Study on the bilinear softening mode and fracture parameters of concrete in low temperature environments

Engineering Fracture Mechanics 211 (2019) 1–16

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Study on the bilinear softening mode and fracture parameters of concrete in low temperature environments Shaowei Hua, Bing Fana,b, a b

T

⁎

Department of Materials and Structural Engineering, Nanjing Hydraulic Research Institute, Nanjing 210024, China College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China

ARTICLE INFO

ABSTRACT

Keywords: Concrete Low temperature Bilinear softening model Fracture energy Characteristic length Fracture toughness

The mode and fracture parameters of concrete at different temperature levels (20 °C, 0 °C, −20 °C, and −40 °C) were investigated using three-point bending fracture experiment in a lowtemperature fracture test system. The three-point bending beams with the same sizes, 750 mm × 150 mm × 70 mm, were divided into two groups according to age: 28 days and 120 days. The bilinear softening parameters for each beam were determined using the inverse analysis; then, the fracture energy GF and characteristic length lch were determined by the calculated softening curve. In addition, the fracture toughness including the initial fracture toughness KIcini , unstable fracture toughness KIcun and cohesive toughness K Icc were also analyzed. The experimental and calculation results show that the softening curve shape and fracture parameters of the concrete change significantly when the temperature decreases. Finally, the double-K fracture model was used to verify the validity of the bilinear softening curve calculated in this paper.

1. Introduction With the popularization of concrete materials, concrete structures have become the most common building type. Worldwide, many concrete structures are built in regions with cold winter temperatures. The temperature greatly influences the mechanical properties of concrete. Thus, investigations of these mechanical properties under low temperatures are necessary. Many experiments have proven that the strength of concrete at low temperatures is significantly stronger than that at normal ambient temperatures [1–6], mainly because of the effect of the water content; a higher water content corresponds to a greater observed increase [3–6]. For instance, Yamane et al. [3] report that the compressive strength of concrete in wet conditions increases rapidly during the initial stage of cooling. However, for concrete in dry conditions, the compressive strength does not clearly change during the initial stage of cooling (approximately 20 °C to −30 °C) but then significantly increases with a further decrease in temperature. Similar results were obtained from tensile strength tests. Since fracture mechanics was first applied to concrete materials [7], research on concrete fractures has made great progress, but most studies have focused on concrete at room temperature. The influence of high temperatures on the fracture parameters, mainly the fracture energy, material brittleness and fracture toughness, has been studied by several researchers [8–12]. Most experimental results show that the fracture energy sustains an increase-decease tendency, whereas fracture toughness decreases steadily with increasing temperature. To the best of the authors’ knowledge, few fracture tests of concrete at low temperatures have been reported. Planas et al. [13] and Maturana et al. [14] observed from three-point bending (TPB) fracture testing that the fracture energy and ⁎

Corresponding author. E-mail address: [email protected] (B. Fan).

https://doi.org/10.1016/j.engfracmech.2019.02.002 Received 17 July 2018; Received in revised form 27 December 2018; Accepted 2 February 2019 Available online 04 February 2019 0013-7944/ © 2019 Published by Elsevier Ltd.

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Nomenclature

a a0 ac b Ci CMOD CMODc CTOD CTODc dmax E ft g GF GFcal GFRILEM h KIcini KIcun K¯Icun

KIcc

l length of TPB beams (mm) lch characteristic length of concrete (mm) m weight of the beam between the supports (kg) m(x,a) weight function M1, M2, M3 parameters of weight function Nmax number of points in numerical simulation load (kN) P Pini initial cracking load (kN) Pmax maximum load (kN) Pexp experimental measured load (kN) Pnum numerical simulation load (kN) S span of TPB beams (mm) w crack opening displacement at the tip of initial notch (mm) ws crack opening at turning point of bilinear softening curve (mm) w0 crack width at stress-free point of bilinear softening curve (mm) W0 area under the P-δ curve cohesive stress at ws (MPa) s (x ) cohesive stress at equivalent-elastic crack length x (CTODc ) cohesive stress at initial crack tip at Pmax (MPa) crack-depth ratio or relative crack length parameter relating to maximum aggregate size F maximum displacement of P-δ curve (mm) c

effective crack length (mm) initial crack length (mm) critical effective crack length (mm) specimen thickness (mm) initial compliance for P-CMOD curve (mm/N) crack mouth opening displacement (mm) critical rack mouth opening displacement (mm) crack tip opening displacement (mm) critical crack tip opening displacement (mm) maximum aggregate size used in tested concrete (mm) Young’s modulus of concrete (GPa) tensile strength of concrete (MPa) acceleration of gravity fracture energy (N/mm) fracture energy calculated by softening curve (N/ mm) fracture energy calculated by the area under P-δ curve (N/mm) height of TPB beams (mm) initial fracture toughness (MPa m1/2 ) unstable fracture toughness (MPa m1/2 ) calculated unstable fracture toughness (MPa m1/2 ) cohesive fracture toughness (MPa m1/2 )

characteristic length of concrete increased clearly with decreasing temperature. Subsequently, a similar increasing tendency of the fracture energy has been reported by Ohlsson et al. [15]. Nevertheless, among the available literature, there is no discussion on the softening curve and fracture toughness of concrete at low temperatures. Moreover, the abovementioned research is far from sufficient; therefore, the low-temperature fracturing of concrete needs to be further studied. The softening curve is an important parameter in terms of the calculation of the fracture energy of the concrete and the cohesive force along the fictitious fracture zone throughout the fracture process. Therefore, it is necessary to accurately determine the softening curve of concrete for low-temperature conditions and to study the effect of temperature on the curve. The double-K fracture model, which was initially proposed by Xu and Reinhardt [16–18], introduced a new method with which to study crack expansion in concrete using two fracture parameters: the initial cracking toughness KIcini and the unstable fracture toughness KIcun . The fracture process can be divided into three stages: crack initiation, stable crack propagation, and unstable crack propagation. Correspondingly, the double-K fracture criterion can be described as follows: when K < KIcini , no crack appears; when KIcini K < KIcun , the crack stably develops; and when K KIcun , the crack unstably develops. To date, the double-K fracture theory has been widely studied by many scholars using experiments and numerical simulations [19–23]. This paper first reports the experimental results of TPB beams at different low temperatures. Then, the softening curve of each beam is determined using the inverse analysis. The effects of the low temperatures on the fracture parameters of the fracture energy GF , characteristic length lch , fracture toughness KIcini and KIcun and cohesive toughness KIcc are reported and analysed. Finally, the validity of the calculated softening curve is proven using the double-K fracture model.

Fig. 1. Geometry of the TPB beam.

2

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2. Experiment program 2.1. Specimen preparation and experimental setup In total, 32 TPB beams with a uniform dimension (750 mm × 150 mm × 70 mm) were cast for mode-I fracture testing. The notch depth a0 of each specimen was 30 mm, and the notch was 3 mm thick (see Fig. 1). All the beams were produced from one batch with the proportions of 0.55:1:1.87:3.48 (water/cement/natural sand/granite); grade 42.5 Portland cement was mixed with medium river sand and graded coarse aggregate with a maximum size of 20 mm. PT100 thermocouples were pre-embedded in each concrete beam before mould vibration to monitor the internal temperature during the fracture experiment. The beams were demoulded 24 h after casting and moved to a curing room at 20 ± 2 °C and a relative humidity of 100% for 28 days. After 28 days of moist curing, all the beams were removed from the curing room to the air environment and divided into two groups based on different ages; the first group was dried in the laboratory for approximately 5 days prior to the test, and the second group was dried in the laboratory for 120 days and then tested. During the testing period, the test beams were covered with plastic film to prevent moisture loss. Four identical beams were prepared for each test age and temperature. Eight standard concrete cubes with dimensions of 150 mm × 150 mm × 150 mm were prepared using the same batch of concrete to evaluate the compressive strength of the material at different ages under room temperature; the average compressive strengths were 30.8 MPa and 46.7 MPa for 28 days and 120 days, respectively. Another eight standard concrete prisms with dimensions of 300 mm × 150 mm × 150 mm were also used to measure the Young’s modulus of the material under room temperature; the average Young’s modulus were 29.6 GPa

Fig. 2. Low-temperature fracture test system. 3

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for 28 days and 33.1 GPa for 120 days. A low-temperature fracture test system was developed to perform the fracture experiment at low temperatures, which consisted of a computer-controlled electro-mechanical servo universal testing machine with a load capacity of 20 tons, two low-temperature boxes, a track and a control station, as shown in Fig. 2. Four target temperatures from 20 °C to −40 °C (20 °C, 0 °C, −20 °C, and −40 °C) were studied using this test system. The experimental process can be divided into the following steps: (1) put the TPB beams in the clamp; put two clip gauges on the mouth and tip of the pre-crack to measure the crack mouth opening displacement (CMOD) and crack tip opening displacement (CTOD), respectively; use two steel plates with thicknesses of 3 mm and 10 mm to fix the clip gauges; and paste PT100 thermocouples on the surface of the beam; (2) close the two low-temperature boxes through the control station, and set the target temperature; and (3) start the electro-mechanical servo testing machine after reaching the cooling time, and apply the load using displacement control with a constant rate of 0.05 mm/min . 2.2. Cooling process before fracture experiment Due to a change in ambient temperature, thermal expansion will occur in concrete. If a fracture test is carried out during the thermal expansion process, the influence of thermal expansion must be considered. Fig. 3(a) presents the tendency of the displacement measurement from the clip gauge at the mouth of the pre-crack during the cooling process. The displacement increases significantly with decreasing temperature and then maintains a constant value. Then, the internal temperature of the beam reaches the target temperature and remains constant (see Fig. 3(b)). Subsequently, a constant temperature was maintained for 8 h to minimize the impact of thermal expansion on the fracture test. Therefore, it is not necessary to consider the effect of thermal expansion caused by temperature inhomogeneity on the fracture mechanism in this study. Notably, each fracture experiment (at 0 °C, −20 °C, and −40 °C) took one day to complete; therefore, the ages of the two groups are approximately 33 days to 45 days and 120 days to 132 days. For convenience, these two groups were named “28 days” and “120 days”, respectively. 3. Experimental results and discussions Similar to the result obtained at room temperature, the P-CMOD curves include three stages with the crack growth at low temperatures. In the first stage, no crack occurs in the beam, and the curve has a linear relationship. In the second stage, the crack stably propagates, and the fracture process zone (FPZ) develops. In the third stage, the crack rapidly propagates and eventually causes the fracture failure of the specimen. Fig. 4 shows the typical complete experimental P-CMOD curve for the 28 day and 120 day beams at several low temperatures from 20 °C to −40 °C (the experimental curves of 0 °C for 120 days are not shown because they are similar to those of −20 °C). The PCMOD curves for the 28 days have greater peaks and become steeper with decreasing temperature. For 120 days, the height of the curve has no obvious change from 20 °C to −20 °C; when the temperature reaches −40 °C, the curve height becomes significantly high. The initial cracking load Pini , peak load Pmax , CMODc , CTODc , modulus of elasticity E and critical effective crack length ac are summarized in Table 1. The average values of Pini and Pmax for 28 days increase with decreasing temperature, from 2.57 kN and 4.64 kN at room temperature to 2.92 kN and 5.94 kN at 0 °C, 3.81 kN and 7.55 kN at −20 °C, and 5.28 kN and 9.48 kN at −40 °C, with a final increase of 117% and 103%, respectively. However, for 120 days, the average Pini and Pmax values first decrease, from 4.44 kN and 6.75 kN at room temperature to 3.37 kN and 6.64 kN at 0 °C and then 3.02 kN and 6.50 kN at −20 °C, subsequently increasing to 3.58 kN and 7.68 kN at −40 °C. The above experimental phenomena can be attributed to the water freezing within the concrete beams. For 28 days, because the

Fig. 3. Variation in the value of clip gauge and thermocouple in the cooling process. 4

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Fig. 4. Typical P-CMOD curves of TPB beams at different temperatures.

Table 1 Experimental results of fracture parameters. Item

Temp

Pini (kN)

Pmax (kN)

CMODc (μm)

CTODc (μm)

E (GPa)

ac (mm)

28 days 20-1 28 days 20-2 28 days 20-3 28 days 20-4 Average 28 days 0-1 28 days 0-2 28 days 0-3 28 days 0-4 Average 28 days -20-1 28 days -20-2 28 days -20-3 28 days -20-4 Average 28 days -40-1 28 days -40-2 28 days -40-3 28 days -40-4 Average 120 days 20-1 120 days 20-2 120 days 20-3 120 days 20-4 Average 120 days 0-1 120 days 0-2 120 days 0-3 120 days 0-4 Average 120 days -20-1 120 days -20-2 120 days -20-3 120 days -20-4 Average 120 days -40-1 120 days -40-2 120 days -40-3 120 days -40-4 Average

20 °C

2.55 2.49 2.57 2.65 2.57 2.85 2.74 3.32 2.76 2.92 3.85 3.83 3.90 3.98 3.89 5.17 5.43 5.03 5.49 5.28 4.30 4.82 4.22 4.41 4.44 3.54 3.66 3.08 3.21 3.37 3.14 2.77 3.23 2.93 3.02 3.62 4.01 3.36 3.32 3.58

4.56 4.74 4.58 4.69 4.64 5.73 5.90 6.29 5.85 5.94 7.67 7.63 7.50 7.38 7.55 9.11 9.57 9.59 9.66 9.48 6.85 7.09 6.55 6.52 6.75 6.97 6.47 6.36 6.75 6.64 6.19 6.61 6.61 6.58 6.50 7.75 7.96 7.56 7.46 7.68

45.47 35.99 45.89 34.79 40.54 62.59 62.51 52.57 45.38 55.76 61.81 56.57 50.16 47.52 54.02 45.87 54.15 45.60 58.99 51.15 40.50 41.99 46.97 46.12 43.90 46.53 45.26 51.89 50.85 48.63 56.93 60.34 53.11 51.01 55.35 49.46 74.29 57.01 68.09 62.21

28.35 25.36 26.40 16.39 24.13 44.27 46.60 32.23 24.70 36.95 38.33 34.46 32.23 30.61 33.91 25.86 33.49 24.33 33.55 29.31 20.33 22.95 30.36 26.16 24.95 24.92 25.39 30.57 25.99 26.72 38.46 40.52 39.02 29.13 36.78 27.23 44.78 32.19 45.65 37.46

24.17 25.26 23.59 25.88 24.73 27.70 31.19 27.11 35.40a 28.67 30.93 30.16 32.96 32.64 31.67 37.47 36.22 33.43 32.25 34.84 28.82 31.31 26.22 26.28 28.16 27.44 28.89 29.31 31.26 29.23 26.32 27.90 28.31 28.89 27.86 28.99 26.33 27.71 27.31 27.59

53.05 46.56 52.50 46.59 49.68 59.54 62.04 51.29 56.78 57.41 54.01 50.91 50.51 49.15 51.15 45.88 48.44 41.33 47.31 45.74 43.25 45.61 46.00 45.68 45.14 45.27 48.04 52.83 52.39 49.63 53.18 54.63 51.40 50.96 52.54 45.73 53.60 48.99 54.01 50.58

a

0 °C

−20 °C

−40 °C

20 °C

0 °C

−20 °C

−40 °C

Not considered for average value. 5

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wet curing process had just finished, the water content in the concrete is relatively high; ice fills the pores and bridges the microcracks while significantly enhancing the compactness of the concrete, leading to an increase of 117% for Pini and 103% for Pmax when the temperature decreases from 20 °C to −40 °C. Meanwhile, with decreasing temperature, the aggregate binding ability is increased and fracturing in the aggregates is more obvious if they are in the FPZ, as indicated in Fig. 5, meaning that the brittleness of concrete increases. In contrast, for 120 days, the moisture content and porosity of the concrete are relatively small (because of the drying process). Water in some pores freezes and expands, leading to slight damage to the concrete, leading to a decrease in Pini and Pmax from 4.44 kN and 6.75 kN at room temperature to 3.02 kN and 6.50 kN at −20 °C. With the further decrease in temperature, the water in the numerous finer pores also begins to freeze, which enhances the compactness of the concrete, leading to a significant increase in Pini and Pmax . On the other hand, the variation law of the peak load Pmax in this study is similar to those of the compressive strength and tensile strength given by Yamane et al. [3]. 4. Determination of softening curve and fracture parameters 4.1. Softening traction-separation law The softening traction-separation law of concrete describes the characteristics of the FPZ and is the key input parameter in both finite element simulations and fracture mechanics calculations. The softening traction-separation law has been studied by many scholars, and several softening curves, such as the linear, bilinear and exponential forms, have been widely used. Because of their simple form, convenient calculation and high accuracy, the bilinear forms are most commonly used to describe the softening tractionseparation law of quasi-brittle materials such as concrete. The shape of the bilinear softening curve, which is shown in Fig. 6, is decided by four parameters, ft , s , ws and w0 . Different researchers obtained different turning points ( s and ws ) and zero stress points w0 in the expression of softening traction-separation law, shown as follows: (a) Model proposed by Peterson [24]: s = ft /3 ws = 0.8Gf /ft

w0 = 3.6Gf /ft

(1)

(b) Model proposed by CEB-FIP [25]: s = 0.15ft ws = 2Gf / ft 0.15w0

w0 =

F Gf / ft

(2)

where ft is the tensile strength, GF is the fracture energy,

F

is a coefficient relating to the maximum coarse aggregate size,

Fig. 5. Crack surface at 20 °C and −40 °C for 28 days. 6

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Fig. 6. Bilinear softening curve. F =9

dmax /8, dmax is the maximum size of aggregate. The abovementioned softening constitutive models are obtained at room temperature and are not necessarily applicable in lowtemperature conditions. Moreover, because the acquisition of softening parameters, including the tensile strength ft , is not easy from the direct tensile test executed in this study, an inverse analysis is used. 4.2. Determination of softening curve using the inverse analysis On the basis of the experimental results from TPB tests or wedge splitting tests, a softening curve can be determined indirectly by an inverse analysis [26–28], which can be used for determining the non-linear fracture parameters of concrete. The principle consists of minimizing the differences between the measured and simulated P-CMOD curve, thus allowing us to define the relationship between the stress and the crack opening w , namely, the softening curve. To simulate crack propagation in concrete, a certain crack propagation criterion is needed; this criterion can be stress-based, energy-based, or stress intensity factor (SIF)-based. Considering that the size of the plastic zone in the FPZ is very small, the maximum tensile stress criterion was proposed as a crack propagation criterion to simulate the fracture process in concrete [29]. Additionally, based on the principle of energy conservation, an energy-based cohesive crack propagation criterion was proposed for concrete [30,31], indicating that the crack can propagate when the strain energy release rate exceeds the energy dissipation rate in the fictitious crack. However, the SIF-based crack propagation criteria are also widely used in numerical simulations of fracture processes for concrete. Considering the variation in FPZ length, an initial fracture toughness KIcini criterion based on SIF was proposed [32,33]. In this criterion, a crack propagates when the difference of SIF, i.e. KI , caused by the external load and the one by the cohesive force exceeds the initial fracture toughness of the concrete, i.e. KI KIcini . Subsequently, Dong et al. [34,35] determined that compared to the nil-SIF and maximum tensile stress criterion, the initial fracture toughness criterion is more suitable for the simulation of crack propagation with increasing concrete strength, particularly for high-strength concrete. It should be noted that the strength of the Table 2 Parameters of bilinear softening curve for each beam. Item

ft (MPa)

28 days 20-1 28 days 20-2 28 days 20-3 28 days 20-4 Average 28 days 0-1 28 days 0-2 28 days 0-3 28 days 0-4 Average 28 days -20-1 28 days -20-2 28 days -20-3 28 days -20-4 Average 28 days -40-1 28 days -40-2 28 days -40-3 28 days -40-4 Average

2.80 3.11 2.80 2.83 2.89 3.50 3.78 4.12 3.72 3.78 5.02 5.05 4.78 4.75 4.90 6.42 6.11 7.63 6.49 6.66

s

(MPa)

0.75 0.97 0.95 0.93 0.90 1.27 1.28 1.26 1.28 1.27 1.45 1.49 1.41 1.47 1.46 1.79 1.78 1.69 1.67 1.73

ws (mm)

w0 (mm)

Item

0.036 0.030 0.035 0.033 0.034 0.049 0.049 0.036 0.039 0.043 0.045 0.045 0.039 0.038 0.042 0.026 0.042 0.024 0.037 0.032

0.239 0.149 0.123 0.151 0.166 0.139 0.132 0.112 0.135 0.130 0.078 0.135 0.102 0.165 0.120 0.121 0.101 0.116 0.114 0.113

120 days 120 days 120 days 120 days Average 120 days 120 days 120 days 120 days Average 120 days 120 days 120 days 120 days Average 120 days 120 days 120 days 120 days Average

7

ft (MPa) 20-1 20-2 20-3 20-4 0-1 0-2 0-3 0-4 -20-1 -20-2 -20-3 -20-4 -40-1 -40-2 -40-3 -40-4

4.11 3.92 3.78 3.61 3.86 4.93 4.04 4.10 4.43 4.38 3.92 4.52 4.21 4.43 4.27 5.71 5.13 5.53 5.01 5.35

s

(MPa)

0.70 0.68 0.78 0.78 0.74 1.62 1.37 1.41 1.44 1.46 1.43 1.55 1.62 1.70 1.58 1.79 2.18 1.95 1.97 1.97

ws (mm)

w0 (mm)

0.033 0.038 0.038 0.036 0.036 0.030 0.035 0.047 0.046 0.040 0.046 0.047 0.046 0.044 0.046 0.036 0.052 0.037 0.051 0.044

0.192 0.223 0.170 0.221 0.202 0.186 0.183 0.156 0.146 0.168 0.175 0.144 0.160 0.162 0.160 0.192 0.165 0.163 0.140 0.165

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beams in this study significantly increases when the temperature decreases to −40 °C, particularly for the 28 day beams. To determine more accurate values of the softening curve’s parameters in the inverse analysis process, the initial fracture toughness criterion is used; a detailed explanation, which includes the calculating process, can be found elsewhere [32,33]. Additionally, an error function was used to determine the softening parameters that optimally fit the experimental P-CMOD curve in an iterative process, shown as follows: Nmax

e=

(Pexp

Pnum (ft ,

s,

ws , w0 ))2

min

(3)

i=1

where ft, s , ws and w0 are the bilinear softening parameters; Pexp and Pnum are the measured and simulated loads at the same CMOD value, respectively; and Nmax is the number of points in numerical simulation. The detailed calculation process of the inverse analysis is presented as follows: 1. Input the specimen geometry (including height, span, thickness and initial crack length), Young’s modulus E and initiation load Pini . 2. Determine the crack length increment Δa and load increment ΔP for the numerical simulation (the values determine the speed and accuracy of the numerical simulation). 3. Input the starting parameters of the bilinear softening curve. 4. Run the simulation to match the P-CMOD curve, adjust the parameters of the softening curve until the optimal fit is obtained (the error function e and the deviation between the measured and calculated values of Pmax and CMODc are minimized). The detailed parameters of the bilinear softening curves obtained from the inverse analysis are listed in Table 2. At room temperature, the ratios of ws· ft and w0· ft to the fracture energy GF (calculated by using the average value of the softening curve equal to 123.83 N/m ) and ft to s for the 28 day beams are 0.79, 3.87 and 3.21, respectively, which are comparable to 0.8, 3.6 and 3, as suggested by Petersson by using Eq. (1). Fig. 7 shows the average values of the obtained bilinear softening curves from the inverse analysis; 28 day beams and 120 day beams have similar trends. The tensile strength ft and stress at the break point s significantly increase with the decrease in temperature. The crack width of the stress-free point w0 at low temperature is lower than that at room temperature. However, the crack width of the break point ws exhibits no obvious trend when the temperature decreases. Fig. 8 shows the experimental and simulated P-CMOD curves for typical beams at different temperatures. The simulated P-CMOD curves agree well with the experimental data, meaning that the obtained softening curves are sufficiently precise for calculating the fracture parameters in the following analysis. Additionally, this also reveals that the bilinear softening curve can be applied to accurately simulate the fracture process in concrete, which is in agreement with the conclusion of Hoover and BaŽant [36], who carried out numerical simulations using the crack band (or cohesive crack) model. 4.3. Determination of fracture energy The fracture energy, GF , defined as the total energy dissipated over a unit area of the cracked ligament, can be determined by the area under the softening curve [37]. Based on the softening curve calculated by the inverse analysis in this paper, GFcal can be obtained using the following equation:

Fig. 7. Average value of bilinear softening curves obtained from the inverse analysis. 8

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Fig. 8. P-CMOD curves obtained from experiment and numerical simulation.

GFcal =

1 (f ws + 2 t

s w0 )

(4)

Meanwhile, the fracture energy can be calculated with the help of RILEM [38] to validate the accuracy of GFcal as follows:

GFRILEM =

W0 + mg A

c

(5)

where W0 is the area under the P-δ curve, m is the weight of the beam between the supports, g is the acceleration of gravity of 9.8 N/kg , c is the maximum displacement, and A is the fracture area and equal to (h a0) b . Fig. 9 shows the tendency of fracture energy GFcal ; the values for both 28 day and 120 day beams increases monotonously with decreasing temperature. For 28 days, the average value of GFcal increases from 120.82 N/m at ambient temperature to 163.85 N/m at 0 °C, 188.67 N/m at −20 °C and 203.70 N/m at −40 °C, with a significant increase of 82.88 N/m or 69%. For 120 days, the average value of GFcal increases from 143.65 N/m at ambient temperature to 208.50 N/m at 0 °C, 224.06 N/m at −20 °C and 278.69 N/m at −40 °C, with a significant increase of 135.04 N/m or 94%. A comparison between the fracture energy calculated based on the optimized softening curve GFcal and the experimentally measured fracture energy GFRILEM is presented in Fig. 10. The calculated and measured values are very similar. Among the 32 beams, the deviations between GFcal and GFRILEM are within 5% for 23 beams and within 10% for 9 beams. Meanwhile, the calculated values are slightly lower than the measured results; this could be explained by considering the following. (1) The bilinear softening curve has limitations, as Fig. 8(a) shows that the area under the measured P-CMOD curve for −20 °C is slightly larger than that of the simulated P-CMOD curve, which leads to a calculated fracture energy that is slightly lower than the measured values; and (2) the measured

Fig. 9. Tendency of fracture energy GFcal at low temperatures. 9

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Fig. 10. Comparison between GFRILEM and GFcal.

fracture energy is calculated from the P-δ curve, while the optimized softening constitutive parameters are determined by fitting the P-CMOD curve [26]. 4.4. Determination of characteristic length Brittleness can be explained as the tendency of a material or structure to fracture suddenly before significant irreversible deformation occurs. Different brittleness parameters are used to assess the brittleness of concrete, such as energy-based, deformationbased and complex parameters. Based on an elastic-brittle approach, Hillerborg et al. [37] proposed that the characteristic length lch can be used as a brittle index of concrete shown as follows:

lch =

EGFcal ft2

(6)

where the material properties E , and ft are the functions of temperature. is the fracture energy and ft is the tensile strength, which are determined by inverse analysis. E is the modulus of elasticity determined by the initial slope of P-CMOD curve. Because the characteristic length includes a combination of energy, stiffness, and strength parameters, it is regarded as a synthetic brittleness parameter, and a smaller value indicates greater brittleness in concrete. The average value of lch in Fig. 11 shows that the brittleness for 120 days is higher than that for 28 days at room temperature, meaning that the concrete brittleness increased with the age. Because of the effect of the humidity and porosity of the concrete, the average value of lch for 28 days decreased monotonically from 360.80 mm at room temperature to 335.37 mm at 0 °C, 248.97 mm at −20 °C and 164.93 at −40 °C, with a significant overall decrease of 195.87 mm, or 54%. For 120 days, lch sustained an increase-

GFcal

GFcal

Fig. 11. Tendency of characteristic length lch at low temperatures. 10

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S. Hu, B. Fan

decrease tendency with decreasing temperature. The average value of lch increases from 273.73 mm at room temperature to 322.53 mm at 0 °C, reaches the maximal value of 344.08 mm at −20 °C by increasing 70.35 mm, or 26%, and finally decreases to 270.83 mm at −40 °C. This conclusion is consistent with the experimental phenomenon, i.e., for a large brittle concrete beam, the load will suddenly drop after reaching the maximum value, emitting a breaking sound. 5. Calculation of fracture toughness and verification of bilinear softening parameters 5.1. Calculation of KIcini and KIcun According to the double-K fracture model proposed by Xu and Reinhardt [17], the initial cracking fracture toughness KIcini in this paper can be calculated by substituting the initial cracking load Pini and initial notch depth a0 into Eq. (7).

KIcini = f (a 0 / h ) =

1.99

3Pini S 2bh2

(a0 / h )(1

a 0 f (a 0 / h ) 3.93a0 / h + 2.7(a0 / h)2]

a0 / h)[2.15

(7)

2a0 / h )3/2

(1 + 2a0 / h )(1

can also be calculated by substituting the peak load Pmax and critical With the above formula, the unstable fracture toughness notch length ac for Pini and a0 , respectively. In this paper, Pini was obtained through the initial point of non-linearity in the P-CMOD curve listed in Table 1. An enlarged part of the P-CMOD curve measured from beam “28 days -20-4” is shown in Fig. 12. It can be seen that the point surrounded by the black circle is the starting point of the observable nonlinear segment of the P-CMOD curve, this point is often regarded as an initial cracking point, and the corresponding load value is initial cracking load Pini . Based on the handbook of stress intensity factors by Tada [39], the P-CMOD relation for the TPB beam of S / h = 4 can be expressed as follows:

KIcun

CMOD = v ( ) = 0.76

24Pa v( BEh

2.28 + 3.87

2

) 2.04

3

+

0.66 (1

(8)

)2

where is equal to (a + h 0/ h + h 0) ; h 0 is the thickness of the knife edge holding the clip gauges and equal to 3 mm in this paper; a is the fictitious crack length; ac can be determined by Eq. (8) when P is equal to Pmax and CMOD is equal to CMODc ; and E is the Young’s modulus, which can be calculated by Eq. (9) with the Ci from the initial linear part of the P-CMOD curve.

E=

24a0 v ( 0) Ci Bh

(9)

in which, 0 = (a0 + h 0 / h + h 0 ), Ci = the initial compliance = CMODi /Pi . It was confirmed by the extensive tests that the value of Young’s modulus E calculated from P-CMOD curve or P–δ was almost no difference as the measured from compression cylinders [40]. The trends in the initial fracture toughness KIcini and unstable fracture toughness KIcun for the 28 day and 120 day beams are presented in Fig. 13. The double-K fracture parameters of concrete beams are significantly affected by the temperature decrease. The tendency of double-K fracture parameters for 28 days is shown in Fig. 13(a). Both KIcini and KIcun increases clearly with decreasing temperature; the low temperature appears to increase the strength of concrete. The average value of KIcini increases from 0.44 MPa m1/2 at room temperature to 0.51 MPa m1/2 at 0 °C, 0.67 MPa m1/2 at −20 °C and 0.92 MPa m1/2 at −40 °C, with a significant increase of 0.48 MPa m1/2 or 109%. The average value of KIcun increases from 1.13 MPa m1/2 at room temperature to 1.66 MPa m1/2 at

Fig. 12. Determination of initial cracking load Pini through P-CMOD curve. 11

Engineering Fracture Mechanics 211 (2019) 1–16

S. Hu, B. Fan

Fig. 13. Tendency of double-K fracture toughness at low temperatures.

0 °C, 1.88 MPa m1/2 at −20 °C and 2.16 MPa m1/2 at −40 °C, with a significant increase of 1.03 MPa m1/2 or 91%. The trend of 120 days is different because of the influence of humidity and porosity of concrete (see Fig. 13(b)). The beam appears to suffer slight damage from 20 °C to −20 °C, and the beam is easier to fracture. The average value of KIcini decreases from 0.77 MPa m1/2 at room temperature to 0.58 MPa m1/2 at 0 °C, reaches the minimal value of 0.52 MPa m1/2 at −20 °C with a loss of 0.25 MPa m1/2 or 32%, and finally slightly increases to 0.62 MPa m1/2 at −40 °C. The average value of KIcun increases steadily with decreasing temperature from 1.52 MPa m1/2 at room temperature to 1.61 MPa m1/2 at 0 °C, 1.66 MPa m1/2 at −20 °C and 1.90 MPa m1/2 at −40 °C, finally increase of 0.38 MPa m1/2 or 25%. On the other hand, by comparing Figs. 13 and 11, it can be found that the characteristic length lch has an inverse relationship with the initial fracture toughness KIcini , meaning that a higher brittleness of concrete makes crack initiation more difficult. 5.2. Calculation of KIcc 5.2.1. Cohesive force distribution of TPB beam To calculate the cohesive toughness KIcc , the form of the cohesive force distribution along the fictitious crack zone of the TPB beam must be determined first. With the calculated bilinear softening constitutive parameters in Table 2, the critical crack tip opening displacement CTODc is less than the break point ws ; accordingly, the cohesive stress distribution in the fictitious fracture zone in a critical situation is approximated to be linear as shown in Fig. 14. The varied cohesive stress along the fictitious fracture zone (x) is calculated in Eq. (10).

(x ) =

(CTODc ) + (ft

(CTODc ))(x

a0)/(ac

(10)

a0 )

where (CTODc) is the critical cohesive stress at the tip of the initial notch. Combine with Fig. 6, (CTODc) can be calculated by substituting the bilinear softening constitutive parameters into Eq. (11).

(CTODc ) =

s

+ (ws

CTODc )(ft

(11)

s )/ ws

The value of CTODc is an important parameter to determine the cohesive force in the fictitious fracture zone; thus, it is necessary to accurately calculate CTODc . 5.2.2. Calculation of CTODc The crack tip opening displacement (CTOD) is an important part of the calculation of the nonlinear fracture of concrete. Jenq et al.

Fig. 14. Cohesive stress distribution in the FPZ at peak load. 12

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S. Hu, B. Fan

[41] proposed a two-parameter fracture model of concrete; in this paper, CTODc can be calculated by substituting CMODc into Eq. (12).

a 0 / ac ) 2 + [1.081

CTODc = CMODc {(1

1.149(ac / h)][a 0 /ac

(12)

(a 0 / ac ) 2]}1/2

where CMODc is the critical crack mouth opening displacement of TPB beam. To date, research on CTOD has generally been performed at normal temperatures, and there are few reports at low temperatures. To verify the applicability of Eq. (12) under low-temperature conditions, this study measured CTOD using a clip gauge fixed at the tip of the pre-crack. Fig. 15 shows the comparison between measured and calculated CTODc for the 28 day and 120 day beams. The calculated values agree well with the measured values. For 28 days, among 16 beams, the deviations between measured and calculated CTODc of 15 beams are below 15%. For 120 days, among 16 beams, the deviations between measured and calculated CTODc of 14 beams are below 15%. Thus, Eq. (12) has good applicability under low-temperature conditions. 5.2.3. Weight function method to calculate cohesive toughness KIcc The use of a weight function offers an efficient analytical technique to calculate the stress intensity factor in a cracked body subject to arbitrarily applied stress conditions, which was originally proposed by Bueckner [42]. The basic principle of this method is that the expression of the weight function depends on the geometry of the specimen, regardless of the form of loading that is used in this geometry. For a cracked body in two different cases r and s of mode I loadings, Kr and Ks are the stress intensity factors. If Kr and the respective crack face displacement ur are known, the value of Ks can be directly obtained as follows:

Ks =

a 0

s (x ) m (x ,

a) dxs

(13)

The term m(x,a) is the weight function and expressed as:

m (x , a ) =

E ur 2K r a

(14)

where s (x) is the cohesive stress distribution along the crack line for the loading case s , which can be experimentally, numerically or 2 for analytically determined. dxs is the infinitesimal length along the crack surface, and E ' is the Young’s modulus, i.e., E or E/1 the plane stress or plane strain, respectively. is the Poisson’s ratio. The weight function m (x , a) can be further expressed as:

m (x , a ) =

2 2 (a

x)

[1 + M1 (1

x / a)1/2 + M2 (1

x / a) + M3 (1

x / a)3/2

+Mn (1

x / a ) n]

(15)

where M1, M2 , M3 , , Mn are the parameters of the weight function. If the first four terms of Eq. (15) are used to approximate the expression weight function m (x , a) , then:

m (x , a ) =

2 2 (a

x)

[1 + M1 (1

x / a)1/2 + M2 (1

x / a) + M3 (1

x / a)3/2]

(16)

To determine the values of three parameters, M1, M2 , and M3 , of the weight function in Eqs. (15) and (16) for a single-edge cracked specimen with a finite-width plate subjected to a pair of normal unit forces, Kumar et al. [20] approximated the Green function given by Tada based on using the regression analysis of least squares. In his study, the values of M1, M2 , and M3 can be expressed as a

Fig. 15. Comparison between measured and calculated CTODc . 13

Engineering Fracture Mechanics 211 (2019) 1–16

S. Hu, B. Fan

function of the a/ h ratio as follows: For i = 1 and 3:

Mi =

1 [ai + bi a /h + ci (a/ h)2 + di (a/ h)3 + ei (a/ h) 4 + fi (a/ h)5] a/ h)3/2

(1

(17)

For i = 2: (18)

Mi = [ai + bi a /h]

where the values of the coefficients ai , bi, ci, di , ei and fi can be found elsewhere [20]. Compared with the standard Tada Green’s function, Eqs. (17) and (18) have the error of less than 3% for 0 a/ h 0.95 in the range of 0 x /a 0.98 [20]. After the parameters of the weight function and cohesive stress distribution in FPZ (see Fig. 14) at the peak load are known and substituted into Eq. (13), the cohesive fracture toughness KIcc is expressed in the form of the four terms weight function that can be expressed as:

KIcc =

ac a0

{ (CTODc ) + (ft

+ M3 (1

(CTODc ))(x

2 2 ac

a 0)} ×

2 2 (a

x)

[1 + M1 (1

x / a)1/2 + M2 (1

x /a) (19)

x / a)3/2] dx

After the integration of Eq. (19),

KIcc =

a0)/(ac

KIcc

A1 ac 2s1/2 + M1 s +

is expressed as follows:

{

2 1 4 3/2 1 4 1 M2 s3/2 + M3 s 2 + A2 ac2 s + M1 s 2 + M2 s5/2 + M3 [1 3 2 3 2 15 6

(a 0 /ac )3

3s (a0 / ac )]

}

(20) ft

(CTODc )

where A1 = (CTODc ) and can be determined by Eq. (11); A2 = ; ft is the tensile strength of concrete determined by the ac a0 inverse analysis as shown in Table 2; and CTODc can be calculated by Eq. (12). Fig. 16 shows the tendency of cohesive fracture toughness at low temperatures. The cohesive fracture toughness for the 28 day and 120 day beams has similar increasing trends with decreasing temperature. For 28 days, the average KIcc significantly increases from 0.68 MPa m1/2 at room temperature to 1.13 MPa m1/2 at 0 °C; subsequently with a slight increase to 1.19 MPa m1/2 at −20 °C and 1.24 MPa m1/2 at −40 °C, with a final increase of 0.56 MPa m1/2 or 82%. For 120 days, the average KIcc significantly increases from 0.75 MPa m1/2 at room temperature to 1.03 MPa m1/2 at 0 °C, 1.13 MPa m1/2 at −20 °C and 1.28 MPa m1/2 at −40 °C, with a final increase of 0.53 MPa m1/2 or 71%. 5.3. Verification of bilinear softening parameters by the double-K fracture method According to the values of the initial fracture toughness KIcini and cohesive toughness KIcc calculated in the previous section, the unstable fracture toughness can also be determined by an analytical back-calculation method expressed as Eq. (21), denoted by K¯Icun . By comparing K¯Icun with the unstable fracture toughness KIcun calculated by Pmax and ac , the accuracy of bilinear softening parameters calculated in this paper can be verified.

K¯Icun = KIcini + KIcc

(21)

Fig. 17 shows the comparison between

KIcun

and K¯Icun for the 28 days and 120 days specimen groups. For 28 days, of all 16 effective

Fig. 16. Tendency of cohesive fracture toughness at low temperatures. 14

Engineering Fracture Mechanics 211 (2019) 1–16

S. Hu, B. Fan

Fig. 17. Comparison between KIcun and K¯Icun .

beams, the deviations between KIcun and K¯Icun of 15 beams are less than 2% and that of one beam is 2.6%. For 120 days, of all 16 effective beams, the deviations between KIcun and K¯Icun of all beams are less than 2%. 6. Conclusions In this study, the three-point bending fracture experiment of concrete under different temperature levels (20 °C, 0 °C, −20 °C, and −40 °C) was performed using a low-temperature test system. According to age, the concrete beams were divided into two groups, 28 days and 120 days. Combined with the measured P-CMOD curves, the bilinear softening curve for each beam was calculated using the inverse analysis. From the calculated softening curve, important parameters such as ft , GF , lch and KIcc can be obtained. The following conclusions are drawn: (1) Low temperatures greatly influence the fracture properties of concrete, and different beam ages lead to different experimental results. For the 28 day beams, the complete P-CMOD curve is higher and steeper at lower temperatures. In contrast, the highest point of the P-CMOD curve for the 120 day beams exhibits no significant changes from 20 °C to −20 °C but significantly increases at −40 °C. (2) The shape of the softening curve significantly changes when the temperature decreases; the 28 days and 120 days changes have similar trends. The tensile strength ft and stress of the break point s significantly increase when the temperature decreases. The crack width at the stress-free point w0 at a low temperature is lower than that of room temperature, whereas the crack width of the break point ws undergoes no obvious systematic change. (3) At room temperature, the characteristic length lch of the 28 day beams is greater than that of the 120 day beams. With the decrease in temperature, lch of 28 days decreases monotonically, while lch of 120 days sustains an increase-decrease tendency and reaches a maximum value at −20 °C. lch has an inverse relationship with the initial fracture toughness KIcini . (4) A decrease in temperature results in a coincident increase in the fracture energy GF , unstable fracture toughness KIcun and cohesive toughness KIcc for both the 28 days and 120 days specimen groups. (5) The relationship between KIcun and K¯Icun was investigated based on the obtained softening curve by the inverse analysis of the measurements from 32 beams. The deviations between KIcun and K¯Icun are less than 2% and 2.6% for 31 beams and one beam, respectively. The calculated softening curve of the concrete studied in this paper has been fully verified by the double-K fracture model. Acknowledgments This research has been financially supported by the National Major Scientific Instruments Development Project of China (No. 51527811) and the National Key Research and Development Program of China (No. 2016YFC0401902). Compliance with ethical Standards Funding: This study was funded by the Program of the National Major Scientific Instruments Development Project of China (grant number 51527811) and the National Key Research and Development Program of China (grant number 2016YFC0401902). Conflict of interest: The authors declare that they have no conflict of interest. 15

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