2013) A multi-scale strain-localization analysis of a layered strip with debonding interfaces. Int J Solids Struct 50(13):2061–2077CrossRefGoogle Scholar
- Bacigalupo A, Gambarotta L (2014a) Computational dynamic homogenization for the analysis of dispersive waves in layered rock masses with periodic fractures. Comput Geotech 56:61–68CrossRefGoogle Scholar
- Bacigalupo A, Gambarotta L (2014b) Second-gradient homogenized model for wave propagation in heterogeneous periodic media. Int J Solids Struct 51(5):1052–1065CrossRefGoogle Scholar
- Bacigalupo A, Morini L, Piccolroaz A (2014) Effective elastic properties of planar sofcs: a non-local dynamic homogenization approach. Int J Hydrogen Energy 39(27):15017–15030CrossRefGoogle Scholar
- Bakhvalov N, Panasenko G (1984) Homogenization: averaging processes in periodic media. Kluwer Academic Publishers, DordrechtGoogle Scholar
- Bazant Z, Jirasek M (2002) Non local integral formulations of plasticity and damage: survey of progress. ASCE J Eng Mech 128(11):1119–1149CrossRefGoogle Scholar
- Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128(11):1119–1149CrossRefGoogle Scholar
- Bensoussan A, Lions J, Papanicolaou G (1978) Asymptotic analysis for periodic structures. North-Holland, AmsterdamGoogle Scholar
- Bigoni D, Drugan W (2007) Analytical derivation of cosserat moduli via homogenization of heterogeneous elastic materials. J Appl Mech 74(4):741–753CrossRefGoogle Scholar
- Bleyer J, Alessi R (2018) Phase-field modeling of anisotropic brittle fracture including several damage mechanisms. Comput Methods Appl Mech Eng 336:213–236CrossRefGoogle Scholar
- Bourdin B, Francfort G, Marigo J (2008) The variational approach to fracture. Springer, BerlinCrossRefGoogle Scholar
- Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826CrossRefGoogle Scholar
- Boutin C (1996) Microstructural effects in elastic composites. Int J Solids Struct 33:1023–1051CrossRefGoogle Scholar
- Camacho G, Ortiz M (1996) Computational modelling of impact damage in brittle materials. Int J Solids Struct 33:2899–2938CrossRefGoogle Scholar
- Carollo V, Reinoso J, Paggi M (2018) Modeling complex crack paths in ceramic laminates: a novel variational framework combining the phase field method of fracture and the cohesive zone model. J Eur Ceramic Soc 38(8):2994–3003 cermodel 2017: Modelling and Simulation Meet Innovation in Ceramics TechnologyCrossRefGoogle Scholar
- Comi C (1999) Computational modelling of gradient-enhanced damage in quasi-brittle materials. Mech Cohes Frict Mater 4(1):17–36CrossRefGoogle Scholar
- De Bellis ML, Addessi D (2011) A cosserat based multi-scale model for masonry structures. Int J Multiscale Comput Eng 9(5):543CrossRefGoogle Scholar
- Del Toro R, Bacigalupo A, Paggi M (2019) Characterization of wave propagation in periodic viscoelastic materials via asymptotic-variational homogenization. Int J Solids Struct 172:110–146CrossRefGoogle Scholar
- Dimitrijevic B, Hackl K (2011) A regularization framework for damage-plasticity models via gradient enhancement of the free energy. Int J Numer Methods Biomed Eng 27:1199–1210CrossRefGoogle Scholar
- Dirrenberger J, Samuel F, Dominique J (2019) Computational homogenization of architectured materials. Architectured materials in nature and engineering. Springer, Cham, pp 89–139CrossRefGoogle Scholar
- Fantoni F, Bacigalupo A, Paggi M (2017) Multi-field asymptotic homogenization of thermo-piezoelectric materials with periodic microstructure. Int J Solids Struct 120:31–56CrossRefGoogle Scholar
- Fantoni F, Bacigalupo A, Paggi M (2018) Design of thermo-piezoelectric microstructured bending actuators via multi-field asymptotic homogenization. Int J Mech Sci 146:319–336CrossRefGoogle Scholar
- Feyel F (2003) A multilevel finite element method (fe2) to describe the response of highly non-linear structures using generalized continua. Comput Methods Appl Mech Eng 192(28–30):3233–3244CrossRefGoogle Scholar
- Forest S (2002) Homogenization methods and the mechanics of generalized continua-part 2. Theor Appl Mech 28(29):113–144CrossRefGoogle Scholar
- Forest S, Sab K (1998) Cosserat overall modeling of heterogeneous materials. Mech Res Commun 25(4):449–454CrossRefGoogle Scholar
- Forest S, Trinh D (2011) Generalized continua and non-homogeneous boundary conditions in homogenisation methods. ZAMM J Appl Math Mech 91(2):90–109CrossRefGoogle Scholar
- Francfort G, Marigo J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46:1319–1342CrossRefGoogle Scholar
- Gambin B, Kröner E (1989) Higher order terms in the homogenized stress-strain relation of periodic elastic media. Physica status solidi (b). Int J Eng Sci 151(2):513–519Google Scholar
- Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond A 221:163–198CrossRefGoogle Scholar
- Guillén-Hernández T, García IG, Reinoso J, Paggi M (2019) A micromechanical analysis of inter-fiber failure in long reinforced composites based on the phase field approach of fracture combined with the cohesive zone model. Int J Fract. https://doi.org/10.1007/s10704-019-00384-8 CrossRefGoogle Scholar
- Gültekin O, Dal H, Holzapfel GA (2018) Numerical aspects of anisotropic failure in soft biological tissues favor energy-based criteria: a rate-dependent anisotropic crack phase-field model. Comput Methods Appl Mech Eng 331:23–52CrossRefGoogle Scholar
- Hansen-Dörr AC, de Borst R, Hennig P, Kästner M (2019) Phase-field modelling of interface failure in brittle materials. Comput Methods Appl Mech Eng 346:25–42CrossRefGoogle Scholar
- Herráez M, Mora D, Naya F, Lopes CS, González C, LLorca J (2015) Transverse cracking of cross-ply laminates: a computational micromechanics perspective. Compos Sci Technol 110:196–204CrossRefGoogle Scholar
- Hubert JS, Palencia ES (1992) Introduction aux méthodes asymptotiques et à l’homogénéisation: application à la mécanique des milieux continus. Masson, ParisGoogle Scholar
- Kaczmarczyk L, Pearce CJ, Bićanić N (2008) Scale transition and enforcement of rve boundary conditions in second-order computational homogenization. Int J Numer Methods Eng 74(3):506–522CrossRefGoogle Scholar
- Khisamitov I, Meschke G (2018) Variational approach to interface element modeling of brittle fracture propagation. Comput Methods Appl Mech Eng 328:452–476CrossRefGoogle Scholar
- Kouznetsova V, Geers M, Brekelmans W (2002) Advanced constitutive modeling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 54:1235–1260CrossRefGoogle Scholar
- Kouznetsova V, Geers M, Brekelmans W (2004) Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng 193(48):5525–5550CrossRefGoogle Scholar
- Kuhn C, Müller R (2014) Simulation of size effects by a phase field model for fracture. Theor Appl Mech Lett 4(5):051008CrossRefGoogle Scholar
- Linder C, Armero F (2007) Finite elements with embedded strong discontinuities for the modeling of failure in solids. Int J Numer Methods Eng 72:1391–1433CrossRefGoogle Scholar
- Linse T, Hennig P, Kästner M, de Borst R (2017) A convergence study of phase-field models for brittle fracture. Eng Fract Mech 184:307–318CrossRefGoogle Scholar
- Maimí P, Camanho P, Mayugo J, Dávila C (2007) A continuum damage model for composite laminates: Part I—constitutive model. Mech Mater 39:897–908CrossRefGoogle Scholar
- Mantič V, García IG (2012) Crack onset and growth at the fibre-matrix interface under a remote biaxial transverse load. Application of a coupled stress and energy criterion. Int J Solids Struct 49(17):2273–2290CrossRefGoogle Scholar
- Martínez-Pañeda E, Golahmar A, Niordson CF (2018) A phase field formulation for hydrogen assisted cracking. Comput Methods Appl Mech Eng 342:742–761CrossRefGoogle Scholar
- Meguid S, Kalamkarov A (1994) Asymptotic homogenization of elastic composite materials with a regular structure. Int J Solids Struct 31:303–316CrossRefGoogle Scholar
- Miehe C, Aldakheel F, Raina A (2016) Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory. Int J Plast 84:1–32CrossRefGoogle Scholar
- Miehe C, Hofacker M, Schänzel L-M, Aldakheel F (2015a) 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 294:486–522CrossRefGoogle Scholar
- Miehe C, Hofacker M, Welschinger F (2010a) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45):2765–2778CrossRefGoogle Scholar
- Miehe C, Hofacker M, Welschinger F (2010b) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45–48):2765–2778CrossRefGoogle Scholar
- Miehe C, Mauthe S, Ulmer H (2014) Formulation and numerical exploitation of mixed variational principles for coupled problems of Cahn-Hilliard-type and standard diffusion in elastic solids. Int J Numer Methods Eng 99:737–762CrossRefGoogle Scholar
- Miehe C, Schänzel L, Ulmer H (2015b) Phase field modeling of fracture in multi-physics problems. Part i. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput Methods Appl Mech Eng 294:449–485CrossRefGoogle Scholar
- Miehe C, Welschinger F, Hofacker M (2010c) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field fe implementations. Int J Numer Methods Eng 83(10):1273–1311CrossRefGoogle Scholar
- Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150CrossRefGoogle Scholar
- Nguyen T-T, Yvonnet J, Bornert M, Chateau C, Sab K, Romani R, Le Roy R (2016a) On the choice of parameters in the phase field method for simulating crack initiation with experimental validation. Int J Fract 197(2):213–226CrossRefGoogle Scholar
- Nguyen TT, Yvonnet J, Bornert M, Chateau C, Sab K, Romani R, Le Roy R (2016b) On the choice of parameters in the phase field method for simulating crack initiation with experimental validation. Int J Fract 197(2):213–226CrossRefGoogle Scholar
- Nguyen T-T, Yvonnet J, Zhu Q-Z, Bornert M, Chateau C (2016c) A phase-field method for computational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography. Comput Methods Appl Mech Eng 312:567–595CrossRefGoogle Scholar
- Oliver J, Huespe A, Blanco S, Linero D (2006) Stability and robustness issues in numerical modeling of material failure with the strong discontinuity approach. Comput Methods Appl Mech Eng 195:7093–7114CrossRefGoogle Scholar
- Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Methods Eng 44:1267–1282CrossRefGoogle Scholar
- Ostoja-Starzewski M, Boccara SD, Jasiuk I (1999) Couple-stress moduli and characteristic length of a two-phase composite. Mech Res Commun 26(4):387–396CrossRefGoogle Scholar
- Otero F, Oller S, Martinez X (2018) Multiscale computational homogenization: review and proposal of a new enhanced-first-order method. Arch Comput Methods Eng 25(2):479–505CrossRefGoogle Scholar
- Paggi M, Corrado M, Reinoso J (2018) Fracture of solar-grade anisotropic polycrystalline silicon: a combined phase field-cohesive zone model approach. Comput Methods Appl Mech Eng 330:123–148CrossRefGoogle Scholar
- Paggi M, Reinoso J (2017) Revisiting the problem of a crack impinging on an interface: a modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model. Comput Methods Appl Mech Eng 321:145–172CrossRefGoogle Scholar
- Paggi M, Wriggers P (2012) Stiffness and strength of hierarchical polycrystalline materials with imperfect interfaces. J Mech Phys Solids 60(4):557–572CrossRefGoogle Scholar
- Panasenko G (2009) Boundary conditions for the high order homogenized equation: laminated rods, plates and composites. Comptes Rendus MEcanique 337(1):8–14CrossRefGoogle Scholar
- Peerlings R, Geers M, De R, Brekelmans W (2001) A critical comparison of nonlocal and gradient-enhanced softening continua. Int J Solids Struct 38:7723–7746CrossRefGoogle Scholar
- Pham K, Amor H, Marigo J, Maurini C (2011) Gradient damage models and their use to approximate brittle fracture. Int J Damage Mech 20(4):618–652CrossRefGoogle Scholar
- Pham K, Marigo J (2013) From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models. Continu Mech Thermodyn 25(2):147–171CrossRefGoogle Scholar
- Quintanas-Corominas A, Reinoso J, Casoni E, Turon A, Mayugo J (2019) A phase field approach to simulate intralaminar and translaminar fracture in long fiber composite materials. Compos Struct 220:899–911CrossRefGoogle Scholar
- Reinoso J, Catalanotti G, Blázquez A, Areias P, Camanho P, París F (2017a) A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3d-version of the puck failure criterion. Int J Solids Struct 126–127:37–53CrossRefGoogle Scholar
- Reinoso J, Paggi M, Linder C (2017b) Phase field modeling of brittle fracture for enhanced assumed strain shells at large deformations: formulation and finite element implementation. Comput Mech 59:981–1001 CrossRefGoogle Scholar
- Rizzi G, Dal Corso F, Veber D, Bigoni D (2019a) Identification of second-gradient elastic materials from planar hexagonal lattices. Part I: analytical derivation of equivalent constitutive tensors. Int J Solids Struct 176–177:1–18CrossRefGoogle Scholar
- Rizzi G, Dal Corso F, Veber D, Bigoni D (2019b) Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation. Int J Solids Struct 176–177:19–35CrossRefGoogle Scholar
- Salvadori A, Fantoni F (2016) Fracture propagation in brittle materials as a standard dissipative process: general theorems and crack tracking algorithms. J Mech Phys Solids 95:681–696CrossRefGoogle Scholar
- Salvadori A, Wawrzynek P, Fantoni F (2019) Fracture propagation in brittle materials as a standard dissipative process: effective crack tracking algorithms based on a viscous regularization. J Mech Phys Solids 127:221–238CrossRefGoogle Scholar
- Sargado JM, Keilegavlen E, Berre I, Nordbotten JM (2018) High-accuracy phase-field models for brittle fracture based on a new family of degradation functions. J Mech Phys Solids 111:458–489CrossRefGoogle Scholar
- Sevostianov I, Giraud A (2013) Generalization of maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape. Int J Eng Sci 64:23–36CrossRefGoogle Scholar
- Sevostianov I, Yilmaz N, Kushch V, Levin V (2005) Effective elastic properties of matrix composites with transversely-isotropic phases. Int J Solids Struct 42(2):455–476CrossRefGoogle Scholar
- Smyshlyaev V (2009) Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization. Mech Mater R59:434–447CrossRefGoogle Scholar
- Smyshlyaev V, Cherednichenko K (2000) On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media. J Mech Phys Solids 48(6):1325–1357CrossRefGoogle Scholar
- Tanné E, Li T, Bourdin B, Marigo J-J, Maurini C (2018) Crack nucleation in variational phase-field models of brittle fracture. J Mech Phys Solids 110:80–99CrossRefGoogle Scholar
- Teichtmeister S, Kienle D, Aldakheel F, Keip M (2017) Phase field modeling of fracture in anisotropic brittle solids. Int J Non Linear Mech 97:1–21CrossRefGoogle Scholar
- Tran T, Monchiet V, Bonnet G (2012) A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media. Int J Solids Struct 49(5):783–792CrossRefGoogle Scholar
- Trovalusci P, Ostoja-Starzewski M, De Bellis ML, Murrali A (2015) Scale-dependent homogenization of random composites as micropolar continua. Eur J Mech A 49:396–407CrossRefGoogle Scholar
- Turon A, González E, Sarrado C, Guillamet G, Maimí P (2018) Accurate simulation of delamination under mixed-mode loading using a cohesive model with a mode-dependent penalty stiffness. Compos Struct 184:506–511CrossRefGoogle Scholar
- Verhoosel CV, de Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96(1):43–62CrossRefGoogle Scholar
- Willis JR (1981) Variational and related methods for the overall properties of composites. In: Advances in applied mechanics, vol 21. Elsevier, pp 1–78Google Scholar
- Wu JY (2017) A unified phase-field theory for the mechanics of damage and quasi-brittle failure. J Mech Phys Solids 103:72–99 CrossRefGoogle Scholar
- Yuan X, Tomita Y, Andou T (2008) A micromechanical approach of nonlocal modeling for media with periodic microstructures. Mech Res Commun 35(1–2):126–133 CrossRefGoogle Scholar
- Zienkiewicz O, Taylor R (1977) The finite element method. McGraw-hill, LondonGoogle Scholar