Shear Modulus Reduction Research Articles

The behavior of cracked cross-ply composite laminates under simple shear loading

An interlaminar shear-stress analysis developed earlier (Tsai et al. 1990, Proc. ASME 1990 Winter Annual Meeting, AMD-Vol. 111, MD-Vol. 22, pp. 9–18. ASME, New York) for a [ Φ m θ n ] s bi-directional composite laminate is combined with the Timoshenko beam theory in this work. A cantilever beam of a [ 0 m 90 n ] s cross-ply laminate with a cracked 90° layer is analyzed. Displacements, strains, forces and laminate shear modulus reduction due to matrix cracking in the 90° layer were obtained. The relationship obtained between the shear modulus and transverse crack density is in complete agreement with predictions from a pure shear analysis (Tsai and Daniel 1991, submitted to Int. J. Solids Structures). This proved that simple shear loading can give the correct shear modulus data as pure shear. Experimental results obtained by a simple shear test agree very well with theoretical predictions.

Fiber Pushout Test: A Three-Dimensional Finite Element Computational Simulation

A fiber pushthrough process was computationally simulated using three-dimensional finite element method. The interface material is replaced by an anisotropic material with greatly reduced shear modulus in order to simulate the fiber pushthrough process using a linear analysis. Such a procedure is easily implemented and is computationally very effective. It can be used to predict fiber pushthrough load for a composite system at any temperature. The average interface shear strength obtained from pushthrough load can easily be separated into its two components: one that comes from frictional stresses and the other that comes from chemical adhesion between fiber and the matrix and mechanical interlocking that develops due to shrinkage of the composite because of phase change during the processing. Step-by-step procedures are described to perform the computational simulation, to establish bounds on interfacial bond strength and to interpret interfacial bond quality.

Network formation and properties: Rate theory description of effects of ring formation on elastic shear modulus of RA2 + RB3 networks

AbstractThe rate theory of RA2+RB3 polymerization has been developed to enable the concentrations of the smallest inelastic loops and the resulting reductions in elastic shear modulus at complete reaction to be evaluated. Reduction in modulus is expressed as Mc/M°c, the effective molar mass of chains between elastically active junction points (Mc) relative to that for the perfect network for given reactants (M°c). Calculations have been carried out for stoichiometric reaction mixtures and the concentration of loops is characterized by λ, a ring‐forming parameter. λ = Pab/CAo, where CAo is the initial concentration of A‐groups and Pab = (3/(2πvb2))3/2/NAv, with v the number of bonds in the chain forming the smallest loop, b the effective bond length of the chain, and NAv Avogadro's number. Pab is equal to the mutual concentration, assuming Gaussian statistics, of the A‐ and B‐groups at the two ends of the linear sub‐chain of v bonds. Loop formation increases as λ increases, that is, as the initial concentration of reactive groups (CAo) decreases, or, at constant CAo, as the molar mass of reactants (v) or their chain stiffness (b) decreases. Comparison with existing experimental data on two series of polyurethane networks formed from hexamethylene diisocyanate and two polyoxypropylene triols at different initial dilutions shows that the values of Pab decrease with increases in initial dilution and molar mass of triol. The decrease with molar mass is entirely accounted for by changes in v and the variation with dilution shows that the approximation of counting only smallest loops improves as dilution increases.

Influence of plastic deformation of the flux line lattice on the 2D pinning

The field and temperature dependences of the volume pinning force were investigeted for amorphous Nb 3 Ge films. The results are discussed in the frameworks of Larkin – Ovchinnikov's collective pinning theory. A discrepancy in the high field range is attributed to the strong plastic deformation of the flux line lattice due to shear modulus reduction near the critical field.

On the Mechanics of Microcracking in Fiber Composite Laminates Under Combined Stress

The formation of matrix microcracks in fiber composite laminates has been widely observed. The present paper addresses the mechanics of this microcracking by means of a shear lag model in conjunction with a Weibull failure criterion for further cracking. This shear lag model considers both shear loading as well as cross-fiber tension in the cracked ply. The reduction in inplane shear and tensile modulus of the cracked ply is computed directly from the model. The increase in applied load required to produce new cracks is computed and shows general agreement with previous experimental results. A general approximate relationship between the applied strain levels and the modulus reduction is presented.

Elastic softening versus amorphization in a simple model of ion-induced radiation damage.

Recent experimental work on irradiated films of ${\mathrm{Nb}}_{3}$Ir have shown that under radiation-induced damage the films show a large (\ensuremath{\approxeq}40%) reduction in an average shear modulus while remaining essentially crystalline. The shear modulus saturated at larger radiation doses. This paper presents a simple 2D central-force model that displays the same qualitative behavior as it is progressively disordered by random displacements of its lattice sites. Bragg peaks are always present at every level of ``damage,'' although modified by an envelope function. The effect of disorder on the Bragg peaks of the model is solved analytically in closed form; the effect on the elastic moduli is found numerically, via computer simulation.

Temperature Dependence of Hardness of Alumina‐Based Ceramics

Hardness was measured as a function of temperature (20° to 1000°C) for several Al2O3 ceramics, including single‐crystal sapphire and polycrystalline aluminas containing different amounts of second phase. Hardness decreased steadily with increasing temperature for all materials tested, in accordance with a semiempirical relation of the form H=H0 (1 – T/T0). This behavior conformed with a thermally activated slip process, limited by Peierls stresses. At lower temperatures, the hardness values for debased aluminas were less (smaller H0) than for the pure materials, consistent with a reduction in shear modulus resulting from the “soft” phase. However, at higher temperatures the hardness values for all the aluminas converged (identical T0, i.e., material‐invariant activation energy). The latter behavior indicated that the temperature dependence of the indentation deformation was controlled predominantly by the Al2O3 component.

Reductions of erythrocyte membrane viscoelastic coefficients reflect spectrin deficiencies in hereditary spherocytosis.

Hereditary spherocytosis is a common hemolytic anemia associated with deficiencies in spectrin, the principal structural protein of the erythrocyte membrane-skeleton. We have examined 20 different individuals from 10 spherocytosis kindreds and 2 elliptocytosis kindreds to determine the effects of different levels of spectrin deficiency on the viscoelastic properties of the erythrocyte membrane. Micropipettes were used to perform single-cell micromechanical measurements of approximately 1,000 individual cells to determine the membrane elastic shear modulus, the apparent membrane bending stiffness, and whole cell recovery time constant for the different cell populations. The membrane viscosity was calculated by the product of the shear modulus and the recovery time constant. Results show correlation between the fractional reduction in shear modulus and the fractional reduction in spectrin content (determined by spectrin radioimmunoassay) and spectrin density (determined by the ratios of spectrin to band 3 on electrophoresis gels) suggesting that membrane shear elasticity is directly proportional to the surface density of spectrin on the membrane (P less than 0.001). The apparent membrane bending stiffness is also reduced in proportion to the density of spectrin (P less than 0.001). The membrane viscosity is reduced relative to control (P less than 0.001), but the nature of the relationship between spectrin density and membrane viscosity is less clearly defined. These studies document striking relationships between partial deficiencies of erythrocyte spectrin and specific viscoelastic properties of the mutant membranes.

Numerical study of some post-cracking material parameters affecting nonlinear solutions in RC deep beams

This paper describes a numerical study of some of the quasi-material parameters which are used to define the post-cracking response in smeared crack models for the nonlinear finite element analysis of reinforced concrete. The two major effects studied were interface shear transfer and tension stiffening. The investigation was carried out on a solid deep beam using a nonlinear plane stress formulation. Parabolic isoparametric elements were used with a standard nonlinear solution procedure.The effect of the quasi-material parameters was found to be significant in predicting the behaviour of the deep beam. A sudden drop in the stress at the cracking load was found to be important in the tension stiffening model. Different values can lead to different solutions for load–deflection curves, ultimate loads, and crack patterns. Also it was found that a large reduction in shear modulus to account for interface shear transfer can affect the nonlinear solution and cause numerical problems when used with a gradual release tension stiffening model with no discontinuity at the cracking load. Key words: deep beams, finite element, load deflection, post-cracking, shear transfer, tension stiffening, ultimate load.

Effects of inherited membrane abnormalities on the viscoelastic properties of erythrocyte membrane

Several workers have identified molecular abnormalities associated with inherited blood disorders. The present work examines how these alterations in molecular structure affect the viscoelastic properties of the red blood cell membrane. Changes in the membrane shear modulus, the membrane viscosity, and the apparent membrane bending stiffness were observed in cells of eight patients having a variety of disorders: Two had reductions in the number of high-affinity ankyrin binding sites, two had abnormalities associated with the protein band 4.1, and six were known to be deficient in spectrin. The data suggest that the membrane shear modulus is proportional to the density of spectrin on the membrane and support the view that spectrin is primarily responsible for membrane shear elasticity. Although membranes having abnormalities associated with the function of ankyrin or band 4.1 exhibited reduced elasticity, the degree of mechanical dysfunction was quantitatively inconsistent with the extent of the molecular abnormality. This indicates that these skeletal components do not play a primary role in determining membrane shear elasticity. The membrane viscosity was reduced in seven of the eight patients studied. The reduction in viscosity was usually greater than the reduction in shear modulus, but the degree of reduction in viscosity was variable and did not correlate well with the degree of molecular abnormality.

Fatigue analysis of offshore platforms with uncertainty in foundation conditions

Foundation flexibility is an important consideration, in designing offshore structures against fatigue, since it has a significant effect on the system's dynamic response. However, a considerable amount of uncertainty is expected in the estimation of foundation flexibility due to the natural nonhomogeneity of the soil as well as laboratory and in-situ testing errors. The proposed method explicitly uses of the statistical nature of uncertainty in the soil stiffness to estimate the variability in the instanteneous and fatigue response using the First-Order Second Moment technique, with emphasis on uncertainties in the dynamic shear modulus along with random wave loading. Uncertainties in the response are estimated considering the randomness in the eigen values, eigen vectors, and the mechanical transfer function. Fatigue life is estimated at different levels of uncertainty for the dynamic shear modulus. It is observed that the uncertainty in the dynamic shear modulus of the soil has more significant effect on the fatigue life of the joints close to the base. The variability in the fatigue response increases with a reduction in the mean dynamic shear modulus. The response is more sensitive to changes in the dynamic shear modulus at its lower values.

Applicability of the Biot theory. I. Low-porosity materials

The Biot theory of wave propagation in porous, saturated materials contains 13 parameters. We discuss empirical and theoretical ways to predict values of these parameters for natural materials. The self-consistent theory of composites is used to predict the elastic moduli of the skeletal frame assuming that the inclusions (pore fluid, sediment grains) are needle-shaped at low concentrations. The Biot theory is then used to predict compressional and shear wave speeds in consolidated materials. In a man-made material (porous sintered glass), the predictions agree with experimental data to within 3% (experimental uncertainty is also 3%); these samples were distilled-water-saturated. The predictions agree with data to within 1% for Berea sandstone, and to within 5% for Bedford limestone; all of these samples were brine-saturated. The predictions agree with data to within 8% for distilled-water-saturated Bedford limestone. In the natural sedimentary rocks, the experimental uncertainty in measured wave speeds is estimated at 5%, and the comparisons apply at high effective pressures (>0.3 kbar), where the experimentally determined wave speeds approach ‘‘limiting velocities.’’ In distilled-water-saturated Massilon sandstone, predicted shear wave speeds are about 25% to 35% higher than measured speeds; this may indicate a reduction of the frame shear modulus due to interaction of the pore fluid with the minerals of the rock.

Magnetostriction of pristine graphite and SbCl 5-graphite intercalation compounds

Magnetostrictive strain in pristine highly oriented pyrolytic graphite (HOPG) and SbCl 5-graphite intercalation compounds (GIC's) has been studied in steady magnetic fields up to 10 tesla, including the quantum oscillation effect. Fourier spectra of the magneto-oscillations are stage dependent and in agreement with those reported by Takahashi et al. For pristine HOPG, the monotonie component of the longitudinal magnetostrictive strain (LMS) parallel to the c-axis is found to be negative and to increase in magnitude with increasing width at half maximum intensity of the (002) line of X-ray diffraction. This result is explained qualitatively within the framework of the Keyes theory by considering the reduction of the static shear modulus of graphite due to the displacement of basal-plane dislocations. The stress coefficient of the majority hole Fermi surface is determined as δ In S h δσ [0001] = 0.034 ± 0.006 kbar − by using magnetostriction and torque de Haas-van Alphen measurements.

Effective Stress Analysis of Seismic Response and Liquefaction: Theory

An effective stress method is presented for analysis of seismic response and liquefaction of horizontally layered saturated sand deposits, based on solution of the fully coupled equations for the granular solid and the pore fluid. A new material model for hysteretic behavior of sands and a new effective stress path rule are presented. The shear modulus reduction and damping of the proposed model are compared with other experimentally based relations. The proposed effective stress analysis method is used to simulate shaking table tests, with results in reasonably good agreement with experimental results. The proposed method of analysis is compared with a number of other well‐known methods.

Effect of hydrostatic pressure on mechanical properties of crystalline polymers under shear

1. Superimposing hydrostatic pressure increases shear strain resistance and leads to a reduction in structural defects. 2. A sharp reduction in shear modulus and yield point at 50–60 °C caused by superimposing hydrostatic pressure is explained by the presence of a relaxation transition at this temperature and is the result of freezing the amorphous phase of the partially recrystallized polymer by the action of pressure. 3. A uniform change in elasticity modulus and yield point with increasing pressure was established in the temperature ranges not containing relaxation transitions. 4. Limits were established for external parameters at which shear diagram similarity is observed.

Shear Moduli of Sands under Cyclic Torsional Shear Loading

To evaluate degree of reduction in shear moduli of sands with an increase in shear strain amplitude, dynamic soil tests were performed with use of hollow cylindrical samples applying torsional shearing forces. Two types of soil testing equipments, a resonant-column apparatus and a torsional shear apparatus, were employed. It is pointed out that shear moduli at shear strain amplitude of 10-4 obtained from these two different equipments agree satisfactorily. Furthermore, extents of reduction in shear moduli with an increase in shear strain amplitude are shown for various sands. These data are compared with those presented by other investigators. On the basis of test results, a simplfied procedure for predicting reduction in shear modulus with an increase in shear strain amplitude is proposed.

Viscous phase in rocks and the low-velocity zone

The presence of partial melt in the low-velocity zone (LVZ) explains the occurrence of basaltic volcanism and serves as a soft layer for plate tectonics. If it is assumed that the melt occurs in inclusions, theoretical and experimental results show that the effective bulk modulus depends mainly on the bulk modulus of the inclusions and their aspect ratio. The effective shear modulus and attenuation in shear depend on the viscosity, aspect ratio, and frequency of applied stress waves. Only a very small amount of melt in narrow inclusions is necessary to cause a large reduction in shear modulus. The results are consistent with observed variations with depth in the LVZ of compressional and shear waves. However, both effective shear modulus and attenuation in shear are dependent on viscosity and frequency. Consequently, if there is partial melt in the LVZ, velocity and attenuation in, and the shape of, the LVZ are frequency dependent. When short-period and long-period seismic velocity data are combined, the ratio of relaxed to unrelaxed shear moduli in the LVZ is ∼0.83, the viscosity of the melt in the LVZ is found to be approximately 107 poises, and the period at which attenuation in shear has a peak is around 20 sec.

Resonant Column And Cyclic Triaxial Testing Of Tailing Dam Material

A series of resonant column and cyclic triaxial tests has been conducted in the frame of the analysis of tailing dam stability during earthquakes. The investigation program for a silty sand from uranium tailings is presented. The paper describes the testing procedures and presents all significant results of these experiments. Single-stage and multi-stage resonant column tests were performed in order to determine the dependence of shear modulus and damping ratio on confining pressure and shear strain amplitude. Stress-controlled cyclic triaxial tests on the same material were conducted to determine the increase in pore pressure and the amount of accumulated strain in dependence of the cycles number and the load configuration. Approximate formulas are given for stiffness and damping to be used in seismic response analyses. INTRODUCTION When applying finite element methods for the seismic response analyses of dams and embankments, e.g. Seed*, the nonlinear soil behaviour can be taken into account by use of equivalent elastic constants which vary as a function of stress and strain level. These parameters are obtained either from empirical relations, e.g. Hardin & Drnevich^, or from laboratory tests on the particular soil material. Much of the published data deal with soils such as poorly graded sands, principal of alluvial origin. However, for soils that are dirty, gravelly, well graded or composed of relatively angular particles only a limited amount of test data is available in the Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509 292 Soil Dynamics and Earthquake Engineering literature. In the frame of the stability analysis of tailing dams located in a moderate seismic risk region in the south-east part of Germany an extensive dynamic in-situ and laboratory investigation testing program was conducted. Since the tailings sand was placed hydraulically with little compaction the strength reduction due to pore pressure increase had also to be considered. In the following, results of resonant column and cyclic triaxial tests are presented which were performed on samples of sand and silty sand from uranium tailings. MATERIAL TESTED The material tested was obtained from borings drilled in the crest or slopes of a major tailings dam. It consisted of sub-angular silty sand and had the following properties: i) grain size distribution: D$Q = 0.2 mm, Dso = 0.08-0.12 mm and £>go = 0-5 mm, ii) particle density 2.77 Mg/nr* and iii) effective strength parameters <^' = 31°, d = 5 kN/m^. The dry density pd varied between 1.59 and 1.82 Mg/m^ corresponding to void ratios e from 0.74 to 0.52. All specimens, 50 mm in diameter and 100 mm in height, were formed in an unsaturated state corresponding to the in situ water content and compacted to the required density. RESONANT COLUMN TESTS The tests were conducted on a Drnevich type apparatus, cf. Drnevich et al according to the specification of the ASTM D4015 87 Standard. All specimens were isotropically consolidated using standard techniques. Multi-stage tests were conducted to determine the shear modulus and damping ratio, Gmax and Dmin, respectively, at very low shear strain amplitudes 7. The test was started at the lowest confining pressure and repeated for intermediate pressures up to a maximum confining pressure of 500 kPa. At the highest confining pressure under consideration the amplitude of excitation was gradually increased to study the shear modulus G as a function of shear strain amplitude. Damping was determined by the steady-state vibration of the specimen. Spot checks were made by the free vibration decay method. In addition, single-stage tests were performed at selected levels of confining pressure by stepwise increasing the excitation force up to a maximum shear strain amplitude of approximately 10~~*%. Densities and confining pressures for each test are summarized in Table 1, where the confining pressure <JQ is normalized with respect to the atmospheric pressure pa = 100 kPa. The measured shear moduli Gmax at small strains of approximately 10~* % are depicted in Figure 1 as a function of confining pressure <JQ. Shear modulus reduction curves at two confining pressures are presented in Figure 2 versus shear strain. Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509

The shear modulus of flowing polyethylene melts

AbstractTests on three low‐density polyethylene melts over a range of temperatures and shear rates, using a Weissenberg rheogoniometer, have enabled the reduced shear modulus to be obtained as a function of shear rate at the initiation of shearing and at various subsequent stages. For the two polymers which gave plateau values of modulus during the course of shearing these were in good agreement with figures quoted in the literature, but the initial modulus (measured on one polymer) was an order of magnitude higher. This supports the view that a network structure exists in the polymer at rest and is disrupted in the early stages of shearing. It was not possible to obtain unequivocal evidence that the equilibrium modulus decreases with amount of shearing, as required by the hypothesis that rheological breakdown is caused by a reduction in molecular entanglements.