A multi-scale approach for mesh size independent finite element analysis of quasi-brittle material is proposed in this article. The homogenization based multi-scale damage representation is first introduced in obtaining the damage law from micro-cell simulation with consideration of microscopic crack propagation. The key idea to remedy the mesh size sensitivity in the 2D macroscopic finite element analysis is to introduce a damage law directly from the micro-cell simulation, where the micro-cell size should be identical to the macroscopic mesh size. The micro-cells with different sizes are generated and the corresponding simulations are presented in the numerical tests to obtain the micro-cell dependent damage law. The mesh independent finite element analysis results of the notched beam and the double-edge notched specimen affirm the strategy of correcting the mesh sensitivity.
Bakhvalov N and Panasenko G (1984) Homogenisation: averaging processes in periodic media. Netherlands: Kluwer Academic Publishers.
2.
BazantZP (1991) Why continuum damage is nonlocal: Micromechanics arguments. Journal of Engineering Mechanics117(5): 1070–1087.
3.
BazantZPJirásekM (1994) Damage nonlocality due to microcrack interactions: statistical determination of crack influence function. Continuum1(2): 3–17.
4.
BažantZPOhBH (1983) Crack band theory for fracture of concrete. Materials and Structures16(3): 155–177.
5.
BazantZPPlanasJ (1997) Fracture and Size Effect in Concrete and Other Quasibrittle Materials, Boca Raton, Florida: CRC Press.
6.
BazantZPPijaudier-CabotG (1989) Measurement of characteristic length of nonlocal continuum. Journal of Engineering Mechanics115(4): 755–767.
7.
CerveraMChiumentiM (2006) Smeared crack approach: back to the original track. International Journal for Numerical and Analytical Methods in Geomechanics30(12): 1173–1199.
8.
CrumpTFertéGJivkovAet al. (2017) Dynamic fracture analysis by explicit solid dynamics and implicit crack propagation. International Journal of Solids and Structures110–111: 113–126.
9.
DascaluCBilbieGAgiasofitouEK (2008) Damage and size effects in elastic solids: A homogenization approach. International Journal of Solids and Structures45(2): 409–430.
10.
Di PriscoMFerraraLMeftahFet al. (2000) Mixed mode fracture in plain and reinforced concrete: some results on benchmark tests. International Journal of Fracture103(2): 127–148.
11.
FariaROliverJCerveraM (1998) A strain-based plastic viscous-damage model for massive concrete structures. International Journal of Solids and Structures35(14): 1533–1558.
12.
FishJYuQShekK (1999) Computational damage mechanics for composite materials based on mathematical homogenization. International Journal for Numerical Methods in Engineering45(11): 1657–1679.
13.
GhoshSBaiJPaquetD (2009) Homogenization-based continuum plasticity-damage model for ductile failure of materials containing heterogeneities. Journal of the Mechanics and Physics of Solids57(7): 1017–1044.
14.
GhoshSLeeKRaghavanP (2001) A multi-level computational model for multi-scale damage analysis in composite and porous materials. International Journal of Solids and Structures38(14): 2335–2385.
15.
GitmanIMAskesHAifantisEC (2005) The representative volume size in static and dynamic micro-macro transitions. International Journal of Fracture135(1–4): 3–9.
16.
GitmanIMAskesHSluysLJ (2007) Representative volume: existence and size determination. Engineering Fracture Mechanics74(16): 2518–2534.
17.
Hill R (1984) On macroscopic effects of heterogeneity in elastoplastic media at finite strain. Mathematical Proceedings of the Cambridge Philosophical Society 95(3): 481–494.
18.
HillerborgAModéerMPeterssonP (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research6(6): 773–781.
19.
HuyseLMaesMA (2001) Random field modeling of elastic properties using homogenization. Journal of Engineering Mechanics127(1): 27–36.
20.
IlangoSJJSarkarSSameenA (2013) Reconstruction of 2-D porous media using Karhunen–Lóeve expansion. Probabilistic Engineering Mechanics32: 56–65.
21.
JainJRGhoshS (2009) Damage evolution in composites with a homogenization-based continuum damage mechanics model. International Journal of Damage Mechanics18(8): 533–568.
22.
JirasekMGrasslP (2008) Evaluation of directional mesh bias in concrete fracture simulations using continuum damage models. Engineering Fracture Mechanics75(8): 1921–1943.
23.
JuJW (1989) On energy-based coupled elastoplastic damage theories: constitutive modeling and computational aspects. International Journal of Solids and Structures25(7): 803–833.
24.
LiJRenXD (2010) Homogenization-based multi-scale damage theory. Science China Physics, Mechanics and Astronomy53(4): 690–698.
25.
LiangSXRenXDLiJ (2013) A random medium model for simulation of concrete failure. Science China Technological Sciences56(5): 1273–1281.
26.
LinSChenJLiangS (2016) A damage analysis for brittle materials using stochastic micro-structural information. Computational Mechanics57(1): 371–385.
27.
LublinerJOliverJOllerSet al. (1989) A plastic-damage model for concrete. International Journal of Solids and Structures25(3): 299–326.
28.
MazarsJ (1986) A description of micro- and macroscale damage of concrete structures. Engineering Fracture Mechanics25(5): 729–737.
29.
MehtaPKMonteiroPJ (2006) Concrete: Microstructure, Properties, and Materials, New York: McGraw-Hill.
30.
MoësNBelytschkoT (2002) Extended finite element method for cohesive crack growth. Engineering Fracture Mechanics69(7): 813–833.
31.
NezhadMMZhuHWoodyJJet al. (2016) A simplified multiscale damage model for the transversely isotropic shale rocks under tensile loading. International Journal of Damage Mechanics25(5): 705–726.
32.
Nooru-MohamedMBSchlangenEvan MierJG (1993) Experimental and numerical study on the behavior of concrete subjected to biaxial tension and shear. Advanced Cement Based Materials1(1): 22–37.
33.
Ostoja-StarzewskiM (2001) Microstructural randomness versus representative volume element in thermomechanics. Journal of Applied Mechanics69(1): 25–35.
34.
PeerlingsRFleckNA (2004) Computational evaluation of strain gradient elasticity constants. International Journal for Multiscale Computational Engineering2(4): 599–619.
35.
Pijaudier-CabotGBazantZP (1987) Nonlocal damage theory. Journal of Engineering Mechanics113(10): 1512–1533.
36.
SmyshlyaevVPCherednichenkoKD (2000) On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media. Journal of the Mechanics and Physics of Solids48(6): 1325–1357.
37.
RenXDChenJSLiJet al. (2011) Micro-cracks informed damage models for brittle solids. International Journal of Solids and Structures48(10): 1560–1571.
RenXDYangWZZhouYet al. (2008) Behavior of high-performance concrete under uniaxial and biaxial loading. ACI Materials Journal105(6): 548–557.
40.
RotsJGDe BorstR (1987) Analysis of mixed‐mode fracture in concrete. Journal of Engineering Mechanics113(11): 1739–1758.
41.
ShinozukaMDeodatisG (1996) Simulation of multi-dimensional Gaussian stochastic fields by spectral representation. Applied Mechanics Reviews49(1): 29–53.
42.
SillingSAAskariE (2005) A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures83(17–18): 1526–1535.
43.
WangDChenJSunL (2003) Homogenization of magnetostrictive particle-filled elastomers using an interface-enriched reproducing kernel particle method. Finite Elements in Analysis and Design39(8): 765–782.
44.
Willam K, Bicanic N and Sture S (1986) Composite fracture model for strain-softening and localised failure of concrete. Computational Modeling of Reinforced Concrete Structures. Swansea: Pineridge Press, pp. 122–153.
45.
WuJYLiJFariaR (2006) An energy release rate-based plastic-damage model for concrete. International Journal of Solids and Structures43(3): 583–612.
46.
XuXNeedlemanA (1994) Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids42(9): 1397–1434.
