In this part II, companion article, we present the numerical review of continuum damage mechanics and plasticity in the context of finite element. The numerical advancements in local, nonlocal, and rate-dependent models are presented. The numerical algorithms, type of elements utilized in numerical analysis, the commercial software’s or in-house codes used for the analysis, iterative schemes, explicit or implicit approaches to solving finite element equations, and degree of continuity of element are discussed in this part. Lastly, some open issues in concrete damage modeling and future research needed are also discussed.
Abu Al-RubRKVoyiadjisGZ (2009)
Gradient-enhanced coupled plasticity-anisotropic damage model for concrete fracture: Computational aspects and applications. International Journal of Damage Mechanics18(2): 115–154.
2.
AhmedBVoyiadjisGZParkT (2020)
Damaged plasticity model for concrete using scalar damage variables with a novel stress decomposition. International Journal of Solids and Structures191–192: 56–75.
3.
AhmedBVoyiadjisGZParkT (2021a)
Local and non-local damage model with extended stress decomposition for concrete. International Journal of Damage Mechanics30(8): 1149–1191.
4.
AhmedBVoyiadjisGZParkT (2021b)
A nonlocal damage model for concrete with three length scales. Computational Mechanics68(3): 461–486.
5.
Al-GadhibAHAsad-Ur-RahmanKBaluchMH (1998)
CDM based finite element code for concrete in 3-D. Computers & Structures67(6): 451–462.
6.
Al-GadhibAHBaluchMHShaalanA, et al. (2000)
Damage model for monotonic and fatigue response of high strength concrete. International Journal of Damage Mechanics9(1): 57–78.
7.
AlamSYZhuRLoukiliA (2020)
A new way to analyse the size effect in quasi-brittle materials by scaling the heterogeneity size. Engineering Fracture Mechanics225: 106864.
8.
AzinpourEFerreiraJPSParenteMPL, et al. (2018)
A simple and unified implementation of phase field and gradient damage models. Advanced Modeling and Simulation in Engineering Sciences5(1): 15.
9.
Ba<ZantZPKazemiMT (1990)
Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. International Journal of Fracture44(2): 111–131.
10.
BaktheerAChudobaR (2019)
Classification and evaluation of phenomenological numerical models for concrete fatigue behavior under compression. Construction and Building Materials221: 661–677.
11.
BažantZPPangS-D (2007)
Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture. Journal of the Mechanics and Physics of Solids55(1): 91–131.
12.
BessonJ (2010)
Continuum models of ductile fracture: a review. International Journal of Damage Mechanics19(1): 3–52.
13.
BiZ-wZhangMJinF (2009)
Non-local damage and fracture model of rock and concrete-like materials. Transactions of Nonferrous Metals Society of China19: s824–828.
14.
BobińskiJTejchmanJ (2016)
A coupled constitutive model for fracture in plain concrete based on continuum theory with non-local softening and eXtended finite element method. Finite Elements in Analysis and Design114: 1–21.
15.
BorinoGFaillaBParrinelloF (2003)
A symmetric nonlocal damage theory. International Journal of Solids and Structures40(13–14): 3621–3645.
16.
BrekelmansWAMSchreursPJGde VreeJHP (1992)
Continuum damage mechanics for softening of brittle materials. Acta Mechanica93(1–4): 133–143.
17.
BrünigMMichalskiA (2020)
Numerical analysis of damage and failure behavior of concrete. International Journal of Damage Mechanics29(4): 570–590.
18.
BurlionNGatuingtFPijaudier-CabotG, et al. (2000)
Compaction and tensile damage in concrete: Constitutive modelling and application to dynamics. Computer Methods in Applied Mechanics and Engineering183(3–4): 291–308.
19.
CarpinteriALacidognaG (2006)
Damage monitoring of an historical masonry building by the acoustic emission technique. Materials and Structures39(2): 161–167.
20.
CarpinteriALacidognaGNiccoliniG (2006)
Critical behaviour in concrete structures and damage localization by acoustic emission. Key Engineering Materials312: 305–310.
21.
CarpinteriALacidognaGNiccoliniG, et al. (2007)
Critical defect size distributions in concrete structures detected by the acoustic emission technique. Meccanica43(3): 349–363.
22.
CerveraMChiumentiM (2006)
Smeared crack approach: Back to the original track. International Journal for Numerical and Analytical Methods in Geomechanics30(12): 1173–1199.
23.
CerveraMChiumentiMAgelet de SaracibarC (2004)
Shear band localization via local J2 continuum damage mechanics. Computer Methods in Applied Mechanics and Engineering193(9–11): 849–880.
24.
CerveraMOliverJManzoliO (1996)
A rate-dependent isotropic damage model for the seismic analysis of concrete dams. Earthquake Engineering & Structural Dynamics25(9): 987–1010.
25.
CerveraMTeseiCVenturaG (2018)
Cracking of quasi-brittle structures under monotonic and cyclic loadings: A d+/d− damage model with stiffness recovery in shear. International Journal of Solids and Structures135: 148–171.
26.
ChenEP (1996) A Nonlocal Damage Model for Dynamic Rock Fracture Simulations, 2nd North American Rock Mechanics Symposium.
Montreal, Quebec, Canada:
American Rock Mechanics Association, p. 8.
27.
ChenEP (1999)
Non-local effects on dynamic damage accumulation in brittle solids. International Journal for Numerical and Analytical Methods in Geomechanics23(1): 1–21.
28.
ChenYChenXBuJ (2018)
Nonlinear damage accumulation of concrete subjected to variable amplitude fatigue loading. Bulletin of the Polish Academy of Sciences-Technical Sciences66: 157–163.
29.
ChernJCMarchertasAHBazantZP, et al. (1987)
Damage-plastic loading surface model for concrete. Nuclear Engineering and Design101(2): 107–125.
30.
CicekliUVoyiadjisGZAbu Al-RubRK (2007)
A plasticity and anisotropic damage model for plain concrete. International Journal of Plasticity23(10-11): 1874–1900.
31.
ClaytonJD (2008)
A model for deformation and fragmentation in crushable brittle solids. International Journal of Impact Engineering35(5): 269–289.
32.
ColpoABKosteskiLEIturriozI (2016)
The size effect in quasi-brittle materials: Experimental and numerical analysis. International Journal of Damage Mechanics26(3): 395–416.
33.
ComiC (1999)
Computational modelling of gradient-enhanced damage in quasi-brittle materials. Mechanics of Cohesive‐Frictional Materials4(1): 17–36.
34.
ComiC (2001)
A non-local model with tension and compression damage mechanisms. European Journal of Mechanics – A/Solids20(1): 1–22.
35.
ComiCMarianiSPeregoU (2007)
An extended FE strategy for transition from continuum damage to mode I cohesive crack propagation. International Journal for Numerical and Analytical Methods in Geomechanics31(2): 213–238.
36.
ComiCPeregoU (2001a)
Fracture energy based bi-dissipative damage model for concrete. International Journal of Solids and Structures38(36-37): 6427–6454.
37.
ComiCPeregoU (2001b)
Numerical aspects of nonlocal damage analyses. Revue Européenne Des Éléments Finis10(2–4): 227–242.
38.
ContrafattoLCuomoM (2006)
A framework of elastic–plastic damaging model for concrete under multiaxial stress states. International Journal of Plasticity22(12): 2272–2300.
39.
Da SilvaGSKosteskiLEIturriozI (2020)
Analysis of the failure process by using the lattice discrete element method in the Abaqus environment. Theoretical and Applied Fracture Mechanics107: 102563.
40.
de BorstR (1997)
Some recent developments in computational modelling of concrete fracture. International Journal of Fracture86(1/2): 5–36.
41.
de BorstR (2002)
Fracture in quasi-brittle materials: A review of continuum damage-based approaches. Engineering Fracture Mechanics69(2): 95–112.
42.
de BorstRGutiérrezMA (1999) A unified framework for concrete damage and fracture models including size effects. In: BažantZPRajapakseYDS (eds) Fracture Scaling.
Dordrecht:
Springer Netherlands, pp.261–277.
43.
de BorstRMühlhausH-B (1992)
Gradient-dependent plasticity: Formulation and algorithmic aspects. International Journal for Numerical Methods in Engineering35(3): 521–539.
44.
de BorstRPaminJ (1996)
Some novel developments in finite element procedures for gradient-dependent plasticity. International Journal for Numerical Methods in Engineering39(14): 2477–2505.
45.
de BorstRRemmersJJCNeedlemanA, et al. (2004)
Discrete vs smeared crack models for concrete fracture: Bridging the gap. International Journal for Numerical and Analytical Methods in Geomechanics28(78): 583–607.
46.
de BorstRSluysLJMuhlhausHB, et al. (1993)
Fundamental issues in finite element analyses of localization of deformation. Engineering Computations10(2): 99–121.
47.
De SciarraFM (1997)
A new variational theory and a computational algorithm for coupled elastoplastic damage models. International Journal of Solids and Structures34(14): 1761–1796.
48.
DesmoratR (2005) Unified framework for damage and fatigue of metals, concrete and elastomers. In: 11th International Conference on Fracture ICF11, Turin, Italy, 20–25 March 2005.
49.
Desmorat R and Gatuingt F (2008)
Anisotropic damage modelling of biaxial behaviour and rupture of concrete structures. Computers and Concrete5(4): 417–434.
50.
DesmoratRGatuingtFRagueneauF (2007a)
Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials. Engineering Fracture Mechanics74(10): 1539–1560.
51.
DesmoratRRagueneauFPhamH (2007b)
Continuum damage mechanics for hysteresis and fatigue of quasi-brittle materials and structures. International Journal for Numerical and Analytical Methods in Geomechanics31(2): 307–329.
52.
DesmoratRGatuingtFRagueneauF (2009)
Nonstandard thermodynamics framework for robust computations with induced anisotropic damage. International Journal of Damage Mechanics19(1): 53–73.
53.
DimitriRTrulloMDe LorenzisL, et al. (2015)
Coupled cohesive zone models for mixed-mode fracture: A comparative study. Engineering Fracture Mechanics148: 145–179.
54.
DingZFengD-CRenX, et al. (2019)
Physically based constitutive modeling of concrete fatigue and practical numerical method for cyclic loading simulation. Engineering Failure Analysis101: 230–242.
55.
DongYWuSXuSS, et al. (2010)
Analysis of concrete fracture using a novel cohesive crack method. Applied Mathematical Modelling34(12): 4219–4231.
56.
DuanKHuXWittmannFH (2006)
Scaling of quasi-brittle fracture: Boundary and size effect. Mechanics of Materials38(1-2): 128–141.
57.
DuanKHuXWittmannFH (2007)
Size effect on specific fracture energy of concrete. Engineering Fracture Mechanics74(1-2): 87–96.
58.
DurandRda SilvaFHBT (2019)
A coulomb-based model to simulate concrete cracking using cohesive elements. International Journal of Fracture220(1): 17–43.
59.
EvangelistaFRoeslerJRProençaSP (2013)
Three-dimensional cohesive zone model for fracture of cementitious materials based on the thermodynamics of irreversible processes. Engineering Fracture Mechanics97: 261–280.
60.
FariaROliverJCerveraM (1998)
A strain-based plastic viscous-damage model for massive concrete structures. International Journal of Solids and Structures35(14): 1533–1558.
61.
FathimaKMPKishenJMC (2015)
A thermodynamic framework for the evolution of damage in concrete under fatigue. Archive of Applied Mechanics85(7): 921–936.
62.
GeersMGDde BorstRBrekelmansWAM, et al. (1998)
Strain-based transient-gradient damage model for failure analyses. Computer Methods in Applied Mechanics and Engineering160(1–2): 133–153.
63.
GiryCDufourFMazarsJ (2011)
Stress-based nonlocal damage model. International Journal of Solids and Structures48(25–26): 3431–3443.
64.
GrasslPJirásekM (2006)
Plastic model with non-local damage applied to concrete. International Journal for Numerical and Analytical Methods in Geomechanics30(1): 71–90.
65.
GrasslPNyströmURemplingR, et al. (2011) A damage-plasticity model for the dynamic failure of concrete. In: Eurodyn (2011) 8th International Conference on Structural Dynamics, Leuven, Belgium, 4–6 July 2011. Leuven: Katholike Universiteit Leuven, p.8.
66.
GrasslPXenosDNyströmU, et al. (2013)
CDPM2: A damage-plasticity approach to modelling the failure of concrete. International Journal of Solids and Structures50(24): 3805–3816.
67.
HanQXuJCarpinteriA, et al. (2015)
Localization of acoustic emission sources in structural health monitoring of masonry bridge. Structural Control and Health Monitoring22(2): 314–329.
68.
HavlásekPGrasslPJirásekM (2016)
Analysis of size effect on strength of quasi-brittle materials using integral-type nonlocal models. Engineering Fracture Mechanics157: 72–85.
69.
HeWWuY-FXuY, et al. (2015)
A thermodynamically consistent nonlocal damage model for concrete materials with unilateral effects. Computer Methods in Applied Mechanics and Engineering297: 371–391.
70.
Hervé-SecourgeonGGatuingtFIbrahimbegovićA (2005)
On numerical implementation of a coupled rate dependent damage-plasticity constitutive model for concrete in application to high-rate dynamics. Engineering Computations22(5/6): 583–604.
71.
Hien PohLSwaddiwudhipongS (2009)
Over-nonlocal gradient enhanced plastic-damage model for concrete. International Journal of Solids and Structures46(25-26): 4369–4378.
72.
HintonE (1979) An Introduction to Finite Element Computations.
Swansea:
Pineridge Press.
73.
HuangFWuHJinQ, et al. (2005)
A numerical simulation on the perforation of reinforced concrete targets. International Journal of Impact Engineering32(1–4): 173–187.
74.
JenabidehkordiA (2019)
Computational methods for fracture in rock: A review and recent advances. Frontiers of Structural and Civil Engineering13(2): 273–287.
75.
Jirasek M (2018) Regularized Continuum Damage Formulations Acting as Localization Limiters. Computational Modelling of Concrete Structures. Euro-C 2018: 25–41.
76.
JohnsonGRHolmquistTJBeisselSR (2003)
Response of aluminum nitride (including a phase change) to large strains, high strain rates, and high pressures. Journal of Applied Physics94(3): 1639–1646.
77.
JouanGKotronisPCollinF (2014)
Using a second gradient model to simulate the behaviour of concrete structural elements. Finite Elements in Analysis and Design90: 50–60.
78.
JuJW (1989)
On energy-based coupled elastoplastic damage theories: Constitutive modeling and computational aspects. International Journal of Solids and Structures25(7): 803–833.
79.
KarihalooBL (1999) Size effect in shallow and deep notched quasi-brittle structures. In: BažantZPRajapakseYDS (eds) Fracture Scaling.
Dordrecht:
Springer Netherlands, pp.379–390.
KindrachukVMThieleMUngerJF (2015)
Constitutive modeling of creep-fatigue interaction for normal strength concrete under compression. International Journal of Fatigue78: 81–94.
82.
KosteskiLEIturriozILacidognaG, et al. (2020)
Size effect in heterogeneous materials analyzed through a lattice discrete element method approach. Engineering Fracture Mechanics232: 107041.
83.
KotronisPMazarsJDavenneL (2003)
The equivalent reinforced concrete model for simulating the behavior of walls under dynamic shear loading. Engineering Fracture Mechanics70(7-8): 1085–1097.
84.
Krayani A, Dufour F and Pijaudier-Cabot G (2007) Elastoplastic nonlocal damage model for concrete and size effect analysis. In: Framcos06. Taylor & Francis Italy, pp.343–349.
85.
KupferHHilsdorfHKRuschH (1969)
Behavior of concrete under biaxial stresses. Journal Proceedings66: 656–666.
86.
Li PianiTWeerheijmJKoeneL, et al. (2019)
The adobe Delta damage model: A locally regularized rate-dependent model for the static assessment of soil masonry bricks and mortar. Engineering Fracture Mechanics206: 114–130.
87.
LiQBLüPZhangL X (2004)
Damage degradation of concrete due to compressive fatigue loading. Key Engineering Materials274–276: 123– 128.
88.
LiZ-XZhongBShiY, et al. (2017)
Nonlocal formulation for numerical analysis of post-blast behavior of RC columns. International Journal of Concrete Structures and Materials11(2): 403–413.
89.
LiangJRenXLiJ (2015)
A competitive mechanism driven damage-plasticity model for fatigue behavior of concrete. International Journal of Damage Mechanics25(3): 377–399.
90.
LiuPFZhengJY (2010)
Recent developments on damage modeling and finite element analysis for composite laminates: A review. Materials & Design31(8): 3825–3834.
91.
LiuWHongJ-W (2012)
A coupling approach of discretized peridynamics with finite element method. Computer Methods in Applied Mechanics and Engineering245–246: 163–175.
92.
LüPLiQSongY (2004)
Damage constitutive of concrete under uniaxial alternate tension–compression fatigue loading based on double bounding surfaces. International Journal of Solids and Structures41(11-12): 3151–3166.
93.
LuYXuK (2004)
Modelling of dynamic behaviour of concrete material under blast loading. International Journal of Solids and Structures41(1): 131–143.
94.
LubinGCheng-Tzu ThomasH (1998)
Fatigue of concrete under uniaxial compression cyclic loading. ACI Materials Journal95: 575–581.
95.
MaiSHLe-CorreFForetG, et al. (2012)
A continuum damage modeling of quasi-static fatigue strength of plain concrete. International Journal of Fatigue37: 79–85.
96.
MauludinLMOucifC (2020)
Computational modeling of fracture in concrete: A review. Frontiers of Structural and Civil Engineering14(3): 586–598.
97.
MazarsJGrangeS (2017)
Simplified strategies based on damage mechanics for concrete under dynamic loading. Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences375: 0170.
98.
MazarsJHamonFGrangeS (2015)
A new 3D damage model for concrete under monotonic, cyclic and dynamic loadings. Materials and Structures48(11): 3779–3793.
99.
MazarsJPijaudier-CabotGSaouridisC (1991) Size effect and continuous damage in cementitious materials. In: BažantZP (ed.) Current Trends in Concrete Fracture Research.
Dordrecht:
Springer Netherlands, pp.159–173.
100.
NegiAKumarS (2019)
Localizing gradient damage model with smoothed stress based anisotropic nonlocal interactions. Engineering Fracture Mechanics214: 21–39.
101.
NguyenNHTBuiHHKodikaraJ, et al. (2019)
A discrete element modelling approach for fatigue damage growth in cemented materials. International Journal of Plasticity112: 68–88.
102.
NguyenTHABuiTQHiroseS (2018)
Smoothing gradient damage model with evolving anisotropic nonlocal interactions tailored to low-order finite elements. Computer Methods in Applied Mechanics and Engineering328: 498–541.
103.
OmidiOLotfiV (2013)
Continuum large cracking in a rate-dependent plastic–damage model for cyclic-loaded concrete structures. International Journal for Numerical and Analytical Methods in Geomechanics37(10): 1363–1390.
104.
OwenDRJHintonE (1986) Finite Elements in Plasticity: Theory and Practice.
Swansea, UK:
Pineridge Press.
105.
PaliwalBHammiYChandlerM, et al. (2020)
A three-invariant cap-viscoplastic rate-dependent-damage model for cementitious materials with return mapping integration in Haigh-Westergaard coordinate space. International Journal of Solids and Structures182-183: 77–99.
106.
PaliwalBHammiYMoserRD, et al. (2017)
A three-invariant cap-plasticity damage model for cementitious materials. International Journal of Solids and Structures108: 186–202.
107.
PapaETaliercioA (1996)
Anisotropic damage model for the multiaxial static and fatigue behaviour of plain concrete. Engineering Fracture Mechanics55(2): 163–179.
108.
PeerlingsRHJde BorstRBrekelmansWAM, et al. (1996)
Gradient enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering39(19): 3391–3403.
109.
PegonPAnthoineA (1997)
Numerical strategies for solving continuum damage problems with softening: Application to the homogenization of masonry. Computers & Structures64(1–4): 623–642.
110.
PereiraLFWeerheijmJSluysLJ (2016)
A new rate-dependent stress-based nonlocal damage model to simulate dynamic tensile failure of quasi-brittle materials. International Journal of Impact Engineering94: 83–95.
111.
PereraRCarniceroAAlarconE, et al. (2000)
A fatigue damage model for seismic response of RC structures. Computers & Structures78(1-3): 293–302.
112.
PicandetVHérissonBChallamelN, et al. (2016)
On the failure of a discrete axial chain using a continualized nonlocal continuum damage mechanics approach. International Journal for Numerical and Analytical Methods in Geomechanics40(3): 436–466.
113.
Pijaudier-CabotGHaidarKDubéJ-F (2004)
Non-local damage model with evolving internal length. International Journal for Numerical and Analytical Methods in Geomechanics28(78): 633–652.
114.
PiroozniaAMoradlooA (2020)
Investigation of size effect and smeared crack models in ordinary and dam concrete fracture tests. Engineering Fracture Mechanics226: 106863.
115.
PohLHSunG (2017)
Localizing gradient damage model with decreasing interactions. International Journal for Numerical Methods in Engineering110(6): 503–522.
116.
PohLHSwaddiwudhipongS (2009)
Gradient-enhanced softening material models. International Journal of Plasticity25(11): 2094–2121.
117.
PoorsolhjouyPGonzalezM (2018)
Connecting discrete particle mechanics to continuum granular micromechanics: Anisotropic continuum properties under compaction. Mechanics Research Communications92: 21–27.
118.
PugliaVBKosteskiLERieraJD, et al. (2019)
Random field generation of the material properties in the lattice discrete element method. The Journal of Strain Analysis for Engineering Design54(4): 236–246.
119.
RabczukT (2013)
Computational methods for fracture in brittle and quasi-brittle solids: State-of-the-art review and future perspectives. ISRN Applied Mathematics2013: 1–38.
120.
RagueneauFGatuingtF (2003)
Inelastic behavior modelling of concrete in low and high strain rate dynamics. Computers & Structures81(12): 1287–1299.
121.
RagueneauFLa BorderieCMazarsJ (2000)
Damage model for concrete-like materials coupling cracking and friction, contribution towards structural damping: First uniaxial applications. Mechanics of Cohesive-Frictional Materials5(8): 607–625.
122.
RenXZengSLiJ (2015)
A rate-dependent stochastic damage–plasticity model for quasi-brittle materials. Computational Mechanics55(2): 267–285.
123.
ResendeL (1987)
A damage mechanics constitutive theory for the inelastic behaviour of concrete. Computer Methods in Applied Mechanics and Engineering60(1): 57–93.
124.
RichardBRagueneauF (2013)
Continuum damage mechanics based model for quasi brittle materials subjected to cyclic loadings: Formulation, numerical implementation and applications. Engineering Fracture Mechanics98: 383–406.
125.
RichardBRagueneauFCremonaC, et al. (2010a)
Isotropic continuum damage mechanics for concrete under cyclic loading: Stiffness recovery, inelastic strains and frictional sliding. Engineering Fracture Mechanics77(8): 1203–1223.
126.
RichardBRagueneauFCremonaC, et al. (2010b)
A three-dimensional steel/concrete interface model including corrosion effects. Engineering Fracture Mechanics77(6): 951–973.
127.
Rossi CabralNInvaldiMABarrios D'AmbraR, et al. (2019)
An alternative bilinear peridynamic model to simulate the damage process in quasi-brittle materials. Engineering Fracture Mechanics216: 106494.
128.
RouquandAPontiroliCCantonE (1996)
An explicit damage model for dynamic concrete behaviour. Numerical simulations and comparisons with experimental results on reinforced concrete plates under blast loading. WIT Transactions on the Built Environment25: 10.
129.
SainTChandra KishenJM (2007)
Energy-based equivalence between damage and fracture in concrete under fatigue. Engineering Fracture Mechanics74(15): 2320–2333.
130.
SánchezPJHuespeAEOliverJ, et al. (2012)
A macroscopic damage-plastic constitutive law for modeling quasi-brittle fracture and ductile behavior of concrete. International Journal for Numerical and Analytical Methods in Geomechanics36(5): 546–573.
131.
SarkarSSinghIVMishraBK, et al. (2019)
A comparative study and ABAQUS implementation of conventional and localizing gradient enhanced damage models. Finite Elements in Analysis and Design160: 1–31.
132.
SaroukhaniSVafadariRSimoneA (2013)
A simplified implementation of a gradient-enhanced damage model with transient length scale effects. Computational Mechanics51(6): 899–909.
133.
ScottaRVitalianiRSaettaA, et al. (2001)
A scalar damage model with a shear retention factor for the analysis of reinforced concrete structures: Theory and validation. Computers & Structures79(7): 737–755.
134.
ShanZYuZLiX, et al. (2019)
A damage model for concrete under fatigue loading. Applied Sciences9(13): 2768.
135.
SilvaCMCastroLMSS (2006)
Hybrid-mixed stress formulation using continuum damage models. Communications in Numerical Methods in Engineering22(6): 605–617.
SimoJCJuJW (1987)
Strain- and stress-based continuum damage models – I. Formulation. International Journal of Solids and Structures23(7): 821–840.
138.
SinaieSNgoTDNguyenVP (2018)
A discrete element model of concrete for cyclic loading. Computers & Structures196: 173–185.
139.
TaylorD (2007) The Theory of Critical Distances.
Amsterdam:
Elsevier.
140.
ThaiTQRabczukTBazilevsY, et al. (2016)
A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering304: 584–604.
141.
Van MierJGMVan VlietMRA (2003)
Influence of microstructure of concrete on size/scale effects in tensile fracture. Engineering Fracture Mechanics70(16): 2281–2306.
142.
Van VlietMRAVan MierJGM (1999) Effect of strain gradients on the size effect of concrete in uniaxial tension. In: BažantZPRajapakseYDS (eds) Fracture Scaling.
Dordrecht:
SpringerNetherlands, pp.195–219.
143.
van VlietMRAvan MierJGM (2000)
Experimental investigation of size effect in concrete and sandstone under uniaxial tension. Engineering Fracture Mechanics65(2–3): 165–188.
144.
VandorenBSimoneA (2018)
Modeling and simulation of quasi-brittle failure with continuous anisotropic stress-based gradient-enhanced damage models. Computer Methods in Applied Mechanics and Engineering332: 644–685.
145.
VoyiadjisGZAbu Al-RubRK (2006)
A finite strain plastic-damage model for high velocity impacts using combined viscosity and gradient localization limiters: Part II – Numerical aspects and simulations. International Journal of Damage Mechanics15(4): 335–373.
146.
VoyiadjisGZTaqieddinZNKattanPI (2009)
Theoretical formulation of a coupled elastic–plastic anisotropic damage model for concrete using the strain energy equivalence concept. International Journal of Damage Mechanics18(7): 603–638.
147.
VoyiadjisGZZhouY (2019)
Numerical modeling of frictional contact between a blunt tool and quasi-brittle rock. Rock Mechanics and Rock Engineering52(10): 3771–3790.
148.
Voyiadjis GZ, Zhou Y and Kattan PI A New Anisotropic Elasto-plastic-damage Model for Quasi-brittle Materials Using Strain Energy Equivalence. Mechanics of Materials “MOM” Journal, 2021, 35 manuscript pages (in press).
149.
WittmannFHMihashiHNomuraN (1990)
Size effect on fracture energy of concrete. Engineering Fracture Mechanics35(1-3): 107–115.
150.
WuJ-Y (2018)
A geometrically regularized gradient-damage model with energetic equivalence. Computer Methods in Applied Mechanics and Engineering328: 612–637.
151.
WuJYLiJFariaR (2006)
An energy release rate-based plastic-damage model for concrete. International Journal of Solids and Structures43(3-4): 583–612.
152.
WuJ-YXuS-L (2011)
An augmented multicrack elastoplastic damage model for tensile cracking. International Journal of Solids and Structures48(18): 2511–2528.
153.
XuJFuZHanQ, et al. (2017)
Micro-cracking monitoring and fracture evaluation for crumb rubber concrete based on acoustic emission techniques. Structural Health Monitoring17(4): 946–958.
154.
YadavINThapaKB (2020)
Fatigue damage model of concrete materials. Theoretical and Applied Fracture Mechanics108: 102578.
155.
YuRCRuizGChavesEWV (2008)
A comparative study between discrete and continuum models to simulate concrete fracture. Engineering Fracture Mechanics75(1): 117–127.
156.
YunKWangZRonaldS, et al. (2017)
An advanced continuum damage mechanics model for predicting the crack progress process based on the consideration of the influence of crack direction under quasi-static load. International Journal of Mechanical Sciences130: 487–496.
157.
ZhangHWeiDM (2009)
Fractal effect and anisotropic constitutive model for concrete. Theoretical and Applied Fracture Mechanics51(3): 167–173.
158.
ZhangJLiJ (2014a)
Construction of homogeneous loading functions for elastoplastic damage models for concrete. Science China Physics, Mechanics and Astronomy57(3): 490–500.
159.
ZhangJLiJ (2014b)
Elastoplastic damage model for concrete based on consistent free energy potential. Science China Technological Sciences57(11): 2278–2286.
160.
ZhangJLiJ (2016)
Semi-implicit algorithm for elastoplastic damage models involving energy integration. Advances in Materials Science and Engineering2016: 1–9.
161.
ZhangJLiJJuJW (2016)
3D elastoplastic damage model for concrete based on novel decomposition of stress. International Journal of Solids and Structures94-95: 125–137.
162.
ZhangJMaLZhangZX (2018)
Elastoplastic damage model for concrete under triaxial compression and reversed cyclic loading. Strength of Materials50(5): 724–734.
163.
ZhangLGaoZHaynesRA, et al. (2020a)
An anisotropic continuum damage model for high-cycle fatigue. Mechanics of Advanced Materials and Structures28(16): 1633–1615.
164.
ZhangSChengMWangJ, et al. (2020b)
Modeling the hysteretic responses of RC shear walls under cyclic loading by an energy-based plastic-damage model for concrete. International Journal of Damage Mechanics29(1): 184–200.
165.
ZhaoBChenJLiuT, et al. (2019)
A new Timoshenko beam model based on modified gradient elasticity: Shearing effect and size effect of micro-beam. Composite Structures223: 110946.
166.
ZhaodongDJieL (2017)
A physically motivated model for fatigue damage of concrete. International Journal of Damage Mechanics27(8): 1192–1212.
167.
ZhuRAlamSYLoukiliA (2020)
An experimental investigation on the correlation between the aggregate size effect and the structural size effect. Engineering Fracture Mechanics234: 107101.
