PSI - Issue 47

Ezio Cadoni et al. / Procedia Structural Integrity 47 (2023) 268–273 Author name / Structural Integrity Procedia 00 (2023) 000–000

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5

a) b) Fig. 5. Stress versus strain curves of the smooth sample at di ff erent temperature @850 1 / s (a); Set-up for high temperature testing (b)

5. E ff ect of triaxiality and failure surface

The stress triaxiality at the centre of net section can be calculated by the formula proposed by Bridgman (1964):

+ ln 1 +

a 0 2 R 0

1 3

(5)

τ =

where R 0 is the notch radius and a 0 is the radius of the specimen cross section. For the notched sample of R2.0mm, R0.8mm and R0.4mm the triaxiality values are 0.56, 0.82 and 1.15, respectively, while for the smooth sample the value is 0.33. Failure surfaces are generally represented as a function of strain rate (˙ ), temperature ( T ) and stress triaxiality ( τ ). The almost totality of the static and dynamic material models uses as failure surface parameters the total equiva lent plastic strain at failure, separating with multiplicative and uncoupled sub-functions its dependencies in the rate, temperature and triaxiality parameters: f = f 1 (˙ ) · f 2 ( T ) · f 3 ( τ ) (6) By identifying equation 6 from an experimentally derived data array, the software can be directly implemented with realistic data, or it can be further interpolated with the characteristic equations from any other material model: e.g., the damage Johnson- Cook material model (equation 7), or any other. f = ( D 1 + D 2 e D 3 n )(1 + D 4 ln ˙ ˙ 0 (1 + D 5 T ∗ ) (7) where: f is the failure strain; n is the triaxiality; ˙ is the strain rate; ˙ 0 is the reference strain-rate; T ∗ is the non dimensional temperature. An array of test results has been obtained from dynamic tests at high strain rates and elevated temperatures on smooth and notched specimens. The five parameters were derived from these data: D 1 = − 0 . 192374, D 2 = 0 . 188010, D 3 = 0 . 093419, D 4 = 0 . 500087, D 5 = 0 . 420759 at 20°C and D 1 = 0 . 036584, D 2 = − 0 . 025180, D 3 = 0 . 082921, D 4 = 0 . 462512, D 5 = 0 . 472262 at 1100°C. According to these parameters, the failure surfaces have been determined and shown in Fig. 6. The dynamic failure surface is strongly dependent on triaxiality, strain rate, and temperature. A quasi-brittle failure strain of 5% is observed in the quasi-static regime for this alloy. As the strain rate increases in the dynamic regime, the strain is significantly increased in the static regime: at true strain rates of 30’000 s − 1 at 20°Cas well as at 800°C when the triaxiality is equivalent to uniform tensile stress, 30% strain is reached. Further, when the stress triaxiality increases as in a hydrostatic stress state, the material decreases its deformation at fracture. With triaxiality equal to 1.15, the maximum failure strain is equal to 15% both at 20°C and 800°C.

6. Concluding remarks

The results of a preliminary experimental campaign on a commercial Tungsten alloy aimed at investigating its me chanical response with increasing temperature, strain rate and triaxiality have been described. Three di ff erent notches