PSI - Issue 33

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Davide Palumbo et al. / Procedia Structural Integrity 33 (2021) 528–543

538

Equation (33) shows the absolute error in Δ T evaluation ε Δ T , depends on:  the material properties b and υ ,  the polar coordinates r and ,  the square of K Imax and the SIF ratio R . To have a comparison between different materials, the use of the relative error can be used: ��

(34)

It is important to highlight that Equations (33) and (34) provide theoretical errors and represent only an analytical tool for estimating the difference between the proposed formulation and the classical one. A complete error analysis should involve the study of the error sources and the uncertainty evaluation. This analysis will be the object of further works. In Fig. 5 and 6, are reported results in terms of thermoelastic temperature variations Δ T and the relative error ε Δ Tr% for three different cases, for the titanium and the aluminium alloy. In Table 1, are reported the material constants, while in Tables 2 and 3 are shown the values of variables R , θ , r and K Imax used for each case. It is important to underline that the values in Tables 2 and 3 were only used to study the behaviour of the new equation (Eq. 29) and to show how this latter differs with respect to the classical TSA equation (Eq. 24). As already said in Introduction, the effect of the plastic conditions has been consciously neglected since the present work is focused on the presenting a new formulation for TSA equations. Fig. 5 and 6, a) and b), show the effect of the polar coordinate r , Fig. c) and d) show the effect of the SIF ratio R and Fig. e) and f) show the effect of the polar coordinate θ on the relative error. In the same way, the effect of θ and R has been investigated by adopting a step of 1° and 0.1 , respectively. In each case, the error can become significant if values of Δ T are considered in the proximity of the crack tip and with a positive SIF ratio away from -1 . In particular, it is interesting to notice that the maximum relative error presents its maximum values around θ =90° with fixed values of r and R .

Table 2. Values of the parameters used for investigating the relative error (Ti6Al4V).

(°)

K Imax Ti6Al4V (MPa(m) 1/2 )

Case

r (mm)

60 60

0.1

70 70 70

1 2 3

(2÷10) step=0.1

2 2

(-1÷0.9) step 0.1

0.1

(0÷180) step 1

Table 3. Values of the parameters used for investigating the relative error (AA6082).

(°)

K Imax AA6082 (MPa(m) 1/2 )