Study on damage and cracking of Mg-Gd-Y-Ag-Zr alloys during rolling based on experimental and finite element method

Edge cracking, a common issue encountered during the rolling of magnesium alloys, holds substantial importance in determining the success of subsequent finishing processes. It serves as a pivotal parameter for evaluating the formability of rolled plates. In this particular investigation, researchers concentrated on understanding the behavior of edge cracks within the solid solution magnesium alloy designated as Mg-10Gd-3Y-2Ag-0.4Zr (expressed in weight percentage as GWQ1032K). To support this analysis, one delved into the thermal rheological characteristics of the magnesium alloy and established a mathematical relationship connecting rheological stress, strain rate, and temperature. This served as the foundation for a constitutive model tailored to the alloy. Furthermore, practical rolling experiments were conducted to examine how reductions in thickness influenced the morphology of edge cracks in rolled plates. The study also explored shifts in stress–strain behavior and microstructural changes during the deformation process. The results highlighted the substantial impact of compression levels on the magnesium alloy’s anisotropic behavior, subsequently influencing the shape of the resultant plate and the stress–strain characteristics observed during deformation. Significantly, as the rolling reduction increased, a notable increase in heat generation due to the plastic deformation of the magnesium alloy plate was observed. This heightened heat played a key role in dynamic recrystallizationand and facilitating the formation of the brittle Mg5(RE, Ag) phase. Consequently, minimizing the generation of this brittle phase emerged as a critical factor in effectively managing and controlling edge cracks in the rolling process.


Introduction
Owing to the growing emphasis on environmental protection and energy conservation, traditional materials like aluminum alloys and steel have been progressively supplanted in many sectors by a diverse range of new materials offering superior comprehensive properties [1].In recent years, the development of lightweight structural materials has spurred significant attention worldwide [2], prompting numerous countries to focus on the research and development of innovative lightweight alloy materials [3].Presently, magnesium alloys find extensive utility in aerospace [4], automobile manufacturing [5], the defense industry [6], and various other fields.Particularly within the realm of 3C (Consumer Electronics, Computers, and Communications) products -including mobile phones, notebook computers, digital cameras, and compact recording devicesmagnesium alloys have garnered substantial attention due to their promising application prospects in product casings.This underscores the appeal of leveraging our available resources for magnesium alloy research [7].However, despite their potential, the practical application of magnesium alloys as structural materials is hampered by constraints related to material preparation, processing technology, and other factors [8].In recent years, researchers have proposed a plethora of innovative processing technologies for magnesium alloy materials, including traditional drawing, stamping, expansion, and rolling techniques [9] .Meanwhile, significant progress has been made in new methods of magnesium alloy design and new technologies for plastic processing, forming important achievements such as new asymmetric processing technologies [10], reciprocating upsetting-extrusion [11], extrusion forging composite forming technology [12], asynchronous rolling [13], lining plate rolling [14], thick plate lateral rolling [15], continuous bending deformation [16], curved extrusion profiles [17][18][19], and sideways extrusion [20,21] etc, providing important scientific basis and technical support for the further development and application of magnesium alloy plastic processing technology.
Notably, rolling forming technology allows for a more extensive size range of magnesium alloy plates, with the potential for varying thickness specifications by adjusting the size of the working roll and the corresponding roll type.Consequently, the resulting deformed magnesium alloy products exhibit improved mechanical properties, including heightened strength and elasticity [22].Nevertheless, achieving precise control in the rolling process of specific products hinges upon the development of high-precision mathematical control models.These models are pivotal in enhancing control accuracy during rolling and elevating the quality of the rolled product.At present, magnesium alloy rolling technology primarily draws from the rolling theories of steel and aluminum, lacking a comprehensive framework tailored specifically to magnesium alloy rolling processes.The preparation of high-quality magnesium alloy sheets typically necessitates multiple rolling processes, with the impact of rolling shape predominantly influenced by the inherent properties of the metal and the specifics of the rolling process [23][24][25].In pursuit of enhancing the mechanical of magnesium alloys, Mg-RE (Magnesium-Rare earth) alloys have emerged as a high-performance category with promising application prospects, capturing considerable research attention in recent years [26][27][28][29][30]. Scholars have delved into various facets of Mg-RE alloy rolling, investigating its processes, scrutinizing the influence of rolling on the alloy's microstructure and mechanical properties, and ultimately yielding high-strength magnesium alloy plates [31].Metal strip rolling embodies an exceedingly intricate thermodynamic coupling process, marked by substantial deformation and nonlinear behavior.This complexity encompasses material nonlinearity, geometric nonlinearity, boundary contact conditions, and temperature-induced nonlinear changes.The deformation mechanism is intricate and challenging to accurately describe, leading to relatively limited research on the thin strip forming of Mg-Gd alloy [32].Furthermore, for newly developed alloy materials, numerical simulation has emerged as an effective means of predicting parameters such as force and energy variations throughout the rolling process.Numerical simulation not only enables the prediction of force and energy parameters during the rolling process but also facilitates equipment and process optimization, reducing the need for costly on-site test rolling.Moreover, it allows for the simulation of microstructure changes in plates and strips during rolling, prediction of mechanical properties after rolling, and anticipation of macroscopic defects, such as cracks, that may arise during the rolling process.Consequently, finite element technology has undeniably emerged as a critical tool in the study of strip rolling [33].In conventional magnesium plate rolling production, techniques involving hot or warm rolling, with multiple passes at small reductions in thickness, and multiple intermediate annealing processes are commonly employed to minimize the occurrence of edge cracks.Researchers in this field have dedicated considerable effort to investigate and regulate the occurrence of rolling-induced edge cracks, leading to noteworthy findings.Kim [34,35], for instance, identified that an asynchronous rolling process for magnesium plate production can weaken its basal texture, refine the grain size to enhance plastic deformation capacity and mechanical properties, all while minimizing edge cracks and improving plate shape.Xin [36] established that the vertical roll pre-rolling process is highly effective in reducing edge cracks in magnesium plates, representing an optimal approach.This reduction in edge crack occurrence can be attributed to the adjustment of basal texture through the formation of tensile twinning (where the grain axis is oriented perpendicular to the plate surface) during the rolling process, thereby enhancing formability.Furthermore, Wang [22] introduced a mid-rolling process for magnesium plates along with the incorporation of hot liners.Results from this study indicated that this approach reduces temperature drops during rolling and, simultaneously, improves formability and surface quality by altering the stress state on the surface of the rolled parts.
As the most common rolling method used in metal plastic processing, further research is needed on how to achieve dynamic and efficient suppression of edge cracks in production when processing metals with poor plasticity such as magnesium alloys, as well as the matching law between the rolling process and material properties required for edge crack suppression.However, there has been a relative scarcity of research regarding magnesium alloy rolling-induced edge crack simulation and its underlying mechanisms.In light of this backdrop, this paper undertakes an investigation.Classically, both temperature and strain rate are factors affecting mode of deformation in forming metals, and the deformation difficulty of metal materials can be characterized by constructing the relationship between these variables, which is the constitutive model [37,38].The study employs the Gleeble thermal simulation experimental machine to conduct thermodynamic tests on the GWQ1032K alloy at varying temperatures and deformation velocities.The experimental results are continuously refined through mathematical inversion to derive constitutive equations that best match the experimental data, thereby enabling an in-depth analysis of the stress-strain behavior of the alloy.Additionally, appropriate numerical simulation software is selected to calculate the temperature, flow, and stress fields experienced by the GWQ1032K alloy during the rolling process.Through simulation and analysis, the paper explores the dynamic changes in temperature, flow, and stress fields during the rolling process of magnesium alloy plates.Furthermore, it delves into the microscopic relationship between the evolution of microstructure and the impact of alloy damage during the rolling process.

Test method
In this study, the stress-strain behavior of the solid-soluble GWQ1032K alloy was investigated using a Gleeble-3000 thermal simulator.This investigation encompassed a range of compression rates (0.01 s −1 , 0.1 s −1 , 1 s −1 , and 5 s −1 ) as well as heating temperatures (648 K, 673 K, 698 K, and 723 K) with the ultimate goal of determining the alloy's stress-strain curve.The experimental procedures can be summarized as follows: To prepare samples for thermal compression, cylindrical samples with dimensions of Ф12 mm × 15 mm were created using the solid solution alloy.During the thermal compression process, the sample underwent rapid heating from room temperature to the desired test temperature within just 1 min.It was then held at that temperature for 3 min to ensure temperature stabilization.The test concluded upon reaching a strain of 0.8, at which point the sample was quenched with water, as shown in figure 1(a).When using the finite element method for analysis, the experimental roller and workpiece dimensions are modeled in UG.Subsequently, the magnesium alloy was divided into finite elements using the automatic meshing method included in DEFORM-3D software.During the experiment, the time between taking out the specimen from the heating furnace and the start of rolling is relatively short, so the heat loss of the alloy before rolling can be ignored.During the simulation process, the temperature of the billet can be set to a stable value.Meanwhile, since the roller does not participate in plastic deformation, it can be set as a rigid material with thermal conductivity properties.In the established model, the block alloy is divided into 20000 tetrahedral element type grids for the calculation of the rolling process.According to Mei's research, the friction category during the rolling simulation process set as shear friction, with a friction factor of 0.3 [39].The alloy rolling process model and the division of the plate alloy grid are shown in figure 1(b).Additionally, rolling experiments were conducted in a dia.400 high rigidity rolling mill with the following parameters: a rolling temperature of 723 K, a rolling speed of 0.5 rad s −1 , a roll diameter of Φ400 mm and the height of the roller is 900 mm (figure 1(c)), and variable rolling reductions of 20%, 40%, 60%, and 80%.Specifically, the solid solution alloy is cut into a rectangular shape ( specimen with size of 30 × 20 × 2 mm is given in the study) by wire cutting, and then preheat the alloy in a 723 K holding furnace for 30 min before rolling.It is worth noting that in the experiment, the down rolling amount of the alloy was controlled by adjusting the spacing between the rolling mill rollers, and each pass 0.2 mm (10%) reduction.After each down rolling pass, an intermediate annealing was performed for 10 min, and then the next rolling was continued.The alloy was gradually rolled down to thickness of 1.6 mm (20%), 1.2 mm (40%), 0.8 mm (60%), and 0.4 mm (80%), as shown in figure 1(d).Then etched with a self-made corrosive agent comprised of nitric acid (10 ml), oxalic acid (10 g), and water (150 mL).The corrosion process was conducted for a duration ranging from 10 to 20 seconds.The microstructure was analyzed with an OLYMPUS-GX71 optical microscope (OM), a VEGA3 TESCAN scanning electron microscope (SEM) equipped with the energy dispersive spectrometer (EDS, Oxford Inca) and a x-ray diffractometer (D/2000X).

Finite element model establishment
The study employed the Deform-3D finite element software to construct a numerical model based on real-world rolling conditions.A comparative analysis was conducted by comparing the outcomes of actual hot-rolled magnesium plate experiments with finite element calculations.The alloy constitutive equation data acquired through hot compression experiments were incorporated into the material library of the DEFORM-3D software, simplifying the definition of material properties for magnesium plates during rolling simulations.The specific rolling parameters utilized in the finite element software are presented in table 1.The magnesium plate was subjected to loading from six different sides, with no consideration given to variations in the temperature field of the rolls.In order to minimize resource wastage during the plastic deformation of materials, numerical simulation techniques were often required to perform failure analyses on metal materials.Such analyses enable the measurement of cracks and, in some cases, even fractures within materials by utilizing damage values as indicators.In accordance with the real-world rolling conditions, the DEFORM finite element software was employed to create a three-dimensional model depicting the rolling of magnesium alloy plates.This allowed for a numerical simulation of the entire magnesium alloy plate rolling process, resulting in the generation of a distribution cloud representing damage values post-rolling.Throughout the simulation, the alloy's damage model adhered to the Crockroft-Latham fracture criterion, which is activated only when the maximum tensile stress experienced by the alloy surpasses a critical void volume fraction threshold [40].The fracture criterion expression is obtained by integrating the maximum tensile stress along the plastic strain path: Where: C is the damage critical value and indicates that fracture occurs when the maximum stress-strain reaches the C value of the material, f e is the fracture tensile principal strain, ⁎ s is the maximum tensile stress, s is the equivalent stress, e is the equivalent strain increment.
Given that the plate exhibits significant differences in the thickness direction compared to its height and width when analyzing sheet rolling, it is often possible to simplify the problem by treating it as a quasi-plane problem.This simplification allows us to disregard the damage values within the plate's thickness direction.To provide a clearer view of the temperature and stress variations within the alloy material under various deformation conditions, the simulation conceals the rolls, focusing solely on the analysis of the threedimensional spatial model of the magnesium alloy plate.Notably, the alloy demonstrates excellent rolling ductility at a temperature of 723 K. Consequently, the analysis centers on the damage progression of the alloy at this specific temperature.

Results and discussion
3.1.Constitutive model establishment Figure 3 illustrates the true stress-strain curve for high-temperature compression tests conducted on the GWQ1032K alloy at varying strain rates.The data in the figure reveal that the alloy's rheological stress is primarily influenced by the deformation temperature and strain rate.Furthermore, a comparison of the true stress-strain curve data highlights the alloy's sensitivity to both strain rate and deformation temperature.Notably, there is a noticeable evolution in the rheological stress, with the initial deformation stress exhibiting  linearity and a rapid increase.As deformation progresses, the rate of rheological stress growth diminishes, eventually reaching its peak stress.Continuing to increase strain results in a phenomenon where softening (dynamic recrystallization) within the alloy surpasses hardening (work hardening), causing the stress to decrease.When comparing the compressive stress-strain curves of the alloy, it becomes evident that, while maintaining a constant deformation temperature, the curve during the steady-state stage at a high strain rate (1 s −1 ) continuously fluctuates, primarily due to the persistent softening effect post dynamic recrystallization.This leads to a reduction in the driving force for recrystallization, resulting in a relatively weaker effect that cannot balance the work hardening stemming from continuous deformation.Consequently, the work hardening effect surpasses the softening effect of dynamic recrystallization, causing the stress value to rise once more.Once dislocations accumulate to a certain extent, dynamic recrystallization once again takes precedence, causing a decrease in the stress curve.This cyclic process, driven by the competing mechanisms of work hardening and dynamic recrystallization softening, continues, gradually attenuating the stress amplitude, resulting in the stress curve assuming a wavy pattern at this high strain rate.Conversely, at a lower strain rate (0.001 s −1 ), the alloy's stress curve continues to decline after reaching its peak, following the initial linear and rapid stress increase during early deformation stages.This suggests that at lower strain rates, the dominant factor in the alloy is the softening effect of dynamic recrystallization.When strain rate is kept constant, achieving the stress peak requires a more substantial strain at lower temperatures (648 K), whereas only a minor strain is needed at higher temperatures (723 K).In this scenario, the true stress-strain curve remains relatively flat, indicating that higher temperatures enable the magnesium alloy to engage a greater number of cone slip systems, making dislocation slip more accessible than at lower temperatures.Consequently, it becomes easier to reach the critical strain conditions necessary for dynamic recrystallization.Thus, the increase in temperature accelerates the occurrence of dynamic recrystallization in magnesium alloys, resulting in a quicker attainment of stress peaks.Comparing the rheological curves of alloys in different states, it can be seen that the peak stress of the alloy decreases with increasing temperature and increases with increasing strain rate, which were in accordance with the regulation of rheological stress as that of other magnesium alloys [41][42][43].
In order to gain insights into the plastic deformation characteristics of GW103K alloys, a crucial step involves the analysis of the constitutive equations governing this material.When investigating the interplay between deformation temperature (T), strain rate (ε), flow stress (σ), and the association between flow stress and strain rate, the predominant and widely employed constitutive relationship is as follows [44]: Where, A is material-related parameters, Q is activation energy, is , and Z is Z-parameter (temperature compensated strain rate).From the above equation, it can be obtained that the deformation activation energy Q of the material in the process of plastic deformation can be calculated by the following equation:  To obtain the constitutive equation of GWQ1032K alloy, the value of n and lnA in equation (2) need to be solved.By using the least squares method, a linear relationship between lnZ and ln[sinh(ασ) can be fitted, as shown in figure 5.
From equation (1), the rheological stress of the alloy can be expressed as equation (3):

Simulation of rolling stress distribution
During the rolling of the magnesium plate, the stress state within the rolling deformation zone assumes a threeway compressive stress configuration [45].In other words, each cube unit experiences compressive stresses both in the vertical direction due to the upper and lower rolls' reduction and in the horizontal direction, extending along both the rolling and width directions.Friction naturally arises between the magnesium plate and the rolls, leading to the generation of compressive stress in both the rolling and width directions.However, it's important to note that the frictional resistance between the rolled magnesium alloy plate is most pronounced at the plate's edge.As one progresses from the center towards the edge along the width direction, the degree of broad extension deformation intensifies, causing a gradual reduction in compressive stress along the width of the rolled plate until it eventually reaches zero at the edge.Following the principle of volume invariance, this broadening effect leads to a decrease in elongation along the rolling direction, resulting in tensile stress manifesting along this direction, the results are shown in figure 6.It's worth highlighting that edge cracking represents a prominent cause of failure during the magnesium alloy rolling process.As a result, it becomes imperative to investigate the stress distribution characteristics along the edges of the rolled plate.Furthermore, Zhi's research delved into the impact of stress on edge cracks in rolled AZ31B magnesium alloy sheets and confirmed the significant conclusion that the reduction in base texture strength could notably enhance sheet formability [46].
The Deform-3D software was employed to simulate the rolling process of GWQ1032K magnesium alloy plates, considering various reduction amounts, all conducted at a rolling temperature of 723 K, as established in prior research [47].The corresponding test results are visually represented in figure 7. The primary aim of this simulation was to investigate how the internal stresses generated by different rolling reduction levels impact the occurrence of edge cracks in the alloy.Upon examining the image, it is evident that the stress distribution within the alloys varies across different reduction conditions.However, a consistent trend in the stress distribution emerges.This uniformity can be attributed to the intricate interplay of factors such as thermal and mechanical nonlinear interactions, frictional forces, metal flow dynamics, and temperature heterogeneity within the system.The figure illustrates that the regions in close contact between the roll and the alloy exhibit relatively  concentrated stress concentrations during the rolling process.This concentration diminishes gradually as the distance between the alloy and the roller increases.Nevertheless, the actual stress experienced by the metal plate during rolling consistently falls within the range of 500-600 MPa.This phenomenon can be attributed to the application of external force through the fast tangent line between the roller and the specimen, causing the specimen to endure higher stress levels due to the smaller contact area between the specimen and the roller.Notably, the section in direct contact with the roll undergoes the most significant stress due to the extrusion force applied during rolling.As the magnesium alloy plate is rolled, the central portion of the plate experiences higher temperatures compared to the edges.Consequently, the flow velocity at the center surpasses that at the edges, resulting in tensile stress at the plate's edges.When examining the equivalent force field under varying levels of reduction, it becomes apparent that the equivalent force does not linearly increase with the amount of reduction.Instead, it exhibits a complex pattern of initially rising and then declining at different rolling temperatures.For instance, at a 20% rolling reduction, the peak equivalent force at the roller-specimen contact position measures approximately 552 MPa.When the rolling reduction is increased to 40%, the peak equivalent force rises to approximately 595 MPa.However, with a 60% rolling reduction, the peak equivalent force drops to around 556 MPa, and when the reduction is further increased to 80%, the peak equivalent force decreases to about 520 MPa.In summary, the equivalent force in the rolling process remains primarily contingent on the thermodynamic conditions during plastic deformation when the rolling equipment and lubrication conditions remain constant.The thermodynamic conditions often manifest through the temperature effect on deformation speed.Therefore, the essential factors influencing the equivalent force encompass the rolling temperature and the compression level.When the rolling temperature remains constant, the equivalent force increases as the compression level rises.Conversely, when all other conditions remain unaltered, the equivalent force decreases as the actual rolling temperature increases.Consequently, it can be deduced that the complex pattern of the equivalent force rising initially and then declining with increasing reduction is attributed to the combined influence of the temperature effect and reduction.

Simulation of rolling strain distribution
The critical cracking strain serves as a pivotal metric for evaluating the operational performance of materials [48].In the realm of fracture mechanics, critical strain is regarded as a direct factor that profoundly influences the service life of components.In the context of metal plate rolling, macroscopic cracks that appear in magnesium alloy plates result from the significant discrepancy in plasticity between impurities, oxides, and other brittle phases within the material and the plasticity of the magnesium alloy matrix.This discrepancy can readily lead to dislocation accumulation and the formation of stress concentrations, microcrack initiation points, voids, and other defects at these locations.As the degree of deformation increases, these defects continue to propagate, and when the strain reaches the threshold for cracking, the material fractures.Hence, it holds significant importance to simulate the strain distribution during alloy rolling and investigate the relationship between critical cracking strain and deformation process parameters.
Figure 7 illustrates the outcomes of simulating the strain within the alloy under various levels of reduction to scrutinize the effect of strain on edge cracks in the GWQ1032K alloy within the rolling deformation zone.The figure reveals that the strain distribution across the alloy after rolling is relatively uniform.As the reduction level increases, the equivalent effect on the rolled plate gradually intensifies, primarily due to the internal stress concentration within the alloy resulting from increased reduction.Morphologically, the plate undergoes substantial changes in length and thickness but experiences only slight alterations in width.Specifically, when the alloy plate is subjected to a 20% reduction, the strain distribution exhibits significant disparities in the rolling direction, as depicted in figure 7(a).Numerically, the maximum strain measures around 0.36, the minimum merely reaches 0.026, and the average strain value hovers around 0.18.In contrast, when the rolling reduction is elevated to 40%, the average strain value of the alloy experiences a notable increase compared to the 20% reduction, with an average value of 0.48.The maximum and minimum strains on the alloy plate are 0.76 and 0.14, respectively, as displayed in figure 7(b).A comparison between figures 7(a) and (b) reveals a distinct difference in the strain distribution pattern between the two indentations, with the middle part of the alloy undergoing a significantly higher strain when subjected to a 40% reduction.At a 60% rolling reduction, the average equivalent effect increases to approximately 0.92, as evidenced in figure 7(c).At this stage, the maximum and minimum strains recorded on the sample are 1.3 and 0.21, respectively.Subsequently, with an 80% rolling reduction, the alloy's average equivalent effect escalates to 1.8, accompanied by maximum and minimum strain values of 4.0 and 0.86, respectively, as presented in figure 7(d).Post-rolling completion, the strain across the plate predominantly exceeds 1.5, with only slight strain variations observed at both ends.
Analyzing the changes in strain data reveals a consistent pattern: as the rolling reduction increases from 20% to 80%, the average strain value of the alloy steadily rises, displaying a single peak.Furthermore, with increasing reduction, the peak shifts towards the right, the peak's width gradually expands, and the peak itself diminishes in magnitude.This trend signifies that after multiple passes of rolling deformation, there is an overall accumulation of strain within the material, accompanied by a gradual reduction in the gap between the maximum and minimum strain values.This analysis underscores the fact that in cumulative rolling, the subsequent strain builds upon the previous strain without disrupting the plate's structural integrity.Through multiple rolling cycles, the internal strain of the alloy increases, ultimately leading to transformations in the microstructure and mechanical properties of the alloy.Throughout the magnesium alloy rolling process, external force is applied through the tangent line connecting the roller and the cube, causing the deformation stress within the alloy plate to be particularly pronounced due to the limited contact area between the alloy and the roller.During the initial stages of rolling, the spacing between grains within the alloy is relatively substantial.The force exerted at this point compels these grains within the magnesium alloy plate to undergo compression.As the amount of compression applied to the alloy increases, the spacing between the internal grains of the metal diminishes.Consequently, when the force acting between the grains reaches a specific threshold, the microscopic forces within the alloy manifest at a macroscopic scale.In essence, this results in the non-uniform distribution of stress within the deformation zone generated by rolling, which subsequently impacts the strain distribution during the later stages of alloy rolling.This phenomenon fosters the development of edge fissures.

Simulation of rolling edge-cracking behavior
Figure 8 illustrates the damage incurred by GWQ1032K magnesium alloy following rolling at 723 K under various reduction conditions.Examining Figures 8(a) and (b), it becomes evident that at lower rolling reductions (20% and 40%), the overall damage to the plate is relatively minimal, with the lowest damage values occurring at the plate's boundary.Conversely, when the rolling reduction of the alloy is heightened (60% and 80%), the damage within the material at all positions increases.Notably, the damage along the rolling direction undergoes marginal alteration, while the damage perpendicular to the rolling direction steadily escalates, as depicted in figures 8(c) and (d).Furthermore, it's worth noting that under all four reduction conditions, the peak damage resulting from rolling occurs at the boundary position.When the rolling reduction is set at 20%, the damage values are distributed more evenly within the alloy, primarily due to the relatively modest force exerted by the roller.However, it's noteworthy that the damage value at the alloy's boundary section stands at 0.330, a significant increase compared to the internal damage value of 0.0122.Upon increasing the rolling reduction to 40%, the damage within the alloy's interior begins to exhibit smaller values, while the damage within its boundary area surges from 0.330 to 0.515.Notably, regions surpassing the critical value for crack initiation begin to emerge at the alloy's boundary position.Additionally, it becomes evident that the alloy experiences the least damage at the outset and conclusion of the rolling stage.Upon further increasing the rolling reduction to 60%, the distribution of damage values inside the alloy mirrors that observed with a 40% reduction, but with notable value shifts.The trend of edge cracking becomes more pronounced.Finally, at an 80% rolling reduction, the maximum damage value within the alloy experiences a substantial increase, soaring from 0.9 to 270 when compared to the 60% rolling reduction.Moreover, both the boundary and interior damage values within the alloy escalate by more than 200 times.
By analyzing the damage incurred by the GWQ1032K alloy during the rolling process, one can discern that during the initial stages of rolling, the spaces between the grains within the alloy are relatively wide, and the force exerted by the roller on the magnesium alloy remains modest.Consequently, the mutual extrusion effect among the grains within the magnesium alloy plate, under the influence of this stress, remains limited.As a result, the extent of damage remains relatively stable during this initial phase of rolling.However, as the reduction increases, the spaces between the internal grains of the metal become more confined, leading to increased mutual extrusion among the grains.When the force acting between these grains reaches a specific threshold, the microscopic forces within the alloy manifest as macroscopic cracks.In other words, the internal damage within the alloy takes the form of cracks, with the most significant damage occurring at the boundary position.Consequently, it becomes more likely for cracks to manifest at the boundary region during the rolling process.

Analysis of edge-cracking of magnesium alloy plate during hot rolling 3.3.1. Effect of rolling reduction on edge cracks and morphology of magnesium alloy
In the case of rolled alloy plates, the presence and extent of edge cracks have a direct impact on the surface quality of the plate.However, owing to the existence of coarse and irregularly distributed rare earth-rich brittle precipitates within the solid solution rare earth magnesium alloy structure, even after undergoing homogenization heat treatment, sizable precipitated phases remain near the grain boundaries of the alloy [47].These residual precipitates can significantly impair the alloy's plastic deformation capability, making it susceptible to edge cracking during the rolling process [49].Consequently, it is imperative to scrutinize the edge cracks and their morphology in GWQ1032K alloy under various rolling conditions.Figure 9 shows the macroscopic morphology of GWQ1032K alloy with different rolling pressures during rolling at 723 K. Observing the figure, it becomes evident that during rolling at 723 K with a 5% reduction when the total alloy mass is 20% and 40%, the plate experiences some warping, yet no apparent cracks manifest along the plate's edges.However, when the overall reduction of the alloy increases to 60%, a few cracks begin to emerge at the edges.As the rolling reduction reaches 80%, the number of edge cracks on the alloy surges significantly, with these cracks extending into the interior of the alloy.Among these cracks, it becomes apparent that lateral expansion occurs at the material's edges, primarily because it experiences the least transverse compressive stress during the rolling process.When the rolling reduction is minimal, the deformation of the plate remains limited, resulting in an absence of edge cracking, although the plate may still exhibit warping along the rolling direction, which is not conducive to subsequent plastic deformation.As the height of the specimen subjected to reduction gradually decreases, the number and depth of cracks within the depressurized alloy continue to intensify.In summary, with increasing levels of rolling reduction, the specimen's surface demonstrates a consistent trend of edge crack width transitioning from absence to emergence and from small to substantial.According to the simulation findings in figure 8, the red region signifies the point of maximum damage in the rolled plate, marking the initial location where the plate undergoes cracking.The red and yellow sections within the cloud image denote the areas of edge cracking on the rolled plate.As the reduction level increases, it is evident that the width of the red and yellow regions in the damage distribution cloud map gradually expands.At a reduction of 60%, visible cracks emerge at the alloy's boundary, surpassing the critical threshold for edge cracking in magnesium alloys (as documented in Huang's research [50], where this threshold is specified as 0.44 at 723 K).Notably, as the reduction reaches 80%, the cracks propagate inward into the alloy.An analysis of the morphology of the actual rolled workpiece reveals a consistent trend: the actual length of edge cracks on the rolled plate increases with the rise in alloy reduction, aligning with the simulation results (figure 9).Furthermore, the width of the rolling crack closely corresponds to the outcomes of the simulations.

Edge-cracking development as a result of the alloy's microstructure changing
To comprehensively investigate the impact of evolving microstructures on edge cracks during rolling deformation of the alloy, both the microstructure and the morphology of the Mg-RE precipitated phase were analyzed in alloys subjected to different levels of rolling reduction.Figure 2 illustrates the microstructure of the GWQ1032K alloy after undergoing rolling reductions of 20%, 40%, 60%, and 80%. Figure 2(a) reveals that at a 20% reduction, the alloy exhibits a limited presence of twin structures, along with some refined recrystallized grains near certain twins.This occurrence may be attributed to the fact that twin deformation intensifies with cumulative reduction, causing the original coarse grains to break down into numerous lamellar twins.Although dynamic recrystallization is not notably prominent, a considerable number of coarse grains still persist.Upon increasing the reduction to 40%, the number of twins within the alloy multiplies, while the size of the original coarse grains doesn't significantly diminish.Nevertheless, the trend of increasing twin structures under rolling deformation becomes more pronounced, as depicted in figure 2(b).Studies have indicated that twin deformation in magnesium alloys is more likely to occur within the initial coarse grains, explaining the presence of coarse twins when the reduction is set at 40%.As the reduction amount rises to 60%, the alloy experiences a substantial increase in the number of giant twins, a significant surge in recrystallization, and a notable reduction in grain size, as shown in figure 2(c).This shift is primarily due to the gradual replacement of most slender striplike twin structures by dynamic recrystallization grains and secondary twins during the rolling deformation.With a further increase in reduction to 80%, the alloy's grain size continues to refine, while the number of twins have some decreases.This phenomenon is primarily attributed to twins storing a substantial amount of deformation energy, which is absorbed as strain energy, effectively promoting dynamic recrystallization.The regions where twins and secondary twins are present become potential nucleation sites for dynamic recrystallization, as demonstrated in figure 2(d).Additionally, it was observed that the preferential initiation of twins and subsequent comprehensive dynamic recrystallization at lower reductions significantly inhibits the precipitation of the second phase, as evident in figures 2(e) and (f).However, as the influence of twins and dynamic recrystallization in inhibiting precipitate formation weakens with increased cumulative reduction, portions of the Mg-RE precipitated phase begin to form aggregations within the matrix, as depicted in figures 2(g) and (h).Zhu's study on the rolling behavior of ZK60 magnesium alloy found that the combined effect of fully refined dynamic recrystallization and twins significantly inhibits the emergence of microcracks during rolling deformation.In this context, during the rolling deformation process, the sequential development of twinning and dynamic recrystallization plays a substantial role in absorbing strain energy, effectively countering microcrack initiation [51].However, it is observed in this study that as the rolling reduction increases, the cumulative deformation energy and stress concentration during rolling significantly escalate.Consequently, the effectiveness of twinning and dynamic recrystallization in competing to absorb strain energy diminishes.In summary, it is apparent that as the rolling reduction increases, the refinement and dynamic recrystallization of twins reduce the likelihood of edge cracks in rolled alloys.However, as twins become depleted, the alloy's propensity for edge cracking significantly amplifies.
Figure 10 presents the x-ray diffraction patterns of the GWQ1032K alloy subjected to rolling reductions of 20%, 40%, 60%, and 80%.A comparison with the standard magnesium pattern reveals that the crystal plane with the highest intensity of the 20% diffraction peak corresponds to {10-11}, with the second strongest peak indicating {0110}.These findings suggest the absence of a base surface texture in the 20% reduced alloy's structure.However, a significant transformation in macroscopic texture is evident from the diffraction intensities of each crystal face after rolling.In the rolled plate, the three dominant peaks are {0002}, {10-11}, and {10-10}, with the {0002} plane exhibiting the highest diffraction peak.This result signifies the development of a noticeable substrate texture on the rolled plate's surface.Additionally, diffraction spectrum peaks at 21.82°a nd 32.01°correspond to the Mg 5 (RE, Ag) phase, indicating significant alterations in the precipitated phase of the magnesium alloy during hot rolling.Comparing these XRD test results, it becomes evident that changes occur in the precipitated phase alongside grain size and twinning alterations within the alloy during the rolling process.Previous research has indicated that the second phase Mg 5 (RE, Ag) of this study's material typically segregates at grain boundary positions, leading to a substantial increase in Gd and Ag atom concentrations.This phenomenon is primarily attributed to Gd and Ag predominating in the Mg 5 (RE, Ag) phase, coupled with stress concentration stemming from Gd and Ag enrichment at grain boundaries.These factors collectively contribute to the occurrence of edge cracks during rolling.

Ratchet effect edge of hot-rolled magnesium alloy plate
Within the aforementioned region where edge cracks are distributed, it is consistently observed that the maximum damage value manifests right at the extreme edge of the magnesium plate.The edge damage in the magnesium plate follows a periodic distribution pattern under each rolling reduction amount, as depicted in figure 11(d).Along the rolling direction, edge damage occurs intermittently.When rolling the magnesium plate, the stress state within the rolling deformation zone adopts a three-way compressive stress configuration, as illustrated in figure 11(a).In this configuration, each cubic unit experiences compressive stress in the direction of the upper and lower rolls, and this compressive stress extends to both the rolling and width directions.Friction between the magnesium plate and the roll generates compressive stress, as depicted in figure 11(b).Notably, in the backslide area's bite angle, due to the plastic deformation of the magnesium plate under the compressive stress from the roll, a raised section forms, resembling a pawl.As the upper and lower working rollers continuously apply cyclic compressive stress to the magnesium plate during rotation, a ratchet effect is observed during the rolling process, as shown in figure 11(c).Through the analysis of the GWQ1032K alloy during the rolling process and simulating the stress pattern on the alloy, several key insights are gleaned.In the rolling process, the plate-like magnesium alloy undergoes significant geometric changes, and as the rolling process advances, the alloy's internal stress-strain follows a similar pattern.Deformation becomes evident in the thickness direction of the alloy plate and the contact surface with the rolling mill's roller.The downward force within the alloy plate changes with the distance between the roller's centerline and the thickness direction of the alloy plate, with closer proximity to the roller resulting in lower stress-strain.As the roller continues to traverse the magnesium metal plate, the extent of alloy rolling reduction becomes more prominent in the direction of roller movement, leading to macroscopic elongation.Experimental measurements of the alloy corroborate these simulation results, aligning closely with the observed outcomes in experimental settings.

Edge cracking model of hot-rolled magnesium alloy plate
In the initial stages of rolling, when the alloy experiences relatively low pressure, the compressive stress transmitted during the roll's interaction with the alloy tends to seek out the weakest fracture strength within the magnesium alloy.This vulnerability typically resides in the edge of the cast magnesium plate, where brittle phases of the Mg-RE class accumulate.As a result, cracks tend to propagate in the direction of the bite angle along the average orientation of the rolling surface, ultimately giving rise to cyclic cracks along the alloy's edge.Additionally, under hot rolling conditions, the edge temperature of the magnesium alloy plate tends to be lower than that of the central region, leading to localized cooling and cracking at the edge, thereby increasing the damage value along the edge.To analyze the alloy's damage mechanism, the model illustrated in figure 12 was established.Figures 2 and 9 reveal that when the cumulative rolling reduction of the alloy is minimal (20%), the   microstructure exhibits a mixture of large and small grains.This outcome arises because the rolling reduction per pass is relatively small, resulting in insufficient plate deformation.As a consequence, some cast coarse grains within the microstructure remain intact, and the number of twins within the alloy is limited.Upon increasing the rolling reduction to 40%, a substantial number of twins appear within the microstructure, with these twin structures interlocking.This phenomenon primarily stems from the magnesium alloy's limited effective slip mechanisms, with twins aiding in deformation coordination.Furthermore, dynamic recrystallization grains between the boundaries of twin structures exhibit significant growth.When the rolling reduction is further increased to 60%, grain refinement becomes more pronounced, particularly in the central region, and microcracks start to appear.Upon reaching a rolling reduction of 80%, the fine recrystallization grain growth disrupts the twin structures, with twin grains breaking down and transitioning into finer recrystallization grains.Consequently, numerous equiaxed grains emerge within the microstructure.It is evident that a more uniform grain structure can be achieved through multi-pass accumulation when rolling the solid-soluble GWQ1032K alloy.However, with higher levels of applied pressure, the rolling process becomes more prone to causing cracks at the edges of the magnesium alloy plate, ultimately affecting its yield rate.

Conclusions
In this paper, the constitutive equation of the GWQ1032K alloy was established through hot compression tests, and simulations were conducted to replicate its rolling process.The primary focus was on analyzing the impacts of various factors, including the equivalent stress field, strain field, temperature field, and boundary damage within the alloy, under varying rolling temperatures and underpressure conditions.Subsequently, the investigation delved into the behavior of edge cracking in the alloy, yielding the following conclusions: (1) Upon calculation, the constitutive equation for thermal compression of GWQ1032K alloy is: (2) Throughout the rolling process of the GWQ1032K alloy, it was determined that the rolling deformation reduction applied to the GWQ1032K alloy should be limited to 80%.Notably, cracks during the rolling process were most likely to manifest on both vertical sides of the rolled sheet.And the rolling reduction became increasingly significant, causing substantial changes in alloy stress.Nevertheless, the maximum stress experienced by the alloy remained within the range of 500 MPa to 600 MPa.Stress simulation field results indicated that high stress levels occurred at the initial 20% deformation, with a pronounced stress concentration at the plate's side edges.To prevent sample breakage during early deformation stages, it was advisable to round the side edges, as this would substantially alleviate stress concentration before rolling deformation.
(3) Analysis of the strain simulation field revealed that as the depression amount increased from 20% to 80%, the non-uniformity of strain within the plate significantly improved.Consequently, the average strain value of the alloy gradually increased, culminating in a single peak.As the reduction increased to 80%, the minimum strain also reached approximately 0.86, which facilitated dynamic recrystallization.Notably, the peak strain shifted to the right, the peak width expanded, and its magnitude decreased.These observations indicated that after multiple rolling deformation passes, the material's overall strain accumulation increased, while the disparity between the maximum and minimum values gradually diminished.
(4) With increasing pressurization, there was a notable escalation in accumulated deformation energy and stress concentration during the rolling process.Simultaneously, the effectiveness of twin formation, dynamic recrystallization, the formation of the brittle precipitate, as well as their ability to compete with cracks in absorbing strain energy, diminished relatively, ultimately leading to an increase in cracks.In summary, it was evident that as depression increased, the refinement, the formation of the brittle precipitate and occurrence of dynamic recrystallization of twins decreased the likelihood of edge cracks in the rolled alloys.However, once the twin formation process was exhausted, the alloy's susceptibility to edge cracking increased significantly.
Meanwhile, the magnitudes of N and S in equation (2) can be obtained from figure 4. From equation (1), one can see the linear relationship between lnε and lnsinh[ασ], and lnsinh[ασ] and T −1 .So the average value of N and S is 4.340 and 4.718, respectively, so the deformation activation Q of GWQ1032K alloy can be calculated to be 170.417KJ mol −1 .

Figure 12 .
Figure 12.Schematic diagram of the mechanism of rolling edge crack generation.

Table 1 .
The EDS analysis of various types of the phases shown in figure 2.