Non-associated Flow Rule Research Articles

Cavity expansion analysis of normal indention of rocks with lateral confinement

Indentation of rocks with lateral confinement is strongly related to rock fragmentation in drilling, mechanized tunnelling, and mining. Yet, the interpretation of the results and its transfer to use in practice is complicated by both the limited specimen size and boundary conditions associated with the application of lateral confinement. In this study, a new semi-analytical solution for the problem of a finite-cavity expansion in rock specimens of finite radial extent is present for the analysis of rock indentation with lateral confinement. To account for possible strain hardening/softening effect, an extended linear strain hardening/softening, i.e. constant plastic tangential modulus, Drucker-Prager model with non-associated flow rule is used to describe the stress-strain behavior of rocks. The model predictions, valid up to crack initiation, agree well with experimental observations that peak indentation pressure increases with lateral confinement, indicating that for a correct interpretation of results, the ratio between specimen size and the tip width of a truncated indenter as well as the boundary conditions shall be considered. The results point to the need for further development of the cavity expansion model by incorporating more features to model the realistic rock behavior and for more experimental studies with lateral confinement using advanced testing technique to monitor the failure process.

A New Numerical Finite Strain Procedure for a Circular Tunnel Excavated in Strain-Softening Rock Masses and Its Engineering Application

The small strain theory underestimates the self-bearing capacity of rock masses, especially for a soft rock tunnel under high geostress. To perform an efficient and accurate calculation and provide a reference for the stiffness design of a tunnel, the finite strain solution for a circular tunnel in Mohr–Coulomb strain-softening rock masses with a non-associated flow rule was derived as three sets of differential equations under the Lagrangian coordinate, which are in the residue region, the softening region, and the elastic region, respectively. Based on the bisection method, an iteration procedure for solving the finite strain solution was proposed to approximate the boundary condition at infinity, the values of two adjacent boundaries, and the initial values on the excavation boundary. This numerical procedure was verified by comparing with self-similar solutions, recursive solutions, and FLAC simulation results. In the calculation example, the relative error on boundaries can be decreased to less than 10−8 after only 10 times iteration and the time for each calculation is less than 15 s. Applying this procedure on the sensibility analysis and stiffness reliability design for the Zhongyi tunnel, a support stiffness of 4.3 MPa/m is recommended to guarantee a tunnel displacement lower than 0.5 m.

A framework for formulating and implementing non-associative plasticity models for shell buckling computations

In modelling the behavior of thick-walled metal shells under compressive loads, the use of J2 flow theory can lead to unrealistic buckling estimates, while alternative ‘corner’ models, despite offering good predictions, have not been widely adopted for structural computations due to their complexity. The present work develops a new and efficient plasticity model for predicting the structural response of compressed metal shells. It combines the simplicity of the Von Mises yield surface, with a non-associative flow rule, mimicking the effect of a yield surface corner. This allows for tracing the equilibrium path of the loaded shell and identifying consistently structural instability, employing a single constitutive model. A robust backward-Euler integration scheme, suitable for both three-dimensional (solid) and shell elements is developed, along with the corresponding consistent algorithmic moduli for nonlinear isotropic hardening materials, accounting rigorously for the nonlinear dependence of plastic straining on the direction of strain increments. The model is implemented in ABAQUS as a user material subroutine. Simulations of thick-walled metal cylinders under compression predict structural instability in good agreement with experimental data.

Evaluation of Tunnel Face Stability Subjected to Seismic Load Based on the Non-associated Flow Rule

Reasonable determination of the stability of the tunnel face under complex conditions is necessary for safe construction. This paper focuses on the seismic stability of the tunnel face in cohesion-frictional soils based on the non-associated flow rule. The pseudo-static approach is adopted to reflect the seismic effect. An analytical model is proposed based on the limit analysis method. A series of numerical simulations are performed to verify the rationality of the theoretical model. Then, parametric analyses are carried out. Results show that the proposed model is reasonable to investigate the tunnel face stability under the seismic load. The study indicates that it is essential to consider not only the seismic load but also the non-associated flow rule when determining the limit support pressure of tunnel face, especially in the soils with high friction angle or unit weight.

A new asymmetric yield criterion based on Yld 2000-2d under both associated and non-associated flow rules: Modeling and validation

A new asymmetric yield criterion is developed by introducing asymmetrical parameters into Yld 2000-2d. It has been proved that the convexity of the proposed yield criterion is guaranteed. A ‘sum’ form of yield functions is used to increase the flexibility of the yield criterion. One of the advantages of the developed yield criterion is that the convexity of yield criterion is independent of the anisotropic parameters. Besides, the proposed yield criterion not only possesses the inherent characteristics of Yld 2000-2d under tensile stress states but also can be used to predict material's strength differential (SD) effect. By applying the constructed yield criterion to FCC and HCP materials, the accuracy and flexibility of the proposed yield criterion are verified. The results show that both the 11 and 16 parameters yield criteria for non-associated and associated flows, respectively, can accurately capture the SD effects of these materials. In addition, the proposed yield criterion not only can predict the tension-compression asymmetry along the rolling and transverse directions but also can accurately predict the SD effect along the diagonal direction.

Counterexample-trained neural network model of rate and temperature dependent hardening with dynamic strain aging

Constitutive models dealing with the thermal and visco-plasticity of metals have seen wide applications in the automotive industry. A basic plasticity and fracture characterization of a 1.5 mm thick DP780 dual phase steel sheet based on uniaxial tensile (UT) experiments with seven distinct material orientations is complemented by low (∼0.001/s), intermediate (∼1/s) and high (∼150/s) strain rate experiments on notched tensile (NT) and shear (SH) specimens at temperatures ranging from 20 °C to 500 °C. At low strain rates, we observe a non-monotonic effect of the temperature on the force-displacement curves, with the highest curve obtained at 300 °C. Contrasting low speed tests, a monotonic effect of temperature is observed for intermediate and high strain rate experiments, with the highest curves obtained at 20 °C for both cases. Strain rate jump tests are performed proving the positive strain rate sensitivity of the steel. A machine-learning based plasticity model is developed to capture the observed complex strain rate- and temperature effect. The material is modeled as elasto-plastic, with a Hill’48 yield surface and a non-associated flow rule. The flow resistance is decoupled into a reference strain hardening term and a neural network term, which is a function of the plastic strain, strain rate, temperature, and an additional dynamic strain aging variable. The plasticity model is implemented into a material user subroutine and identified using a counterexample-guided hybrid experimental-numerical approach. The extracted loading paths reveal a complex rate and temperature effect on the ductility of DP780. A neural network based fracture initiation model is therefore adopted to describe the fracture onset across various stress states, strain rates and temperatures considered.

Anisotropic distortional hardening based on deviatoric stress invariants under non-associated flow rule

An anisotropic distortional hardening (ADH) model is developed under non-associated flow rule by evolving the analytical asymmetric Yoon2014 yield criterion (Hu and Yoon, 2021). The deviatoric stress invariants in the proposed hardening model are analytically expressed. Then, the evolving equations are imposed to describe Bauschinger effect and transient behavior. The anisotropic parameters in yield function can be directly expressed with the four hardening curves along 0°, 45°, 90° and equi-biaxial directions under the proportional loadings. Most importantly, nonlinear strain paths are incorporated in the model. Permanent softening & strengthen, work-hardening stagnation & overshooting behaviors during strain path changes have also been taken into account in the proposed model. The corresponding parameters for these characteristics are relatively independent, although optimization is required for the parameter identifications. The effectiveness and accuracy of the proposed ADH model have been compared with the kinematic hardening models for SPCC material. The constitutive description for 780R AHSS from the proposed model has also been compared with the YU model (Yoshida et al., 2015) under tension-compression along RD, DD and TD. The accuracy of the proposed model under complex strain paths has also been verified by comparing with the results predicted from the eHAH model (Barlat et al., 2013) for EDDQ and DP780. In addition, a plastic potential function is developed based on the form of the proposed model for non-associated flow rule. The accuracy has been verified by applying it to 780R AHSS, EDDQ and DP780.

Компьютерное моделирование напряженнодеформированного состояния Тунгусской синеклизы и Якутско-Вилюйской крупной изверженной провинции

The states of stress and strain are numerically analyzed under conditions of the geodynamic process of tension in the south part of the Vilyui syneclise and the middle part of the Tunguska syneclise. Two-dimensional models of geological structures of the south part of the Vilyui syneclise and the middle part of the Tunguska syneclise are constructed based on the Kimberlit-1981 geological profile obtained using deep seismic sounding. To describe the plastic strain in the geomedia, the model of elastic-plastic media with the non-associated plastic flow rule based on the Drucker-Prager-Nikolaevskii model is used. In this model, the “jelly sandwich” strength model is adopted for the analysis of the stress state of the lithosphere. The results of the numerical modeling of the state of the stress and strain of the chosen parts of the Siberian Craton are presented. The localization of plastic strain, the region of positive values of horizontal stresses, and the negative deviation of the calculated pressure from the lithostatic pressure correspond to mineral deposits.

An Evolutionary Anisotropic Behavior for DC04 Sheet Using Hill48 Function under Non-Associated Flow Rule Hypothesis

An Evolutionary Anisotropic Behavior for DC04 Sheet Using Hill48 Function under Non-Associated Flow Rule Hypothesis

Efficient non-iterative modelling of pressure-dependent plasticity using paraboloidal yield criterion

A full understanding of the non-linear mechanical response of the polymer is essential for fibre-reinforced polymer composite design because an explicit definition of constitutive material models for the constituents (fibres, matrix, and interface) are prerequisite in micromechanical simulations. Unlike ductile metals, the material behaviour of polymer matrix is characterised by plasticity theories influenced by a combination of distortional and spherical energy dissipation. In this respect, an elastoplastic thermodynamic continuum model derivation is proposed using the paraboloidal yield criterion under isothermal conditions. A non-iterative scheme is developed for the numerical computation of the plastic strain increment multiplier. Both associated and non-associated flow rules are investigated following classical plasticity loading–unloading conditions. It thereby, evades conventional computationally demanding iterative process by replacing it with an exact determination of plastic strain increment. This novel approach highly improves the computational efficiency algorithmically. The real-sized numerical models are investigated and the comparison between simulated and experimental results shows the reliability and unprecedented accuracy of the proposed elastoplastic mathematical model.

Interpretation of Cone Penetration Test Data of an Embankment for Coupled Numerical Modeling

The Nerang Broadbeach Roadway (NBR) embankment in Australia is founded on soft clay deposits. The embankment sections were preloaded and surcharged-preloaded to limit the post-construction deformation and to avoid stability failure. In this paper, we discuss the NBR embankment’s geology, geotechnical properties of the subsurface, and long-term field monitoring data from settlement plates and piezometers. We demonstrate a comparison of cone penetration test (CPT) and piezo cone dissipation test (CPT-u) interpreted geotechnical properties and the NBR embankment’s foundation stratification with laboratory and field measured data. We also developed two elasto-viscoplastic (EVP) models for long-term performance prediction of the NBR embankment. In this regard, we considered both the associated and the non-associated flow rule in the EVP model formulation to assess the flow rule effect of soft clay. We also compared EVP model predictions with the Modified Cam Clay (MCC) model to evaluate the effect of viscous behavior of natural Estuarine clay. Both EVP models require six parameters, and five of them are similar to the MCC model. We used the secondary compression index of clay in the EVP model formulations to include the viscous response of clay. We obtained numerical models’ parameters from laboratory tests and interpretation of CPT and CPTu data. We observed that the EVP models predicted well compared with the MCC model because of the inclusion of soft clay’s viscosity in the EVP models. Moreover, the flow rule effect in the embankment’s performance predictions was noticeable. The non-associated flow rule EVP model predicted the field monitoring settlement and pore pressure better compared to the MCC model and the associated flow EVP model.

A Proposed Algorithm to Compute the Stress-Strain Plastic Region and Displacement of a Deep-Lying Tunnel Considering Intermediate Stress and Strain-Softening Behavior

Past studies on deep-lying tunnels under the assumption of plane strain have generally neglected the influence of intermediate principal stress even though this affects the surrounding rocks in the plastic zone. This study proposes a finite difference method to compute the stress strain plastic region and displacement of a tunnel based on the Drucker–Prager (D–P) yield criterion and non-associated flow rule and considering the influences of intermediate principal stress and the strain-softening behavior of surrounding rock. The computed results were compared with those of other well-known solutions and the accuracy and validity of the method were confirmed through some examples. Parameter analysis was conducted to investigate the effects of intermediate principal stress on stress-strain, the plastic region, the ground response curve, and the dilatability of surrounding rock. The results showed that the plastic radius Rp, the residual radius Rd, and radial displacement of surrounding rock first decreased and then increased with increasing intermediate principal stress coefficient b from 0 to 1, with the minimums occurring at b = 0.75. On the contrary, the peak and rate of variation of the dilatancy coefficient first increased and then decreased with increasing b and the dilatancy coefficient Kψ gradually transitioned from nonlinear to linear variation. Meanwhile, the inhibition of the plastic radius and radial displacement gradually weakened with increasing support pressure, whereas the dilatancy coefficient of the tunnel opening gradually increased.

A Micromechanical Anisotropic Damage Model for Brittle Rocks With Non-Associated Plastic Flow Rule Under True Triaxial Compressive Stresses

A micromechanical anisotropic damage model with a non-associated plastic flow rule is developed for describing the true triaxial behaviors of brittle rocks. We combine the Eshelby’s solution to the inclusion problem with the framework of irreversible thermodynamics. The main dissipative mechanisms of inelastic deformation due to the frictional sliding and damage by microcrack propagation are strongly coupled to each other. A Coulomb-type friction criterion is formulated in terms of the local stress applied onto the microcracks as the yielding function. The back-stress term contained in this local stress plays a critical role in describing the material’s hardening/softening behaviors. With a non-associated flow rule, a potential function is involved. Some analytical analysis of the non-associated micromechanical anisotropic damage model are conducted, which are useful for the model parameters calibration. The proposed model is used to simulate the laboratory tests on Westerly granite under true triaxial stresses. Comparing the numerical simulation results provided by the models with associated/non-associated plastic flow rule and experimental results, it is clear that the proposed non-associated model gives a better prediction than the previous associated model.

Influence of non-associated flow rule on face stability for tunnels in cohesive–frictional soils

This paper aims at analyzing the influence of non-associated flow rule on face instability of a circular tunnel in cohesive–frictional soils. A series of three-dimensional numerical simulations are performed to analyze the influence of dilatation angle on the limit support pressures and the failure zones. The dilatation angle significantly influences the limit support pressure and the failure zone. Then, a three-dimensional failure mechanism that consists of the ellipsoid and the multi-block is proposed based on the upper-bound theorem. Finally, the limit support pressures and failure zones obtained from the new mechanism are compared with those given by existing approaches of limit analyses, numerical simulation and centrifuge tests. The results indicate that the presented model is more effective in analyzing the face stability of tunnels.

User-friendly anisotropic hardening function with non-associated flow rule under the proportional loadings for BCC and FCC metals

This paper presents a user-friendly function under non-associate flow rule to capture anisotropic hardening (or evolution of yield surface) and the curvature of yield loci for body-centred cubic (BCC) and face-centred cubic (FCC) under the proportional loading conditions. The calibrated model for BCC and FCC metals is applied to illustrate the hardening behavior of uniaxial tension along different loading directions and of the equibiaxial tension for several steels and aluminum alloys (718AT, QP980, AA5182-O and AA 6xxx-T81) to evaluate its accuracy. The prediction is compared with the experimental results and modeling results of several popular plasticity models. The comparison demonstrates that the proposed model is pressure-insensitive, relatively simple for numerical implementation, capable of illustrating the different yielding behaviors between BCC and FCC metals and, capable of accurate modeling of flow curves under directional loading directions and equibiaxial tension both under the plane stress and triaxiality loading conditions. The proposed model is also flexible on adjusting the curvature of the yield surface between uniaxial tension and equibiaxial tension. Accordingly, the proposed anisotropic hardening model is useful to illustrate anisotropic hardening behaviors under uniaxial and equibiaxial tension.

How fines content affects granular plasticity of under-filled binary mixtures

In recent years, binary mixtures have been the subject of numerous experimental and numerical studies. However, few attempts have been made on investigating the effect of fines content (fc) on the non-associated plasticity of binary mixtures, which is significant for constitutive modelling of such material. Thanks to 2D DEM simulations, this study aims to provide an understanding of how fc affects the non-associated character of the flow rule and the resulting material instability in binary mixtures. For under-filled materials (where coarse grains constitute most of the load-bearing skeleton), fine grains help to stabilize the granular assembly (1) by limiting macroscopic plastic deformations, which results in strain hardening, and (2) by reducing contractive microstructure reorganizations, which reduces the gap between the associated and non-associated flow rule directions. Fines content influences the plastic flow direction but has no influence on normal direction of yield surface. Eventually, perspectives on mesoscale mechanisms are given to highlight the role of fine grains in the geometrical and mechanical properties of granular materials.

A bounding surface plasticity model for expanded polystyrene-sand mixture

This study presents a bounding surface plasticity model to capture the stress-strain responses of expanded polystyrene (EPS)-sand mixture with various EPS content levels. Based on the mechanical responses in isotropic and triaxial tests, a generalized void ratio concept is applied to the potential deformation characteristics of EPS beads. Subsequently, the void ratio-isotropic stress curve can be expressed in a semi-log space, which shows a three-linear-segments manner corresponding to three typical mechanisms (elastic compression, normal compression and sand particle embedding in EPS beads). The plasticity model is formulated within the framework of critical state concept. For the sake of simplicity, the raindrop shaped yield function has been applied to describe the bounding surface, with a non-associate flow rule being adopted to capture the plastic deformation. Thus, this model requires fewer model parameters, most of which can be calibrated through conventional isotropic and triaxial test. By considering the weakening effect caused by including EPS, special treatment has been applied to EPS proportion dependent parameters. The capability of the proposed model has been demonstrated through a model simulation on EPS-sand mixtures with different EPS proportions and various confining pressures.

Random failure mechanism method for assessment of working platform bearing capacity with a linear trend in undrained shear strength

A bearing capacity evaluation for the surface strip foundation on a working platform modelled on a two-layered substrate is considered in the study. The upper layer is assumed as man-made and well-controlled and thus non-variable. The lower layer modelling natural cohesive soil is subjected to spatial variability of undrained shear strength. The random failure mechanism method (RFMM) is used to evaluate the bearing capacity. This approach employs a kinematic assessment of the critical load and incorporates the averaging of three-dimensional (3D) random field along dissipation surfaces that result from the failure mechanism geometry. A novel version of the approach considering an additional linear trend of undrained shear strength in the spatially variable layer is proposed. The high efficiency of the RFMM algorithm is preserved. The influences of foundation length, trend slope in the spatially variable layer, fluctuation scales, and thickness of the homogenous sand layer on the resulting bearing capacity evaluations are analysed. Moreover, for selected cases, verification of the RFMM based assessment obtained using random finite difference method (RFDM) based on 3D analysis is provided. Two types of analyses are performed using RFDM based on associated and non-associated flow rules. For associated flow rule which corresponds to RFMM, the RFMM is conservative and efficient and thus it seems preferable. However, if RFDM employs non-associated flow rule (much lower dilation angle for sand layer), the efficient RFMM is no longer conservative. For this situation, a combined approach that improves the efficiency of the numerical method is suggested.

Analytical evaluation of the effect of failure mechanisms on the bearing capacity factor Nγ for smooth strip footings

ABSTRACT This paper describes an analytical approach to investigating the role of failure mechanism in the bearing capacity of a smooth strip footing located on a -- soil. The approach combines the plane strain equilibrium condition and Mohr-Coulomb failure criterion to obtain the stress distribution along a translational, rotational and composite failure path. It is shown that the exact solutions for cohesion factor and surcharge factor for a -- soil do not require the superposition assumption and that both these factors are path-independent. A general closed-form path-dependent expression for the soil weight factor is also derived in terms of internal friction angle of the soil and the geometry of the failure path. The application of proposed expression on two-line and three-line failure modes as well as the well-known Hill mechanism are evaluated and discussed. Due to the overestimation of the soil weight factor by two-line and three-line modes, and by Hill mechanism, geometric modifications on the failure paths are proposed. The proposed mechanism and its bearing capacity are verified using the method of the stress characteristics. The effect of a non-associated flow rule on the reduction of the soil weight factor is also investigated quantitatively.

Influence of non-associated flow rule on passive face instability for shallow shield tunnels

This paper aims at analyzing the influence of non-associated flow rule on passive face instability for shallow shield tunnels. The influence exerted by dilation angle on the limit support pressures, the failure zones and partial failure ratios is investigated by numerical simulations. Based on the failure zone obtained from numerical simulation, a log-spiral mechanism is proposed. The upper-bound theorem is employed to acquire the limit support pressures, failure zones and partial failure ratios. The analytical results indicate that dilation angle has a significant influence on the limit support pressure and failure mechanisms. Finally, by comparison with the numerical models and existing analytical models, the results indicate that the proposed log-spiral model has a certain accuracy in analyzing the face instability of blow-out.