Application of Bilinear Softening Laws and Fracture Toughness of Foamed Concrete

1. Introduction

Building structural design entails determining the best way to transmit loads to the ground. However, utilization of concrete is inevitable for construction due to its moldability, impermeability and favorable compression condition; somehow, the weight of normal concrete is relatively hefty and creates a huge dead load for the upper structure. Regarding sub-structure such as the foundation, a massive load, which is transferred, impacts foundation design more than if the soil bearing capacity is low. This becomes a high cost. Introducing lightweight materials such as foamed concrete is promising as a solution to counter this problem. Nevertheless, incorporating void into concrete limits the strength properties but still, has potential as a structural material. Incorporating pozzolanic material as cement replacement not only reduces the production of CO₂ but also enhances the strength material properties of foamed concrete. Several studies utilized silica fume as pozzolanic materials by: Lee et al. (2018)[1], Gökçe et al. (2019)[2], Ahmad et al. (2019)[3] and Wang et al. (2020)[4] with density 1700, 1424, 1300 and 1400 kg/m³ have compression strength with 27.12, 26.8, 24.3 and 20 MPa, respectively.

In conventional theory (stress-strain equilibrium), concrete tensile resistance is negligible, however knowing the fracture behavior of brittle material which can lead to catastrophic failure is essential, especially in critical buildings such as dam, tunnel and nuke power plant. The drawback of conventional theory is not able to predict the fracture behavior of material especially after it reached the material’s resistance. The characteristic of fracture process zone (FPZ) ahead of a crack tip prior to fracture[5], the shape and dimension within concrete material result in improper methods of predicting failure process using Linear Elastic Fracture Mechanic (LEFM)[6]. The stress-transfer capability of the material, sometimes referred to as the softening features of concrete, is compromised by the presence of FPZ. Various studies on fundamental models were approached, such as the fictitious crack model (also known as a cohesive crack model)[5], crack band model[7], two-parameters fracture model, effective crack band model[8], size-effect model[9] and double-K fracture model[10].

Fracture energy is the amount of energy used to open a unit area of a crack surface, and this is one of the parameters governing the damage and fracture mechanism. A study conducted by Jaini et al. (2015)[11] investigated fracture energy on foamed concrete with different densities (1400-1600 kg/m³). A similar investigation by Kozlowski et al. (2015)[12] with density (488-1024 kg/m³). Both studies showed that an increased density of foamed concrete density results in increased fracture energy. According to Falliano et al. (2019)[13], fracture energy in foamed concrete is influenced by curing method selections. While Xu et al. (2018)[14] investigated fracture energy considering the boundary effect. It was found that the smaller the local fracture energy was, the closer it was to the element boundary. Ding et al. (2020)[15] observe fracture energy on slag-based geopolymer (SG) and Portland cement (PC) with various compressive strengths. Similar to the compressive strength, SG developed higher fracture energy compared to PC.

Non-contact monitoring methods, including acoustic emission (AE) and digital image correlation (DIC), have become widely employed to better understand the fracture process occurring before the crack tip [16], [17] and [18]. Ohno et al. (2014)[19] investigated FPZ in a notched concrete beam under three-point bending (3PBT) with various-size aggregates by applying acoustic emission (AE). It found that the fracture energy correlates with the width of the AE cluster, as the energy increase when the width of FPZ expands. Alam et al. (2014) [20] observed FPZ of the notched beam under 3PBT by using DIC and AE. It concluded that DIC is better compared to AE due to DIC is based on crack opening, while AE may cause a loss of information as it is not possible to know exactly the crack tip. While Wu et al. (2011)[21] observed FPZ by DIC and stated the length of FPZ increased during crack propagation and decreased after FPZ is fully developed.

Finite element (FE) has been extensively utilized in previous studies to investigate the damage and fracture mechanisms in structural engineering [22], [23], [24], [25] and [26]. There are three well-known methods within Traction Separation Law (TSL): Extended Finite Element Method (XFEM), Cohesive Zone Model (CZM) and Virtual Crack Closure Technique (VCCT). However, the selection method within TSL will determine the behavior and structural response [27]. An investigation by Yu et al. (2021) [28] observed a notched graphite nuclear beam using XFEM, CZM and VCCT. The result showed VCCT is more sensitive compared to XFEM and CZM. While Omar et al. (2021) [29] worked on foam concrete beams using XFEM and CZM and found that there was less agreement within CZM due to the simplification of adopting a failure path. However, the LEFM theory adopted within ABAQUS software is only well applicable for materials that have a relatively small plastic process zone [30].

The novelty within this study is that the material constituent of foamed concrete is different compared to normal concrete, such as no coarse aggregate, and a more void presence, which affects in fracture process zone length (or critical crack). As far as our best knowledge, there has been no previous study done before. The inverse analysis is adopted to estimate critical crack length, which will describe in the following section. DIC was also carried out to observe critical crack length at the ultimate load. In addition, TSL incorporated with XFEM and CZM within two-dimensional (2D) FEA Modelling is adopted. Later crack propagation at ultimate load and predicted results were also observed.

1.1. Bilinear Softening Law

The classical bilinear softening law is extensively used in fracture mechanics and used as a cohesive zone model [5], [31] and [32]. The model is based on the idea that concrete softens gradually owing to microcracking and other energy dissipation processes in an extended fracture process zone (FPZ) prior to a real traction-free crack. As with a real traction-free crack, this portion of the crack cannot be continuous with complete separation of its faces. The fictitious crack faces have certain residual stress that can be transferred across them that are not consistent along their length. An investigation by S. Hu & Fan (2019) [33], as depicted in Figure 1 a) stated that when critical tip opening displacement reaches its value${(CTOD}_c)$, $\sigma \left({CTOD}_c\right)$, which is the cohesive stress at the fracture tip is reached. Figure 1 b) is a condition where crack is already formed. This condition is described in Figure 1 c). Meanwhile, according to Roelfstra & Wittmann (1986)[34] emphasized that the determined kink point (σ₁, w₁) of bilinear softening law is the most essential.

1.2. Fracture Toughness

Prior to reaching the ultimate load, crack initiation is already formed within the concrete material[33]. However, this condition of crack propagation under applied load can be explained by utilized fracture toughness (K). Since using LEFM theory is not relevant due to FPZ presence, the double-K method was adopted by several researchers [35], [36], [37], [38] and [17]. The important parameters of fracture toughness are Kⁱⁿⁱ and K^un, which are the initial cracking and unstable cracking toughness, respectively. From this case, it is clear that in quasi-brittle, the fracture crack developed in three stages: initial cracking, stable development and failure development. The corresponding fracture criteria are that when ${K<K}_{IC}^{ini}$ no crack appears. When $K_{IC}^{ini}{\le K<K}_{IC}^{un}$ the crack developed stably and when ${K\ge K}_{IC}^{un}$ the crack develops unstably, the specimen is in the failure stage. These parameters have the following relationship: $K_{IC}^{un}{=K}_{IC}^{in}+K_{IC}^C$, ${\Delta a}_c=a_c-a_0$. $K_{IC}^{ini}$ and $K_{IC}^{un}$ can be calculated using Eqs. 1 and 3, respectively.

$$K_{IC}^{ini}=\frac{3P_{ini}\mathrm{S}\sqrt{a_0}}{2H^{2}B}F\left(\frac{a_0}{H}\right)$$

(1)

$$F\left(\frac{a_0}{H}\right)=\frac{1.99-\left(\frac{a_0}{H}\right)\left(1-\frac{a_0}{H}\right)\left[2.15-3.93\left(\frac{a_0}{H}\right)+2.7{\left(\frac{a_0}{H}\right)}^2\right]}{\left(1+2\left(\frac{a_0}{H}\right)\right)\left(1-{\left(\frac{a_0}{H}\right)}^{1.5}\right)}$$

(2)

$$K_{IC}^{un}=\frac{3P_{ult}\mathrm{S}\sqrt{a_c}}{2H^{2}B}F\left(\frac{a_c}{H}\right)$$

(3)

$$F\left(\frac{a_c}{H}\right)=\frac{1.99-\left(\frac{a_c}{H}\right)\left(1-\frac{a_c}{H}\right)\left[2.15-3.93\left(\frac{a_c}{H}\right)+2.7{\left(\frac{a_c}{H}\right)}^2\right]}{\left(1+2\left(\frac{a_c}{H}\right)\right)\left(1-{\left(\frac{a_c}{H}\right)}^{1.5}\right)}$$

(4)

$$a_c=\frac{2}{\pi }\left(H+H_0\right)arctan\sqrt{\frac{B.E}{32.6P_{ult}}{CMOD}_c-0.1135-H_0}$$

(5)

$$E=\frac{6S{a}_0}{C_{i}B{H}^2}\left(0.76-2.28{\alpha }_0+3.87{\alpha }_0^2-2.04{\alpha }_0^2+\frac{0.66}{{\left(1-{\alpha }_0\right)}^2}\right)$$

(6)

$${CTOD}_c={CMOD}_c{\left\{{\left(1-\raisebox{1ex}{$a_0$}\!\left/ \!\raisebox{-1ex}{$a_c$}\right.\right)}^2+\left[1.081-1.14\left(\raisebox{1ex}{$a_c$}\!\left/ \!\raisebox{-1ex}{$H$}\right.\right)\right]\left[\raisebox{1ex}{$a_0$}\!\left/ \!\raisebox{-1ex}{$a_c$}\right.-{\left(\raisebox{1ex}{$a_0$}\!\left/ \!\raisebox{-1ex}{$a_c$}\right.\right)}^2\right]\right\}}^{1/2}$$

(7)

Where P_ult is the ultimate load, and P_ini was obtained through the initial point of non-linearity in the P-CMOD curve. B, H and S are the width, height and span of the beam, respectively; H₀ is clip gauge holder thickness (this study is 2 mm). In addition, elastic modulus can be predicted using inverse analysis from the P-CMOD curve. Where ${\alpha }_0=(a_0+H_0)/(H+H_0)$ and C_i is the initial compliance of the P-CMOD. The sum of the initial pre-cut crack length ($a_0$) and fictitious crack extension length ($\Delta a_c$) equals the critical crack length ($a_c$).

2. Experimental Investigations

2.1. Experimental Testing

A specific gravity mix design was utilized for batching the lightweight foamed concrete. The component of lightweight foam concrete was river sand with a maximum 3 mm grain size, Portland Cement I-42.5R, superplasticizer (SP), water pH 7, and silica fume (SF). SF was manufactured by Shinjiang Shengping Minerals Co. Ltd as a partial cement replacement. A synthetic-based foaming agent Sika Aer 50/50, with a ratio of 1:20 was used to generate foam. According to Table 1, water to binder, sand to binder, and SP to binder ratio, SF replacement were 0.42, 3, 0.01 and 5%, respectively. Prior to foam addition, the base mix density was measured to be 1995 kg/m^3, and the addition of foam was intended to achieve a density of 1820 kg/m³. Casting was carried out when concrete achieved desired density. The concrete was set for 24 hours before demold and water cured for 28 days.

Figure 2. Fabrication of foamed concrete and density checking each stage.

Figure 2. Fabrication of foamed concrete and density checking each stage.

After reaching the 28^th day, to obtain dry conditions, specimens were put inside the oven for 24 hours. Prior to testing, painting and marking were conducted. Contrast color white and black spackles, smooth surface, were applied in order for DIC software (GOM Suite) able to detect. Three specimens of each notched beam and cylinder were tested and recorded by two configured cameras (camera 1 was to detect deformation, and camera 2 was to show applied load, Figure 3 (d)). Un-notched concrete beams were tested under a four-point bending test following ASTM-C78-02 [39]. The purpose of an un-notched beam is to determine tensile strength without being affected by stress concentration (or un-notched strength) which this value is required in FEM later. While notched beams were tested under a three-point bending test based on JCI-S-001[40] standard with a speed rate of 0.1 mm/min. The purpose of this notched beam is to observe fracture characteristics and fracture energy (G_F). The schematic test and geometry of specimens for the un-notched and notched concrete beam are shown in Figure 3. While cylinder specimens were tested under compression test following ASTM C469-02 [41] standard, which is to determine elastic modulus and poison’s ratio required for FEM. Table 2 shows the testing series for each specimen

Figure 3. (a) Un-notched beam, (b) Notched beam, (c) Cylinder of foamed concrete and (d) UTM testing.

Figure 3. (a) Un-notched beam, (b) Notched beam, (c) Cylinder of foamed concrete and (d) UTM testing.

Figure 4. Schematic test of (a) Un-notched beam, (b) Notched beam (with GF-30) and (c) Cylinder of foamed.

Figure 4. Schematic test of (a) Un-notched beam, (b) Notched beam (with GF-30) and (c) Cylinder of foamed.

2.2. Experimental Output

The shape of the bilinear softening law is determined by four parameters f_t, σ₁, w₁, w₂ (see. Figure 3 (a)). Compared to the previous study [15], this study obtained different kink points (σ₁, w₁) and zero stress points (w₂) in the expression of cohesive law (see Table 3). Regarding kink point (σ₁, w₁), according to [42], it was assumed to occur at 0.15-0.33 and are similar within this study. Nevertheless, zero stress point (w₂) was compared to other studies[15,33] (with values 0.284 and 0.239 mm, respectively) on normal concrete, but foamed concrete developed a value of more than 2.0 mm. This condition was due to void presence, which allowed to arrest of the crack. Regards to an initiation crack obtained from a second non-linear load-CMOD [16], [33], the initiation within foamed concrete is relatively close to the ultimate load. This is because the foamed concrete does not contain coarse aggregate to provide bridging. Stress concentration such as notch influenced height of FPZ above crack tip (Figure 10 (a)) which reduces tensile strength (f_t) and also at kink point (σ₁, w₁) see Table 3.

Figure 5. a) Bilinear softening law and (b) P-d curve within 3PBT.

Figure 5. a) Bilinear softening law and (b) P-d curve within 3PBT.

In this study, fracture energy (G_F) was estimated using the classical Hillerborg [43] model (based on load-displacement) (Eq 8). The previous study used the Hillerborg model to estimate fracture energy of foamed concrete material [11], [12] with values 0.018-0.025 N/mm for density 1400-1600 kg/m3 and 0.001-0.012 N/mm for density 488-1024 kg/m³, respectively. While Falliano et al. (2019) [13] estimated the fracture energy according to JCI-S-001[40] equation (based on load-CMOD) with a value of 0.003-0.010 N/mm for density 800 kg/m3. Figure 3(b) depicts the softening area (U₀) within this model, while m_g is specimen weight, do is deflection at the failure, W is specimen width, H is specimen height, and a0 is notch height. Compared to other studies working on normal concrete, foamed concrete fracture energy exhibited relatively lower. In addition, inverse analysis (from the notched beam) and conventional analysis (from the cylinder) are carried out; however, both analyses showed good agreement in terms of elastic modulus (E), as mentioned in Table 4. Double-K was used to observe fracture toughness in this study, and critical tip opening displacement (CTOD_c) was computed from critical mouth opening displacement (CMOD_c) at ultimate load.

$$G_F=\frac{\left(U_0+m_g{d}_0\right)}{W(H-a_0)}$$

(8)

2.3. Digital Image Correlation

In this observation, the deformation represented by micro-crack was analyzed using the DIC software GOM Suit. Figure 4 (a) depicts the deformation of each stage within foamed concrete by referring to Figure 3. During the initiation crack, the micro-crack above the notch tip is started. The microcrack develops and gets longer until at ultimate load. At this stage, the length of the micro-crack is assumed to be a critical crack. However, both inverse analyses and the DIC method showed good agreement. After reaching the ultimate load, the crack starts visibly with the naked eye within softening phase (SF). During the first softening (SF1), the traction starts losing from the notched tip. The traction disappears along the crack propagation, as seen during the second softening (SF2). Nevertheless, the traction length gets shorter when the crack is propagated until fully separated. Figure 4 (b) shows the opening distance along the notch height. This figure can be as a verification related to inverse analysis (or superposition method) for CTOD_c. The value of CTOD_c from DIC observation was slightly lower (see Figure 4 (b) at ultimate load = 0.07 mm) compared to CTOD_c from inverse analysis (see Table 3 = 0.075 mm). In addition, the value from CMOD_c from experimental observation (Table 3 = 0.123 mm) is slightly higher compared to DIC (Figure 4 (b) at ultimate = 0.120 mm).

Figure 6. Deformation observation using DIC (with testing series GF-30).

Figure 6. Deformation observation using DIC (with testing series GF-30).

3. Finite Element Works

In this study, XFEM and CZM are adopted to perform strength prediction and crack propagation at ultimate load, which will be described in the following section. In this study, ABAQUS software is utilized to carry out FEM. Table 5 describes the material (elastic modulus (E), Poisson’s ratio (v), un-notched strength (σ₀), fracture energy mode I (G_I)) and geometry properties of foam concrete beam used in FEA Modelling. While Figure 5 shows the implementation of the XFEM and CZM area (red zone) within the notched concrete beam.

Figure 7. Schematic and implementation of XFEM and CZM in FEM.

Figure 7. Schematic and implementation of XFEM and CZM in FEM.

3.1. Sensitivity Study

Sensitivity analysis was carried out to evaluate the effects of mesh and damage stabilization cohesive dependence on FEA Modelling. Munjiza et al. (2001) [44] stated that the size of a finite element near the crack tip needs to be smaller than the actual plastic zone size. Nevertheless, shell-based formulation within this study is modelled using a 4-noded shell element named CPE4R. More & Bindu., (2015) [45] stated that an FE model with a larger mesh might lead to less accurate results but faster computational time. Finer mesh yields high accuracy but takes more computational effort. Therefore in this study, the global mesh size range from coarse to fine mesh size (i.e., mesh density 5-0.5) was observed. Figure 6 shows the global mesh size 2 and damage stabilization cohesive with a value 1 × 10^-5. The model showed good consistency at ultimate load for both XFEM and CZM methods. Similar findings with global mesh size 2 were also reported in [46], while [47], [48] and [49] with global mesh sizes 0.01mm, 0.125 mm and 1-0.5 mm, respectively. Meanwhile, damage stabilization cohesive with a value of 1 × 10^-5 was also reported in [28], [50] and [51].

Figure 8. A sensitivity study on (a) Global mesh size, (b) Damage stabilization cohesive.

Figure 8. A sensitivity study on (a) Global mesh size, (b) Damage stabilization cohesive.

3.2. Damage Plot

Figure 7 depicts a schematic damage plot of each stage of damage from the beginning until it completely failed. At Stage 1 the crack starts to initiate after reaching its maximum principal tensile stress of the material. Nevertheless, compared to crack initiation exhibited in the experiment, FEM results indicate propagation at a lower load. By increasing the load, the crack is propagated until Stage 2, where the ultimate load is reached. At this stage, the length of the crack is named a critical crack. However, compared to experimental (DIC observation) or inverse analysis, within FEM, results showed shorter. In Stage 3, the beam is under softening conditions, and the crack still propagates until the beam is fully separated, as shown in Stage 4. While Table 6 shows the bilinear parameter from FEM (referred to as load-CMOD in Figure 6). Compared to the experimental output in Table 2, it was found that the characters within FEM were not able fully to represent the experimental. Nevertheless, related to critical crack length (a_c) at the ultimate load both experimental (DIC and FEM) showed a good agreement. It can be seen the presence of notch height influenced the height of critical crack length (known as fracture process zone at ultimate load). However, CZM provides better agreement with the experimental compared to XFEM related to critical crack length. Again, the LEFM theory adopted within ABAQUS software is only well applicable for materials with relatively small plastic process zones [30] while concrete material has a large plastic process zone or commonly named as fracture process zone.

Figure 9. Damage plot (a) Load-displacement, (b) Tensile-CMOD.

Figure 9. Damage plot (a) Load-displacement, (b) Tensile-CMOD.

Figure 10. Critical crack at ultimate load between (a) DIC deformation and (b) XFEM deformation.

Figure 10. Critical crack at ultimate load between (a) DIC deformation and (b) XFEM deformation.

Figure 11 showed output FEM for both XFEM and CZM related to load prediction and opening crack. It showed that in terms of load prediction compared to CZM, XFEM exhibited better (see Table 6). Both experimental and FEM (XFEM and CZM) exhibited a similar trend in terms of tensile strength, in which the higher notch, the less material to resist the load due to stress concentration. This stress concentration is the outcome of fracture process zone length which is influenced by notch height. Meanwhile, during softening after reaching ultimate load, CZM is more representing experimental compared to XFEM in the crack opening. Regards to kink points (σ₁, w₁) in Table 6, CZM provides relatively higher (for σ₁) and larger (for w₁). Nevertheless, zero stress point (w₂) compare to experimental, both XFEM and CZM provide a good agreement for series GF-0 and GF-30.

Figure 11. Load-displacement and Tensile strength (a) XFEM and (b) CZM.

Figure 11. Load-displacement and Tensile strength (a) XFEM and (b) CZM.