Properties Of Heterogeneous Materials Research Articles

Biomechanical analysis of various internal fracture fixation devices used for treating femoral neck fractures: A comparative finite element analysis

IntroductionSeveral internal fixation devices are available for treating Pauwels type I, II and III femoral neck fractures. The present study compared various fixation implants for all Pauwels fracture types using a CT-based subject-specific finite element model of the femur and determined the most effective implant for each fracture type. Materials and MethodsThe analysis included four different configurations of cannulated screw models, Femoral Neck System, Dynamic Hip Screw and Dynamic Condylar Screw (with and without anti-rotational screw). Ti-alloy was considered as the implant material. Heterogeneous bone material property was assigned based on CT grey value. Frictional contact was assumed in the contact interfaces. Peak loading corresponding to normal walking and stair-climbing were considered. Equivalent strain in bone, equivalent stress in the implants, femoral head deformation and rotation, micromotion in the contact interfaces, and strain-shielding in bone were evaluated for each implanted model. ResultsStresses generated in the implants were within the yield limit of the implant material. In Pauwels I and II, the micromotion predicted at the contact regions in all the implanted models was within 100 µm, which is suitable for bone integration. However, in Pauwels III fracture, most of the implanted models other than DHS with AR-screw model exhibited micromotion of more than 150 µm in the contact regions, which is expected to inhibit bone growth. ConclusionsThe DHS with AR-screw implanted model was identified as the most effective in treating Pauwels I and III fractures. However, for Pauwels type II, DCS with an AR-screw implant was deemed superior to the other configurations.

Effect of Thermal Shock Conditions on the Low-Cycle Fatigue Performance of 3D-Printed Materials: Acrylonitrile Butadiene Styrene, Acrylonitrile-Styrene-Acrylate, High-Impact Polystyrene, and Poly(lactic acid).

3D printing technology is becoming a widely adopted alternative to traditional polymer manufacturing methods. The most important advantage of 3D printing over traditional manufacturing methods, such as injection molding or extrusion, is the short time from the creation of a new design to the finished product. Nevertheless, 3D-printed parts generally have lower strength and lower durability compared to the same parts manufactured using traditional methods. Resistance to the environmental conditions in which a 3D-printed part operates is important to its durability. One of the most important factors that reduces durability and degrades the mechanical properties of 3D-printed parts is temperature, especially rapid temperature changes. In the case of inhomogeneous internal geometry and heterogeneous material properties, rapid temperature changes can have a significant impact on the degradation of 3D-printed parts. This degradation is more severe in high-humidity environments. Under these complex service conditions, information on the strength and fatigue behavior of 3D-printed polymers is limited. In this study, we evaluated the effects of high humidity and temperature changes on the durability and strength properties of 3D-printed parts. Samples made of commonly available materials such as ABS (Acrylonitrile Butadiene Styrene), ASA (Acrylonitrile-Styrene-Acrylate), HIPS (High-Impact Polystyrene), and PLA (Poly(lactic acid)) were subjected to temperature cycling, from an ambient temperature to -20 °C, and then were heated to 70 °C. After thermal treatment, the samples were subjected to cyclic loading to determine changes in their fatigue life relative to non-thermally treated reference samples. The results of cyclic testing showed a decrease in durability for samples made of ASA and HIPS. The ABS material proved to be resistant to the environmental effects of shocks, while the PLA material exhibited an increase in durability. Changes in the internal structure and porosity of the specimens under temperature changes were also evaluated using microcomputed tomography (microCT). Temperature changes also affected the porosity of the samples, which varied depending on the material used.

Brillouin spectroscopy for accurate assessment of morphological and mechanical characteristics in micro-structured samples

Brillouin spectroscopy has recently attracted attention as a powerful tool for the characterization of the mechanical properties of heterogeneous materials, particularly in the biological and biomedical domains. This study investigates the procedure to use Brillouin data to provide relevant morphological parameters of micro-structured samples. When acquiring Brillouin spectra at the interface between two regions of the sample, the spectrum shows signatures of both regions. This feature can be used to precisely identify the position of the interfaces by analyzing the evolution of the fitting parameters of the Brillouin spectra acquired by performing a linear scan across the interface. This concept has been demonstrated by measuring the thickness of adherent HEK293T cells. The results are validated using fluorescence microscopy, showing an excellent agreement. The present analysis showcases the wealth of information present in the Brillouin spectrum and the potentiality of Brillouin spectroscopy not only for mechanical characterization but also for label-free, high-resolution imaging of sample morphology. The study introduces the possibility of correlating mechanical properties and shape of biological samples using a single technique.

The property and mechanism of peroxymonosulfate activation boosted carbon cage encapsulated Mo2C/Bi2O2CO3 Z-scheme heterojunction for enhanced catalytic degradation and antibacterial activity

The regulation of heterogeneous material properties for the synergistic degradation of pollutants remains a challenge. Herein, cobalt-nitrogen co-doped carbon cage (NCC) encapsulated Mo2C nanoparticles boosting Bi2O2CO3 photocatalyst (Mo2C@NCC/BOC) we designed to achieve photoexcitation synergistic PMS activation. The degradation results revealed Mo2C@NCC/BOC could achieved 96.4 % tetracycline (TC) removal after only 5 min. The degradation rate was 3.1 times and 5.9 times higher than those of pure Bi2O2CO3 and pure Mo2C@NCC, respectively. The improved TC degradation performance could be attributed to the Z-scheme heterostructure, resulting in a higher conversion rate of photo-generated electrons into radicals. In addition, Co2+/Co3+ and Mo5+/Mo6+ dual ions redox cycle system effectively triggers the activation of HSO5− under visible light. The TC degradation pathways and intermediate toxicity, including the photocatalyticmechanism were evaluated in detail. This study provides novel perspectives on the rational design of atom co-sharing heterojunction photocatalysts in activating PMS for water remediation.

Coexisting single-atomic Fe and Fe2O3 on nitrogen-deficient g-C3N4 with enriched Fenton-like oxidation and photocatalytic performances for tetracycline degradation: DFT calculation, degradation mechanism and toxicity evaluation

The regulation of heterogeneous material properties to strengthen the peroxymonosulfate activation to destroy refractory pharmaceuticals remains a challenge. Herein, a promising nitrogen-deficient catalyst (FeCN) with single atom Fe and Fe2O3 nanoparticles dispersing on g-C3N4 matrix was synthesized and applied as a competent peroxydisulfate (PDS) activator for tetracycline (TC) degradation under visible light. The K-edge X-ray absorption fine structures and 57Fe Mössbauer measurements confirmed the unique configuration of Fe2O3 and single atom Fe active sites (pyrrolic and pyridinic FeN4) induced by the neighbouring nitrogen vacancies. The theory calculations and experiments provided a deep insight into the PDS activation and TC degradation via radical and non-radical pathways. The pyrrolic FeN4 active sites dominated the electron transfer, while the pyridinic FeN4 and Fe2O3 decomposed PDS to produce free radicals. The synergistic effect between the Fenton-like oxidation and photocatalysis is responsible for efficient TC degradation. This study offers a new strategy by integrating novel catalysts and advanced oxidation technology for effective wastewater purification.

The Potential for Material Property Characterization Using Physics-Informed Neural Networks and Ultrasonic Wave Data

ABSTRACT There is a need for reliable nondestructive test methods that can collect data from structural members and analyze the results in a rapid and efficient manner. Large amounts of test data are needed to achieve such characterization, which provides additional challenges because of their heterogeneity and complexity. Advances in machine learning, in particular physics-informed neural networks (PINN), offer potential to address these problems. PINN is a particular form of artificial neural networks (ANN) and portends notable advantages over traditional measurand analysis or purely data-driven approaches. Here, we explore the potential of heterogeneous material property characterization using PINN and ultrasonic wave data. First, several types of 1-D ultrasonic wave data are numerically simulated for a spatially heterogeneous material, and then PINN is applied to predict wave velocity, defect location, and energy dissipation. Then, three different types of defects are simulated and all defects are detected using the corresponding 2-D ultrasonic wave data and PINN. The presented results demonstrate the promise of PINN to assist with heterogeneous material characterization methods.

R‐Vine Copulas for Data‐Driven Quantification of Descriptor Relationships in Porous Materials

AbstractLocal variations in the 3D microstructure can control the macroscopic behavior of heterogeneous porous materials. For example, the permittivity through porous sheets or membranes is governed by local high‐volume pathways or bottlenecks. Due to local variations, unfeasibly large amounts of microstructure data may be needed to reliably predict such material properties directly from image data. Here it is demonstrated that a vine copula approach provides parametric models for local microstructure descriptors that compactly capture the 3D microstructure including its local variations and efficiently probe it with respect to selected, measurable properties. In contrast to common methods of complexity reduction, the proposed approach creates parametric models for the multivariate probability distribution of high‐dimensional descriptor vectors that inherently contain the complex, nonlinear dependencies between these descriptors. Therein, material properties are offered in physically motivated distributions of microstructure descriptors rather than as normally distributed data. Applied to porous fiber networks (paper) before and after unidirectional compression, it is shown that the copula‐based models reveal material‐characteristic relationships between two or more microstructure descriptors. In this way, the presented modeling approach can provide deeper insight into the microscopic origin of effective macroscopic properties of heterogeneous porous materials.

Application of Installations with Uniform-Strength Beams as Working Deformation Standards

A method for obtaining homogeneous deformation along the length of the measuring section of a uniform-strength beam and the possibility of its application as part of a working deformation standard are considered. The results of the analysis of the bending model and the design of a uniform-strength beam are presented. In the focus of attention are the parameters included in the measurement equation and related to methodological factors, as well as influencing the result of relative deformation measurements. The subjects of research are the presence of contact friction forces, heterogeneity of material properties, the special nature of the load application (bending, torsion, the presence of residual stresses in the body, geometrical parameters of the calibration beam, orientation of primary transducers on the beam, application of bending load, measurement of deflection and displacement of the neutral layer). The advantages and disadvantages of using a uniform-strength beam for determining the characteristics of primary strain transducers during testing, calibration, and verification were established. The deviation of signals of primary transducers located outside the axial cross-section when oriented along the beam axis and along the force lines converging at the point of load application was experimentally revealed. The error due to the orientation of the primary transducers on the beam depending on the angle between the lateral faces can range from 0.15 to 0.23 %. The study adds to the theoretical knowledge base on the possibility of using a cantilever uniform-strength beam as a load-bearing element in calibration installations. The conclusions may be useful for testing, calibrating, and verifying primary strain transducers.

Modelling a gas injection experiment incorporating embedded fractures and heterogeneous material properties

This study focuses on the modelling of a gas injection experiment to assess the effects of incorporating heterogeneous material properties. The numerical model considers a two-phase flow coupled hydro-mechanical problem, and includes embedded fractures that open with deformation, thereby enhancing permeability. The approach used is integrated in the CODE_BRIGHT software, which allows for the consideration of geomaterials with a spatially correlated heterogeneous field of porosity that follows a normal distribution. This spatial correlation can be either isotropic or anisotropic. A key aspect of this approach is that material properties such as intrinsic permeability, diffusivity or cohesion are defined as a function of porosity. Consequently, these properties also exhibit heterogeneity with spatial correlation and, eventually, anisotropy. The results derived from the numerical model align well with in-situ measurements. The study also includes sensitivity analyses to the variation of critical variables. The calibration of the model has been validated through a similar experiment. The findings indicate that the consideration of heterogeneous material properties can have a significant influence on gas injection problems, particularly when a hydraulic fracture is formed.

A statistical analysis of human talar shape and bone density distribution.

The shape of the talus, its internal structure, and its mechanical properties are important in determining talar behavior during loading, which may be significant for the design of surgical tools and implants. Although recent studies using statistical shape modeling have described quantitative talar external shape variation, no similar quantitative study exists to describe the density distribution of internal talar structure. The goal of this study is to quantify statistical variation in talar shape and density to benefit the design of talar implants. To this end, weight-bearing computed tomography (CT) scans of the ankle were collected in neutral, bilateral standing posture, and three-dimensional models were generated for each talus. Local density derived from the Hounsfield unit of each CT voxel was extracted. A weighted spherical harmonic analysis was performed to quantify the talar external shape. One hundred and seventy-nine volumes of interest were placed in the same relative position within each talus to quantify the talar density. Additionally, a finite element analysis (FEA) was conducted on a talus with both heterogeneous and homogeneous material properties to observe the effect of these properties on the stress and strain response. Significant differences were found in the talar density in sex and age, as well as in the stress and strain response between homogeneous and heterogeneous FEA. These differences show the importance of considering heterogeneity when examining the load response of tarsal bones.

Study on meso‑mechanical properties and failure mechanism of soil-rock mixture based on SPH model

This study adopts the Smoothed Particle Hydrodynamics (SPH) technique to accurately and efficiently replicate and forecast the mesoscopic behavior of soil-rock mixtures (SRM). It introduces a novel approach for generating rock blocks within the SRM, utilizing a method that randomly selects angles and lengths. In addition, this research proposes a method for discretizing any shaped region into free particles with specific material attributes, named the regional medium particle discretization method. It incorporates the Drucker-Prager constitutive model to develop the SPH numerical model for SRM. Furthermore, it examines the effects of different rock sizes and rock contents on the SRM's failure characteristics and mechanical properties. The findings revealed that, for identical rock contents, smaller rock samples exhibit a more dispersed failure surface with numerous secondary shear bands, whereas larger rock samples display a smoother and more concentrated failure surface. As the rock content decreases, shear bands typically form in the sample's center and are relatively straight. However, as the rock content increases, the shear bands' configuration becomes more intricate, often featuring multiple shear bands. This method offers a fresh perspective for exploring the mechanical properties of heterogeneous materials.

A learned born series for highly scattering media

The Helmholtz equation with heterogeneous material properties is essential in various fields requiring single-frequency wave simulations, including optics, seismology, acoustics, and electromagnetics. Traditional methods for solving this equation, such as the Born Series, face limitations in converging for high-contrast scattering potentials. To address this challenge, we introduce a novel method called the Learned Born Series (LBS). The LBS is derived from the convergent Born Series but employs components that are learned through training. It demonstrates significantly improved accuracy compared to the conventional convergent Born Series, especially in scenarios with high-contrast scatterers, while maintaining similar computational complexity. The LBS can rapidly generate reasonably accurate predictions of the global pressure field with only a few iterations, and the errors decrease as more iterations are learned. We show its effectiveness through experiments on simulated datasets. The LBS offers promising prospects for accelerating simulations in scenarios with strong sound speed contrasts, potentially revolutionizing applications in transcranial treatment planning and full waveform inversion.

Yield strength evaluation of 3D-printed Ti–6Al–4V components based on non-contact eddy-current measurement

The demand for metal additive manufacturing (AM), also known as three-dimensional (3D) printing, is increasing owing to its ability to produce components with complex shapes, heterogeneous material properties, low material waste, etc. However, for broader acceptance of metal AM, quality assurance of mechanical properties, such as elastic (Young's) modulus, Poisson's ratio, and ductility, is becoming critical in additively manufactured components. This study proposes a non-contact, non-destructive and automated eddy-current technique for yield strength estimation of additively manufactured Ti–6Al–4V components with the potential for online operation during metal AM. After conducting the eddy-current measurement at a specific inspection point within the Ti–6Al–4V component, the relationship between the eddy-current phase value and electrical conductivity is established. Subsequently, the yield strength of the Ti–6Al–4V component is estimated by utilizing the analytical relationship between yield strength and electrical conductivity, which is based on Andrew's research and the Hall–Petch relationship. For experimental validation, Ti–6Al–4V test specimens with different electrical conductivities (yield strengths) were fabricated by adjusting the cooling rate during metal direct energy deposition AM. Then, the yield strengths estimated using the proposed technique were compared with those obtained using conventional destructive tensile tests. The results show that the proposed technique can estimate the yield strength with an error rate of less than 8%. The uniqueness of this study lies in (1) the theoretical derivation of the relationship between the eddy-current phase value and yield strength, (2) non-destructive, non-contact, and automated estimation of the yield strength using eddy-current measurements, and (3) performance evaluation using additively manufactured Ti–6Al–4V specimens with different yield strengths.

A comparison of brain retraction mechanisms using finite element analysis and the effects of regionally heterogeneous material properties

Finite element (FE) simulations of the brain undergoing neurosurgical procedures present us with the great opportunity to better investigate, understand, and optimize surgical techniques and equipment. FE models provide access to data such as the stress levels within the brain that would otherwise be inaccessible with the current medical technology. Brain retraction is often a dangerous but necessary part of neurosurgery, and current research focuses on minimizing trauma during the procedure. In this work, we present a simulation-based comparison of different types of retraction mechanisms. We focus on traditional spatulas and tubular retractors. Our results show that tubular retractors result in lower average predicted stresses, especially in the subcortical structures and corpus callosum. Additionally, we show that changing the location of retraction can greatly affect the predicted stress results. As the model predictions highly depend on the material model and parameters used for simulations, we also investigate the importance of using region-specific hyperelastic and viscoelastic material parameters when modelling a three-dimensional human brain during retraction. Our investigations demonstrate how FE simulations in neurosurgical techniques can provide insight to surgeons and medical device manufacturers. They emphasize how further work into this direction could greatly improve the management and prevention of injury during surgery. Additionally, we show the importance of modelling the human brain with region-dependent parameters in order to provide useful predictions for neurosurgical procedures.

Assessment of the nano-mechanical properties of healthy and atherosclerotic coronary arteries by atomic force microscopy.

Nano-indentation techniques might be better equipped to assess the heterogeneous material properties of plaques than macroscopic methods but there are no bespoke protocols for this kind of material testing for coronary arteries. Therefore, we developed a measurement protocol to extract mechanical properties from healthy and atherosclerotic coronary artery tissue sections. Young's modulus was derived from force-indentation data. Metrics of collagen fibre density were extracted from the same tissue, and the local material properties were co-registered to the local collagen microstructure with a robust framework. The locations of the indentation were retrospectively classified by histological category (healthy, plaque, lipid-rich, fibrous cap) according to Picrosirius Red stain and adjacent Hematoxylin & Eosin and Oil-Red-O stains. Plaque tissue was softer (p < 0.001) than the healthy coronary wall. Areas rich in collagen within the plaque (fibrous cap) were significantly (p < 0.001) stiffer than areas poor in collagen/lipid-rich, but less than half as stiff as the healthy coronary media. Young's moduli correlated (Pearson's ρ = 0.53, p < 0.05) with collagen content. Atomic force microscopy (AFM) is capable of detecting tissue stiffness changes related to collagen density in healthy and diseased cardiovascular tissue. Mechanical characterization of atherosclerotic plaques with nano-indentation techniques could refine constitutive models for computational modelling.

GLARE deformation in low-restraint state: Tensile and bending behavior

Fiber metal laminates (FMLs) possess a better weight reduction effect than traditional metal materials due to the coupling properties of heterogeneous materials, showing promise in the areas of aviation and aerospace. At present, the research on FMLs mainly focuses on the deformation properties after curing while the mechanical properties in the low-restraint state, i.e. the laminates are temporarily not cured and are formed under lower constraint stresses, are not studied in detail. Herein, the low-restraint GLARE is taken as a research object and its deformation characteristics and failure mode are studied using uniaxial tensile, bending experiments and numerical simulation in this paper. Moreover, the influence of key parameters, such as temperature, laminate structure and span length, is investigated, as well as the tensile and bending deformation behavior of GLARE in a low-restraint state is obtained. In addition, based on experimental observations, it is found that the low-restraint GLARE produces the phenomenon of interlayer slip, and the interlaminar residual stress is much smaller than the cured laminate, whereas the deformation limit and performance are improved. This paper provides theoretical bases for the in-depth analysis of deformation properties and forming laws of low-restraint GLARE, guiding further applications of laminates.

Segmenting mechanically heterogeneous domains via unsupervised learning.

From biological organs to soft robotics, highly deformable materials are essential components of natural and engineered systems. These highly deformable materials can have heterogeneous material properties, and can experience heterogeneous deformations with or without underlying material heterogeneity. Many recent works have established that computational modeling approaches are well suited for understanding and predicting the consequences of material heterogeneity and for interpreting observed heterogeneous strain fields. In particular, there has been significant work toward developing inverse analysis approaches that can convert observed kinematic quantities (e.g., displacement, strain) to material properties and mechanical state. Despite the success of these approaches, they are not necessarily generalizable and often rely on tight control and knowledge of boundary conditions. Here, we will build on the recent advances (and ubiquity) of machine learning approaches to explore alternative approaches to detect patterns in heterogeneous material properties and mechanical behavior. Specifically, we will explore unsupervised learning approaches to clustering and ensemble clustering to identify heterogeneous regions. Overall, we find that these approaches are effective, yet limited in their abilities. Through this initial exploration (where all data and code are published alongside this manuscript), we set the stage for future studies that more specifically adapt these methods to mechanical data.

Imposing different boundary conditions for thermal computational homogenization problems with FFT‐ and tensor‐train‐based Green's operator methods

AbstractTo compute the effective properties of random heterogeneous materials, a number of different boundary conditions are used to define the apparent properties on cells of finite size. Typically, depending on the specific boundary condition, different numerical methods are used. The article at hand provides a unified framework for Lippmann–Schwinger solvers in thermal conductivity and Dirichlet (prescribed temperature), Neumann (prescribed normal heat flux) as well as periodic boundary conditions. We focus on the explicit jump finite‐difference discretization and discuss different techniques for computing the discrete Green's operator. These include Fourier‐type methods, that is, based on discrete sine, cosine and Fourier transformations, as well as low‐rank tensor methods, that is, the tensor‐train approximation, whose computational prowess was demonstrated recently. We use the developed computational technology to investigate a number of interesting random materials and assess different microstructure‐generation techniques in terms of their compatibility with the prescribed boundary conditions.

Effect of thermal poling in sodium tantalum phosphate glass-ceramics

Eu3+-doped sodium tantalum phosphate glass-ceramics containing bronze -like perovskite crystalline phase Na2Ta8O21 were obtained with different crystalline states (degree of crystallinity and average crystallite size) by a suitable control of the heat-treatment time and temperature of the precursor glass. Luminescence properties of Eu3+ in these materials are in agreement with a progressive insertion of the rare earth ions in the sodium tantalate Na2Ta8O21 phase, as probed by a clear decrease of the intensity ratio between transitions [5D0 →] 7F2/7F1 and narrowing of characteristic emission bands. Thermal poling insights were performed on these glassy and glass-ceramic materials and the experimental conditions had to be carefully optimized in the glass-ceramics to avoid electrical sparks and sample break associated with dielectric breakdown of the samples. Suitable and stable thermal poling treatments could be performed at 250 °C and 900 V under N2 atmosphere on both glass and glass-ceramics. These first insights pointed out a lower migration kinetic of sodium ions when compared to the precursor glass as well as a lower content of depleted ions, since part of sodium is supposed to be “frozen” inside the crystalline phase in glass-ceramics. Despite these lower kinetic poling conditions, a poled sodium depleted layer whose thickness depends on the poling time was successfully formed at the anode side. Reflectance infrared spectroscopy was also used to investigate the structural changes in the poled layer. These IR data pointed out that structural changes induced by thermal poling are mainly promoted in the phosphate network rather than tantalate units in both glass and glass-ceramics in agreement with a selective sodium migration from the remaining glassy phase. Macroscopic induced SHG measured by the Maker fringes technique is also consistent with the EFISH model and quantitative simulations of these data allowed to estimate second order nonlinear optical susceptibilities χ(2) ten to twenty times lower in the glass-ceramics than in the precursor glass. Such electric behavior is discussed in terms of dielectric properties of heterogeneous materials and interfacial polarization mechanisms between crystallites and glassy phase.

A thermodynamic model for predicting the stress–strain relation of stochastic heterogeneous materials with experimental verification

Materials are intrinsic stochastic heterogeneity regardless of their natural or artificial origin. Traditional models for predicting mechanical properties of materials typically involve homogenization, which exhibits certain limitations in practical applications, such as investigating the size effect in stress–strain relationships. In this study, a random field thermodynamic theory for predicting the mechanical properties of stochastic heterogeneous materials was proposed, and its applicability and reliability were verified by testing hard-drawn 304 stainless-steel wires with gauge lengths of 10–500 mm. In these wires composed of the lamellar martensite with a thickness of ∼47.1 nm, it is found that failure strains decreased significantly with increasing gauge length, whereas fracture stresses show little dependence on the gauge length leading to a great increase of the effective modulus. Digital image correlation investigations revealed a significant strain heterogeneity during the tensile process, even at the elastic deformation stage. Nanoindentation tests show the non-uniformly distributed reduced modulus and hardness, which can be attributed to the intrinsic stochastic heterogeneity of materials. Taking Young's modulus as a one-dimensional stationary random field along the longitudinal direction, our random field model can well demonstrate the variation of failure strain and effective modulus with changing gauge lengths. The size effect on the macroscopic mechanical properties mainly originated from the microscopic mechanical property correlation of different points within the materials. These findings could provide new insights into the size effect on the mechanical properties of materials.