Stress Space Research Articles

Formulation and numerical implementation of a state parameter-based generalized plasticity model for mine tailings

Mine tailings are rock flour that is generally deposited in tailings dams. When deposited in a loose and saturated state, the tailings are prone to flow liquefaction. Simulating the onset of liquefaction and the post-liquefaction behavior of tailings has therefore become an essential aspect in geotechnical engineering of tailings dams. Several analysis procedures have been developed to address this problem, most of them employing the state parameter ψ as the key indicator of material behavior. The use of numerical deformation modelling for addressing the onset of flow liquefaction is increasingly practiced. Therefore, constitutive models, which are based on the state parameter and can simulate a wide range of stress paths, are demanded by the industry. This paper presents the formulation of an extension of a generalized plasticity model that is based on the state parameter and that simulates liquefaction within a critical state framework. As such, it is attractive for modelling loose tailings. The model is extended to the general 3D stress space and implemented by using an implicit integration scheme as a user model in Plaxis. This paper describes the set up and calibration of the model based on a set of laboratory tests on tailings and includes a numerical exercise proving the consistency and robustness of the formulation.

Strain Localization of Orthotropic Elasto-Plastic Cohesive-Frictional Materials: Analytical Results and Numerical Verification.

Strain localization analysis for orthotropic-associated plasticity in cohesive–frictional materials is addressed in this work. Specifically, the localization condition is derived from Maxwell’s kinematics, the plastic flow rule and the boundedness of stress rates. The analysis is applicable to strong and regularized discontinuity settings. Expanding on previous works, the quadratic orthotropic Hoffman and Tsai–Wu models are investigated and compared to pressure insensitive and sensitive models such as von Mises, Hill and Drucker–Prager. Analytical localization angles are obtained in uniaxial tension and compression under plane stress and plane strain conditions. These are only dependent on the plastic potential adopted; ensuing, a geometrical interpretation in the stress space is offered. The analytical results are then validated by independent numerical simulations. The B-bar finite element is used to deal with the limiting incompressibility in the purely isochoric plastic flow. For a strip under vertical stretching in plane stress and plane strain as well as Prandtl’s problem of indentation by a flat rigid die in plane strain, numerical results are presented for both isotropic and orthotropic plasticity models with or without tilting angle between the material axes and the applied loading. The influence of frictional behavior is studied. In all the investigated cases, the numerical results provide compelling support to the analytical prognosis.

Tests and Finite Element Simulation of Yield Anisotropy and Tension-Compression Strength Difference of an Extruded ZK60 Mg Alloy

Multiroute yield experiments were carried out on extruded ZK60 magnesium alloy samples under 200 °C and strain rate of 10−2 s−1 by a thermomechanical simulator. A hypoelastoplastic large deformation constitutive relationship was employed to simulate large deformation yield subjected to uniaxial loading, biaxial loading, loading-reverse loading, and orthogonal loading in the finite element (FE) software with user subroutine. The results showed the following: (1) As the accumulative plastic strain increased, the convex yield surface in the 2D stress space gradually expanded or shrank, rotated, and distorted from the approximate ellipse at low accumulative plastic strain. At the same accumulative plastic strain, uniaxial tensile or compressive yield values along different directions were not the same. In addition, the tensile yield value differed considerably from the compressive one. The large deformation yield of ZK60 magnesium alloy showed significant anisotropy, tension-compression strength difference, and evolution effect. (2) FE simulations based on the hypoelastoplastic constitutive relationship could accurately capture the strongly evolving asymmetric yield behavior under complex loading routes. The stress-strain relationships and hardening rates were in better accordance with the experimental results and reflected the yield behavior more realistically compared to simulations without the evolution effect or with traditional linear interpolation. Deformation at low temperature and high strain rate is important to save process costs and improve processing efficiency, but it may cause evolving asymmetric yield during large deformation. It is believed that the simulation approach used herein is reliable for the prediction and optimization of severe plastic deformation processes of hexagonal close-packed (HCP) alloys.

Experimental study and failure criterion analysis on combined compression-shear performance of rubber concrete (RC) with different rubber replacement ratio

In order to study the combined compression-shear mechanical properties of rubber concrete (RC), the compression-shear tests of RC under varying axial compression ratios and rubber content were carried out using compression-shear hydraulic servo machine. The shear failure modes and shear load–displacement curves of RC were generated. The characteristics points of load–displacement curves, peak shear strength, residual strength, shear failure displacement, for instance, were also extracted under different loading conditions. The effect of the axial compression ratios and replacement level of rubber particles on the shear mechanical properties of concrete was evaluated. It was reported that shear failure of RC developed slowly while shear failure plane was relatively flat by an increase in axial compression ratio. The friction traces on shear failure plane got more evident when the law of change was not associated to the replacement ratio of rubber particles. Research results also indicated that with successive increase in replacement ratio, the shear strength of concrete gradually decreases while the shear displacement gradually increases. Simultaneously, the increase coefficient of shear load of RC shows a decreasing trend, influenced by the axial compression ratio. The shear strength, residual load and shear displacement were enhanced as the axial compression ratio increased. While the reduction degree of shear load is larger due to the rubber replacement ratio. The cohesion and internal friction angle of concrete decrease by an increase in the rubber replacement ratio while the friction coefficient of the shear failure plane is relatively stable. The failure criterion models of concrete associated with the rubber replacement ratio under combined compression-shear stress are derived from the theory of plane stress space and octahedral stress space. Model results show favorable consistence with experimental data. The manuscript discussed and analyzed the mechanism of concrete under combined compression-shear. The research of this paper is of great significance for scientific research and engineering applications of RC.

A 3D non-orthogonal elastoplastic constitutive model for transversely isotropic soil

The significant influence of transverse isotropy on mechanical behaviours of soil has long been recognised, but it is still difficult to capture well under 3D stress conditions. In this study, a method of establishing 3D non-orthogonal elastoplastic (NOEP) constitutive models for transversely isotropic soil (TI-soil) is proposed. The effect of transverse isotropy is considered by extending the characteristic stress with the fabric tensor of TI-soil. The extended characteristic stress can be used to treat the mechanical behaviour of TI-soil from the perspective of isotropic soil. Therefore, a unified strength criterion for TI-soil is developed in the same form as the Drucker–Prager criterion. In order to reflect the non-orthogonality of the plastic flow direction with respect to the yield surface, the non-orthogonal plastic flow rule is adopted, which obviates the tedious work of constructing the plastic potential function or the dilatancy relation. Based on the yield function and hardening parameter expressed in the extended characteristic stress space, specific formulations of the NOEP model for TI-soil can be derived. Comparisons between model predictions and test results show that the developed model can well reflect the effects of transverse isotropy and intermediate principal stress on the strength and deformation behaviour of TI-soil.

Investigation on three-dimensional failure criterion of asphalt mixture considering the effect of stiffness

In order to reveal the effect of complex stress state on the characteristics of strength and stiffness of asphalt mixture, the self-developed double confining pressure triaxial test method was used in this study. The one-time failure and resilient modulus tests in complex stress state were carried out on the SMA-13 and AC-13 asphalt mixtures widely used in China's asphalt pavement, especially in the combined tensile-compressive stress state. The failure principal stress of material and its stress-strain data under the step loading condition were obtained. The calculation model of resilient modulus in complex stress state and the failure criterion I1-J2 in the principal stress space were established. On this basis, based on the generalized Hooke's law, a three-dimensional strain failure criterion model for asphalt mixture considering the effect of stiffness was also established by the conversion method. The research results can provide experimental and theoretical support for the reasonable selection and resistance calculation of material parameters of asphalt mixture in complex stress state.

Non-coaxial plastic flow of granular materials through stress probing analysis

The current study investigates the non-coaxial plastic flow in granular media where the principal directions of incremental plastic strain differ from those of the current stress (and of the current fabric tensor). The stress probing technique is used with Discrete Element Method simulations to capture the plastic response of a 2D granular assembly to loading increments along different directions in stress space, including non-coaxial ones. A general form for the plastic flow is suggested, based on multi-mechanism plasticity theory which is demonstrated to be capable of capturing the essential characteristics of the non-coaxial plastic flow. Deviations between DEM results and model predictions are associated with the incrementally nonlinear nature of the response. The analysis of the results indicate that, at a given stress state, the plastic yielding direction is not coaxial with the current stress. The significance of the non-coaxial plastic flow is shown to be around 20% for lower deviatoric stresses, but this value diminishes as the peak stress is achieved, hinting to the dominance of a single mechanism at failure. The study concludes by comparing the developed model with two well-known non-coaxial models in the literature, where the performance of the developed model is found to be close to the double-shearing theory.

Loss of ellipticity analysis in non-smooth plasticity

This work is concerned with the influence of non-smooth yield functions on material instabilities. The loss of ellipticity condition is examined in the case of a family of non-smooth yield functions in principal stress space. A procedure for the numerical detection of loss of ellipticity in multisurface plasticity is proposed. An explicit expression of the subdifferential of the non-smooth yield function is obtained, thereby extending existing results in the literature for the Tresca and Mohr–Coulomb criteria. Structural computations show that the use of non-smooth yield functions can lead to much earlier failure prediction than in the case of a commonly used von Mises criterion.

Multi-level deep drawing simulations of AA3104 aluminium alloy using crystal plasticity finite element modelling and phenomenological yield function

This paper proposed a hierarchical multi-level model to study the crystallographic texture induced mechanical anisotropy of AA3104-H19 aluminium sheet from mesoscale to continuum scale. In the mesoscale, full-field crystal plasticity finite element method (CPFEM) was used to provide both in-plane and out-of-plane yield stresses and plastic potential points in various deformation modes. In the continuum scale, these materials sampling points were used to determine the parameters of two phenomenological yield functions (Yld2000-2d in plane stress space and Yld2004-18p in 3D stress space) using associated flow rule (AFR) and non-associated flow rule (non-AFR). The results indicate that higher accuracy obtained by Yld2000-2d and Yld2004-18p yield functions associated with non-AFR in comparison with AFR. These phenomenological models were successfully implemented into finite element (FE) code using an explicit integration scheme to simulate sheet metal forming. It is found that the 3D Yld2004-18p model involved with both in-plane and out-of-plane anisotropies is superior to 2D Yld2000-2d model which only accounts for in-plane anisotropy.

Evolution of subsequent yield surfaces with plastic deformation along proportional and non-proportional loading paths on annealed AA6061 alloy: Experiments and crystal plasticity finite element modeling

This work presents an experimental and numerical investigation of annealed aluminum alloy AA6061 subjected to proportional and non-proportional loading paths at various finite levels of pre-strain. A careful mechanical characterization program is conducted for subsequent yield surfaces in the σ11−3σ12 stress space using an equivalent 10με offset linearity definition of yield. Electron backscatter diffraction (EBSD) is also used to characterize the microstructure of the material in the annealed condition. Each subsequent surface shows a very well-known shape of being nearly flat in unloading and having a prominent “nose” in the loading direction that deviates substantially from a generalized isotropic behavior. All proportional loading conditions show a rapid evolution of the mixed hardening effect that was characterized by the first load reversal in the first yield surface probe. Differences associated with the path-dependent effect in shifting the non-proportional yield surfaces are also observed in experiments. The experimental program is followed by Taylor-type crystal plasticity finite element (CPFE) simulations of the observed mechanical phenomenon. The Taylor-type CPFE model is calibrated using only the proportional uniaxial tension response and the EBSD imaging to predict all subsequent proportional and non-proportional loading paths. All CPFE show good predictive capabilities in capturing the experimental yield surface measurements for both proportional and non-proportional loading conditions. The CPFE simulations predict both the resulting shape and magnitude of the “nose” in the yield surfaces, as well as the mixed hardening, distortion, and yield surface translations. A systematic study is presented to investigate the effects of latent hardening on the yield surface predictions. This work highlights the promising results of advanced material model developments for larger-scale component simulations of lightweight aluminum alloys.

Effects of Porosity and Water Saturation on the Yield Surface of Upper Cretaceous Reservoir Chalks From the Danish North Sea

AbstractThe locus and shape of the yield surface of chalk in the stress space represent key input parameters in geomechanical models, as it dictates the onset and amount of plastic deformation. The study aims at reconstructing for the first time the initial end cap and shear failure line of intact reservoir chalk in plots of mean‐deviatoric stress versus porosity. One hundred and thirty‐seven stress‐strain measurements describe the effects of initial porosity between 30% and 45% and initial water saturation on the yield surface in order to cover a wide spectrum of reservoir rocks. The physico‐chemical processes taking place at pore collapse under oil‐ and water‐saturated conditions are discussed. The experimental data collected from previous works quantify the hardening and softening effects observed with decreasing porosity and increasing water saturation. The results also indicate that the stress interval characterizing the elastic to elasto‐plastic transition zone and the curvature of the end cap are dependent on porosity and quartz content. These observed trends suggest changes in the sequence of local failures acting in the rock matrix at the onset of compaction. Comparisons with previous studies suggest that the water weakening effect differs between outcrop and reservoir chalk likely due to a change in mineralogy. A yield function is proposed to build the yield surface and to capture the porosity‐dependent change in the shape of end cap and the water weakening effect. The outcomes provide key input data describing the mechanical behavior and properties of chalk under laboratory conditions required for compaction simulation studies.

Validity of continuous-failure-state unloading triaxial tests as a means to estimate the residual strength of rocks

The residual strength of rocks and rock masses is an important parameter to be constrained for analysis and design purposes in many rock engineering applications. A residual strength envelope in principal stress space is typically developed using residual strength data obtained from compression tests on many different specimens of the same rock type. In this study, we examined the potential for use of the continuous-failure-state testing concept as a means to constrain the residual strength envelope using a limited number of specimens. Specifically, cylindrical specimens of three rock types (granodiorite, diabase, and Stanstead granite) were unloaded at the residual state such that a full residual strength envelope for each individual specimen was obtained. Using a residual strength model that introduces a single new strength parameter (the residual strength index, or RSI), the results of the continuous-failure-state unloading tests were compared to conventionally obtained residual strength envelopes. Overall, the continuous-failure-state residual strength data were found to be consistent with the conventional residual strength data. However, it was identified that the primary factor limiting an accurate characterization of the residual strength for a given rock type is not the amount of data for a given specimen, but the variety of specimens available to characterize the inherent variability of the rock unit of interest. Accordingly, the use of continuous-failure-state testing for estimation of the residual strength of a rock unit is only recommended when the number of specimens available for testing is very limited (i.e. < 5).

Mesoscale Modelling of Concretes Subjected to Triaxial Loadings: Mechanical Properties and Fracture Behaviour.

The mechanical properties and fracture behaviour of concretes under different triaxial stress states were investigated based on a 3D mesoscale model. The quasistatic triaxial loadings, namely, compression–compression–compression (C–C–C), compression–tension–tension (C–T–T) and compression–compression–tension (C–C–T), were simulated using an implicit solver. The mesoscopic modelling with good robustness gave reliable and detailed damage evolution processes under different triaxial stress states. The lateral tensile stress significantly influenced the multiaxial mechanical behaviour of the concretes, accelerating the concrete failure. With low lateral pressures or tensile stress, axial cleavage was the main failure mode of the specimens. Furthermore, the concretes presented shear failures under medium lateral pressures. The concretes experienced a transition from brittle fracture to plastic failure under high lateral pressures. The Ottosen parameters were modified by the gradient descent method and then the failure criterion of the concretes in the principal stress space was given. The failure criterion could describe the strength characteristics of concrete materials well by being fitted with experimental data under different triaxial stress states.

Tsai–Wu based orthotropic damage model

This paper presents a novel approach concerning the development of an orthotropic damage model for composite materials, based on the original plane stress Tsai–Wu failure criterion. In its original formulation, the Tsai–Wu is a mode-independent criterion only capable of acknowledging the existence of damage in a certain point of a composite material. It is not capable of identifying if the damage is located in the fibre, matrix or interlaminar zone. This study aims at filling this gap in knowledge by providing a simple method, based on equivalent stresses and strains, that identifies the relevant failure modes when the Tsai–Wu failure criterion is at the onset of damage. Using this novel methodology, it is possible to implement classical damage evolution constitutive laws based on the fracture energy regularization. At present, the proposed damage formulation is based on the consideration of a plane stress space and Mode I fracture, but its generalization to a full 3D damage model is expected to be defined in the near future. The damage model is implemented in the commercial finite element software ABAQUS through a user-defined material (UMAT) subroutine, and all numerical models are compared with experimental results available in the literature.

Linking bulk modulus to an unilateral damage yield criterion: A thermodynamic modeling approach

This work presents a new damage criterion suitable for elastic, elastic-plastic/viscous or elastic-viscous-plastic materials involving rupture effects. Its derivation, made here within a thermodynamic framework, follows previous scalar-valued damage mechanics approaches. Such approaches are appropriate to many geophysical problems involving quasi-brittle materials for which there is no clear physical justification for the level of complexity of a tensorial damage variable. Distinction between the mechanical response to compressive and tensile stresses is therefore not introduced by the damage itself but via a special definition of the Helmholtz free energy. This scheme differs from previous ones in that it combines with an evolution of Poisson’s ratio with the level of damage, which allows expressing the damage criterion in the principal stresses space. Moreover, there is no need to compute the stress eigensystem, which makes it simpler to implement than the Mohr–Coulomb damage criterion. Here we derive this damage criterion and compare it to observations of the variations of the bulk modulus in damaged geomaterials. We also compare it to in-situ stress measurements and find a good agreement in terms of the shape of the criterion in the stress space. We tentatively interpret the results in the context of previous studies of rock and ice mechanics.

Exploration of subsequent yield surfaces through unit cell simulations

Subsequent yield surfaces of ductile solids, pre-strained in shear and/or tension, exhibit various characteristic features that depend on the proof strain used to detect yield. These features include kinematic hardening when the proof strain denoting yield is low, isotropic hardening when it is high, formation of a ‘nose’ in the loading direction and flattening of the rear part of the yield surface. With the aid of a computational homogenisation scheme that allows application of arbitrary macro stress or macro deformation gradient to a unit cell, we show that essential features of the subsequent yield surfaces owe their origin to the extent of plasticity accumulated within the unit cell. For instance, a simple unit cell containing a spherical void, where the void serves to create a heterogenous distribution of plastic strain, suffices to computationally reproduce the experimentally observed features. Moreover, in contrast to experiments which generally allow plotting of the yield surface in a normal stress – shear stress plane, the computational scheme allows fuller exploration of the subsequent macro yield surface in the macro stress space. We find that what seems to be a translation and distortion of the yield surface in experiments, is actually most pronounced near some critical octahedral planes characterised by levels of triaxiality induced by the pre-straining process. The evolution of the yield surface in regions where the triaxiality is higher is considerably different. Moreover, as homogeneous yielding is approached with loading of the unit cell, the material seems to forget the nature of the pre-straining process and undergoes isotropic hardening.

Cyclic and Dynamic Behavior of Sand–Rubber and Clay–Rubber Mixtures

In this paper, the possibility of using fine scrap tyre rubber to improve the mechanical properties of soil subjected to cyclic loading is addressed. Ground rubber (0.1–0.8 mm) in various proportions (0, 9, 33% and 100% by weight) was mixed with a uniform river sand and a lean clay. Cyclic triaxial tests with bender elements were executed to observe the behaviour of the materials and also to determine damping and shear stiffness parameters. The results have shown that the addition of rubber has significantly decreased the density and shear stiffness of both types of soils, which favours mitigation of vibrations. The shear stiffness degradation at shear strains higher than 10−3 was lower in specimens containing more rubber. Within this strain range, addition of rubber decreased the damping ratio, but increased the normalized accumulated absolute strain energy absorbed by the material. Higher rubber content in sandy specimens resulted in more elastic behaviour, with lower strain accumulation in each loading cycle, eventually resulting in a higher number of loading cycles before failure. The positive effect of rubber presence was not observed in compacted clay–rubber mixture, which sustained less loading cycles than clay alone. The influence of rubber addition in the p′-q stress space was expressed in the form of lower pore pressure generation which shifted the stress path further from the failure envelope.

Structured UH model considering induced anisotropy

Based on the structured unified hardening model (structured UH model), introducing rotational hardening, a structured UH model considering induced anisotropy is presented. Applying the transformed method, the presented model is extended in three-dimensional stress space. Comparing the test data with the model simulations, the structured UH model considering induced anisotropy is qualified to smoothly and continuously describe the behaviors of natural soils in cyclic loading.

Effect of plastic straining on the remanent magnetization of ferritic-pearlitic steel: Experimental and modeling aspects

The remanent magnetization of a carbon steel which has been plastically deformed by uniaxial tensile test at different strain levels is measured along the tensile direction during unloading and at the unloaded state. It is shown that plastic strain considerably reduces the remanent magnetization but that remanence is recovered under applied stress, leading to the conclusion that behavior of the deformed material seems to be able to be obtained by a simple translation in the stress space of the behavior at the undeformed state. This result is in accordance with the results of a modeling approach considering the deformed material as a two phase material submitted to a strong internal stress level.

CYCLIC TORSION WITH INTERNAL PRESSURE OF PRE-STRETCHED THIN-WALLED CYLINDERS

The article considers the following two tasks. 1) Strength calculations are carried out for a long thin-walled cylinder, the ends of which are not closed during reloading. At the first loading, the thin-walled cylinder is stretched by a longitudinal force leading to longitudinal plastic deformations. When reloaded, the cylinder twists and at the same time uniform pressure acts from the inside. It is noted that, in particular, the shaft of wind generators is subjected to such loading. It is proved that the equation of the yield curve in the plane of the normal and tangential stresses of the stress space is an ellipse, whose semi axes are a function of the mechanical characteristics of the material and the residual stresses after the first loading. 2) We consider cyclic twisting with internal pressure of pre-stretched thin-walled cylinders. The yield surface equation is obtained for a thin-walled cylinder under repeated loading τ + p. which stretched upon first loading. It is proved that the yield surface equations also yield the fatigue surface equations if the yield stress σ_(T.) is replaced by the fatigue stress σ_(0.) Expressions are found for the number of cycles required for fatigue failure under an asymmetric loading cycle under repeated loading. It is proved that in order to increase the number of cycles required for fatigue fracture, during the first loading, when the material is strengthened due to plastic deformation, and with repeated cyclic loading, the same stress components must take place. It is proved that if during repeated loading the end of the loading path is inside the endurance surface, then the number of cycles required for fatigue failure in an asymmetric loading cycle tends to infinity. Keywords: Initial loading, reloading, cyclic loading, strength, fatigue failure, plastic deformation, residual stress, yield surface.