Variation Of Shear Modulus Research Articles

Characterization of the Functionally Graded Shear Modulus of a Half-Space

In this article, a method is proposed for determining parameters of the exponentialy varying shear modulus of a functionally graded half-space. The method is based on the analytical solution of the problem of pure shear of an elastic functionally graded half-space by a strip punch. The half-space has the depth-wise exponential variation of its shear modulus, whose parameters are to be determined. The problem is reduced to an integral equation that is then solved by asymptotic methods. The analytical relations for contact stress under the punch, displacement of the free surface outside the contact area and other characteristics of the problem are studied with respect to the shear modulus parameters. The parameters of the functionally graded half-space shear modulus are determined (a) from the coincidence of theoretical and experimental values of contact stresses under the punch and from the coincidence of forces acting on the punch, or (b) from the coincidence of theoretical and experimental values of displacement of the free surface of the half-space outside the contact and coincidence of forces acting on the punch, or (c) from other conditions. The transcendental equations for determination of the shear modulus parameters in cases (a) and (b) are given. By adjusting the parameters of the shear modulus variation, the regions of “approximate-homogeneous” state in the functionally graded half-space are developed.

Variation of shear moduli across superconducting phase transitions

We study how shear moduli of a correlated metal change across superconducting phase transitions. Using a microscopic theory we explain why for most classes of superconductors this change is small. The Fe-based and the A15 systems are notable exceptions where the change is boosted by five orders of magnitude. We show that this boost is a consequence of enhanced nematic correlation. The theory explains the unusual temperature dependence of the orthorhombic shear and the back-bending of the nematic transition line in the superconducting phase of the Fe-based systems.

Transient interaction of edge interfacial cracks in bonded dissimilar strips with a functionally graded interlayer: Antiplane deformation

Transient stress intensification due to the interaction of two parallel, edge interfacial cracks in bonded media with a functionally graded interlayer is investigated under the condition of antiplane shear impact. The graded interlayer is assumed to exhibit power-law variations in shear modulus and mass density between dissimilar, homogeneous semi-infinite strips. Laplace and Fourier integral transforms are employed to derive a system of singular integral equations with generalized Cauchy kernels, which is solved in the Laplace domain. Mode III stress intensity factors are defined and evaluated as a function of time. Numerical results include variations of dynamic stress intensity factors, addressing the impact-induced interaction between two nearby edge interfacial cracks affected by geometric and material parameters of the bonded system. More specifically, this study elaborates on the effects that layer thickness, crack length, shear modulus, and mass density have on the dynamic overshoot characteristics of transient behavior in the near-tip regions.

Inverse coefficient problem of the variable properties reconstruction for the viscoelastic cylindrical waveguide

AbstractAn effective technique for inverse coefficient problem solving of the shear modulus reconstruction for both inhomogeneous elastic and viscoelastic cylindrical waveguide is proposed. The identification is carried out using the displacement field data measured on the finite part of the outer boundary of the cylindrical waveguide in the torsional vibrations mode. An iterative scheme for solving the inverse problem is developed. At each iteration step, the correction to the shear modulus distribution function is calculated using the Fredholm integral equation of the first kind with a smooth kernel. Numerical results are obtained for both direct and inverse problems for different laws of the shear modulus variation.

Does the stacking fault energy affect dislocation multiplication?

This paper investigates the interdependence of the stacking fault energy (SFE) and the stress required for dislocation multiplication. For this purpose copper-aluminum (Cu-Al) alloys with varying Al content are investigated by spherical nanoindentation. The load-displacement curve shows a characteristic pop-in, which is interpreted as elastic-plastic transition and correlated to the dislocation source nucleation and/or activation stress. The statistical analysis of the pop-in behaviour documents a strong dependence of the maximum shear stress beneath the indenter on the Al content, which cannot be explained by a variation in shear modulus. Instead, the pop-in stress clearly increases with increasing stacking fault energy.

Shearing resistance of tailing sand waste pollutants mixed with different contents of fly ash.

The co-placement of mine tailings and fly ash (CMF) can reduce acid mine drainage (AMD) production and decrease metal mobilization. This aids in waste management construction. However, few people have studied a large number of tailing sand-fly ash mixtures under the condition of neutral saturated solution in tailing ponds, wherein the pozzolanic reaction is highly gradual. In this study, a series of tests were conducted to determine the monotonic and cyclic shear characteristics of a mixture of fly ash and tailings. In particular, the effects of the fly ash content on the monotonic shear peak, shear strength parameters, dynamic modulus, and damping ratio of the mixture were analyzed. The results reveal that in a monotonic shear test, the peak shear strength of the saturated CMF mixture decreases as the fly ash content increases. The shear strength parameters (cohesion c and internal friction angle φ) were observed to increase and decrease linearly, respectively, as the fly ash content increased. Furthermore, the maximum dynamic shear modulus was observed to decrease by 41.4% as the fly ash content increased from 0 to 50%, during the cyclic cutting process. Moreover, the experimental results fit well with the fitting formula for the variation in shear modulus in the cyclic shear process of the saturated CMF mixture with varying fly ash content. Meanwhile, the initial damping ratio of the cyclic shear was observed to increase from 10.3 to 13.6% as the fly ash content increased. Therefore, when the CMF method is used to treat AMD waste, it is necessary to consider the extent to which the design stability of the tailing pond may be reduced. These experimental results can be used as a reference for similar CMF projects.

Plane contact problem between a rigid punch and a bidirectional functionally graded medium

Analytical and computational solutions are presented for the sliding frictional contact problem between a bidirectional functionally graded material (BFGM) half-plane and an arbitrarily shaped rigid punch. The plane-strain elasticity is regarded in the solutions. Independent shear modulus variations are imposed through the lateral and the thickness directions of the elastic medium. Fourier transformation techniques are utilized in the derivation of the field quantities. The quartic characteristic equation is solved using the Ferrari's method. A singular integral equation (SIE) of the second kind is then formulated considering a known displacement gradient on the contact surface. Solution of the SIE is performed for the flat and triangular punch profiles via an expansion-collocation method, quadrature integration techniques and a recursive integration method for the Cauchy integral. Finite element analyses (FEA) the same contact problems are also implemented choosing the augmented Lagrange method as the iterative algorithm. The parametric analyses indicate the reliability and validity of both procedures developed in this study. Extra numerical results are generated to show the influences of the problem parameters on the surface stresses and contact forces. It is observed that the contact responses can be influentially controlled upon imposing a bidirectional gradation in an elastic material.

The mode III problem of a Dugdale-Barenblatt interfacial crack in a FGM strip bonded to a homogeneous strip by the integral method

The elasto-static anti-plane problem of a Dugdale-Barenblatt interfacial crack in a functionally graded strip bonded to a homogeneous substrate is formulated in term of a singular integral equation. The functionally graded strip is modeled with an exponentially variable shear modulus. The singular integral equation is solved using Tchebychev polynomials. The influence of the non-homogeneity parameter, the shear modulus of the homogeneous strip, the widths of the functionally graded strip and the homogeneous strip, and the critical sliding displacement jump in the Dugdale-Barenblatt cohesive model on the fracture load are discussed.

A Green operator-based elastic modeling for two-phase pantographic-inspired bi-continuous materials

We propose an estimate for the effective elastic properties of a new imagined two-phase bi-continuous composite material type with a so-called ``pantographic-inspired” (P-I) architecture in the sense of a matrix reinforcement which is a 3D fiber network capable of large, pantographic-like, deformations, owing to particular properties carried by the fiber interconnections. Such fiber networks co-continuous with the matrix are described from an assemblage of inter-penetrated planar alignments of parallel infinite, identical and equally distant, rods or beams that we call fiber layers or FPAs for ``fiber planar alignments”. Piece-wise linear deformation of such a structure is constrained, as 2D pantographs are, by the specific relations that link the evolutions of the FPA misorientations, of the fiber inter distance and concentration within each FPA, and of the given characteristics to the through-layer interconnections. We derive effective properties within a well defined first gradient elastic homogenization scheme for macro-homogeneous composite structures, based on explicating the involved mean Green operator (mGO) for the representative pattern of such a P-I networked reinforcement in an isotropic matrix. This mGO for P-I composites is built in making use of previously derived and presented mGOs for rod and beam FPAs. Comparisons with Young and shear modulus variations during numerical homogeneous extension simulations show that the proposed modeling provides relevant effective property evolutions with the advantage of being analytical. Comparing so estimated force-displacement curves with numerical ones for 2D pantographs enables to identify ways to further account in the modeling for the specific strengthening effects of the FPA interconnections in pantographs. This tends to prove such P-I composite structures, still to be manufactured, to possibly also behave similarly to pantograph ones.

Mechanical Property Variation of a Rotating Cantilever FGSW Beam under Parametric Excitation

We report here the dynamic stability of functionally graded sandwich (FGSW) rotating cantilever Timoshenko beams under parametric excitation. Power law with various indices as well as exponential law were used to find out the properties along the thickness of the FGSW beam. The stability boundaries were established using Floquet’s theory. The equation of motion was governed by Hamilton’s principle and solved by Finite element method. The power index was optimized for uniform variation of shear modulus along the thickness of FGSW beam.The shear modulus variation along the thickness of the FGSW beam was compared both by power law and exponential law.It has been confirmed that the Exponential distribution of constituent phases renders better stability compared to power law distribution of the phases in the functionally graded material(FGM).

Axisymmetric adhesive contact problem for functionally graded materials coating based on the linear multi-layered model

The contact problem with adhesion for a functionally graded coated half-space indented by a rigid spherical punch is considered. The whole contact region is stick assuming the interfacial friction sufficient to prevent any slip taking place between the indenter and the coated half-space. The linear multi-layered model is used to model functionally graded materials with arbitrarily varying shear modulus and constant Poisson’s ratio under axisymmetric loads. By using the transfer matrix method and Hankel integral transform technique, the problem is reduced to a set of Cauchy singular integral equations. An iterative method is developed to numerically solve singular integral equations. The contact stress and indentation are calculated for different variations in elastic modulus. It is found that appropriate gradual variation in the shear modulus can improve the resistance to contact damage at the contact surfaces.

Reducing stress concentration factor by strengthening circular hole with functionally graded incompressible material layer

Abstract The reduction in the stress concentration factor (SCF) around a circular hole in an infinite panel by adding an isotropic functionally graded incompressible material (FGIM) lining is analytically investigated. We assume that the shear modulus varies only in the radial direction either as a power-law function or an exponential function. For a panel subjected to far field uniaxial tension, the problem is solved by superposing the solution of two independent problems – one of biaxial tension and the other of pure shear loading. The former has a closed-form solution and latter's solution is either closed-form or in the form of the Frobenius series which depends on the functional form of the shear modulus. We also study tailoring the FGIM shear modulus in the radial direction to achieve a desired value of the SCF. For the biaxial tension loading, the required shear modulus variation is explicitly obtained, but for the uniaxial tension loading the approximate variation is numerically found. Numerical results for a few example problems reveal that the SCF strongly depends upon the sign of the shear modulus slope at the hole periphery. The FGIM lining alleviates the SCF at the hole periphery and in some cases migrates the point of the maximum hoop stress to the panel interior. Thus not only the SCF but also the maximum hoop stress in the panel interior must be considered in the panel design. The analytical results presented here should serve as benchmarks for verifying the accuracy of approximate/numerical solutions for a panel with a circular hole.

Seismic Velocity Change Patterns Along the San Jacinto Fault Zone Following the 2010 M7.2 El Mayor‐Cucapah and M5.4 Collins Valley Earthquakes

AbstractWe study temporal changes of seismic velocity (dv/v) in the crust around the central section of the San Jacinto fault zone (SJFZ), Southern California. Focusing on a 200‐day‐long period around April 2010, our analysis resolves two tens‐of‐days‐long successive episodes of reduced velocities that are compatible with signals from the long base strainmeter at the Piñon Flat Observatory. The imaged dv/v sequences are proxies for evolving material properties in the crust surrounding the SJFZ. The temporal and the spatial coincidence of the observed dv/v patterns with the occurrence of two proposed creep episodes suggest that the relative velocity changes reflect the response to deep creep events that follow the M7.2 El Mayor‐Cucapah earthquake and the M5.4 Collins Valley earthquake that occurred 94 days later on the San Jacinto fault. The main slip during the creep events was proposed to occur below 10‐km depth. Wavefield properties suggest sensitivity to medium changes above this source zone, in the top 10 km. The distribution of the obtained dv/v reductions shows a strong difference between large values to the west of the SJFZ and significantly smaller amplitudes to the east. The similarity to the seasonal velocity change pattern implies that the results are likely controlled by the contrast of mechanical properties across the fault, such as fault‐perpendicular shear modulus variations. Our analysis extends the spectrum of methods that can be used to study earthquake interaction, fault zone rheology and dynamics, triggering, and the interplay between creep episodes and earthquakes.

Study of the anti-plane problem of a Dugdale-Barenblatt crack in a welded strip using the integral equation method

The elasto-static anti-plane problem of a Dugdale-Barenblatt crack in a welded infinite strip is formulated in terms of a singular integral equation (SIE). The weld joint is modeled as a tri-material structure: the weld metal (WM), the base metal (BM) and the heat-affected zone (HAZ). The HAZ is modeled with an exponentially variable shear modulus. The SIE is solved using Tchebychev polynomials. The influence of the elastic mismatching (the ratio between the shear modulus of WM and BM) and the HAZ width on the fracture load and on the crack propagation is investigated.

Laboratory and field evaluation of modulus-suction-moisture relationship for a silty sand subgrade

Abstract A change in soil modulus in response to suction-moisture variation after compaction plays a vital role on modulus-based compaction control during earthwork construction. This study investigated the small-strain modulus-suction-moisture relationship for a silty sand subgrade using the laboratory free-free resonant frequency (FFR) and the in-situ spectral analysis of surface waves (SASW) in a trial section of highway construction project in Thailand. The post-compaction soil water retention curves (SWRCs) at different compaction states were investigated over the entire range of suction. Relationships between the SWRC parameters, i.e., air-entry suction, water-entry suction, residual suction etc., and as-compacted dry density and water content were obtained using the multiple-linear regression analysis. Such relationships are useful for the prediction of post-compaction suction in the field. The variation of SASW and FFR shear moduli with the suction stress, inferred from SWRCs, was found to be linear. A modified model was proposed, considering the combined effect of void ratio and the suction stress on the shear modulus, represented as Parameter A which was found to decrease as suction increased. The inclusion of suction stress and void ratio within the model variables yields better accuracy of prediction as compared to those models that considered either only suction or water content.

An analytical continuum model for axially loaded end‐bearing piles in inhomogeneous soil

SummaryAn approximate static solution is derived for the elastic settlement and load‐transfer mechanism in axially loaded end‐bearing piles in inhomogeneous soil obeying a power law variation in shear modulus with depth. The proposed generalised formulation can handle different types of soil inhomogeneity by employing pertinent eigenexpansions of the dependent variables over the vertical coordinate, in the form of static soil “modes”, analogous to those used in structural dynamics. Contrary to available models for homogeneous soil, the associated Fourier coefficients are coupled, obtained as solutions to a set of simultaneous algebraic equations of equal rank to the number of modes considered. Closed‐form solutions are derived for the (1) pile head stiffness; (2) pile settlement, axial stress, and side friction profiles leading to actual, depth‐dependent Winkler moduli, (3) displacement and stress fields in the soil; and (4) average, depth‐independent Winkler moduli to match pile head settlement. The predictive power of the model is verified via comparisons against finite element analyses. The applicability to inhomogeneous soil of an existing regression formula for the average Winkler modulus is explored.

Shear modulus and damping ratio of a nonplastic silt at large shear strains

Site response analyses and solution of dynamic soil-structure interaction problems need determination of variation of shear modulus and damping ratio with shear strain. Since many studies in literature concern evaluation of behavior of sands and silty sands, a series of cyclic triaxial tests were performed to determine the variation of shear modulus and damping ratio of a nonplastic silt with shear strain. Stress controlled cyclic triaxial tests on silt specimens of initial relative densities ranging among 30%, 50% and 70% were performed. Tests were carried out on identical samples under different CSR levels, and the confining pressure was selected as 100 kPa. Variation of shear modulus and damping ratio of silts with cyclic stress ratio amplitude, relative density and number of cycles were investigated. It was understood that soil relative density and cyclic stress ratio amplitude has a significant influence on shear modulus and damping ratio of silts. It was also observed that, as the cyclic stress ratio amplitude is increased, greater shear modulus and lower damping ratio values were obtained.

Torsion of bi-directional functionally graded truncated conical cylinders

Analytical solutions are presented for the torsion of bi-directional functionally graded (FG) linearly elastic truncated conical cylinders for six functional forms of the shear modulus varying in both the radial and the axial directions. Furthermore, for an arbitrary variation of the shear modulus along the two directions, we employ a weighted residual approach (WRA) to obtain accurate numerical solutions for the linear elastic boundary value problem. The influence of the variation of the shear modulus and of the cone angle on the stress distribution in the cylinder is discussed through numerical examples. The analytical solutions presented herein can serve as benchmarks for ascertaining the accuracy of approximate or numerical solutions. The WRA combined with an optimization algorithm can be used to find the shear modulus variation to maximize the torsional stiffness.

Geosynthetic-reinforced cushioned piles with controlled rocking for seismic safeguarding

In this study, a cushioned pile foundation reinforced with geosynthetics is proposed to protect buildings and foundations from seismic energy. This composite foundation utilises piles to control foundation settlement while the geosynthetic-reinforced cushion modifies the dynamic structural characteristics and the load transfer mechanism. The seismic performance of this proposed foundation system is evaluated numerically using FLAC3D software. A fully coupled nonlinear dynamic analysis was conducted in the time domain. The variation of shear modulus corresponding to shear strains in the soil is used to simulate the dynamic behaviour of the soil, while the influence of the plasticity index is also captured. The soil-geosynthetic interface utilises the Mohr-Coulomb failure criterion to capture possible sliding and pull-out of the reinforcement layers. 3D numerical predictions of the tensile forces mobilised in the geosynthetic layers, the shear forces, the lateral deformations and maximum and residual inter-storey drifts in the building are presented and discussed in this paper, as well as how the shear forces and bending moments develop in the piles, and the lateral pile displacements. The results indicate that the proposed geosynthetic-reinforced cushioned pile foundation can provide design engineers with an alternative solution for safeguarding buildings constructed on soft soils in earthquake-prone regions.

Shear modulus of sand–tyre chip mixtures

This paper presents experimental results on the shear modulus of sand–tyre chip (STCh) mixtures. A series of bender element tests was carried out on specimens of sand mixed with varying proportions of tyre chips (TCh). Tests were carried out on STCh mixtures at a constant initial relative density of 50% for different initial effective confining pressures. The bender element test results indicate that the maximum shear modulus of the STCh mixtures increases with effective confining pressure and decreases with the gravimetric proportion of TCh, which ranged from 23 to 138 kPa and 0 to 40%, respectively. However, the strain-controlled cyclic triaxial test results show that the shear modulus at large shear strains decreases with increasing single-amplitude shear strain as a function of the proportion of TCh in the mixture. The lower the proportion of TCh, the larger the degradation observed. The modified Hardin and Drnevich mathematical formulation adequately captures the variation in shear modulus of the STCh mixtures for a wide range of shear strain amplitudes.