Unloading Branches Research Articles

On flutter instability in elastoplastic constitutive models

Two conditions of non-propagation of wave modes are analyzed: flutter instability as described by Rice (1976) and non propagation due to different algebraic and geometric multiplicity in the eigenvalues of the acoustic tensor. Explicit reference is made to elastoplastic constitutive operators at finite strains. Both loading and unloading branches of the constitutive operator are analyzed, but they are treated independently (we disregard the interaction between loading and unloading). The spectral analysis of Bigoni and Zaccaria (1994) is generalized to examine an unsymmetric acoustic tensor for the unloading branch of the constitutive operator. Two constitutive laws for finite-strain elastoplasticity are considered, one of which is widely in use (Rudnicki and Rice 1975). In both constitutive laws, unloading of the material follows a specific grade 1-hypoelasticity, lacking in any stress-rate potential. For these materials, we show that instabilities are excluded in the unloading branch, whereas they remain possible in the loading branch of the elastoplastic constitutive operator. Therefore, the geometrical terms of the constitutive equations (when small compared to the elastic shear modulus) provide examples of perturbations which induce flutter and non-propagation instability in elastoplasticity, yet have no effect on infinitesimal, three-dimensional, isotropic elasticity (where two wave speeds always coincide).

Development of anisotropy in the incremental shear moduli for rock undergoing inelastic deformation

Abstract New experimental data on the evolution of two incremental shear moduli have been obtained for rock undergoing inelastic deformation. Changes in the incremental shear moduli during increasing axial straining probably reflect the development of a deformation-induced, anisotropic fabric resulting from accumulation of oriented damage. These incremental moduli occupy a prominent position in a constitutive equation for the plane deformation of inelastic solids that has evolved over nearly 30 years to include pressure-sensitivity, compressibility and plastic non-normality. This equation is often used in analyses of strain localization. One of the moduli, G, governs shearing parallel to the coordinate axes, and the other one, G ∗ , governs shearing at 45° to the coordinate axes. They both were measured on the loading branch of the constitutive equation. The modulus for shearing at 45° was also measured on the unloading branch, where it is called G ∗ . Though the predicted onset of strain localization for this type of constitutive model is a strong function of G ∗ /G , there appears to have been no such data collected for rock. Experiments to obtain estimates of these incremental moduli were conducted on solid cylinders and thin-walled tubes of Tennessee marble during axisymmetric compression under fluid confining pressures of 60 MPa. All three moduli were equal to the isotropic elastic value of E/2(1 + ν) ≃ 30 GPa in the elastic range of axisymmetric compression, but then the moduli decreased with increasing axial strain up to 0.01, always with G ∗ ≤ G ∗ ≤ G , and the ratio G ∗ /G decreased markedly. The decreased ratio, G ∗ /G , allows strain localization to occur earlier in a program of axial straining than would be predicted for a similar material that did not exhibit such a change in moduli.

Consideration of the effect of internal cracking in concrete on its deformation under short- and long-time tension

1. Discontinuity of the cross sections has a direct effect on the coefficient of alteration of the secant modulus of deformation and leads to the development of unique nonlinear (pseudoplastic) deformations, which are partially recoverable. 2. The permanent part of deformations in the case of short-time unloading depends on the occurrence of a secondary field of stressesσ on the sides of closed cracks (squeezing stresses), which, along with the effect of some growth of cracks, determine the character of curvature of theσ-e diagram on the unloading and subsequent loading branches. 3. Discontinuity especially affects the development of creep, leading to violation of the principle of superposition of influences with the use of stressesσ (traditionally referred to the total cross-sectional area), whereas with the use of real stressesσ referred to the continuous part of the section this principle can be preserved. The uncoordinated scheme of formation and closing of cracks is the most exact for concrete. 4. Consideration of the indicated factors permits constructing a more accurate model of deformation of concrete in short- and long-time tension regimes.

Cyclic Characterization of Liquefiable Sands

The method for cyclic characterization of liquefiable sands, which employs the concept of dynamic backbone curve in conjunction with the Masing criteria for construction of unloading and reloading branches of cyclic loops, is refined and verified. The method also encompasses subsequent cyclic loading, when soil undergoes significant degradation due to pore‐water pressure buildup. To accurately describe the initial backbone curve and associated initial cyclic loop, and subsequent degraded backbone curves and associated subsequent cyclic stress‐strain loops, a well‐known hyperbolic model and a degradation model are modified. Although relatively simple, such modifications enable accurate description of the stress‐strain loops of the first and subsequent cycles, even in cases when soil rapidly degrades and liquefies in just a few cycles. Consequently, a much better agreement between the measured and calculated damping values can be also obtained. The investigation is based on test results obtained on five dif...

Hysteretic Behavior of Anchorage Slip in R/C Members

Large-scale reinforced concrete columns were tested to investigate hysteretic behavior of anchorage slip in reinforced concrete structures. The columns were subjected to constant axial compression and unidirectional and bidirectional lateral-deformation reversals. The results indicate a significant increase in column rotation due to anchorage slip. Penetration of yielding into the column footing was observed in the tension reinforcement, while the compression yielding was localized at the column-footing interface. Axial compression resulted in early closure of the crack associated with anchorage slip, reducing related deformations. An analytical model was developed for hysteretic moment-anchorage slip-rotation relationship. The model consists of a primary curve and a set of rules defining unloading and reloading branches. The primary curve is constructed by computing the extension and slippage of tension reinforcement in the adjoining member. This is accomplished by considering inelastic strain distribution along the embedment length of reinforcement, as well as the local bond-slip relationship. Hysteretic rules are obtained from experimental observations, and incorporate pinching of hysteresis loops as well as the effect of axial compression. Comparisons of experimental and analytical hysteretic relationships indicate excellent agreement.

Development of Strain During Monotonic Shear of Soft Clay

The development of shear strain in a soft, marine clay undergoing monotonic shear in torsional and simple shear equipment has been studied. The ratio between the irrecoverable, or plastic, shear strain to the overall shear strain that develops on a particular plane was found to be related to the overall shear strain developed on that plane by a continuous, increasing function. The function was found to be linear on a semilogarithmic scale up to a shear strain of the order of 25%, and it appears to be relatively unaffected by the previous development of shear strains on other planes. A similar functional relationship was found to be applicable, also, for shear strain increments. The Masing model was found to reasonably predict the shape of the unloading branch of the shear stress‐strain curve under various boundary conditions, making it possible to estimate plastic strains from monotonic loading tests. On the basis of observations made in the study, it appears that kinematic rather than isotropic work‐hard...

Determination of dynamic characteristics and relaxation and creep functions for elastically damping elements

Elastically damping elements made of metal-rubber composites have a parabolic hysteresis curve. Thus, the loading and unloading branches of the curve are nonlinear. The stiffness of the elements ― defined as the tangent on the angle between a tangent line to some point of the loading (or unloading) branch and the displacement axis ― changes with a broad range of values (0, ∞). The forms of the hysteresis loops of structural elements in compression-unloading cycles differ under static and dynamic conditions

Crack Band Based Model for FEM Analysis of Concrete Structures

Over the last 20 years several finite element models for cracked concrete have emerged [e.g., Nilson (1982)]. In early models tensile behavior of concrete was described by a linear relationship. To improve realism and to account for the influence of reinforcement and post-peak softening of concrete in tension, a strain-softening branch in the relationship was introduced [e.g., Lin and Scordelis (1975)]. The adoption of an unloading branch while still using a tensile strength criterion to control crack propagation, created a problem of mesh sensitivity. It was recognized (Bazant and Cedolin 1979; Hillerborg et al, 1976) that the fracture mechanics criterion of constant energy dissipation in the process zone during crack propagation had to be maintained to achieve objectivity of the model. Several models based on this approach were proposed [e.g., Hillerborg et al. (1976) and Bazant and Oh (1983)]. The aim of this paper is to present a new formulation capable of handling strain-softening processes both in tension and compression as well as nonlinear, pre-peak behavior of concrete in compression. The model does not place restrictions on crack directions or number of cracks that can appear at a single integration point. Further, a method of including effects of local bond slip of reinforcement, compatible with the approach to fracture of concrete, is described.

Hysteretic Shear Model for Reinforced Concrete Members

A hysteretic model is proposed for shear response of reinforced concrete members subjected to shear force and bending moment reversals. The model consists of a primary shear force–shear displacement curve and unloading and reloading branches under cyclic loading. The primary curve is the envelope curve for the hysteretic relationship, and can be considered the same as the force displacement relationship under monotonic loading up to the onset of strength decay. A primary curve established by any acceptable procedure, with well-defined cracking and yield points, can be used in constructing the model. The rules proposed for unloading and reloading branches of the model were obtained from a large number of test data. Full-scale column tests, under inelastic load reversals, were used for this purpose. These rules were developed to predict experimentally observed stiffness degradation and pinching of hysteresis loops. Empirically derived expressions were included to introduce the effects of axial load, deformation level, and the number of deformation cycles. The model was employed to reproduce the results of large scale tests of reinforced concrete elements. The comparisons include columns, beams, and structural walls, tested by different investigators. Individual hysteresis loops of small and large deformation cycles are compared with those produced by the model. The results show a good agreement between the model and the experimental data.

Deformation and elastic properties of WC-Co system sintered carbides in micro- and macroindentation

1. The work of deformation of WC-Co system alloys in micro- and macroindentation was determined and the character of its change in relation to the binder phase content was shown. The work of deformation with a constant depth of indentation of the indentor is a more sensitive characteristic relative to the structure of tungsten sintered carbides than with a constant load. 2. The share of elastic deformation and of the work of elastic deformation of tungsten carbide is very significant in relatively brittle alloys with a low cobalt content while, on the other hand, in high-cobalt alloys the total work of deformation in indentation of the indentor is basically determined by the work of plastic deformation of the binder phase. 3. Determination of the modulus of elasticity of WC-Co system alloys from the unloading branch of the indentor indentation curves showed the possibility of establishment of this characteristic by that method. The divergence from the values obtained by the ultrasonic method is related to relaxation processes at comparatively low loading rates. 4. The deviation of the change in the work of elastic deformation in indentation of the indentor and of the modulus of elasticity of the alloys from linearity is related to features of their structure and primarily to the high degree of proximity of the carbide phase in the low-cobalt sintered carbides. 5. The increase in hardness with low loads for tungsten carbide and the relatively brittle low-cobalt sintered carbides in relation to the force in indentation of the indentor is the same as for many brittle refractory compounds.

Modelling hysteretic behaviour of coupled walls for dynamic analysis

AbstractModelling techniques for dynamic inelastic response analysis of coupled wall structures are investigated. Emphasis is placed on effects of parameters defining the force‐displacement hysteresis loop. Specifically, effects of axial force‐moment interaction, strength reduction, shear yielding, pinching, reloading and unloading branches of hysteresis loops are considered. Effects of modelling parameters on selected response quantities are investigated and discussed in detail.A 20‐storey coupled wall structure was selected for dynamic analysis. Ranges of parameters characterizing force‐displacement hysteresis loops were obtained from laboratory tests under slowly reversed static loading. Previously recorded strong motion accelerograms were used as input motions.Results indicate that wall axial forces and beam strength reduction can have significant effects on response envelopes. Moderate variations in unloading and reloading branches of hysteresis loops and pinching appear to have little effect on dynamic response.

Inelastic stability of diagonal braces in building frames

The inelastic behavior of diagonal braces in building frames subjected to relative horizontal end displacements is investigated both theoretically and experimentally with particular reference to the post-buckling behavior. The analysis is based on commonly adopted piecewise linear approximate moment-thrust interaction curves and bilinear elastic-plastic moment-curvature relations. The maximum loads as well as the horizontal load-displacement curves observed in the experimental investigation agree well with the theoretical prediction for short braces. In the medium range, the stable branches of the horizontal load-displacement1 curves are followed by a steep unloading branch, with initially positive slope which becomes infinite at a point beyond which the curves become unstable. The maximum experimental loads are smaller than the theoretical predictions and occur consistently in the vicinity of the point of infinite slope in the theoretical horizontal load-displacement curves. The experimental curves start to deviate from the theoretical curves shortly before reaching this point, showing rapid increase in the displacement. Finally, long braces behave elastically until the maximum loads are attained at the termination of the elastic state. For practical purposes, a simple means for predicting the carrying capacity of diagonal braces is recommended in view of these findings.

Tension and compression diagrams for sandstone

1. Tension and compression diagrams show that sandstones from the “Zagadka” bed and Arkagalinsk deposit, with different petrographic structure, have deformation curves which express different relations between E and σ. 2. For the specimens which we tested, repeated loading and unloading increases the steepness of the loading branch and has little effect on the unloading branch. This effect is very marked for Zagadka sandstone, but less so for Arkagalinsk sandstone. 3. Under loading, the deformation curves for Zagadka sandstone under tension and compression tend in opposite directions; the tension curve is convex towards the σ axis, while the compression curve is convex towards the e axis. It follows that under tension E for the Zagadka sandstone decreases with rise of stress, but under compression it increases. For Arkagalinsk sandstone the loading branches of the tension and compression diagrams are almost straight. 4. During unloading, the deformation curves for tension and compression are convex towards the e axis. Thus the deformation coefficient increases with the stress. This relation is less marked for Arkagalinsk sandstone.