Shear Modulus Of Material Research Articles

Spatial and temporal changes in small-strain shear modulus of geogrid-stabilized crushed aggregate materials

Geogrid stabilization can be used by transportation agencies to build durable roadways over soft subgrade soil. The performance of geogrid stabilization is highly dependent on the properties of the geogrid material, the aggregate material, and the interaction between the two materials when combined. Therefore, transportation agencies need to perform their own studies to assess the performance of geogrid stabilization for their local materials. Additionally, the current knowledge base needs to be continually expanded to a variety of aggregate-geosynthetic composites to develop performance-based design methods for geosynthetic-stabilization. This study evaluates the long-term performance of two geogrids used to stabilize a crushed aggregate material. A full-scale traffic loading system was built to simulate a full half-axle (40kN) traffic load for thousands of load cycles. Two trials were completed using the same crushed aggregate material. For each trial, two geogrid-stabilized sections, and one control section were evaluated. The performance of the test sections was monitored for 4000 load cycles by measuring surface rutting and completing multichannel analysis of surface waves (MASW) to measure aggregate stiffness. Results showed that the geogrid-stabilized sections had better long-term performance than the control sections with lower degradation of the as-built aggregate stiffness. There was good agreement amongst the MASW results, rutting measurements, and Shakedown analysis. It has been concluded that MASW is an effective method for evaluating the long-term performance of geogrid stabilization in aggregate layers with a customized instrumentation plan according to the targeted measurements.

Rigidity coefficients of rubber belts for dynamic testing of modulus of elasticity and shear modulus of non-wood engineered board

To determine the appropriate stiffness coefficient k values for rubber belts used in dynamic testing of the elastic modulus and shear modulus of timber and solid wood composite materials, this study employed three different thicknesses of rubber belts. Dynamic tests were conducted on straw boards, Laminated Veneer Lumber (LVL), and Spruce-Pine-Fir (SPF) materials, and the results were validated and analyzed using static four-point bending tests. The conclusions drawn from this research indicate that the range of stiffness coefficient k values for the rubber belts obtained through dynamic testing fell between 0.05 and 0.28 N/m. The correctness of the dynamic testing method was verified through static four-point bending tests. The error levels for elastic modulus E and shear modulus G of the same type of board measured using the three different rubber belts were below 9.5% and 9.8%, respectively.

Effects of fabric anisotropy on the small-strain shear modulus of granular materials

Effects of fabric anisotropy on the small-strain shear modulus of granular materials

A Constitutive Law Modeling the Mixed Hardening Behavior of Particle-Reinforced Composites

Abstract A homogenized elasto-plastic constitutive law (including both constitutive equations and inversed constitutive equations) is proposed in an incremental form for the particle-reinforced composites based on the flow theory of plasticity and the asymptotic homogenization method. The constitutive law can be used to predict the mixed hardening behavior of particle-reinforced composites under arbitrary loading conditions if the uniaxial tension test curve of matrix materials is known. It is found that the constitutive law of particle-reinforced composites is similar in form to the law of matrix materials. There is a simple proportional relationship between the yield stress, the plastic modulus, and the deviatoric back stress of particle-reinforced composites and the corresponding parameters of matrix materials, which is equal to the ratio of the shear modulus of composites to the shear modulus of matrix materials. The tangent modulus of particle-reinforced composites can be calculated using a simple arithmetic formula according to the tangent modulus of matrix materials. A numerical algorithm is suggested.

Characterization of Potting Epoxy Resins Performance Parameters Based on a Viscoelastic Constitutive Model.

To describe the evolution of residual stresses in epoxy resin during the curing process, a more detailed characterization of its viscoelastic properties is necessary. In this study, we have devised a simplified apparatus for assessing the viscoelastic properties of epoxy resin. This apparatus employs a confining cylinder to restrict the circumferential and radial deformations of the material. Following the application of load by the testing machine, the epoxy resin sample gradually reduces the gap between its surface and the inner wall of the confining cylinder, ultimately achieving full contact and establishing a continuous interface. By recording the circumferential stress-strain on the outer surface of the confining cylinder, we can deduce the variations in material bulk and shear moduli with time. This characterization spans eight temperature points surrounding the glass transition temperature, revealing the bulk and shear relaxation moduli of the epoxy resin. Throughout the experiments, the epoxy resin's viscoelastic response demonstrated a pronounced time-temperature dependency. Below the glass transition temperature, the stress relaxation response progressively accelerated with increasing temperature, while beyond the glass transition temperature, the stress relaxation time underwent a substantial reduction. By applying the time-temperature superposition principle, it is possible to construct the relaxation master curves for the bulk and shear moduli of the epoxy resin. By fitting the data, we can obtain expressions for the constitutive model describing the viscoelastic behavior of the epoxy resin. In order to validate the reliability of the test results, a uniaxial tensile relaxation test was conducted on the epoxy resin casting body. The results show good agreement between the obtained uniaxial relaxation modulus curves and those derived from the bulk and shear relaxation modulus equations, confirming the validity of both the device design and the testing methodology.

Systematic quantification of differences in shear wave elastography estimates between linear-elastic and viscoelastic material assumptionsa).

Quantitative, accurate, and standardized metrics are important for reliable shear wave elastography (SWE)-based biomarkers. For over two decades, the linear-elastic material assumption has been employed in SWE modes. In recent years, viscoelasticity estimation methods have been adopted in a few clinical systems. The current study aims to systematically quantify differences in SWE estimates obtained using linear-elastic and viscoelastic material assumptions. An acousto-mechanical simulation framework of acoustic radiation force impulse-based SWE was created to elucidate the effect of material viscosity and shear modulus on SWE estimates. Shear modulus estimates exhibited errors up to 72% when a numerical viscoelastic phantom was assessed as linearly elastic. Shear modulus estimates of polyvinyl alcohol phantoms between rheometry and SWE following the Kelvin-Voigt viscoelastic model assumptions were not significantly different. However, the percentage difference in shear modulus estimates between rheometry and SWE using the linear-elastic assumption was 50.1%-62.1%. In ex vivo liver, the percentage difference in shear modulus estimates between linear-elastic and viscoelastic methods was 76.1%. These findings provide a direct and systematic quantification of the potential error introduced when viscoelastic tissues are imaged with SWE following the linear-elastic assumption. This work emphasizes the need to utilize viscoelasticity estimation methods for developing robust quantitative imaging biomarkers.

Emerging technologies and advanced analyses for non-invasive near-surface site characterization

The in-situ small-strain shear modulus of soil and rock materials is a parameter of paramount importance in geotechnical modeling. It can be derived from non-invasive geophysical surveys, which provide the possibility of testing the subsurface in its natural and undisturbed condition by inferring the velocity of propagation of shear waves. In addition, for soil dynamics and earthquake engineering applications, the small-strain damping ratio plays a relevant role, yet its estimation is still challenging, lacking consolidated approaches for its in-situ evaluation. Recent advancements in instrumentation, such as distributed acoustic sensing (DAS), combined with advanced analysis methodologies for the interpretation of seismic wave propagation (e.g., machine learning and full waveform inversion), open new frontiers in site characterization. This paper presents and compares some advanced applications of measuring 1D and 2D variations in shear wave velocity and attenuation in-situ with reference to a specific case history.

Stress-strain state of composite multi-layer stay rope considering breakages in reinforcing elements and nonlinear distribution of mechanical properties

Purpose. Formulation of an algorithm for considering the influence of continuity breakages in fiber reinforcing elements on stay rope strength. Methods. Construction and analytical solution of a mathematical model of interaction for parallel fiber reinforcingelements connected through elastic material, in a case of continuity breakage of individual elements in reinforcement. Findings. A static strength calculation algorithm for a multi-layer stay rope with breakages in one cross-section of reinforcing elements is developed. It is established that a continuity breakage of an arbitrary element of a stay rope reinforcement leads to a significant change in internal loads of only the adjacent reinforcing elements and is practically independent of a nonlinear deformation character of cable components. Greater loads on reinforcing elements occur in a case of breakage of a corner reinforcing element, and the smallest loads – in a case of breakage of a central one.It is established that a number of rows of reinforcing elements in a stay rope and location of a damaged reinforcing element do not significantly affect displacement of cable end and do not affect distribution of loads among reinforcing elements in a damaged cross-section. Displacements depend on a ratio of shear modulus of elastic material and Young's modulus of reinforcing elements; the ratio is varied along cable length. It is established that a reinforcing element location with discontinuity does not significantly affect the character of relative growth of stay rope deformations in a case of nonlinear character of elastic shell deformation. Deformation nonlinearity of cable components does not affect redistribution of forces in stay rope with damaged reinforcing elements. Maximum relative displacements of reinforcing elements and the resulting maximum shear angles of material located between reinforcing elements are smaller than a value of the assumed nonlinearity coefficient. Characters of displacements of reinforcing elements are qualitatively similar. Scientific novelty. An analytical algorithm is developed for calculating a stress-strain state of a multilayer stay rope considering its design, nonlinearly of mechanical properties of its components distributed along the cable with damage to an arbitrary group of reinforcing elements in one cross-section. Practical significance. The developed algorithm allows considering a nonlinear deformation character of stay rope components on its stress state in a case of breakages in an arbitrary number of reinforcing elements arbitrarily located in a stay rope with a continuity breakage in one cross-section. The algorithm can be applied to determine a stress-stress state of a stay rope with breakage in a cross-section infinitely distant from rope ends. The algorithm allows considering the influence of breakages in reinforcing elements on rope strength, which increases rope reliability in a structure.

Stress-strain state of a composite tractive element with a broken structure due to elastomer shell rheology

Purpose. Establishing a dependency of change in a stress-strain state for a rubber-cable tractive element with broken structure due to rheology of a rubber shell. Methods. Analytical solution of a model of a rubber-cable tractive element with a broken structure due to rheology of a rubber shell. Results. An algorithm for determining a stress-strain state of a rubber-cable tractive element with a broken structure due to rheology of a rubber shell is developed. The suggested algorithm involves splitting the rope into two parts, which allows considering the influence of a dependency of shear modulus on deformations in a form of a broken line constructed of two segments. Splitting the rope in the area of integrity breakage into three or more parts allows considering a more complex dependency of shear modulus on shear deformations. A mechanism of changing a stress-strain state of a rubber-cable rope due to rheology of a rubber shell is established. The local influence of changes in properties of elastic material interacting with a damaged cable on a stress-strain state of a rubber-cable tractive element with a broken structure is analyzed. Scientific novelty. A mechanism of influence of a shear modulus of rubber shell material in the rubber-cable rope (belt) with a cable integrity breakage on stress-strain state of composite tractive element is established to be time-varying and nonlinearly dependent on deformations. Practical significance. Consideration of shell rubber rheology makes it possible to predict a rope stress state considering a non-linear law of changes in properties of rubber during its use and to increase safety and operational reliability of rubber-cable tractive elements. A local change in mechanical parameters does not increase the danger of using a rope with continuity breakages of cables. An increase in lengths of redistribution of forces and displacements requires an increase in length of steps in butt-joint connections.

Methodological Foundations Protective Structures Development for Shielding Electromagnetic and Acoustic Fields

In many cases the simultaneous protection against electromagnetic and acoustic fields of wide frequency ranges is required. The complexity of such a task lies in the impossibility of significantly reducing low-frequency sound levels with traditional sound-absorbing materials, as well as the different requirements for materials that shield low-frequency and high-frequency electromagnetic fields. It is proposed to solve this problem by creating a two-layer structure, the front surface of which is solid, and the inner surface is perforated. The front panel can be made of non-conductive material. It is a variant of the Bekeshi panel and is mainly intended to reduce the level of low-frequency sound of a certain frequency with high amplitude. The perforation of the internal (metal) panel is chosen based on the need to provide shielding of electromagnetic and acoustic fields of certain frequencies or frequency bands. A calculator is provided for calculating the required panel parameters (linear dimensions, thickness, hole diameters, their number per unit of surface area). It is advisable to fill the space between the panels with a sound-absorbing material, for example, granulated polystyrene. It provides sound absorption of medium and high frequencies and makes the design broadband. The use of ferromagnetic material of the inner panel provides protection against the magnetic component of the ultra-low frequency electromagnetic field (mainly industrial). The perforation of the panel is calculated based on the waveguide theory of high-frequency electromagnetic waves. The results of tests of the effectiveness of facing metal-polymer material in the form of tiles are presented. The performance of the material is satisfactory for most industrial and domestic conditions. Determination of the mechanical properties of the material (Young's modulus, shear modulus and Poisson's ratio) showed that it is not inferior to known materials even with a large content of shielding substance. In order to rationalize the design of the structure, a priority factor and frequency (frequency band) is selected on the basis of live measurements of the electromagnetic and acoustic spectra. It is shown that the calculations have large volumes, and the solution of two-factor optimization problems is not always possible. Appropriate creation of application software to simplify the process of designing protective structures and rationalizing panel parameters based on the principles of reasonable sufficiency.

Water impact of deformable spheres with vertical and oblique entries

At low Froude numbers, vertical and oblique impacts of deformable spheres into water are performed. The dependence levels of the hydrodynamics, cavity formations, and stress responses on the oblique angle are investigated. Based on theoretical and numerical analyses, the differences between the vertical and oblique impacts in sphere deformations and cavity formations are clarified. Elastic forces inside the sphere induce different models of the deformation behaviors during oblique water entry, and the sphere with a smaller oblique angle extends the formation time of the oblate deformation and nested cavity. Over a single deformation period, the vertical hydrodynamic forces do not continuously increase with the oblique angle, and they depend on the deformation behavior. Additionally, the evolution in the high-stress area at the bottom of the sphere caused by the oblique motion is similar to a solar eclipse. Over the first deformation period, the energy change in the sphere with different oblique angles can be predicted from the dimensionless ratio of the material shear modulus to the impact hydrodynamic pressure.

Characterization of the healing potential of asphalt binders with separation of thixotropy after cyclic loading

Considering the self-healing property of asphalt binders in material selection and pavement design can help improve pavement performance and reduce maintenance costs. The study of self-healing of asphalt materials gradually is becoming an important step in the design of durable pavements. Traditionally, the performance recovery of asphalt binders has been attributed only to microcrack healing, which may lead to an overestimation of the self-healing ability. This is because an increasing number of studies have shown that the driving mechanism of asphalt binder performance recovery includes not only self-healing but also thixotropy. In order to accurately quantify the self-healing ability of asphalt binders without the influence of thixotropy, a novel test procedure and a three-step analysis method are proposed. The novel test procedure includes the large amplitude time sweep test and the small amplitude time sweep test. The three-step analysis method consists of establishing a thixotropy relationship between the dynamic shear modulus and the phase angle, calculating the effective modulus from the phase angle data via the thixotropy relationship, and determing the healing degree of asphalt binders by separating the thixotropy. The results show that the recovery of the dynamic shear modulus of asphalt materials results from the combined action of thixotropy and self-healing. And thixotropy plays a prominent role in the dynamic shear modulus recovery at the initial resting period. The contribution of self-healing to dynamic shear modulus recovery gradually increases with increasing resting time. Without excluding the influence of thixotropy, the self-healing ability of asphalt binders is overestimated, especially the short-term self-healing ability.

A time-temperature-ageing shift model for bitumen and asphalt mixtures based on free volume theory

ABSTRACT This paper aims to develop a general form of the time-temperature-ageing shift model for bituminous materials based on free volume theory and validate its application in the Finite Element (FE) modelling for predicting aged asphalt mixtures’ mechanical responses. The frequency sweep, dynamic modulus, and Fourier Transform Infrared Spectroscopy (FTIR) tests were used to validate the proposed time-temperature-ageing shift model. A generalised Maxwell model incorporating the proposed shift model of asphalt mixtures was used to predict the mechanical responses of aged asphalt mixtures using FE modelling, verified via laboratory cyclic load tests. Results indicate that the time-temperature-ageing shift function can model the complex shear modulus and dynamic modulus master curves of bituminous materials, addressing the modulus dependency of temperature and ageing. Temperature dominates the modulus change of bituminous materials in the ageing process. The coefficients of the time-temperature-ageing shift model have clear physical meanings and can potentially quantify the effects of asphalt mixtures’ air voids on their temperature and ageing susceptibilities. The time-temperature-ageing shift model can be incorporated into the constitutive relations of the FE model to effectively couple and efficiently predict the effects of temperature and ageing on the mechanical responses of aged asphalt mixtures.

An exact expression of three-body system for the complex shear modulus of frictional granular materials.

We propose a simple model comprising three particles to study the nonlinear mechanical response of jammed frictional granular materials under oscillatory shear. Owing to the introduction of the simple model, we obtain an exact analytical expression of the complex shear modulus for a system including many monodispersed disks, which satisfies a scaling law in the vicinity of the jamming point. These expressions perfectly reproduce the shear modulus of the many-body system with low strain amplitudes and friction coefficients. Even for disordered many-body systems, the model reproduces the results by introducing a single fitting parameter.

Research progress on dynamic testing methods of wood shear modulus: A review

Wood is a non-homogeneous and orthotropic natural polymer material. It is important to test the wood shear modulus and elastic constants accurately and reliably using dynamic methods. Based on the introduction of the advantages of six common methods for dynamic testing of wood shear modulus, such as free plate torsional mode method, free bar torsional vibration method, and Timoshenko beam iterative method, issues associated with the applicability and accuracy of these methods are also pointed out. Recent methods, such as the free square plate torsional mode method and the square plate static torsional strain method, that were developed to dynamically test the shear modulus of wood and wood composite materials, are presented as effective ways to tackle these issues. These new approaches are expected to provide beneficial technical support for using small specimens, overcoming the size effect of specimens, simplifying the testing procedures, improving the test accuracy, and expanding the application range in the dynamic testing of wood shear modulus. These approaches have practical significance in promoting the industrialization and development of structural engineering, furniture and interior decoration, transportation, military, and musical instrument industries.

Послойный динамический модуль сдвига в поперечном сечении древесностружечной плиты

The research results are aimed at the formation of ideas concerning the features of molecular motion of the components of the outer and inner layers of the particle board material in the temperature range from room temperature to 275–300 °C, as well as at revealing the fact of binder undercuring in the inner layers. The paper provides data on the temperature dependences of the dynamic shear modulus of the outer and inner layers of the particle board material, obtained by dynamic mechanical analysis using a torsion pendulum. We found significant differences in the nature of the dependencies for samples taken at different distances from the surface layers. The material of the outer layers is characterized by the typical pattern of a continuous irregularly consistent decrease in the dynamic shear modulus with increasing temperature, which is common to most polymeric and composite materials. A short-term intermittent increase in the dynamic shear modulus relative to neighboring areas was detected in the inner layers at 140±5 °C, which is not typical for materials in a stable state. There is also a tendency for the dynamic shear modulus of the particle board material to decrease at room temperature with distance from the outer layers, due to the heterogeneous fractional composition and differences in the nature of chemical cross-linking during hot pressing. It has been assumed that the detected anomalous increase in the dynamic shear modulus in the inner layers of the material at 140±5 °С is a symptom of the binder post-treatment process directly in conditions of its heating when measuring the dynamic shear modulus by dynamic mechanical analysis. Thus, it is concluded that the particle board sample shows maximum curing of the thermosetting binder in the layers, which are 4.5–5.0 mm distant from both surfaces. Partial undercuring of the binder occurs in the inner layers, which are more than 5.5 mm away from the surfaces. Thus, it is shown that the method of dynamic mechanical analysis can be used as a tool to control the presence in the inner layers of the particle board material of the components of thermosetting binder, undercured in the hot pressing process, which will help to obtain a material with more stable characteristics.

Experimental Study on the Gmax Characteristics of the Sand-Silt Mixed Soil Materials Using Bender Element Testing.

To study the small strain shear modulus (Gmax) of saturated sand-silt mixed soil materials, a series of tests were conducted using the bender element apparatus, and the influences of fines content (FC), relative density (Dr), and effective confining pressure () were taken into consideration. The test results indicate that the Gmax of the mixed soil materials decreases first and then increases with the FC up to 100% with Dr = 35% and 50%, while the Gmax decreases with the increasing FC when Dr = 60%. Moreover, for a given Dr, the Gmax increases with the increasing , and the increase rate keeps constant under various FCs. The Gmax of specimens under various FCs decreases with the increase of the void ratio (e). The decrease rate between the Gmax and e differs when the is given, which is influenced by the FC. The Gmax of the mixed soil materials can be evaluated by the Hardin model when the FC is determined. The best-fitting parameter A of the Hardin model first decreases and then increases as FC increases. The revised Hardin model, considering the influence of FC, , and e, can be used to evaluate the Gmax for different types of sand-silt mixed soil materials. The error between the evaluated and tested Gmax is less than 10%.

The Torsional Rigidity of a Rectangular Prism

Using the membrane analogy, in 1934 Timoshenko derived the torsional rigidity of a rectangular prism of isotropic material as a function of its material shear modulus, width and thickness. However, he did not consider the energy conservation criterion, as it could be either unnecessary or replaced by other criteria in Timoshenko’s process. To confirm the correctness of Timoshenko’s solution, this work re-derives the torsional rigidity by considering all the equilibrium conditions, boundary conditions, symmetric and anti-symmetric conditions of displacement and stress, the energy conservation criterion, and even the energy minimization criterion. Using the TSAI technique, exact solutions for the displacements, strains, stresses and the torsional rigidity are derived perfectly. The derived torsional rigidity is in a completely different form from that derived by Timoshenko and is numerically identical. Interestingly, the solutions derived in this work verify that, when the values of the width and thickness of the rectangular prism are swapped, the value of the torsional rigidity remains the same, which makes perfect sense physically but is not discussed in Timoshenko’s process or any other research. This work presents a procedure considering all the mathematical details and the results remain correct when the width and thickness of the prism swap. This fact makes perfect sense physically, though has never been expounded before in Timoshenko’s or other researchers’ solutions, either for torsional rigidity or for the induced shear stresses and displacements.

Highly transparent MgAl0.5Ga1.5O4 ceramic for overcoming the trade-off between infrared transmittance and mechanical properties

A spinel-type MgAl0.5Ga1.5O4 transparent ceramic was prepared by pressureless sintering with hot isostatic pressing (HIP) post-treatment. The fully dense transparent ceramic with an average grain size of 47.5 μm exhibits excellent mechanical properties and wide transmission range. The transmission wavelength covers a range of 0.23–7.4 μm. The in-line transmittance is above 84.3% in the 3–5 μm atmospheric window. The thermal conductivity is 11.46 W·m−1·K−1 at ambient temperature. Benefiting from the introduction of stronger Al–O bonds in the crystal lattice, the hardness, Young's modulus, and shear modulus of ceramic material were enhanced to 12.2, 269.8, and 105.1 GPa, respectively. The superior overall performance of the MgAl0.5Ga1.5O4 transparent ceramic offers a chance to overcome the trade-off between infrared transmittance and mechanical properties.

Characterizing the complex shear modulus of materials with an iPhone

Dynamic mechanical analysis (DMA) is a technique that accurately measures the mechanical properties, such as stiffness and damping, of a material as a function of frequency, temperature, and time. However, DMA is expensive, requires significant training to operate, and must be performed on-site, lending itself impractical for hands-on classroom activities or research during the early stages of a global pandemic. To characterize the complex shear modulus of seagrass during the COVID-19 shelter-in-place order in May 2020, an iPhone XS Max was used as the tail mass of a freely oscillating torsional pendulum. The Vibrations application (Diffraction Limited Design, LLC) acquired the angular velocity of the underdamped system during oscillation, and the damped natural frequency and decay rate were later extracted from the time series. By measuring the cross-section and length of the material under test, the natural frequency and decay rate were used to determine the complex shear modulus. A strip of 1-cm-wide polyethylene film (McMaster Carr #85655K13) was first used to test the measurement technique. iPhone-obtained shear modulus of the strip was within the range of tabulated values, and was in agreement with DMA-obtained shear modulus measured after COVID-19 restrictions were relaxed. Uncertainty analysis and instructional applications will be discussed.