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

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scheme (Murakami et al. (1981)) and explicit multiplicative scheme (Cordebois et al. (1982)). It allows to obtain the relationships between the principal values of damage tensor and the velocities ¯ v 1 , ¯ v 2 , ¯ v 3 of ultrasonic waves in a material with a system of orthogonal microcracks. Let us consider the case of propagation of an ultrasonic wave along the third principal axis n 3 of damage tensor D along the thickness of the material and the coincidence of the axes of anisotropy of the material n 1 , n 2 with the other two principal axes of damage tensor. For this case, the principal values of damage tensor of an initially orthotropic material have the following form: For the case of explicit additive symmetrization:   D 3 = 1 − (¯ v 3 / v 3 ) 2 D 2 = 1 − 2 (¯ v 2 / v 2 ) − 2 − (¯ v 3 / v 3 ) − 2 − 1 D 1 = 1 − 2 (¯ v 1 / v 1 ) − 2 − (¯ v 3 / v 3 ) − 2 − 1 (4) For the case of explicit multiplicative symmetrization:   D 3 = 1 − (¯ v 3 / v 3 ) 2 D 2 = 1 − (¯ v 2 / v 2 ) 4 / (¯ v 3 / v 3 ) 2 D 1 = 1 − (¯ v 1 / v 1 ) 4 / (¯ v 3 / v 3 ) 2 (5) For the case of implicit additive symmetrization:   D 3 = 1 − (¯ v 3 / v 3 ) 2 D 2 = 1 − 2 (¯ v 2 / v 2 ) 2 − (¯ v 3 / v 3 ) 2 D 1 = 1 − 2 (¯ v 1 / v 1 ) 2 − (¯ v 3 / v 3 ) 2 (6) where v 3 = E (1 − ν ) (1 + ν )(1 − 2 ν ) ρ is the velocity of the longitudinal wave in the undamaged material, v 1 = v 2 = E 2(1 + ν ) ρ are the velocities of the transverse waves in the undamaged material, E is the Young’s modulus, ν is the Poisson’s ratio, ρ is the density of the material. Relations for the velocities of ultrasonic waves can be obtained for each scheme of symmetrization of the tensor of e ff ective stresses ¯ σ from (4),(5) and (6). The relationship between acoustic anisotropy and damage can be obtained by substituting velocities into formula (1) and simplified for the case of small damages D 1 << 1 , D 2 << 1 , D 3 << 1. The relation for acoustic anisotropy obtained by asymptotic analysis has the following form:

D 2 − D 1 4

∆ a =

(7)

.

The principal value D 3 of damage tensor does not influence on acoustic anisotropy in the linear approximation.

3. Experiment

We carried out mechanical tests for uniaxial rigid loading of aluminum specimens from commercial alloy on the hydraulic tensile machine INSTRON-8801 for the experimental investigation of the relationship between acoustic anisotropy and damage. The specimens were prepared normal to the rolling direction (cf. Fig. 1a). Acoustic anisotropy was measured at the central point of each specimen by using a serial certified IN-5101A ultrasonic device. The device is used for measuring mechanical stresses of engineering structures in the nuclear power industry and in the oil and gas industry by the acoustoelasticity method. Acoustic anisotropy was measured in terms of the time delay between the probing ultrasonic pulse and its reflections from the opposite surface of the specimen. Measurements of acoustic anisotropy of one of the specimens were carried out at each of the eight loading stages. Measurements of the initial acoustic anisotropy were also carried out to assess the contribution of the texture to acoustic anisotropy. We noted (Belyaev et al. (2016b)) that the Portevine-Le Chatelier e ff ect (Le Chatelier (1909), Chihab et al. (1987),Franklin et al. (2000), Jiang et al. (2007)) has a significant influence on acoustic anisotropy. The e ff ect consists