Elastoplastic Deformation Research Articles

Towards a real-time simulation of elastoplastic deformation using multi-task neural networks

This study introduces a surrogate modeling framework merging proper orthogonal decomposition, long short-term memory networks, and multi-task learning, to accurately predict elastoplastic deformations in real-time. Superior to single-task neural networks, this approach achieves a mean absolute error below 0.40% across various state variables, with the multi-task model showing enhanced generalization by mitigating overfitting through shared layers. Moreover, in our use cases, a pre-trained multi-task model can effectively train additional variables with as few as 20 samples, demonstrating its deep understanding of complex scenarios. This is notably efficient compared to single-task models, which typically require around 100 samples. Significantly faster than traditional finite element analysis, our model accelerates computations by approximately a million times, making it a substantial advancement for real-time predictive modeling in engineering applications. While it necessitates further testing on more intricate models, this framework shows substantial promise in elevating both efficiency and accuracy in engineering applications, particularly for real-time scenarios.

Large elastoplastic deformation prediction for stiffened plate based on incremental absolute nodal coordinate formulation

Large elastoplastic deformation prediction for stiffened plate based on incremental absolute nodal coordinate formulation

Dynamic properties of alkali residue-based foamed concrete under dry-wet cycles

Alkali residue-based foamed concrete (A-FC) is a lightweight and sustainable building material made from cement, alkali residue, blast furnace slag and foam. The dynamic properties under dry-wet cycles and confining pressure are essential considerations for the engineering application of A-FC. Dynamic triaxial tests were used in this study to explore the influence of dry-wet cycles and confining pressure on the backbone curves, damping ratio, and dynamic elastic modulus of A-FC. A calculation model was established to illustrate the combined effects of confining pressure and dry-wet cycles on the maximum dynamic elastic modulus. Test results indicate that during dry-wet cycles, the coupling effects of temperature, expansion force, and osmotic pressure cause vulnerable pore walls to rupture, pores to become rougher, and microcracks to develop. The backbone curves demonstrate elastoplastic deformation, with no significant change in shape caused by the dry-wet cycles. As the number of dry-wet cycles increases, internal damage accumulates, resulting in a weakening of dynamic deformation ability and a decline in the dynamic elastic modulus. The damping ratio rises with the number of dry-wet cycles once the axial dynamic strain surpasses the transitional strain. Increasing confining pressure can mitigate the adverse impacts of dry-wet cycles on dynamic performance. A-FC demonstrates remarkable durability and favorable dynamic performance, supporting its widespread application in diverse environments.

Bending Properties of Finger-Jointed Bamboo Scrimber Composite Beams

The finger-joint technique is an effective and economical method for producing bamboo scrimber composites for structural engineering and construction applications. This study investigates the failure modes and mechanical strength of finger-jointed bamboo scrimber specimens and composite beams loaded parallel and perpendicular to the finger profile orientation. Results indicate that the primary failure mode in finger-jointed bamboo scrimber specimens is damage to the finger-joint area. In V-type composite beams, primary failure was observed as the separation of laminated boards and finger joints, while in H-type beams, large cracks formed and expanded alongside finger joint damage. No statistically significant difference was observed in the modulus of elasticity (MOE) and modulus of rupture (MOR) between the two types of finger-jointed bamboo scrimber. However, the MOR of the finger-jointed bamboo scrimber specimens decreased significantly, by more than 50% compared to the control, while the MOE increased. The ultimate load capacity and displacement of the V-type beams were higher. Under bending, the V-type beams demonstrated elastic deformation, whereas the H-type beams exhibited initial elastic deformation followed by elasto-plastic deformation. Strain distribution along the height of both beam types remained linear, consistent with the plane-section assumption.

On the multi-physics elastoplastic electrical contact of rough surfaces

Multi-physics coupling effect and elastoplastic deformation of rough surface asperities on contact interfaces are the two critical factors influencing the electrical contact properties. It is challenging to consider both factors in the modeling and efficient computation of rough surface electrical contact problems. To address this challenge, this study proposes a novel and efficient method for the behavior investigation of multi-physics elastoplastic electrical contact of rough surfaces. The electro-thermal and thermo-mechanical coupling equations are derived by incorporating the Joule heating and resultant expansion of material. The J2 plasticity criterion, with both the isotropic and kinematic hardening laws, describes the plastic deformation of contact components. The displacement field caused by the contact pressure of the asperities is calculated using the boundary integral method, while the stresses due to plastic and thermal strains are treated as bulk stresses, and their contribution is solved using the volume integral method. An iterative algorithm is applied to obtain the electrical contact results including electric potential, electrical contact resistance, temperature rise, contact area, contact pressure, displacement, stress, elastic and plastic strains. This method is validated compared with a finite element model. The influence of plastic deformation and multi-physics coupling on the behavior of rough surface electrical contact is investigated.

Surface texture transfer in skin-pass rolling under mixed lubrication

This paper presents a novel integrated approach to modeling surface texture transfer in skin-pass rolling under mixed elasto-plasto-hydrodynamic lubrication (EPHL). The innovation lies in combining discrete fast Fourier transform (DC-FFT) for precise characterisation of elastically deformed asperities on the roll surface, dynamic explicit finite element analysis (FEA) for capturing cross-scale deformations, and a transient average Reynolds equation for governing the lubrication flow. By integrating these methods, the model addresses the complex interplay between elastic roll deformation, microscale asperity-lubricant interactions, and elastoplastic strip deformation, providing a more comprehensive understanding of texture transfer mechanisms. In addition, the model predictions are validated by experimental results. Furthermore, this study investigates the effects of rolling speed and surface pattern orientation, revealing that higher speeds reduce texture transfer while surface patterns aligned with the rolling direction enhance it. These insights demonstrate the potential of this integrated modeling approach for advancing the field of skin-pass rolling.

Mechanism of Internal Crack Formation in All-Solid-State Batteries Due to Expansion and Shrinkage and Its Effects

Si anode, a high-capacity active material, is expected to be used in all-solid-state batteries, which are expected to be put into practical use as soon as possible in terms of safety and output characteristics. However, Si expands about three times during charging, which may cause cracks between the solid electrolyte (SE) and active material (AM). Cracks can cause a break in the Li+ conduction path and a decrease in the interfacial area and can also lead to Li metal precipitation (dendrites), which is a cause of short circuits (1). The mechanism of crack formation due to AM expansion and contraction during charging and discharging and its effects have not been fully elucidated, so in this study, we calculated the void formation in a particle-deposited structure using the discrete element method (DEM). The conductivity and interfacial area that change dynamically due to crack formation were determined, and their effects on battery performance were investigated quantitatively.DEM was used to account for the elastoplastic deformation of sulfide-based SE. Simulations were performed using the equations from previous studies (2), (3). DEM is a method for calculating the forces acting between individual particles and the stress distribution in a particle-filled structure, assuming springs for elasticity and dashpots for viscous damping between contacting particles. SE is a single particle of 75Li2S-25P2S5 and AM is an aggregate consisting of several primary particles of Si with an expansion coefficient of 3.08. The particle diameters are 2 μm and 10 μm, respectively. The spring constants between contacting particles were calculated from the effective Young's modulus and effective radius of each particle. Ion conductivity and coverage changes during anode expansion and contraction were calculated for three phases of porous electrode layers: AM, SE, and void, and their frequency distributions were obtained. The calculations assume that the Si particles expand and contract uniformly. The periodic boundary condition was applied, and calculations were performed for the top and bottom surfaces under three conditions of constant confining pressure (5, 100, and 340 MPa) and a total of four conditions of constant thickness and varying confining pressure.At a constant pressure, a void was formed during the process of AM expansion at the end of charging, while at a constant thickness, a void was formed during the process of AM contraction at the end of discharging. Although these two crack morphologies have been reported in previous studies (4), (5). the formation mechanism, which has not been clarified, can now be clarified by numerical calculations. Based on this structure, the relative conductivity of the SE phase and the coverage between SE and AM were calculated and frequency distributions were obtained. It was found that the pore formation mechanism differs depending on the restraint conditions, and that the effects on the relative conductivity and coverage are different.The mechanism of crack initiation due to AM expansion and contraction during charging and discharging was clarified by DEM calculations for each constraint condition. The influence of cracks was replaced by relative conductivity and coverage, and battery performance could be predicted. In the future, it is necessary to improve the accuracy of the calculation prediction by comparing with the results of model experiments and actual measurements. Through these efforts, we aim to establish the basic technology for the early commercialization of all-solid-state batteries. Furthermore, this calculation technique should be applied not only to all-individual batteries but also to all lithium-ion batteries.This study was supported by JST-Mirai Program (JPMJMI24G1), Japan.References(1) G. McConohy et al., Nature Energy (2023).(2) M. So and G. Inoue et al., MethodsX, 9, 101857 (2022).(3) M. So and G. Inoue et al., J. Power Sources, 546, 231956 (2022).(4) Xiaohan Wu et al., Adv. Ene. Mate., 9, 1901547 (2019).(5) Yamamoto et al., JPS., 473, 228595 (2020).

Local stress/strain field analysis of die-casting Al alloys via 3D model simulation with realistic defect distribution and RVE modelling

The deformation and fracture behavior of die-casting aluminium (Al) alloys is very complex. Due to local variations in properties, the microstructure and mechanical behavior of materials are highly anisotropic. In this paper, an attempt has been made to quantitatively study the defect characteristics of high pressure cast Al alloy parts using experimental and finite element calculation methods, and to analysis the effect of local porosity and pore size on plasticity. Three-dimensional solids with real defect distribution are obtained by using 3D X-ray computed tomography, and used as an input for building a finite element model. The damage initiation of cast Al alloys under complex stress states is analyzed from micro to macro scales. Crack propagation occurs through two modes of microporous agglomeration: aggregated pores produce cracks from internal necking and stress concentration. Then, they expand in the same direction, agglomerate in a specific direction and eventually fracture. Subsequently, the effect of porosity on non-homogeneity was elucidated by obtaining local stress/strain behavior through digital image correlation measurements. Additionally, a theoretical framework of elasto-plastic deformation of microstructures and a 3D representative volume element model are developed to simulate the deformation and damage processes under cyclic boundary conditions of materials. The simulation results show that the local stress/strain around the pores evolves gradually with deformation. During the die casting process, the method demonstrates the ability to predict the mechanical behavior of Al alloys.

Remarkable wear resistance in ceramic-metal composites with composition gradients and regionalized heterostructures

Ceramic–metal composites undergo a predominant mode of wear failure known as brittle intergranular fracture caused by strain localization. In this work, we report for the first time the employment of heterostructures to alter the failure mode of frictional interfaces within such composites. We achieve this by integrating fine-grained tungsten carbide (WC)/graphene nanoplates (GNPs) into the surface layer of coarse-grained WC–11 wt%Co cemented carbides, constructing a gradient heterostructure that simultaneously possessed fine-grained, binder phase, GNPs composition gradients, and regionalized heterostructure of coarse and fine grains. This structure yielded extraordinary wear resistance, outperforming gradient-structured cemented carbides and decreasing the wear rate of the WC-11 wt% Co cemented carbides surface by an order of magnitude. The multi-typed gradient structure improved the resistance of the composite to frictional loads along with fracture resistance. During dry sliding, the heterostructure facilitated strain hardening and transfer by activating dislocations and stacking fault networks. Moreover, the pilling ups of high-density geometrically necessary dislocations of types a, c, and a+c within the constrained coarse-grained WC increased the strain tolerance and promoted plastic deformation uniformity. This synergistic effect enabled the frictional interface to accommodate elastoplastic deformation induced by frictional stresses, thereby preventing localized brittle fractures. The proposed approach paves a novel pathway for designing wear-resistant ceramic–metal composites.

Microstructure-sensitive crystal plasticity and phase-field modeling of deformation and fracture in polycrystalline ice

The formation of crevasses in Greenland and Antarctica is primarily driven by the brittle failure of ice at low strain rates. Understanding this phenomenon requires exploring the effect of microstructural heterogeneity and anisotropic elastoplastic deformation on the fracture behavior of ice. Here, we have developed a microstructure-sensitive coupled crystal plasticity and phase-field model. This model allows us to study the plastic deformation, damage initiation and propagation under various strain rates in hexagonal open-packed polycrystalline ice, the prevailing phase on Earth. By employing a Bayesian optimization method, we derived the material parameters for the micromechanical model based on experimental results. The modeling results reveal that the activation of basal dislocations results in a brittle to ductile transition in ice at a low strain rate of 10-7s-1 under tension. At higher strain rates, the intrinsic plastic anisotropy of ice leads to heterogeneous deformation among grains and stress concentration near grain boundaries, triggering crack initiation and exacerbating the brittleness of ice. Under compression, cracks usually do not propagate throughout the specimen due to a significant decrease in stress around the crack tip upon propagation. Moreover, we found that the formation of the shear-induced basal texture at the bottom of ice layers and along glacier valley sides intensifies the brittleness of ice. This study provides insights into the micromechanical deformation and fracture mechanisms of ice, elucidating the intricate process of crevasse development in polar ice sheets.

Numerical Simulation of Electromagnetic Nondestructive Testing Technology for Elasto-Plastic Deformation of Ferromagnetic Materials Based on Magneto-Mechanical Coupling Effect.

A numerical tool for simulating the detection signals of electromagnetic nondestructive testing technology (ENDT) is of great significance for studying detection mechanisms and improving detection efficiency. However, the quantitative analysis methods for ENDT have not yet been sufficiently studied due to the absence of an effective constitutive model. This paper proposed a new magneto-mechanical model that can reflect the dependence of relative permeability on elasto-plastic deformation and proposed a finite element-infinite element coupling method that can replace the traditional finite element truncation boundary. The validity of the finite element-infinite element coupling method is verified by the experimental result of testing electromagnetic analysis methods using TEAM Problem 7. Then, the reliability and accuracy of the proposed model are verified by comparing the simulation results under elasto-plastic deformation with experimental results. This paper also investigates the effect of elasto-plastic deformation on the transient magnetic flux signal, a quantitative hyperbolic tangent model between Bzpp (peak-peak value of the normal component of magnetic flux signal) and elastic stress, and the exponential function relationship between Bzpp and plastic deformation is established. In addition, the difference and mechanism of a magnetic flux signal under elasto-plastic deformations are analyzed. The results reveal that the variation of the transient magnetic flux signal is mainly due to domain wall pinning, which is significantly affected by elasto-plastic deformation. The results of this paper are important for improving the accuracy of quantitative ENDT for elasto-plastic deformation.

Prediction of huge earthquake-induced deformation of in-service embankments using crushed mudstone as a soil material with slaking and proposal of countermeasures

Evaluating the seismic resistance of embankments using crushed mudstone as a geomaterial is an urgent and crucial requirement. In this study, a ground investigation was conducted on the actual embankment. Based on the results, the seismic response of the embankment, simulating progressed slaking, was conducted by elastoplastic finite deformation analysis using two major types of earthquake motion: epicentral and subduction zone earthquakes. Based on the results of the geotechnical investigation, the embankment could be divided into three layers owing to differences in physical properties, and slaking progressed below the groundwater level in the embankment. The embankment did not exhibit large deformation during the epicentral earthquake owing to the short duration. For the subduction zone earthquake, the developed shear strain was from the two large acceleration groups and the subsequent smaller accelerations, resulting in large deformation. Seismic loading caused the gradual loss of the overconsolidation and decay of the structure which reduced the embankment strength. This analysis revealed that shear strain developed at the slope toe and the lower part of the embankment. Furthermore, the analysis after the earthquake was also conducted to examine whether or not countermeasure method is feasible for emergency restoration. The seismic resistance was greatly improved when a combination of ground improvement and replacement/counterweight fill methods were used to reinforce these areas, which is not only during but also after the earthquake. This study can contribute to the understanding of the seismic behavior of soil structures using materials undergoing internal deterioration and to the development of countermeasure methods for such structures.

Unified hardening (UH) model for saturated frozen soils

The mechanical behavior of frozen soils is highly sensitive to temperature variations due to their complex micro-mechanisms. In regions with seasonal freeze-thaw cycles, the transition of soils between frozen and unfrozen states is significantly influencing their mechanical properties. This study synthesizes existing theories and data to categorize the effects of subzero temperatures on the properties of frozen soils. By introducing temperature and unfrozen water saturation, the phase change component of void ratio—decoupled from stress—is distinguished from actual voids, enabling the definition of the equivalent void ratio. Nonlinear relationships between temperature and other mechanical properties including elastoplastic deformation, cryogenic cohesion and ice segregation are established. Through the derivation of a loading-temperature yield equation, an elastoplastic constitutive model within a dual stress-variable framework of effective stress and temperature is established. This model captures key aspects of the behavior of frozen soils, including strength weakening due to ice segregation and temperature-induced strength changes. Applicable to conditions at or below the pore water melting point, the model's predictions align well with experimental observations.

Anisotropy and symmetry in the elastoplastic deformation of single crystals under scratching: Unravelling the microscopic deformation mechanisms

The nanoscratch test, as an established technique for assessing material tribological properties has received significant attention. However, the symmetry and anisotropy in scratching performances as well as the quantitative correlation between the orientation-dependent deformation and inherent microscopic deformation mechanism remain unexplored. Herein, crystal plasticity simulations can quantitatively capture scratching forces, elastic recovery, and surface pile-ups, as well as accurately describe inner deformation fields and lattice rotation patterns, as confirmed by experimental results. The simulation results reveal that surface pile-up and elastic recovery mappings on (001)-, (011)-, and (111)-oriented samples exhibit eight-fold, four-fold, and six-fold symmetries, respectively. The orientation-dependent location and intension of both slip activities and lattice rotation, determine the features of macroscopic elastoplastic deformation under scratching.

Inertia effect of deformation in amorphous solids: A dynamic mesoscale model

Shear transformation (ST), as the fundamental event of plastic deformation of amorphous solids, is commonly considered as transient in time and thus assumed to be an equilibrium process without inertia. Such an approximation however poses a major challenge when the deformation becomes non-equilibrium, e.g., under the dynamic and even shock loadings. To overcome the challenge, this paper proposes a dynamic mesoscale model for amorphous solids that connects microscopically inertial STs with macroscopically elastoplastic deformation. By defining two dimensionless parameters: strain increment and intrinsic Deborah number, the model predicts a phase diagram for describing the inertia effect on deformation of amorphous solids. It is found that with increasing strain rate or decreasing ST activation time, the significant inertia effect facilitates the activation and interaction of STs, resulting in the earlier yield of plasticity and lower steady-state flow stress. We also observe that the externally-applied shock wave can directly drive the activation of STs far below the global yield and then propagation along the wave-front. These behaviors are very different from shear banding in the quasi-static treatment without considering the inertia effect of STs. The present study highlights the non-equilibrium nature of plastic events, and increases the understanding of dynamic or shock deformation of amorphous solids at mesoscale.

Elastoplastic plate shakes down under repeated impulsive loadings

When subjected to repeated dynamic impacts at identical load level, a metallic monolithic beam/plate may reach a stable state wherein measurable deformation ceases (i.e., shakedown in elastic state) after undergoing a sequence of elastoplastic deformations, which has been termed as “pseudo-shakedown” (P-S) (Jones, 1973; Shen and Jones, 1992). While the response of a single beam/plate under repeated low-velocity impacts has been thoroughly studied, its dynamic behavior under high-velocity impacts, such as explosive or alternating impulsive loads, is difficult to measure experimentally, due mainly to high costs and setup challenges. In the current study, the method of metallic foam projectile impact was employed to produce repeated impulsive loadings on a fully-clamped elastoplastic monolithic plate made of L907A (a Chinese standard shipbuilding steel). Its dynamic responses, including mid-point deflection versus time histories, final deflections, and deformation modes after each impact, were systematically measured. The phenomenon of dynamic shakedown was observed. To further explore this phenomenon, the method of finite elements (FE) was employed to simulate the repeated impulsive impact test, and its prediction accuracy was validated against experimental results. Unlike an elastoplastic (e.g., steel) monolithic plate subjected to repeated low-velocity impacts, which exhibits zero plastic energy dissipation in the P-S (pseudo-shakedown) state, the same plate under repeated high-velocity impacts shows a small level of plastic energy dissipation in the P-S state, mainly due to more extreme loading conditions. The initial impact momentum, yield strength, and tangent modulus of the material the plate is made of significantly affect both the stable deflection in the P-S state and the number of impacts needed to reach it, while the elastic modulus has limited influence. A modified dimensionless impulse loading number, accounting for the strain-hardening effect, is proposed. An approximately linear relationship between stable deflection in the P-S state and impulsive loading is found in dimensionless form.

A Novel Normal Contact Stiffness Model of Bi-Fractal Surface Joints

The contact stiffness of the mechanical joint usually becomes the weakest part of the stiffness for the whole machinery equipment, which is one of the important parameters affecting the dynamic characteristics of the engineering machinery. Based on the three-dimensional Weierstrass–Mandelbrot (WM) function, the novel normal contact stiffness model of the joint with the bi-fractal surface is proposed, which comprehensively considers the effects of elastoplastic deformation of asperity and friction factor. The effect of various parameters (fractal dimension, scaling parameter, material parameter, friction factor) on the normal contact stiffness of the joint is analyzed by numerical simulation. The normal contact stiffness of the joint increases with an increase in the fractal dimension, normal load, and material properties and decreases with an increase in the scaling parameter. Meanwhile, the fractal parameters of the equivalent rough surface of the joint are calculated by the structural function method. The experimental results show that when the load is between 14 and 38 N∙m, the error of the model is within 20%. The normal contact stiffness model of the bi-fractal surface joint can provide a theoretical basis for the analysis of the dynamic characteristics of the whole machine at the design stage.

Low-dose neutron irradiation effects on the elastoplastic deformation mechanisms of aluminum-doped gallium nitride under contact loading

The elastoplastic deformation mechanisms of irradiated aluminum (Al)-doped gallium nitride (GaN) under contact loading are investigated in this work using the nanoindentation simulations, which is of great significance for understanding the mechanical properties of the Al-doped GaN and guiding the design of durable and high-performance GaN-based devices. The mechanical behaviors of the Al-doped GaN with different doping concentrations are analyzed, including the indentation hardness, Young's modulus, elastic recovery rates, phase transformations, and stress distribution. It is found that Al doping increases their hardness, Young's modulus, and elastic recovery rates, and leads to an enlargement of the phase transformation regions, which is dominated by the high coordination number (CN) phase transformations. Furthermore, the effects of low-dose neutron irradiation on their elastoplastic deformation mechanisms are studied by triggering cascade collisions within the structure. When subjected to such irradiation, structural changes occur in the Al-doped GaN, their indentation hardness, Young's modulus, and elastic recovery rates increase remarkably, and its phase transformation mechanism is changed remarkably. The dislocation behaviors of the doped and undoped GaN are different under neutron irradiation. This study is important for capturing the mechanical stability and integrity of Al-doped GaN in an irradiation environment, as well as developing GaN-based devices with superior irradiation resistance.

Plastic deformation and damage modeling of AA7075 synthetic 3D microstructure created using generative AI

3D microstructures provide valuable insight into material behavior which is essential in elucidating microstructural phenomena, such as particle morphology and void damage, and consequent macroscopic material response. However, creating 3D microstructures is extremely laborious and expensive, requiring complex microstructural characterization and imaging techniques such as Focussed-Ion Beam based Scanning Electron Microscopy (FIB-SEM) or X-ray Computed Tomography (XCT). To this end, synthetic 3D microstructures were rapidly generated from orthogonal 2D images using SliceGAN, which proved a practical and cost-effective method. In this study, multiple synthetic microstructures of AA7075-O, a complex microstructure of various strengthening precipitates within a softer aluminum matrix, were post-processed, meshed, and modeled for different damage behavior in FEA using advanced constitutive material models. Subsequently, the synthetic and real microstructures were qualitatively and quantitatively analyzed for their elastoplastic deformation and ductile void damage responses.This study illustrates the viability of an integrated AI-FE methodology in studying microstructural micromechanics, demonstrating that synthetic microstructures exhibited a very similar stress-strain response, especially when using a free boundary condition, and comparable stress distribution and void damage, albeit with some discrepancies. Also, it emphasizes the influence of particle morphology on strength and damage, where highly irregular particles play a dual role in increasing strain hardening by restricting matrix flow at the cost of increased ductile damage induced by decohered particles. Lastly, the more advanced FE models, with multiple voiding mechanisms, reduced the discrepancy between real and synthetic microstructures compared to simpler models

Numerical analysis of small-strain elasto-plastic deformation using local Radial Basis Function approximation with Picard iteration

In this paper, we discuss a von Mises plasticity model with nonlinear isotropic hardening assuming small strains in a plane strain example of internally pressurised thick-walled cylinder subjected to different loading conditions. The elastic deformation is modelled using the Navier-Cauchy equation. In regions where the von Mises stress exceeds the yield stress, corrections are made locally through a return mapping algorithm. We present a novel method that uses a Radial Basis Function-Finite Difference (RBF-FD) approach with Picard iteration to solve the system of nonlinear equations arising from plastic deformation. This technique eliminates the need to stabilise the divergence operator and avoids special positioning of the boundary nodes, while preserving the elegance of the meshless discretisation and avoiding the introduction of new parameters that would require tuning. The results of the proposed method are compared with analytical and Finite Element Method (FEM) solutions. The results show that the proposed method achieves comparable accuracy to FEM while offering significant advantages in the treatment of complex geometries without the need for conventional meshing or special treatment of boundary nodes or differential operators.