Drucker-Prager Criterion Research Articles

Accelerating fracture simulation with phase field methods based on Drucker-Prager criterion

The paper presents a framework for accelerating the phase field modeling of compressive failure of rocks. In this study, the Drucker-Prager failure surface is taken into account in the phase field model to characterize the tension-compression asymmetry of fractures in rocks. The degradation function that decouples the phase-field and physical length scales is employed, in order to reduce the mesh density in large structures. To evaluate the proposed approach, four numerical examples are given. The results of the numerical experiments demonstrate the accuracy and efficiency of the proposed approach in tracking crack propagation paths in rock materials under Drucker-Prager criterion.

Numerical study of rock-breaking mechanism in hard rock with full PDC bit model in compound impact drilling

Geothermal reservoir rocks have the characteristics of high stress, high hardness and high abrasiveness, which makes the drilling rate low and the drilling cost high, and hinders the application of geothermal energy. Improving drilling efficiency in hard rock strata is the prerequisite for large-scale application of geothermal energy. Impact drilling is considered to be the most effective drilling method in hard formations. PDC (Polycrystalline diamond Compact) bit compound impact drilling is one of the impact drilling methods, which has been widely concerned in recent years. In order to further improve the drilling efficiency of this method, the mechanism of PDC bit compound impact rock breaking is studied. Drucker–Prager criterion was used as rock constitutive relation, equivalent plastic strain was used as rock failure criterion, MSE (mechanical special energy) was used as evaluation method of rock-breaking efficiency, and a three-dimensional simulation model of full-size PDC bit for compound impact rock-breaking was built. The rock breaking law in PDC bit compound rock breaking process and the impact vector angle and impact duration on rock breaking efficiency are analyzed. The results show that the compound impact drilling can effectively alleviate the stick–slip vibration phenomenon, and improve drilling efficiency. In the compound impact mode, compressive stress failure and tensile stress failure occur at the same time, and the tensile stress failure is the main failure mode. The peak stress of the rock is 5.49% larger, and the residual plastic deformation is about 4.28% smaller than that of non-impact mode. With the increase of impact angle, the MSE decreases first and then increases a little. The optimal impact vector angle is in the range of 30°–50°. With the increase of the impact time, the MSE decreases in a wavy manner. In the range of 0.2 ms–1.0 ms, the optimal impact time is 0.8 ms–1.0 ms.

Numerical simulation the fracture of rock in the framework of plastic-bond-based SPH and its applications

In this paper, a two-dimensional numerical approach named plastic-bond-based smoothed particle hydrodynamics (PB-SPH) is proposed to simulate the fracture of rock with pre-existing flaws. Under the PB-SPH framework, interaction particles are interconnected through bonds. The Drucker–Prager criterion of a tensile zone cut-off coupled maximum tensile stress criterion is applied to judge shear failure and tensile failure. To verify the capability of the PB-SPH method in simulating rock fracture behavior, two-dimensional numerical simulations of a single flaw, two coplanar flaws, and multiple flaws of uniaxial compression are carried out. Compared with previous experimental and numerical results, the present numerical results show good agreements on crack initiation positions, crack morphology and crack patterns. The numerical simulation of the progressive failure process of rock slope with multiple flaws is carried out, which shows that the PB-SPH method can be well applied to rock mechanics engineering. The present results show that the PB-SPH method has a certain reference in the study of rock fracture mechanics and the application of rock mechanics engineering.

Analytical Model of Wellbore Stability Analysis of Inclined Well Based on the Advantageous Synergy among the Five Strength Criteria

The proper drilling mud density is vital for wellbore stability maintenance, which relies on both stress distribution on the borehole wall and rock strength. Thus, the selection of rock strength criterion is vital for wellbore stability analysis (WSA), and several strength criteria have been developed and employed to perform WSA, but the real collapse pressure does not comply with any single strength criterion. Therefore, to accurately predict the critical mud weight against wellbore collapse, an analytical model of WSA was proposed for inclined wells based on the advantageous synergy among the five types of strength criteria, such as the Mohr-Coulomb criterion, Mogi-Coulomb criterion, Drucker-Prager criterion, modified Wiebols-Cook criterion, and modified Lade criterion. The predicted results among these analytical and five conventional models were compared under three kinds of typical stress regimes, such as normal, strike-slip, and reverse faults. Finally, five kinds of typical oil and gas field data were collected to further verify the accuracy of this new model. The results indicated that different models predicted different collapse pressure, owing to the Mohr-Coulomb criterion ignoring the intermediate principal stress ( σ 2 ), so as to always give the greatest collapse pressure, followed by the present analytical model and the Mogi-Coulomb criterion, while the other three types of strength criteria gave different results, because of the difference in the influence of σ 2 , while the present analytical model integrated the advantageous synergy among the five strength criteria. The prediction error of the conventional analytical model ranges from 9.4 to 34.2% with an average of 22.1%, while the prediction error of this new model ranges from 2.3 to 12.5% with an average of 7.1%, so that the present analytical model has much higher accuracy than that of any single strength criterion. In addition, this new analytical model can be simplified as the conventional analytical model by adjusting the weight coefficients.

Ordinary state-based peridynamic plastic model with Drucker-Prager criterion considering geometric nonlinearity

In this paper, an ordinary state-based peridynamic (OSB-PD) plastic model is proposed to simulate the large deformation of geomaterials. The PD dilatation, which is equivalent to the volumetric strain in terms of the logarithmic strain in classical continuum mechanics (CCM), is proposed. The PD material parameters are obtained by equaling the assumed PD energy density to the energy density in terms of the Kirchhoff stress and logarithmic strain in CCM. The PD force, which is obtained by calculating the Frechet derivative of PD energy density, is naturally collinear with the deformed bond. The OSB-PD plastic model with the Drucker-Prager criterion considering geometric nonlinearity is subsequently established. Benchmark problems including the plate under large deformation and the l-shaped panel test are simulated, which has verified the proposed model.

Elastic-plastic analysis of circular tunnel based on unified strength theory and considering multiple plastic zones

Considering that the internal friction angle of the surrounding rock is not a constant but a function of the mean normal stress, and the unified strength theory (UST) is used as the plastic condition of the tunnel surrounding rock, the plastic zone of the circular tunnel surrounding rock is divided into multiple annular regions. For different plastic zones, there are different yield conditions represented by different stress functions, the elastic-plastic analysis of circular tunnels is performed by using the piecewise linearization of nonlinear yield function, stress and displacement in the elastic-plastic zone and the radius of the plastic zone are obtained by combining the equilibrium equations. It is shown that the radius of the plastic zone increases, the radial stresses in the elastic-plastic zone and the circumferential stresses in the plastic zone become smaller and the circumferential stresses in the elastic zone becomes larger compared with that of only one plastic zone. By using single factor analysis method, the calculated values of the tunnel displacements and plastic radii by UST were compared with those obtained by the Mohr-Coulomb (M-C) criterion, Drucker-Prager (D-P) criterion, and Zienkiewicz-Pande (Z-P) criterion. The analysis shows that under the same conditions, the solution of UST is smaller than M-C criterion and D-P criterion, therefore, the selection of rock strength criterion has great influence on the calculation of rock mechanics and engineering, the proper application of UST will guarantee the safety of engineering practice and have more practical value.

Constitutive Model of Rock, Nonlinearity and Localization

The paper deals with the key features of rock behavior and their manifestations in loading diagrams. Mathematical models are proposed for describing elastic-plastic deformation, creep, and fracture of rocks. Deformation beyond the elastic limit is described using a model based on the combined yield surface and the nonassociated flow rule. The yield surface consists of tension cut-off segment, modified linear segments of the Drucker–Prager criterion and cup. The dilatancy coefficient depends on pressure and volumetric deformation. Model equations are derived to describe deformation in the dilation and compaction modes, as well as strain localization and fracture with consideration for damage kinetics. Several examples of numerical modeling are given to illustrate these phenomena.

Evolution of temperature and damage in rock mass after heat shock

Heat shock is a feasible means to break rock mass in engineering duo to the large additional thermal stress caused by thermal expansion. This paper established a coupled thermal transfer and rock deformation model based on the energy conservation and the elastic deformation theory of rock. In the model, the failure and damage of rock are judged according to the Drucker-Prager criterion. A finite element method of COMSOL Multiphysics to study the characteristics of temperature, stress distribution and damage zone on a rock surface is proposed. Results show that the failure of rock mass occurs at the cusps of the heater first duo to stress concentration and then grows at both sides of the heater greatly.

Study of the influence of the surface characteristics of steel bars on the steel-concrete interaction

The steel-concrete adherence is of fundamental importance for the operation of reinforced concrete elements since they provide the transfer of stresses between the materials. The geometry of the steel bars is one of the main factors that interfere with adherence behavior. This paper makes an experimental study of adherence using steel bars with different surface characteristics, applying direct pull-out tests. From the experimental investigation, it was concluded that the steel bars' geometry interferes with the steel-concrete interface's performance. After identifying the necessary parameters, numerical simulations were performed using the Finite Element Method to verify the numerical results with the experimental ones and thus validate the modeling. Furthermore, the Mohr-Coulomb and Drucker-Prager criteria made it possible to visualize the stresses in the concrete. Finally, after the analyses, it was found that the numerical model developed was close in parts to the experimental model and that further refinement is needed to accurately represent the physical effects present in direct pull-out tests.

GPU-accelerated SPH modeling of flow-driven sediment erosion with different rheological models and yield criteria

Flow-driven sediment erosion exhibits multiphase characteristics and involves reversible solid-liquid mass exchange, wherein various rheological models and yield criteria are available, but their cooperative performance remains unclear. Here, six combinations of popular rheological models and yield criteria with varying physics are implemented in the Smoothed Particle Hydrodynamics (SPH) to reproduce four classic erosion experiments. We propose a GPU-accelerated Effective Interface Searching Technique (EIST) that reduces computational complexity from O(N2) to O(1). Our results suggest that the combination between Herschel-Bulkley-Papanastasiou (HBP) model and Mohr-Coulomb (MC) criterion best captures the erosion and waterfront, whilst the Drucker-Prager (DP) criterion appears to over-predict the erosion. The HBP-MC model reveals that velocity decreases exponentially and sediment concentration increases linearly before saturation concentration along with the depth in the movable sediment layer. This work provides a robust GPU-based EIST method and a useful reference for identifying appropriate adoptions of rheological models and yield criteria that vary in physics.

Three-Dimensional Finite Element Analysis for Pressure on Flexible Wall Silos

A 3D finite element model (FEM) for predicting the distribution of lateral pressure in a square flexible walled steel silo during the filling phase was analyzed in this study. The numerical approach, developed using Abaqus software, predicts the stress state in bulk solids, as well as the pressures exerted on the silo walls. An elasto-plastic model using the Drucker–Prager criterion was employed to simulate the behavior of the granular materials. The FEM simulates the behavior of the bulk solid and its interaction with the silo’s wall and base using a surface-to-surface discretization model. The model’s predictions were validated by previous experimental measurements. The results revealed good agreement between the FEM predictions and the experimental measurements. The research confirms that the lateral pressure distribution is not uniform at any silo level. This highlights the fact that many available theories and current design codes are not accurate for flexible steel walls. As a result of the wall’s deformability, pressure regimes on the silo wall change significantly in the horizontal direction at any level. The results showed that the horizontal variations of lateral pressure change drastically with regard to wall stiffness. The FEM has been used to investigate the effect of critical parameters on wall pressure distribution, such as properties of bulk solids, wall thickness, and silo type, whether deep or flat.

Experimental and numerical study on fracture characteristics and constitutive model of sandstone under freeze-thaw-fatigue

The present paper proposes a whole-process constitutive model of rock mass under freeze-thaw-fatigue based on the Drucker-Prager criterion and the Lemaitre strain equivalence hypothesis and by using a Weibull distribution. The fracture characteristics of rock mass under freeze-thaw-fatigue are studied from a macro and mesoscopic perspective, by using the discrete element method and acoustic emission technology. The results show that as the number of freeze-thaw treatments increases, the b value fluctuation amplitude of the rock mass at fatigue failure decreases, the shear crack increases, the anisotropy of the mesoscopic force field enhances, and the required energy decreases, and the rock burst tendency weakens.

Research on Mechanical Properties and Damage Constitutive Model of Water-Bearing Coal

Many water-related problems are confronted in coal mining, and the mechanical properties of coal inevitably deteriorate due to water–rock interactions. Therefore, it is necessary to study the macroscopic mechanical properties and damage constitutive model of water-bearing coal for safe mining. The uniaxial compression tests of raw coal samples with five moisture contents under four loading rates were carried out. Based on the test, the Drucker–Prager criterion is introduced to describe the failure law of micro-elements; assuming that the strength of micro-elements obeys a two-parameter Weibull distribution, a three-stage damage constitutive model is established. The model, existing model, and test curves were compared, and four indicators were introduced to evaluate the fitting effect. The results demonstrate that the stress–strain curve has a near horizontal step before the elastic stage, and its length is positively correlated with the moisture content ω. With the increase in ω, the peak strain increases linearly, and the compressive strength and Young’s modulus first increase and then decrease. The loading rate does not change the type of the fitting function between the mechanical parameters and ω. The three-stage model is more universal and can better fit the full stress–strain curve of water-bearing coal under uniaxial compression.

Modeling of soil–rock mixture landslides with the generalized interpolation material point method

The new numerical model for studying the dynamic evolution of soil–rock mixture landslides is presented in this article. The numerical model based on the generalized interpolation material point method analyzes a simplified slope. The gravity is linearly loaded, and the linear elastic model is used to update the stress to obtain the initial state of the slope. A small soil cohesion is set to trigger the slope sliding until the equilibrium state is reached again. During this period, the elastic–plastic material model based on the Drucker–Prager criterion is adopted for soil and stones. The differences in dynamic evolution between the homogeneous soil slope and soil–rock mixture slope are studied. Under the same stone content, the influence of the size and shape of stone on the dynamic evolution of slope is studied.

Experimental analysis of the multiaxial failure stress locus of commercially pure titanium at low and high rates of strain

The mechanical response and failure mechanism of commercially pure titanium subjected to combined tension-torsion loading are studied experimentally at strain rates ranging from 10−3s−1 to 103s−1. A novel tension-torsion split Hopkinson bar (TTHB) equipped with a high speed camera was utilised during high-rate experiments, while quasi-static tests were conducted using a universal screw-driven machine. The multiaxial dynamic experiments demonstrate the ability of the developed TTHB apparatus to achieve synchronisation of longitudinal and torsional waves upon loading the specimen, to satisfy the dynamic equilibrium of the specimen and to attain constant strain rate loading. The failure envelope of commercially pure titanium was analysed over a wide range of stress states including pure torsion, shear-dominated combined tension-shear, tension-dominated combined tension-shear, and plain tension. The analyses of the loading paths show that these were nearly proportional in terms of strain. The multiaxial failure stress locus was constructed in the normal versus shear stress space from experiments conducted at low and high rates of strain.The Drucker-Prager criterion was employed to approximate the failure envelope and to assess its rate sensitivity. The failure stress locus of commercially pure titanium and its rate dependence are reported for the first time. The TTHB apparatus developed allows the definition of the failure stress locus of aerospace materials directly from experiments and, therefore, the evaluation of the existing failure/yielding criteria.

Numerical estimate of critical failure surface of slope by ordinary state-based peridynamic plastic model

In this paper, two-dimensional ordinary state-based peridynamic (OSB-PD) plastic model is coupled mechanically with the Drucker-Prager (D-P) criterion, aiming at numerically reproducing localized deformation and locating the critical failure surface of slopes. A non-physical (or negative) incremental plastic energy can be avoided under extreme non-uniform deformation in this model. The strength reduction method is adopted to estimate the critical failure surface and factor of safety. For PD strength reduction method, three evaluation criteria are introduced to estimate the critical damage state of slopes. Two numerical simulations including a classical slope model and centrifuge tests of sand slopes are performed by the proposed PD method. The critical failure surfaces of slopes are obtained by PD and compared with the results obtained by (i) the finite element method (FEM), (ii) the simplified Bishop method, and (iii) the centrifuge tests. The numerical, analytical and experimental results have collectively verified the proposed PD method.

The mechanical response of commercially pure copper under multiaxial loading at low and high strain rates

In this paper, we present the dynamic response of commercially pure copper subjected to combined tension-torsion loads representative of real case impact scenarios. Experiments were conducted both quasi statically, at a strain rate equal to 10−3 s−1, and dynamically at strain rates in the region between 500 s−1 and 1000 s−1. All high rate experiments were conducted using a novel Split Hopkinson Tension-Torsion Bar instrumented with high-speed photographic equipment. The dynamic combined loading experiments demonstrate the capability of the apparatus to generate longitudinal and torsional stress waves which are synchronised upon loading of the specimen. The presented data show that dynamic equilibrium conditions and nearly steady strain rates were achieved during the experiments. Additionally, the analyses of the loading paths show that nearly proportional strain loading was attained during testing.The measured experimental results illustrate, for the first time, the failure stress locus of the material over a wide range of stress states including pure torsion, shear-dominated combined tension-shear, tension-dominated combined tension-shear and plain tension. The quasi-static and dynamic failure envelopes are herein presented in the normal stress vs shear stress space to motivate the development of accurate and effective constitutive models. To conclude, the Drucker-Prager criterion was employed to approximate the failure loci and to assess the rate sensitivity of the material. A moderate asymmetry of the uniaxial ultimate stresses in tension and compression is predicted both at quasi-static and dynamic strain rates.

Mathematical modelling, numerical analysis and damage of dams subjected to hydrodynamic pressure

Mathematical modelling and numerical analysis of a concrete dam with combined effects of water-sediment-foundation-earthquake are the main and novel contributions of this study. For this purpose, a concrete dam is modeled by exponential shear deformation beam theory (ESDBT). The hydro-dynamic pressure of water due to earthquake load is obtained by Helmholtz equation assuming various boundary conditions of reservoir and wave reflection coefficient. In addition, the forces of foundation and sediment are assumed by vertical and axial springs. On the basis of Hamilton's principle, the final motion equations are derived and solved by Harmonic differential quadrature (HDQ), Integral Quadrature (IQ) and with Newmark methods for calculating the dynamic deflection of the concrete dam. The Drucker–Prager criterion is applied for damage analysis of the concrete dam. The influences of various important parameters such as wave reflection coefficient, sediment type and its depth as well as foundation are investigated on the dynamic deflection, hydrodynamic pressure as a function of water depth, time history of hydrodynamic pressure and Drucker–Prager damage index of the concrete dam. The results indicated that with enhancing the sediment depth, the dynamic deflection and hydrodynamic pressure are decreased. In addition, the hydrodynamic pressure increases the dynamic deflection of the concrete dam about 35% with respect to hydrostatics pressure. With increasing the wave reflection constant, the absolute of Drucker–Prager damage index and the safety of concrete dam are reduced at the crest, middle and heel of the concrete dam.

Approximate plastic yield criteria of geomaterials with pores and grains embedded in a porous matrix

Mineral grains and porosity are two common properties of geomaterials. This study focuses on establishing a macroscopic yield criterion for geomaterials having pores and mineral grains at the mesoscale which is embedded in a porous matrix. At the microscale, the solid phase is pressure-sensitive and obeys to the Drucker–Prager criterion which considers the asymmetric property between tension and compression. Based on a two-step homogenization procedure and the modified secant method, a macroscopic yield criterion is firstly derived which takes into account the effects of microstructure on the macroscopic mechanical behaviors of the studied material, such as the meso-porosity and mineral grains content at the mesoscale, the micro-porosity and the pressure-sensitive of the solid phase at the microscale. However, it overestimates the material strength in compressive loading by comparing with the Fast Fourier Transform (FFT) based numerical results. Then, this criterion is improved by considering the exact solution in pure hydrostatic loading of a porous medium with a Drucker–Prager matrix. By comparing to the numerical results with different microstructures and to experimental data, the new macroscopic yield criterion is estimated and validated. The main features of the studied geomaterial are well captured by this criterion.

Borehole stability analysis: Considering the upper limit of shear failure criteria to determine the safe borehole pressure window

The issue of borehole stability has attracted great attention worldwide but serious downhole problems still exist. The influences of various factors on borehole stability have been extensively studied in the literature. It is usually believed the lower limit of safe borehole pressure window is determined by collapse failure while the upper limit is determined by fracture failure of rocks. However, it is proved in this paper the upper limit can also be decided by collapse failure. A new model of determining the safe borehole pressure window is built and the new upper limit based on shear failure criteria is proposed, therefore, a more precise safe borehole pressure window can be determined. The effects of borehole orientation, structural weak planes, in-situ stress and pore pressure are then thoroughly analyzed. The results show that the two limits defined by shear failure criteria are resulted from principal stress alteration and usually featured by change of failure mode. The upper limit of shear failure criteria is quite critical in determining safe mud pressure under various conditions. The increase of mud pressure within boreholes may not alleviate collapse failures. The Mohr-Coulomb criterion confines a narrower window while the Drucker-Prager criterion predict a wider one. For determining the safe borehole pressure window, a uniform in-situ stress and a low pore pressure will make it more necessary to consider the upper limit of shear failure criteria. The model built and conclusions drawn can provide new guidance for mud weight selection and well trajectory design in drilling practice.