Fracture Mechanics of Concrete on The Basis of Finite Element by Applying InverseAnalysis Method

This study is to evaluate the effect of compressive strength of the concrete on the fracture mechanics of the concrete through finite element (FE) analysis. Cohesive crack and crack band model were employed to observe the significance of increasing compressive strength of the concrete on the value of fracture energy and propagation of crack, respectively. A set of notched beams specimens manufactured with compressive strength concrete of 36, 45, 53 and 95 MPa were experimentally investigated. On the basis of invers analysis, the tensile softening curve of the concrete was intended to create for a specific compressive strength of the concrete which was then used as an input parameter for FE model. The value of fracture energy was calculated based on FE result based on the load against crack opening displacement curve. Comparison was carried out between FE and experimental result. This study was found that increasing compressive strength of the concrete markedly influenced on the fracture energy of the concrete and induced the cracking path in the concrete. Keyword-Concrete strength, Fracture energy, Fracture mechanics, Inverse analysis


IV. EXPERIMENT WORKS A. Concrete Mix Proportions
Four different compressive strengths were manufactured in order to produce notched beam specimens as discussed in the next section, Table II.

B. Specimen Preparations
The compressive strength of concrete was measured using cubes of 100 x 100 x 100 mm regarding to BS EN 12390: Part 3(2000) [10]. The softening curves was defined using beam specimens with size of 45 mm width, 100 mm depth, and 500 mm length tested in three-point bend (TPB). All of the specimens were applied water cured method for 28 days. A water-cooled diamond rotary cutter with 2.5 mm width was used to create 40 mm notch depth (a o ), Fig. 3.

C. Three-point Bend (TPB) Test Set-up
The geometry of TPB test set up is shown in Fig. 3. The testing was carried out using a servo-hydraulic closed-loop testing machine, Fig. 4. A vertical displacement of 0.01 mm/s [11] was applied to control subjected load. The crack opening displacement was measured by a calibrated linear variable differential transducer (LVDT)with 7.5 mm capacity and ± 0.0007 mm linearity.Before submitting your final paper, check that theformat conforms to this template. Specifically, check theappearance of the title and author block, the appearance ofsection headings, document margins, column width, column spacing and other features.

V. DETERMINATION OF FRACTURE ENERGY
Fracture process of material consumes amount of energy until totally failure. The amount of energy absorbed of material to fracture is used to classify fracture resistance of material. Fracture toughness and dissipated fracture energy of material have been extensively employed to characterise the fracture resistance of material. Concrete as a quasi-brittle material is proposed to use the fracture energy (G F ) in observing the degree of the fracture resistance. In this study the fracture energy was calculated based on the RILEM TC50-FCM recommendation (1985) [5]: In Eq. (1), G F is the total fracture energy, W o is the area under the load-deflection curve, m is the total mass of specimen between supports, g is gravity, δ o is the end deflection at P=0, and D,a o and t are the depth of the beam, depth of the notch, and width of the beam, respectively. Fig. 5 shows the beam responses on applied load by the means of load against crack opening displacement for various concrete strengths. The average peak loads of beams manufactured with compressive strength of 95, 53, 45 and 36 MPa are 4170, 2448, 2510 and 2140 N respectively. As the compressive strength of concrete controlling the quality of interfacial zone matrix which is the weakest link in the concrete [12], increasing compressive strength of concrete by reducing ratio of water to cement as well as adding up supplementary cementitious materials such as fly ash and microsilica, can reduce considerably the degree of porosity in concrete. Consequently, the strength of interfacial zone matrix rises as the compressive strength of the concrete rises and thus applied load subjecting on the beam is increased as shown in Fig. 5.

VI. RESULTS AND DISCUSSIONS
The characteristic of the concrete after the peak load can be derived on the slop of descending curve. Albeit it is only descriptive analysis over the curve, it assists in describing the effect of compressive strength on the post-peak behavior of concrete. Fig. 5 shows that the descending curve slope of compressive strength of 95 MPa is moderately steep compared to others in which the descending curve of compressive strength of 36, 45, and 53 MPa tend to have a relatively same slope. However, it is a descriptive analysis over the load-crack opening displacement curve. The area under the curve representing the total energy dissipated during fracture process [5] can be engaged to calculate the fracture energy of concrete for evaluating sincerely the effect of compressive strength of concrete on the characteristic of concrete.   Table III. The worst error given by FE results for initial crack opening width is of compressive strength of 36 MPa, i.e. 15.7%, whereas the peak load gives an error of 8.4%. Hence, the variation of result given by FE using invers analysis in creating the tensile softening parameter as fracture parameter input for FE will not be more than by 16% when analyzing the concrete behavior. The value of fracture energy is then engaged to analyze the influence of concrete strength on the concrete behavior. The comparison of experiment and FE results for the value of fracture energy is shown in Table IV. The FE gives the worst variation of 9.4% on predicting the value of fracture energy compared to experimental results. Increasing compressive strength noticeably affects increasing energy needed in fracture process, see Fig.  7.

VII. CRACK PROPAGATION OF CONCRETE
Fracture energy of the concrete as discussed in the previous section is then employed as fracture parameter input for finite element modeling by applying crack band models in observing the crack propagation of concrete. Fracture process of the concrete is observed at three points representing propagation of crack in the concrete,  Fig. 9 shows that the fracture process zone at the notch for concrete strength of 53 and 95 MPa tends to be wider but smaller than that of 36 MPa, and vice versa. The width of fracture process zone is controlled by constituents of materials in the concrete such as water, cement and maximum particle size [13]. The compressive strength of concrete influenced by water/cement ratio, dictates the strength of interfacial zone matrix. The higher strength of interfacial zone matrix and maximum particle size contribute significantly to the distribution of applied stress amongst particles. Hence, higher strength of concrete and larger maximum particle size tends to have a wider fracture process zone. Increasing applied load after peak load propagates the apparent crack as shown in Fig. 10. Crack propagation of the concrete with the strength of 36 MPa demonstrates a more tortuous cracking path compared to concrete strength of 53 and 95 MPa. As a consequence of the degree of interfacial matrix strength, crack propagation of low strength concrete tends to develop surrounding the particles and ends up with a tortuous cracking path. The final failure as shown in Fig. 11 describes the effect of concrete strength on the residual strain (marked as the red element) along the cracking path. The residual strain value of concrete strength of 36 MPa is not existed along the cracking path. However, in the concrete strength of 53 and 95 MPa is obviously occurred and having a relatively long zone. This indicates that tensile strength of concrete still contribute in resisting crack opening and therefore concrete strength will involve in post-peak behavior of concrete.