Effect Of Triaxiality Research Articles

Polar motions equivalent to high frequency nutations for a nonrigid Earth with anelastic mantle

The coefficients of polar motions of the rigid/nonrigid Earth in frequency bands other than the retrograde diurnal one are systematically computed using general expressions, derived here for the first time, for the prograde and retrograde torques exerted on the Earth by lunisolar potentials of arbitrary spherical harmonic type. Taken together with the already known coefficients of low frequency nutations and UT1 variations, they provide a complete characterization, with high precision, of the motions of the pole of the terrestrial reference frame in space; this is needed for high precision studies in astronomy and space geodesy. The inputs used for our computations are a table of tidal amplitudes, and values of the geopotential coefficients of degrees up to 4 and of other relevant basic Earth parameters. General relations which connect the coefficients of high frequency nutations to those of the equivalent polar motions are established and used for deducing the former. The Chandler resonance plays a significant role in low frequency polar motions. In this context, the role of mantle anelasticity and the nature of the Earth's deformational response to zero frequency forcing are given special consideration. The free core nutation (FCN) resonance of low frequency nutations is shown to affect the prograde semidiurnal nutations through the coupling produced between the nutations in the two frequency bands by triaxiality terms in the angular momenta of the whole Earth and of its fluid core. It is shown in a transparent fashion that the effect of the core triaxiality arises almost exclusively from the huge FCN-related resonance in the wobble of the core. The magnitude of the effect is found to be a few times smaller than reported in a recent paper; it is also found, unlike in that paper, that the changes in the eigenfrequencies due to trixiality are only of the second order in the triaxiality parameter. Numerical results for the polar motions of the nonrigid Earth in different frequency bands, as well as for the elliptical nutations of the rigid Earth, are tabulated and compared with available numbers from earlier works.

Modeling of Notch Tensile Behavior of Martensitic Steels

Notch tensile tests have been carried out on six grades of a high-strength martensitic steel at different hardness levels to investigate the effects of stress triaxiality at the net section and uniaxial tensile properties on fracture behavior. Cylindrical V-notched specimens were used in these tests with the notch-root radius, ρ, ranging from 0.03–1.4 mm, and with the value of the net-to-gross diameter ratio being 0.6. The notch strength ratio (NSR) was found to attain its maximum value at ρ=0.38 mm. The test results were used successfully for screening the fracture toughness behavior of the martensitic steel by extrapolating the corresponding ρ-NSR curve to ρ=0.0. Further, finite element computation of the average stress triaxiality factor ( $$\bar \eta $$ ) enabled the development of a model more accurate than that reported in the literature for estimating NSR for a ductile material as a function of $$\bar \eta $$ and its uniaxial tensile properties.

高力アルミニウム合金の延性破壊に及ぼす応力三軸度の影響と破壊じん性における拘束効果

Constraint effect on the ductility and the fracture toughness of Al-alloys were investigated to examine a modelling of the fracture toughness and to confirm the applicability of unique parameter fracture mechanics to the mircovoid coalescence type of fracture. Stress triaxiality effect on the ductility was investigated in notched round bar specimens. Toughness testing was performed for three points bend and CT specimens with different thickness and notch depth ratio. Stress triaxiality effect on the ductility is significant in low strength materials. The fracture toughness, JIC of high strength Al-alloys, 7075-T6 is varied with the specimen thickness even in the specimens that satisfies the prescription on the size requirements for plane strain condition in the toughness testing method, ASTM E1820. Morphology of the fracture surface composed with the dimple pattern and the intergranular fracture is closely related to the degradation in the toughness. Correlation between the toughness and the ductility in notched specimens demonstrates the validity of the characteristic distance model that describes the toughness in terms of the flow stress and the ductility of the materials.

On the influence of stress triaxiality and strain rate on the behaviour of a structural steel. Part I. Experiments

Abstract Notched axisymmetric specimens of the structural steel Weldox 460 E have been tested at high strain rates in a Split Hopkinson Tension Bar. The aim was to study the combined effects of strain rate and stress triaxiality on the material's behaviour. It is further considered important to obtain experimental data that can be used in validation of constitutive relations and fracture criteria. The force and elongation of the specimens were measured by strain gauges during the high strain rate tests, while the fracture strain was calculated based on measurements of the fracture area in the microscope. Optical recordings of the notch deformation were obtained using a digital high-speed camera system. Using image processing of the digital images, it was possible to estimate the true strain against time at minimum cross-section. The strength of the material was found to increase with increasing strain rate, while for the ductility no significant effect of strain rate could be ascertained from the notched specimen tests. The ductility of the material was found to depend considerably on the stress triaxiality.

An experimental and theoretical investigation of the influence of stress state on cavitation during hot working

The effect of stress triaxiality on cavity growth during necking in the hot tension test was investigated and quantified. Experimentally, Ti–6Al–4V specimens deformed over a range of temperatures and strain rates developed necks of different sharpness and thus of varying levels of the ratio of hydrostatic to effective stress. The magnitude of these stresses was calculated using the classical work of Bridgman. Metallographic examination revealed that the average cavity size, cavity population, and cavity volume fraction were higher in the central region of each specimen as compared to that in the edge region. An analytical model was developed to separate the effects of cavity nucleation and coalescence from the apparent cavity growth rate and thus to determine the individual cavity growth rate at the two locations. The ratio of the cavity growth rate at the center of the specimen to that of the edge was correlated to the stress state (i.e., ratio of mean to effective stress) and compared to predictions derived from the work of Rice and Tracey and of Pilling and Ridley. These comparisons demonstrated that model predictions based on the Pilling-and-Ridley work provided good agreement with observations at low levels of hydrostatic stress, while the Rice-and-Tracey approach was better at higher levels. In addition, it was established that the analysis could be applied over a wider range of processing and material parameters than that originally assumed by Pilling and Ridley and Rice and Tracey.

Elastoplastic modeling of metal matrix composites with evolutionary particle debonding

A microstructural-overall two-level elastoplastic and damage model is proposed to predict the overall mechanical behavior of particle-reinforced metal matrix composites. Unidirectionally aligned spheroidal elastic particles, some of which are partially debonded from the matrix, are randomly distributed in the ductile medium. These imperfect particles are modeled by fictitious orthotropic inclusions without debonding. An ensemble-volume averaged homogenization procedure is employed to estimate the effective yield function of the said composites. The associative plastic flow rule and the hardening law are postulated based on the continuum plasticity theory. The evolution of volume fraction of debonding particles is considered in accordance with Weibull’s statistical function to characterize the varying probability of reinforcement debonding. The uniaxial elastoplastic stress–strain behavior of particle composites is investigated as the first application. In particular, comparisons between the proposed uniaxial stress–strain predictions and experimental data are performed to illustrate the capability of the proposed method. Furthermore, the effect of stress triaxiality is discussed under either the purely hydrostatic or axisymmetric loading on the overall elastoplastic behavior of composites.

Seismic damage evaluation of a steel building using stress triaxiality

This paper presents a procedure for the evaluation of the seismic damage in steel moment frame buildings. The rotational capacities, of typical connections of a building in the Los Angeles area, which suffered extensive damage during the Northridge earthquake, were determined using stress triaxiality as an indicator for fracture. The rotational demands of the same connections were obtained from a three-dimensional finite element analysis of the building model. Comparisons of the rotational demands at several locations in the building to the rotational capacities determined by the stress triaxiality analysis showed that fracture of steel moment connections of the building are governed by the triaxiality of stresses. Based on a number of moment connections analyzed, the paper suggests a moment capacity equation for the pre-Northridge welded steel moment connections taking into account the fracture incidence due to stress triaxiality effect. The moment capacity of such connections as dictated by stress triaxiality compared favorably to FEMA-350 requirements.

Influence of triaxiality and second-order terms in flattenings on the rotation of terrestrial planets

Unlike for the Earth, the equatorial flattening of Mars is important and almost of the same magnitude as the polar flattening. The classical semi-analytical model for the rotation of an ellipsoidal rotating planet with an elastic mantle and incompressible fluid core is therefore extended to incorporate the effects of the planet’s triaxiality. As triaxiality effects are nevertheless small, other second-order effects in the small parameters not related to triaxiality have also been taken into account. The absolute values of the frequencies of two rotational normal modes: (1) the free core nutation (FCN); and (2) the Chandler wobble (CW), are found to be smaller than the corresponding frequencies for a biaxial planet. The period change is larger for the CW than for the FCN, for which the triaxiality effect is comparable to the effect associated with the other second-order terms, and amounts to about 1 day for the CW of Mars.

Deviations from axial symmetry in181Os

Lifetimes of rotational states built on the ${7/2}^{\ensuremath{-}}[514]$ and ${1/2}^{\ensuremath{-}}[521]$ Nilsson single-particle orbitals in ${}^{181}\mathrm{Os}$ have been measured using the coincidence recoil-distance method. The experimentally determined quadrupole moment, assuming axial symmetry, is $\ensuremath{\approx}20%$ larger for the ${7/2}^{\ensuremath{-}}[514]$ band compared with the ${1/2}^{\ensuremath{-}}[521]$ band. The difference can be understood as an effect of triaxiality. Total Routhian surface calculations show deviations from axial symmetry for both bands, leading to good agreement between theory and experiment.

Constitutive Relation And Failure Criterion For Three Structural Steels At High Strain Rates

The effects of strain rate, temperature and stress triaxiality on strength and ductility are studied for the three structural steels Weidox 460 E, Weldox 700 E and Weldox 900 E. The material properties are obtained from three types of tensile tests: quasi-static tests with smooth and notched specimens, quasi-static tests at elevated temperatures and dynamic tests over a wide range of strain rates. The test data are used to determine material constants for constitutive relations and fracture criteria for the three steels.

α／β型チタン合金の延性に及ぼす応力三軸度とミクロ組織の影響

The effects of microstructure and triaxiality on ductility were investigated for α/β Titanium alloys with equiaxed α phase. The materials tested were Ti-6Al-4V, Ti-4.5Al-3V-2Fe-2Mo and Ti-10V-2Fe-3Al, and several different heat treatments were conducted for these materials to obtain the different grain size and morphology. It was shown that the most of micro-voids in α/β titanium alloys were nucleated at α/β interface, and the microstructural parameter defined as the average distance between α/β interfaces was closely related to the dimple size on the fracture surface. The refinement of microstructure leads to the reduction of ductility, though near-β titanium alloys with ultra-fine particle have exceptionally a different tendency from other materials. The dependency of triaxiality on fracture strain obtained in the round bar specimens were also investigated. Significant reduction of strain with the increase of triaxiality is observed, but the reduction due to the high triaxiality is different in each materials. It is shown that the void growth strain in higher strength materials is relatively small, and it results small dependency of the triaxiality on the ductility.

Yield behavior of rubber-modified epoxy adhesives under multiaxial stress conditions

In rubber-modified epoxy resins, a damage zone is generated in the vicinity of the crack tip due to the cavitation of rubber particles, which improves fracture toughness dramatically. Hence, in evaluating the stress distribution in adhesive joints with rubber-modified adhesives, the void formation and growth should be taken into account. In most studies, however, the adhesive layer is still considered as a continuum material governed by the von Mises yield criterion. For many ductile materials, Gurson's model is used for the stress analysis, in which the void formation and growth is taken into account. In a previous study, using adhesively bonded scarf and torsional butt joints, the effect of stress triaxiality on the yield stress in the adhesive layer was investigated. In this study, these experimentally-obtained yield stresses were compared with those obtained by a finite element method, where Gurson's constitutive equations were applied to the adhesive layer. As a result, the calculated yield stresses agreed well with the experimentally-obtained yield stresses. This indicates that Gurson's model is a useful tool for estimating stress distributions in adhesive joints with rubbermodified adhesives.

Micromechanical modelling of porous materials under dynamic loading

The behaviour of porous material under dynamic conditions is assessed by a micromechanical approach. By averaging, a general form for the dynamic macrostress is proposed which recovers the static definition when inertia effects are neglected. In this work, a representative volume element for the porous material is defined as a hollow sphere. Using an approximation of the velocity field and the principle of virtual work, an explicit relationship is found between the macroscopic stress and strain rate. The macrostress tensor is proved to be symmetric, in the present formulation proposed for porous materials. Illustrations are shown for hydrostatic tension or compression and also for axisymmetric loading. In the latter case, the effect of stress triaxiality is captured.

Adhesive Strength of Rubber Bonded to Al2O3 Surfaces: The Role of Chemistry and Morphology

Tensile pull tests were performed on polydimethylsiloxane (PDMS) rubber bonded to single-crystal and ceramic Al2O3 substrates prepared using several surface treatments. An accurate and reliable mechanical test method was developed using a frustro-conical geometry to eliminate the influence of tri-axial stress effects at the periphery. Properties of the sample surface were quantified prior to encapsulation using surface science techniques. Chemical cleaning of atomically-smooth sapphire resulted in a significantly larger joint strength (1 MPa) than attained from as-received substrates (0.448 MPa). The introduction of a significant amount of contamination (∼ 25 nm thick) from isopropyl alcohol (IPA) residue resulted in even weaker adhesion (0.241 MPa). Abrasive cleaning of sapphire using fine (∼ 100 micron) Al2O3 beads created more 0.5–1.0 micron topographic features and significantly higher joint strengths (1.103 MPa) than for similarly-prepared surfaces using coarse (∼ 350 micron) Al2O3 beads (0.621 MPa). The application of a titanate-modified silicone primer resulted in the greatest joint strength encountered (> 1.207 MPa), even when significant levels of surface contamination were present prior to priming. Adhesion to ceramic Al2O3 substrates was systematically stronger than to sapphire for as-received samples (0.965 MPa), abrasively-cleaned surfaces (1.034 MPa), and surfaces containing IPA residue (0.827 MPa) as a result of the rougher topography.

Effect of stress triaxiality corrected plastic flow on ductile damage evolution in the framework of continuum damage mechanics

Most of the damage models derived from the theory of Lemaı̂tre have been developed using the hypothesis based on “proportional loading” for the determination of some material parameters. This hypothesis is adopted in such a way that a material coefficient (a scale factor for damage values) can be evaluated from tensile tests on unnotched specimens, considering the triaxiality of the stress state to be that corresponding to perfect uniaxiality, over the entire load history, and so ignoring necking induced variations in the triaxiality. The present paper analyses the response of some continuum damage mechanics models which incorporate a correlation between the triaxiality factor and logarithmic plastic strain. These correlation laws, affecting also the true stress–true strain material curve, were derived from experimental data processed using the Bridgman method. Due to the lack of experimental data specifically related to damage evolution, the aim of the study was to evaluate the sensitivity of these models to the variability of the triaxiality parameter during the deformation history.

突合せおよびスカーフ接着継手の繰返し荷重下における接着層の損傷挙動

Rubber modified epoxy adhesive has been widely used to structure fields due to its high toughness. The importance of fracture mechanism of the rubber-toughened epoxy is the void formation by rubber particles, where damage zone enclosing the voids propagates in the vicinity of the crack tip. Hence, in considering strength of the rubber modified epoxy, the damage behavior should be taken into account. However, there are few studies about the fatigue behavior of the rubber modified epoxy adhesive discussed from the viewpoint of damage mechanics.In this study, the damage behavior of the adhesively bonded butt and scarf joints under cyclic loading conditions has been studied using a rubber modified epoxy adhesive, where stress triaxiality in the adhesive layer can be varied in a wide range by changing the scarf angle. The strain and crack length of the adhesive layer were monitored by using strain and crack gages pasted on the adhesive layer. A damage parameter is introduced to depict the degree of fatigue damage. Based on these parameters, the effect of stress triaxiality on damage development behavior was discussed.

The effect of stress triaxiality on tensile behavior of cavitating specimens

A force-equilibrium approach was utilized to simulate the tensile behavior of sheet specimens under cavitating conditions. Unlike previous work, the current model incorporated the effect of stress state on the cavity growth rate parameter η. It was found that stress triaxiality that develops after quasistable deformation has a relatively small effect on η. Thus, at a given level of true strain, the increased value of η leads to higher cavity volume fraction inside the specimen. Simulation results revealed that tensile elongation is not affected by the higher cavity growth rate parameter when failure is localization controlled; however, in cases in which failure is fracture (cavitation) controlled, the overall elongation decreases.

Effective elastoplastic behavior of metal matrix composites containing randomly located aligned spheroidal Inhomogeneities. Part II: applications

Based on the proposed formulation in Part I of this sequel (Ju, J.W., Sun, L.Z., Int. J. Solids Struct. 38, 183–201), effective elastoplastic constitutive relations are implemented in this article for metal matrix composites (MMCs) with randomly located and unidirectionally aligned spheroidal particles. First, we investigate the uniaxial elastoplastic stress–strain behavior of MMCs. In particular, we perform comparisons among the theoretical uniaxial stress–strain predictions, existing finite element results and experimental data for MMCs to illustrate the capability of the proposed method. Furthermore, the effect of stress triaxiality is discussed under either the purely hydrostatic or axisymmetric loading on the overall elastoplastic behavior of composites. The proposed initial effective yield surfaces for composites are demonstrated and compared with those of the experimental data. As a special case of the incompressible ductile material containing aligned spheroidal voids, the initial effective yield criterion is studied and compared with that of mathematical upper bound. Finally, viscoplastic extension is briefly presented.

A photodiode optical deformation-measurement system for high strain-rate tensile tests

, This paper describes a novel approach developed at Leeds University to measure deformation during high strain-rate tensile tests using a Photodiode Optical Deformation-measurement System (PODS). The System is developed to obtain true stress-strain data from the dynamic tensile testing rig known as the flying wedge. PODS consist of an array of photodiode cells which is placed on one side of a cylindrical specimen whilst a collimated light beam is placed on the other side. As the specimen deforms, the voltage output from each element of the photodiode array changes. This change is directly related to the change in diameter of the specimen. Each element of the photodiode array is connected to a purpose built low noise high frequency amplifier. Output from each element is recorded simultaneously using multi-channel digital oscilloscopes and a complete profile of the deforming specimen then constructed by post-processing the recorded signals. The paper describes the careful calibration of the system and its validation for dynamic tests using the flying wedge tensile test machine in conjunction with high-speed video recordings of the deforming specimen. Typical results are presented in terms of true stress-strain curves for thermoplastic polymers (ultra high molecular weight polyethylene) measured using the above system at high strain rates. This new system can be used to study the effect of strain-rate and stress triaxiality and to derive constitutive and fracture models for different materials. Not only that PODS cost significantly less than a high speed video camera but it also allows for the complete deformation history up to fracture to be recorded.

Camage leading to ductile fracture under high strain-rate conditions

Quantitative metallographic studies of damage evolution leading to ductile fracture under high strainrate loading conditions are presented. A model material is considered, namely, leaded brass, which contains a dispersed globular lead phase that acts as void nucleation sites. Interrupted tensile split Hopkinson bar tests have been performed to capture the evolution of porosity and void aspect ratio with deformation at strain rates up to 3000 s−1. Both uniaxial and notched specimen geometries were considered to allow the effects of remote stress triaxiality to be investigated. Plate impact testing has also been performed to investigate the evolution of damage under the intense tensile triaxiality and extremely high rates of deformation (105 s−1) occurring within a spall layer. Quantitative metallographic measurements of damage within deformed specimens are used to assess predictions from a Gursonbased constitutive model implemented within an explicit dynamic finite element code. A stresscontrolled void nucleation treatment is shown to capture the effect of triaxiality on damage initiation for the range of experiments considered.