Fracture Mechanics

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CHAPTER – 4 THREE PARAMETER FRACTURE CRITERION
4.1 MODIFIED TWO-PARAMETER FRACTURE CRITERION K max = K F �(1 − m) � The two – parameter fracture criterion as per Newman is, σf �� σu (4.1)

Here KF and m are two fracture parameters evaluated from base line test data. σf is the failure stress normal to the direction of the crack in a body σu is the nominal stress required to produce a plastic hinge on the net section. The modified two-parameter fracture criterion is an empirical relation developed by Rao and Acharya between the failure stress and the elastic stress intensity factor at failure which is given by, K max σf σf p = K F �(1 − m) � � − (1 − m) � � � σu σu (4.2)

Here there are three parameters KF, m and p, evaluated from the baseline test data, σf is the failure stress, σu is the nominal stress and Kmax is the elastic stress intensity factor at failure. The two parameter fracture criterion of Newman explained above applies relations derived within the scope of Linear Elastic Fracture Mechanics (LEFM). In this criterion, the two fracture parameters take account of the deviation of the stress to failure from the stress calculated pursuant to LEFM principles. These parameters have to be calculated earlier in pretests known as base line tests to be conducted under identical conditions of the material. It was possible neither to find

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the failure stress of pressure vessels by means of the fracture parameter obtained from fracture mechanics specimen nor to find the failure load of the pressure vessel using the fracture parameters of pressure vessel. Rao and Acharya then developed an empirical relationship between the failure stress and the elastic stress intensity factor at failure. The results obtained using this three parameter fracture criterion is found to be in reasonable agreement with the test results. The significant parameters affecting the size of a critical crack in a structure are the applied stress levels, the fracture toughness of the material, the location of the crack and its orientation. Because the stress orientation factor, K is a function of load, geometry and crack size, it will be more useful to have relationship between the stress intensity factor at failure (Kmax) and the failure stress from the fracture data of cracked of cracked specimens for the estimation/prediction of the fracture strength to any cracked configuration. The relationship between Kmax and σf from equation (4.2) is given as, K max σf σf p = K F �(1 − m) � � − (1 − m) � � � σu σu

Here, σf is the failure stress normal to the direction of a crack in a body and σu is the nominal stress required to produce a fully plastic region (or hinge) on the net section. For the centre CCT, σu is equal to the strength (σult) of the material. For cylindrical pressure vessels, σu is the hoop stress at the burst pressure level of the un-flawed thin cylindrical shell. KF, m and p are the three fracture parameters to be determined from the fracture data. Figure.4.1shows a Centre Cracked Tensile Specimen and a cylindrical vessel containing an axial surface crack. Stress intensity factor expressions for these cracked configurations are available based on finite

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element solutions. Using the value of failure stress (σf) in the stress intensity factor expression, the stress intensity factor at failure (Kmax) for the cracked configuration can be obtained.

(a) Center Cracked Test Specimen

(b) Axial Surface crack in a cylindrical shell

Figure 4.1 Cracked Configurations Stress Intensity Factor for the cracked configurations in Fig (4.1) is given by, K max = σf (πa) Where, = Pmax tW σF = σmax = 1� 2

M ϕ

(4.3)

pbf R for cylindrical pressure vessels t
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for CCT Specimens

(4.4)

c a √π M = �M1 + �ϕ� − M1 � � � a t = Me fs =

σu = σult

a M1 = 1.13 − 0.1 � � c π 1�2 M2 = � � 4

a 1.65 ϕ2 = 1 + 1.454 � � c

pb R t

for cylindrical pressure vessels for CCT Specimens for cylindrical pressure vessels for...
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