Issue 39

J. Klon et alii, Frattura ed Integrità Strutturale, 39 (2017) 17-28; DOI: 10.3221/IGF-ESIS.39.03 18 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 [5–14]; 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– 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]. C ONCEPTUAL APPROACH 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), W a W a W a f f,b f,fpz d ( ) d ( ) d ( )   [J] (1) which represents the area under the P – d diagram. Here, d W f,b (subscript ʻ b’ denotes ʻ brittle’) is the energy release connected with creation of new fracture surfaces and d W f,fpz (with ʻ fpz’ denoting ʻ fracture 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. d a 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 well- known definition of the fracture characteristics (of LEFM), the toughness, W a G a B a f,b f d ( ) 1 ( ) d  [Jm −2 ] (2) 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 d W 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, T