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Fracture Mechanics and Size Effect of Concrete

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1 Fracture Mechanics and Size Effect of Concrete
KAIST Civil and Environmental Engineering Jin-Keun KIM

2 Necessity of fracture mechanics in concrete
Index Introduction Necessity of fracture mechanics in concrete Fracture mechanics in concrete Size effect of concrete Researches in KAIST concrete lab.

3 Introduction Fracture mechanics ? Fracture Mechanics (LEFM)
Applied mechanics combined with material science to study structure with discontinuity like holes or cracks Fracture Mechanics (LEFM) 1921 Griffith : Stress cannot be used as a standard of fracture if there’s a crack, Suggestion of energy criterion. ~1940 Fracture mechanics is applicable to brittle materials only. 1940’s Development of F.M. after the accident of ‘Liberty’ ships. 1957 Irwin : Introduction of stress-intensity factor 1968 Rice : stress and strain field are related with energy release rate by Using J-integral 1971 Begley : Experimental standards of J-integral 1978 Shih : Suggestion of theoretical background to apply J-integral into the fracture design

4 Introduction Fracture Mechanics (NLFM) Fracture Mechanics of Concrete
1958 Irwin : Error occurs when using LEFM because of the size of plastic region at the crack tip Suggestion and development of R-curve concept 1959, 62 Barenblatt : cohesive crack model Relationship between resistance of crack growth and atomic bond energy 1960 Dugdale : Suggestion of linear fracture model in plastic region having constant plastic stress Fracture of ductile material with adequate plastic region can be modeled well Fracture Mechanics of Concrete 1976 Hillerborg : Suggestion of ‘fictitious crack model’ 1983 Bazant : Development of ‘crack band model’ 1984 Bazant : Suggestion of size effect law

5 Introduction-definitions of terminologies
Fracture Mode (파괴 모드) Stress Intensity Factor (응력 확대 계수) Energy Release Rate (에너지 해방률) R-Curve J-Integral Fracture Energy (파괴 에너지) Surface Energy (표면 에너지) Characteristic Length (특성 길이) Fracture Process Zone (파괴 진행 영역) Brittleness Number (취성 계수) Quasi-Brittle Material (준취성 재료) Fictitious Crack Model (가상 균열 모델) Smeared Crack Model (분산 균열 모델) Crack Band Model (균열 띠 모델) Equivalent Crack Length (등가 균열 길이) Crack Opening Displacement (균열 개구 변위)

6 Fracture mode Mode 1 (Opening mode) Mode 2 (Sliding mode)
Mode 3 (Tearing mode) Separating chopstick Tearing paper

7 Linear fracture mechanics
Stress Intensity Factor Elastic theory can not explain the fracture of the material because the stress at the tip of crack goes to infinity according to the elastic theory. Stress field can explain by introducing the stress intensity factor Crack propagates when KI≥KIC(crack initiation toughness) Stress near the crack tip 𝜎 𝑖𝑗 𝑟,𝜃 = 1 2𝜋𝑟 [ 𝐾 𝐼 𝑓 𝑖𝑗 𝐼 𝜃 + 𝐾 𝐼𝐼 𝑓 𝑖𝑗 𝐼𝐼 𝜃 + 𝐾 𝐼𝐼𝐼 𝑓 𝑖𝑗 𝐼𝐼𝐼 𝜃 ] 𝐾 𝐼 = Stress intensity factor in mode I 𝐾 𝐼𝐼 = Stress intensity factor in mode II 𝐾 𝐼𝐼𝐼 = Stress intensity factor in mode III 𝑓 𝑖𝑗 𝜃 = dimensionless function depending on the geometry

8 Linear fracture mechanics
Energy release rate The crack propagates when the energy release rate exceeds the critical energy release rate, the fracture toughness. In other words, crack propagates when G≥GC Potential Energy Π=𝑊−𝑈−𝑆 where Π is the potential energy W is the work done by external forces U is the strain energy stored in the body S is the surface energy using for crack occurrence For equilibrium, δΠ=0 𝜕𝑆 𝜕𝑎 = 𝜕 𝜕𝑎 (𝑊−𝑈) (for K=0) R(𝑎)= 𝜕𝑆 𝜕𝑎 : R-curve (material resistance to crack extension) g(𝑎)= 𝜕 𝜕𝑎 (𝑊−𝑈) : energy release rate

9 Nonlinear fracture mechanics
Necessity of nonlinear fracture mechanics Fracture of brittle material can be explained by energy intensity factor or energy release rate. Linear fracture mechanics is not enough to explain the fracture of material in which the plastic region near the tip of the crack exists. R-Curve, J-Integral, CTOD plastic zone micro-crack zone macro-crack (a) metal (b) glass (c) concrete

10 Nonlinear fracture mechanics
R-Curve Potential Energy Π=𝑊−𝑈−𝑆 where Π is the potential energy W is the work done by external forces U is the strain energy stored in the body S is the surface energy using for crack occurrence For equilibrium, δΠ=0 𝜕𝑆 𝜕𝑎 = 𝜕 𝜕𝑎 (𝑊−𝑈) (for K=0) R(𝑎)= 𝜕𝑆 𝜕𝑎 : R-curve (material resistance to crack extension) g(𝑎)= 𝜕 𝜕𝑎 (𝑊−𝑈) : energy release rate (Condition for y-intercept) (Condition for slope)

11 Nonlinear fracture mechanics
Method for obtaining the R-Curve (Load and the equivalent crack length should be measured) (Five kinds of method depending on the decision of the equivalent crack length) (Decision of R-curve for the size effect)

12 Nonlinear fracture mechanics
J-Integral (Rice) δ𝑊=𝑏 Γ 𝑡 𝑖 𝛿𝑢 𝑖 ds δ𝑊 : work done by externally (by traction at boundary) Γ : contour(subbody) which surrounds the crack tip 𝑡 𝑖 : traction vector gδ𝑎= Γ 𝑡 𝑖 𝛿𝑢 𝑖 ds−δ[ 𝐴(Γ) 𝑈 dA When the crack increases as 𝛿𝑎 parallel to the crack δ𝑈=𝑈 𝐴 ′ Γ ′ −𝑈 𝐴 Γ =𝑏δ 𝐴 Γ 𝑈 dA =−𝑏 Γ 𝑈 d 𝑥 2 𝛿𝑎 𝑈 : Elastic strain energy density of subbody defined as Γ J=g= Γ ( 𝑈 d 𝑥 2 − 𝑡 𝑖 𝑢 𝑖,1 ds) : It can be used for the cases when all the points of Γ are elastic

13 Fracture mechanics in concrete
Mechanism of crack development in concrete Interface between mortar matrix and aggregate is weak for crack When concrete is under uni-axial compressive stress, tensile strain occurs orthogonally Crack is developed at the interface between mortar and aggregate by tension

14 Fracture mechanics in concrete
Mechanism of crack development in concrete Cracks around aggregates

15 Fracture mechanics in concrete
Mechanism of crack development in concrete Stress and strain distribution around the aggregates Stress distribution Strain distribution

16 Fracture mechanics in concrete
Mechanism of crack development in concrete stress near the aggregates Concrete is a material containing lots of voids and flaws If there is a void, the higher stress than the outside occurs If the crack is very sharp, very large stress at the crack tip is developed

17 Fracture mechanics in concrete
Fracture Process Zone(FPZ) in concrete Micro-cracks Applied load, P Initial crack Fracture process zone ft Aggregates Long and slender shape Length is relatively long Proportional to da (width and length) Independent on the size of structures Strongly affected by concrete strength

18 P Fracture mechanics in concrete a Fracture criterion of concrete CTOD
Fracture mech. criterion CMOD P initial crack CTOD a GF P - CMOD curve fracture mechanical criterion - KIC : KI at maximum load - CTODc : CTOD at maximum load - GF : area of P - CMOD curve

19 Fracture mechanics in concrete
Fracture behavior of concrete

20 Fracture mechanics in concrete
Necessity of crack models for concrete 콘크리트 구조물의 파괴내력이나 외력에 대한 응답을 역학적 문제로 보고 해결하려고 하면 그 정확해를 구하기가 매우 어려움 재료비선형성, 구조물의 기하 형상, 하중 조건, 경계 조건 등을 고려하고, 특히 파괴 거동에 관한 역학적인 해를 얻기 위해 수치 해석 방법과 계산 역학적 방법 적용 이 때 균열 해석 모델 필요 콘크리트 균열 해석 모델 (Crack Models for Concrete) 가상 균열 모델 (Fictitious Crack Model) 분산 균열 모델 (Smeared Crack Model) 균열 띠 모델 (Crack Band Model)

21 Fracture mechanics in concrete
가상 균열 모델 (Fictitious Crack Model) Is suggested by Hillerborg Supposes the fracture process zone at crack tip as a fictitious crack Simulates the nonlinearity of fracture process zone from the relationship between the crack opening of the fictitious crack and cohesive stress Defines the fracture energy by the cohesive stress-crack opening curve (Cohesive stress near crack tip) (Cohesive stress-crack opening curve)

22 Fracture mechanics in concrete
분산 균열 모델 (Smeared Crack Model) Is suggested by Rashid. Substitutes a continuum having variable properties for crack. Regards the band width as a material property Has characteristic that the model is very sensitive to the meshes. Micro-structure Stress distribution at crack boundary Discrete crack model Smeared crack model

23 Fracture mechanics in concrete
균열 띠 모델 (Crack Band Model) Is suggested by Bazant. Idealizes the fracture region as a band of micro -cracks with constant width. Explains the nonlinearity of fracture region by the relationship between the stress and strain of the band of micro-cracks. Defines the fracture energy by multiplying the area under the stress-strain curve of the band and band width.

24 Fracture mechanics in concrete
Size effect of concrete (Independence of size -> strength criterion) (Proportionality to 1/ 𝑑 -> linear elastic fracture mechanics)

25 Necessity of fracture mechanics in concrete
Size Effect Primary reason of introducing fracture mechanics in concrete

26 Necessity of fracture mechanics in concrete
Energy concept Necessity of energy criterion (energy for crack growth) Objectivity of analysis Define the energy dissipated per unit crack length or crack band (Independent of division of the element)

27 Necessity of fracture mechanics in concrete
Limit on yielding area

28 Necessity of fracture mechanics in concrete
Energy absorption and ductility Fracture energy decides ductility of the structure. By performing plastic limit analysis, we cannot know the stress and energy after the maximum stress point.

29 Size effect of concrete
Why is it observed? Stress concentration on the inherent or developed crack Factors affecting size effect Microcrack development area (FPZ area) Ratio between the area of FPZ and the size of the member

30 Size effect of concrete
Size effect law

31 Modification of size effect law with dissimilar initial crack
 Modified size effect law  Prediction of strength of structures with no initial crack or dissimilar initial cracks  Add size independent term, to the size effect law of Bazant =fracture strength =characteristic length of member =maximum size of aggregate =tensile strength =experimental coefficient =size independent strength

32 Modification of size effect law with dissimilar initial crack
Dissimilar crack

33 Modification of size effect law with dissimilar initial crack

34 Size effect of concrete
Shear strength Shear stress with the effective depth

35 Researches in KAIST concrete lab.
Prediction of shear strength Size effect for compressive strength of concrete Tensile cracking behavior at early ages

36 Prediction of shear strength
Comparison of various equations with increasing d (a/d≥3.0)

37 Size effect for compressive strength of concrete
Size effect for axial compressive strength Size effect for flexural compressive strength

38 Size effect of concrete
Axial compressive strength

39 Size effect of concrete
Flexural compressive strength

40 Tensile cracking behavior at early ages
Wedge splitting test specimens were manufactured for three types of concrete strength such as LS, NS, and HS concrete. Concrete mix proportions Type W/C (%) unit weight (kgf/m3) W C S G LS 70 185 268 726 1002 NS 55 342 727 1030 HS 30 160 533 712 1090 . This is for water cement ratio 50%, this is for water cement ratio 60%. From these results, we can see that the effect of autogenous shrinkage should be considered in current basic creep model especially at early ages even for ordinary strength concrete. Specimen geometry

41 Tensile cracking behavior at early ages
Gage setting Specimen setting Vertical loading Tests performed at 1, 3, 7, 14 and 28 days Measurements from two COD gages along the crack and one CMOD gage at the location of roller axis . This is test results of basic creep test for four applied load levels. This figure shows total strain including autogenous shrinkage, and this figure shows total strain excluding autogenous shrinkage. Application of splitting force

42 Tensile cracking behavior at early ages
Test results Load-CMOD curves characterize tensile cracking behaviors. Obtained curves are analyzed in development of age-dependent model. Horizontal load-CMOD curve . This is test results of basic creep test for four applied load levels. This figure shows total strain including autogenous shrinkage, and this figure shows total strain excluding autogenous shrinkage. LS concrete NS concrete HS concrete

43 Tensile cracking behavior at early ages
Test results COD at peak load decreases as age and strength increase. More brittle behavior is shown as age and strength increase. Crack opening displacement at peak load . This is test results of basic creep test for four applied load levels. This figure shows total strain including autogenous shrinkage, and this figure shows total strain excluding autogenous shrinkage. LS concrete NS concrete HS concrete

44 Thank you 


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