Homogeneous Material Research Articles

Technological and Mechanical Studies on Additively Manufactured Cellular Structures Made of the Ultrafuse 316L Composite Filament

Regular cellular materials produced using additive manufacturing (AM) techniques represent a significant development trend in modern engineering materials. Depending on the applied elementary unit cell of topology, it is possible to define the course of the structure deformation process in order to obtain the highest possible energy absorption effectiveness. This feature makes regular cellular structural materials particularly attractive for a number of industrial branches. This paper presents the results of technological trials and the results of experimental studies on the mechanical properties of regular cellular structures made using 3D printing technology under compression test loading conditions. The material used for their production was the Ultrafuse 316L filament and the fused filament fabrication (FFF) technique. The proposed material is a polymer-metal composite with a manufacturing process similar to metal injection moulding (MIM). The adopted research methodology included a feasibility study aimed at achieving material homogeneity and compression tests of the developed and manufactured regular cellular structures under quasi-static loading conditions. Several structural topologies were analysed. Experimental results indicated that regular cellular structures made from 316L steel exhibit high mechanical strength and extensive plastic deformation capabilities. The utilized in this work manufacturing technology used combines the advantages of 3D printing process with the potential of powder metallurgy. The proposed technique of structure manufacturing process is much cheaper than other based on melting metallic powders.

Enforcing interface field continuity to advance finite element approximations in heterogeneous materials

To investigate the discrete field continuity at the interface of inhomogeneous materials, this paper investigates the scattering of electromagnetic waves interacting with a perfect electrical conductor object coated with dielectric materials, employing the finite element method. In the derivation process of the variational formulation, the tangential continuity of electric fields eliminates any discrepancies occurring along the internal interfaces. However, while it is important to maintain tangential continuity of electric and magnetic fields at the interface between different dielectrics, this requirement is not met precisely within discrete space. As a result, approximations can exhibit inaccuracies and potentially fail to fully capture the underlying physical phenomena. To alleviate these issues, we propose an imposition of tangential continuity of the magnetic field within the minimizing functional, thereby ensuring adherence to interface conditions between two dielectric layers. This approach can be naturally applied to a variety of interface problems to enhance the approximation accuracy of interaction models in real-world environments.

Multiscale modelling strategy for predicting fatigue performance of welded joints

This study predicts the fatigue performance of welded joints through a multiscale modelling strategy accounting for material and structural inhomogeneities. An S-N curve and detailed fracture surfaces with distinct beach marks were first derived by uniaxial fatigue tests utilising the designed cruciform welded joints. Considering the intrinsic features of welded joints, a multiscale modelling strategy was proposed to integrate multiple factors, including microstructural variations, strength distributions within the heat-affected zone (HAZ), and the diversity of three-dimensional weld toe shapes. Significantly, a modelling strategy was presented for the first time to simulate the simultaneous initiation, growth, and coalescence of multiple cracks, and was validated against experimental evidence. The results indicate that the proposed strategy can accurately predict both the fatigue strength and the overall crack growth process. Additionally, comparative assessments of single-crack and multiple-crack modelling strategies revealed notably shorter predicted fatigue lives when considering crack coalescence. Overall, this work establishes a multiscale framework for assessing the fatigue performance of welded joints considering both microscopic and macroscopic factors, offering substantial practical implications for engineering applications.

The Effect of the Loading Frequency on the Dynamic Bending Strength of Spruce Wood.

Wood is increasingly being used in construction as an alternative to steel and concrete. As wood is an inhomogeneous material, this has a strong effect on its static and dynamic properties. When timber is used as a load-bearing component, there is a possibility that it will be exposed to unfavourable weather conditions (wind) or dynamic environments (vibrations), leading to fatigue of the material. In this article, the effects of load frequency and load magnitude on the durability of Norway spruce wood (Picea abies) were investigated. The frequencies of 5 and 10 Hz were compared at three load levels of 70%, 80% and 90% of the static breaking force. The research results show that the load magnitude has a major influence on the dynamic strength at the same fatigue frequency. Each increase in load means a lower dynamic strength of the spruce, which is reflected in the load cycles achieved. In addition, the dynamic properties of spruce wood deteriorate with an increasing loading frequency, which is more pronounced at higher loading forces. These test results are the basis for determining the Wöhler curve, which is required as input data for the material properties in numerical calculations to determine the durability of the material.

Homogenization of quasi-periodic conformal architectured materials and applications to chiral lattices

In this study, we propose to extend asymptotic periodic homogenization for non-periodic continuous microstructured media, assuming that the non-periodic geometry (called quasi-periodic) can be designed by a conformal planar transformation of a periodic parent domain architectured media with periodically disposed unit cells. Conformal transformations are shown to play a privileged role in the design of circular macroscopic heterogeneous domains tessellated with non-periodic unit cells, obtained from a periodic parent domain architectured with these unit cells. The conditions for conformal invariance are established, leading to the general form of conformal transformation in their dependencies upon the periodic coordinates. It is shown that any conformal map can be decomposed into the product of an isotropic dilatation function of the first periodic spatial position of decreasing exponential type and a rotation characterized by an angular function linear in the second periodic position. A general theory of quasi-periodic homogenization in the framework of conformal transformations is established for the first time, leading to an expression of the tensor of quasi-periodic moduli which is fully evaluated from the solution of the elasticity boundary value problem posed over the periodic unit cell. The influence of microcurvature distortion of individual unit cells on their effective properties is evaluated. Closed-form solutions are confronted to numerical examples issued from the implementation of circular periodicity in a finite element solver, showing overall a good agreement with the identified homogenized moduli.

Establishing a CAD-CAM workflow for stabilization appliances (Michigan splints).

Stabilization appliances (Michigan splints) are considered well-studied and widely adopted for managing bruxism and temporomandibular disorders (TMDs). Traditionally, these appliances have been fabricated by wax modeling and pressing resin onto casts made from irreversible hydrocolloid or silicone impressions. This article provides a detailed description of an all-digital workflow that uses intraoral scanning and computer-aided design (CAD) software to design a stabilization splint on a digital cast that can be manufactured autonomously by a computer-aided manufacturing (CAM) grinding machine in a subtractive procedure. The workflow is applicable to both dentists and technicians. Special attention is given to aspects and procedures that are important for the successful fabrication of the splint. Working without a cast can save time and money, and the use of CAD-CAM technology provides a homogeneous splint material quality.

Study on forming process and performance of nano-La2O3 reinforced tungsten-based composites by selective laser melting

In this study, La2O3 nanoparticles were incorporated into a tungsten (W) matrix to enhance the microstructure and properties of pure W. The selective laser melting (SLM) process for W-2wt% La2O3 was optimized, and the influence of SLM processing parameters on the relative density and internal defects of the samples was examined. Challenges such as uneven powder layering and layer-by-layer thickening during the SLM process were addressed by incorporating a homogeneous material support formed with low laser energy density beneath the sample. This approach widened the process window for sample formation, resulting in a 2.38 % increase in the average relative density of supported samples compared to unsupported ones.With optimized parameters of 240W laser power, 200 mm/s scanning speed, and 80 μm hatch spacing, the sample achieved a relative density of 98.32 % with minimal internal pores, though some microcracks remained. Further investigation revealed that surface roughness decreased with increasing laser power and scanning speed. High laser power combined with low scanning speed yielded good surface quality. Under conditions of high relative density (over 96 %), samples processed with low laser power and high scanning speed (200W, 200 mm/s) exhibited fine and uniform grain morphology, few internal defects, and excellent mechanical properties, with a microhardness of 474HV and a compressive strength of 2237 MPa.

On anisotropic local bifurcation based on hypoplastic model

The incipience of shear band in homogeneous granular materials is well captured by the bifurcation analysis. Most bifurcation analyses are based on isotropic plastic constitutive models. In this paper, a hypoplastic constitutive model is presented by considering a fabric tensor for inherent anisotropy. Based on this model, we carry out bifurcation analysis for the plane strain case, and then extend the analysis to consider general three-dimensional stress state. The theoretical results are compared with experiments on sand conducted using a plane strain device and a true triaxial device. It's indicated the salient features of stress-strain behaviour and shear band formation are well captured by our analyses.

Why heterogeneous cloud particles matter. Iron-bearing species and cloud particle morphology affect exoplanet transmission spectra

The possibility of observing spectral features in exoplanet atmospheres with space missions like the James Webb Space Telescope (JWST) and Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL) necessitates the accurate modelling of cloud particle opacities. In exoplanet atmospheres, cloud particles can be made from multiple materials and be considerably chemically heterogeneous. Therefore, assumptions on the morphology of cloud particles are required to calculate their opacities. The aim of this work is to analyse how different approaches to calculate the opacities of heterogeneous cloud particles affect the optical properties of cloud particles and how this may influence the interpretation of data observed by JWST and future missions. We calculated cloud particle optical properties using seven different mixing treatments: four effective medium theories (EMTs; Bruggeman, Landau-Lifshitz-Looyenga (LLL), Maxwell-Garnett, and Linear), core-shell, and two homogeneous cloud particle approximations. We conducted a parameter study using two-component materials to study the mixing behaviour of 21 commonly considered cloud particle materials for exoplanets. To analyse the impact on observations, we studied the transmission spectra of HATS-6b, WASP-39b, WASP-76b, and WASP-107b. Materials with large refractive indices, like iron-bearing species or carbon, can change the optical properties of cloud particles when they comprise less than 1<!PCT!> of the total particle volume. The mixing treatment of heterogeneous cloud particles also has an observable effect on transmission spectroscopy. Assuming core-shell or homogeneous cloud particles results in less muting of molecular features and retains the cloud spectral features of the individual cloud particle materials. The predicted transit depths for core-shell and homogeneous cloud particle materials are similar for all planets used in this work. If EMTs are used, cloud spectral features are broader and the cloud spectral features of the individual cloud particle materials are not retained. Using LLL leads to fewer molecular features in transmission spectra than when using Bruggeman.

Optimising structural health monitoring systems: A 2D numerical investigation on impedance matching for FML with integrated sensors

AbstractFibre metal laminates (FML) represent an innovative class of advanced composite materials that integrate the mechanical properties of both metals and fibre‐reinforced composites (FRP). Combining the strength and ductility of metals with the lightweight and high stiffness of FRP and FMLs have emerged as new material compositions for applications in chemical, nuclear, automobile, and aerospace engineering disciplines. Structural health monitoring (SHM) using guided ultrasonic waves (GUW) is the state‐of‐the‐art for non‐destructive testing of thin‐walled structures. When applied to FML, SHM plays a crucial role in monitoring the integrity over time and detecting potential damage such as delamination, fibre breakage, or other structural anomalies. In SHM with GUW, a wave‐field is emitted by actuators. This wave‐field can be affected by damage in the structure, thereby changing its propagation characteristics. Sensors monitor the interaction between damage and GUW, which can be utilized to locate and classify the damage and ascertain the overall health state of the structure. In this study, an advanced integration of measurement hardware, that is, sensors and actuators, within the laminate structure is investigated. Sensor integration into FML allows for improved and more sophisticated monitoring capabilities in comparison to measuring techniques like laser vibrometers, which are limited to measuring displacements on the surface of the structure. However, the integration of sensors and actuators yields the technical difficulty of distorting the wave‐fields and may result in an over‐ or underestimation of the damage. Similar to damage, the distortion of the wave‐field is caused by the changes in acoustic impedance resulting from different material properties. In a previous study, incorporating a functionally graded artificial interphase through acoustic impedance matching between the sensor and host material showed notable and significant outcomes. The current contribution extends the prior graded artificial interphase for an isotropic homogeneous material to an FML structure. This paper presents a comprehensive numerical simulation study on a two‐dimensional model of FML with integrated sensors. The interphases are designed based on impedance matching, which improves signal transmission and reduces disturbing reflections. The conducted investigations hold for several interphase configurations for a wide frequency range. The optimised integration of sensors demonstrates promising results for enhancing the reliability and accuracy of SHM systems. This research serves as a foundation for further experimental validation and the development of advanced sensor‐integrated FML structures with improved monitoring capabilities.

Influence of MXene Interlayer Spacing on the Interaction with Microwave Radiation

AbstractThe origin of MXene's excellent electromagnetic shielding performance is not fully understood. MXene films, despite being inhomogeneous at the nanometer scale, are often treated as if they are compared to bulk conductors. It is reasonable to wonder if the treatment of MXene as a homogeneous material remains valid at very small film thickness and if it depends on the interlayer spacing. The goal of the present work is to test if the homogeneous material model is applicable to nanometer‐thin Ti3C2Tx MXene films and, if so, to investigate how the model parameters may depend on variations in MXene interlayer spacings. MXene films containing flakes with interlayer spacing between 1.9 and 5.5 Å have been prepared using various intercalating agents. It is shown that modeling the films as being homogeneous agrees with experimental tests in the microwave frequency range. Microwave conductivity and dielectric constant parameters are estimated for the homogeneous film model by fitting measured results. The direct current (DC) conductivity matches the estimated microwave conductivity on the order of magnitude. A highly effective dielectric constant provides a good fit for the lower conductivity MXene films. Optical absorption agrees with the homogeneous material model of thin films as well.

Lithium Volatilization and Phase Changes during Aluminum-Doped Cubic Li6.25La3Zr2Al0.25O12 (c-LLZO) Processing

Stabilized Li6.25La3Al0.25 Zr2O12 (cubic LLZO or c-LLZO) is a Li+-conducting ceramic with ionic conductivities approaching 1 mS-cm. Processing c LLZO so that it is suitable for use as a solid state electrolyte in all solid state batteries, however, is challenging due to the formation of secondary phases at elevated temperatures. The work described in this manuscript examines the formation of one such secondary phase La2Zr2O7 (LZO) formed during sintering c LLZO at 1000 °C. Specifically, spatially resolved Raman spectroscopy and X-ray Diffraction (XRD) measurements have identified gradients in Li distributions in the Li ion (Li+)-conducting ceramic Li6.25La3Al0.25 Zr2O12 (cubic LLZO or c-LLZO) created by thermal processing. Sintering c-LLZO under conditions relevant to solid state Li+ electrolyte fabrication conditions lead to Li+ loss and the formation of new phases. Specifically, sintering for 1 h at 1000 °C leads to Li+ depletion and the formation of the pyrochlore lanthanum zirconate (La2Zr2O7 or LZO), a material known to be both electronically and ionically insulating. Circular c-LLZO samples are covered on the top and bottom surfaces, exposing only the 1.6 mm-thick sample perimeter to the furnace’s ambient air. Sintered samples show a radially symmetric LZO gradient, with more LZO at the center of the pellet and considerably less LZO at the edges. This profile implies that Li+ diffusion through the material is faster than Li+ loss through volatilization, and that Li+ migration from the center of the sample to the edges is not completely reversible. These conditions lead to a net depletion of Li+ at the sample center. Findings presented in this work suggest new strategies for LLZO processing that will minimize Li+ loss during sintering, leading to a more homogeneous material with more reproducible electrochemical behavior.

Elastic properties identification of a bio-based material in tertiary packaging: Tools and methods development

This study focuses on the use of bio-based materials for structural purposes in the packaging field, which requires the identification of their mechanical properties at a representative scale. The mechanical properties of bio-based materials are more variable than those of traditional composite materials. In a standard characterization approach using elemental coupons under uniaxial loading, the variability depends on the chosen representative elementary volume (REV), free edges, boundary conditions, etc. for elastic properties that are not identified for representative working conditions; this could lead to the ineligibility of these bio-based materials as structural materials. This paper contributes to the debate on how to study the response of bio-based materials within a structure, here a packaging structure as a logistic unit (LU) subjected to a compressive load simulating storage and stacking conditions. In the set of tools and methods for the design of packaging materials made of bio-based materials, an elastic nonlinear geometric finite element model (FEM) and an experimental approach are presented. The FEM allows the numerical identification of zones of interest within the LU. Inevitably, the FEM classically requires input data which are elastic properties of the equivalent homogeneous material. The design of the FEM is based on a calculation-test approach using an existing reference LU and it can be summarized in two main steps. The first step concerns the development of a FEM able to restore the experimental conditions of vertical compression imposed by transport standards for packaging. The second step is based on updating the input properties of the FEM by reverse identification, to achieve the representative working condition properties, using experimental results obtained on the existing reference LU. For the reverse identification a multi-scale investigation is mandatory. For this purpose, the linear elastic part of the load/vertical displacement curves (at the LU stiffness scale) and the displacement and strain fields measured (at the local LU scale) by 3D digital image correlation (3D DIC) are evaluated. Then, FEM property updating is carried out by reducing the deviation of displacement/strain fields between FEM and experimentally measured results (3D DIC). Finally, we explain how FEM and 3D DIC help in decision-making by allowing the recognition of zones of interest in a phase of design of new LUs with the concept of Multi-Instrumented Technological Evaluator (MITE).

Understanding why restrictive trial eligibility criteria are inappropriate

BackgroundAn important difference between explanatory and pragmatic clinical trials concerns eligibility criteria. Eligibility criteria are restrictive in explanatory trials, while pragmatic trials are more inclusive or even all-inclusive. MethodsTo better understand the diverging views regarding eligibility criteria, we examine the contrast between theoretical and clinical medicine, and 3 different research contexts: laboratory research, population studies and clinical trials. In each context we review the purpose for selecting study subjects or research material, as well as the type of inductive inference or generalization that is sought by such selection. ResultsIn each context, selection concerns different things and serves different purposes: In the laboratory, selection concerns the homogenous research material that will help isolate a causal signal. In the epidemiological context selection concerns the (random) sampling method, designed to produce a representative sample of the population. In the clinical trial setting, selection concerns patients in need of care. Restrictive eligibility criteria become inappropriate in the care setting because the aim of the trial is not to represent a population nor to isolate a causal signal, but to find out which patients benefit from treatment. ConclusionThe idea of selecting patients comes from methods that belong to theoretical medicine. In the care setting, most clinical trials should be pragmatic and as inclusive as possible.

An investigation of the thermal performance of functionally graded annular fins on a horizontal cylinder under natural convection

ABSTRACT In industrial applications, annular fins are commonly made of homogeneous materials such as aluminum, struggling to provide uniform temperature distribution along their length. Recognizing the potential of Functionally Graded Materials (FGMs) to enhance thermal performance by combining two different materials with a gradual variation, this study proposes their use as fin materials to achieve optimum performance. The study consists of three major components: (1) numerical analyses to determine the optimum volume composition of FG fins, (2) the fabrication process of aluminum and FG fins utilizing powder metallurgy and the hot-pressing technique, and (3) experiments to evaluate the thermal performance of FG and aluminum fin arrays attached to a horizontal cylinder of 250 mm length under natural convection. Heat ranging from 25 W to 150 W is applied to the cylinder during these experiments. Based on the experiments, the thermal performances of fins are evaluated in terms of net free convection heat transfer rate, the Nusselt number, and fin effectiveness. Overall, experimental results demonstrate that the net convection heat transfer rate depends on fin spacing, material, and the base-to-ambient temperature difference. Specifically, FG fin arrays enhance the net heat transfer rate by 59%, while aluminum fin arrays increase it by 33% compared to finless cylinders. Moreover, FG fins outperform aluminum fin arrays by 40% in convective heat transfer coefficient h. Due to being the first experimental study, this study sets itself apart.

Tensile test and finite element simulation of coarse-grained thin aluminium sheets

Abstract The aim of this work was to investigate the ability of finite element simulations applied to specimens containing only a few grains for tensile testing. For this purpose, two different finite element models were created: one with individual grains with individual flow curves generated by viscoplastic self-consistent (VPSC) calculations and one with uniform material properties throughout the cross-section of the sample. In comparison of the two models with experimental tensile tests, the following conclusions can be yielded. The measurement and the simulation with individual grains have the same deformation shape. Also, the mechanical properties (yield strength, tensile strength and elongation at break) were within 10% error. However, if the measurement results were compared to the simulation with homogeneous material, it showed an error up to 53%. Therefore, it can be stated that the tensile tests of these types of specimens can be simulated, but only if the grains are modelled individually.

An approximation method to determine the travel time in total focusing imaging of carbon fiber-reinforced laminates

Abstract It is critical to accurately determine the propagation time in testing objects for ultrasonic array imaging using the Total Focusing Method (TFM). However, it poses challenges for the calculation of wave propagation time in carbon fiber reinforced polymer (CFRP) laminates because of the material anisotropy and inhomogeneity. This paper introduces a novel approach for computing ultrasonic wave travel time to enhance the TFM image of CFRP laminates. This method, termed the layer-by-layer acoustic travel time approximation (L-L TravelTimeAppro), considers the anisotropy within each monolayer and the heterogeneity of CFRP. It approximates the propagation path as straight throughout the material and computes the wave propagation duration utilizing the group velocity in each ply. The application of this method to TFM imaging is referred to as TravelTimeAppro-TFM. Experiments were conducted in a specially designed CFRP laminate. Subsequently, TFM imaging was processed by using propagation time calculated by isotropic, Dijkstra, and L-L TravelTimeAppro methods, respectively. The results indicate that TravelTimeAppro-TFM outperforms isotropic-TFM in terms of imaging amplitude gain with a maximum gain of 4.67 dB. Furthermore, it offers superior computational efficiency compared to Dijkstra-TFM.

The spermatheca ultrastructure of the ground beetle Clinidium canaliculatum (Costa) (Carabidae, Rhysodinae)

The ground beetle Clinidium canaliculatum is a member of Rhysodinae, a taxon with still discussed systematic position. The spermatheca of this species is a small cylindrical structure connected to the common oviduct by a thin duct. The ultrastructure of the organ has revealed that the apical receptacle is provided with an epithelium lined by a thick cuticle from the deeper region of which several finger-like cuticular structures extend into the cytoplasm. On these structures adhere microtubule bundles that cross the whole cytoplasm to anchor on short densities along the basal plasma membrane. These specializations are strongly reminiscent of the hemidesmosomes, possibly playing a mechanical role enabling the cells to resist to the muscle contractions pushing the sperm towards the spermathecal duct. The cells are rich in mitochondria and glycogen granules and they are possibly involved in fluid uptake from the spermathecal lumen. The spermathecal duct has a simple epithelium lined by a soft cuticle. The sperm present in the apical receptacle and in the duct lumen maintain the structure described in the male genital apparatuses. They are generally free and embedded in a homogeneous electron-dense material. Occasionally, a sperm bundle, still with an apical cap, was visible in the spermathecal receptacle.

Expansivity of Fused Quartz Glass Measured Within 6×10-10K-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6 \ imes 10^{-10}\\,\ ext{K}^{-1}$$\\end{document}

A method is described to measure the thermal expansion coefficient of fused quartz glass. The measurement principle is to monitor the change in resonance frequency of a Fabry–Perot cavity as its temperature changes; the Fabry–Perot cavity is made from fused quartz glass. The standard uncertainty in the measurement was less than 0.6 (nm·m-1)·K-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ extrm{nm}{\\cdot } \ extrm{m}^{-1}){\\cdot }\ extrm{K}^{-1}$$\\end{document}, or 0.15 %. The limit on performance is arguably uncertainty in the reflection phase-shift temperature dependence, because neither thermooptic nor thermal expansion coefficients of thin-film coatings are reliably known. However, several other uncertainty contributors are at the same level of magnitude, and so any improvement in performance would entail significant effort. Furthermore, measurements of three different samples revealed that material inhomogeneity leads to differences in the effective thermal expansion coefficient of fused quartz; inhomogeneity in thermal expansion among samples is 24 times larger than the measurement uncertainty in a single sample.

Optimization of functionally graded materials to make stress concentration vanish in a plate with circular hole

This paper is devoted to the minimization of the stress concentration factor in infinite plates with circular hole made of functionally graded materials and subjected to a far-field uniform uniaxial tension. Despite the vast literature on the versatility of these materials, the novelty of the results is that the material distribution is not limited to prefixed laws, as in many works available in the literature. Instead, it is assumed to be an unknown piecewise constant function, thus aiming to derive the material distribution by exploiting, at best, the inhomogeneity concept associated with functionally graded materials. After a brief review of the governing equations, the motivation, the statement and the mathematical formulation of the optimization problem are given under the hypothesis of axisymmetric material distribution. Still, the problem could not be solved analytically, therefore a direct transcription approach by the aid of finite difference method has been followed to convert it into a nonlinear programming problem, whose solution has been obtained numerically by dedicated gradient-based solvers. Numerical optimal solutions are reported in graphical forms, thoroughly discussed and validated by means of the finite element method. The developed numerical approach yields a material inhomogeneity obeying a sigmoid-like function and a uniform hoop stress along the radial direction, thus making the stress concentration factor at the rim of the circular hole vanish.