Strain-dependent Shear Modulus Research Articles

Shear modulus of sand-rubber mixtures: element testing and constitutive modeling

In this study, a series of resonant column tests was conducted to measure the shear modulus of sand-rubber mixtures at small strain amplitudes (i.e. between 10−4% and 10−2%), considering different rubber percentages and confining stress levels. The results were then combined with data obtained by dynamic hollow cylinder tests to investigate shear modulus degradation of the mixtures over a wider shear strain range. Based on the test results, a new expression was proposed to improve the prediction of maximum shear modulus of sand-rubber mixtures using the modified equivalent void ratio concept. A new constitutive model was also developed for estimation of strain-dependent shear modulus of the mixtures based on the modified hyperbolic framework. The shear modulus of the mixtures was found to be a function of rubber percentage, confining stress, the modified equivalent void ratio and the relative shear stiffness of rubber and sand. The experimental data and the developed models showed that the shear modulus decreased with rubber percentage and increased with confining stress. Moreover, the reference shear strain of the modified hyperbolic model increased with both rubber percentage and confining stress while its curvature coefficient increased more considerably with rubber percentage compared to the confining stress.

Effects of consolidation conditions on the dynamic shear modulus of saturated coral sand over a wide strain range

The utilization of coral sand as a construction material for ports and other infrastructure systems in coral reefs is prevalent, wherein the soil elements within slopes and embankments typically experience an anisotropic stress state. It is imperative to accurately characterize the degradation behavior of the stratigraphic material's stiffness across a wide range of strains. In this study, the maximum and strain-dependent shear modulus (G0 and G) characteristics of saturated coral sand under isotropic and anisotropic consolidations were investigated by performing multi-stage stress-controlled undrained cyclic torsional shear tests with cyclic hollow cylinder shear apparatus. The experimental results indicated that G0 and G generally rose with increasing initial effective consolidation pressure but decreased with increasing inclination of initial major principal stress from the vertical (αc). In contrast, the effect of consolidation stress ratio (kc) on G0 and G was nonmonotonic, and that of kc and αc on the strain-dependent shear modulus reduction (G/G0) curve was slight. By defining anisotropic consolidation indexes δ0 and δr used for quantifying the effect of consolidation-induced anisotropy on the G0 and the G/G0, respectively. A unified formula of G0 obtained by multiplying Hardin's model by δ0, as well as a unique formula of G/G0 obtained by correcting γr in the Davidenkov skeleton curve expression as a linear function of δr were established under various consolidation conditions, respectively. The proposed G formula over a wide strain range is well validated by the experimental data for siliceous sandy soils from the literature.

Shear modulus and damping ratio of saturated coral sand under generalised cyclic loadings

The influence of complex cyclic loadings associated with earthquakes on the shear modulus and damping properties of saturated coral sands has yet to be established. This paper presents comprehensive data sets from multi-staged strain-controlled (denoted Mγ), constant-amplitude strain-controlled (Cγ) and variable-amplitude strain-controlled (Vγ) undrained cyclic triaxial (UCTX) tests on a coral sand. The strain-dependent shear modulus and damping ratio from the Mγ UCTX tests and the cyclic stiffness degradation behaviour in the Cγ and Vγ UCTX tests are investigated. A correlation-function-based method is proposed to calculate the secant shear modulus and damping ratio at different strain levels, which shows higher precision at both small and large strain levels compared with the conventional methods. An important finding of the study is that the shear modulus reduction rate of the coral sand is significantly lower than that of siliceous sands, while the damping ratio for shear strain amplitude between 0·01% and 0·1% is well below that of siliceous sands. Two quantities: one is referred to as the strain ratio (ΓN) and used for characterising the normalised cycle-dependent strain amplitude, and the other referred to as the damage energy measure (δN) and used for characterising the cumulative normalised elastic strain energy absorption, are proposed such that a unique formula can be established for predicting the cyclic stiffness degradation in the UCTX tests with different loading patterns. This energy-strain-dependent formula is further validated using independent data from stress- and strain-controlled UCTX tests involving four patterns of cyclic loadings, and a satisfactory performance is demonstrated.

Localized stationary seismic waves predicted using a nonlinear gradient elasticity model

This paper aims at investigating the existence of localized stationary waves in the shallow subsurface whose constitutive behavior is governed by the hyperbolic model, implying non-polynomial nonlinearity and strain-dependent shear modulus. To this end, we derive a novel equation of motion for a nonlinear gradient elasticity model, where the higher-order gradient terms capture the effect of small-scale soil heterogeneity/micro-structure. We also present a novel finite-difference scheme to solve the nonlinear equation of motion in space and time. Simulations of the propagation of arbitrary initial pulses clearly reveal the influence of the nonlinearity: strain-dependent speed in general and, as a result, sharpening of the pulses. Stationary solutions of the equation of motion are obtained by introducing the moving reference frame together with the stationarity assumption. Periodic (with and without a descending trend) as well as localized stationary waves are found by analyzing the obtained ordinary differential equation in the phase portrait and integrating it along the different trajectories. The localized stationary wave is in fact a kink wave and is obtained by integration along a homoclinic orbit. In general, the closer the trajectory lies to a homoclinic orbit, the sharper the edges of the corresponding periodic stationary wave and the larger its period. Finally, we find that the kink wave is in fact not a true soliton as the original shapes of two colliding kink waves are not recovered after interaction. However, it may have high amplitude and reach the surface depending on the damping mechanisms (which have not been considered). Therefore, seismic site response analyses should not a priori exclude the presence of such localized stationary waves.

Dynamic Deformation Characteristics of Soil in the Tasks of Seismic Micro Zoning

The article discusses a technique for determining the nonlinear characteristics of the layers of computational models of the soil profile for the equivalent linear and nonlinear modeling of its response to seismic effects. The results of studying the factors influencing the curves of the strain-dependent shear modulus G (γ) and damping ratio D (γ) are analyzed. Based on the results of the analysis, the main parameters have identified that control the shape of the curves and allow you to quickly select the corresponding curves from the existing database with acceptable accuracy for each layer of the soil profile model. For clayey rocks: this is the plasticity index PL and the depth of occurrence; for sandy rocks: particle size, percentage, and depth. The paper presents the results of studying the effect of relative errors that arise when choosing the curves G (γ) and D (γ) for the soil layers of the computational seismic-geological model on the parameters of the frequency response of the soil, calculated using the equivalent linear modeling of ground vibrations during earthquakes. It was found that errors in determining the strain characteristics of soil layers in the calculation model lead to a shift in the maxima of the amplitude-frequency characteristic, to a change in the amplification factors of oscillations, as well as to the appearance of "false" maxima at high frequencies.
 The methodological approach to the formation of computational seismic-geological models of soil strata, by introducing the curves G (γ) and D (γ), which reflect the nonlinear properties of the soil, makes it possible to improve the computational methods for determining the resonance properties of soils under construction sites. The most accurate values of the frequency characteristics of the soil strata under construction sites are necessary for the development of effective measures to ensure the seismic resistance of the designed and existing facilities.
 As a result, the validity and accuracy of determining the quantitative parameters of seismic hazard at the construction and operational sites under study are increased.

ВПЛИВ ЛОКАЛЬНИХ ҐРУНТОВИХ УМОВ НА СЕЙСМІЧНІ КОЛИВАННЯ МАЙДАНЧИКА ТАШЛИЦЬКОЇ ГАЕС

Метою роботи є дослідити вплив фільтруючих властивостей осадової товщі на сейсмічні коливання будівельних або експлуатаційних майданчиків, розміщених на території України. Проаналізовано вплив фізико-механічних властивостей осадової товщі на сейсмічний ефект на поверхні в межах території Ташлицької гідроакумулювальної електростанції (ГАЕС) при можливих сейсмічних впливах з різними максимальними піковими прискореннями, що з імовірністю 99% не будуть перевищені за найближчі 50 років. Перевірено твердження, згідно з яким, зменшення товщини осадового шару завжди покращує сейсмічні умови будівництва. Результати отримано шляхом моделювання реакції ґрунтової товщі на сейсмічні впливи з використанням програмного продукту ProShake. При моделюванні поведінка кожно-го шару сейсмогеологічної моделі ґрунтової товщізадавалася моделлю Кельвіна-Фойгта (в‘язкопружною). Кожний шар сейсмогеологічної моделі ґрунтової товщі характеризувався такими параметрами, як: товщина шару, густина, швидкості поздовжніх і поперечних хвиль, нелінійні залежності модуля зсуву і коефіцієнта поглинання від зсувної деформації. Використання при розрахунках залежностей модуля зсуву і коефіцієнта поглинання від зсувної деформації дозволяють врахувати нелінійну реакцію ґрунтової товщі на сейсмічні впливи. Показано, що зменшення товщини осадового шару під будівельним майданчиком не завжди зменшує значення параметрів сейсмічних впливів. Рентабельність з усунення верхнього пухкого осадового шару слід оцінювати у кожному конкретному випадку. Проектувальникам слід враховувати інформацію про фільтруючі властивості ґрунтової товщі під будівельним майданчиком, вибираючи параметрипроектованих споруд такими, що забезпечують їх максимальну стійкість при сейсмічних впливах. Зміна параметрів ґрунтових умов на будівельному майданчику може істотно вплинути на сейсмічний ефект на його поверхні. Отримані дані про фільтруючі властивості ґрунтової товщі на кожній із ділянок досліджуваної території, для якої визначаються кількісні характеристики сейсмічної небезпеки, дозволяють одночасно забезпечити стійкість проектованих об‘єктів та істотно зменшити вартість сейсмостійкого будівництва за рахунок уникнення резонансного підсилення осадовою товщею сейсмічних коливань на власних періодах проектованих споруд.

Dynamic Properties of Clean Sand Modified with Granulated Rubber

Waste automobile tires are used as additives or replacements instead of traditional materials in civil engineering works. In geotechnical engineering, tires are shredded to certain sizes and mixed with soil, especially used as backfill material behind retaining walls or fill material for roadway embankments. Compared to soil, rubber has high damping capacity and low shear modulus. Therefore, it requires the determination of the dynamic characteristics of rubber/soil mixtures. In this paper, the cyclic behavior of recycled tire rubber and clean sand was studied, considering the effects of the amount and particle size of the rubber and confining stresses. A total of 40 stress-controlled tests were performed on an integrated resonant column and dynamic torsional shear system. The effects of the relative size and proportion of the rubber on the dynamic characteristics of the mixtures are discussed. The dynamic properties, such as the maximum shear modulus, strain-dependent shear modulus, and damping ratio, are examined. For practical purposes, simple empirical relationships were formulated to estimate the maximum shear modulus and the damping ratio. The change in the shear modulus and damping ratio with respect to shear strain with 5% of rubber within the mixture was found to be close to the behavior of clean sand.

Dynamic Properties of Sand–Fines Mixtures

Dynamic properties namely shear modulus (G) and damping (h) are input parameters for predicting the ground response of natural soil deposits. The dynamic properties of sands and clays are dependent on many factors such as void ratio, confining pressure, strain amplitude, morphology, and plasticity characteristics. Considerable research has been reported in the recent past by Seed and Idriss (Soil moduli and damping factors for dynamic response analysis, 1970), Hardin and Drnevich (J Soil Mech Found Div ASCE 98(7):667–692, 1972); Seed and Idriss (Ground motions and soil liquefaction during earthquakes. Earthquake Engineering Research Institute, Berkeley 1982), Vucetic and Dobry (J Geotech Eng Div ASCE 117(1):89–107, 1991) and many others on dynamic properties of sands and clays. The factors affecting the dynamic properties of sands and clays are well documented, but soils generally, occur as mixtures of sand, silt and clay in various proportions. There is very little effort made in understanding the dynamic behavior of sand–fines mixtures and therefore there is a need to investigate the dynamic properties of sand–fines mixtures. In the present paper, the results of the behavior of sand–fines mixtures under cyclic loads are discussed. The experimental program consisted of a series of cyclic triaxial shear tests conducted as per ASTM D 3999 (Standard Test Methods for Determination of the Modulus and Damping properties of soils Using the cyclic Triaxial Apparatus. American Society for Testing and Materials 1991) to investigate the strain dependent shear modulus and damping of sand–fines mixtures of different plasticity. Sand was obtained from the Sabarmati river and three types of locally available fines i.e. low-plastic, medium- plastic and bentonite fines were used. Fines content was varied from 15 to 60%. The effect of fines content, plasticity, and relative density on shear modulus and damping is reported. Shear modulus at 0.005% strain level correlates well with the plasticity index for sand–fines mixtures of low to medium plasticity i.e. 0 < IP < 40.

Cyclic Triaxial Test to Measure Strain-Dependent Shear Modulus of Unsaturated Sand

Cyclic Triaxial Test to Measure Strain-Dependent Shear Modulus of Unsaturated Sand

비틂전단시험에 의한 서해안 새만금 모래의 동적특성 연구

The dynamic characteristics of west coast sand were investigated in order to evaluate the design properties of the offshore wind turbine foundations to be constructed in the West Sea. Torsional shear tests were performed at different confining pressures and densities on specimens constituted by the dry fluviation method. The strain-dependent shear modulus and damping curves were obtained, together with modulus degradation curves. The results show that the confining pressure is more influential on the dynamic characteristics of the sand than the density. It was also found that the dynamic curves from this study were similar to those proposed by others. The modulus degradation ratio G/G 1st varies slightly at a small strain level, but increases significantly once beyond the intermediate strain level.

MODULUS REDUCTION AND DAMPING CURVES FOR SAND OF SOUTH-EAST COAST OF INDIA

Adequate information on dynamic soil properties, especially strain dependent shear modulus (G) and damping ratio (ξ) for each soil layer are the essential input data for seismic ground response analysis and soil-structure interaction studies. In the present study, the shear modulus and damping ratio of sand are estimated for a wide range of strains based on undrained strain-controlled cyclic triaxial tests. The bender elements are also utilized in the cyclic triaxial test to estimate the low strain shear modulus. For this purpose, the soil samples are taken from a nuclear power plant site located at the south-east coastal region of India. Based on the experimental results, an empirical expression is developed to calculate the maximum shear modulus, Gmaxas function of void ratio and effective confining stress. Predictive relationships are also developed for estimating normalized shear modulus and damping ratio curves for the sand. The predictive relationships are based on the hyperbolic model and cyclic triaxial test results. The developed modulus reduction and damping ratio curves from the predictive relationships are compared with the previously available curves in the literature.

FLOCCULATION, REINFORCEMENT, AND GLASS TRANSITION EFFECTS IN SILICA-FILLED STYRENE-BUTADIENE RUBBER

Abstract The introduction of silanes to improve processability and properties of silica-reinforced rubber compounds is critical to the successful commercial use of silica as a filler in tires and other applications. The use of silanes to promote polymer–filler interactions is expected to limit the development of a percolated filler network and may also affect the mobility of polymer chains near the particles. Styrene-butadiene rubber (SBR) was reinforced with silica particles at a filler volume fraction of 0.19, and various levels of filler–filler shielding agent (n-octyltriethoxysilane) and polymer–filler coupling agent (3-mercaptopropyltrimethoxysilane) were incorporated. Both types of silane inhibited the filler flocculation process during annealing the uncured rubber materials, thus reducing the magnitude of the Payne effect. In contrast to the significant reinforcement effects noted in the strain-dependent shear modulus, the bulk modulus from hydrostatic compression was largely unaltered by the silanes. Addition of polymer–filler linkages using the coupling agent yielded bound rubber values up to 71%; however, this bound rubber exhibited glass transition behavior which was similar to the bulk SBR response, as determined by calorimetry and viscoelastic testing. Modifying the polymer–filler interface had a strong effect on the nature of the filler network, but it had very little influence on the segmental dynamics of polymer chains proximate to filler particles.

Elasto-viscoplastic modeling of Osaka soft clay considering destructuration and its effect on the consolidation analysis of an embankment

A numerical modeling of Osaka soft clay was carried out using an elasto-viscoplastic constitutive model. The effect of destructuration, demonstrated by the shrinkage of the yield and the overconsolidation boundary surfaces and the strain-dependent elastic shear modulus, were studied through a comparison of the simulations with the experimental results of undrained triaxial compression tests. Although consideration of the structural degradation in the modeling of soft soil behavior leads to a substantial improvement, in terms of strain softening and post-peak responses, the strain-dependent shear modulus was introduced to reproduce more precise behavior, particularly before the peak stress. In order to evaluate the effect of these two aspects in a boundary value problem, a two-dimensional consolidation analysis of an embankment construction on a soft clay layer was conducted for three different cases. The deformations and the excess pore pressure responses for each case were presented and discussed. The strain localization, the consequent large ground displacement, and the temporary increase in pore pressure during the consolidation were observed in the cases with structural degradation. Considering the strain-dependent shear modulus, however, larger strain localization and displacement were predicted even in the early stages of loading.

Influence of constitutive model on numerical simulation of EPS seismic buffer shaking table tests

The paper describes two constitutive models for the simulation of expanded polystyrene (EPS) seismic buffer shaking table tests previously reported by the writers. The first model is linear elastic–plastic with Mohr–Coulomb criterion and Rayleigh damping. The second is called the equivalent-linear method (ELM) which incorporates hysteretic load–unload cycles using Masing-type functions and strain-dependent shear modulus and damping ratio functions. The constitutive models were applied to both the geofoam buffer and sand backfill in both approaches. Both models are shown to capture qualitative features of measured peak load-time response of the physical tests. The advanced ELM model gave predictions that were closer to the measured results for the seismic buffer test with the most compressible EPS when model accelerations and total wall forces were greatest. However, the simpler model is judged to be sufficiently accurate for practical purposes when the dynamic force increment is considered only.

Seismic stability and permanent displacement of landfill along liners

The three-part wedge limit equilibrium method for seismic stability analysis of the landfill along liners is presented. The approximate solutions of the factor of safety and the yield acceleration coefficient are obtained. Parametric studies show that the interface strength of liners, the shear strength of waste and the height of re- taining wall can influence the seismic stability of landfill along liners. The density and the shear wave velocity of the field waste are obtained by the borehole inves- tigation and the spectral analysis of surface wave (SASW), respectively. The strain-dependent shear modulus and damping ratio of the artifical waste are ob- tained by the moderate-scale dynamic triaxial tests. The one- dimensional (1D) equivalent linear dynamic response analysis is used to calculate the horizontal equivalent seismic coefficient-time history of the sliding landfill during earthquake. The seismic permanent displacement of the landfill along liners with different site conditons and heights is evaluated by the Newmark method. The catculated results show that ratio of ky /kmax, site conditions, the amplitude and frequency content of the bedrock motion can affect the seismic permanent displacement of the landfill along liners in some degree. Finally, the seismic stability and permanent dis- placements of three expanded configurations of a certain landfill case are analyzed.

Nonlinear Rheology of CTAB/NaSal Aqueous Solutions: Finite Extensibility of a Network of Wormlike Micelles

The nonlinear rheology of aqueous solutions of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) was investigated. The concentration of CTAB was fixed at 0.1 mol L(-1), and the concentration of NaSal was varied from 0.07 to 0.4 mol L(-1). For all test solutions, dynamic moduli were described with the Maxwell model having a single relaxation time, tau. Time evolutions of the shear stress, sigma, and the first normal stress difference, N(1), after inception of the steady shear flow were measured. For solutions having low NaSal concentrations, strain-hardening was observed and sigma and N(1) diverged at a certain strain when the shear rate, , exceeded tau(-1). For solutions with high NaSal concentrations, stress overshoot similar to that of ordinary entangled polymer solutions was observed at between tau(-1) and a certain critical rate, (C), while the strain-hardening was observed at > (C). A simple relationship for elastic solids, N(1)/sigma = gamma with gamma being the strain imposed by shear flow, held for all the solutions in the strain-hardening regime. The strain-hardening was attributable to the strain-dependent shear modulus and well described with the network theory considering the finite extensibility of network strands. The segment size of network strands was successfully determined. Thus, the stress-strain relationship obtained after the inception of fast flows is useful for characterizing the network properties.

Use of uniaxial compression test results in stress modelling around openings in nonlinear geomaterials

A phenomenological approach extending linear elastic constitutive equations to nonlinear materials is presented and applied to stress analysis around circular openings. This extension uses stress-dependent Lamé parameters instead of fixed values: a mean effective stress-dependent compressibility modulus C( σ′) and a shear stress-dependent inverted shear modulus, D( τ 2)=1/ G( τ 2). Both C( σ′) and D( τ 2) are expressed as power series, allowing several typical pre-peak material nonlinearities to be described with reasonable accuracy. A similar, strain-dependent model is also provided. In the strain-dependent version, the model uses volumetric strain-dependent bulk modulus K( ε) and shear strain-dependent shear modulus, G( γ 2). Material parameters of these nonlinear models can be identified from uniaxial compression tests. Numerical examples showing linear and nonlinear stresses around openings in different rocks are presented. It is shown that a model without yield does not necessarily have to underpredict opening stability. Underpredictions depend on the convexity of the rock compression curve and on the degree of rock nonlinearity.

Dynamic Response of Concrete-Faced Rockfill Dams to Strong Seismic Excitation

A dynamic plane-strain finite-element study of the response of a typical 100-m-tall concrete-faced rockfill (CFR) dam to strong seismic shaking is presented. The rockfill is modeled as an equivalent-linear material, whose strain-dependent shear modulus is proportional to the square root of the confining pressure. Coulomb's friction law governs the behavior of the interface between face slab and dam, and slippage is allowed to occur whenever the seismic shear tractions exceed the pertinent frictional capacity. Two sets of historic accelerograms, with peak ground accelerations (PGA) of about 0.40 g and 0.60 g , respectively, are used as excitation. Numerical results highlight key aspects of the seismic response of CFR dams with emphasis on the internal forces developing in the slab. It is shown that slab distress may be produced only from axial tensile forces, developing due mainly to the rocking component of dam deformation. For the 0.60 g shaking tensile stresses that are much higher than the likely tensile strength of concrete develop in the slab. The presented results do not account for potentially detrimental three-dimensional (3D) narrow-canyon effects.