Issue 39

J. Klon et alii, Frattura ed Integrità Strutturale, 39 (2017) 17-28; DOI: 10.3221/IGF-ESIS.39.03 20 (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 a 0 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   K R P a Y E E 2 2 Ic 1 ( ) ( )     [J] (4) 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 G f −a curve. Transformation of the P−d diagram into the R−  curve with the indication of meanings of G f and R, W f,b , W f,fpz and W f is shown in Fig. 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). 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 G f -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, W a w a a f,fpz f,fpz d ( ) ( ) d  [Jm −1 ] (5) To facilitate the differentiation, the W f,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 H f is uniform over the FPZ (i.e. also CDZ), the FPZ width can be expressed as w a t a BH f,fpz f ( ) ( )  [m] (6) The procedure is illustrated in Fig. 3 top left, where the W f ( a ) and W f,b ( a ) curves are plotted, Fig. 3 top right, where the W f,fpz ( a ) curve is determined as the subtraction of W f ( a ) and W f,b ( a ), and Fig. 3 bottom, where the cumulative FPZ extent is plotted as the t ( a ) function (displayed along the beam ligament).