Numerical study and pilot evaluation of experimental data measured on specimen loaded by bending and wedge splitting forces

The fracture mechanical properties of silicate based materials are determined from various fracture mechanicals tests, e.g. threeor fourpoint bending test, wedge splitting test, modified compact tension test etc. For evaluation of the parameters, knowledge about the calibration and compliance functions is required. Therefore, in this paper, the compliance and calibration curves for a novel test geometry based on combination of the wedge splitting test and three-point bending test are introduced. These selected variants exhibit significantly various stress state conditions at the crack tip, or, more generally, in the whole specimen ligament. The calibration and compliance curves are compared and used for evaluation of the data from pilot experimental measurement.


INTRODUCTION
or evaluation of fracture mechanical properties of materials like concrete, standardized methodology is not published yet.There is only recommendation for measurement of properties by RILEM [18].In the literature, researchers used various specimen geometries for experimental measurement of fracture properties of concrete, F see e.g.wedge splitting test (WST) [4,5,7,14,15,21,23,24,31], three-point bend [10,17], comparison between data from WST and three-point bend (3PB) tests are introduced in [12], modified compact tension (MCT) test [6] and another configurations can be found in handbooks e.g.[11,27].Note that from various geometry, the different values of fracture parameters can be obtained for the same material.Therefore the combined WST/3PBT geometry has been investigated, the variants are proposed in [29] Fig. 1 (Variant I represents the classical WST [7,24] and is included in the study as a reference case), see in [29].In all these cases the crack propagates from a notch provided on the top side of the specimen (in the groove for inserting of the WST loading fixtures.Finally, the variant IIIb differs from the variant III by the central notch provided also from the bottom surface.),provides a wide range of various stress distributions in the specimen ligament -from bending to tension -which is expected to result in the desired variety e.g. in the fracture process zone size and shape, fracture energy or fracture toughness, etc.In the paper, the numerical support (calibration and compliance curves) for evaluation of the experimentally obtain data is shown/introduced.The pilot numerical study of the selected shape of specimens by using Williams expansion was introduced in [22,25,28,30].The values of stress intensity factor (SIF), T-stress and crack opening displacement (COD, see Fig. 2) at the load line for load P sp = 1000 N = 1 kN are introduced.The changes of properties are compared and discussed.These changes could be obtained modifying the specimen length to width and the span to length ratios (and/or simultaneously the wedge angle).At the end of the contribution, examples of the evaluation of experimental data measured by using the studied combination [29] have been presented.

variant I variant III variant II variant IIIb
Figure 1: Considered variants for the combined bending/splitting configurations, taken from [29].
Dimensions of specimens for all four geometry variants are summarized in Tab. 1 (dimensions common to all variants) and in Tab.

THEORETICAL BACKGROUND
ccording to the two-parameter fracture mechanics approach which uses T-stress as a constraint parameter [1,11,13,24,34], the stress field around the crack tip of a two-dimensional crack embedded in an isotropic linear elastic body subjected to normal mode I loading conditions is given by the following expressions [33]: A where r and θ are the polar coordinates and x and y are the Cartesian coordinates, both with their origins at the crack tip.KI is the stress intensity factor for mode I and T is the T-stress.Thus, in two-parameter based fracture mechanics, the stress field is expressed by means of the two parameters, the stress intensity factor K I and the T-stress (see e.g.[1,11,24]).The values of crack opening displacement at load line is measured in the axes of roller bearings through which the load of specimens is applied (see e.g. in [4,14] and sketch of forces in Fig. 2).The applied load ratio between forces is following, [4,14,19,24]: where where α w is the angle of the slope of the wedge and µ c refers to friction in the roller bearings.

MODELING IN ANSYS
he finite element software ANSYS [2] is used for numerical calculation of mentioned fracture parameters (K, Tstress and COD).Plots of variants of the test geometry selected for the experimental study are shown in Fig. 1.Note that geometries are symmetric for all considered specimen shapes (including boundary conditions); therefore, only one half of the problem was modelled like in [21,22,26].The size of the smallest element in the crack tip is 5 × 10 -5 mm.The specimen geometry IIIb could leads to crack closure, therefore the whole body of the specimen was modeled and the example of numerical model with applied boundary conditions is shown in Fig. 3.The crack length to depth ratio a/W eff varies from 0.1 to 0.9.The thickness B is taken as unity in the computations, conditions of plane strain was applied.The material input data for the concrete used in the numerical simulations were as follows: E c = 33 000 MPa and ν c = 0.2; and for the steel: Es = 210 000 MPa and νs = 0.3.For good comparison of numerically obtain results for all cases, the load was applied as splitting force P sp = 1 000 N = 1 kN.

NUMERICAL RESULTS
he numerically calculated values of K I and T-stress are given in Figs. 4 and 5, respectively.In the present paper, four cases of the specimens' shapes/arrangements on the calibration curves were investigated, see Fig. 1 (I, II, III and IIIb) for wedge angle: 15, 20 25, 30.On the left side of Fig. 4, the wedge splitting test configuration for α w =15º is compared with bending/splitting combination as II and III and IIIb.All four studied cases show similar trend of the SIF results, when the configuration is changed the SIF value for the same load decrease.On the right side of Fig. 4, the IIIb configuration with various wedge angles 15º, 20º, 25º and 30º are shown.Up to a/W eff = 0.6 the values have a smooth character, however for the a/W eff reaches 0.6 till 0.9 the values change unpredictably, there is a dominant effect of the 3PB loading.The displacement, COD, in the line of the load is shown in Fig. 6.On the left side of the Fig. 6, the values of WST are compared with variants II and III for wedge angle 15°, where the splitting loading is dominant.It can be seen that for longer specimens the value of COD decreases.On the right side of the Fig. 6, it can be seen the COD for configuration IIIb and various wedge angles 15°, 20°, 25°, 30° when the 3PB load is dominant and the crack start from bottom side.

EVALUATION OF EXPERIMENTAL DATA AND DISCUSSION
t should be mentioned that quasi-brittle fracture of concrete is akin to elastic-plastic fracture of metals.The ASTM standard on fracture toughness KIC [3] has clearly specified the conditions to avoid elastic-plastic or in study case quasi-brittle fracture, i.e. crack 10 times of characteristic crack etc.The simple methodology presented in this example is consistent with the ASTM standard for linear-elastic fracture, but can also cover the first part of the quasibrittle diagram.The material and experimental procedure are described in [29].For evaluation of data in our paper, knowledge about selected input data are needed, therefore the relative initial notch length a/W eff and the maximal value of splitting force Pspmax are shown in Tab. 3. Note that for I variants the values of SIF were evaluated according to [14] for wedge angle 30° and for all others the new calibration curves were used, see Fig. 4. Using the classical linear elastic fracture mechanics, the fracture toughness K IC can be worked out on the basis of the initial relative notch length.The following expressions are used: Using the results (calibration curves for 1000N) presented in Fig. 4, relation between the maximum splitting force Eq. ( 4) for relative notch length a/W eff, we obtain results of fracture toughness of concrete, the results are shown in Tab. 4. It can be seen, that the values of fracture toughness are within the interval 0.2÷1.4MPam 1/2 , see in [1,9].The values of fracture toughness have decreasing tendency when the relative notch length grow, this is in accordance with the results in [34], where for this effect explanation the changes of T-stress values is used. The values of the T-stress increase with the distance of the two supports on the bottom side of the specimen, varies from negative to positive values with increasing relative crack length (a/Weff), for specimen variants I, II and III. The values of COD increase in the whole range of the relative crack length (a/Weff) for all variants of the boundary conditions for specimen variants I, II and III. The variant IIIb has a crack from the bottom part of the specimens, the crack growth is influenced by combination of the wedge splitting force which in turn leads to crack closure during the load of specimen, see in Fig. 6.The obtain calibration and compliance functions could be especially usefull for the published advanced model e.g.[8,16,20,32].

Figure 2 :
Figure 2: Detail of boundary conditions, see the half of specimen and the load application (Psp and Pv/2) with crack opening displacement (COD), position at load line.

Figure 3 :
Figure 3: Example of numerical model, where boundary conditions are shown, with detail in the vicinity of the crack tip.

Figure 4 :
Figure 4: Stress intensity factor (KI) as a function of the relative crack length, α/Weff, for the combined bending/splitting variants defined in Fig. 1, loaded by P sp = 1000 N, on the left side for wedge angle =15° and on the right side for variant IIIb for various wedge angles.On the left side of the Fig.5, the values of the T-stress for the WST configuration are compared with combination of WST and 3PB as variants II, III and IIIb for angles 15º.As we suppose according to the reference[23] the function for WST vary from negative values for very short cracks to positive values for relative crack larger than 0.2.When the

Figure 5 :
Figure 5: T-stress as function of the relative crack length, , for the combined bending/splitting variants defined in Fig. 1, loaded by P sp = 1000 N, on the left side for wedge angle =15° and on the right side for variant IIIb for various wedge angles.

Figure 6 :
Figure 6: COD as a function of the relative crack length, α, for WST and combined bending/splitting variants defined in Fig. 1 (as II, III and IIIb), loaded by Psp = 1000 N, on the left side for wedge angle =15° and on the right side for variant IIIb for various wedge angle.

ICONCLUSIONSn
this paper, the combinations of wedge splitting and three-point bending load applied on beam-shaped notched specimens are numerically analyzed.The numerically obtain data could be used for evaluation of experimentally obtain data as is shown in example.Based on the numerical results presented here, the following conclusions can be drawn: The values of the stress intensity factor (KI) have the same trend in the whole range of the relative crack length  for specimen variants I, II and III, see Fig.1.

Table 1 :
[29]here are unique dimension of all studied variants (I, II, III and IIIb) with angles.Nominal variant dimensions and test geometry parameters, taken from[29].

Table 3 :
[29]ants of evaluated experiments and corresponding wedge angle, and the relative notch length with maximal value of splitting force (both values are mean values from 3 up to 6 measurements, see detail in[29].

Table 4 :
Variants of evaluated experiments and relative notch length with corresponding value of fracture toughness.