Modelling of size and shape of damage zone in quasi-brittle notched specimens – analytical approach based on fracture-mechanical evaluation of loading curves

An analysis focused on capturing the phenomenon of quasibrittle fracture is presented. Selected parameters relevant for quasi-brittle fracture are evaluated and an assessment of their dependence on the size and shape of the test specimen is studied. Determination of these fracture characteristics is based on the records of the fracture tests on notched specimens, particularly from recorded loading diagrams. A method of separation of the energy amounts released for the propagation of the (effective) crack and that dissipated within the volume of a large nonlinear zone at the crack tip – the fracture process zone – is introduced and tested on selected data from experimental campaigns published in the literature. The work is accompanied with own conducted numerical simulations using commercial finite element code with implemented cohesive crack model. Results from three-point bending tests on specimens of different sizes and relative notch lengths are taken into account in this study. The proposed model has only two parameters whose values are constant for all specimen sizes and notch lengths.


INTRODUCTION
racture properties of quasi-brittle materials, corresponding to commonly used fracture models relevant to this type of fracture behaviour, appear to be dependent on the test specimen size, shape and boundary conditions.This fact has been well recognized in the last five or six decades (summarized in e.g.[1][2][3][4]).The reason for that is in the existence of a large zone of nonlinearly behaving material around the tip of the propagating crack.The major part of this zone is referred to as the fracture process zone (FPZ), in which mechanisms of material toughening take a place.These F mechanisms cause the so-called tension softening of the material and can be particularly regarded as interpretation of the crack bending around aggregates, friction of cracks face and aggregate interlock, blinding of crack tip in pores, crack branching and others [1,3].There were many attempts to capture/deal with the phenomena of the size/shape/boundary effect on fracture properties of quasi-brittle materials proposed with success in the last more than twenty years [5][6][7][8][9][10][11][12][13][14]; however, the applicability/validity of the suggested remedies are usually rather limited and not general.This is true also for the standardized work-of-fracture method for determination of fracture energy of concrete [15].The value of fracture energy determined by this method is strongly dependent on the specimen size and geometry [9,10,12] This phenomenon is caused by the change in the size and shape of the FPZ during crack propagation, from which the change of energy dissipated in this area results.This change is determined by the distance and the position of the crack tip and the FPZ in relation to free surfaces of the specimen [12].Therefore, the existence of the FPZ begins to be taken into account recently in the context of models describing quasi-brittle fracture, which is also the aim of this study.Thus, also own attempts to capture the abovementioned effects, in this case via an incorporation of the real size and shape of the FPZ at the crack tip to the method of evaluation of the fracture characteristics, have been introduced [16][17][18].To identify the properties of FPZ, a combination of adaptation of linear elastic fracture mechanics (LEFM) (which is used to express the amount of energy released for creation of the new surface of an effective crack without considering the existence of FPZ) and cohesive-crack-based fracture models for concrete and other quasi-brittle materials [19,20] (which enable the modelling of the FPZ extent) was proposed.This method models the FPZ in detail during the whole fracture process along the specimen ligament.However, its relation to the energetic evaluation of the fracture process was suggested only schematically.Basically, it divides the energy dissipated for the fracture into the part released for propagation of the effective crack and that released in the FPZ volume corresponding to the current stage of fracture process [21].Therefore, further analyses have been conducted [22,23].This present paper follows and extends on those mentioned works.It is focused on the evaluation of selected fracture parameters of concrete and particularly on the assessment of their dependence on the size and shape of the test specimen.The envelope of FPZ extents evolving during fracture and the fracture resistance are among the investigated parameters.The whole analysis regards two sets of three-point bending tests on notched beams of varying sizes and notch lengths published in [24] and [25,26].

Modelling of energy dissipation during quasi-brittle fracture process through the specimen ligament
he nonlinear fracture is treated in this work via a division of the energy released for the fracture propagation into two parts.This separation is performed on the level of processing of the recorded P-d diagram, where P is the applied load and d is the load-point displacement.It is treated on the level of evaluation of the work of fracture (its infinitesimal increment), which represents the area under the P-d diagram.Here, dW f,b (subscript ʻb' denotes ʻbrittle') is the energy release connected with creation of new fracture surfaces and dW f,fpz (with ʻfpz' denoting ʻfracture process zone') is the energy dissipated in the volume of the current FPZ.The energy released for the creation of the new fracture surface increment can be specified by its area, i.e. da ef •B, where a ef is the effective crack length (marked simply as a in the further text) and B is the specimen thickness.This leads to the wellknown definition of the fracture characteristics (of LEFM), the toughness, The term fracture energy (instead of toughness) is used for this quantity mainly in the field of quasi-brittle cementitious composites like concrete.Accordingly, the dW f,fpz part should be specified by the FPZ volume, which leads to the expression of a quantity describing the density of the energy dissipation in the FPZ, The symbol A fpz represents the cumulative area of the FPZ, referred to as the cumulative damage zone (CDZ) here.It means that a unification of all FPZ's extents from the beginning of the crack propagation from the initial notch to the current fracture stage.It is assumed in the approach that FPZ extent doesn't vary over the specimen thickness B. In addition, the parameter G f is considered to be constant during the fracture propagation.
In this work, however, the model is even much more simplified.The parameter Hf is supposed to be of uniform distribution over the FPZ extent, which is definitely an unrealistic assumption.However, it is taken into account for its simplicity as an initial step towards verification/assessment of the proposed concept.Then, the real area of the current CDZ (of the volume Afpz•B) doesn't enter the procedure, since the current shape of that zone is uneasy to express.
Instead, a rectangular increment of the cumulative FPZ area expressed as a•t is considered.Here, t is the FPZ width at the current position of the equivalent elastic crack tip.The model is sketched in Fig. 1; axonometric view on single edge notched beam subjected to three-point bending (SEN-TPB) with the initial crack of length a 0 , from where the effective crack (of current length ai, the blue line) propagates during further loading.FPZ evolves around the crack tip; the envelope of FPZ's from all stages of fracture process from its beginning to its current point is indicated by the red line.

Construction of R-curves
As was noted above, the estimation of the Wf,b and Wf,fpz parts of the total amount of dissipated energy is performed from the recorded P−d curves.The technique of separation of the whole area under the loading curve (interpreting the work of fracture W f ) into these two parts is following [23].A specific load level is assigned to be the initial load by which the propagation of the effective crack begins (basically, in accordance with the double-K model [27,28]).The energy release rate G corresponding to this load is kept constant for the further effective crack propagation and designated as G f .Thus, the area under the loading curve is divided into two regions by this curve corresponding to G f : W f,b (under that curve) and Wf,fpz (above it), see Fig. 2 left (the curve corresponding to the constant value of Gf is marked as "LEFM" there).
The practical implementation of this separation might be in many cases more convenient after a transformation of the analysed P−d curve into the R−a curve (or R−a, where a is the effective crack increment, or possibly R−, where  =a/W is the relative crack length).The equivalent elastic crack model [1] is employed for estimation of the current (effective) crack length a at arbitrary stage of the fracture process.Determination of a is based on the difference between the initial compliance of the specimen with the crack of length a0 and the specimen compliance at the current point of the P−d diagram.Then, the value of fracture resistance R is calculated from the current load and effective crack length, most conveniently as

Estimation of process zone width.
Based on the constructed R−a curve, the work of fracture dissipated in the FPZ, W f,fpz , can be expressed from two monotonically increasing functions of the effective crack length a, i.e.W f and W f,b (obtained by simple integration of the R-curve and Gf-curve for the quasi-brittle and the brittle fracture propagation, respectively), see Fig. 3 top left, as their subtraction, Fig. 3 top right.
After its differentiation with regard to a, the energy dissipated in the increment of the FPZ volume corresponding to effective crack increment equal to a is obtained, To facilitate the differentiation, the Wf,fpz(a) function can be approximated by a polynomial function of a reasonable order (typically from 3 rd to 6 th , with a sufficient accuracy), see Fig. 3 top right.Under the above-mentioned assumptions, that the FPZ area increment is of the rectangular shape expressed as a•t and the energy dissipation density Hf is uniform over the FPZ (i.e. also CDZ), the FPZ width can be expressed as The procedure is illustrated in Fig. 3

APPLICATION OF THE MODEL -EVALUATION OF EXPERIMENTAL DATA
Experiment by Hoover et al.
n the comprehensive experimental campaign reported in [24], four beam sizes of widths W = 500, 215, 93 and 40 mm (marked as A to D), with three relative notch lengths 0 = 0.075, 0.15, and 0.3, were tested.Ratio of the smallest and the largest tested specimen is remarkable, namely 1:12.5.Several samples were tested for each W and  0 , for details see [24].P-d diagrams were not measured properly in many cases; therefore, reconstructions of these diagrams based on the correctly recorded P-CMOD curves (CMOD stands for crack mouth opening displacement) were conducted with the help of numerical simulations in ATENA FEM programme [29] (parameters of the fracture-plastic material model used for concrete were identified according to the recorded P−CMOD diagrams) and subsequent data corrections were performed (details see in [30]).Nominal dimensions of the test specimen are shown in Tab.These R-curves are shown also in Fig. 5 due to better understanding of their connection with the determined cumulative FPZ extents.Diagrams with the R-curves are plotted as  vs. R, where the axis of the relative crack length  is oriented along the specimen width W. In these graphs, a constant red curve is plotted, that corresponds to the specific energy released for propagation of the effective crack G f .This value was estimated at 30 Jm −2 for all specimen sizes and notch lengths.This value was selected as the "optimum" from previously performed parametric study [30], where also other values of Gf were considered, see Fig.

Experiment by Vidya Sagar & Raghu Prasad.
Extensive experimental study on fracture characteristics performed on notched beams of three sizes and notch lengths was published in [25,26].Fracture tests on SEN-TPB specimens were accompanied also with acoustic emission (AE) scanning.Nominal dimensions of the tested specimens are shown in Tab. 2. Three specimens from each size and notch length configuration were tested, i.e. altogether 27 tested beams.These tests were conducted under the CMOD control, similarly as those in [24].Again, the P-d diagrams were recorded simultaneously in order to evaluate fracture energy.However, no pure deflection of the beam was recorded but a quantity including also crushing and plastic deformation at supports, deformation of the loading machine frame etc., i.e. evaluation of the recorded P-d diagrams was not appropriate.Therefore, the ATENA FEM code [29] was utilized to simulate the fracture test according to the correctly recorded P-CMOD curves, similarly as in the previous experimental campaign.
Again, a simple trial and error optimization technique was performed for identification of the fracture-plastic (cohesive crack-based) material model in the employed code, details can be found in [31].Reconstructed P-d curves (via numerical simulations) with corresponding R- curves for all beam depths and relative notch lengths are shown in Fig. 6.It can be seen that in this case, contrary to the previous experimental set, the tails of the R-curves grow up, which is most likely caused by the accumulated numerical error in the area of tails of the simulated P-d diagrams.Therefore, estimates of the R-curve ends (horizontal lines) are used for further analysis of the FPZ extent.They are depicted in the R-curve plots by dotted lines (for relative crack lengths  0 = 0.15 and 0.3) in Fig. 7.
Rel. crack length   The envelopes of FPZ extents corresponding to these beams are displayed in Fig. 7.In this case, they are plotted in three graphs for the small, middle and large beams, respectively.All three notch lengths for each beam size are included in one diagram.The diverging branches of the FPZ envelopes are displayed in gray, those corresponding to the constant R estimates are displayed as dotted lines in the colours matching the related R-curve.The two parameters of this tested model, i.e. the specific energy for the effective crack propagation Gf and the space density of energy dissipation in FPZ H f , were estimated at values 40 Jm − 2 and 215 Jm −3 , respectively, for the experiment [25].

DISCUSSION OF RESULTS
Experiment by Hoover et al.
he compliance-based constructed R-curves from experiment [24] exhibit rather different trends for the small and large beams.Small samples (mainly D, also C with the longest notches) are characteristic by increasing trend of R along the specimen ligament, which is even more pronounced close to its end.A steady progress of R is observed for large samples (A and B) with eventually decreasing trend near the end of the ligament (mainly for the shortest notches).The FPZ extent seems to be described by the constructed envelope curves reasonably well except the very beginning and the end of the ligament.The FPZ extent estimation is corrupted at the ligament ends in the cases where significant increases in the R-curve trends are observed.This increase in R in that area is caused by an error of measurement/modelling accumulated at the ends of descending branches of the recorded/simulated loading curves.The odd shape of the FPZ envelope at the beginning of the fracture propagation from the notch (the "fish tail" shape) that is observed mainly for small sizes (D, also C and B for the longest notches) corresponds actually to negative value of Wf,fpz at these points.This is caused by too high value of G f at these stages of fracture.This feature could be eliminated using nonconstant value of Gf (modelled e.g. by some increasing function with a plateau), which was undesirable in this study.Both parameters of the models were considered as constants.Based on Fig. 5, it can be concluded that the introduced simple model captures the size/geometry effect on the fracture parameters reasonably well.The size of FPZ for smaller specimens takes a relatively larger part of the total volume of the specimen than it occurs in the case of larger specimens.The large beams exhibit much more brittle behaviour; therefore, the energy released for effective crack propagation W f,b occupies a larger portion of the whole energy W f released during fracture.
Maximum widths of the FPZ tmax shown in Tab. 1 are plotted for the individual notch lengths as functions of the beam size in Fig. 8 left.The increasing trend is obvious.After relating t max to the specimen size, all the data were fitted.Based on the fact that for an infinitely large specimen the ratio tmax/W must tend to zero, a power function was chosen.The regression is plotted in Fig. 8 right in the bi-logarithmic scale.
As is evident from Eq. 6, the FPZ width is inversely proportional to the space density of energy dissipation in FPZ H f .Thus, the tmax value is dependent on the Hf value considered in the study.The authors must admit, that the value was guessed in this case.However, its much proper estimation can be possibly performed based on experimental observations of FPZ using e.g.AE scanning (see below) or/and radiography [32][33][34].Alternatively, analytical estimation of the FPZ extent based on matching the crack-tip stress state description and strength properties of the material (via proper failure criteria) might be also helpful for this task, see e.g.[16,17,[35][36][37].Nevertheless, the power law proposed for the FPZ width estimation would apply also for different values of H f as the value of t max would be scaled by the same factor for all sizes.Experiment by Vidya Sagar & Raghu Prasad.
The range of specimen sizes tested in this campaign was considerably smaller than that in [24]; however, extents of cumulative FPZ estimated from localized AE events were reported.Fig. 9 shows displays of AE events locations at the end of the fracture test.Unfortunately, only very limited amount of these results from the whole campaign is available in the published papers [25,26].Thus, the comparison of the cumulative FPZ extents from AE measurement with that of the proposed model is limited here only to the cases with the shortest notches.Similarly to Fig. 7, the diverging parts of the FPZ envelopes are in gray and the estimates of these parts based on the constant R in this portion of the specimen ligament are displayed by the red dotted line.A relatively good agreement was obtained.
 0 = 0.15 S M L Figure 9: Notched beams of all tested sizes with the shortest relative notch length with the indication of the envelope of the FPZ for the FEM simulated response compared to localized AE events (top figure adopted from [26], middle and bottom from [25]).

Figure 1 :
Figure 1: Sketch of the used model of the (effective) crack propagation and the cumulative FPZ extent advancement during the quasibrittle fracture process in the SEN-TPB test specimen.
) where (P) is the nominal stress in the line of the crack in the specimen due to the load P, Y() is the corresponding geometry function.Thus the value of W f,b is equal to the area under the G f −a curve and W f,fpz to the area under the R−a curve minus that under the Gf−a curve.Transformation of the P−d diagram into the R− curve with the indication of meanings of Gf and R, Wf,b, Wf,fpz and Wf is shown in Fig. 2.

Figure 2 :
Figure 2: Indication of the individual portions of work of fracture at the current stage of fracture process in the loading diagram (left) and R-curve (right).
top left, where the W f (a) and W f,b (a) curves are plotted, Fig. 3 top right, where the Wf,fpz(a) curve is determined as the subtraction of Wf(a) and Wf,b(a), and Fig. 3 bottom, where the cumulative FPZ extent is plotted as the t(a) function (displayed along the beam ligament).

Figure 3 :
Figure 3: Plots of work of fracture due to quasi-brittle fracture W f (a) and brittle effective crack length propagation W f,b (a) (top left), their subtraction resulting in plot of cumulative energy dissipated in FPZ Wf,fpz(a) (top right), and plot of FPZ envelope (bottom).

I
Load-deflection diagrams for all beam depths and relative notch lengths in are shown Fig.4.Recorded P-d curves (with some adjustments regarding the ascending part of the curve and unification of the elastic stiffness) are plotted by gray lines.Each diagram is complemented by a red line which corresponds to the FEM simulated response calibrated by recorded P−CMOD curve.In most of the cases, the simulated curve represents an average response of the individual specimen set very well.Each cell of the array of plots in Fig.4contains also a diagram with R- curves corresponding to the P-d curves in the diagrams on the left.

Figure 4 :
Figure 4: P-d diagrams and corresponding R- curves for each specimen size W and relative notch length  0 ; red curves correspond to simulated "average" response.The value of the space density of energy dissipation in the FPZ was estimated at H f = 215 Jm − 3 .The ratio of the beam length and width is not displayed in real value in Fig.5due to better visual comparison FPZ extents in all beams.The graph is complemented with blue dimension lines above each beam which indicate the mean values of estimated maximal FPZ widths (calculated from the black parts of the envelopes).These values are added to the right column of Tab. 1.

30 WFigure 5 :
Figure 5: Scheme of the notched beam (the width W is scaled by a factor of 1.34 with respect to the length L; in reality L/W = 2.4) with the indication of the envelope of the FPZ for each tested sample and the FEM simulated response (green line); each figure is complemented with R-curve plots (grey lines, the red line corresponds to the Gf value released for the effective crack propagation).

Figure 6 :
Figure 6: Simulated P-d diagrams and corresponding R- curves for each specimen size W and relative notch length 0 set.

Figure 7 :
Figure 7: Scheme of the notched beam (the width W is scaled by a factor of 2.42 with respect to the length L; in reality S/W = 3.3) with the indication of the envelope of the FPZ for the FEM simulated response for each relative crack length; each figure is complemented with R-curve plots (the red line corresponds to the G f value released for the effective crack propagation).

TFigure 8 :
Figure 8: Maximum width of the FPZ extent envelope for all specimen sizes: Dependencies plotted individually for each notch length (left), regression of all the data by power function.
[24]nal dimensions of specimens from experiment[24]; estimated maximal widths of FPZ are shown in the right column.