Page 178 - Handbook of Materials Failure Analysis
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174    CHAPTER 7 Investigation of failure behavior of tubular components




                         The elastic part is calculated from the expression of linear elastic stress intensity factor
                         K I (thecrack-tipissubjectedtomode-Iloading)fortheplanestressconditionasfollows.

                                                            2
                                                       J el ¼ K =E                       (7.3)
                                                            I
                         where E is the Young’s modulus of elasticity (90 GPa for this material at room tem-
                         perature). The stress intensity factor K I depends upon applied load P, thickness t,
                         crack length a, width W, and geometry of the specimen and is written as
                                                         P
                                                    K I ¼ p ffiffiffiffiffi fa=Wð  Þ               (7.4)
                                                       2t W
                         The effect of geometry is accommodated through the term f(a/W) and these are taken
                         from Refs. [21–24] for the axially cracked thin-walled Zircaloy fuel-clad tubes. The
                         variation of J-integral (i.e., elastic part J el , plastic part J pl and the sum total value of J)
                         with crack extension for the axially cracked fuel-clad specimen with a 0 /W ratio of 0.5
                         is shown in Figure 7.12. It can be observed that the loading process with conical man-
                         drel involves large-scale plastic deformation as the material is highly ductile and the
                         component is thin-walled. The extent of elastic deformation energy in the total
                         energy required for crack growth is very limited and it is of the order of 10% as
                         can be seen from Figure 7.12. Similar observations were made for other specimens
                         with a 0 /W ratios varying from 0.1 to 0.4, respectively.
                            From the load-displacement data and crack-size estimates at each data point, the
                         plastic part of J-integral J pl at each data point i is evaluated using the following
                         expression
                                                 η                       a  a
                                                                   "      0    0  #
                                                              ð
                                             Þ +  ð i 1Þ A pl iðÞ  A pl i 1Þ  1 γ  i ðÞ  ð i 1Þ  (7.5)
                                   J pl iðÞ ¼ J pl i 1ð               ð i 1Þ
                                                 b i 1Þ    2t               b i 1Þ
                                                  ð                         ð
                                      1000
                                              J pl
                                              J
                                       800    el
                                              J
                                     J (N/mm)  600

                                       400

                                       200

                                        0
                                          0    0.25   0.5   0.75    1    1.25   1.5
                                                       Crack length (mm)
                         FIGURE 7.12
                         Variation of J-integral (elastic part J el , plastic part J pl , and total J) with crack extension for
                         the axially cracked fuel-clad specimen with a 0 /W ratio of 0.5.
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