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The influence of polygonal cavity on fracture behaviour of concrete
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Structuralonline Integrity 00 (2019) 000–000 Available atProcedia www.sciencedirect.com Structural Integrity Procedia 00 (2019) 000–000
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Procedia Structural Integrity 17 (2019) 690–697
ICSI 2019 The 3rd International Conference on Structural Integrity ICSI 2019 The 3rd International Conference on Structural Integrity
The influence of polygonal cavity on fracture behaviour of concrete The influence of polygonal cavity on fracture behaviour of concrete Michal Vyhlídala,a,*, Jan Klusákbb Michal Vyhlídal *, Jan Klusák
Brno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 602 00 Brno, Czech Republic Technology, Faculty of Civil of Engineering, Veveří 331/95, Czech University Academy ofofSciences, Institute of Physics Materials, Žižkova 513/22,602 61600 62Brno, Brno,Czech CzechRepublic Republic b Czech Academy of Sciences, Institute of Physics of Materials, Žižkova 513/22, 616 62 Brno, Czech Republic a
baBrno
Abstract Abstract In this work, the influence of polygonal cavity on fracture behaviour of cement composite (concrete) is investigated. Specimens In thisnominal work, the influence 40 of polygonal on fracture behaviour composite is investigated. Specimens of the dimensions × 40 × 160cavity mm with polygonal cavity ofof8cement × 8 × 40 mm were(concrete) provided with an initial central edge of the with nominal dimensions × 40was × 160 mmby with polygonal cavity × 8 × 40 mm provided with an initial edge notch a depth 12 mm,40 which made diamond blade saw.of To8 determine the were influence of polygonal cavitycentral on fracture notch with afracture depth 12 mm, which was made diamond blade saw.The To determine influence polygonal cavity on of fracture behaviour, tests were conducted viabythree-point bending. aim of thisthework is to of analyse the behaviour such behaviour, wereelement conducted via (FEM) three-point bending. The aim this work to analyse behaviour such specimen byfracture means tests of finite method principles in Ansys, Inc.ofsoftware. Foristhis reason, athe simplified 2Dofmodel specimen byfor means finiteconditions element method (FEM) principles Inc. software. For this reason, aassessment simplified was 2D model was created planeofstrain and based on the fracture in testAnsys, configuration. The crack propagation based wasgeneralized created forfracture plane strain conditions and based on athe fractureoftest configuration. The crack propagation assessment was based on mechanics approaches using criterion an average value of tangential stress determined in dependence on generalized fracture mechanics approaches using a criterion of an average value of tangential stress determined in dependence on the polar angle θ. Results of numerical analysis indicates that the actual crack depth must be greater than it supposed. In other on the polar angle θ. Results of numerical analysis indicates that the actual crack depth must be greater than it supposed. In other words, diamond blade saw damage the specimen more than expected. words, diamond blade saw damage the specimen more than expected. © 2019 The Authors. Published by Elsevier B.V. © 2019 Published by Elsevier B.V. © 2019The TheAuthors. Authors. Published by Peer-review under responsibility of Elsevier the ICSIB.V. 2019 organizers. Peer-review under responsibility of the ICSI 2019 organizers. Peer-review under responsibility of the ICSI 2019 organizers. Keywords: Average tangential stress; Finite element method; Fracture mechanics; Polygonal cavity. Keywords: Average tangential stress; Finite element method; Fracture mechanics; Polygonal cavity.
1. Introduction 1. Introduction Silicate-based materials, mainly concrete as a representative of such composites, belong to the widely used Silicate-based materials, concrete of such composites, belong the widely used building materials. Concretemainly structures suchasas:a representative highway bridges, cooling towers of powerto plants, hydraulic building materials. Concrete structures such as:infrastructures highway bridges, cooling serve towersmany of power plants,since hydraulic structures (dams), etc. are parts of important and should generations their structures (dams), etc. are parts important are infrastructures serve many generations their construction. Traditionally, theseof structures designed and usingshould a procedure mentioned in since standards, construction. Traditionally, these structures are designed using a procedure mentioned in standards,
* Corresponding author. Tel.: +420 541 147 131; * Corresponding Tel.: +420 541 147 131; E-mail address:author. [email protected] E-mail address: [email protected] 2452-3216 © 2019 The Authors. Published by Elsevier B.V. 2452-3216 2019responsibility The Authors. of Published Elsevier B.V. Peer-review©under the ICSIby 2019 organizers. Peer-review under responsibility of the ICSI 2019 organizers.
2452-3216 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ICSI 2019 organizers. 10.1016/j.prostr.2019.08.092
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e.g. EN 1992-1-1 (2004), which typically provide concise information on materials and structural behaviour based on experimental and empirical experience (elasticity and plasticity theory). However, structures made of these composites show nonlinear, more precisely, quasi-brittle behaviour – the ability to carry load continues even after the deviation from the linear branch of load-displacement diagram until the peak point and then the decrease of loading force follows until the failure, so called tensile softening Karihaloo (1995). The reason for this behaviour is, except of strong heterogeneity, the existence of internal defects (pores, cracks, transition zones, etc.), which work as obstacles to or promoters of crack propagation and are not taken into consideration by standards. When the structural design of these structures is more difficult (e.g. new materials, difficult geometry), it is suitable to apply the principles of fracture mechanics. Fracture mechanics together with a finite element method (FEM) is a powerful tool for the assessment of concrete structures behaviour which determines the durability within the structure’s lifetime. Nomenclature a0 An B B1 B2 CMOD d E F F1 F2 Fapp 𝑓𝑓ij (n, θ) g Ki KI,c L r, θ S W α ν 𝜎𝜎ij 𝜎𝜎𝜃𝜃𝜃𝜃 (𝑟𝑟, 𝜃𝜃) 𝜎𝜎̅𝜃𝜃𝜃𝜃 (𝜃𝜃) 𝜎𝜎̅𝜃𝜃𝜃𝜃,𝑐𝑐 𝛷𝛷
initial crack length terms of Williams’s expansion thickness of the specimen real thickness of the specimen model thickness of the specimen crack mouth opening displacement averaging distance Young’s modulus loading force force applied on the real specimen force applied in the model maximum applied force shape function maximum grain size of the aggregate stress intensity factor in loading mode I, II, III fracture toughness (under pure mode I) specimen’s length polar coordinates span specimen’s width relative crack length Poisson’s ratio stress tensor component tangential stress average tangential stress critical value of the average tangential stress Airy’s function
2. Theoretical background Most of the building materials contain internal defects or material discontinuities – cracks, cavities, pores, inclusions, etc. These discontinuities form stress concentrators and serve as potential weak elements which determine the structure’s lifetime. Generalized fracture mechanics deals with the influence of these stress concentrators and derive closed-form solutions for the stress and strain fields in the vicinity of crack tip.
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2.1. Linear elastic fracture mechanics Linear elastic fracture mechanics (LEFM) deals with the study of stress and displacement field in the vicinity of cracks for the case of homogenous, isotropic and linearly elastic material, i.e. Hook’s law is valid. Another assumption is that the size of the plastic zone is small compared to the body dimensions and crack length – small scale yielding. Stress and strain fields can be obtained by solving the biharmonic partial differential equation (1) including the conditions of stress-free edges, see e.g. Anderson (2005).
(
𝜕𝜕2
𝜕𝜕𝑟𝑟 2
1
+ ⋅ 𝑟𝑟
𝜕𝜕
𝜕𝜕𝜕𝜕
+
1
𝑟𝑟 2
⋅
𝜕𝜕2
𝜕𝜕𝜃𝜃 2
)(
𝜕𝜕2 𝛷𝛷 𝜕𝜕𝑟𝑟 2
1
+ ⋅ 𝑟𝑟
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
+
1
𝑟𝑟 2
⋅
𝜕𝜕2 𝛷𝛷 𝜕𝜕𝜃𝜃 2
)=0
(1)
where Φ is Airy stress function and r, θ are the polar coordinates. Best known solutions are by the complex variable technique Westergaard (1939) and by infinite power series Williams (1957). Stress field in the close vicinity of the crack tip according to Williams (1957) is described by following equation (Williams’s expansion): 𝑛𝑛
n
−1 ⋅ 𝑓𝑓ij (𝑛𝑛, 𝜃𝜃), 𝜎𝜎ij = ∑∞ 𝑛𝑛=1 (𝐴𝐴n ⋅ ) ⋅ 𝑟𝑟 2 2
(2)
where σij is the stress tensor component, An are terms of Williams’s expansion and fij (n, θ) is shape function. In the vicinity of crack tip (𝑟𝑟 → 0) higher order terms of the infinite series can be neglected and the stress and strain fields are represented only by the first (singular) term and the second (constant) term. Components of the stress tensor are given by the superposition of three basic failure modes – opening (mode I), in-plane shear (mode II) and out-of-plane shear (mode III) – Irwin (1957):
𝑙𝑙𝑙𝑙𝑙𝑙 𝜎𝜎ij = 𝑟𝑟→0
𝐾𝐾I
√2𝜋𝜋𝜋𝜋
𝑓𝑓ijI (𝜃𝜃) +
𝐾𝐾II
√2𝜋𝜋𝜋𝜋
𝑓𝑓ijII (𝜃𝜃) +
𝐾𝐾III
√2𝜋𝜋𝜋𝜋
𝑓𝑓ijIII (𝜃𝜃),
(3)
where 𝐾𝐾i for i = I, II, III are stress intensity factor in loading modes I, II, III. The values Ki are ascertained from a numerical solution of the studied geometry, materials and boundary conditions. For cracks in homogeneous media, procedure for Ki determination is usually included in the FEM software (e.g. Ansys, Inc. Software), while for general singular stress concentrators various direct or integration methods are used, see Ping et al. (2008), Klusák et al. (2008) or Profant et al. (2008). The most important and predominant failure mode in engineering experience is the opening mode (mode I). This dominance is due to several reasons, but the most important is that, unlike other two failure modes, the crack loaded by mode I propagates in its own plane and there are also no frictional forces between crack surfaces – see Karihaloo (1995). According to the above reasons, we will pay attention to pure mode I. 2.2. Criterion of stability based on average stress ahead of the crack tip One of the well-known LEFM conditions of stability says that a crack initiation occurs if the stress intensity factor KI reaches its critical value KI,c. Critical value KI,c is also called fracture toughness and is the material constant, see e.g. Anderson (2005). Another theory says that a crack will propagate in the direction where the tangential stress σθθ is maximal. This maximum tangential stress (MTS) criterion was established by Erdogan and Sih (1963) and seems to be suitable especially in the case of brittle fracture failure.
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LEFM studies and describes behaviour of cracks in homogenous media. Generalized LEFM is able to describe also crack initiation in sharp notches, bi-material notches, bi-material inclusions etc. Knésl et al. (2007), Klusák et al. (2007), Náhlík et al. (2007) modified MTS criterion for general singular stress concentrators (bi-material notch, free edge singularity, etc.). The stability condition is related to the average stress σ ̅θθ (θ) calculated across a distance d ahead of the crack tip. The distance d is usually chosen in dependence on the mechanism of a rupture (dimension of a plastic zone or material grain size). The average stress σ ̅θθ (θ) ahead of the crack tip is given by expression: 1
𝑑𝑑
𝜎𝜎̅𝜃𝜃𝜃𝜃 (𝜃𝜃) = ∫0 𝜎𝜎𝜃𝜃𝜃𝜃 (𝑟𝑟, 𝜃𝜃)𝑑𝑑𝑑𝑑 𝑑𝑑
(4)
The crack propagation direction is determined from the maximum of the average value of tangential stress, so the following conditions (5) have to be satisfied:
(
𝜕𝜕𝜎𝜎 ̅ 𝜃𝜃𝜃𝜃 𝜕𝜕𝜕𝜕
)
𝜃𝜃0
= 0∧(
𝜕𝜕2 𝜎𝜎 ̅ 𝜃𝜃𝜃𝜃 𝜕𝜕𝜃𝜃 2
)
𝜃𝜃0
< 0.
(5)
A crack propagation direction is clear. Now it is necessary to determine the critical value of tangential stress 𝜎𝜎̅θθ,c corresponding to the crack initiation. For a crack in homogeneous material under fracture mode I we obtain equation (6). For more details, see also Klusák et al. (2013), Klusák et al. (2016), Knésl et al. (2007), Klusák et al. (2007) or Náhlík et al. (2007).
𝜎𝜎̅𝜃𝜃𝜃𝜃,𝑐𝑐 =
2𝐾𝐾Ic
√2𝜋𝜋𝜋𝜋
(6)
The material ahead of the crack tip fractures when the mean value of tangential stress σ ̅θθ (θ0 ) exceeds its critical value σ ̅θθ,c defined in (6): 𝜎𝜎̅𝜃𝜃𝜃𝜃 ≥ 𝜎𝜎̅𝜃𝜃𝜃𝜃,𝑐𝑐
(7)
This generalized crack initiation criterion of average tangential stress (7) will be in the following used for complete study of crack propagation process from the initial notch ahead to the polygonal cavity and then for further crack propagation through the ligament of the specimen above the cavity. 3. Experimental part To determine the influence of polygonal cavity on the fracture behaviour of cement composite, fracture tests were conducted on specimens for three-point bending (3PB). These specimens, with the nominal dimensions 40×40×160 mm and an internal inclusion placed at midspan with the dimensions 8×8×40 mm, were provided with an initial central edge notch with a depth of 12 mm by diamond blade saw, see Fig. 1. Specimen geometry was designed according to the following recommendations – Karihaloo (1995) or Nallathambi and Karihaloo (1986): • The least dimension of specimen should be greater than 5g, where g is the maximum grain size of the aggregate, • Specimen’s width should be in the range of 40 – 100 mm, • Specimen’s length should be 4 times greater than specimen’s width,
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• Ratio 𝛼𝛼 = 𝑎𝑎0 /𝑊𝑊 should be in the range of 0.2 – 0.6.
Fig. 1. Specimen geometry and three-point bending (3PB) fracture test configuration.
3.1. Materials and specimens Due to the dimensions of the test specimens and the need for their compaction, the concrete was prepared from a fine-grained cement-based composite mixture consisting of Portland cement CEM I 42.5 R from the Mokrá cement plant, standard quartz sand with a maximum grain size of 2 mm and water. The ratio was 3:1:0.35. To increase the processability of the fresh mixture, the superplasticizer SIKA SVC 4035 was used at an amount of 1 % by cement mass. The individual components were mixed under laboratory conditions at the Institute of Chemistry (CHE), Faculty of Civil Engineering, Brno University of Technology (FCE BUT) under the supervision of Associate Professor Pavel Rovnaník using an automatic laboratory mixer. The moulds with compacted fresh mixture were sealed with thin PE foil and stored under stabilized laboratory conditions for 3 days. After this period, the specimens were stored in a water bath for until testing. After 28 days, the specimens were removed from the water bath, provided with an initial central edge notch with the depth a0 = 12 mm by diamond blade saw and subjected to fracture testing in threepoint bending at the AdMaS research centre operated by FCE BUT under supervision of Dr. Barbara Kucharczyková and Ing. Iva Rozsypalová. See Vyhlídal et al. (2019) for more details. 3.2. Fracture tests The fracture tests were carried out on a LabTest 6-1000.1.10 multi-purpose mechanical testing machine. The incremental displacement loading of the specimen was performed and F–CMOD (force vs. crack mouth opening displacement) diagrams were recorded. 4. Numerical model A simplified 2D model was created in Ansys software. Plane strain condition were adopted. The geometry corresponds to the real specimen’s dimensions and boundary conditions, see Fig. 2. Materials were modelled as linear, elastic and isotropic, which are represented by their elastic constants, i.e. Poisson’s ratio ν and Young’s modulus E (Table 1). In order to determine critical value of tangential stress 𝜎𝜎̅θθ,c one more material parameter – fracture toughness KI,c – is required (Table 1). See Vyhlídal et al. (2019) for more details.
Fig. 2. Simplified 2D model of the cracked specimen created in software ANSYS.
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Because of high stress gradients surrounding crack tips and edges, highly refined meshes near the crack tip (Fig. 3a) and the bottom peak of cavity (Fig. 3b) with small elements in all directions are required. The crack was modelled as ideally sharp. Distance d was considered as the length between crack tip and bottom corner of the cavity, which are both potential stress concentrators. According to theories of finite fracture mechanics, see Taylor et al. (2005), Cornetti et al. (2006), Taylor (2017), we suppose, that the region under the distance d corresponds to current crack extension increment of final length. In our case we suppose that the bridge between initial crack (edge notch) and the cavity breaks at once.
Fig. 3. (a) Radial mesh around crack tip; (b) radial mesh around bottom peak of the cavity. Table 1. Overview of the material’s parameters used in the numerical model. Layer
E [GPa]
ν [–]
KI,c [MPa∙m1/2]
Matrix
44.04
0.20
0.50
210
0.30
–
Steel plates
The force loading was applied to the top plate. When applying this force, it must be considered the thickness of the model which is equal to 𝐵𝐵 = 1.0 m. To achieve the same stress field in model as in real experiments, the equality of stress acting on the top plate must be satisfied, see Equation (8). The relationship between the force 𝐹𝐹2 applied in the model and the force 𝐹𝐹1 applied on the real specimen is as follows (B1 is the real thickness and B2 is the model thickness of the specimen): 𝐹𝐹1
𝐵𝐵1
=
𝐹𝐹2
𝐵𝐵2
𝐹𝐹2 = 𝐹𝐹1 ⋅
5. Results
(8) 𝐵𝐵2 𝐵𝐵1
= 𝐹𝐹1 ⋅
1000 40
= 25 ⋅ 𝐹𝐹1
(9)
It is obvious from the results of finite element analysis (FEA) that the polygonal cavity serves as another stress concentrator and influences the stress state ahead of the crack tip, see Fig. 4. It is necessary to consider this fact when choosing the distance d. Otherwise, we will get a significantly higher applied force which is very dangerous in the case of structural design. From this reason d has been chosen as the whole length between crack tip and the bottom corner of the cavity. By means of the average tangential stress criterion (7), the critical values of the applied force Fapp were calculated. The first crack extension was evaluated considering varying initial crack (edge notch) length a0 in the range 12 – 13.5 mm. The results are introduced in the Table 2. As you can see, the value of Fapp for crack
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length 𝑎𝑎0 = 12 mm is much higher than the experimental results introduced in Fig. 5. The applied forces Fapp are equal when the crack length a0 is approximately 13.5 mm (for specimens 1 and 2) or 13 mm (for specimen 3). Table 2. Maximum applied force in dependence on crack length Crack length a0 [mm]
12
12.5
13
13.5
Distance d [mm]
2.34
1.84
1.34
0.84
Maximum applied force Fapp [kN]
0.671
0.595
0.509
0.406
Fig. 4. Von Mises Stress between the initial crack and the cavity
Thus, the calculated F–CMOD diagram for crack length 𝑎𝑎0 = 13.5 mm in comparison with the experimental measurements of fracture tests can be found in Fig. 5. As you can see, the ascending branch is similar to the measured ones, while the descending branch is different. Reason for this behaviour is an application of LEFM, which do not take tensile softening into consideration. However, the most important variable for the design of structures is still the maximum load. Thus, the modified MTS criterion (7) has shown that the initial crack length a0 must be greater than it is supposed. In other words, in case of the notch tip near the cavity, the diamond blade saw introduces cracks ahead of the notch tip and it is necessary to account for this fact.
Fig. 5. Calculated F–CMOD diagram with the experimental results at background.
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6. Conclusion Stability criterion based on the average value of tangential stress ahead of the crack tip seems to be suitable for determination of the maximum load, while the value of crack mouth opening displacement is, due to tensile softening, underestimated. However, the most important variable for the design of structures is still the maximum load thus from that point of view is this criterion sufficient. From the detailed numerical analysis of the described fracture test, we concluded that the actual crack length 𝑎𝑎0 must be greater than it was supposed. In other words, diamond blade saw damaged the specimen more than expected. This damage is due to the small distance between the bottom corner of the cavity and the crack tip, which is approximately 2.34 mm. It is obvious that such a small area above the crack tip cannot resist the load that is caused by cutting the specimen by diamond blade saw and must inevitably lead to its partial failure. Real crack length a0 is approximately 13.5 mm (for specimens 1 and 2) and 13 mm (for specimen 3). Acknowledgements This outcome has been achieved with the financial support of the Brno University of Technology under project No. FAST-J-19-6079. References Anderson, T. L., 2005. Fracture mechanics: fundamentals and applications (3rd ed). CRC Press, Boca Raton, FL, pp. 630. Cornetti P., Pugno N., Carpinteri A., Taylor D., 2006. Finite fracture mechanics: a coupled stress and energy failure criterion. Engineering Fracture Mechanics 73: 2021–33. EN 1992–1–1, 2004. Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings. European Committee for Standardization. Erdogan, F., Sih, G. C., 1963. On the Crack Extension in Plates Under Plane Loading and Transverse Shear. Journal of Basic Engineering 85, 519–525. Irwin, G., 1957. Analysis of Stresses and Strains near the End of a Crack Traversing a Plate. Journal of Applied Mechanics 24, 361–364. Karihaloo, B. L., 1995. Fracture Mechanics and Structural Concrete. Longman Scientific & Technical, New York, pp. 165. Knésl, Z., Klusák, J., Náhlík, L., 2007. Crack initiation criteria for singular stress concentrations: Part I: A universal assessment of singular stress concentrations. Engineering mechanics 14, 399–408. Klusák, J., Knésl, Z., Náhlík, L., 2007. Crack initiation criteria for singular stress concentrations: Part II: Stability of sharp and bi-material notches. Engineering mechanics 14, 409–422. Klusák, J., Profant, T., Kotoul, M., 2008. A comparison of two direct methods of generalized stress intensity factor calculations of bi-material notches, Key Eng Mater. 385-387, 409-412. Klusák, J., Profant, T., Knésl, Z., Kotoul, M., 2013: The influence of discontinuity and orthotropy of fracture toughness on conditions of fracture initiation in singular stress concentrators, Engineering Fracture Mechanics 110, 438–447 Klusák J., Krepl O., Profant T., 2016: Behaviour of a crack in a corner or at a tip of a polygon-like particle, Procedia Structural Integrity 2, 19121919. ISSN: 2452-3216 Náhlík, L., Knésl, Z., Klusák, J., 2008. Crack initiation criteria for singular stress concentrations: Part III: An Application to a Crack Touching a Bimaterial Interface. Engineering mechanics 15, 99–114. Nallathambi, P., Karihaloo, B. L., 1986. Determination of specimen-size independent fracture toughness of plain concrete. Magazine of Concrete Research 38, 67–76. Ping, X.C., Chen, M.C., Xie, J.L., 2008. Singular stress analyses of V-notched anisotropic plates based on a novel finite element method. Eng Fract Mech. 75, 3819–3838 Profant, T., Ševeček, O., Kotoul, M., 2008. Calculation of K-factor and T-stress for cracks in anisotropic bimaterials, Eng Fract Mech. 75, 3707– 3726. Taylor, D., Cornetti, P., Pugno, N., 2005: The fracture mechanics of finite crack extension Engineering Fracture Mechanics, 72, 1021-1038. Taylor, D., 2017: The Theory of Critical Distances: A link to micromechanisms, Theoretical and Applied Fracture Mechanics, 90, 228-233. Vyhlídal, M., Rozsypalová, I., Majda, T., Daněk, P., Šimonová, H., Kucharczyková, B., Keršner, Z., 2019. Fracture Response of Fine-Grained Cement-Based Composite Specimens with Special Inclusions. Solid State Phenomena 292, 63–68. Westergaard, H. M., 1939. Bearing Pressures and Cracks. Journal of Applied Mechanics 61, 49–53. Williams, M. L., 1956. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics 24, 109–114.
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Procedia Structural Integrity 17 (2019) 690–697
ICSI 2019 The 3rd International Conference on Structural Integrity ICSI 2019 The 3rd International Conference on Structural Integrity
The influence of polygonal cavity on fracture behaviour of concrete The influence of polygonal cavity on fracture behaviour of concrete Michal Vyhlídala,a,*, Jan Klusákbb Michal Vyhlídal *, Jan Klusák
Brno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 602 00 Brno, Czech Republic Technology, Faculty of Civil of Engineering, Veveří 331/95, Czech University Academy ofofSciences, Institute of Physics Materials, Žižkova 513/22,602 61600 62Brno, Brno,Czech CzechRepublic Republic b Czech Academy of Sciences, Institute of Physics of Materials, Žižkova 513/22, 616 62 Brno, Czech Republic a
baBrno
Abstract Abstract In this work, the influence of polygonal cavity on fracture behaviour of cement composite (concrete) is investigated. Specimens In thisnominal work, the influence 40 of polygonal on fracture behaviour composite is investigated. Specimens of the dimensions × 40 × 160cavity mm with polygonal cavity ofof8cement × 8 × 40 mm were(concrete) provided with an initial central edge of the with nominal dimensions × 40was × 160 mmby with polygonal cavity × 8 × 40 mm provided with an initial edge notch a depth 12 mm,40 which made diamond blade saw.of To8 determine the were influence of polygonal cavitycentral on fracture notch with afracture depth 12 mm, which was made diamond blade saw.The To determine influence polygonal cavity on of fracture behaviour, tests were conducted viabythree-point bending. aim of thisthework is to of analyse the behaviour such behaviour, wereelement conducted via (FEM) three-point bending. The aim this work to analyse behaviour such specimen byfracture means tests of finite method principles in Ansys, Inc.ofsoftware. Foristhis reason, athe simplified 2Dofmodel specimen byfor means finiteconditions element method (FEM) principles Inc. software. For this reason, aassessment simplified was 2D model was created planeofstrain and based on the fracture in testAnsys, configuration. The crack propagation based wasgeneralized created forfracture plane strain conditions and based on athe fractureoftest configuration. The crack propagation assessment was based on mechanics approaches using criterion an average value of tangential stress determined in dependence on generalized fracture mechanics approaches using a criterion of an average value of tangential stress determined in dependence on the polar angle θ. Results of numerical analysis indicates that the actual crack depth must be greater than it supposed. In other on the polar angle θ. Results of numerical analysis indicates that the actual crack depth must be greater than it supposed. In other words, diamond blade saw damage the specimen more than expected. words, diamond blade saw damage the specimen more than expected. © 2019 The Authors. Published by Elsevier B.V. © 2019 Published by Elsevier B.V. © 2019The TheAuthors. Authors. Published by Peer-review under responsibility of Elsevier the ICSIB.V. 2019 organizers. Peer-review under responsibility of the ICSI 2019 organizers. Peer-review under responsibility of the ICSI 2019 organizers. Keywords: Average tangential stress; Finite element method; Fracture mechanics; Polygonal cavity. Keywords: Average tangential stress; Finite element method; Fracture mechanics; Polygonal cavity.
1. Introduction 1. Introduction Silicate-based materials, mainly concrete as a representative of such composites, belong to the widely used Silicate-based materials, concrete of such composites, belong the widely used building materials. Concretemainly structures suchasas:a representative highway bridges, cooling towers of powerto plants, hydraulic building materials. Concrete structures such as:infrastructures highway bridges, cooling serve towersmany of power plants,since hydraulic structures (dams), etc. are parts of important and should generations their structures (dams), etc. are parts important are infrastructures serve many generations their construction. Traditionally, theseof structures designed and usingshould a procedure mentioned in since standards, construction. Traditionally, these structures are designed using a procedure mentioned in standards,
* Corresponding author. Tel.: +420 541 147 131; * Corresponding Tel.: +420 541 147 131; E-mail address:author. [email protected] E-mail address: [email protected] 2452-3216 © 2019 The Authors. Published by Elsevier B.V. 2452-3216 2019responsibility The Authors. of Published Elsevier B.V. Peer-review©under the ICSIby 2019 organizers. Peer-review under responsibility of the ICSI 2019 organizers.
2452-3216 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ICSI 2019 organizers. 10.1016/j.prostr.2019.08.092
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e.g. EN 1992-1-1 (2004), which typically provide concise information on materials and structural behaviour based on experimental and empirical experience (elasticity and plasticity theory). However, structures made of these composites show nonlinear, more precisely, quasi-brittle behaviour – the ability to carry load continues even after the deviation from the linear branch of load-displacement diagram until the peak point and then the decrease of loading force follows until the failure, so called tensile softening Karihaloo (1995). The reason for this behaviour is, except of strong heterogeneity, the existence of internal defects (pores, cracks, transition zones, etc.), which work as obstacles to or promoters of crack propagation and are not taken into consideration by standards. When the structural design of these structures is more difficult (e.g. new materials, difficult geometry), it is suitable to apply the principles of fracture mechanics. Fracture mechanics together with a finite element method (FEM) is a powerful tool for the assessment of concrete structures behaviour which determines the durability within the structure’s lifetime. Nomenclature a0 An B B1 B2 CMOD d E F F1 F2 Fapp 𝑓𝑓ij (n, θ) g Ki KI,c L r, θ S W α ν 𝜎𝜎ij 𝜎𝜎𝜃𝜃𝜃𝜃 (𝑟𝑟, 𝜃𝜃) 𝜎𝜎̅𝜃𝜃𝜃𝜃 (𝜃𝜃) 𝜎𝜎̅𝜃𝜃𝜃𝜃,𝑐𝑐 𝛷𝛷
initial crack length terms of Williams’s expansion thickness of the specimen real thickness of the specimen model thickness of the specimen crack mouth opening displacement averaging distance Young’s modulus loading force force applied on the real specimen force applied in the model maximum applied force shape function maximum grain size of the aggregate stress intensity factor in loading mode I, II, III fracture toughness (under pure mode I) specimen’s length polar coordinates span specimen’s width relative crack length Poisson’s ratio stress tensor component tangential stress average tangential stress critical value of the average tangential stress Airy’s function
2. Theoretical background Most of the building materials contain internal defects or material discontinuities – cracks, cavities, pores, inclusions, etc. These discontinuities form stress concentrators and serve as potential weak elements which determine the structure’s lifetime. Generalized fracture mechanics deals with the influence of these stress concentrators and derive closed-form solutions for the stress and strain fields in the vicinity of crack tip.
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2.1. Linear elastic fracture mechanics Linear elastic fracture mechanics (LEFM) deals with the study of stress and displacement field in the vicinity of cracks for the case of homogenous, isotropic and linearly elastic material, i.e. Hook’s law is valid. Another assumption is that the size of the plastic zone is small compared to the body dimensions and crack length – small scale yielding. Stress and strain fields can be obtained by solving the biharmonic partial differential equation (1) including the conditions of stress-free edges, see e.g. Anderson (2005).
(
𝜕𝜕2
𝜕𝜕𝑟𝑟 2
1
+ ⋅ 𝑟𝑟
𝜕𝜕
𝜕𝜕𝜕𝜕
+
1
𝑟𝑟 2
⋅
𝜕𝜕2
𝜕𝜕𝜃𝜃 2
)(
𝜕𝜕2 𝛷𝛷 𝜕𝜕𝑟𝑟 2
1
+ ⋅ 𝑟𝑟
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
+
1
𝑟𝑟 2
⋅
𝜕𝜕2 𝛷𝛷 𝜕𝜕𝜃𝜃 2
)=0
(1)
where Φ is Airy stress function and r, θ are the polar coordinates. Best known solutions are by the complex variable technique Westergaard (1939) and by infinite power series Williams (1957). Stress field in the close vicinity of the crack tip according to Williams (1957) is described by following equation (Williams’s expansion): 𝑛𝑛
n
−1 ⋅ 𝑓𝑓ij (𝑛𝑛, 𝜃𝜃), 𝜎𝜎ij = ∑∞ 𝑛𝑛=1 (𝐴𝐴n ⋅ ) ⋅ 𝑟𝑟 2 2
(2)
where σij is the stress tensor component, An are terms of Williams’s expansion and fij (n, θ) is shape function. In the vicinity of crack tip (𝑟𝑟 → 0) higher order terms of the infinite series can be neglected and the stress and strain fields are represented only by the first (singular) term and the second (constant) term. Components of the stress tensor are given by the superposition of three basic failure modes – opening (mode I), in-plane shear (mode II) and out-of-plane shear (mode III) – Irwin (1957):
𝑙𝑙𝑙𝑙𝑙𝑙 𝜎𝜎ij = 𝑟𝑟→0
𝐾𝐾I
√2𝜋𝜋𝜋𝜋
𝑓𝑓ijI (𝜃𝜃) +
𝐾𝐾II
√2𝜋𝜋𝜋𝜋
𝑓𝑓ijII (𝜃𝜃) +
𝐾𝐾III
√2𝜋𝜋𝜋𝜋
𝑓𝑓ijIII (𝜃𝜃),
(3)
where 𝐾𝐾i for i = I, II, III are stress intensity factor in loading modes I, II, III. The values Ki are ascertained from a numerical solution of the studied geometry, materials and boundary conditions. For cracks in homogeneous media, procedure for Ki determination is usually included in the FEM software (e.g. Ansys, Inc. Software), while for general singular stress concentrators various direct or integration methods are used, see Ping et al. (2008), Klusák et al. (2008) or Profant et al. (2008). The most important and predominant failure mode in engineering experience is the opening mode (mode I). This dominance is due to several reasons, but the most important is that, unlike other two failure modes, the crack loaded by mode I propagates in its own plane and there are also no frictional forces between crack surfaces – see Karihaloo (1995). According to the above reasons, we will pay attention to pure mode I. 2.2. Criterion of stability based on average stress ahead of the crack tip One of the well-known LEFM conditions of stability says that a crack initiation occurs if the stress intensity factor KI reaches its critical value KI,c. Critical value KI,c is also called fracture toughness and is the material constant, see e.g. Anderson (2005). Another theory says that a crack will propagate in the direction where the tangential stress σθθ is maximal. This maximum tangential stress (MTS) criterion was established by Erdogan and Sih (1963) and seems to be suitable especially in the case of brittle fracture failure.
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LEFM studies and describes behaviour of cracks in homogenous media. Generalized LEFM is able to describe also crack initiation in sharp notches, bi-material notches, bi-material inclusions etc. Knésl et al. (2007), Klusák et al. (2007), Náhlík et al. (2007) modified MTS criterion for general singular stress concentrators (bi-material notch, free edge singularity, etc.). The stability condition is related to the average stress σ ̅θθ (θ) calculated across a distance d ahead of the crack tip. The distance d is usually chosen in dependence on the mechanism of a rupture (dimension of a plastic zone or material grain size). The average stress σ ̅θθ (θ) ahead of the crack tip is given by expression: 1
𝑑𝑑
𝜎𝜎̅𝜃𝜃𝜃𝜃 (𝜃𝜃) = ∫0 𝜎𝜎𝜃𝜃𝜃𝜃 (𝑟𝑟, 𝜃𝜃)𝑑𝑑𝑑𝑑 𝑑𝑑
(4)
The crack propagation direction is determined from the maximum of the average value of tangential stress, so the following conditions (5) have to be satisfied:
(
𝜕𝜕𝜎𝜎 ̅ 𝜃𝜃𝜃𝜃 𝜕𝜕𝜕𝜕
)
𝜃𝜃0
= 0∧(
𝜕𝜕2 𝜎𝜎 ̅ 𝜃𝜃𝜃𝜃 𝜕𝜕𝜃𝜃 2
)
𝜃𝜃0
< 0.
(5)
A crack propagation direction is clear. Now it is necessary to determine the critical value of tangential stress 𝜎𝜎̅θθ,c corresponding to the crack initiation. For a crack in homogeneous material under fracture mode I we obtain equation (6). For more details, see also Klusák et al. (2013), Klusák et al. (2016), Knésl et al. (2007), Klusák et al. (2007) or Náhlík et al. (2007).
𝜎𝜎̅𝜃𝜃𝜃𝜃,𝑐𝑐 =
2𝐾𝐾Ic
√2𝜋𝜋𝜋𝜋
(6)
The material ahead of the crack tip fractures when the mean value of tangential stress σ ̅θθ (θ0 ) exceeds its critical value σ ̅θθ,c defined in (6): 𝜎𝜎̅𝜃𝜃𝜃𝜃 ≥ 𝜎𝜎̅𝜃𝜃𝜃𝜃,𝑐𝑐
(7)
This generalized crack initiation criterion of average tangential stress (7) will be in the following used for complete study of crack propagation process from the initial notch ahead to the polygonal cavity and then for further crack propagation through the ligament of the specimen above the cavity. 3. Experimental part To determine the influence of polygonal cavity on the fracture behaviour of cement composite, fracture tests were conducted on specimens for three-point bending (3PB). These specimens, with the nominal dimensions 40×40×160 mm and an internal inclusion placed at midspan with the dimensions 8×8×40 mm, were provided with an initial central edge notch with a depth of 12 mm by diamond blade saw, see Fig. 1. Specimen geometry was designed according to the following recommendations – Karihaloo (1995) or Nallathambi and Karihaloo (1986): • The least dimension of specimen should be greater than 5g, where g is the maximum grain size of the aggregate, • Specimen’s width should be in the range of 40 – 100 mm, • Specimen’s length should be 4 times greater than specimen’s width,
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• Ratio 𝛼𝛼 = 𝑎𝑎0 /𝑊𝑊 should be in the range of 0.2 – 0.6.
Fig. 1. Specimen geometry and three-point bending (3PB) fracture test configuration.
3.1. Materials and specimens Due to the dimensions of the test specimens and the need for their compaction, the concrete was prepared from a fine-grained cement-based composite mixture consisting of Portland cement CEM I 42.5 R from the Mokrá cement plant, standard quartz sand with a maximum grain size of 2 mm and water. The ratio was 3:1:0.35. To increase the processability of the fresh mixture, the superplasticizer SIKA SVC 4035 was used at an amount of 1 % by cement mass. The individual components were mixed under laboratory conditions at the Institute of Chemistry (CHE), Faculty of Civil Engineering, Brno University of Technology (FCE BUT) under the supervision of Associate Professor Pavel Rovnaník using an automatic laboratory mixer. The moulds with compacted fresh mixture were sealed with thin PE foil and stored under stabilized laboratory conditions for 3 days. After this period, the specimens were stored in a water bath for until testing. After 28 days, the specimens were removed from the water bath, provided with an initial central edge notch with the depth a0 = 12 mm by diamond blade saw and subjected to fracture testing in threepoint bending at the AdMaS research centre operated by FCE BUT under supervision of Dr. Barbara Kucharczyková and Ing. Iva Rozsypalová. See Vyhlídal et al. (2019) for more details. 3.2. Fracture tests The fracture tests were carried out on a LabTest 6-1000.1.10 multi-purpose mechanical testing machine. The incremental displacement loading of the specimen was performed and F–CMOD (force vs. crack mouth opening displacement) diagrams were recorded. 4. Numerical model A simplified 2D model was created in Ansys software. Plane strain condition were adopted. The geometry corresponds to the real specimen’s dimensions and boundary conditions, see Fig. 2. Materials were modelled as linear, elastic and isotropic, which are represented by their elastic constants, i.e. Poisson’s ratio ν and Young’s modulus E (Table 1). In order to determine critical value of tangential stress 𝜎𝜎̅θθ,c one more material parameter – fracture toughness KI,c – is required (Table 1). See Vyhlídal et al. (2019) for more details.
Fig. 2. Simplified 2D model of the cracked specimen created in software ANSYS.
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Because of high stress gradients surrounding crack tips and edges, highly refined meshes near the crack tip (Fig. 3a) and the bottom peak of cavity (Fig. 3b) with small elements in all directions are required. The crack was modelled as ideally sharp. Distance d was considered as the length between crack tip and bottom corner of the cavity, which are both potential stress concentrators. According to theories of finite fracture mechanics, see Taylor et al. (2005), Cornetti et al. (2006), Taylor (2017), we suppose, that the region under the distance d corresponds to current crack extension increment of final length. In our case we suppose that the bridge between initial crack (edge notch) and the cavity breaks at once.
Fig. 3. (a) Radial mesh around crack tip; (b) radial mesh around bottom peak of the cavity. Table 1. Overview of the material’s parameters used in the numerical model. Layer
E [GPa]
ν [–]
KI,c [MPa∙m1/2]
Matrix
44.04
0.20
0.50
210
0.30
–
Steel plates
The force loading was applied to the top plate. When applying this force, it must be considered the thickness of the model which is equal to 𝐵𝐵 = 1.0 m. To achieve the same stress field in model as in real experiments, the equality of stress acting on the top plate must be satisfied, see Equation (8). The relationship between the force 𝐹𝐹2 applied in the model and the force 𝐹𝐹1 applied on the real specimen is as follows (B1 is the real thickness and B2 is the model thickness of the specimen): 𝐹𝐹1
𝐵𝐵1
=
𝐹𝐹2
𝐵𝐵2
𝐹𝐹2 = 𝐹𝐹1 ⋅
5. Results
(8) 𝐵𝐵2 𝐵𝐵1
= 𝐹𝐹1 ⋅
1000 40
= 25 ⋅ 𝐹𝐹1
(9)
It is obvious from the results of finite element analysis (FEA) that the polygonal cavity serves as another stress concentrator and influences the stress state ahead of the crack tip, see Fig. 4. It is necessary to consider this fact when choosing the distance d. Otherwise, we will get a significantly higher applied force which is very dangerous in the case of structural design. From this reason d has been chosen as the whole length between crack tip and the bottom corner of the cavity. By means of the average tangential stress criterion (7), the critical values of the applied force Fapp were calculated. The first crack extension was evaluated considering varying initial crack (edge notch) length a0 in the range 12 – 13.5 mm. The results are introduced in the Table 2. As you can see, the value of Fapp for crack
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length 𝑎𝑎0 = 12 mm is much higher than the experimental results introduced in Fig. 5. The applied forces Fapp are equal when the crack length a0 is approximately 13.5 mm (for specimens 1 and 2) or 13 mm (for specimen 3). Table 2. Maximum applied force in dependence on crack length Crack length a0 [mm]
12
12.5
13
13.5
Distance d [mm]
2.34
1.84
1.34
0.84
Maximum applied force Fapp [kN]
0.671
0.595
0.509
0.406
Fig. 4. Von Mises Stress between the initial crack and the cavity
Thus, the calculated F–CMOD diagram for crack length 𝑎𝑎0 = 13.5 mm in comparison with the experimental measurements of fracture tests can be found in Fig. 5. As you can see, the ascending branch is similar to the measured ones, while the descending branch is different. Reason for this behaviour is an application of LEFM, which do not take tensile softening into consideration. However, the most important variable for the design of structures is still the maximum load. Thus, the modified MTS criterion (7) has shown that the initial crack length a0 must be greater than it is supposed. In other words, in case of the notch tip near the cavity, the diamond blade saw introduces cracks ahead of the notch tip and it is necessary to account for this fact.
Fig. 5. Calculated F–CMOD diagram with the experimental results at background.
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6. Conclusion Stability criterion based on the average value of tangential stress ahead of the crack tip seems to be suitable for determination of the maximum load, while the value of crack mouth opening displacement is, due to tensile softening, underestimated. However, the most important variable for the design of structures is still the maximum load thus from that point of view is this criterion sufficient. From the detailed numerical analysis of the described fracture test, we concluded that the actual crack length 𝑎𝑎0 must be greater than it was supposed. In other words, diamond blade saw damaged the specimen more than expected. This damage is due to the small distance between the bottom corner of the cavity and the crack tip, which is approximately 2.34 mm. It is obvious that such a small area above the crack tip cannot resist the load that is caused by cutting the specimen by diamond blade saw and must inevitably lead to its partial failure. Real crack length a0 is approximately 13.5 mm (for specimens 1 and 2) and 13 mm (for specimen 3). Acknowledgements This outcome has been achieved with the financial support of the Brno University of Technology under project No. FAST-J-19-6079. References Anderson, T. L., 2005. Fracture mechanics: fundamentals and applications (3rd ed). CRC Press, Boca Raton, FL, pp. 630. Cornetti P., Pugno N., Carpinteri A., Taylor D., 2006. Finite fracture mechanics: a coupled stress and energy failure criterion. Engineering Fracture Mechanics 73: 2021–33. EN 1992–1–1, 2004. Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings. European Committee for Standardization. Erdogan, F., Sih, G. C., 1963. On the Crack Extension in Plates Under Plane Loading and Transverse Shear. Journal of Basic Engineering 85, 519–525. Irwin, G., 1957. Analysis of Stresses and Strains near the End of a Crack Traversing a Plate. Journal of Applied Mechanics 24, 361–364. Karihaloo, B. L., 1995. Fracture Mechanics and Structural Concrete. Longman Scientific & Technical, New York, pp. 165. Knésl, Z., Klusák, J., Náhlík, L., 2007. Crack initiation criteria for singular stress concentrations: Part I: A universal assessment of singular stress concentrations. Engineering mechanics 14, 399–408. Klusák, J., Knésl, Z., Náhlík, L., 2007. Crack initiation criteria for singular stress concentrations: Part II: Stability of sharp and bi-material notches. Engineering mechanics 14, 409–422. Klusák, J., Profant, T., Kotoul, M., 2008. A comparison of two direct methods of generalized stress intensity factor calculations of bi-material notches, Key Eng Mater. 385-387, 409-412. Klusák, J., Profant, T., Knésl, Z., Kotoul, M., 2013: The influence of discontinuity and orthotropy of fracture toughness on conditions of fracture initiation in singular stress concentrators, Engineering Fracture Mechanics 110, 438–447 Klusák J., Krepl O., Profant T., 2016: Behaviour of a crack in a corner or at a tip of a polygon-like particle, Procedia Structural Integrity 2, 19121919. ISSN: 2452-3216 Náhlík, L., Knésl, Z., Klusák, J., 2008. Crack initiation criteria for singular stress concentrations: Part III: An Application to a Crack Touching a Bimaterial Interface. Engineering mechanics 15, 99–114. Nallathambi, P., Karihaloo, B. L., 1986. Determination of specimen-size independent fracture toughness of plain concrete. Magazine of Concrete Research 38, 67–76. Ping, X.C., Chen, M.C., Xie, J.L., 2008. Singular stress analyses of V-notched anisotropic plates based on a novel finite element method. Eng Fract Mech. 75, 3819–3838 Profant, T., Ševeček, O., Kotoul, M., 2008. Calculation of K-factor and T-stress for cracks in anisotropic bimaterials, Eng Fract Mech. 75, 3707– 3726. Taylor, D., Cornetti, P., Pugno, N., 2005: The fracture mechanics of finite crack extension Engineering Fracture Mechanics, 72, 1021-1038. Taylor, D., 2017: The Theory of Critical Distances: A link to micromechanisms, Theoretical and Applied Fracture Mechanics, 90, 228-233. Vyhlídal, M., Rozsypalová, I., Majda, T., Daněk, P., Šimonová, H., Kucharczyková, B., Keršner, Z., 2019. Fracture Response of Fine-Grained Cement-Based Composite Specimens with Special Inclusions. Solid State Phenomena 292, 63–68. Westergaard, H. M., 1939. Bearing Pressures and Cracks. Journal of Applied Mechanics 61, 49–53. Williams, M. L., 1956. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics 24, 109–114.