Plastic Volumetric Strain Research Articles

Experimental Study on Evolution of Rock Damage Rule in the Condition of Conventional Three Axis Compression

Based on continuum damage mechanics, by defining the initial and critical damage value, because of the plastic volumetric strain of the specimen, we established the calculation methods of the rock specimen’s damage value. Through the conventional triaxial compression test of the shale, we find that damage value D changes as axial strain changes during compression, and summarizes its variation.

Study on Rheological Behavior of Undisturbed Soft Clay in Wenzhou Considering Structural Property

Based on boundary surface elastoplastic theory, a boundary surface elastoplastic constitutive model is established which is considered both structural property and rheological behavior of soft clay with the method of taking Cam-Clay yield surface as boundary surface. The structural parameters are introduced to calculate structure-damage strain. Plastic volumetric strain is regarded as a function timetso that a rheological parameter is introduced. The constitutive model is validated by numerical analysis of triaxial rheological tests.

Pore pressure response of seabed in standing waves and its mechanism

Experiments were carried out in a newly-developed rectangular flume to investigate the pore pressure response of silty seabed in standing waves, and to emphatically discuss the physical mechanism of wave-induced pore pressure. We first analyzed the features of wave-induced stress, laid out the experimental setup, and particularly monitored the temporal and spatial variation of pore pressure. Then, we summarized the physical essence of wave-induced pore pressure, discussed the applicability of the existing models, and proposed several key points about developing new models. Results indicate that regular standing waves were obtained in the rectangular flume. The oscillating pore pressure (OPP) and the residual pore pressure (RPP) were observed simultaneously in the seabed under wave nodes or antinodes, and were found with coupling effect. The OPP and RPP were related to the elastic and plastic volumetric strains, respectively. Both elastic and plastic volumetric strains may be caused by the wave-induced cyclic spherical stress or deviatoric stress. The elastic model could only simulate the OPP and the pore pressure development mode could only simulate the RPP. The yielding under isotropic compression and the shear expansion of geotechnical materials, as well as the wave-induced stress paths, should be considered in the elastoplastic model to simulate the wave-induced pore pressure accurately.

Numerical modeling of rapid impact compaction in loose sands

A three dimensional finite element model was used to simulate rapid impact compaction (RIC) in loose granular soils using ABAQUS software for one impact point. The behavior of soil under impact loading was expressed using a cap-plasticity model. Numerical modeling was done for a site in Assalouyeh petrochemical complex in southern Iran to verify the results. In-situ settlements per blow were compared to those in the numerical model. Measurements of improvement by depth were obtained from the in-situ standard penetration, plate loading, and large density tests and were compared with the numerical model results. Contours of the equal relative density clearly showed the efficiency of RIC laterally and at depth. Plastic volumetric strains below the anvil and the effect of RIC set indicated that a set of 10 mm can be considered to be a threshold value for soil improvement using this method. The results showed that RIC strongly improved the soil up to 2 m in depth and commonly influenced the soil up to depths of 4 m.

UNDRAINED SHEAR STRENGTH OF K0CONSOLIDATED SOFT CLAYS UNDER TRIAXIAL AND PLANE STRAIN CONDITIONS

Theoretical formulas for predicting the undrained shear strength of K0consolidated soft clays under the stress path related to triaxial and plane strain tests are presented within the framework of critical state soil mechanics. An inclined elliptical yield surface is adopted to take account of the initial anisotropic stress state. The undrained strength is determined by combining the undrained stress path in the volumetric stress–strain space and the initial yield surface in the deviator-mean stress space. The derived mathematical expressions are functions of the critical state frictional angle, the plastic volumetric strain ratio and the overconsolidation ratio, which can be simplified into the solutions for isotropically consolidated clays under triaxial tests or under plane strain tests. The results calculated by using the theoretical formulas obtained in this paper are in good agreement with the available collected test results. It indicates that these new formulas are applicable to triaxial and plane strain tests on normally and lightly to moderately overconsolidated soft clays.

Predicting fault damage zones by modeling dynamic rupture propagation and comparison with field observations

We use a two-dimensional plane strain dynamic rupture model with strongly rate-weakening friction and off-fault Drucker-Prager plasticity to model damage zones associated with buried second-order thrust faults observed in the SSC reservoir. The modeling of ruptures propagating as self-sustaining pulses is performed in the framework of continuum plasticity where the plasticity formulation includes both deviatoric and volumetric plastic strains. The material deforming inelastically due to stress perturbations generated by the propagating rupture is assumed to be the damage zone associated with the fault. Dilatant plastic strains are converted into a fracture population by assuming that the dilatant plastic strain is manifested in the form of fractures. The cumulative effect of multiple slip events is considered by superposition of the plastic strain field obtained from individual slip events. The relative number of various magnitude slip events is chosen so as to honor the Gutenberg-Richter law. Results show that the decay of fracture density (F) with distance (r) from the fault can be described by a power law F = F0r− n. The fault constant F0 represents the fracture density at unit distance from the fault. The decay rate (n) in fracture density is approximately 0.85 close to the fault and increases to ~1.4 at larger distances (>10 m). Modeled damage zones are approximately 60–100 m wide. These attributes are similar to those observed in the SSC reservoir using wellbore image logs and those reported in outcrop studies. Considering fault roughness affects local damage zone characteristics, these characteristics are similar to those modeled around planar faults at a scale (~10 m) that affects bulk fluid-flow properties.

Combined effects of structure and partial saturation in natural soils

The mechanical behaviour of natural clays is significantly affected by their in situ or initial structure in the form of cementation or interparticle bonding. This behaviour can differ substantially from the behaviour of reconstituted clays. Suction as well as plastic volumetric strains drive isotropic hardening/softening as this is a simple way to account for the phenomenon of volumetric collapse upon wetting and the stiffening effect that suction has on the soil skeletal response. A model that combines unsaturated and structured behavior is presented and then used to simulate stress strain behaviour observed for an unsaturated natural clay subjected to isotropic load paths. A parametric analysis is performed to observe the influence suction hardening has on mobilized strengths. It is also shown that the model can predict the maximum of collapse of unsaturated soils.

Non-isothermal unified hardening model: a thermo-elasto-plastic model for clays

A constitutive, non-isothermal unified hardening (UH) model is presented to interpret the thermo-elasto-plastic behaviours of normally consolidated and overconsolidated clays. Two yield surfaces are adopted in the proposed model: the current yield surface and the reference yield surface. A UH parameter (H) is developed to describe the evolution of the current yield surface, and the plastic volumetric strain is employed to quantify the hardening of the reference yield surface. The similarity ratio (RT) between the current yield surface and the reference yield surface, which is a function of the temperature and the plastic volumetric strain, is developed to govern the volume change behaviour and the shear strength of soils with different stress histories and at varying temperatures. The performance of the proposed model is then discussed in five typical scenarios: isotropic heating and cooling, drained/undrained triaxial compression with constant temperatures, and heating under constant non-isotropic states (drained/undrained). The mechanisms for thermal contraction/swelling and thermal failure are interpreted within the framework of the proposed non-isothermal UH model. Finally, the proposed model is validated through test results in the literature: heating/cooling tests, temperature-controlled drained triaxial compressions, and temperature-controlled undrained triaxial compressions.

A new generalized Drucker–Prager flow rule for concrete under compression

In this paper, a new generalized Drucker–Prager flow rule for concrete under compression with infinitesimal deformation is proposed, where the plastic volumetric strain is taken as the hardening parameter. The flow rule is proposed based on two fundamental phenomenological deformation properties of concrete, and on the fact that the deviatoric part of the plastic strain has no contribution to plastic volumetric strain. There is only one material parameter needing calibration for each grade of concrete in the flow rule. This parameter has a clear physical meaning to reflect the brittleness of concrete, and is defined as the brittleness index of concrete. The flow rule has a concise and simple form similar to the traditional Drucker–Prager flow rule, and for the case of uniaxial compression, the proposed flow rule is identical to the traditional Drucker–Prager flow rule. The effect of confinement stress is well considered by multiplying the hydrostatic part of the flow rule with a simple piecewise function, which differentiates the proposed flow rule from the traditional Drucker–Prager flow rule. An iterative method is proposed to determine the incremental stress–strain relationship when determining the flow rule. The correctness and reliability of the suggested flow rule are is validated using uniaxial, biaxial, and triaxial experimental results.

Formulation of a sand plasticity plane-strain model for earthquake engineering applications

The formulation of a sand plasticity model for geotechnical earthquake engineering applications is presented. The model follows the basic framework of the stress-ratio controlled, critical state compatible, bounding surface plasticity model for sand presented by Dafalias and Manzari, Journal of Engineering Mechanics, ASCE, 2004;130(6): 622–634. . Modifications to the model were developed and implemented to improve its ability to approximate the stress–strain responses important to geotechnical earthquake engineering applications. These constitutive modifications included: revising the fabric formation function to depend on plastic shear rather than plastic volumetric strains; adding fabric history and cumulative fabric formation terms; modifying the plastic modulus relationship and making it dependent on fabric; modifying the dilatancy relationships to provide more distinct control of volumetric contraction versus expansion behavior; providing a constraint on the dilatancy during volumetric expansion so that it is consistent with Bolton′s Géotechnique 1986; 36(1): 65–78. dilatancy relationship; modifying the elastic modulus relationship to include dependence on stress ratio and fabric history; modifying the logic for tracking previous initial back-stress ratios (i.e., loading history effect); recasting the critical state framework to be in terms of a relative state parameter index; and simplifying the formulation by restraining it to plane strain without Lode angle dependency for the bounding and dilation surfaces. Model responses to various loading conditions, including drained and undrained monotonic and cyclic loading, are used to illustrate the efficacy of the model modifications. Calibration and implementation of the model for practice are described in a companion paper.

Effect of the Triaxiality in Plane Stress Conditions. Triaxiality Effect in a PVC Material

Polymer materials are gaining more and more importance in engineering applications. A new methodology of analysis is required in order to assess the capability of such material in withstanding complex loads. Therefore, the behavior of these materials currently arouses a great research interest. The use of PVC plastic pipes in pressure vessels and pipelines has increased rapidly in the last decade. In order to determine the plastic behavior of PVC, an experimental method is presented. Through the results obtained from experimental tests, in the first part of this paper, we investigate the use of a phenomenological model proposed by G’Sell and Jonas. The true stress-strain response under large plastic deformation was investigated in different stress triaxiality frameworks. Particular attention was given to volumetric strain evolution, separation resulting from elastic volumetric strain, plastic volumetric strain and pure shear. The effect of stress triaxiality on plastic instability and fracture strain was also examined. The deformation process should be considered as explained, and the anisotropic plastic response induced by the deformation could be introduced in constitutive equations of G’Sell.

Plasticity based stress–strain model for concrete confinement

Concrete confinement has an important role in reinforced concrete columns, especially for strengthening applications and in hybrid construction. It is well known that confinement increases axial strength and ultimate strain of concrete. However, existing confinement models are extremely dependent on the type of confinement, i.e., passive or active; and the materials of the confining jacket; i.e., steel or fiber reinforced polymers (FRPs). In this paper, a unified stress–strain model is proposed to evaluate any type of uniform confinement. The model is based on the theory of plasticity, with a key feature on the use of plastic volumetric strain as the hardening parameter. New equations are proposed to define the hardening function in the ascending and descending branches. Initially, a three-dimensional plasticity algorithm was implemented in a nonlinear finite element analysis. The algorithm was validated against a series of test data with different states of stresses, including uniaxial or biaxial compression and uniform confinement. Upon validation of the plasticity model, its equations were rearranged in such a way that the behavior within a strain increment could be described totally using simple equations for uniaxial compression and uniform confinement. Consequently, an explicit stress–strain model was developed and verified against test data for any type of uniform active or passive confinement with any confining material. The procedure is incremental, while no iterative process is necessary. Due to the simplicity of this model, it can be easily implemented in a spreadsheet, so as to consider many parts of the response curve, if so desired. Finally, the model was validated against a representative set of experiments with favorable results.

Modeling the inelastic deformation and fracture of polymer composites – Part I: Plasticity model

A new transversely-isotropic elastic–plastic constitutive model for unidirectional fiber reinforced polymers (FRP) is presented. The model is able to represent the fully nonlinear mechanical behavior under multi-axial loading conditions and under triaxial stress states prior to the onset of cracking. Since associated flow rules often give a wrong prediction of plastic Poisson coefficients, a non-associated flow rule is introduced to provide realistic predictions of the volumetric plastic strains. This paper focusses on the simulation of triaxiality dependent plasticity based nonlinearities of FRP until failure occurs. The onset and propagation of failure is predicted by a new smeared crack model presented in an accompanying paper (Camanho et al., 2012). In order to demonstrate the capabilities of the new material model, a yield surface parameter identification for IM7-8552 carbon epoxy is presented and simulations of quasi-static transverse and off-axis compression tests and of uniaxial compression tests superimposed with various values of hydrostatic pressure are shown as a model verification.

On the effects of deformation induced anisotropy in isotropic materials

SUMMARYEffects of recoverable deformation induced anisotropy in the elastic stiffness of isotropic materials are described. In isotropic materials, thermodynamics predicts coupling of hydrostatic and deviatoric responses. It is shown that the coupling of the two responses is more significant than previously recognized in the literature. Properly accounting for the coupling of hydrostatic and deviatoric responses requires re‐evaluating elastic materials characterization data, allowing for the coupled response. The result is an apparent decrease in the pressure sensitivity of the elastic shear modulus. The decrease in the pressure sensitivity of the shear modulus leads to stress paths that are more tangential to the yield surface in stress space, resulting in an increase in predicted elastic strain at each step of an elastic–plastic stress update. Consequently, predicted plastic strains and, in particular, volumetric plastic strains, are smaller than if recoverable deformation induced anisotropy had been neglected. The result is an associated plasticity model, which appears to be non‐associated. Copyright © 2012 John Wiley & Sons, Ltd.

A two‐mechanism elastoplastic model for shakedown of unbound granular materials and DEM simulations

SUMMARYAn elastoplastic model has been developed for the finite elements modelling of repeated load triaxial tests. This model is based on the shakedown theory established by Zarka for metallic structures. To the previous works, which were based on the Drucker–Prager yield surface and the plastic potential of Von Mises, a compression cap has been added to each one. The model straightforwardly determines the purely elastic state or the elastic shakedown state or the plastic shakedown state and calculates the deviatoric and the volumetric plastic strains. The calibration of the elastoplastic model has been carried out with DEM simulations and an unbound granular material for roads under repeated load triaxial tests using finite element method. The calculations underline the capabilities of the model to take into account, with a unique formalism, the accumulation of the deviatoric and volumetric plastic strains along the loading cycles. Copyright © 2011 John Wiley & Sons, Ltd.

Fabric dilatancy and the plasticity modeling of granular media

SUMMARYA novel conceptual model of the mechanics of sands is developed within an elastic–plastic framework. Central to this model is the realization that volume changes in anisotropic granular materials occur as a result of two fundamentally different mechanisms. The first is purely kinematic, dilative, and is the result of the changes in anisotropic fabric. There is also a second volume change in granular media that occurs as a direct response to changes in stress as in a standard elastic/plastic continuum. The inclusion of the two sources of volume change results in three important datum states. When subjected to isotropic strains, the resulting stress state in granular materials is not isotropic but lies upon the kinematic normal consolidation line. There exists a state at which the fabric‐induced volumetric strain rate becomes equal to the stress‐induced volumetric strain rate making the total plastic volumetric strain rate equal to zero. Granular response changes from contractive to dilative at this phase transformation line. The third datum state is the one in which the stress‐induced volumetric strain rate is zero. The sand, however, continues to dilate at this state with the difference between stress and dilation ratio a constant as predicted by Taylor's stress–dilatancy rule. These predictions are shown in accordance with experimental data from a series of drained tests and undrained on Ottawa sand. Copyright © 2011 John Wiley & Sons, Ltd.

Generalized Gradient-Dependent Plasticy Theory

Based on the Generalized Plastic Mechanics (GPM) and strain gradient-dependence of yield surfaces, the differential expressions of the volumetric and shear plastic strains are constructed, then the Generalized Gradient-dependent Plastic Mechanics (GGPM) is established to describe the strain localization characteristics and the basic mechanics characteristics of geo-materials. In order to discretize the controlling equations and the boundary conditions of analyses filed, an increment functional in which the displacement and plastic multipliers are independent variables was constructed to build up the variational principle of GGPM. Both the displacement and the plastical factor are discretized in space. A set of non-linear equations taking the displacements and the plastic factors of nodes as the unknown variables are obtained to solve the new proposed mechanic model. The boundary condition and the increment algorithm are also given. The numerical examples indicate that: (1) when strain localization appears,the plastic shear strain and the plastic volumetric strain mainly concentrate in localization band; (2) the mesh- dependency does not appear any more when the size of mesh changes; (3) when the parameters reflecting localization increase,the width of the strain localization band will be wider obviously.

Consistent Modelling of Expansive and Collapsive Response of Unsaturated Soils

In unsaturated soil modelling, collapse is viewed as an ordinary response, whereas swelling upon an increase of moisture is considered as a special class of problems. However, an investigation based on fundamentals of plasticity indicates that the two seemingly opposite responses are unified within the same ordinary elastoplasticity framework. In fact, similar to the well-known contractive and dilative responses in the critical state framework, the collapsive and expansive responses of unsaturated soils differ only in the sign of the dissipative hydrostatic stress. For a soil with a low applied net stress, the dissipative mean normal stress during wetting is likely to be negative, and the plastic volumetric strain is kept negative (swelling) as well, such that the dissipation is always positive, complying with the second law of thermodynamics. When the applied net stress is high, the dissipative mean normal stress may turn to be positive, resulting in a positive plastic volumetric strain (collapse). This paper explains the concept of this ‘ordinary’ modelling framework, and uses a bounding surface triaxial model to demonstrate the concept. The numerical results are compared with experimental observations reported in the literature.

Mechanics of pressure solution seam growth and evolution

Pressure solution seams (PSSs) can be idealized as localized volume reduction structures (LVRSs) in terms of their mechanics. Previous mechanical analyses of LVRSs, including compaction bands, showed that the normal stresses at the tips of LVRSs are compressive and significantly amplified with respect to the remote stresses, whereas on the flanks they are slightly reduced. These results can be used to rationalize the in‐plane growth of PSSs for a certain distance, however, based on these stress conditions alone, it is not possible to explain the widening and transverse coalescence of PSSs. In this study, based on laboratory and field observations that the PSS surfaces are extremely rough, we introduced asperities with triangular, semicircular, and rectangular geometries on the flanks of the LVRSs into our mechanical model to see if these asperities can raise stresses in these locations, which may rationalize the transverse growth of PSSs. It is found that these asperities can produce strong stress perturbations on the LVRS flanks thereby inducing a significant increase in the compressive normal stresses. In addition, to account for the rate factor of the pressure solution process, a creep law was adopted to simulate growth and coalescence of the LVRSs. Using the calculated normal compressive stresses and volumetric plastic strains as proxy for the growth, we show that (1) a single LVRS is able to grow both laterally and transversely for a short distance and (2) two in‐plane aligned neighboring LVRSs with a short distance between the adjacent tips and two parallel echelon and overlapped LVRSs with a small spacing may be able to link and coalesce to form a longer and wider LVRS, respectively. The influences of the LVRS geometric configurations, the material properties within the LVRSs, and the distance and spacing of the aligned in‐plane, echelon, and overlapping neighboring parallel LVRSs on the growth and coalescence of the LVRSs are investigated and their implications specifically for the PSSs are also discussed.

Formulation and validation of a constitutive model for sands in monotonic shear

A constitutive model for sands in monotonic shear is presented. The model is designed to simulate the behavior of sands in the whole stress and strain range of engineering interest with enough accuracy for practical usage. Material parameters were chosen to be state independent and easy to calibrate using conventional testing procedures. The formulation is based on effective stresses, pressure-dependent hyperelasticity, non-associative elastoplasticity, an isotropic hardening law and Rowe’s stress-dilatancy theory. The implementation of Rowe’s stress-dilatancy theory within the framework of elastoplasticity theory is discussed. It is found that Rowe’s theory produces a volumetric plastic strain rate function that has a discontinuity in its first derivative w.r.t. stress, and a smoothed form is proposed instead. Finally, some experimental tests are simulated and the results are briefly discussed.