Microscale Plastic Deformation Research Articles

Unveiling orientation dependence of anisotropic plasticity in polycrystalline copper by nanoindentation.

The mechanical properties of crystalline materials are influenced by their deformation behavior, which is associated with their microstructural characteristics. Specifically, crystallographic orientation greatly affects the microscale plastic deformation of individual grains. In this study, experiments and finite element simulations of Berkovich nanoindentations are conducted to investigate the impact of crystallographic orientation in polycrystalline copper. A crystal plasticity constitutive model is developed for copper materials, which accurately captures their indentation mechanical response. The results showed that the indentation behavior of polycrystalline copper exhibits a high degree of anisotropy due to significant variation in slip systems for different crystallographic orientations. This results in different mechanical responses of individual grains and distinct material pileup morphologies on the indented surface. Additionally, the study revealed that crystallographic orientation plays a critical role in determining the indentation size effect. These findings have important implications for the design of materials where plasticity is a crucial factor.

The modified GTN model for fracture of nickel-based superalloys considering size effect and healing effect in pulsed current assisted deformation

Gurson-Tvergaard-Needleman (GTN) model is one of the most famous and classical criterions widely used in modelling the ductile fracture process of metal. However, the existence of grain size effect affects the ductile fracture prediction in micro-scaled plastic deformation of nickel-based superalloy ultrathin sheet. In the pulsed current assisted (Pca) forming process, the current has a healing effect on the micro-void of metal due to the existence of local Joule heat, yet the strain has a promotion on the micro-void growth, and the coupled effect between grain size and electrical field is complicated, which makes it difficult to predict the fracture for the traditional GTN model in Pca forming process. To solve this problem, the authors added pulsed current parameters to the GTN model to describe the healing effect of pulsed current on the voids in the material. The parameters of the GTN model were calibrated by the finite element inverse identification method. The modified GTN model was embedded into Abaqus by VUMAT, and the Backward Euler algorithm was employed to integrate the constitutive equation. The validity of the model was verified by comparing the true stress-strain curves with the simulation results and comparing the void volume fraction (VVF) measured by three-dimensional X-ray computerized tomography (3DCT). The model provides a new method for ductile fracture prediction of multiphase alloys under a discontinuous energy field. Meanwhile, it is confirmed that the behaviour of fracture and necking in the Pca forming process is affected by the coupling effect of pulsed current and grain size effect, which provides a new idea for studying the Pca forming process.

Significant improvement in surface hardness of Zr-based metallic glass by nanosecond pulsed laser irradiation in graphite powder water suspension

Metallic glasses (MGs) have shown promising application prospects as structural materials due to their unique mechanical properties. While, for broadening their application as functional materials, significant improvement in surface hardness of MGs is required. Accordingly, in this study, laser carbonization of MGs was attempted by irradiating the Zr-based MG surface immersed in graphite powder water suspension using a nanosecond pulsed laser. Nanoindentation response and surface morphologies of the initial and irradiated surfaces were characterized in detail. The results indicated that the hardness of the laser-irradiated surfaces was increased by 130 %–200 % (from 6.407 GPa to 19.193 GPa), and as well, the serrated flows in load-depth curves and the surface shear bands around the residual indent disappeared, suggesting the change in micro-scale plastic deformation behaviors of the laser-irradiated surfaces. Furthermore, the chemical composition of the initial and irradiated surfaces was analyzed by X-ray diffraction (XRD) and energy dispersive spectrometer (EDS), which showed that the irradiated surfaces were carbonized in the graphite powder water suspension. Accordingly, the increase in surface hardness was due to the formation of the hard crystalline phases.

Influence of silicide and texture on the room temperature deformation size effect of Ti65 alloy sheets/foils and its constitutive model modification

In micro-scaled plastic deformation process such as micro-forming, material size effect is challenging to investigate and reveal using traditional material models. Since high-temperature titanium alloys have become the critical materials of micro-parts depending on outstanding mechanical properties, studying the effect of texture and silicide precipitate on size effect in microtensile deformation of a high-temperature titanium alloy is of great significance because the size effect of highly alloyed titanium alloys has been less studied. In the present study, various heating treatments were performed to regulate grain size and texture and obtain silicide precipitate. The room temperature tensile tests were carried out to investigate the effect of the texture and silicide precipitate on the size effect of plastic deformation behaviour. It is revealed that silicide precipitation in the heat-treated specimens can lead to a more obvious “smaller is weaker” phenomenon and the specimens with transverse textures show a better strength–ductility match. In addition, considering solution strengthening, grain boundary strengthening and precipitation strengthening, the material constitutive model is modified by the effect of silicide and texture on the size effect, allowing for an accurate description of the tensile deformation characteristics at the microscopic scale. The results of the physical experiments and the hybrid constitutive model provide a basis for understanding and further exploration of the microscale plastic deformation behaviour of high-temperature titanium alloys.

Experimental characterization and theoretical modeling of size-dependent distortional hardening behavior of ultrathin metal sheets under multi-axial loading

The research on distortional hardening (DH) behavior of ultrathin metal sheets under multi-axial proportional loading affected by size effect is still missing and existing constitutive models cannot accurately predict the size-dependent distortional hardening (SDDH) behavior of ultrathin metal sheets. To solve the problem and fill the gap, experimental investigation and theoretical modeling of SDDH behavior of ultrathin SUS304 sheets under biaxial proportional loading is conducted in this paper. Firstly, the experimental yield loci of the ultrathin SUS304 sheets with different thicknesses and grain sizes are characterized through biaxial and uniaxial tensile tests. Experimental results show that DH behavior exists during the micro-scale plastic deformation of the ultrathin SUS304 sheets and that the degree of DH is amplified with the increase of the grain size. Subsequently, a mixed hardening constitutive model considering SDDH is proposed based on the Yld2000-2d yield criterion and its effectiveness to predict the evolving yield behavior of ultrathin SUS304 sheets with different grain sizes is verified. In addition, the capability of the developed model to predict the plastic flow behavior of the ultrathin SUS304 sheets under biaxial loading cases is explored. It is found that the prediction accuracy by the developed model is higher than that by the Yld2000-2d yield criterion. The developed constitutive model has a stronger capability than the Yld2000-2d yield criterion combined with isotropic hardening, extensively implemented in commercial software, to enhance the prediction precision of plastic deformation behavior of the ultrathin sheets in micro forming.

The modified GTN-Thomason criterion for modelling of ductile fracture considering shear factor and size effect in micro-scaled plastic deformation

Coarser grains lead to fewer voids on the fracture surfaces of miniaturized workpiece via affecting the void nucleation, while finer grains result in a higher flow strength by influencing void growth. Furthermore, finer grains make the occurrence of normal tensile fracture difficult, but they accelerate the happening of shear-dominated fracture. These insights help the development of ductile fracture criterion by considering the fracture micro-mechanism in micro-scaled plastic deformation. Shear factor and size effect are thus introduced into the newly modified GTN-Thomason criterion to predict the ductile fracture in micro-scaled plastic deformation of copper sheets with different microstructural grain sizes and deformation stress states. The developed criterion is implemented in simulation of deformation process and well validated to be efficient by corroboration of the experimental results and the simulated load-stroke responses and deformation profiles of five sheets with different grain sizes and stress states. According to the accurate simulation results, the influences of stress triaxiality, the normalized third invariant and grain size on void evolution are further analysed. The results show that the increase of the normalized invariant prevents void shear and leads to a better ductility, while the increase of stress triaxiality facilitates the growth of voids to result in a worse ductility. When the deformation with a relatively high normalized invariant has an ignored void shear, the increase of grain size accelerates the onset of void coalescence and the rate of void growth, leading to a rapid failure. However, when the deformation with a relatively low normalized invariant has an unneglected void shear, finer grains increases the rates of void nucleation, growth and shear, causing a smaller fracture strain. These findings and the modified GTN ductile fracture criterion enhance the understanding and prediction of ductile fracture in micro-scaled plastic deformation, respectively, and further facilitate the development of microparts using by deformation-based micro-manufacturing process.

A multiscale constitutive model coupled with martensitic transformation kinetics for micro-scaled plastic deformation of metastable metal foils

The mechanical behavior of metastable austenitic foils at the size scale from micron to submillimeter is strongly affected by the coupling between size effect and strain-induced martensitic transformation (SIMT), which remains to be a pressing issue to be explored. In this research, the focus is on developing a multiscale constitutive model to reveal the mechanical behavior of metastable foils and more accurately predict the size effect on SIMT. In tandem with this, the martensitic transformation and hardening behavior of SUS304 foils with different thicknesses and grain sizes were explored. The results figured out that the SIMT is promoted by the increase in grain size and foil thickness. Furthermore, the onset and end of stages II of work-hardening behavior are advanced and the work-hardening rate in stage II increases faster with increasing grain size and foil thickness. The SIMT kinetic model was coupled with the intermediate mixture law and the iso-work hypothesis to identify the stress-strain relationship of individual austenite and martensite at the surface and interior layers, which was used to construct the multiscale constitutive model. The multiscale model was developed based on the framework of the surface layer model and the intermediate mixture law to represent the coupling between the size effect and the SIMT. Through finite element simulation by using the proposed multiscale constitutive model, the dispersion hardening mechanism in micro-scaled deformation of metastable austenitic foils caused by the non-homogeneous plastic deformation at the interface between austenite and martensite was revealed. The multiscale model was validated via the corroboration of finite element simulation with experiments and therefore can provide a robust analysis of the micro-scaled deformation behavior of metastable austenitic foils.

Surface texture transfer in skin-pass rolling with the effect of roll surface wear

This paper presents a novel three-dimensional characterisation of the wear effects of a roll surface on the texture transfer in skin-pass rolling. The method integrates the interactions of the microscale elastic deformation of roll surface asperities with the microscale plastic deformation of strip surface asperities. An elasticity theory and a discrete fast Fourier transform were used to rationally predict the microscale elastic deformation of work roll in the rolling system. The microscale plastic deformation of the asperities on the soft strip surface was calculated by the material redistribution process. The Archard wear law was used to consider the wear effects with the random contact pressure, relative sliding and surface hardness. A multiscale analysis was carried out by the dynamic explicit finite element method (FEM), which fully coupled microscale surface contacts, macroscale elastic deformation of the work roll and the bulk plastic deformation of the metal strip. Due to the complexity associated with numerical simulation, a novel modularised approach was established to complete the calculation in a feasible and efficient manner. The study reveals that the roughness scale of a rolled metal strip decreases gradually due to the roll surface wear and that the rolling force, relative sliding and rolling speed at the roll-strip interface significantly affect the roll surface wear. It is expected that the method established can be used to control the surface texture transfer in skin-pass rolling under the continuous wear of roll surfaces.

A multiscale experimentally-based finite element model to predict microstructure and damage evolution in martensitic steels

The objective of this work is to investigate the plastic deformation and associated microstructural evolution and damage in a martensitic steel at multiple length scales, using a combination of finite-element (FE) modelling and experimental measurements. A multiscale model is developed to predict damage evolution in the necked region of a uniaxial tensile test specimen. At the macroscale, a von Mises plasticity FE model in conjunction with a Gurson-Tvergaard-Needleman damage model is used to predict the global deformation and damage evolution. A physically-based crystal plasticity model, incorporating a damage variable is used to investigate the microscale plastic deformation behaviour and the changes in crystal orientation under large strains. The model predicts that slip bands form at the onset of plastic deformation and rotate to become almost parallel to the loading direction at large strain. In the necked region, the initially randomly orientated microstructure develops texture, brought about by inelastic deformation and lattice rotation towards the stable [011] orientation. The predicted crystal orientations and misorientation distribution are in good agreement with measurements obtained through electron backscatter diffraction in the centre of the necked region of the tensile test specimens. The experimental and modelling techniques developed in this work can be used to provide information on the evolution of plastic deformation and damage as well as the orientation-dependent crack initiation and microstructural evolution during large deformation of engineering materials.

Effect of the elastic deformation of rolls on the surface texture transfer in skin-pass rolling

This paper investigates the surface texture transfer with elastically deformed work rolls in skin-pass rolling. A numerical approach was developed to efficiently overcome the multiscale challenges associated with the numerical characterisations. To couple the microscale plastic deformation of strip surface asperities with the microscale elastic deformation of roll surface asperities, a specific method was first developed to generate the characteristics between normal contact pressure and interface separation. To deal with the three-dimensional deterministic rough surfaces, a material redistribution scheme was used to predict the plastic deformation of the strip surface asperities and a discrete fast Fourier transform algorithm was employed to efficiently calculate the elastic deflection of the roll surface asperities. These enabled the subsequent multiscale analysis in coupling the microscale interface contact with the macroscale deformation of the metal strip and work roll. As such, the stick/slip transition in the rolling bite due to the elastic deformation of the roll as well as the surface topography of the rolled strip have been quantified. The new method has successfully revealed the effects of elastic roll deformation, roll radius, and strip thickness/roughness on the resultant surface texture of strips in skin-pass rolling.

Size effect on the shear damage under low stress triaxiality in micro-scaled plastic deformation of metallic materials

Micro-forming is one of the major micro-manufacturing methods with promising application potentials, in which the damage response and fracture behavior need to be insightfully addressed. Among all the damage criteria to predict fracture, GTN model is a widely-used one and able to predict void-dominated fracture in micro-scale deformation. However, it is not very applicable under low stress triaxiality and shear-dominated condition. An in-depth understanding of shear damage and its potential size effect are crucial to explore the micro-scaled damage and fracture mechanisms. This research characterizes the size effect on flow stress via employing a combined constitutive model, and an approach for applying a phenomenological shear damage evolution law to the GTN-Thomason model via considering the size effect is developed. The prediction of micro-scaled fracture in a wide stress triaxiality range is thus enabled. Through simulation and experiment, the proposed model is validated and verified. In addition, stress state parameters including stress triaxiality, Lode parameter, and weight function, are also discussed, and the two damage parameters are analyzed quantificationally to reveal different fracture mechanisms occurring in different stress states and grain sizes. The research thus facilitates the physical insight and in-depth understanding of the size effect on damage evolution and fracture formation in micro-scaled plastic deformation of materials.

Surface morphology of the bulk tungsten divertor tiles from JET ITER-like wall

Surface characterization of bulk tungsten tiles (W lamellae) used during the first campaign of JET with the ITER-Like Wall (JET-ILW) was performed by means of microscopy and tritium imaging techniques. This is the first report regarding very detailed structural studies of W lamellae from the JET-ILW divertor. A special feature of the W lamellae installed in JET is the intrinsic network of micro-cracks detected on surfaces of the as-manufactured material. Analyses of different W lamellae samples on the plasma-facing surface reveal two types of surface structures caused by plasma impact: areas with strong erosion and regions of mild plasma interaction. In regions of strong erosion, a thin modified layer (thickness of ∼20 nm) with a high density of defects including bubble-like structures has been formed. In addition, features indicating melting along edges of micro-cracks with the micro-scale plastic deformation have been identified.

Feasibility evaluation of failure models for predicting forming limit of metal foils

Forming limit of metal foil is an important index to evaluate its formability, and is of considerable significance to improve the quality of products. The ductile fracture (DF) behavior in microscale plastic deformation is remarkably affected by the geometry and grain size. To explore the size-dependent forming limit curve (FLC), the Holmberg and Marciniak tests of SUS304 foils with the thicknesses of less than 0.1 mm and diverse grain sizes were performed. In addition, the validity and feasibility of three types of existing failure models including Swift/Hill, Marciniak-Kuczynski (M-K) and DF criteria for predicting the micro-scaled FLCs were discussed. It is found that the Swift/Hill model possesses the worst accuracy with predicting deviation above 50%. Four classical DF criteria including Freudenthal, Ayada, Brozzo and Oh show great difference, and Oh model considering plastic anisotropy presents the best precision. The predicted deviation of M-K model is aggravated with increasing grain size and decreasing foil thickness, which is attributed to the intensified free surface roughening and transformation of fracture mechanism with miniaturization. This research thus provides a deeper understanding and valuable reference for the widespread application of FLC in microforming.

Influence of dead metal zone on dislocation strengthening effect during micro-progressive forming

Micro-scale plastic deformation (microforming) is a promising manufacturing technology in modern industries. To put microformed parts in a more competitive position compared with micro-products made by other micro-manufacturing methods (such as micromachining and 3D printing), micro-progressive forming must be fully studied for the mass production of micro-parts to obtain good grain texture and decrease product costs. Simultaneously, the undiscovered dislocation strengthening effect of micro-scale progressive forming should be investigated. Using micro-universal joint progressive forming as a case study, the influence of the dead metal zone–induced strengthening effect on ductile fracture in micro-progressive forming is discussed. Based on the results of upsetting experiment and finite element simulation, it was found that the specimen produced via metal foil forward-extrusion-blanking process eliminated the ductile fracture when applying the same experiment condition to the specimen produced by micromachining. Based on the micro-indentation and X-ray diffraction results, a dead metal zone–induced dislocation strengthening model was established, and the mechanism for the strengthening effect during micro-progressive forming was revealed. This research thus provides an in-depth understanding of the micro-scale progressive forming process and takes an important step in the field of ductile fracture in multi-stage plastic deformation.

Influence of microstructures on the tensile and low-cycle fatigue damage behaviors of cast Al12Si4Cu3NiMg alloy

The high-performance Al12Si4Cu3NiMg alloy with massive transition elements has been developed recently for piston of diesel engines. The influences of microstructures on the tensile and low-cycle fatigue (LCF) behaviors were investigated with conventional and in situ testing technologies in this study. Upon the tensile loading, the microcracks initiate from the primary Si near the surface when the tensile stress reaches approximately 280 MPa (close to the tensile strength); then, the microcracks grow rapidly through eutectic Si and intermetallics (Al3CuNi) for the gravity casting (GC) alloy. With regard to the LCF, the microcracks initiate from the shrinkage pores regularly because of the localization of fatigue damage in the GC alloy. The shrinkage pores, usually accompanied by the brittle intermetallics (Al3CuNi), typically exhibit more complex geometry. The lamellar Al3CuNi provides a preferred path for the crack propagation after fatigue crack initiation. The ultrasonic melt treatment (UT) was used to optimize the microstructures and reduce casting defects (shrinkage pores), thereby resulting in improvement in tensile and LCF properties. For the UT alloy (pore-free), the microcrack initiates from some coarser primary Si rather than from the shrinkage pores because of the accumulation of micro-scale plastic deformation. Furthermore, higher resistance of crack propagation can also be realized with refinement of microstructures in the UT alloy. Increase in crack initiation and propagation resistances is the main reason for improvement in the LCF life of the UT alloy.

Characterization of the microscale forming limit for metal foils considering free surface roughening and failure mechanism transformation

The failure behavior of metal foils is greatly influenced by size effect and surface roughness. To explore and characterize the effects induced by the two factors on the formability and failure mechanism in microscale plastic deformation, the micro-scaled forming limit of metal foils (μ-FLC) was investigated by physical experiment and theoretical modeling. A sectionalized failure criterion was proposed according to the modified Considère and Parmar-Mellor-Chakrabarty (PMC) models, which considers the coupled effects of free surface roughening and failure mechanism transformation at microscale. In detail, the modified Considère criterion coupled with the free surface roughening was proposed to predict the right-hand-side μ-FLC, while the PMC model by adjusting the surface roughness parameter was developed to construct the left-hand-side μ-FLC. The physical experiment suggests that the magnitude of surface roughness caused by free roughening can be up to 10˜20% of foil thickness until the fracture of the sample. The failure mechanism of metal foils in microscale deformation changes from localized necking to diffuse instability with the transformation of stress state from uniaxial to equi-biaxial tension. Furthermore, the increasing grain size or the decreasing foil thickness promotes the transformation of failure pattern from the localized necking to diffuse instability. The experimentally determined μ-FLC descend with the increasing grain size and the decreasing foil thickness. The original Considère criterion and Marciniak-Kuczynski (M-K) model are inappropriate for the determination of μ-FLC as they do not consider the significant effects of free surface roughening and failure mode transformation on the micro-formability of metal foils. The developed criterion was validated and corroborated by comparing the theoretically determined μ-FLC with the experimental one. All of these findings advance the insight into the ductile fracture and formability of metal foils influenced by size effect and surface roughness and help to improve the microformed product quality and facilitate the applications of microforming technologies.

Plastic deformation and material transfer on steel gage blocks during low force mechanical probing

Contact probing of gaging surfaces is used throughout dimensional metrology. Probe tips such as ruby, sapphire, or diamond are commonly employed as styli for universal length measuring machines (ULMs) and coordinate measuring machines (CMMs) due to the hardness, durability, and wear resistance. Gaging surfaces of gage blocks are precision ground or lapped, with very low surface roughness to enable wringing. Damage or contamination of these surfaces can prevent wringing and lead to measurement error. Experimental investigations using a horizontal ULM and CMM have revealed that even at low force settings (≤0.16 N), probe materials such as ruby and sapphire can cause plastic deformation to hardened carbon chrome steel (such as AISI 52,100) gage block surfaces at the microscale, likely attributed to fretting-associated wear. Under some conditions, permanent transfer of material from the probe stylus to the gaging surface is possible. Results demonstrate irreversible changes and damage to gaging surfaces with repeated probe contact on a ULM and CMM. Optical microscopy, optical profilometry, and scanning electron microscopy (SEM) provide a semi-quantitative assessment of microscale plastic deformation and material transfer. X-ray photoelectron spectroscopy (XPS), Auger electron spectroscopy (AES), and Raman techniques confirm chemical constituency of reference materials used (gage blocks and probes) and also identify makeup of deposits on gaging surfaces following probe contact.

Size effect on plastic anisotropy in microscale deformation of metal foil

Metal foils are extensively used for manufacturing microparts with a high depth-to-thickness ratio by microforming processes. Since there are few grains participating in the deformation of metal foil, the mechanical response of each individual grain intensively affects the deformation behavior and the quality of the formed micropart, which leads to strong anisotropy in microforming. To clarify the size effect on material anisotropy and the impact of scale-dependent plastic anisotropy on the formability of metal foil, a series of tensile experiments and geometrically scaled-down deep-drawing tests of SUS304 foil with different thicknesses of 50–200 μm and grain sizes of 10.2–80.5 μm were performed. The experimental results indicate that the anisotropy of flow stress, yield tensile ratio (i.e., ratio of the yield strength to the tensile strength) and elongation at angles of 0°, 45° and 90° to the rolling direction is prominently enhanced with decreasing thickness and that the anisotropy generally shows a substantial sensitivity to grain size. In addition, the abnormally increased hardening rate of flow behavior for metal foil with few grains in thickness direction arises from the deformation-induced transformation from austenite to martensite. The Lankford values in different orientations, the planar anisotropy and normal anisotropy coefficients decrease with increasing grain size and decreasing foil thickness. The foil manifests strong plastic anisotropy when the grain size is comparable to the foil thickness due to the dominant response of individual grains in terms of crystal orientation, texture, size and shape at a low value of t/d ratio (i.e., the ratio of the thickness to the grain size). Furthermore, several macroscale yield criteria are adopted to characterize the anisotropic phenomenon in the multi-scaled deep drawing process. The results suggest that the Yld2000-2d criterion is more suitable than Hill’48 and Von Mises functions to predict the anisotropic behavior of metal foil with a large number of grains involved in microforming. However, the prediction precision is worsened with decreasing t/d ratio, and the use of Yld2000-2d becomes problematic when the foil thickness is of the same magnitude as grain size, which is closely linked to the interactive effects of grain size, foil thickness and crystallographic texture in microscale plastic deformation.

Micro-scaled plastic deformation behavior of biodegradable AZ80 magnesium alloy: experimental and numerical investigation

Microforming is a promising technology in manufacturing micro parts through plastic deformation. In manufacturing of micro part, grain size and material behavior plays a very important role in material flow, yield strength, surface roughness, and microhardness of the micro part. In this work, a generic microextrusion tooling system is developed and biodegradable AZ80 magnesium with four different grain sizes is experimented to analyze the material behavior and grain size effect based on the punch force, microhardness, and interfacial friction. It is attempted to minimize the interfacial friction using two different die coatings. It is found that size, boundary, and distribution of grains have severe effect in the microextrusion process. Microhardness values at the surface layer and radial location show an adverse effect on overall hardness distribution of the micro part due to inhomogeneous deformation. The quantitative variation in interfacial friction is determined with different friction coefficient by numerical evaluation. This research provides the fundamental understanding about the microextrusion process on AZ80 magnesium and facilitates the development of microimplant.

Determination of optimal residual stress profiles for improved rolling contact fatigue resistance

A theoretical framework is developed for the evaluation of favorable residual stress profiles, suppressing fatigue damage initiation in rolling contact fatigue. Non-metallic inclusions at the microstructure of bearings are one of the most important reasons for fatigue damage initiation since they act as stress risers. In order to evaluate the stress state around such inclusions at the micro-scale, macroscopic stress histories are determined by Hertzian contact theory at different depths below the raceway for a typical roller bearing. These stress distributions are then used as far-field stresses for a micro-scale model accounting for single inclusions of different geometries and orientations. Eshelby’s method is used to relate far-field and local stresses in the vicinity of inclusions. The von Mises stress criterion is then used as a conservative estimator of crack initiation due to micro-scale plasticity. The effect of compressive residual stresses added to the axial and circumferential normal stress components at different depths is analyzed. The von Mises stress field around different inclusions at different depths is investigated in order to determine the most critical case in terms of micro-scale plastic deformation. Finally, an optimization process is carried out in order to determine the residual stresses that minimize the maximum observed von Mises stress as a function of depth.