Estimation of Stresses in a Plate with a Concentrator through Ultrasonic Measurements of Acoustic Anisotropy

Introduction. In recent decades, domestic nondestructive testing technologies have been developed to determine the stress-strain state of elements of energy systems, railway and pipeline transport [1]. These approaches involve solving inverse problems in the case of a nonuniform stress state [2], noncontact electromagnetic acoustic strain measurement [3], taking into account the degradation of elastic properties in combined nondestructive testing [4], as well as precision measurement of time delays in the propagation of elastic waves in the material [5]. The technologies have been verified in industrial measurements1 [6], their application is accompanied by the operation of modern diagnostic equipment [7]. However, the task of testing existing approaches in the diagnostics of anisotropic structural materials, which include industrial rolled products, remains relevant [8].

This work is devoted to the study of acoustic anisotropy [9] of an initially inhomogeneous material. In this case, it is commercially pure aluminum of the AMc brand (AW-3003 according to ISO). Samples cut from a rolled sheet were not prepared to remove the initial internal stresses caused by plastic deformation during rolling [10]. The task is to establish the possibility of estimating the difference in the magnitude of biaxial stresses in a metal with inhomogeneous initial acoustic anisotropy [11].

Materials and Methods. In the experiments, we used the IN-5101A ultrasonic acoustic anisotropy analyzer². This is a certified device for measuring mechanical stresses by the acoustoelasticity method [12]. The stresses were calculated from the formula:

(1)

The current value of the acoustic anisotropy parameter a_σ [14] was determined from formula (2) as the relative difference in the propagation time of transverse ultrasonic waves of mutually orthogonal polarization [15]:

(2)

Here, t₁, t₂ — current values of time delays during the passage of transverse wave packets through the thickness of the material after their multiple reflection, whose velocities are directed along and across the line of uniaxial loading.

The measurements were performed using acoustic sensors with a pulse emission frequency of 5 MHz. The accuracy of the measurements of time delays t₁, t₂ was 3 ns.

The low-alloy corrosion-resistant aluminum-manganese alloy of the AMc brand, close in its properties to commercially pure aluminum (97/99% Al in composition), was studied. Aluminum was a model material for ultrasonic research. It was with it that the basic results in the acoustoelasticity were obtained [1].

Mechanical tests were performed on a 510×120×15 mm rectangular plate, cut from a rolled aluminum sheet across the rolling direction. The sample had a stress concentrator in the form of a central circular hole with a diameter of 40 mm (Fig. 1).

The direction of rolling affects the sign of the initial acoustic anisotropy a₀ [16]:

(3)

Here, t₀₁, t₀₂ — initial time delays of transverse waves. In the case of samples cut across the rolled product, the dimensionless parameter of the initial acoustic anisotropy , calculated from formula (3), is negative: a₀ < 0.

For rigid step loading of the plate, uniaxial tension was applied in an Instron-8850 hydraulic machine (Fig. 3). The stress-strain state was considered at three loading stages with tensile load values F₁ = 30 kN, F₂ = 50 kN and F₃ = 70 kN. For the study, n = 8 sections (points) of the sample were selected. The diagram of their location is shown in Figure 2.

Ultrasonic testing was performed along three rows of points: 1–5, 2–4 and 6–8, symmetrically located relative to each other around the stress concentrator (Fig. 2). The characteristic size of the plates of the piezoelectric elements of the acoustic sensor was 12×12 mm. That means that each point was associated with its own local representative volume of material. The points were selected so that the representative volumes differed significantly in their stress-strain state [17].

Mechanical tensile testing in an Instron-type machine involved different boundary conditions for the sample. Its lower part (points 4–6 in Figure 2) was rigidly fixed in a stationary grip of the testing machine (Fig. 3). In the upper part (points 1, 2 and 8 in Figure 2), the sample, fixed in a movable grip, was stretched at a low constant strain rate (Fig. 3). Points 3 and 7 were located on the central transverse axis of the sample (Fig. 2).

The asymmetry of the problem conditions was taken into account in numerical modeling using the finite element method. The actual dimensions of the sample, its elastic and mechanical properties obtained during testing of samples from the same batch (Young's modulus E, yield strength σ_y, elasticity modulus H), as well as known data for commercially pure aluminum (density ρ, Poisson's ratio ν) were used. It was taken into account that the sample was fixed in the testing machine over a wide surface, and the tangential tensile load was applied to it, and not to the end of the sample, as in the case of two-dimensional models.

The stress-strain state was calculated in the ANSYS finite element modeling package. Taking into account the area of application of loads F₁, F₂, F₃, the values of tensile stresses were determined: σ₁ = 16.67 MPa, σ₂ = 27.78 MPa, and σ₃ = 38.89 MPa. They were used in the numerical solution. Figure 4 shows a model of the sample with the display of the finite element mesh and boundary conditions. It consists of 936,152 elements and 3,981,073 nodes.

The analytical solution was based on the use of formulas (4) and (5), which are the solution to the Kirsch problem in polar coordinates [18]:

(4)

(5)

where S — load applied to the plate; a — radius of the hole in the plate; r — distance from the center of the hole to the stress calculation point; θ — angle corresponding to the point being calculated.

The formulation of the problem in this work assumes the presence of only elastic deformations of the sample. Its further development, taking into account the influence of inelastic deformations, is associated with the consideration of the solution to the plane elastic-plastic problem of stretching a plate with a circular hole (an ideal plastic body), obtained by L.A. Galin in 1946 [19].

The principal stresses co-directed with the longitudinal axis x (σ_θθ) and the transverse axis y of the sample (σ_rr) were calculated for all n = 8 points under study. The components σ_θθ, σ_rr — functions of the radial distance to the hole center r and of the angle θ, measured relative to the reference axis. Relations (4) and (5) were obtained under the assumption that the circular hole was located at the center of an infinite isotropic linear elastic plate subjected to uniform plane loading. They were previously used in [20] to determine the stresses in two regions on the boundary of the hole located along (θ = 0°) and across (θ = 90°) the line of action of the load.

Ultrasonic measurements were performed at points 1–3 (Fig. 2) in the sample before loading (F₀ = 0 kN), as well as upon reaching elastic deformation levels corresponding to loads F₁, F₂, F₃. The finite size effect of the plate (mainly, in its cross-section) on the discrepancy between the stress values obtained experimentally, numerically and analytically, was investigated. The values of acoustic anisotropy a₀, % and a_σ, % were calculated using formulas (2) and (3). To take into account the effects associated with stress relaxation, control measurements were performed at the start and end of each stage. A total of 198 time delays (t₀₁, t₀₂) and (t₁, t₂) were measured during the propagation of transverse waves along and across the sample.

Research Results. Table 1 shows the values of internal stresses σ_xx (analog σ_θθ in the polar coordinate system) and σ_yy (analog σ_rr) for n = 8 points around the plate hole (Fig. 2). They were obtained as a result of finite element modeling and calculations using formulas (4) and (5) with external tensile stresses σ₁, σ₂, σ₃ (correspond to loads F₁, F₂, F₃).

The conditions of fixing the sample are different: the left part is fixed, the right one is movable. Hence the asymmetry in the pattern of the sample deformation field, which is preserved for all values of loads in the elastic region (an example for F₁ = 30 kN is shown in Figure 5).

Due to the asymmetry noted above, when solving the problem using the finite element method, the stress values in pairs of points:

• coincide if they are obtained through mirroring relative to the central longitudinal axis of the sample;
• differ if they are obtained by the reflection relative to the transverse axis of the sample.

In the first case, pairs 2, 8; 3, 7 and 4, 6 can serve as an example, in the second case — 1, 5; 2, 4 and others. Note that even if there is a difference in stresses, it does not exceed 1 MPa. This corresponds to the engineering error level ±5% (Table 1). The stress values calculated for the specified pairs of points using formulas (4) and (5) are the same.

Finally, close values between the results of analytical and numerical calculations are observed only for points {3, 7}, located near the region with the highest stress concentration (Table 1). At the other points, there is a multiple difference not only to modulo (2, 8 and 4, 6), but also in sign (1, 5). This is due to the fact that calculations using formulas (4) and (5) for a plate of equal (infinite) dimensions assume the presence of a compressive-stress zone in the region of the location of points 1, 5. The situation is different with a real plate (Fig. 1 and 2), whose dimensions are limited and differ from each other in the ratio of length and width by 4.25 times. In this case, there is no compressive-stress zone (Table 1 and Fig. 5).

The comparison of the results suggests that the effect of the finite dimensions of the plate does not allow for the use of relations (4) and (5) to estimate the stresses in this problem. This contradicts the conclusions of [20]. Note that the presence of a correlation for points with the maximum value of the stress concentration coefficient is confirmed by the results for pair 3, 7.

Table 2 shows the initial a₀ and current a_σ values of acoustic anisotropy in the region of the locations of points 1, 2 and 3, calculated from formulas (2) and (3) in the no-load state (F₀) and at three load values F₁, F₂, F₃.

It follows from Table 2 that the difference in the values of the initial acoustic anisotropy parameter a₀ at points 1, 2 and 3 is largely determined by the initial heterogeneity of the material. This is evident from the variation of parameter a₀ before loading. In addition, with the application of an external load, the discrepancy between the values of the acoustic anisotropy parameter a_σ at the points under study increases.

Based on the ultrasonic measurement data (Table 3), the values (σ_θθ – σ_rr) of the difference in the principal stresses in the polar coordinates were calculated using the acoustoelasticity relations (1) [21]. They are presented in Table 3 in comparison with the results of finite element modeling and calculations using formulas (4) and (5). The stress values that differ for pairs of symmetrically located points obtained by reflection relative to the transverse axis of the sample are given in parentheses.

Table 3 shows that the data from acoustoelastic measurements and the finite element method solutions correlate qualitatively and quantitatively with each other. Consequently, we can talk about mutual verification of the results of full-scale and numerical experiments. The highest correlation is observed for points 2, 8; (4, 6) and 3, 7, located near the region of maximum tensile stresses.

Different results were obtained for points 1 and (5), where the presence of compressive stresses was assumed according to formulas (4), (5). Here, the differences in the principal values (σ_θθ – σ_rr), obtained according to the ultrasonic measurements, were on average two times smaller than the predicted numerical values. Thus, calculations based on the acoustic anisotropy values provide a lower estimate of the stresses for the studied sections of the sample.

Discussion and Conclusion. The principal biaxial stresses in an aluminum plate around a concentrator — a central circular hole — were investigated. The values obtained as a result of natural ultrasonic measurements, numerical experiments using the finite element method, and analytical calculations using the Kirsch relations were compared. The effect of the asymmetry of stress and strain fields, arising due to different conditions of sample fixation and reflecting the process of uniaxial elastic deformation under tension in a testing machine, was taken into account.

It was noted that similar results of the analytical and numerical calculations were observed only for points located near the region of the greatest stress concentration. In all other cases, the values differed several times both to modulo and in sign. This is explained by the presence or absence of compressive stresses. The analytical approach assumes that they exist. There are none in a real plate. Thus, the Kirsch relations for stresses of a uniaxially stretched infinite isotropic linear-elastic plate cannot be correctly applied in the case under consideration.

The acoustoelasticity method has established a correlation between the results of numerical modeling and ultrasonic measurements of biaxial stresses. This is specifically noticeable in relation to points located near the zone of maximum tensile stresses.

The research results can be used in industrial nondestructive testing for stress diagnostics in rolled metal objects.