Evaluation of mixed mode I/II fracture toughness of C 50/60 from Brazilian disc test

Durability and sustainability of structures made from concrete like materials is getting more into an interest of civil engineers. Structures are subjected not only to uniaxial load, this means if there is a crack inside the load could be divided into mode I and shear mode II, but also to a mixed mode I/II. Therefore, it is necessary to perform test, which covers mixed mode loads and could be used for specimen made from concrete core-drill. Recommended test specimens used for evaluation of fracture mechanical parameters has usually a prismatic or rectangular shape. The cost of reshaping cylinder into rectangular is expensive and not very efficient, therefore it is very effective to use Brazilian disc test specimen with central notch to obtain mixed mode fracture parameters. The contribution deals with numerical support for Brazilian disc test to obtain calibration curves for mode I and mode II, evaluation of experimental results and compare them with data adapted from literature.


INTRODUCTION
oday's trends in civil engineering pushes investors and owners of concrete structures more often into renovation, than into demolition of structures.This fact leads into need for knowledge of material characteristics such as bulk density, Young's modulus of elasticity [1], compressive strength [2], flexural strength [3], etc.Some types of civil engineering structures are subjected to mixed mode loading I and II, therefore the knowledge of fracture mechanical parameters is necessary to predict life-time of concrete structures (bridges, cooling towers).To obtain material sample from considered structure, a core-drill is used, which drills out a cylindrical sample from a structure.This specimen is tested to obtain mechanical properties, but not for fracture mechanical parameters.

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Fracture mechanical parameters of cementitious materials are usually obtained from recommended tests such as: three-point (3PBT) and four-point (4PBT) bending test with notch in tested specimen [4], for mixed mode load [5], wedge splitting test (WST) [6][7][8][9], or a combination of WST/3PBT [10] and modified compact tension test (MCT) [11,12].All tests have a predefined prismatic geometry and using them on specimens made from the core-drill is expensive and not very efficient, therefore it is very appropriate to use a Brazilian disc test specimen with central notch (circle cut from the core-drill cylinder) [13][14][15][16] to determinate fracture parameters of building materials see Fig. 1.The main advantage of Brazilian disc is, that it could be used for investigation of fracture toughness for mode I, mode II and mixed mode by rotating the cracked against the load positions.This article compares the measured experimental data with data presented in [17].

Mechanical properties
he unnotched Brazilian disc is very often used as an indirect test to determinate tensile strength of rock materials, therefore it is very valuable to use it to obtain tensile strength of concrete [18].The tensile strength can be evaluated from following equation: where  t is tensile stress, P is compressive load, D is diameter of disc and B is specimen's thickness.

Fracture Mechanics
This contribution is based on a linear elastic fracture mechanics.The linear elastic fracture mechanics concept uses the stress field in the close vicinity of the crack tip described by Williams's expansion [19].This expansion is an infinite power series originally derived for a homogenous elastic isotropic cracked body, which can be described by a following equation: , ( ) ( ) ( , ) where  ij represents the stress tensor components, K I , K II is the stress intensity factor for mode I respectively mode II, f I ij , f II ij , are known shape functions for mode I and mode II, O ij represents higher order terms and r, θ are the polar coordinates (with origin at the crack tip; crack faces lie along the x-axis).

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The values of the stress intensity factor (SIF) for a finite specimen and the polar angle θ = 0° can be expressed in the following form [20][21][22]: ( / , ) ( / , ) where P is compressive load, a is a crack length, R is radius of the disc (D/2), B is disc thickness and fI(a/R, ), fII(a/R, ) are dimensionless shape functions (calibration curves) for mode I and mode II, see Figs. 2 and 3.

Calibration curves
o obtain right calibration curves for each mode of SIF a numerical simulation was necessary.The numerical simulation was performed in finite element (FE) software Ansys 17.2 [23].A two-dimensional (2D) numerical model with plane strain boundary condition were used to calculate SIF (K I and K II ).The numerical model was meshed with element type PLANE183 taken from ANSYS's elements library and command KSCON was used to take into account the crack tip singularity.Input material properties of concrete used in FE software are following: Young's modulus and Possion's ratio, E = 40 GPa and ν = 0.2, respectively.The geometry of disc has radius R = 50 mm and the relative crack length a/R varied from <0.1; 0.9>, notch angle α varied <0°; 90°> and was loaded with constant force P = 100 N in all calculated cases.In FE software ANSYS, the following equations are used for calculation of the SIF K I and K II for θ = ±180.
where u, v are nodal displacements, G shear modulus,  is Kolosov's constant for plane strain respectively plane stress conditions and r is coordinate in cylindrical coordinate system.From Brazilian disc geometry, mention above a calibration curves for mode I and II for various notch angle  and a/R ratio can be determined as a following functions [25,26]: ( / , ) 1 From Fig. 2, it can be noticed, that for some angle  and a/R, the value of calibration curve for mode I equals zero (fI(a/R,) = 0).This means, that there is only mode II (pure shear mode).Therefore, the fracture toughness for mode II could be evaluated. T

Specimen's geometry
Brazilian disc specimens were prepared from standardized cylindrical specimens used for evaluation of cylindrical compressive strength of concrete [2].Notches were prepared by using water jet cutter.The dimensions of specimens are introduced in Tabs.

Experimental procedure
The machine for tests was Seidner D7940 with maximum loading capacity 4 000 kN.The load rate was 0.01 kN/s.Brazilian disc specimens without notch were tested to obtain the tensile strength.Brazilian disc specimens with the notch were tested under the selected angles inclined against loading positions see Tab. 3.

Selected material and mechanical properties
nnotched Brazilian disc specimens were subjected to compressive load to evaluate tensile strength from Eq. 1.
The measured forces and calculated tensile strength are showed in Tab.The properties of used concrete are compared with material characteristics of mortar and concrete from Hou [17], for detailed information about composition of these materials see [17].Comparison of selected mechanical properties is showed in Tab. 5.

Bulk density ρ
Mortar from [

Fracture mechanical properties
Fracture mechanical properties of structural concrete were evaluated from Eq. ( 2) and ( 3).The values of stress intensity factors were evaluated for mode I (specimens 05_4,09_6 and 01_6), for mode II (specimens 04_4 and 07_6) and for mixed mode I/II.The evaluated fracture mechanical properties are summed up in Tab.

Comparison of experimental data
Experimental results from Hou [17] were digitalized and measured forces were used to be compared with data measured in the presented study.The comparison of results is done in two ways.The first one compares fracture resistance values of KII/KIIC (y axis) against KI/KIC (x axis) which is valuable to see pure mode I and pure mode II fracture resistance.An envelope curve (a circle with radius K I /K IC = 1 and K II /K IIC = 1) is plotted in Fig. 5. to see effect of mixed mode I/II failure for each tested specimen (different inclination angle  of notch).The comparison made in Fig. 4. is not very often, therefore it is being necessary to made comparison which can be usually seen in literature.Fig. 5 compares normative values of K II /K IC (y axis) against K I /K IC (x axis) which is valuable to see mode II effects fracture resistance.An envelope curve (an ellipse with radius KII/KIC = mean value and KI/KIC = 1) is plotted in Fig. 6. to see effect of mixed mode failure for each tested specimen (different inclination angle of notch).

Figure 1 :
Figure 1: Geometry of a typical Brazilian disc specimen with load position alongside crack.

Figure 6 :
Figure 6: Comparison of fracture resistance K II /K IC (y axis) against K I /K IC (x axis) behavior with data from literature [17].

Table 2 :
Dimensions of Brazilian disc specimens with a center notch.

Table 3 :
The relative length of notch in Brazilian disc specimens and angle of used angle position.

Table 5 :
[17]arison of selected mechanical characteristics of used concrete with data adapted from literature[17]

Table 6 :
6. Measured experimental load P and calculated fracture mechanical parameters KI and KII for each specimen.