Mao KURUMATANI, Tatsuya YUMOTO, Kenjiro TERADA, Takashi KYOYA and Mitsuyoshi AKIYAMA

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1 応用力学論文集 Vol.3, pp.33-3 ( 年 8 月 ) Vol. 3 8 土木学会 Study for evaluating fracture-toughness in quasi-brittle materials based on the energy valance of structures Mao KURUMATANI, Tatsuya YUMOTO, Kenjiro TERADA, Takashi KYOYA and Mitsuyoshi AKIYAMA In this paper, we study the fracture toughness of quasi-brittle materials by applying the analysis of crack propagation with the cohesive crack model based on the energy balance between deformation and cracking in a specimen. Owing to this feature, we can not only evaluate fracture energy as an intrinsic material property, but also simulate the size/scale effect behavior as a structural response automatically. After explaining the analysis method for crack propagations incorporated with the cohesive crack model, several numerical examinations for validation and verification are carried out. irst, we demonstrate with both theoretical and numerical studies that macroscopic behavior of quasi-brittle materials is characterized by the energy balance of its structure. Second, we illustrate that the fracture energy is uniquely determined and independent of specimen size by evaluating the energy balance for 3-point bend test on differently-sized beam with a single-edge notch. Key Words : energy balance, fracture process zone, fracture energy, cohesive crack model, size/scale effect in quasi-brittle materials. f t f t f t f t f t f t 3 ),),3) RILEM ) 4) 9) J ) ) f t ),3)

2 ),4) Hillerborg et al. 5) Cohesive crack model Cohesive crack model EM Cohesive crack model. Cohesive crack model racture Process Zone; PZ Hillerborg et al. 5) Cohesive crack model PZ PZ PZ t coh Elastic domain Γ EL racture Process Zone Γ PZ f t t coh ( κ ) κ Traction-free crack u [b] u [a] Cohesive crack model κ (x) : g EM 6) 7),8) I t coh ft exp ( f t κ ) on Γ PZ () Γ PZ t coh f t κ f t f t RILEM f t PZ f t PZ () G Cohesive crack model g g u [a] u [b] on Γ PZ (Γ EL ) ()

3 t coh t t Γ EL Γ t n m coh g t coh Γ u t coh u [a] u [b] u u Γ PZ Γ R g u [a] u [b] Linear elasticity Ω b 3 t coh p coh g Cohesive crack model u [a], u [b] g t coh t coh t coh g g Cohesive crack model () 5),7),8) Cohesive crack model 9) (3) t coh p coh g on Γ PZ (4) p coh g 9) p coh (4) t coh p coh (5) g () κ g. 3 Cohesive crack model (4) Γ EL p Γ PZ Cohesive crack model p coh 3 Cohesive crack model δu : σ dω Ω + p δg g dγ + p coh δg g dγ Γ } EL Γ {{ }} PZ {{ } Penalty term Ω Cohesive crack term δu b dω + δu t dγ δu ( δg) (6) Γ t σ Cauchy Ω Γ EL Γ t Neumann b t p 4.3 (6) (4) λ p λ p g on Γ EL (7)

4 Γ EL b ractured surface γ b ractured surface Notch Γ R Γ PZ Element length h S β h L l α l (a) Direct tension tests of defferently-sized models (a) Crack propagations in case the fracture energy is small Load W S W L Load / Area W S W L Displacement Displacement / Length (b) Two types of energy-evaluation for theoretical analysis 5 (b) Crack propagations in case the fracture energy is large 4 λ n > and λ f t (8) n Γ EL, Γ PZ EM Γ EL Γ PZ f t (8) Cohesive crack model.4 Cohesive crack model Cohesive crack model Cohesive crack model (6) ),),3) 4 9) II (a) S L

5 S 3 mm L 4 mm Load / Area (MPa) : Model S : Model L : Model L 4 mm Model S Model L Model L. Displacement / Length Young's modulus (MPa) Poisson's ratio... 7 Tensile strength (MPa) racture energy (N/mm) Load or load/area f n f n f nstp S ΔS nstp ( f n + n S ΔS Δu Displacement or displacement/length f n ( Δu 3. S L 5(b) 5(b) W S bhl W S, W L αβγbhl W L (9) W W () G S f, GL f W S bh G S f, W L βγbh G L f () (9) () W S l GS f, W L αl GL f () 8 G S f G L f W L α W S (3) W S W L G L f α G S f (4) α (3), (4) l α /α l α α (l, h, b) (,, ) (α, β, γ) (, 3, 4) S L L mm 7 S L

6 ) l α /α S L 8 S S S L S L D Width B mm orce displacement: 4 mm Ligament C Span L Model C4 L 8, D, C 4 (mm) Model C5 L 8, D, C 5 (mm) Model C6 L 8, D, C 6 (mm) S L S S α S V S, A S L V L, A L S S V S A S N/mm S L V L A L N/mm S L S L L Young's modulus Poisson's ratio Tensile strength racture energy E ν MPa. 3. MPa. N/mm J ) 3 J 3 () 9 RILEM ) 3 L D B 3 () (a) (b) 8 f t

7 Load (kn) (a) : Model C4 : Model C5 : Model C6 D orce displacement: 5 mm Width B mm Ligament C Span L Model 5 L 3, D 5, C 5 (mm).5 Displacement (mm).5 Model L 6, D, C 5 (mm) Load / Ligament-area (MPa).3.. (b) : Model C4 : Model C5 : Model C6.. Displacement / beam-span (a) G G () G C G C G C N/mm.985. N/mm N/mm G. N/mm G 3 G J C5 C4 G 5 4 G 6 5 C6 C5 Model L, D, C (mm) Young's modulus Poisson's ratio Tensile strength racture energy E ν MPa. 3. MPa. N/mm 3 G (5 4).35. N/mm G (6 5).. N/mm J 4. () 3 B L D C 3 4),7) () (a) (b) f t

8 Load (kn) (a) : Model 5 : Model : Model Load / Ligament-area (MPa) Displacement (mm) (b) : Model 5 : Model : Model.. Displacement / beam-span (b) 3 9) (a) G G 5 G G N/mm.9. N/mm.87. N/mm 5. 3 I. f (x) A B x f (x) Ae ( Bx) A exp ( Bx) (I.) 8-3 -

9 Width B mm Height: mm Span: 8 mm orce displacement Ligament: 5 mm Load (kn)..6 : Plain mortar (Exp.) : Plain concrete (Exp.) : Present analysis Plain mortar Plain concrete Polymer (%) mortar Polymer (5%) mortar Steel fiber (%) concrete Steel fiber (%) concrete E (kgf/cm ) 5 3 f T (kgf/cm ) G (kgf/cm) Load (kn) Displacement (mm)..6 : Polymer mortar % (Exp.) : Polymer mortar 5% (Exp.) : Present analysis f t f () f t A f t [ f (x) dx A ] B exp ( Bx) (I.) B f t / f (x) ( f (x) f t exp f ) t x (I.3) Load (kn) Displacement (mm) : Steel fiber concrete % (Exp.) : Steel fiber concrete % (Exp.) : Present analysis II. 6) II. 3 6) 3 RILEM ) % 5 % % % 6 E f T G 6) kg/m 3 II. 6) 5 Displacement (mm) ) f t

10 Plain mortar Plain concrete Polymer (%) mortar Polymer (5%) mortar Steel fiber (%) concrete Steel fiber (%) concrete E (MPa) 3 3 f t (MPa) (N/mm) ) P.E. Petersson: racture energy of concrete: Method of determination, Cem. Concr. Res., Vol., pp.78 89, 98. ) RILEM Draft Recommendation (5-MC): Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams, Mater. Struct., Vol.8, pp.85 9, ) JCI JCI S 3 4) S. Mindess: The effect of specimen size on the fracture energy of concrete, Cem. Concr. Res., Vol.4, pp , ) A. Hillerborg: Results of three comparative test series for determining the fracture energy G of concrete, Mater. Struct., Vol.8, pp.47 43, ) V. Guinea, J. Planas and M. Elices: Measurement of the fracture energy using three-point bend tests: Part Influence of experimental procedures, Mater. Struct., Vol.5, pp. 8, 99. 7) Q. Jueshi and L. Hui: Size effect on fracture energy of concrete determined by three-point bending, Cem. Concr. Res., Vol.7, pp.3 36, ) S.H. Kwon, Z. Zhao and S.P. Shah: Effect of specimen size on fracture energy and softening curve of concrete: Part II. Inverse analysis and softening curve, Cem. Concr. Res., Vol.38, pp.6 69, 8. 9) RILEM Vol.585, pp. 6, 4. ) V.C. Li, C.-M. Chan and C.K.Y. Leung: Experimental determination of the tention-softening relations for cementitious composites, Cem. Concr. Res., Vol.7, pp.44 45, 987. ) Vol.453, pp.5 5, 993. ) K. Duan, X.-Z. Hu and.h. Wittmann: Thickness effect on fracture energy of cementitious materials, Cem. Concr. Res., Vol.33, pp , 3. 3) K. Duan, X. Hu and.h. Wittmann: Size effect on specific fracture energy of concrete, Engng. ract. Mech., Vol.74, pp.87 96, 7. 4) A. Hillerborg: The theoretical basis of a method to determine the fracture energy G of concrete, Mater. Struct., Vol.8, pp.9 96, ) A. Hillerborg, M. Modéer and P.-E. Petersson: Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cem. Concr. Res., Vol.6, pp , ) Vol., No., pp.47 5, ) G.N. Wells and L.J. Sluys: A new method for modelling cohesive cracks using finite elements, Int. J. Numer. Meth. Engng., Vol.5, pp ,. 8) J. Mergheim, E. Kuhl and P. Steinmann: A finite element method for the computational modelling of cohesive cracks, Int. J. Numer. Meth. Engng., Vol.63, pp.76 89, 5. 9) Cohesive crack A, Vol.64, No.3, pp , 8. ) A Vol.65, No., pp.6 7, 9. ).H. Wittmann, H. Mihashi and N. Nomura: Size effect on fracture energy of concrete, Engng. ract. Mech., Vol.35, pp.7 5, 99. ) H. Mihashi and N. Nomura: Microcracking and tentionsoftening properties of concrete, Cem. Concr. Compos., Vol.7, pp.9 3, 99. 3) K. Otsuka and H. Date: racture process zone in concrete tention specimen, Engng. ract. Mech., Vol.65, pp. 3,

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