True Modulus Research Articles

Correction of substrate-induced thin-film modulus deviations in nanoindentation measurements

The influence of substrate effects on the accurate measurement of mechanical properties of thin films remains to be clarified. Therefore, in this study, (AlCrNbSiTi)N thin films were deposited on four substrates—Si, SiO2, sapphire (Al2O3), and Ni—with varying thicknesses. Nanoindentation testing was conducted to measure the mechanical properties, revealing variations in the modulus of the thin film depending on the substrate and film thickness. To address these variations, we introduced the concept of two springs connected in series, following Hooke's law, to determine the elastic deformations of the thin film and its underlying substrate. By applying linear regression to four datasets corresponding to equivalent film thicknesses but different substrates, substrate-induced measurement errors were effectively mitigated, and the true modulus of the thin film was determined. Furthermore, we introduced a k value to represent the ratio of the actual force on the substrate to the force exerted by the indenter. Our findings indicate that even when the indenter contact depth is less than 1% of the film thickness, the substrate is still subjected to 7.6% of the applied force. This underscores the non-negligible nature of substrate elastic deformation during the measurement process and importance of correcting for substrate-induced variations in thin-film moduli. The findings can contribute to advancing the understanding of mechanical properties in ceramics and related materials.

How to Make the Stress Relaxation Experiment for Polymers More Informative.

Different viscoelastic models and characteristics are commonly used to describe, analyze, compare and improve the mechanical properties of polymers. A time-dependent linear relaxation modulus next to frequency-domain storage and loss moduli are the basic rheological material functions of polymers. The exponential Maxwell model and the exponential stretched Kohlrausch-Williams-Watts model are, probably, the most known linear rheological models of polymers. There are different identification methods for such models, some of which are dedicated to specific models, while others are general in nature. However, the identification result, i.e., the best model, always depends on the specific experimental data on the basis of which it was determined. When the rheological stress relaxation test is performed, the data are composed of the sampling instants used in the test and on the measurements of the relaxation modulus of the real material. To build a relaxation modulus model that does not depend on sampling instants is a fundamental concern. The problem of weighted least-squares approximation of the real relaxation modulus is discussed when only the noise-corrupted time-measurements of the relaxation modulus are accessible for identification. A wide class of models, that are continuous, differentiable and Lipschitz with respect to parameters, is considered for the relaxation modulus approximation. The main results concern the models that are selected asymptotically as the number of measurements tends to infinity. It is shown that even when the true relaxation modulus description is completely unknown, the approximate optimal model parameters can be derived from the measurement data that are obtained for sampling instants that are selected randomly due to the appropriate randomization introduced whenever certain conditions regarding the adopted class of models are satisfied. It is shown that the most commonly used stress relaxation models, the Maxwell and Kohlrausch-Williams-Watts models, satisfy these conditions. Since the practical problems of the identification of relaxation modulus models are usually ill posed, Tikhonov regularization is applied to guarantee the stability of the regularized solutions. The approximate optimal model is a strongly consistent estimate of the regularized model that is optimal in the sense of the deterministic integral weighted square error. An identification algorithm leading to the best regularized model is presented. The stochastic-type convergence analysis is conducted for noise-corrupted relaxation modulus measurements, and the exponential convergence rate is proved. Numerical studies for different models of the relaxation modulus used in the polymer rheology are presented for the material described by a bimodal Gauss-like relaxation spectrum. Numerical studies have shown that if appropriate randomization is introduced in the selection of sampling instants, then optimal regularized models of the relaxation modulus being asymptotically independent of these time instants can be recovered from the stress relaxation experiment data. The robustness of the identification algorithm to measurement noises was demonstrated both by analytical and numerical analyses.

Analytical Models for Predicting the Formation Resistivity Factor and Resistivity Index at Overburden Conditions

In this study, new parameters referred to as rock resistivity modulus (RRM) and true resistivity modulus (TRM) were defined. Analytical models were developed based on RRM, TRM, and Archie’s equation for predicting formation resistivity factor (FRF) and resistivity index (RI) under overburden pressure conditions. The results indicated that overburden FRF is dependent on FRF at initial pressure (ambient FRF), RRM, and net confining pressure difference. RRM decreases with cementation factor and rock compressibility. The proposed FRF model was validated using 374 actual core data of 79 plug samples (31 sandstone and 48 carbonate plug samples) from three sandstone reservoirs and four carbonate reservoirs, measured under four to six different overburden pressures. The developed FRF model fitted the experimental data with an average relative error of 2% and 3% for sandstone and carbonate samples, respectively. Moreover, the applications and limitations of the models have been investigated and discussed. Further theoretical analysis showed that overburden RI is a function of RI at initial pressure, TRM, and net confining pressure difference. The developed models supplement resistivity measurements and can be applied to estimate FRF, RI, and saturation exponent (n) variations with overburden pressure.

Distances to Local Group Galaxies via Population II, Stellar Distance Indicators. I. The Sculptor Dwarf Spheroidal* *Based in part on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. These observations are associated with program No.

We determine the distance to the Sculptor Dwarf Spheroidal via three Population II stellar distance indicators: (a) the Tip of the Red Giant Branch (TRGB), (b) RR Lyrae variables (RRLs), and (c) the ridgeline of the blue horizontal branch (HB). High signal-to-noise, wide-field VI imaging that covers an area and reaches a photometric depth approximately 2 mag fainter than the HB was acquired with the Magellan-Baade 6.5 m telescope. The true modulus derived from Sculptor’s TRGB is found to be mag. Along with periods adopted from the literature, newly acquired RRL phase points are fit with template light curves to determine mag. Finally, the HB distance is found to be mag. Absolute calibrations of each method are anchored by independent geometric zero-points, utilize a different class of stars, and are determined from the same photometric calibration.

Development of new mathematical models and their comparison with existing models for the prediction of compression pressure using the cut-strip method

Compression socks are a highly acclaimed textile garment for pressure exertion on the lower part of the leg (ankle). This part of the research explains the theoretical investigation of compression pressure by the modelization technique that helped to develop two new mathematical models by incorporating a few missing parameters for the measurement of compression pressure (kPa) and compared with existing models simultaneously. These new parameters including deformed width ( wf), true stress ( σT), logarithmic strain ( εT), true modulus (ET), and engineering stress ( σE)/strain ( εE)/modulus (EE) were incorporated and compared with Hook’s law ( k = fx) used for elastic materials at the ankle position. Various brands of compression socks composed of similar fibrous combinations as well as the knit type were purchased. Initially they were hand-washed, put on the leg for marking, a square was marked, circular cut-strips were detached from the ankle portion of the compression socks, and they were then cut to a linear shape for all 13 sock samples. For this, all linear cut-strips were installed on a Testometric tensile testing machine and were extended to a fixed elongation of 65% of gauge length five times simultaneously, and were then extracted for different extension (mm) values considering the requisite practical elongation values (circumferential difference between leg and socks at the ankle portion). The models developed were compared with Hook’s law and Laplace’s law. The developed models were compared with compression pressure values measured using existing models statistically.

Knot reversal acts non-trivially on the concordance group of topologically slice knots

We construct an infinite family of topologically slice knots that are not smoothly concordant to their reverses. More precisely, if \(\mathcal {T}\) denotes the concordance group of topologically slice knots and \(\rho \) is the involution of \(\mathcal {T}\) induced by string reversal, then \(\mathcal {T}/ \text {Fix}(\rho )\) contains an infinitely generated free subgroup. The result remains true modulo the subgroup of \(\mathcal {T}\) generated by knots with trivial Alexander polynomial.

Indentation modulus extrapolation and thickness estimation of ta-C coatings from nanoindentation

Coatings used in tribological applications often exhibit high hardness and stiffness to achieve high wear resistance. One coating characterization method frequently used is nanoindentation which allows the determination of indentation hardness and indentation modulus among other material properties. The indentation modulus describes the elastic surface behavior during indentation and is, among hardness, a direct indicator for wear resistance. To obtain the true indentation modulus of a coating, it must be measured with varying loads and then extrapolated to zero load. Current recommendation of the standard ISO 14577-4:2016 is a linear extrapolation which fits poorly for nonlinear curves. Such nonlinear curves are commonly found for high hardness mismatches between coating and substrate, for example, superhard tetrahedral amorphous carbon coatings (ta-C) on a steel substrate. In this study, we present a new empirical fit model, henceforth named sigmoid. This fit model is compared to several existing fit models described in the literature using a large number of nanoindentation measurements on ta-C coatings with wide ranges of indentation modulus and coating thickness. This is done by employing a user-independent and model agnostic fitting methodology. It is shown that the sigmoid model outperforms all other models in the combination of goodness of fit and stability of fit. Furthermore, we demonstrate that the sigmoid model’s fit parameter directly correlates with coating thickness and thus allows for a new approach of determining ta-C coating thickness from nanoindentation.

Nanomechanical properties of the ZEP-520 electron beam resist film

The ZEP-520 electron beam resist has been widely used in electron beam lithography and mask fabrication owing to its high resolution and good sensitivity. The stiffness and stability of the ZEP-520 resist are related to its mechanical properties, i.e., its hardness and elastic modulus. A higher hardness and elastic modulus can endow the ZEP-520 resist with higher stiffness, thereby allowing it to overcome the issue related to the collapse of the electron beam resist pattern. However, to date, only few studies have reported the mechanical properties of the ZEP-520 resist, and the relationship between the thickness of the ZEP-520 resist and its hardness and elastic modulus has not yet been determined. To obtain this information, in the present work, silicon is used as a substrate to analyse the hardness and elastic modulus of the ZEP-520 resist with different film thicknesses via nanoindentation. Additionally, a local area function method is proposed, through which the hardness of thin films can be conveniently and accurately obtained without being affected by the substrate effect. It is found that the true hardness increases with the decrease of the thickness of the ZEP-520 resist film; however, the true elastic modulus has no clear relationship with the film thickness. For a film thickness of 412 nm, the true hardness and elastic modulus of the ZEP-520 resist film are found to reach the maximum values of 0.52 and 11 GPa, respectively.

Stochastic fatigue damage in viscoelastic materials using probabilistic pseudo J-integral Paris' law

Fatigue damage of engineering materials severely affects the serviceability of their structures. It is impracticable to accurately predict fatigue damage process of engineering materials due to the variability of material properties, microstructure heterogeneity and others. This study aims to track the entire fatigue damage process of viscoelastic materials by coupling the variability with the fatigue damage mechanism. A typical viscoelastic material, asphalt binder, widely used in pavement engineering is selected for investigation in this study. A pseudo J-integral Paris' law model and probability theory are combined to establish a stochastic fatigue damage model for viscoelastic materials. Results show that the damage density can be determined by the apparent shear modulus and true shear modulus. The damage evolution rate is a function of material parameters (Paris’ law coefficients), apparent shear modulus, apparent shear strain amplitude and apparent phase angle. Then, a cumulative distribution function of loading time (TCDF) and damage density exceedance probability (DDEP) are derived and experimentally verified. Next, the stochastic fatigue damage model is proposed, which can track the entire fatigue damage process for viscoelastic materials, and it depends on the damage density, material parameters and variability parameters. The variability of minor damage can be used to predict the variability of severe damage based on the stochastic fatigue damage model.

Numerical inverse engineering as a route to determine the dynamic mechanical properties of metallic cellular solids

Metallic cellular solids are recognized for their high strength to weight ratio, being desirable for applications in the transportation industries to reduce fuel consumption and emissions. Although, there have been significant developments in the manufacturing of these materials and in the prediction of their static mechanical properties, their dynamic behavior (e.g. harmonic response and damping ratio) is still difficult to characterize by direct methods. This is due to the usual complex shape and low damping of metallic cellular solids and the inclusion of external damping by the instrumentation in vibration tests. This study presents an indirect method, based on a numerical inverse engineering approach, to determine the true apparent modulus and damping ratio of metallic cellular solids. For validation purposes, the proposed method was applied to Al-based non-stochastic cellular solids manufactured by investment casting with different architectures, being determined that their dynamic apparent modulus (1.4–217.2 MPa) is significantly different from the static structural results. The damping ratios of these samples were also successfully determined (0.002–0.011) and it is shown that these values are within the expected range for this aluminum based cellular solids.

Mathematical Underpinnings of the Multiwavelength Structure of the Tip of the Red Giant Branch

Abstract We consider the application of the tip of the red giant branch (TRGB) in the optical and in the near-infrared for the determination of distances to nearby galaxies. We analyze ACS VI (F555W and F814W) data and self-consistently cross-calibrate WFC3-IR JH (F110W and F120W) data using an absolute magnitude calibration of M I = −4.05 mag as determined in the Large Magellanic Cloud using detached eclipsing binary star geometric parallaxes. We demonstrate how the optical and near-infrared calibrations of the TRGB method are mathematically self-consistent, and illustrate the mathematical basis and relations among these multiwavelength calibrations. We go on to present a method for determining the reddening, extinction, and the true modulus to the host galaxy using multiwavelength data. The power of the method is that with high-precision data, the reddening can be determined using the TRGB stars themselves, and decreases the systematic (albeit generally small) uncertainty in distance due to reddening for these halo stars.

Q-Analogues of Dwork-type supercongruences

In 1997, Van Hamme conjectured 13 Ramanujan-type supercongruences. All of the 13 supercongruences have been confirmed by using a wide range of methods. In 2015, Swisher conjectured Dwork-type supercongruences related to the first 12 supercongruences of Van Hamme. Here we prove that the (C.3) and (J.3) supercongruences of Swisher are true modulo p3r (the original modulus is p4r) by establishing q-analogues of them. Our proof will use the “creative microscoping” method, recently introduced by the author in collaboration with Zudilin. We also raise conjectures on q-analogues of an equivalent form of the (M.2) supercongruence of Van Hamme, partially answering a question at the end of [Adv. Math. 346 (2019), 329–358].

Performance Characterization and Pressure Prediction of Compression Socks

Objective of the current research is to develop the mathematical model for the prediction of compression pressure by incorporating some of new but missing variables; circumferential aspect of leg (Lc) and socks (Sc), true stress (σT), true/logarithm strain (eT), true modulus of elasticity (ET), deformed width (Wf) at b ankle position. Socks samples were purchased, hand washed, donned, marked, sliced to loops and then cut to rectangular strips to obtain force-elongation curve at different extensions context to practical elongation. Additionally, all samples were evaluated to analyze the influnce of multiple-machine washes as well as for identification of most influential fabric parameter on compresson pressure, statistically. Standard sized wooden leg (S9=24 cm) attached with pneumatic based sensors (Salzmann MST MK IV) to medial side was used. Force-extension diagram was obtained using Testometric tensile tester. Mathematical models were developed on the basis of principle of Laplace’s law. Developed two models were analyzed and comapred with two existing recently developed models statitistically using MINTAB 17 software.

New approach for the prediction of compression pressure using the cut strip method

The main objective of the current research is the development of a new mathematical model for the prediction of compression pressure based on the incorporation of some new parameters. These new parameters include deformed width (wf), true stress ([Formula: see text]), true/logarithm strain ([Formula: see text]), true modulus of elasticity ([Formula: see text]), along with measurement of engineering stress ([Formula: see text]), engineering strain ([Formula: see text]) and engineering modulus of elasticity ([Formula: see text]) at ankle position. Various brands of compression socks comprising similar fibrous combinations, as well as knit type, were purchased. Initially they were hand washed, put on the leg for marking, marked in a square, sliced, and cut into rectangular strips. The rectangular cut strips were evaluated for force–elongation characterization at different strain values considering the requisite practical elongation values (circumferential difference between leg and sock at ankle portion). For pressure measurement, a Salzmann MST MK IV pressure measuring device using a standard-sized wooden leg (circumference = 240 mm) was used. For tensile evaluation, a Testometric tensile tester was used. In this research we developed the two mathematical models based on true Young’s modulus and engineering Young’s modulus were compared with Hooke’s law and Laplace’s law. The developed models were also compared with previously existing models statistically.

Investigation of mechanical properties of jointed granite under compression using lattice-spring-based synthetic rock mass modeling approach

Joint waviness is one of the important parameters of joint that need to be considered in civil and mining engineering design of structures built in or on jointed rock masses. However, this parameter is often overlooked by designers. In this study the mechanical properties of laboratory scale jointed granite with both planar and non-planar joints of different joint configurations are investigated using the lattice-spring-based synthetic rock mass (LS-SRM) modeling approach. Jointed granite models are calibrated using the laboratory test results and the calibrated lattice model parameters are used to perform both conventional triaxial and true triaxial compression tests. In addition, the influences of joint orientation, intensity, and persistence on the strength and deformability of jointed granite with both planar and non-planar joints are investigated. The simulation results show that the strength and deformation modulus of the jointed rock decrease with the increase of joint intensity and persistence. The strength and deformation modulus of the jointed rock are higher for the ones with non-planar joints. Under the true triaxial loading condition, the strength and deformation modulus of the jointed granite depend strongly on the magnitude and orientation of the intermediate principal stress.

Adjoint Hamiltonian Monte Carlo algorithm for the estimation of elastic modulus through the inversion of elastic wave propagation data

SUMMARYEfficient inversion of noisy seismic waveform data produced due to elastic wave propagation for the estimation of a high‐dimensional elastic modulus vector is achieved. Estimation is carried out in a Bayesian framework using Hamiltonian Monte Carlo (HMC) that enables efficient statistical estimation over high‐dimensional parameters. The truncated Karhunen‐Loève (K‐L) expansion is introduced to reduce the dimensionality of the elastic modulus vector. Expensive computations of the gradient of the state vector with respect to the parameter vector at every step are also eliminated through the adjoint method, which is developed from a general one‐step discretization of the governing second‐order ordinary differential equations (ODEs). An Adjoint HMC algorithm that employs a truncated K‐L expansion of the elastic modulus vector is presented. The efficacy of the algorithm is investigated with respect to two representative problems with varying geometric complexity. Adjoint HMC offers a significant speed up in gradient calculation time over the direct differentiation counterpart as the number of terms in the K‐L expansion increases. The algorithm is able to estimate the true elastic modulus within the credible intervals for both cases.

Characterizing Transformation Phenomena and Elastic Moduli of Austenite and Oriented Martensite of Superelastic Thin NiTi Wire through Isothermal Dynamic Mechanical Analysis

In this paper, superelastic behavior of Nickel Titanium thin wires is characterized using the method of dynamic mechanical analysis. Nominal dynamic storage modulus ′ is measured as function of nominal strain and stress during isothermal superelastic tensile tests at three testing temperatures above the reverse martensitic transformation finish temperature. The method brings new information on deformation mechanisms compared to the consideration of only tensile stress-strain curves. It is shown that determination of the elastic moduli E, especially at high strain, requires to calculate the true storage modulus ′. Using ′ and not ′, elastic modulus of oriented martensite is determined equal to 73 GPa of the same order than the elastic modulus of austenite equal to 70 GPa. Two models are then proposed to simulate experimental storage moduli evolution during the tests. A first model explains the ′ evolution during stress plateau by the localization phenomenon ; it leads to express ′ as function of the nominal strain. A second model describes the evolution of ′ after the stress plateau as function of true stress and test temperature. This model permits to determine the Clausius-Clapeyron coefficient of the forward transformation.

An Improved Technique for Prebored Pressuremeter Tests

The prebored pressuremeter (PBP) test results depend on the quality of borehole, including its diameter, verticality, effectiveness of cleaning, and degree of wall disturbance. The true elastic modulus cannot be reasonably determined using the cavity pressure-strain curves measured in a borehole of poor quality. In this research, a stainless-steel casing (SSC) was devised to improve the quality of borehole. SSC of 1 to 4 m length was conjugated with a mechanical drilling system (MDS). At a test site, an auger and the MDS with or without the SSC were attempted to drill boreholes for the PBP tests. Among the three techniques, the MDS with SSC produced the reliable values of the initial elastic modulus at a low strain range (0.003% to 0.089%).

Elastic Modulus of Nanofiller in Polymer-Matrix Composites

The paper explores the stiffness of the nanofiller particle aggregates in polymer matrix composites, such as particulate reinforced, carbon nanotube polymer and graphene polymer composites, i.e. composites reinforced with 0D, 1D and 2D nanofillers. It is shown that independently of the nanofiller, the structure of its particle aggregates is characterized by the fractal dimension as a physically accurate parameter. The true elastic modulus of the nanofiller particle aggregates is determined by two factors, namely the aggregates structure and the stiffness of the polymer matrix. The idea about the reinforcement degree control of polymer-matrix composites by the nanofiller structure is proved in this paper.

Experimental Research on Deformation Characteristics of Waste-Rock Material in Underground Backfill Mining

Waste-rock material used in underground backfill mining has a granular texture and acquires non-linear deformation characteristics when compressed. The deformation modulus of waste-rock measured by a laboratory compression test is significantly different from the true deformation modulus in the field, due to the complete confining effect of the loading steel cylinder. In this study, we performed a series of laboratory-based compression tests on waste-rock samples. The results showed that lab-acquired deformation modulus variations of waste rock could be divided into three stages: slow increase, accelerated increase, and rapid increase. We also measured the true deformation modulus of backfill waste rock by conducting a field test in gob areas of the Tangshan coal mine, China. The hardening process of backfill waste rock during the field test was analyzed, and could be divided into four stages: roof contact, rapid compression, slow compression, and long-term stable. With the increase of axial strain, the lab- and field-measured deformation moduli of waste rock both increased exponentially. A correction parameter was proposed to investigate the relationship between the field-generated true deformation modulus and the lab-tested deformation modulus. The correction parameter k positively correlated with the axial strain, in the form of an exponential function. The magnitude of k was between 0.5616 and 0.6531.