The use of magnesium (Mg) alloy has been continuously on the rise with numerous expanded application in transportation/aerospace industries due to their lightweight and other areas, such as biodegradable medical implants. It was shown recently that machining can be used to improve the functional performance of Mg-based products/components, such as corrosion resistance, through engineered surface integrity. In this paper, the behavior of AZ31B Mg alloy in cryogenic machining was discussed firstly. The surface integrity can be significantly improved by introducing the ultrafine grained (UFG) layer due to the severe plastic deformation (SPD) effect during cryogenic machining. The mechanisms of microstructure evolution and plastic deformation were analyzed based on the experimental findings in literature. A physics-based constitutive model involving material plasticity and grain refinement is developed based on both slip and twinning mechanisms and successfully implemented in a finite-element (FE) analysis with multiple cutting passes to predict the microstructure evolution by nanocrystalline grain refinement and other improvement of the surface integrity in the cryogenic machining of AZ31B Mg alloy. With a more quantitative assessment, the FE model results are further discussed for grain refinement, changes in microhardness, residual stresses, and slip/twinning mechanism with the apparent SPD taking place due to rapid cryogenic cooling.

## Introduction

Magnesium (Mg) alloys have in recent times attracted great attentions for their low density and good strength-to-weight ratio, with the ever-increasing environmental concerns on increasing the fuel efficiency and reducing emissions by reducing weight [1]. In addition, Mg alloys are persistently explored for clinical use of biodegradable medical implants for their good biocompatible physical properties [2]. The machinability of Mg and its alloys is good in terms of high material removal rate, good surface finish, and extremely low tool wear rate [35]. However, machined and extruded Mg alloys generally suffer from low strength, which mainly results from the coarse grained structure associated with high-temperature processing [6]. The low resistance to wear, fatigue, and corrosion due to undesirable machined surface integrity significantly limit the large application of Mg alloys in component manufacture for use in industrial machines and equipment [7,8].

Cryogenic machining firstly has been considered as an eco-friendly and pollution-free machining technology of difficult-to-machine materials, in order to significantly increase the tool life and reduce tool wear due to the reduction of tool-tip temperature [9,10]. With further studies of this process, the improved surface integrity has been found in the machined surface of Inconel 718, AISI 4140, and 52100 steel [1113]. During cryogenic machining, an UFG surface layer is often formed in the machined surface with the increase of surface hardness and compressive residual stresses due to the SPD. Recently, cryogenic machining has also been conducted on AZ31B Mg alloy to improve the machined surface integrity of mechanical components made by this material [14]. Pu et al. [15] unveiled the significant effect of cryogenic cooling on the surface integrity in the machined surface of AZ31B Mg alloy, in terms of better surface finish, the formation of UFG surface layer, increased microhardness and compressive residual stress. These attributes are critical for the performance on corrosion resistance of Mg alloys [1619]. These past researches indicated that the optimum surface integrity for Mg alloy components can be achieved by cryogenic machining. Hence, accurate process simulations are critically needed to understand the cryogenic machining process, determine the optimum process conditions, and achieve the most desirable surface integrity characteristics.

FE analysis has been performed for cryogenic machining of AZ31B Mg alloys to predict the residual stress in the machined surface layer [14]. Phenomenological material constitutive models, such as Johnson–Cook model, have been mostly used to model the flow stress during FE machining simulation. The Zener–Hollomon parameter-based empirical relationship was calibrated using the experimentally measured grain size. Then, it was embedded in a two-dimensional (2D) FE analysis to simulate the change of grain size during turning of AA7075-T651 Alloy with cutting speed of 180–720 m/min [20,21], Inconel 718 alloy [22], AZ31B Mg alloy [23], and AISI 52100 steel [24]. However, a physics-based grain refinement model is critically needed to better understand the change of grain size during cryogenic machining of AZ31B Mg alloy. Also, a large amount of twinning lamellas have been frequently observed for Mg alloys undergoing SPD, particularly at the early stage of the process [25,26], or in low-strained domain below the machined surface [15,26,27]. However, the predominant twinning response of Mg alloys was rarely accounted in numerical analysis of machining and other SPD processes. With the development of material constitutive model, metallo-thermomechanically coupled material model has been developed and implemented in FE analysis to solve the evolution of microstructure and phase constituents, cutting temperature, chip morphology, and cutting force simultaneously [28].

Dislocation density-based approaches have been applied for constitutive modeling of metals to couple the material dynamic response with the microstructural evolution during complex dynamic deformation processes. Estrin et al. [29] and Tóth et al. [30] presented a set of differential equations to evaluate the dislocation density evolution rates and applied the dislocation density-based material plasticity model to grain refinement in the equal channel angular pressing process. Ding et al. [3133] have adapted this plasticity model in the FE analysis for deformation-based processes. Accurate quantitative predictions have been achieved on the grain size and microstructure evolution of metal materials. The nucleation of dislocations due to deformation, annihilation of dislocations due to dynamic recovery, and interaction of dislocations between the dislocation cell interiors and cell walls were evaluated based on the deformation process state variables.

In this work, a physics-based constitutive material model is developed for AZ31B Mg alloy by considering both slip and twinning responses during cryogenic machining. The material model is implemented in 2D FE analyses with multiple cutting passes to predict the final distributions of various surface integrity attributes under cryogenic cooling. A comprehensive analysis of surface integrity induced by cryogenic machining is performed by a quantitative assessment based on the predicted changes in grain size, microhardness, and residual stresses.

## Principle of Improved Surface Integrity

### Review of the Experimental Analysis.

The principle of the surface integrity change by cryogenic machining is discussed in this section based on a systematic experimental study carried out by Pu et al. from the University of Kentucky [15]. In their study, orthogonal cutting experiments were performed using uncoated C-2 carbide tool inserts. The rake and clearance angle were −7 deg and 7 deg, respectively. The geometry of the tool insert is TNMG 432. The parameters of the grooves are groove width, 1.84 mm; land length, 0.135 mm; groove radius, 1.068 mm; and backwall height, 0.042 mm. Two different edge radii of tool inserts (re) were used in the experiments: 30 μm and 70 μm. When the cryogenic cooling was applied, an Air Products ICEFLY® system was used to inject liquid nitrogen to the machined surface from the clearance side of the cutting tool. The temperature of the AZ31B Mg alloy discs during machining was measured by an infrared (IR) camera. The effect of cryogenic cooling was investigated by performing the orthogonal cutting tests with or without the cryogenic cooling, i.e., cryogenic machining or dry machining. A constant cutting speed of 100 m/min and a feed of 0.1 mm/rev were used for all the test conditions. Table 1 listed all the experimental conditions.

Some of the critical experimental results were summarized in Table 1. For dry machining with an edge radius of 30 μm, the maximum machined surface temperature, Tmax, was found to be 125 °C, while a significant decrease in Tmax was obtained by cryogenic cooling. In the cryogenic condition Cryo-Re30 using the same cutting process parameters, Tmax dropped from 125 °C to 52 °C by 60%. Using a larger tool cutting edge radius of 70 μm, Tmax slightly increased due to more plastic deformation, but it was still as low as 71 °C under cryogenic cooling. Thus, the cooling effect of the application of cryogenic coolant was very significant. In Table 1, the UFG surface layers were also observed in the machined surface due to the SPD effect under cryogenic cooling. With the cutting edge radius of 30 μm, the thickness of UFG layer was about 10–14 μm under cryogenic cooling, while the UFG layer can be barely seen without the cryogenic coolant. Due to the enhanced SPD effect using a larger cutting edge radius of 70 μm, the thickness of UFG layer in cryogenic machining even increased by 50–65% from that of Cryo-Re30.

Figure 1 shows the microstructure below the machined surface (Figs. 1(a)1(c)) and in the chip (Fig. 1(d)) created with a cryogenic machining condition Cryo-Re70. The effect of grain refinement near the machined surface compared with the unaffected area was well characterized by the optical microscope in Fig. 1(a). A featureless microstructure of ultrafine grains can be seen in the topmost layer of about 20 μm in thickness below the machined surface. Below the UFG layer, grain structure was sheared to some extent but quickly recovered to the unaffected microstructure of the bulk material with clear grain boundaries. However, a large amount of twinning lamellas can be found within these coarse grains, located 20 μm below the surface corresponding to a domain not severely strained. Twinning lamellas were observed in the less strained areas for all the other experiment conditions as well. Figure 1(b) shows an SEM image (5000×) of the UFG microstructure, which is a zoom-in view of the location boxed in Fig. 1(a) and shows no obvious grain boundaries. Below the UFG layer, grains were strained and elongated along the cutting direction. To examine the microstructure more closely, an area about 2 μm in width within the UFG layer was studied using AFM as can be seen in Fig. 1(c). Ultrafine grains of 30–60 nm are revealed in this micrograph, which indicates that a machined surface of nanocrystalline grain structure was achieved by cryogenic machining. Comparison of the grain refinement data in Table 1 shows that cryogenic machining using a large edge radius tool induces a thicker UFG layer with nanocrystalline grains than dry machining. Grain refinement to a nanocrystalline grain structure enhances the functional performance of the components such as fatigue life and wear/corrosion resistance. The microstructural alteration inside the serrated chip was also studied as shown in Fig. 1(d). A noticeable fairly thick UFG layer was found near the tool–chip interface, which has a UFG layer thickness comparable to that on the machined surface. Coarse grains with clear grain boundaries still remained in the interior of the chip sawtooth. These structures inside the chip correspond to the less strained area between two shear bands during the formation of serrated chip during cutting.

### Microstructural Evolution.

The microstructural evolution of Mg alloys has been extensively studied for the plastic deformation processes at elevated temperatures above 200 °C and low/quasi-static strain rates. It can significantly affect the material mechanical behavior under different loading conditions. Wang et al. [34] and Liu et al. [35] studied the mechanical behavior of AZ31B Mg alloy at strain rates lower than 5 s−1 and temperature higher than 250 °C. Drastic work hardening can be usually observed firstly, while dynamic recovery was relatively slow. With the increase of true strain, sufficient strain energy was stored to enable dynamic recrystallization (DRX) due to deformation. The onsite of DRX, even the potential subsequent grain growth, provided a softening mechanism after a certain level of strain and resulted in a significant drop of the flow stress to a steady-state level.

To investigated the microstructural evolution of AZ31B Mg alloy at higher strain rate and lower temperature, Ulacia et al. [36] characterized the microstructural evolution under a series of tensile tests over wide range of strain rates (10−3–1.5 $×$ 103 s−1) and temperatures (20–400 °C). For high strain rates in their experiments (5 $×$ 102–1.5 $×$ 103 s−1), generalized DRX was not observed at any temperature. When temperature was greater than 250 °C, only a small amount of recrystallized grains can be observed [37]. These small amounts of recrystallized grains were suggested to be the result of an increasing contribution of rotational recrystallization mechanism in the recrystallized grains due to the limited time for diffusion [36,37]. Li et al. [38] also found the strain softening due to DRX did not occur in the tensile testing of AZ31B Mg alloy at 50–150 °C with strain rates lower than 0.1 s−1.

To study the flow stress behavior of AZ31B Mg alloy for cryogenic machining, Giraud et al. [39] conducted dynamic shear tests on 4 mm thick disks of this alloy using a Gleeble machine below 0 °C. These tests were performed at temperatures ranging from −25 °C to 400 °C and strain rates ranging from 10 s−1 to 5 $×$ 104 s−1. Their results show that for strain rates less than or equal to 4 $×$ 103 s−1, the mechanical behaviors for temperatures less than 200 °C differed from those obtained at over 200 °C. For temperatures less than 200 °C, the alloy deformed by slip and twinning, stress was not sensitive to strain rate, and the softening behavior due to DRX did not occur. For temperatures higher than 200 °C, the alloy exhibited a softening behavior due to DRX, and stress was found to be sensitive to strain rate. However, when strain rate increases to 5 $×$ 104 s−1, the alloy does not exhibit any softening behavior through the whole temperature range from −25 °C to 400 °C. Under such high-strain rate conditions, the time duration during plastic deformation was considered to be significantly shorter than what is required for DRX to occur and develop even at elevated temperatures [36,40].

During the cryogenic machining process, the process temperatures were less than 70 °C and 105 °C on the machined surface and in the chip, respectively; under the dry machining condition, the process temperatures increase to 125 °C and 140 °C on the machined surface and in the chip, respectively. These process temperatures are much lower than the typical DRX starting temperatures of 203–298 °C (0.5–0.6 Tm, melting point) [41,42]. The strain rates during the machining tests with cutting speed of 100 m/min (1.67 m/s) are typically on the order of 105–106 s−1, which are much higher than the strain rates for DRX not to occur found by Giraud et al. [39] and Ulacia et al. [36]. Therefore, it was considered that DRX did not occur during the cutting tests studied in this work.

### Plastic Deformation.

Slip and twinning are the two basic mechanisms of plastic deformation. Two material-dependent factors mainly contribute to determine the mechanism of plastic deformation: the number of slip systems in the crystalline and stacking fault energy (SFE) [43,44]. Mg, a hexagonal closed packed (HCP)-structured metallic material, has a slip plane at {0001} and a slip direction of <1120>, which results in a total of three slip systems. Compared with FCC metals such as copper, due to the lack of slip systems, Mg usually requires higher driven force or elevated temperature to activate the operation of nonbasal systems [25]. Mg also has a relatively low SFE in the range of 60–78 mJ/m2 in comparison with the SFE of greater than 300 mJ/m2 for HCP-structured Titanium. Thus, at room temperature, the low SFE of Mg leads to much more frequent twinning lamellas, which form at the low-strain stage of the SPD process before further strain increase and drastic temperature raise due to heat generation [25,26].

A large amount of twinning lamellas were frequently observed for Mg alloys undergoing SPD, particularly at the early stage of the process [25,26], or in low-strained domain below the machined surface [15,26,27]. Watanabe et al. [40] found the deformation of AZ31B Mg alloy at a high strain rate of ∼103 s−1 proceeds by conventional plastic flow of slip and twinning even at elevated temperatures. With a dynamic strain rate, an increase of twinning has been observed at both room and elevated temperature [26,36]. In cryogenic machining, the process temperature is low due to the cryogenic cooling, and the strain rate is very high due to the cutting speed of 100 m/min. Thus, it is required to model the twinning response of the Mg alloys in the plastic deformation during cryogenic machining.

Besides twinning, the prismatic slip has been found as the predominant deformation mechanism after the early stage of the plastic deformation in the plastic deformation at room temperature with high strain rate greater than 103 s−1 [37,4549]. The occurrence of prismatic slip maintains intergranular compatibility between deforming grains [37,45]. It also carries a large amount of the total strain, especially when basal planes are not favorably oriented for slip [37,47,48]. Therefore, both twinning and slip response should be accounted in the material flow stress model.

## Modeling

### Material Constitutive Modeling.

Mayer et al. developed constitutive models to capture the flow stress behavior of different materials with different crystal structures, taking both twinning and slip response accounted [43,44]. For HCP material, the flow stress when twinning is dominant can be represented as [44]
$σT=σT0+kTd−0.5$
(1)
where σT0 and kT are the athermal portion of the twinning stress and Hall–Petch slope for twinning, respectively. These material parameters were determined by Barnett et al. [50], where kT is a constant of 9.5 MPa mm1/2 and σT0 (MPa) can be fitted as
$σT0=−3.326×10−4T2−4.026×10−2T+41.61,for °C≤T≤200 °C$
(2)

In this model, twinning response is applied when the twinning stress of σT becomes less than or equal to the slip stress of σS [43,44]. Otherwise, the slip response dominates.

The slip response also plays an important role after the early stage of the SPD via cryogenic machining. The slip constitutive response can be calculated with a dislocation density-based material constitutive model developed by authors in this work, which has been successfully used to predict microstructure evolutions and improvement of surface integrity due to SPD for orthogonal cutting processes [31,32]. The model for dislocation density evolution and grain refinement is briefly presented in this section and a more detailed description can be found in Ref. [30]. In the model, a dislocation cell structure is assumed to form during deformation, which consists of two parts, dislocation cell walls and cell interiors. The dislocation cell structure obeys a rule of mixtures. The following describes the dislocation density evolution rates in cell interiors and cell walls, respectively:
$ρ˙c=α*13bρwγ˙wr−β*6bd(1−f)1/3γ˙cr−ko(γ˙crγ˙o)−1/nρcγ˙cr$
(3)
$ρ˙w=β*3(1−f)fbρwγ˙cr+β*6(1−f)23bdfγ˙cr−ko(γ˙wrγ˙o)−1/nρwγ˙wr$
(4)
where the cell interior dislocation density is defined as ρc, and the dislocation density on the cell walls is defined as ρw, which is a sum of statistical dislocations and geometrically necessary dislocations. The first terms on the right side correspond to the generation of dislocations due to the activation of Frank–Read sources. The second terms denote the transfer of cell interior dislocations to cell walls where they are woven in. The last terms in each of the evolution equations represent the annihilation of dislocations leading to dynamic recovery in the course of straining. The dynamic coefficients of dislocation generation (α*), interaction between the cell walls and interiors (β*), and dislocation annihilation (ko) are dislocation evolution rate control parameters for the material. n is a temperature sensitivity parameter, $f$ is the volume fraction of the dislocation cell wall, b is the magnitude of the Burgers vector of the material, d is the dislocation cell size,$γ˙wr$ and $γ˙cr$ are the resolved shear strain rates for the cell walls and interiors, respectively, and $γ˙o$ is the reference resolved shear strain rate. It is assumed that the resolved shear strain rate across the cell walls and cell interiors are equal, i.e., $γ˙wr=γ˙cr=γ˙r$, which satisfies the strain compatibility along the interface between interiors and boundaries. The resolved shear strain rate $γ˙r$can be calculated by the von Mises strain rate $ε˙$ by using $γ˙r=Mε˙$, where M is the Taylor factor.
$τcr=αGbρc(γ˙crγ˙o)1/m$
(5)
$τwr=αGbρw(γ˙wrγ˙o)1/m$
(6)
$τr=fτwr+(1−f)τcr$
(7)
where G is the shear modulus and m is the strain rate sensitivity of the material. Volume fraction f of the dislocation cell wall and total dislocation density ρtot are given as follows:
$f=f∞+(fo−f∞)e(−γrγ̃r)$
(8)
$ρtot=fρw+(1−f)ρc$
(9)

where f0 and $f∞$ are the initial and saturation volume fractions of cell walls, respectively.

As aforementioned, the prismatic slip is the predominant deformation mechanism after the early stage of the plastic deformation at high strain rate and low temperature. In addition, from the microstructure characterization analysis based on Fig. 1, the significant grain refinement occurred within the largely strained top layer. Thus, the grain refinement in this domain can be modeled based on the dislocation density-based grain refinement mechanism. The dislocation cell size, d, is given as
$d=Kρtot$
(10)

where K is a material constant and adopted to be 40 in this study based on the experimental measurement of grain structure generated by the cryogenic machining of AZ31B Mg alloy [15].

The strengthening of material is characterized by the microhardness change (Δh, in GPa), which can be predicted based on the dislocation density evolution due to SPD [51,52]. It is given as
$Δh=khMtαoGbρtot$
(11)

where kh is a constant slope of 1.5 [50], αo is a constant of 0.25 [53], and G is the shear modulus.

To determine the other material model parameters, which are unknown in literature for this material, the dislocation density-based material plasticity model was programmed in matlab to solve Eqs. (3)(11) and the stress–strain curves were simulated at different temperatures and strain rates. The dislocation density evolution rate control parameters, α*, β*, and ko were determined by calibrating the model using the material flow stress experimental data in literature [39,54]. With adapting tunable parameters (α*, β*, and ko) starting from certain initial values, a global minimum in the parameter space was achieved by comparing experimental and calculated data. The optimized material parameters of the dislocation density-based material plasticity model should be able to replicate the stress–strain curves obtained by the material mechanical tests. Flow stress data for Mg alloy AZ31B under the shear tests [39], tensile tests [54], and tensile split Hopkinson bar (TSHB) tests [54] were obtained over a large range of strain rates, from 0.003 s−1 to 1500 s−1, and the temperature range from room temperature to 300 °C, which covers the ranges of temperatures in this cryogenic machining study. The temperature sensitivity coefficients, m and n, were determined to be temperature-dependent as follows [55]:
$m=AT$
(12)
$n=BT$
(13)

The dislocation density model parameters are given in Table 2. The dislocation density-based material plasticity model predicts the stress–strain relationships at different temperatures and strain rates and the simulation results are compared with the experimental data as shown in Fig. 2. The high strain rate experimental data were collected using a TSHB apparatus. Elevated temperature experiments (up to 300 °C) were performed at high strain rates using a radiative furnace mounted on the TSHB apparatus [54]. As can be seen in Fig. 2, the identified model parameters have been proved accurate enough to replicate the stress–strain relationships obtained under the material mechanical tests at different temperatures and strain rates, which validates the dislocation density-based material plasticity model for Mg alloy AZ31B. Although Mg alloy sheets prepared by the rolling process show strong anisotropic behaviors, the isotropic constitutive model was considered in this study because the anisotropy diminished when the strain rate is greater than 1000 s−1 [37]. The material mechanical and thermal properties of Mg alloy AZ31B used in the models are given in Table 3.

### FE Analysis

#### Process Setup.

The FE simulations of the cryogenic machining process were performed with a commercial machining simulation software package advantedge 6.4. The material constitutive model considering the microstructural evolution was implemented using the user-defined subroutines. Figure 3 illustrates the modeling configuration developed in advantedge 6.4. The experimental configuration of orthogonal cutting was modeled using the option of 2D Turning in advantedge.

Advanced features of residual stress and multipass analysis in advantedge were implemented in this model. This residual stress technique implements post-cut analysis, i.e., the relaxation step, after a single-pass or multipass cutting simulation is done. During the relaxation step, the heat is dissipated, the stress fluctuations die out, and then a final state of stress in the workpiece is computed [57]. To predict the residual stresses and microstructural attributes on machined surface accurately, a very fine mesh was generated below the machined surface and in the shear zone, which uses a fully automatic adaptive remeshing algorithm during the cutting simulation. A depth of mesh refinement (dfm) was specified in the simulation setting to determine the thickness of fine-mesh domain below the machined surface. Due to the computation cost, dfm was specified as 0.3 mm which is sufficient for a depth of cut 0.1 mm in the cutting experiments, with a minimum element size constrained as 5 μm. After cutting pass finished, the chip will be automatically removed, followed by thermomechanical relaxation of the workpiece. The maximum number of cutting passes was limited to two for a multipass residual stress analysis. Two-pass cutting simulation was considered to be necessary to predict the final surface integrity accurately, because the final machined surface was produced after multiple cuts experiments. All the solution fields including residual stress, hardness, and microstructural attributes were simulated simultaneously in the machined surface with the fine mesh after multiple cuts.

#### Material Subroutines.

A user-defined yield surface material plasticity model was developed for AZ31B Mg alloy with fortran in advantedge. The constitutive model consisted of subroutines of the dislocation density-based slip response in companion with the twinning response as well as the grain refinement mechanism. Figure 4 shows the algorithm flowchart of the user-defined subroutine used in the 2D cryogenic machining simulations in advantedge. In the beginning of the simulation, the initial process parameters and initial conditions were assigned by the input script of advantedge. The material subroutine would initialize the workpiece domain with the model parameters and initial values of internal state variables. During the simulation, the material subroutines were called at each nodule within the workpiece and chip domain to update the deformation and user-defined solution fields of all internal state variables. The constitutive plasticity model defined before in Sec. 3.1 was implemented to determine the twinning-slip transition by comparing the calculated slip stress with twinning stress. The chip formation and solution fields of strain, strain rate, temperature, etc., would be simulated with the updated material flow stress and thermal properties. Based on the previous analysis, the material can be classified into three states: slip predominant, twinning predominant, and resolved twinning. The resolved twinning, as previously discussed, is considered as the material with twinning predominant at the early stage of plastic deformation, but finally governed by significant prismatic slip. Thus, a deformation mechanism index Mindex (0, 1, or 2) was defined to identify the current material state with different deformation mechanisms at each nodule in each time-step. The index of 0 is defined as slip, 1 corresponds to twinning, and 2 corresponds to resolved twinning. When slip response was firstly determined to be predominant and equivalent plastic strain was further increased to above a critical strain, the deformation mechanism index would be set to 2. The critical strain εr of 0.15 was assumed in this model [39]. The nodules carrying the index of 1 by the end of the simulation would represent the domain with a large amount of twinning structures remained below the machined surface after the whole cutting process was done. The solutions of the dislocation-based grain refinement model were solved over the whole workpiece and chip domain, in the sequence of deformation solution-dependent model parameters, dislocation density evolution rate, dislocation density, microhardness change, and grain size. However, the grain size was only to be calculated and updated at the domain with the Mindex as 2 (predominant slip response by prismatic slip, greater strain with twinning structure resolved). After the current time-step, values of dislocation density, grain size, and other internal state variables would be saved in the register array for next time-step.

#### Two-Pass Simulation.

Figure 5 shows the schematic of a multipass (2-pass) residual stress analysis with advantedge. When the simulation started, the workpiece was initially meshed with coarse elements as can be seen in Fig. 5(a). As the tool started to contact the workpiece during the first cutting pass simulation, a fine mesh was generated by the adaptive remeshing algorithm from the initial coarse mesh within the depth of mesh refinement dfm. The fine-mesh layer kept growing as the tool advanced. As the first pass simulation completed, such a fine mesh was generated along the whole cutting distance, and the chip would be removed for the relaxation step. The simulation results would be then carried over to the second cutting pass simulation. During the second cutting pass simulation, the fine mesh from the previous cut was maintained through the whole cutting distance, with one additional depth-of-cut (0.1 mm) layer of dynamic remeshing moving as the tool advanced as shown in Fig. 5(b). The final machined surface with chip removed after the relaxation steps was also illustrate in Fig. 5(b). Figure 5(c) shows the adaptive remeshing and solution mapping in the chip formation. The mesh was fine inside the shear zone during the chip formation. However, as the sawtooth chip fully formed, the mesh inside the chip became coarser to reduce the computation cost. A solution mapping algorithm was used to map the solution of the state variable fields from a fine mesh to coarse mesh. As demonstrated in Fig. 5(c), result of solution mapping to a coarse mesh lost its high fidelity achieved at a fine mesh to some extent. Hence, extra care should be taken when considering the deformation or microstructural attribute distribution inside the chip.

To investigate the effect of multiple passes on the simulation results on surface integrity, steady-state profiles of shear strain, total dislocation density, grain size, and hardness change were extracted from five paths along the depth from machined surface as shown in Fig. 5(b). Since the simulations reached steady state after about 15% of the total cutting length, five extraction paths were defined at 90%, 75%, 60%, 45%, and 30% of the total cutting length, respectively. The average of these profiles is discussed for one-pass and two-pass simulations. Figure 6 shows the comparison of profiles of equivalent plastic strain, total dislocation density, grain size, and hardness change, which were obtained from a single pass and a two-pass simulation after stress relaxation for the cutting condition Dry-Re70. It can be seen that all the simulation results after two-pass simulation are very close to those from one-pass simulation. The peak values of these state variables are almost the same and the variation in these profiles are very small. These comparisons proved that the simulation results on the machined surface have converged after two cutting pass simulation, and more cutting passes were not necessary to simulate the final surface integrity.

## Simulation Results and Discussions

### Cutting Temperature and Force.

Figure 7(a) shows the simulated temperature contour of cryogenic machining condition Cryo-Re30, in which the IR measured peak surface temperature is labeled. The simulated maximum surface temperature of 59 °C matches well with the IR measurement of 52 °C peak temperature. For the dry machining condition with the same tool edge radius (Dry-Re30), the simulated maximum surface temperature was 131 °C which also agrees well with the measured 125 °C by IR camera. It can be seen in Fig. 7(a) that cryogenic cooling has a limited effect on the chip, which was quite uniform and much higher than the machined surface temperature. This is because the liquid nitrogen coolant was just applied within the cavity between the tool back relief face and machined surface. Figure 7(b) shows the comparison of simulated cutting force and thrust force history compared to the measured average forces for both components. A cyclic cutting force was predicted due to the formation of serrated chips with varying chip thickness. The experimental measured average cutting force was just slightly higher than the medium level of the predicted cyclic force, and the experimental measured average thrust force was almost the same level of the prediction. The simulation results in the process temperature and cutting forces validated the numerical solution developed for the cryogenic machining process.

### Grain Refinement.

Figure 8 shows the predicted deformation and dislocation fields of the 2D cutting simulation of the experiment of condition Cryo-Re30, in terms of (a) equivalent plastic strain, (b) total dislocation density, and (c) grain size. In Fig. 8(a), high strain gradients can be observed near the machined surface. The strain inside the chip is not uniform. The strain along the tool–chip interface is as high as 4. The peak strain along the shear plane where the serrated chip initiated was 2.5, which is much higher than those between the adjacent sawtooth chips. Dislocations accumulate more as strain increases, as can be seen in Fig. 8(b), which shows a similar pattern to the strain not only along the machined surface but also in the chip. Beneath the machined surface, the grain size largely reverses the pattern of the dislocation density distribution, with finer grain sizes near the machined surface and coarser grains in the unaffected bulk material. In the chip, the grain size along chip–tool interface was refined to submicron, similar to that on the machined surface, but quickly increased to the level of the initial grain size in each tooth. This predicted grain size distribution agrees with the grain size change as shown in Fig. 1(d). A similar trend of grain refinement and strain distribution was simulated along the machined surface and chip–tool interface for all the four experimental conditions in this study.

Quantitative assessment of the simulation results was performed for the profiles of equivalent plastic strain, total dislocation density, and grain size along the depth from machined surface. Figure 9 shows the simulated averaged profiles with the cross-sectional microstructure of the machined surface. For cryogenic machining conditions of Cryo-Re70 and Cryo-Re30, the simulation results show high equivalent plastic strains of about 4.2 and 4, high dislocation densities of 4.15 and 3.8 × 1015 m−2, and refined grain sizes of 62 and 64 nm, respectively. The simulated UFG layer thickness is 18 and 14 μm, respectively, which agree well with the measured UFG layer thickness from the experiments. The refined grain size was measured to be 30–60 nm by AFM in the UFG layer on the machined surface for Cryo-Re70. In comparison, the model simulated a grain size was refined to a minimum of 62 nm on the machined surface, which matches well with the experimental measurement. For materials below the UFG layer, the simulated equivalent plastic strain and dislocation density diminished quickly and grain size maintained unchanged as the initial size of 12 μm. These findings are very similar to those observed microstructures in Fig. 9(d), which clearly shows the UFG layer above the unaffected bulk material.

The effects of cryogenic cooling and tool cutting edge on the machined surface integrity are also examined through the comparison of the simulation results and experimental measurements. For the case of Dry-Re30, a slim UFG layer of 2 μm was predicted for a tool edge radius of 30 μm; for the case of Cryo-Re30 with the aid of cryogenic cooling, the UFG layer thickness was predicted to increase drastically to 14 μm. Similarly, for the tool edge radius of 70 μm, cryogenic cooling helps increase the UFG layer thickness from 11 to 18 μm. In addition, by the comparison of equivalent plastic strain and dislocation density, it can be easily found while the greater edge radius introduces more plastic deformation for a thicker affected layer, the cryogenic cooling was the dominant factor of the grain refinement in this machining configuration by introducing more SPD and dislocations.

### Microhardness.

Figure 10 shows the simulated microhardness distribution beneath the machined surface and in the chip for the cryogenic machining condition of Cryo-Re30. Hardness increases greatly on the machined surface and along the tool–chip interface, which has a distribution similar to those for strains and dislocation densities, as can be seen in Figs. 8(a) and 8(b). The initial microhardness of the bulk matrix material was 0.513 ± 0.039 GPa. The predicted microhardness within the top 10 μm was increased to the level of 0.9 GPa (an increase by 75%) on both machined surface and tool–chip interface.

Figure 11 shows the variation of microhardness change along the depth below the machined surface for the four different experimental conditions. The simulated UFG layers under different conditions were highlighted by the rectangular shade in blue or red, corresponding to cryogenic cutting and dry cutting, respectively. Under all conditions, the surface hardness was significantly increased within the UFG layer, while it gradually diminished at a depth of 100 μm. The largest surface hardness of 0.947 ± 0.055 GPa was measured under the condition of Cryo-Re70. The experimental measurements also showed surface hardness could be increased by applying cryogenic cooling or a larger edge radius. Obviously, the cryogenic cooling had a more significant effect on the increase of surface hardness. Compared with the experimental data, all the simulation results successfully captured the trend of microhardness depth profile from the machined surface, as well as the greater effect of cryogenic cooling on the increase of surface hardness.

### Residual Stresses.

Figure 12 shows the simulated circumferential residual stress distribution over the whole cutting length of Cryo-Re70. It was found that the residual stress was compressive on the machined surface. The magnitude of the residual stress continuously decreased along the depth. The darker blue domains at both left and right ends correspond to the stress concentration at the edge corners of the workpiece during relaxation. Quantitative assessments were carried out with profiles extracted within the boxed domain as shown in Fig. 12.

Figure 13 shows the measured and simulated residual stresses in the circumferential and axial directions after both dry and cryogenic machining. Due to a large penetration depth (about 25 μm) of X-ray in Mg alloy, the residual stresses measured on the surfaces were corrected by shifting 12.5 μm below the machined surface. For a 30 μm edge radius tool, compressive circumferential residual stresses were induced as deep as 150 μm below the machined surface under both dry and cryogenic conditions. The peak compressive stress in dry machining was about 40 MPa at a depth of 30 μm below the surface, which was increased to about 50–70 MPa within the same depth by cryogenic machining. For axial residual stress, tensile residual stress was predicted for the dry machining condition of Dry-Re30, which agreed well with the measurement. Tensile residual stress of 37 MPa was generated at 12.5 μm deep in the axial direction. By implementing cryogenic machining, the residual stress turned to compressive with a magnitude of 39 MPa. When the cutting edge radius was increased to 70 μm, the circumferential residual stresses were also compressive for both dry and cryogenic machining, but penetrated 120 μm deeper than those using smaller tool edge radius. For the axial direction, cryogenic cooling made the residual stress more compressive by 50 MPa. Comparing the cryogenic machining results in axial stresses, a much thicker penetration of compressive residual stress was obtained using a larger edge radius. The results revealed a great opportunity to use cryogenic cooling combined with large edge radius tools to induce large and deep compressive residual stresses on the machined surface.

### Discussions on Slip/Twinning Transition.

The transition between slip and twinning during cryogenic machining was investigated with the model. The effect of twinning response can be demonstrated by comparing the subsurface microstructure and texture before and after the cryogenic machining experiments. Figure 14(a) shows the initial microstructure of the specimen before cryogenic machining. Twinning lamellas were found in a surface layer above the dashed line, while no twinning can be seen for the area below. These twinning lamellas were induced by the sample preparation process of milling. Figure 14(b) shows the microstructure produced by cryogenic machining test of Cryo-Re30. A featureless UFG layer existed 10–14 μm just below the machined surface, and no signs of twinning can be found for the topmost 15 μm. However, a large amount of twinning lamellas can be seen from 15 to 80 μm below the machined surface. Comparing the micrographs, cryogenic machining induced a topmost UFG layer accompanied by resolving the twinning lamellas and introduced twinning lamellas in the region from 15 to 80 μm below the surface.

Figure 14(c) further examined the evolution of the crystallographic orientations on the disc surface before (initial) and after machining (Dry-Re30 and Cryo-Re30). The highest peak for all the conditions corresponds to the plane of $(101¯1)$, one of the two major twinning systems of Mg alloys [26]; both of the plane of (0002) and $(101¯0)$, two slip systems of Mg alloys [26]. After the cryogenic machining, the intensity of plane of (0002) was drastically increased, while the intensity decrease in $(101¯1)$ plane from the initial specimen to the cryogenic cut surface indicated the twinning lamellas were reduced by cryogenic machining in the surface layer. The intensity increase in the plane of (0002) and $(101¯0)$ indicated that more slip systems were activated due to high strain rate SPD in cryogenic machining. Also, as the large penetration depth of the X-ray in Mg, the great amount of twinning system signals were picked from the lower less strained domain, making the constant high-intensity peak for the plane of $(101¯1)$ for the condition of cryogenic machining.

Figure 14(d) shows the simulated twinning distribution by the aforementioned deformation mechanism index within an 80 μm thick subsurface domain. Model predicted a 20 μm thick layer with twinning lamellas resolved, while twinning lamellas remained from 20 to 80 μm from the machined surface. The simulated transition level of 20 μm deep was very close to the transition level shown in Fig. 14(b). Therefore, this model can capture the twinning generation and twinning/slip transition very well in the plastic deformation of AZ31B Mg alloy.

## Conclusions

In this study, the behavior of AZ31B Mg alloy in cryogenic machining was discussed firstly. The surface integrity can be significantly improved by introducing the UFG layer due to the SPD effect during cryogenic machining. The mechanisms of microstructure evolution and plastic deformation were analyzed based on the experimental findings in literature. To understand the changes in surface integrity and microstructure, a dislocation density-based constitutive material model of AZ31B Mg alloy was developed to model microstructure evolution and slip and twinning responses. The materials model was implemented in two-pass cryogenic machining FE simulations using a commercial machining simulation software advantedge 6.4, to simulate the microstructural evolution on the surface and in the chip of AZ31B Mg alloy. Compared with the micrographs and experimental measurement from Ref. [15], quantitative assessments were carried out with simulated in-depth profiles for all the solution fields such as temperature, plastic strain, dislocation density, grain size, UFG layer thickness, microhardness change, and residual stress distribution. The peak temperature on the machined surface was below 70 °C with the cryogenic cooling applied. High dislocation densities of 4.15 and 3.8 × 1015 m−2was simulated for the two cryogenic machining conditions. The simulated grain sizes were about 62–64 nm on top of an UFG layer as thick as 18 μm for cryogenic machining condition of Cryo-Re70. Microhardness was significantly increased within the topmost 50 μm layer by as high as 0.4 GPa (∼75%), while it gradually diminished at a depth of 100 μm. Compressive residual stresses were induced along both circumferential and axial directions, as high as 88 MPa and as deep as 0.3 mm from surface for the cryogenic machining condition of Cryo-Re70. Compared with the microstructure and texture below the machined surface before and after cryogenic machining, the FE model also accurately predicts the slip/twinning transition and twinning lamellas resolving in the topmost surface layer of about 20 μm in thickness. The experimental results in previous study [15] and numerical investigation from this study revealed a great opportunity to use cryogenic cooling combined with large edge radius tools to improve surface integrity and microstructure.

## Acknowledgment

The authors gratefully acknowledge the financial support provided for the part of the study carried out at the University of Iowa by the National Science Foundation under Grant No. EPS-1101284 and State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, China under Grant No. MSV201514.

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