Estimation of the ultimate tensile strength and yield strength for the pure metals and alloys by using the acoustic wave properties

In this paper, the acoustic impedance property has been employed to predict the ultimate tensile strength (UTS) and yield strength (YS) of pure metals and alloys. Novel algorithms were developed, depending on three experimentally measured parameters, and programmed in a MATLAB code. The measured parameters are longitudinal wave velocity of the metal, density, and crystal structure. 19-samples were considered in the study and divided into 3-groups according to their crystal structure; 7-FCC, 6-BCC, and 6-HCB. X-ray diffraction was used to examine the crystal structure of each sample of each group, while longitudinal wave velocity and metals’ density were measured experimentally. A comparison between mechanical properties predicted by the model and the ASTM standards was done to investigate the validity of the model. Furthermore, predicted stress–strain curves were compared with corresponding curves in the pieces literature as an additional validation check. The results revealed the excellence of the model with 85–99% prediction accuracy. The study also proved that if metals are grouped according to their crystal structure, a relation between UTS, YS, and modulus of elasticity (E) properties and wave pressure transmission coefficient (Tr) could be formulated.

Scientific RepoRtS | (2020) 10:12676 | https://doi.org/10.1038/s41598-020-69387-z www.nature.com/scientificreports/ applications 11,12 such as testing the strength of pipeline 3 and boiler tubes 13 , which face difficulty in testing directly by using normal tensile tests. Many studies have tried to consider this technique (the acoustic wave's tests) in recent years, though there was always a margin of miss-match between the predicted and actual values. Many studies have referred to this discrepancy and tried to justify it by naming the E value calculated from the tensile test as static modulus of elasticity (E s ) and E calculated from acoustic wave's techniques as dynamic modulus of elasticity (E D ), where E D = C 2 L ρ(1+ν)(1−2ν) (1−ν) . Ciccotti and Mulargia 14 compared between E s for seismogenic rocks (in the Italian Apennines) with E D for the same material and found a 10% difference between them. Further, Builes et al. 15 reported that the difference between the E s and E D increases with the density of the specimens. In the same context, many other published papers focused on this difference [16][17][18][19][20][21] . Mohammed et al. 22 reported that the refractory metals have body force, therefore the equation of E D cannot be used in the calculation of the modulus of elasticity because this equation neglected the term of body forces in the origin of this equation, where the origin of this equation is Navier governing equation. It is worth mentioning that, many of the offered devices in the global markets that used acoustic waves techniques such as 38DL PLUS 23 , 58-E4800, MATEST (C372M)… etc., still use E D equation in finding some of the mechanical properties such as E, ν and other several mechanical properties for solid industry materials. This because these devices give correct results for normal materials such as Al, Fe, Zn.etc., however, for metals such as refractory metals, the results are incorrect 22 . Mohammed et al. 24 succeeded in the calculation of the E for many metals and alloys, through finding a relationship between E × ρ and the pressure transmitted coefficient. Even though of this succeeding in finding E however, Mohammed et al. 24 did not mention anything about the estimation of the YS and UTS for these metals, despite its importance. This study avoided the problem of equation E D through proposing a method that doesn't depend on Navier governing equation, then developed new algorithms to estimate YS and UTS for many pure metals and alloys. Also, the proposed method provides another advantage which the use of a single probe to generate and receive C L , while the other studies, which use E D equation [16][17][18][19][20][21]23 , need two types of probes one for generating and receiving C L and the other is for generating and receiving shear velocity . And this means that the proposed technique could efficiently reduce the random error, cost, besides the main goals which are the calculation of YS and UTS and drawing the stress-strain curve by using acoustic wave properties.
This study proposed four steps to achieve targeted goals. In the first one, the pure metals and alloys specimens were classified according to their crystal structure (BCC, FCC, and Hex). Secondly, Tr was calculated by measuring C L and ρ. Then, a polynomial relation between the Tr and each of E, YS, and UTS (for each crystal structure type) was individually constructed. In the final step, E, YS, and UTS were calculated and graphed using the programmed algorithms.

Methods
Analysis model. In the acoustic tests, Tr is an important parameter between any two connected elements.
This parameter depends on the acoustic impedances of these connected materials where Z 1 and Z 2 are the acoustic impedances for any two connected materials. According to Eq. (1), this study selected the magnesium (Mg) to be Z 1 because Mg has the lowest acoustic impedance among the sold metals (Z Mg = Z 1 = 9.9761 × 10 6 Kg m 2 s ) , while Z 2 is the acoustic impedance of any other test specimen (Z 2 = Z sp ) . Therefore Eq. (1) becomes: This study found there is auniform relationship between YS and UTS: with Tr if the metals were classified according to their crystal structure.
In Fig. 1 the values of Tr, of metals that have FCC crystal structure, calculated from Eq. (2), while the values of the YS and UTS for these metals werecollected from references [25][26][27] . Figure 1  Also, the values of Tr of Fig. 2 were calculated by using Eq. (2) and the values of YS and UTS were collected from the same references [25][26][27] . Figure 2 illustrated the relationships between YS and UTS from the side and their values of Tr from another side for metals that have BCC crystal structure. The Eqs. (5) and (6) represent the mathematical expression of these two relationships shown in Fig. 2 (1) www.nature.com/scientificreports/ Figure 3 was divided into two parts A1 and A2 to find the equivalent relationship between Tr and the YS for HCP metals. In the same figure and same context, there is a uniform relationship between UTS and Tr for HCP metals. Equation (7a) represents the part A1in the Fig. 3  Equation (8) indicates the relationship between UTS and Tr, as shown in green color in Fig. 3, for HCP metals: The collected data in this section (Table 1) were gathered from authorized sources 25,28,29 . The Eq. (9) calculates the convergence (Conv1) between the values of the standard YS coming from tensile tests, according to ASTM (YS ASTM ), and the YS values calculated from the proposed method (YS cal ) in this study (Eqs. 3, 5, 7).    The convergence ratio between YS ASTM and YS cal is shown in column Conv1% in the Table 1, while the convergence between UTS ASTM and UTS cal is as shown in a column Conv2% in Table 1. The last two columns in Table 1 calculated from Eq. (8) in the references 22,24 .
The equations from (3)(4)(5)(6)(7)(8) were programmed as shown in the index (A), where this program has three input values (C L , ρ, and the type of crystal structure (Cy)). The values in the red color in Table 1 represented the faults of this program (in index A).
To compare the results of this program with the other works, and to show the program's ability to draw a stress-strain curve not only for pure metals but also for alloys that have purity more than 99.95% therefore, more information should be added. This information is the elongation of these metals. Figure 4 shows the undisciplined relationships between elongation values with Tr contrary to YS and UTS which have disciplined relations with Tr. It is worthily mentioned, that Tr values, in Fig. 4, calculated from Eq. (2); while the elongation values were collected from references 25,28,29 .

experimental model
The practical methodology of this research depends on three steps: 1. Make sure from crystal structure by using the X-Ray diffraction (XRD) test for each specimen. 2. Calculation Tr for each sample through measurement the C L and ρ for each sample, then, calculate YS cal and UTS cal for these samples by using the Eqs. In this research, the three specimens were selected to prove the theoretical part. Mg specimen was as a sample for HCP crystal structure. And, Ni specimen was selected as a sample of FCC crystal structure; while Nb specimen was as a sample for BCC crystal structure ( Supplementary Information 1).
Step-1: XRD test Small samples of Mg, Ni, and Nb were prepared to be suitable for XRD 6,000 SHIMDZU device for doing the XRD test. The results of these tests, as shown in Fig. 5, proved the purity of these samples and proved identical to the crystal structure of these specimens with the crystal structure of these specimens in Table 1.
Step-2: Measuring C L and ρ then calculation Tr where t TOF is the time of flight of the wave in the specimen, L is the specimen length, and t o is the wedge delay of the used probe 32,33 . The t o for the used probe in this research equals 9µsec.This value (9µsec) was calibrated by using the digital ultrasonic thickness tester GM100 where the natural frequency of this probe equals 2.5 MHz.
The Ultrasonic Pulse UP200 (OSUN) is used to generate the electric pulses at a frequency equal to 1,000 Hz, the duration of the mode is 0.1 and the output voltage equals to 200 V in to excite the probe to generate ultrasonic waves. The oscilloscope type of DSEX1102A (100 MHz) oscilloscope was used in this study. Before starting, an ultrasonic gel is placed to get a good connection between the probe and sample to maintain signal strength.
It is worth mentioning, that this oscilloscope has a high sampling rate (2G samples/sec). This advantage gives the system the ability to detect the echo signal for the specimen with thickness 1 mm and this advantage gives preference over the device 38DL PLUS, which cannot detect the thickening less than 4 mm 23 . Also, this oscilloscope gives directly the delay in time between the electric excitation signal and the returned signal (the returned signal from the test specimen) and this is another feature for this oscilloscope.
Before putting the probe on the specimen, the ultrasonic gel was put on the surface of this specimen to avoid the effect of the air blanks between the probe and the specimen and to guarantee good contact between them. After putting the probe on the specimen, directly the t TOF appears in the middle of the oscilloscope screen as shown in Fig. 6. The red circle in this figure shows the value of t TOF equals 50 µs for the Mg specimen while the values of t TOF are 9.57 µs and 11.6 µs for Nb and Ni respectively as show.
It is worth mentioning the beginning of calculation of the t TOF is from the moment of receiving the echo signal as shown in all Fig. 5. Table 2 involved the details and description of the used specimens in this study. Also, the values of Tr for all these specimens were calculated in this table too. Also, this table refers to the equations that calculate the values of YS and UTS. The previous studies proved that the crystal structure of iron (Cry-Stru-Fe) has different behavior depending on the environments surrounding him. At room temperature the Cry-Stru-Fe is BCC, but when the temperature is rising until reaches 1,200 °C, the Cry-Stru-Fe changes from BCC to FCC. In the same context 34,35 , had succeeded in changing the Cry-Stru-Fe to HCP at a pressure equals to 360 GPa. On the other hand, the pure titanium crystal structure (Cry-Stru-Ti) exists in two crystallographic forms, first, one as HCP at the room temperature, and BCC at 883 °C and this what is known as beta (β) phase 36 .

Results and discussion
This study found these two behaviors (phase transition of crystal structure) for Fe and Ti exists not only at increasing the pressure and temperature, however, these behaviors also exist in the ultimate tensile point. The comparison between the Figs. 7 and 8 illustrates this behavior of Cry-Stru-Fein the tensile testing, where the Cry-Stru-Fe is BCC at the yield point, as shown in Fig. 7, however, when the applied load is increased until reaches to the ultimate load point the Cry-Stru-Fe become so close to HCP group as shown in Fig. 8. In the same context and same comparison, the pure titanium at yield point Cry-Stru-Ti, in Fig. 7, is HCP, however, it becomes so www.nature.com/scientificreports/ close to BCC when the applied load is increased to the ultimate load point as shown in Fig. 8. And this is one of the new findings of this study about the behavior of Fe and Ti. The tensile tests are energy added to the sample, this energy compels the crystal structure to distortion and if the metal has the ability on phase transition such as Fe and Ti, this may be caused by changing the crystal structure of Fe from BCC to HCP and Ti from HCP to BCC. Figure 9a,b shows a good match in the behavior of curves coming from the proposed program and experimental stress-strain curves of the previous studies. Also, there is an excellent match in stress and strain curves values (E, YS, UTS, and elongation) for curves included in Fig. 9a,b. This match in values of E, YS, UTS, and the elongation is within range of (88 ~ 95)%. Figure 9c includes two curves first one belongs to the stress-strain curve for Ni. Even though the match in values of E, YS, and UTS for this curve (Ni curve) is quite good, as shown, however, the divergence of these two curves from yield point to ultimate point is around 30%. This divergence of this area of the curve belongs to the mathematical approximation that was used to draw the plastic deformation as shown in equations below:  , are constant for all ductile metals. This approximation does not affect the significant values in the stress-strain curve such as E, YS, and UTS, therefore it was regarded as an acceptable result. In the same figure, the match of the stress-strain curve of AL1100-O is around 90%. The 10% difference of stress-strain curve may happen from using different types of instruments or random error 40 .
The stress-strain curve can be estimated by knowing specific points in a stress-strain curve such as (0,0) and point of the elastic limit (ɛ = YS/E, YS), UTS, and the elongation. As was mentioned before, three variables must be entered into the program; these are C L , ρ, and type of crystal structure (Cy) to estimate the stress-strain curve. To check the ability and accuracy of this program, the stress-strain curves, for three high purity alloys, published in other papers 20,21,38 were selected to show the identical among of those curves and the curves producing from this program. It worth mention, these alloys (AL1100-O, Gray cast Iron and Mg-0.5Zr) do not exist in Table 1. Table 3 includes C L , ρ, and type of crystal structure for these three alloys. Also, this table includes three alloys that were not drawn however; they match with ASTM in E, YS, and UTS. Figure 9 illustrated these curves and the matching among them. Also, three stress-strain curves for the metals (Ni, Mo, and Zr), from papers 20,37,39 already exist in Table 1, were drawn in this figure too. Anyone can check the ability of this program for the alloys: Ni 233, Ni200, AL1199-O, and AL2014-O by using the data of them (C L , ρ, and Cy) and draws the stress-strain curves for these alloys.
It is worth mentioning that some studies have pointed out a relation between the plastic deformation in the tensile test, the crystal structure, the grain size, and the surface energy of the material, especially in nanomaterials [42][43][44][45] . In the same context, Lu et al. 46 linked between these properties and attenuation of the acoustic waves. The view of 46 and view of this study, refer to the possibility of using the acoustic wave properties especially the attenuation in nanostructure materials field and its applications. This study has put the advance step for studying the changing of acoustic impedance with changing crystal structures. And this will open the door for future studies to use the acoustic impedance to study the relation among grain size, grain boundary types, toughness, YS and UTS for a model of the alloys with complex phases or complex compositions.