Unloading Paths Research Articles

FEM analysis of concrete structures subjected to mode-I and mixed-mode loading conditions

In this study, the finite element method (FEM) for a body containing displacement discontinuity is used for the investigation of tensile fracture behavior under mode-I and mixed-mode loading conditions in concrete structures. A mechanical model for the tensile fracture behavior is reduced to a mathematical problem, and the analysis method is proposed. With the aid of this method, several factors which govern tensile fracture are examined, such as the unloading path in the tension-softening behavior and the transmission of shear stresses across crack surfaces. A plain concrete beam without a notch is analyzed by first neglecting and then taking into account the unloading path in the tension-softening behavior to demonstrate the phenomenon of cracking localization in mode-I crack growth. Pullout test specimens of practical significance are analyzed in order to study the crack growth phenomenon under mixed-mode loading conditions. Cases with and without lateral confinement are considered and the results obtained from the present analysis are compared with those obtained from available experimental data. A simple model for shear transfer across crack surfaces is established. By incorporating this model in the program, a pullout test specimen with lateral confinement is analyzed to examine the influence of shear transfer across crack surfaces on cracking localization.

A comparison of the fabric and permeability anisotropy of consolidated and sheared silty clay

This paper reports the geotechnical aspects of an experimental programme investigating the permeability of shear zones in clay-rich sediments. Oedometric and ring shear permeameters were used to investigate and compare permeability anisotropy in consolidated and sheared silty clay, to simulate the behaviour of wall-rock sediments and shear zone sediments respectively. In line with other studies, consolidated silty clay was found to have no significant permeability anisotropy, although clay fabric anisotropy was well developed. However, sheared silty clay showed an increase in permeability anisotropy ( r k = 3−16) with decreasing void ratio ( e = 0.8−0.4), corresponding to effective stresses of 100 kPa to 4 MPa. This level of anisotropy was retained during shearing along the unloading path, and no significant dilation or enhancement of permeability was observed.

Influence of Elastic Energy on the Unloading Behaviour of NiTi Shape Memory Alloys

A non-linear unloading path of deformed NiTi shape memory alloys was observed during mechanical testing at various temperatures, giving a strain recovery larger than what expected from elasticity. Whilst the non-linear unloading path at above the A s temperature may be rationalised by a simultaneous reverse transformation from the oriented martensite to austenite, the non-linear strain recovery during unloading at below the A s temperature must have a different mechanism, because the thermodynamic driving force for a reverse transformation is negative. This second mechanism is ascribed to a 'self-deorientation' process of the fully oriented martensite and the driving force for this process is attributed to the elastic energy stored in the matrix during the forward deformation process.

Unloading characteristics of anti-plane shear crack in softening materials

The local characteristics of the anti-plane shear stress and strain field are determined for a material where the stress increases linearly with strain up to a limit and then softens nonlinearly. Two unloading models are considered such that the unloading path always returns to the origin while the other assumes the unloading modulus to be that of the initial shear modulus. As the applied shear increases, an unloading zone is found to prevail between a zone in which the material softens and another zone in which the material is linear-elastic even though the crack does not propagate. The divisions of these zones are displayed graphically.

Nonproportional loading effects on elastic-plastic behavior based on stress resultants for thin plates of strain hardening materials

A stress resultant constitutive law in rate form is constructed for power-law hardening materials. The change of plate thickness is considered in the constitutive law. The elastic-plastic behavior of a plate element based on the stress resultant constitutive law under uniaxial combined tension and bending is determined under a limited number of nonproportional and unloading paths. The results based on the stress resultant constitutive law and the through-the-thickness integration method are compared within the context of both the small-strain and finite deformation approaches. The results indicate that the selection of the normalized equivalent stress resultant and the corresponding work-conjugate normalized equivalent generalized strain is appropriate for describing the hardening behavior in the stress resultant space. However, the hardening rule in a power law form must be modified for low hardening materials at large plastic deformation when finite deformation effects are considered.

Shock wave equation of state of muscovite

Shock wave data to provide an equation of state of muscovite (initial density: 2.835 g/cm3) were determined up to a pressure of 141 GPa. The shock velocity (Us) versus particle velocity (Up) data are fit with a single linear relationship: Us=4.62(±0.12) +1.27(±0.04)Up (km/s). Third‐order Birch‐Murnaghan equation of state parameters (isentropic bulk modulus and isentropic pressure derivative of bulk modulus) are Kos=52±4GPa and K'os=3.2±0.3. The pressure‐temperature relation along the Hugoniot suggests that muscovite may dehydrate to KAlSi3O8 (hollandite), corundum, and water, with a small volume change, above 80 GPa. Thermodynamic calculations of the equilibrium pressure for the dehydration yields a significantly lower value. Observed unloading paths from shock pressures up to about 80 GPa are steeper in a density‐pressure plane than the Hugoniot and become shallower with increasing shock pressure above that pressure. The changing slope may indicate that devolatilization occurs during unloading above 80 GPa. The present equation of state data for muscovite are compared with results of previously reported recovery experiments.

BUMPER DEBRIS CLOUD STRUCTURE ESTIMATED BY SHOCK CALCULATIONS

BUMPER DEBRIS CLOUD STRUCTURE ESTIMATED BY SHOCK CALCULATIONS

Borehole yield and hydraulic fracture initiation in poorly consolidated rock strata—part I. impermeable media

The stress distributions around a circular well in granular elastoplastic impermeable media are calculated. Yielding may develop during drilling as deviatoric stress in increased during cavity creation. Hydraulic injection imposes an unloading action on the existing yielded medium because of reduction of deviatoric stress. Thus, it is unlikely that the stress-strain path returns along the original elastoplastic trajectory; instead, it follows an unloading path which is different from the active loading path followed during drilling. The effects of initial yielding due to excavation or drilling disturbance on the local stress state complicate stress measurements, and may even change hydraulic fracturing mechanisms in such media.A non-associated flow rule is used to calculate the plastic deformation in order to obtain a reasonable deformation field. The final volumetric strain around the borehole is solved as the summation of the incremental strains due to the injection preceding hydraulic fracture, and the existing plastic strains created during borehole excavation.

On two-dimensinal self-consistent micromechanical damage models for brittle solids

Two-dimensional self-consistent micromechanical damage models are presented for microcrack-weakedned brittle solids under “cleavage l” deformation processes. The proposed frame-work basically follows the previous work of Horii and Nemat-Nasser (1983. J. Mech. Phys. Solids31 (2). 155–171) and Sumarac and Krajcinovic (1987, Mech. Mater. 6. 39–52). Thermodynamics basis, microcrack opening displacements and damage-induced inelastic compliances are derived. Microcrack evolutions (growth) are characterized through the use of fracture mechanics stability criteria and microstructural microcrack geometry. Mode I, mode ll and mixed mode microcrack growth are considered. Simple and efficient computational algorithms as well as three detailed numerical simulations are also presented to illustrate the potential capability of the proposed micromechanical damage models. In particular, no fitted “material parameters” are needed. More-over loading unloading stress paths and microcks status changes in opening closing are trivially accommodated in this work.

P‐Delta Effect Under Repeated Loading

Recently, a novel P‐Delta method for the design of sway frames was introduced by the writer and demonstrated in conjunction with static proportional loading conditions. The method incorporates the rigid‐plastic collapse load and the elastic buckling load of the structure as the two boundary conditions and does not require iterations or a precise trace of the load history. In this paper it is argued and verified by experiments that the P‐Delta effect for sway frames can also be predicted by the same method if loadings vary independently. Such loadings can lead to a reduced failure load compared with the static load case due to premature incremental collapse if the so‐called shakedown load is exceeded. The presented approach is a simple alternative to much more complex analysis procedures which would have to follow repeatedly the loading and unloading paths until failure is indicated. This has to be done under changing axial forces which requires iterations until the structure stiffness and the axial force level match.

Simulation of downhole and crosshole seismic tests on sand using the hydraulic gradient similitude method

A method of simulating downhole and crosshole seismic shear-wave tests in a model under controlled stress conditionsis described. The downhole and shear wave in horizontal plane (SH) crosshole shear waves are generated and received along the principal stress axes using piezoceramic bender elements. The K0in situ stress conditions, including loading and unloading stress paths, are simulated by the hydraulic gradient similitude method, which allows high stresses simulating field conditions to be obtained. The horizontal stress during the tests is directly measured by a lateral total-stress transducer. The test data are used to evaluate various published empirical equations that relate shear-wave velocity and soil stress state. It is found that although the various empirical equations can predict the in situ shear-wave velocity profile reasonably well, only the equation that relates the shear-wave velocity to the individual principal stresses in the directions of wave propagation and particle motion can predict the variation of the velocity ratio between the downhole and SH crosshole tests. It was also found that the stress ratio has some effects on the downhole (or shear wave in vertical plane (SV) crosshole) shear-wave velocity, but not on the SH crosshole shear-wave velocity. This indicates that it is only the stress ratio in the plane of wave propagation that is important to the shear-wave velocity. Comparison between the downhole and SH crosshole shows that structure anisotropy is in the order of 10%. In addjtion, K0 values are predicted from shear-wave measurement and compared with measured ones. The difficulties in obtaining K0 values from shear-wave measurement are also discussed. Key words: hydraulic gradient, model tests, downhole and crosshole shear-wave tests, sand.

A study of the postbuckling path of cylindrically curved panels of laminated composite materials during loading and unloading

In this paper, Dynamic Relaxation Method is applied to study the postbuckling path of cylindrically curved panels of laminated composite materials during loading and unloading. The phenomenon that loading paths do not coincide with unloading paths has been found. Numerical results are given for cylindrically curved cross-ply panels subjected to uniform uniaxial compression under two types of boundary conditions. The influence of the number of layers, the panels curvature and the initial imperfection on the postbuckling paths is discussed.

Verification of Simplified Inelastic Models of 3‐D Behavior

This paper investigates the accuracy of a simplified model for the description of elastic‐plastic response of wide flange shapes under three‐dimensional loading. The model uses the average effect of moment‐axial force curvature equations in each bending direction to compute the terms in the element stiffness matrix. While this approach is exact for planar response computations, it is an approximation in describing three‐dimensional response. Another problem that arises for rolled shapes is that the model must describe the effect of differential bending on the member's response. This is accomplished by using the elastic stress variation to measure the extent of flange yielding. A large deflection, displacement‐control procedure is used to compute the elastic‐plastic‐failure load of the system and trace the unloading path. Comparison with analytical and experimental work on flexural‐torsional buckling of columns is used to verify the accuracy of the proposed model. Agreement with previously published result...

Dynamic unloading behavior of soda lime glass

An experimental study was conducted to see if the reverberation plate technique, using commercial manganin stress gauges, could be used to determine the dynamic unloading behavior of brittle materials. Using a 50-mm powder gun, plate impact experiments were conducted on soda lime glass, providing unloading data from stress states from below the Hugoniot elastic limit to 17.0 GPa. The slopes of the unloading paths were found to increase as the peak compressive stress was increased. Such unloading behavior is similar to that of quartz, suggesting that the technique is viable.

鋼構造柱材に対する対称限界理論 : その1:一般的定式化

1. INTRODUCTION A symmetry limit theory has been presented recently by the present authors [1] for a new branching phenomenon in the steady states of a linear strain-hardening cantilever beam-column subjected to an idealized completely reversed tip deflection cycling. The purpose of this paper is to develop an extended theory of the symmetry limit for a steel beam-column with an arbitrary cross-section subjected to the same idealized deflection cycling program. The present theory is, however, not a straightforward extension of the previous theory bul deals with more general types of nonlinear hysteretic stress-strain relations. A general formulation on the basis of virtual work equations is presented so that it can be further extended to a more general form applicable to complex structural systems. 2. FUNDAMENTAL CONCEPTS FOR AN IDEALIZED PROBLEM A cantilever beam-column is subjected to an idealized tip deflection cycling program referred to as COIDA under a constant axial compressive dead load IV as shown in Fig. 1. COIDA is an idealized completely reversed tip deflection cycling program of stepwisely increasing amplitude Ψ consisting of infinitely many infinitesimal increments, such that every after an infinitesimal increment of Ψ, the deflection cycling at that increased amplitude is repeated as many times as is necessary for the beam-column to attain a steady state. The following hypotheses have been introduced in [1]. (H 1) A beam-column exhibits a sequence of symmetric steady states during some range of Ψ and then transition from a symmetric steady state to an asymmetric steady state at some value of Ψ. Once the transition occurs, the beam-column exhibits a sequence of asymmetric steady states. (H 2) All the state variables are continuous and piecewisely differentiable functions of Ψ. Sequences of steady states can be visualized as steady state paths in a "steady state space" as shown in Fig. 2. It has been pointed out in Ref. [1] that any two different steady state paths, say OSB and OSF in Fig. 2, can never be the possible responses of a beam-column to one and the same COIDA program and that the emanation of asymmetric branches at S is not a branching phenomenon on equilibrium paths, but may be interpreted as a segregation phenomenon in bundles of equilibrium paths and at the same time as a branching phenomenon in steady state paths. 3. BASIC EQUATIONS FOR STEADY STATES The strain-displacement relation, geometrical boundary conditions and equation of equilibrium are given by Eqs. (2), (3), and (4), respectively. The stress-strain relations for the virgin, unloading and loading paths shown in Fig. 4 can be written as Eqs. (8), [(9), (10)], and (11), respectively, in dimensionless forms. 4. FUNDAMENTAL HYPOTHESES The steady-state paths are computed here by means of an incremental procedure. The three fundamental hypotheses (H 3), (H 4) and (H 5) introduced, together with (Hi) and (H 2) stated above, in [1] are also introduced here in the forms suitable for the nonlinear hysteretic stress-strain relations. (H 3) The strain direction of a state point may be reversed only when the controlled tip deflection v (I) is reversed. For the strain responses during a convergent process to a steady state, however, this hypothesis does not exclude a possibility of strain reversal without reversal of υ (l). (H 4) At every symmetric steady state, the state point of an element of the beam-column travels along a constant closed locus that would reach the compressive virgin curve at the reversal of compressive strain. (H5) In the moment of transition from a symmetric steady state to an asymmetric steady state, no variation toward a steady-state locus whose compressive reversal point is away from the compressive virgin curve as shown in Fig. 7(b), will appear in any finite region in the beam-column. In other words, there occurs only such a variation toward a steady-state locus

Evidence for the Presence of a Sucrose Carrier in Immature Sugar Beet Tap Roots

The objectives of this work were to determine the path of phloem unloading and if a sucrose carrier was present in young sugar beet (Beta vulgaris L.) taproots. The approach was to exploit the characteristics of the sucrose analog, 1'-fluorosucrose (F-sucrose) which is a poor substrate for acid invertase but is a substrate for sucrose synthase. Ten millimolar each of [(3)H]sucrose and [(14)C]F-sucrose were applied in a 1:1 ratio to an abraded region of an attached leaf for 6 hours. [(14)C]F-sucrose was translocated and accumulated in the roots at a higher rate than [(3)H]sucrose. This was due to [(3)H]sucrose hydrolysis along the translocation path. Presence of [(3)H]hexose and [(14)C]F-sucrose in the root apoplast suggested apoplastic sucrose unloading with its subsequent hydrolysis. Labeled F-sucrose uptake by root tissue discs exhibited biphasic kinetics and was inhibited by unlabeled sucrose, indicating that immature roots have the ability for carrier-mediated sucrose transport from the apoplast. Collectively, in vivo and in vitro data indicate that despite sucrose hydrolysis by the wall-bound invertase, sucrose hydrolysis is not entirely essential for sugar accumulation in this tissue.

Steel column strength models

Abstract This paper uses a simple model to unify the two traditional approaches to steel column strength based either on the buckling of straight columns with residual stresses or on the first yield of crooked columns. The inelastic buckling and post-buckling loads of elastic-plastic-strain hardening steel columns with residual stresses are determined using the tangent, reduced, and Shanley theories. These are compared with the maximum load capacities of crooked columns which fail by inelastic bending. It is concluded that the strengths of real steel columns should be analysed by considering the inelastic bending of crooked elastic-plastic-strain hardening columns with residual stresses. It is suggested that this will be most conveniently done by using a tangent modulus theory (which assumes that the loading and unloading paths coincide) to determine the maximum load capacities.

Remaining at Yield During Unloading and Other Unconventional Elastic-Plastic Response

A rather special time-independent or elastic-plastic response is proposed in which, although there is elastic response to unloading, the material remains at yield for all or a significant portion of the unloading path following plastic deformation. In the most elementary form, the material exhibits no memory of prior plastic deformation; the current state of the material is given solely by the current state of stress. A simple but unconventional field of plastic moduli then can be chosen to produce a limit surface that cuts through a nested set of yield surfaces and to model critical aspects of the behavior of sand.

Use of load path dependent material fracture toughness values (WPS) in the safety analysis of RPV stade

The effect of warm prestressing (WPS) has been investigated for the specific material and loading conditions of the central circumferential weld of the reactor pressure vessel of KKS under emergency core cooling. Warm prestressing results in a significant rise of effective fracture toughness. The magnitude of the WPS-effect as a function of warm prestressing, path of unloading, amount of cooling (e.g. change of yield stress) can be predicted on the basis of a theoretical model of Chell et al.. Crack initiation can be excluded for emergency core cooling (ECC)-loading and for material conditions beyond end of life for the central circumferential weld of the pressure vessel of KKS.

Large deformation in polycrystalline copper under combined tension-torsion, loading, unloading and reloading or reverse loading: A study of two incremental theories of plasticity

Experimental data from combined tension-torsion of thin walled tubes of annealed polycrystalline copper subjected to various non-proportionate loading, unloading and reverse loading paths are presented. The measurements are compared with predicted values from classical incremental theory of plasticity in terms of true stress and true strain and a recently developed incremental theory of plasticity by Bell in terms of nominal stress and nominal strain. These experimental data reveal that the plastic strain produced by the various proportionate and non-proportionate loading, unloading and reverse loading paths are in better agreement with Bell's incremental theory of plasticity as compared to classical incremental theory.