PSI - Issue 6

Alexander K. Belyaev / Procedia Structural Integrity 6 (2017) 201–207 A.K.Belyaev et al. / Structural Integrity Procedia 00 (2017) 000–000

205

5

(a)

(b)

−0.004

0.14

a

− explicit additive scheme

D 1 − explicit additive scheme D 1 1 − implicit additive scheme D 2 − explicit additive scheme D 2 2 − implicit additive scheme D D

theoretical

a

− explicit multiplicative scheme

−0.006

− explicit multiplicative scheme

theoretical

0.12

a

− implicit additive scheme

theoretical

−0.008

a

0.10

experimental

− explicit multiplicative scheme

−0.010

0.08

a

D

−0.012

0.06

−0.014

0.04

−0.016

0.02

−0.018

0

0.05

0.10

0.15

0.20

0.25

0.30

0

0.05

0.10

0.15

0.20

0.25

0.30

ε

ε

local

local

Fig. 3. (a) Dependence of the principal values D 1 and D 2 of the damage tensor on local deformations; (b) Dependence of experimental and theoretical acoustic anisotropy on local deformations ε local .

(a)

(b)

−0.004

−0.004

−0.006

−0.008

−0.008

−0.010

−0.012

a

a

−0.012

−0.014

−0.016

−0.016

−0.020

−0.018

0

0.05

0.10

0.15

0.20

0.25

0.30

0

20

40

60

80 100 120 140 160 180

σ ,MPa

ε

local

Fig. 4. Fig. 4. (a) Dependence of acoustic anisotropy a on axial stresses σ in MPa; (b) Dependence of acoustic anisotropy a on local deformations ε local .

The principal values of damage tensor, calculated according to an explicit additive scheme, an explicit multiplica tive scheme and an implicit additive scheme of symmetrization agree with each other (cf. Fig. 3a). The experimental value of the component D 1 exceeds the value of the component D 2 for several times. It is caused by the fact that uniaxial loading was applied along the axis of the specimen. A nonlinear increase in the values of the components with increasing local strains of the specimen is revealed. The correlation between acoustic anisotropy and damage was experimentally confirmed (cf. Fig. 3b). The maxi mum error of measurements for an explicit additive scheme is equal to 4.88%, for an explicit multiplicative scheme is equal to 6.24%, for an implicit additive scheme is equal to 11.97%. Thus, it is possible to estimate the accumulation of damages along the principal axes of the anisotropy of the material by using the explicit additive symmetrization scheme (4), which has the best correlation with the experimental values of acoustic anisotropy. Formula (2) describes the behavior of acoustic anisotropy in the region of elastic and small plastic deformations. The nonlinear dependence of acoustic anisotropy on the stresses (cf. Fig. 4a) and on the deformations (cf. Fig. 4b) in the region of large plastic deformations can be described by formula (7).

5. Concluding remarks

We developed a model of material with anisotropic damage (Semenov (2017)) to describe the influence of micro cracks on acoustic anisotropy. An experimental verification of the relations for the principal values of damage tensor and the velocities of ultrasonic waves according to the explicit additive (4), explicit multiplicative (5), and implicit additive (6) schemes was carried out. Measurements of local deformations of the specimen were carried out by a high-