Homogeneous Material Research Articles

Investigation of effective parameters on Fe/Ta thin films by plasma focus device: number of shots and distance from tip anode

This experimental investigation is the first to generate a surface iron-tantalum (Fe/Ta) alloy as a sublayer-layer using a plasma focus device. Examining how ion beams from a plasma focus device alloy iron and tantalum with varying melting points is one of the key objectives of this study. Fe/Ta thin film nanostructure and surface morphology were also examined. The distance from the tip anode and the varied number of shots are the experimental variables. Although tantalum's melting point (3020 ∘C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^\\circ{\\rm C}$$\\end{document}) is generally known to be near to that of iron (2862 ∘C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^\\circ{\\rm C}$$\\end{document}), it is possible that iron vaporizes and partial alloying of iron with tantalum occurs before tantalum reaches its melting point. Fe/Ta thin film identification techniques include scanning electron microscopy, mapping of cross-section, energy dispersive X-ray spectroscopy, and X-ray diffraction pattern. Additionally, the composition of multilayer structures is examined using EDS. In conclusion, the results of the X-ray diffraction pattern showed that the number of shots had a significant impact on the residual strain degree of the thin films that were deposited. Furthermore, structures made of FeTa and Fe2Ta were produced. Additionally, photos from scanning electron microscopy and cross-section mapping verify that the sample with five shots at an 8 cm distance from the tip anode formed a uniform Fe/Ta alloy structure. The sample with five shots at a distance of 4 cm from the tip anode formed micro-island structures, as seen by scanning electron microscopy, with decreasing distance. Furthermore, depth elemental distribution revealed that the optimal depth of penetration in a homogenous material to develop alloying is best determined by number of PF shots.

On density functional theory models for one-dimensional homogeneous materials

This paper studies Density Functional Theory (DFT) models for homogeneous 1D materials in the 3D space. It follows the previous work [Gontier et al., Commun. Math. Phys. 388, 1475–1505 (2021)] about DFT models for homogeneous 2D materials in 3D. We show how to reduce the problem from a 3D energy functional to a 2D energy functional. The kinetic energy is treated as in the 2D material case by diagonalizing admissible states, and writing the kinetic energy as the infimum of a modified kinetic energy functional on reduced states. Besides, we treat here the Hartree interaction term in 2D, and show how to properly define the mean-field potential, through Riesz potential. We then show the well-posedness of the reduced model and present some numerical illustrations.

Compositional effects in potassium metakaolin geopolymers containing alumina and glass frit

Geopolymers represent a distinct class of materials characterised by their X-ray amorphous nature and nanoporous, nanoparticulate structure. Geopolymers can be conveniently mixed, poured, and cured under ambient conditions. This makes this class of materials an interesting alternative to ordinary Portland cement for structural processes. Additionally, the addition of alumina can improve mechanical properties, while the addition of glass can form an impermeable glaze which could be useful for molten salt containment. Therefore, in this investigation, potassium metakaolin-based geopolymer composites with varying proportions of glass particles and alumina platelets were fabricated, cured, heat-treated, and analyzed to study the effects of composition on material properties. Various attributes including rheological properties, densities, mass loss, shrinkage, and porosities were compared. It was observed that certain compositions exhibited high viscosities, making high shear mixing challenging, while also displaying significant permeability that would hinder their ability to contain liquids without leakage. Additionally, certain samples showed reduced densities, suggesting potentially weaker mechanical properties; however, the investigation did not include a direct assessment of mechanical properties. The most promising candidates for containing liquids at high temperature contained 50 wt% KGP, 25 or 35 wt% glass powder, and 25 or 15 wt% alumina platelets, respectively. ASH-G slurries required a minimum of 65 vol% KGP to produce a homogenous material compatible with additive manufacturing. The minimum amount of glass phase to form surface glazes was 16 vol%. Only samples containing more glass phase than alumina phase produced glazed composites.

Proposed experiment to measure nonlinear optical susceptibilities in the saturated regime.

When an optical beam passes through a thin slice of a homogeneous material, the change of its phase and amplitude is characterized by the material's linear and nonlinear susceptibility, the latter also known as the hyperpolarizability. The standard method for measuring the nonlinear susceptibility is the scan. This widely used method is sometimes applied outside of its range of validity, leading to systematic errors. These errors are illustrated for a two-level system with parameters taken from atomic rubidium. The present paper proposes a method called the phase retrieval of modes to determine the nonlinear susceptibility without an assumption about its functional form, in contrast to both the -scan method and variants intended to apply in cases of saturation. In brief, a Gaussian beam passes through a thin sample and is detected on three planes in a focal scan. Phase retrieval methods are used to find coefficients of the modes which in turn determine the optical nonlinear susceptibility. Nearly exact recovery of the nonlinear susceptibility is shown numerically in the no-noise case. Additionally, two types of noise are considered: shot noise on the detector and intensity fluctuations of the input.

EIT probe based intraoperative tissue inspection for minimally invasive surgery

Abstract Electrical impedance tomography (EIT) has become an integral component in the repertoire of medical imaging techniques, particularly due to its non-invasive nature and real-time imaging capabilities. Despite its potential, the application of EIT in minimally invasive surgery (MIS) has been hindered by a lack of specialized electrode probes. Existing designs often compromise between invasiveness and spatial sensitivity: probes small enough for MIS often fail to provide detailed imaging, while those offering greater sensitivity are impractically large for use through a surgical trocar. Addressing this challenge, our study presents a breakthrough in EIT probe design. The open electrode probe we have developed features a line of 16 electrodes, thoughtfully arrayed to balance the spatial demands of MIS with the need for precise imaging. Employing an advanced EIT reconstruction algorithm, our probe not only captures images that reflect the electrical characteristics of the tissues but also ensures the homogeneity of the test material is accurately represented. The versatility of our probe is demonstrated by its capacity to generate high-resolution images of subsurface anatomical structures, a feature particularly valuable during MIS where direct visual access is limited. Surgeons can rely on intraoperative EIT imaging to inform their navigation of complex anatomical landscapes, enhancing both the safety and efficacy of their procedures. Through rigorous experimental validation using ex vivo tissue phantoms, we have established the probe’s proficiency. The experiments confirmed the system’s high sensitivity and precision, particularly in the critical tasks of subsurface tissue detection and surgical margin delineation. These capabilities manifest the potential of our probe to revolutionize the field of surgical imaging, providing a previously unattainable level of detail and assurance in MIS procedures.

Об идентификации характеристик неоднородных вязкоупругих тел в рамках модели дробного порядка

Nowadays, the development of models of viscoelastic materials with complex inhomogeneous structure is one of the hottest problems of continuum mechanics. Along with classical models, fractional-differential models of viscoelasticity have become increasingly popular. In this paper, we present a model for describing steady-state oscillations of inhomogeneous viscoelastic bodies in terms of fractional-order differential operators. Taking into account the fractional order of the model operators, the corresponding form of the complex module describing the material properties is constructed. The module includes four characteristics: instantaneous and long-term elastic moduli (in the case of material inhomogeneity, they are the functions of coordinates), relaxation time and fractionality parameter. The properties of the complex module are studied, and the ranges of the model parameters, at which the rheological properties are most pronounced, are identified. A general formulation of the inverse problem (IP) on the identification of function-parameters of the model based on the acoustic sounding data is proposed. In the framework of this formulation, the inverse problems for specific objects , namely, an inhomogeneous rod and a round plate, are considered. In both model problems, the influence of the fractionalization parameter on the amplitude-frequency characteristics is analyzed. It has been found that in the vicinity of viscoelastic resonances the fractionality parameter affects the parameters of the oscillatory process most significantly, which is typical for such problems. A solution to the nonlinear IPs under consideration is constructed based on the linearization method, which also serve as basis for the iterative processes supplemented with the elements of the projection approach. This allows one to determine corrections to the required functions in the specified classes of functions using regularization. For both IPs, a series of computational experiments were carried out. The analysis of the experimental results made it possible to formulate recommendations for the selection of optimal sensing modes.

Establishment and Characterization of a Chicken Myoblast Cell Line.

Skeletal muscle, which is predominantly constituted by multinucleated muscle fibers, plays a pivotal role in sustaining bodily movements and energy metabolism. Myoblasts, which serve as precursor cells for differentiation and fusion into muscle fibers, are of critical importance in the exploration of the functional genes associated with embryonic muscle development. However, the in vitro proliferation of primary myoblasts is inherently constrained. In this study, we achieved a significant breakthrough by successfully establishing a chicken myoblast cell line through the introduction of the exogenous chicken telomerase reverse transcriptase (chTERT) gene, followed by rigorous G418-mediated pressure screening. This newly developed cell line, which was designated as chTERT-myoblasts, closely resembled primary myoblasts in terms of morphology and exhibited remarkable stability in culture for at least 20 generations of population doublings without undergoing malignant transformation. In addition, we conducted an exhaustive analysis that encompassed cellular proliferation, differentiation, and transfection characteristics. Our findings revealed that the chTERT-myoblasts had the ability to proliferate, differentiate, and transfect after multiple rounds of population doublings. This achievement not only furnished a valuable source of homogeneous avian cell material for investigating embryonic muscle development, but also provided valuable insights and methodologies for establishing primary cell lines.

Controlling Hybrid Polyhydroxyurethane Adhesive and Rheological Properties by Partial Carbonation of Biobased Epoxy Monomer.

Controlling hybrid material properties by simple monomer design offers an elegant pathway to prepare thermoset adhesives with tunable properties. Herein, biobased hybrid polyhydroxyurethane/polyepoxy is prepared starting from partially carbonated cashew nut shell epoxy derivatives (NC514) and m-xylene diamine (MXDA). The curing reactions, that is, epoxy-amine and cyclic carbonate aminolysis, monitored by ATR-IR spectroscopy at 50°C are found to be concomitant yielding highly homogeneous materials. Hybrid networks are extensively characterized by swelling tests, TGA, DMA, DSC, tensile tests, rheology, and lap-shear-test on aluminum substrates. The introduction of hydroxyurethane moieties within the epoxy-amine networks enhanced the adhesion properties (up to 20% compare to neat epoxy resins) by combining hydrogen bonding capability and vitrimeric properties (thermoset able to flow). Rheological characterizations and reprocessing tests demonstrated that hybrid adhesives with up to 47 mol% of cyclic carbonate groups are capable of covalent exchange (internally catalyzed by tertiary amine) while keeping similar thermomechanical properties and enhanced adhesion strength compare to the permanent epoxy network.

Experimental and computational investigation of FGM coated cylinder liners prepared using HVOF technique

In the present work, to improve the wear resistance of the cylinder liner, without affecting the heat dissipation, cast steel cylinder liners are coated with five-layer Functionally Graded Material (FGM) coatings prepared using High-velocity oxy-fuel (HVOF) technique. The powders were designated as C1 (WC-10Co-4Cr) which is Tungsten Carbide with 10% Cobalt and 4% Chromium, C2 (WC-12Co) which is Tungsten Carbide with 12% Cobalt and C3 (Ni-25Cr) which is Nickel with 25% Chromium. The results revealed that the FGM coating designated as Functionally Graded Coated Sample-1 (FGCS-1) wears less compared to Functionally Graded Coated Sample-2 (FGCS-2) and Functionally Graded Coated Sample-3 (FGCS-3), due to the presence of wear resistive elements in the composition and uniform material homogeneity. Numerical analysis of FGCS-2, conducted using Ansys Fluent software, revealed a negligible temperature gradient across the coating thickness. This is due to the highly conductive coating material and the extremely thin coating layers, making it suitable for engine cylinder liner applications where less wear and high heat transfer rate is desirable.

Variable-order fractional diffusion: Physical interpretation and simulation within the multiple trapping model

The physical interpretation of a variable-order fractional diffusion equation within the framework of the multiple trapping model is presented. This interpretation enables the development of a numerical Monte Carlo algorithm to solve the associated subdiffusion equation. An important feature of the model is variation in energy density of localized states, when the detailed balance condition between localized and mobile particles is satisfied. The variable order anomalous diffusion equations under consideration can be applied to the description of transient subdiffusion in inhomogeneous materials, the order of which depends on the considered spatial and/or time scale. Examples of numerical solutions for different situations are demonstrated. Considering variable-order fractional drift, we calculate and analyze the transient current curves of the time-of-flight method for samples with varying density of localized states.

A Survey on Fused Filament Fabrication to Produce Functionally Gradient Materials.

Fused filament fabrication (FFF) is a key extrusion-based additive manufacturing (AM) process for fabricating components from polymers and their composites. Functionally gradient materials (FGMs) exhibit spatially varying properties by modulating chemical compositions, microstructures, and design attributes, offering enhanced performance over homogeneous materials and conventional composites. These materials are pivotal in aerospace, automotive, and medical applications, where the optimization of weight, cost, and functional properties is critical. Conventional FGM manufacturing techniques are hindered by complexity, high costs, and limited precision. AM, particularly FFF, presents a promising alternative for FGM production, though its application is predominantly confined to research settings. This paper conducts an in-depth review of current FFF techniques for FGMs, evaluates the limitations of traditional methods, and discusses the challenges, opportunities, and future research trajectories in this emerging field.

Uncertainty quantification for locally resonant coated plates and shells

The effect of uncertainty in design parameters on the acoustic performance of coated plates and coated shells for maritime applications is presented. The locally resonant coatings are designed using a viscoelastic material with an impedance similar to water and embedded with layers of inclusions. Both voids and hard inclusions, which respectively exhibit monopole and dipole resonance scattering, are considered. Effective medium approximation theory is employed to characterise the layers of inclusions as homogenised layers with effective material and geometric properties. The coating is then modelled as a multilayered equivalent fluid composed of alternating layers of the homogeneous viscoelastic material and the homogenised layers of inclusions. The coating is bonded to a rigid backing plate and the acoustic response is calculated using the transfer matrix method. The coating is also externally applied to an elastic cylindrical shell. The acoustic response is calculated by expressing the shell displacements and acoustic pressures in the coating and exterior domain in terms of Fourier series expansions, and applying continuity equations at each interface between the shell surface, coating layers and the surrounding water. Efficient stochastic models based on the non-intrusive polynomial chaos expansion (PCE) method are developed by transforming the analytical models for the coated plates and shells into computationally efficient surrogate models using point collocation. Uncertainty in dominant design parameters associated with the geometry of the inclusions and material properties of the coating is examined. The influence of the design parameters for inclusions tuned to different resonance frequencies and for multiple layers of inclusions is also reported. Strong acoustic performance of coating designs occurs in a broad frequency range around local resonance of the inclusions. For all coating models, uncertainty in parameters which predominantly influence the local resonance of the inclusions were observed to yield the greatest variation in the acoustic responses of the coated plates and shells.

A Comparative Study of Methods for Calculating the Dislocation Density in GaN-on-Si Epitaxial Wafers.

A series of characterization methods involving high-resolution X-ray diffraction (HR-XRD), electron channel contrast imaging (ECCI), cathodoluminescence microscopy (CL), and atomic force microscopy (AFM) were applied to calculate the dislocation density of GaN-on-Si epitaxial wafers, and their performance was analyzed and evaluated. The ECCI technique, owing to its high lateral resolution, reveals dislocation distributions on material surfaces, which can visually characterize the dislocation density. While the CL technique is effective for low-density dislocations, it is difficult to accurately identify the number of dislocation clusters in CL images as the density increases. The AFM technique analyzes surface dislocation characteristics by detecting surface pits caused by dislocations, which are easily affected by sample and probe conditions. A prevalent method for assessing the crystal quality of GaN is the rocking curve of HR-XRD (ω-scan), which calculates the dislocation density based on the FWHM value of the curves. By comparing the above four dislocation characterization methods, the advantages and limitations of each method are clarified, which also verifies the applicability of DB=β29b2 for GaN-on-Si epitaxial wafers. This provides an important reference value for dislocation characterization in GaN-on-Si materials. The accuracy evaluation of dislocation density can truly and reliably reflect crystal quality, which is conducive to further optimization. Furthermore, this study can also be applied to other heterogeneous or homogeneous epitaxial materials.

A 3D finite strain constitutive model for shape memory polymers combined viscoelasticity and storage strain

A 3D finite strain constitutive model combined viscoelasticity and storage strain for shape memory polymers (SMPs) is proposed. SMPs are phenomenally regarded as a homogeneous material of the viscoelastic glassy phase and hyperelastic rubbery phase in this model. Based on energy decomposition, the constitutive equation is derived using Clausius inequality and unified Helmholtz free energy. Then, an analytical expression for storage strain is presented, significantly simplifying the prediction of shape memory recovery process. The model is implemented in MATLAB and the commercial finite element software package ABAQUS using the UMAT subroutine. Our additional DMA tests and experimental data from the literature for various SMPs with different shape structures are used to verify the proposed model. All simulation results demonstrate the effectiveness of the proposed model in predicting the shape memory effect and complex thermomechanical behaviors of various SMPs.

Fully Consolidated Deposits from Oxide Dispersion Strengthened and Silicon Steel Powders via Friction Surfacing

Abstract The objective of this work is to study the ability of friction surfacing to deposit metal alloys that are difficult to process with traditional methods. Creep and neutron irradiation resistant oxide dispersion strengthened (ODS) materials cannot be produced via conventional casting route due to insolubility of the oxidic and metallic alloy constituents, causing unintended inhomogeneous oxide dispersion and material behavior. Increasing silicon content of Iron-Silicon (Fe-Si) improves electromagnetic properties but embrittles the material significantly and the fusion based manufacturing methods unable to process this steel. The solid-state nature of the friction surfacing process offers a potential alternative processing route to enable wider usage of difficult to process alloy systems. Both ODS and Fe-Si materials are available in powder forms. While the existing literature in friction surfacing focuses on depositing composites by incorporating small quantities powders through holes in consumable rod, this is a first study that shows a large charge of powder can be converted to a homogeneous fully consolidated deposit in friction surfacing. A novel methodology is used that incorporates the high portion of powder feedstock into hollow consumable friction surfacing rods (up to 35% volume fraction). It was found that fully consolidated deposits can be produced with powder feedstocks using proposed methodology. A recrystallized, homogeneous, equiaxed microstructure was observed in Fe-Si 6.8 wt% and a new generation FeAlOY ODS alloy deposits processed with hollow stainless-steel friction surfacing rods. Both powder and rod material plasticize and deposit without bulk intermixing.

A Novel and Reliable Pixel Response Correction Method (DAC-Shifting) for Spectral Photon-Counting CT Imaging.

Spectral photon-counting cone-beam computed tomography (CT) imaging is challenged by individual pixel response behaviours, which lead to noisy projection images and subsequent image artefacts like rings. Existing methods to correct for this either use calibration measurements, like signal-to-thickness calibration (STC), or perform a post-processing ring artefact correction of sinogram data or scan reconstructions without taking the pixel response explicitly into account. Here, we present a novel post-processing method (digital-to-analogue converter (DAC)-shifting) which explicitly measures the current pixel response using flat-field images and subsequently corrects the projection data. The DAC-shifting method was evaluated using a repeat series of the spectral photon-counting imaging (Medipix3) of a phantom with different density inserts and iodine K-edge imaging. The method was also compared against polymethyl methacrylate (PMMA)-based STC. The DAC-shifting method was shown to be effective in correcting individual pixel responses and was robust against detector instability; it led to a 47.4% average reduction in CT-number variation in homogeneous materials, with a range of 40.7-55.6%. On the contrary, the STC correction showed varying results; a 13.7% average reduction in CT-number variation, ranging from a 43.7% increase to a 45.5% reduction. In K-edge imaging, DAC-shifting provides a sharper attenuation peak and more uniform CT values, which are expected to benefit iodine concentration quantifications.

Homogenization Treatment and Plastic Deformation Improve the Carbide Homogeneity of M42 High‐Speed Steel

The mechanical properties of high‐speed steel (HSS) are significantly influenced by the distribution of carbides formed during solidification. Despite extensive research, the nonuniform cooling in large ingots remains an area requiring further investigation. In this study, the microstructure inhomogeneity in M42 HSS with a diameter of 330 mm is examined. Models are developed to predict the dissolution of carbide networks during homogenization, achieving a 39% reduction in hardness variation (Δr) between the center and surface. Further improvements are attained through hot forging and rolling, leading to the fragmentation of the carbide network into a banded structure. Additionally, a detailed carbide tip dissolution model is established, providing a reliable basis for optimizing annealing processes. As a result, the carbide distribution converges significantly, with hardness variations reduced by 88% to only 0.3 hardness Rockwell C scale, showcasing substantial enhancement in material homogeneity.

Behavior of uranium(VI) in the presence of a geomaterial: an aquifer sediment

Sediments from aquatic environments can be contaminated with uranium from natural and anthropogenic sources; the behavior of U(VI) was investigated in two natural (SPT and DS) and gamma-irradiated (SPT-I) aquifer sediments in batch systems. Sediments were characterized, and quartz, albite, and kaolinite were mainly found in both sediments by X-ray diffraction. The points of zero charge were 5.4, 5.3, and 5.5, and the organic matter percentages were 48.0, 46.0, and 4.7% for SPT, SPT-I, and DS, respectively. Uranyl nitrate solutions were put in contact with each sediment considering different parameters (pH, contact time, initial concentration of uranium, and temperature). DS sediment did not adsorb any quantity of uranium at different pH values; SPT and SPT-I, attained the highest q e values at pH 3.5–4. The results show chemisorption on homogenous materials and endothermic processes. Desorption behavior of U(VI)/SPT showed a hundred percent desorption with nitric acid and humic acid solutions; in the last case, it is due to the high stability constant of the complex of uranium with the humic acids. The results help to understand the contamination and decontamination processes of sediments with uranium, the bioavailability of uranium, the ecological risk of U-contaminated sediments, and remediation. Doses of 1 MGy of Co-60 gamma radiation did not affect the uranyl ions behavior in the aquifer sediment.

Consideration of special effects for the application of an optimized fracture mechanics approach for the RPV safety assessment (CAMERA)

A comprehensive fracture mechanics test program called CAMERA was carried out for seven RPV base and weld materials in unirradiated and irradiated conditions. The objective was to establish an application-oriented completion of the fracture mechanics approach for the RPV safety assessment over the entire relevant temperature range up to operating temperature. For this purpose, fracture toughness tests in the ductile-brittle transition region with and without warm pre-stress (WPS), and in the ductile region (upper shelf) were performed in order to complete the fracture toughness curves of the material concerned. The test program was completed by various analytical and numerical calculations and microstructural analyses. The benefit of the WPS effect was confirmed for both the material (higher apparent fracture toughness values) and the load path (no initiation after maximum loading). By fracture toughness tests in the ductile regime the fracture toughness curve could be completed up to the operating temperature and the T0 based criterion temperature TUS above which no brittle fracture occurs was determined and successfully verified. Based on the results obtained by the Master Curve, WPS and crack resistance tests it was demonstrated how to integrate the criterion temperature TUS in the fracture toughness-temperature diagram including the loading transient to an application window for the RPV safety assessment that is based on T0 only. For three out of nineteen data sets the homogeneity screening procedure described in ASTM E1921 did indicate a macroscopically inhomogeneous material for that a generally conservative reference temperature T0IN was calculated that is on average 11 K higher than the reference temperature T0. A higher susceptibility of weld materials for macroscopic material inhomogeneity compared to base materials was found. The suitability of the applied SE(B) and C(T) specimens for the specific fracture mechanics tests (WPS and/or crack resistance in ductile region) was proven. Micromechanical Local Approach damage models (Bordet and Gurson) were successfully applied for test design and prediction of fracture toughness. The overall results reveal even so some open gaps remaining for future work that are addressed as well.

Indirect Tensile Strength Test on Heterogeneous Rock Using Square Plate Sample with a Circular Hole

Abstract An indirect testing method for determining the tensile strength of rock-like heterogeneous materials is proposed. The realistic failure process analysis method, which can consider material inhomogeneity, is applied to model the failure process of the square plate containing a circular hole under uniaxial compression. The influence of plate thickness and applied loads on the maximum tensile stress is investigated, and the tensile strength equation is deduced. Meanwhile, the initial cracking loads are obtained by the corresponding physical tests, and the tensile strengths are determined by substituting the initial cracking loads into the developed tensile strength equation. The values predicted by the newly proposed method are almost identical to those of the direct tensile tests. Furthermore, the proposed method can give the relatively small tensile strength error with the direct tensile test in comparison to the other test methods, which indicates that the proposed method is effective and valid for determining the tensile strength of rock-like heterogeneous materials.