A martensite stainless steel 30Cr2Ni4MoV is used in the study, whose nominal chemical compositions in wt% are listed in Table 1. The weld joint mockup is heat treated at a temperature of (580±10) °C for 10 h and then cooled with a furnace. Flat dog-bone specimens with the welds located in the center are machined from the weld joint mockup, whose nominal dimensions are shown in Fig. 1. From the optical microscopy analysis, it can be identified that the widths of the weld and the heat affected zone (HAZ) are about 20 mm and 2 mm, respectively.

Table 1.

Table 1.

Table 1. Chemical compositions of 30Cr2Ni4MoV stainless steel in wt%. . C Si Mn Cr Ni Mo V Cu As Sn Fe ≤0.35 ≤0.10 0.20∼0.40 1.50∼2.0 3.25∼3.75 0.25∼0.60 0.07∼0.15 ≤0.15 ≤0.02 ≤0.015 balance

Table 1. Chemical compositions of 30Cr2Ni4MoV stainless steel in wt%. .

Fig. 1.

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	Fig. 1. Dimensions of flat dog-bone specimen of weld joint. All dimensions are in unit mm.

Two sets of plastic deformed specimens are made for experiments using a 100-kN MTS servo hydraulic testing system. One group with 7 specimens are fabricated under stress-controlled mode loading from 774 MPa to 869 MPa (yield strength is 785 MPa), and another one with 5 specimens are fabricated under strain-controlled mode from strain levels of 1.14% to 3.89%. The nonlinear ultrasonic measurements are made off-line after unloading the specimens.

Nonlinear ultrasonic measurements are performed on three locations of the specimens, i.e., weld region (about 20 mm in the center), HAZ region (about 2 mm nearing the weld), and the base region. For nonlinear ultrasonic measurement, a Ritec SNAP system with model of RAM-5000 is used to generate the ultrasonic signal. The experimental setup for longitudinal wave measurement is illustrated in Fig. 2,^[26] which is primarily composed of the Ritec SNAP system, a high power attenuator, a pre-amplifier, low and high pass filters, transducers, oscilloscope and computer. Amplitudes of the fundamental and second harmonic wave are measured by the transmission method in the present measurements. The cut-off frequencies for the low-pass and high-pass filters are about 6.3 MHz and 9.5 MHz, respectively. A narrow-band longitudinal piezoelectric transducer with a nominal frequency of 5.0 MHz is used to generate the fundamental wave, marked by T. A broad-band 10.0 MHz transducer is used to pick up the receiving signals, denoted by R. The diameters of transmitting and receiving transducers are both 8.6 mm. The transducers T and R are coupled to the specimen by using silicon oil. A special fixture is designed to maintain alignment parallelism between the transmitting and receiving transducers during the whole measurement. A tone burst of ten cycles with a frequency of 5.0 MHz is used to excite the transmitter. The received signal is filtered by a Hanning window and captured by a 300-MHz oscilloscope, and then processed by using an amplitude spectrum fast Fourier transformation (FFT) to extract the amplitudes of the fundamental and second harmonic signals in the frequency domain.

Fig. 2.

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	Fig. 2. Experimental setup for nonlinear ultrasonic measurements.[3]

When exciting a sample by using a single frequency excitation, the second-order nonlinear parameter, known as β, generated by the presence of damaged structure can be expressed as

Figures 3(a) and 3(b) show the variations of the with loading stress and the plastic strain, respectively. It shows from Fig. 3(a) that the responses of the versus the loading stress in the three detection regions being measured have the similar tendencies. The measured acoustic nonlinearity exhibits a slight increase in the elastic stage, i.e., the loading stress is less than the yield stress, then it drops as the loading stress slightly exceeds the yield stress, and finally the shows a monotonic rise with the increase of loading stress. Although the material of the weld region is the same as that of the base, the variation of the on the location of weld is maximum, nearly three times bigger than the initial value, which means that the plastic-induced deformation of weld would be much more serious than those of HAZ and base regions. Figure 3(b) shows that the monotonically increases with the plastic strain, which is similar to the change of in the last stage of Fig. 3(a). Some other previous studies have also reported the same results of the acoustic nonlinearity versus the fatigue or plastic deformation in the engineering materials, such as AA7175-T7351 aluminum alloy^[11] and A36 steel.^[12]

Fig. 3.

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	Fig. 3. Variations of the RANP with (a) loading stress and (b) plastic strain.

Microstructure changes of the martensite steel subjected to plastic deformation are thought to originate from the variation of acoustic nonlinearity.^[17] Therefore, TEM are used to observe the microstructural evolutions and understand the change of the with loading stress or plastic strain. Note that the evolution of dislocation in the process of plastic deformation or fatigue damage plays a critical role in the response of acoustic nonlinearity. It is helpful to pay more attention to observing the dislocation multiplication, annihilation and rearrangement during damage. In the following TEM micrograph observation, a Philips CM200 transmission electron microscope is used to take the micrographs. Dislocation density and length are evaluated from the TEM images, in which the density of dislocations is measured by the method of Ref. [27] and the pinning dislocation length is obtained by analyzing the TEM micrographs in computer.

Figure 4 shows the TEM micrographs of 30Cr2Ni4MoV weld joint specimens with different plastic strain levels corresponding to virginal specimen, 1.14% plastic strain, 1.73% plastic strain, and 3.89% plastic strain.^[28] Table 2 gives the values of microstructure parameters of the 30Cr2Ni4MoV specimens with different plastic strains, where the dislocation density, length, and dislocation cell size are presented. Note that each mean value in Table 2 related to dislocations is determined by an average of three to five micrographs from different locations of the specimens. Also, it should be pointed out here that the dislocation cell structures (i.e., the dislocation pile-up) observed in Figs. 4(b)–4(d) would also exert an important influence on the material nonlinearity and whose dislocation density is not easily determined by the method proposed in Ref. [27]. It can be indicated that from Figs. 3 and 4 the evolution of dislocation during the plastic deformation, especially the variation of dislocation density, plays a dominant role in the changing of acoustic nonlinearity. For ease of analysis, it is supposed that the influence of grain or sub-grain evolution of martensite structures on the acoustic nonlinearity is ignored.

Table 2.

Table 2.

Table 2. Microstructure parameters of 30Cr2Ni4MoV weld joint specimens with different plastic strains. . 30Cr2Ni4MoV specimens with different plastic strains/% 0 1.14 1.73 3.89 Dislocation density Λ/×1014 m−2 0.35±0.15 1.30±0.26 1.94±0.73 3.70±1.24 Pinning dislocation segment L/μm 0.38±0.042 0.24±0.032 0.20±0.030 0.16±0.010 Applied longitudinal stress σ0/MPa 60 115 150 200

Table 2. Microstructure parameters of 30Cr2Ni4MoV weld joint specimens with different plastic strains. .

Fig. 4.

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	Fig. 4. TEM micrographs of 30Cr2Ni4MoV weld joint specimens with different plastic strain levels: (a) virginal specimen; (b) 1.14% plastic strain; (c) 1.73% plastic strain, and (d) 3.89% plastic strain.[3]

Considering the longitudinal waves in the medium and ignoring the dispersion and attenuation, the equations of motion in the Lagrangian coordinate Y can be written as

The stress tensor in Eq. (3) may be expanded in terms of the particle displacement gradients as^[29]

Considering a one-dimensional medium such as a rod, the wave solution of Eq. (3) can be obtained by a simple perturbation analysis as

For a medium experiencing plastic deformation, localized plasticity damage would appear in the form of microstructural defects such as dislocations, which could be a source of nonlinearity for the wave propagation. In contrast to β_I, another nonlinearity parameter β_D is defined to describe the plasticity-induced nonlinearity for the damage state of the material. For ease of numerical simulation, here we restrict discussion to the dislocation monopoles contributing to the nonlinear parameters of the material, which can be expressed as^[22,30]

Therefore, the acoustic nonlinearity parameter β of Eq. (1) which can be calculated from the experimental measurement or numerical simulation usually comes from the total contributions of β_I and β_D.

Based on the above analyses of the nonlinear elastodynamics, a finite element simulation is conducted in an attempt to explain the variation of acoustic nonlinearity in the plasticity deformed material, which is developed and carried out on the basis of the commercial FEM software Abaqus/EXPLICIT. The intact state and plastic-damaged state with dislocations in materials are taken into account in the following FEM models.

The numerical simulation of material, geometric, and plasticity-induced nonlinearities is performed by introducing the nonlinear stress-strain constitutive relation (i.e., Eq. (4)) to Abaqus/EXPLICIT through the interface of user subroutine VUMAT, which can be used for material definition and called in the iteration of the program. A two-dimensional (2D) model is established to simulate the longitudinal wave propagating in the weld joints, whose schematic of 2D model is shown in Fig. 5 with the same dimensions as the experimental samples. The wave generation is excited by imposing longitudinal (y axis) stress loadings with a ten-cycle Hanning-windowed sinusoidal tone burst at 5.0 MHz and an input acoustic pressure amplitude of 20 MPa, whose loading size is 8.6 mm, similar to the diameter of the transducer. The signal (longitudinal stress, S22 in Abaqus system) is picked through the middle node of the bottom edge as shown in Fig. 5. The second-order elastic constants of Lamé parameters λ and μ in 30Cr2Ni4MoV are assumed to be 116 GPa and 84 GPa, respectively. The TOE constants of the material used here are as follows: A = 1100 GPa, B = −1580 GPa, and C = 1230 GPa.^[31]

Fig. 5.

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	Fig. 5. Schematic diagram of the simulation model for longitudinal wave propagating in weld joint.

The four-node reduced-integration plane strain elements (CPE4R) in Abaqus are used to mesh the rectangular model through a structured meshing algorithm, whose elements are equally sized at 0.05 mm (Δy) with a spatial resolution of λ_L/24 where λ_L is the wavelength of the fundamental wave (about 1.2 mm). This spatial resolution is chosen to warrant simulation precision and error convergence. According to the Fourier or von Neumann stability analysis,^[32] the time step Δt should be smaller than the time required for the longitudinal wave to travel across the element length, that is,

A snapshot of the ultrasonic wave (in the form of stress contour) propagating in the weld joint is shown in Fig. 6. The results for three simulations corresponding to three variants of the stress-strain constitutive relation (Eq. (4)) are presented as follows.

Fig. 6.

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	Fig. 6. A snapshot of the ultrasonic wave propagating in the weld joint (in the form of stress contour).

According to the above different conditions of (i)–(iii), figure 7 shows the signals of frequency domain for the three cases, in which it can be seen that there is very little difference for the fundamental wave at 5.0 MHz but an obvious change for the second harmonic wave (or nonlinearity). Note that the ‘LEM’ case shows no second harmonic component as Eq. (4) describes (the first term on the right-hand side in the equation) and the other two cases have evident second harmonic components. It is also interesting to see that the amplitude of the second harmonic generated in the ‘NLM’ case is about 4.3 times greater than that generated in the ‘GNM’ case, as also reported in Ref. [23], which means that the material nonlinearity may play a dominant role in the generating of the second harmonics. In the following studies, only the ‘NLM’ case will be considered in all of the simulation implementations.

Fig. 7.

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	Fig. 7. Frequency domain signals obtained using LEM, GNM, and NLM stress–strain constitutive models.

Prior to the numerical simulations, some assumptions should be made for the simulations of plastic-damaged state with dislocations in material. First, only plastic-induced edge dislocations are taken into account in the following simulations just as shown in Fig. 4. According to the dislocation string model presented by Hikata et al.,^[22,30] the change of the acoustic nonlinear parameter resulting from the bowing out of a pinned dislocation under a stress σ₀ can be given by Eq. (8). So, a pinned string (or line) is used to simulate the monopole dislocation in Abaqus system. Second, the pinned string is perpendicular to the direction of wave motion and located in the middle of the simulation area. In addition, the dislocation density is small so that the interaction between dislocations can be ignored.

It can be seen from the microstructural analysis of Subsection 2.3 and Table 2 that the lengths of the pinned dislocations vary from 0.38 μm to 0.16 μm when the 30Cr2Ni4MoV is subjected to plasticity deformation. Therefore, in order to approximately simulate monopole dislocation in Abaqus, the element length should be reduced at least to 0.05 μm that is smaller than dislocation length. For saving computing time and space, localized refined mesh (LRM) with an area of 20 μm× 10 μm and an element length of 0.05 μm is adopted in FEM model described in Subsection 3.2 (here it is called regular model) to model the damaged state with dislocation in the following simulations as shown in Fig. 8. The transition grid is used for the LRM to reduce the element length from 0.05 mm to 0.05 μm as shown in the circled region in Fig. 8. Note that the time step Δt should also satisfy the condition of Eq. (9), and then an auto-increase increment of time step from 5.25 × 10⁻¹²s to 7.19 × 10⁻¹² s is set to guarantee a stable calculation in the simulations for the damaged material.

Fig. 8.

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	Fig. 8. Meshed structure with localized refined mesh (LRM) and a minimum element length of 0.05 μm formed by using transition element.

Three examples of numerical simulations for dislocations with different pinned lengths such as 0.2 μm, 0.3 μm, and 0.4 μm are demonstrated, corresponding to the really measured dislocation lengths as given in Table 2.

Fig. 9.

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	Fig. 9. configurations of the dislocation located in the middle of the LRM model, in which pinned strings are used for modeling different dislocations in the damaged states: (a) 0.2 μm, (b) 0.3 μm, and (c) 0.4 μm. The solid circles represent the pinned points of dislocations with a zero displacement.

Figure 9 shows the configurations of the dislocations located in the middle of the LRM model, in which the displacement of the two pinned points is fixed and set to be zero and the different values of internal stress σ₀ are applied to the nodes of the dislocation line (the values of applied stress can be found in Table 2). All of the other set-up parameters such as excitation condition, loading area and receiving point are the same as those set in Subsection 3.2. Then, the influences of dislocations with different lengths on the response of wave motion can be seen from the frequency domain as shown in Fig. 10. It is noted that although the variation of the receiving signal caused by one dislocation is tiny, the corresponding monotonically increases with σ L⁴ as expected in Eq. (8), which is listed in Table 3.

Fig. 10.

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	Fig. 10. Amplitudes of the fundamental (A1) and second-harmonic (A2) in the frequency domain for simulation of dislocations with different lengths.

Table 3.

Table 3.

Table 3. Values of dislocation simulation for the plastic deformed materials. . The results obtained by LRM model simulation for one dislocation with different lengths L/μm 0.4 0.3 0.2 0.2 Fundamental A1/MPa 0.470 0.489 0.516 0.518 Second harmonic A2/MPa 0.004662 0.001645 0.000765 0.000787 0.02111 0.00686 0.002873 0.002933 Corresponding plastic strain in experiments 0% 1.14% 1.73% 3.89% Corresponding Λ in experiments by TEM/1014 m−2 0.35 1.30 1.94 3.70 Normalization of nonlinearity parameter 1 1.207 0.754 1.468 caused by plastic strain, Eq. (10)

Table 3. Values of dislocation simulation for the plastic deformed materials. .

From Figs. 9 and 10 and Table 3, it can be seen that although the numerical calculation of monotonically increases with dislocation length, the results are not exactly proportional to the value of σL⁴ as described in Eq. (8). In fact, compared with the sophisticated analytical models, here the simple approximation of modeling a single pinned monopole dislocation as a pinned string perpendicular to the direction of wave motion seems to yield the results of proportional to σLⁿ, where n = 3–5.

Note that the above established LRM model represents the damaged state only with one dislocation. Here, in order to extend a one-dislocation result into multi-dislocation state to simulate the practical specimens conducted in the experiments, we can make a correlation of β_D with the simulation result of one-dislocation and the corresponding dislocation density Λ as

Then, the term on the right-hand side of Eq. (10) is normalized by the maximum value, as shown in the last line of Table 3, which can be used to describe the variation of nonlinearity response due to the plasticity damage.

Figure 11 shows the values of normalized by their corresponding initial values, each as a function of the plastic strain for experimental measurement (the measured data of weld zone under stain-controlled mode) and FEM simulation, respectively. The error bars of the standard deviation are determined by repeating the measurements five times. It is worth noting that the result of FEM in Fig. 11 represents the acoustic nonlinear response caused by plastic damage as described by Eq. (8), whereas the experimental result refers to the total acoustic nonlinearity, i.e., β. According to the previous studies,^[8–11] the damage-induced material nonlinearity generally plays a dominant role in the acoustic nonlinearity response. Therefore, it is reasonable to make an approximation i.e., β ≈ β_D in the process of damage loading, especially in the late stage. So, it leads to a good comparability between the experimental result and FEM simulation result after being normalized. From Fig. 11, it is clear that the change tendency of the normalized nonlinear parameter obtained from experimental measurement is in good accordance with that calculated by FEM model, except the third solid circle.

Fig. 11.

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	Fig. 11. Normalized values each as a function of plastic strain for experimental measurement and FEM simulation respectively.

However, the calculated results from the FEM model are underestimated compared with those measured results, especially in the late stage. In this perspective, there may be other microstructures that would contribute to the acoustic nonlinearity during the plastic strain loading, such as the dislocation cells or walls, as marked by arrows in Fig. 4. However, the influence of a dislocation cell on the acoustic nonlinearity has not been considered in this FEM simulation because of lacking proper models to measure the dislocation cell density and to describe the correlation with the acoustic nonlinearity. Another reason for the underestimation of FEM results maybe is due to the fact that the contributions of the mixed dislocations including edge dislocation and screw dislocation to the acoustic nonlinearity have not been considered here in FEM model, which as reported^[19,28] plays a positive influence on the increase of nonlinear parameter β.