Plastic Radius Research Articles

A simple analytical model of wellbore stability considering methane hydrate saturation-dependent elastoplastic mechanical properties

Methane hydrates (MHs) have received widespread attention as a new clean energy resource. However, because of the sharp reduction/change in the mechanical/physical properties of methane hydrate bearing sediments (MHBS) after hydrate dissociation, as well as the strong multi-field coupling, wellbore instability is more likely to occur during drilling in MHBS under the seafloor. To fast predict the wellbore stability in MHBS, an elastoplastic analytical model is proposed in this study that fully considers the saturation-dependent mechanical properties and water/heat production/absorption due to hydrate dissociation. Axisymmetric problems at the horizontal cross-sections of the vertical wellbore are concerned. Based on an analytical model of pore pressure, temperature and salt concentration for drilling in MHBS that considers both conduction/diffusion and convection effects, the produced/absorbed water/heat at the dissociation front related to the MH dissociation rate are additional considered in the compatibility conditions in the modified models. Compared with the results of previous study, the improved solutions show a much lower temperature and slightly higher pressure around the dissociation front. Furthermore, the saturation-dependent mechanical properties of MHBS are simulated by an elastic-brittle-plastic constitutive model with unified strength criterion, where Young's modulus and cohesion are functions of hydrate saturation. The analytical solutions for the stresses, displacement and plastic radius are finally addressed under various locations of the plastic zone and principal stress orders. The analytical models are verified by their good agreement with the results from the FEM model under the same assumptions, and they are consistent with the results from complex numerical models. Through numerous calculations, the influences of the drilling fluid properties and initial conditions on the plastic radius and displacement at the wellbore wall are investigated in detail, and the analytical formula for the safest drilling fluid pressure is presented as a function of the initial pore pressure and in situ stress.

Displacement process analysis of deep tunnels with grouted rockbolts considering bolt installation time and bolt length

This study investigates the interaction process between grouted bolts and tunnel surrounding rock, and establishes an analytical model for deep tunnels reinforced with active grouted rockbolts of the convergence-confinement type. The rock mass is assumed to be elastic-perfectly plastic, obeying the Hoek–Brown criterion. The longitudinal displacement profile is introduced to represent the space effect of the tunnel face, based on which the fictitious pressure along the tunnel axis is determined. Because of delayed installation and limited length of the rockbolts, a rock mass may behave elastically or plastically at bolt installation with the rockbolts anchored in different zones. Six different evolution paths possibly encountered on the basis of the relative magnitude between the anchorage and plastic zones are proposed. The tunnel displacement, plastic radii, and stresses are determined. The analytical results are validated by comparing with those obtained from numerical simulations and an existing model. A parametric analysis shows that doubling bolt length from 2 to 4 m in a 14-m-diameter tunnel reduces the tunnel wall displacement up to 20%. Installing rockbolts close to the tunnel face (less than 1 m) can reduce tunnel wall displacement up to 51%. Doubling the pretension load can reduce tunnel wall displacement up to about 20%. Increasing the support density can reduce the tunnel wall displacement by up to about 50%. The analysis shows that the proposed model outperforms Fahimifar’s model, which deals with infinite rockbolts, in solving tunnel responses under grouted rockbolts, particularly in the case of large cross-sectional tunnels excavated in poor geological conditions.

ANALYSIS OF THE WHOLE-PROCESS MECHANICAL RESPONSE OF TUNNEL SURROUNDING ROCK UNDER THE EFFECT OF ANCHORAGE SYSTEM

The tunnel anchoring system is an important means to control the deformation of surrounding rock, whose scientific design lies in the full understanding of its mechanism. In this paper, the whole process of the response of tunnel surrounding rock with the action of the anchorage system is analyzed. Based on the space effect of the tunnel face, the displacement releases of surrounding rock at bolt and anchor cable installation and the effect of their relative lengths are considered. According to the relationship between the anchorage zone and the plastic zone of the interaction between the anchorage system and the surrounding rock, six mechanical analytical models of the whole process evolution mechanism of anchorage system are proposed and analyzed respectively. The solutions of tunnel deformation, plastic radius and stress field of surrounding rock under the action of anchoring system are obtained. The correctness and reliability of the proposed model are verified through a comparison with numerical simulations using typical examples. Furthermore, the deformation control effect of the anchorage system and its influencing factors are analyzed, and the respective tasks and objectives of rockbolts and anchor cables are defined. The safety reserve function of anchor cable is clarified, giving a new understanding of deformation control principle of anchorage system. The research results lay a theoretical foundation for the quantitative design of tunnel anchorage system and provide a new idea for the control of deformation of tunnel surrounding rock.

Analytical assessment of coupled influences of surrounding rock reinforcement and deformation release on tunnel response

Attempt has been proved to be effective that surrounding rock reinforcement is emphasized simultaneously considering displacement release in weak rock tunnels. In this study, the calculation formulas for mechanical parameters of bolt-reinforced rocks are provided using the homogenization method and the supporting characteristic curve is divided into three stages with proposing the corresponding stiffness equations. The mechanical model for bolted-reinforced rock-yielding supports interaction is then established and coupled solutions for displacements and stresses around tunnels considering bolt reinforcement and yielding effects are provided. Furthermore, parametric investigations on the influences of rockbolts and yielding supports are carried out. Results show that (1) rock displacement gradually decreases as the rockbolt length increases. However, when rockbolt length becomes large enough, the further improvement of rock displacement will not be obvious by still increasing their length. (2) Both rock displacement and plastic zone of tunnel decrease with the increase of rockbolt radius. There exists the highest utilization of rockbolts corresponding to a certain rockbolt radius. (3) Also, the rock displacement and plastic zone of tunnel decrease as installation density of rockbolts including circumferential space and longitudinal space increases. Under the condition prescribed, this decreasing trend becomes sharper and the improvement is more evident. (4) Larger yielding displacement or stiffness parameter leads to smaller support pressure, but greater plastic radius of tunnel. The optimal yielding displacement and stiffness parameters need to be determined through a comprehensive investigation combining rock properties, support characteristics and tunnel design requirements.

Laboratory investigation of cracking resistance performance of the cold‐mixed epoxy resin

AbstractCold‐mix epoxy resin (CER) is an excellent binder material but with weak crack resistance and brittle failure risk. Therefore, it is of significance to find a feasible parameter to constrain the failure potential. To this end, the dynamic mechanical test, tensile test, volume shrinkage test, fracture test, and scanning electron microscopy (SEM) were used to analyze and evaluate mechanical properties, curing shrinkage, and fracture toughness of the CER. Subsequently, the bivariate correlation analysis was used to feature the relationship among different indicators. The results indicated that the amount of curing agent has a significant influence on the tensile strength, elongation at break, volume shrinkage, fracture toughness, and plastic radius of CER. Meanwhile, there is a close correlation between tensile strength, elongation at break, and fracture toughness. Fracture toughness can be used as an evaluation index to represent the crack resistance of CER, and tensile test can be used as a confirmatory parameter. However, the volume shrinkage of CER cannot be ignored. Small voids distributed on the fracture surface of the CER can increase the toughness and then improve the crack resistance of the CER through SEM analysis.

A closed-form hydraulic-mechanical coupling solution of a circular tunnel in elastic-brittle-plastic rock mass

Considering the changes of permeability and Biot’s coefficients, the closed-form elastic-brittle-plastic solution of lined circular tunnels was proposed employing a Mohr-Coulomb non-associated flow rule. The influence of the reserved deformation of lining structure, permeability and Biot’s coefficients on the plastic radius, displacement and water inflow was studied. With an increase in permeability of the plastic region, the plastic radius and displacement decrease, and the water inflow increases. With an increase in Biot’s coefficient, the plastic radius, displacement and water inflow approximately increase in a linear way. The pore-water pressure in the surrounding rock increases with decreasing permeability of lining structure, and the water inflow has a contrary evolution. The increasing reversed displacement of lining structure increases the plastic radius, and lining structure reduces the water inflow but increases the pore-water pressure of the surrounding rock.

Confining stress effect on the elastoplastic ground reaction considering the Lode angle dependence

The convergence confinement methods are solutions employed to estimate convergence in circular tunnels. They are mostly based on constitutive equations governed by the Mohr-Coulomb and Hoek-Brown yield criteria. However, the solutions based on these criteria neglect the intermediate principal stress confining effect on the ground reaction estimation. Therefore, in this paper, a Drucker-Prager yield criterion governed solution integrated with the Lode angle parameter is employed. It considers the intermediate principal stress influence and the critical effect of the parameter on failure characterization. Subsequently, it is verified with results attained from numerical simulations which consider an elastic-perfectly plastic constitutive law with a non-associative flow rule within FLAC3D. It was drawn from the results that the ground reaction and plastic evolution are influenced by the confining stress. Furthermore, considering a suitable yield criterion leads to realistic convergence and plastic evolution estimation. The circumscribed DP criterion governed solution with Lode angle parameter value (0.8) is considered appropriate for the realistic ground reaction estimation in the three-dimensional (3D) stress state rock mass. It estimates approximately 3.4% of tunnel convergence as compared to the classic solutions (5%) and plastic radius estimated to be approximately 2.45 m compared to 2.84 m.

A fully coupled hydraulic-mechanical solution of a circular tunnel in strain-softening rock masses

This paper presents a fully coupled hydraulic-mechanical strain-softening model considering Biot’s effective stress. Both the hydraulic and mechanical parameters are considered as functions of the confining pressure and plastic shear strain. Using the proposed fully coupled hydraulic-mechanical model, the displacement, stress and pore-water pressure around a circular tunnel in both Mohr-Coulomb and Hoek-Brown strain-softening rock masses are derived using a non-associated flow rule. The proposed solutions are validated by the conventional hydraulic-mechanical elasto-brittle-plastic and strain softening rock mass using analytical and numerical methods, and the examples of the fully coupled hydraulic-mechanical strain-softening rock mass are investigated. The results show that the seepage force enlarges the displacement, residual and plastic radii. With the increase in the initial pore-water pressure, the displacement, water inflow, residual and plastic radii increase in an approximately linear manner. The increasing bulk modulus of the solid constituent increases the above four variables, which reach the approximate limit values, corresponding to the Darcy’s effective stress solutions, when Ks = 10 K0. The increasing permeability of the plastic rock mass reduces the increasing rate of the pore-water pressure but increases the water inflow. For the proposed fully coupled examples, both the hydraulic parameters of permeability and Biot’s coefficient show a non-linear decrease with the increasing radius.

Solution for a circular tunnel in strain-softening rock with seepage forces

In this study, a simple numerical approach for a circular tunnel opening in strain-softening surrounding rock is proposed considering out-of-plane stress and seepage force based on Biot\'s effective stress principle. The plastic region of strain-softening surrounding rock was divided into a finite number of concentric rings, of which the thickness was determined by the internal equilibrium equation. The increments of stress and strain for each ring, starting from the elastic-plastic interface, were obtained by successively incorporating the effect of out-of-plane stress and Biot\'s effective stress principle. The initial value of the outmost ring was determined using equilibrium and compatibility equations. Based on the Mohr–Coulomb (M–C) and generalized Hoek–Brown (H–B) failure criteria, the stress-increment approach for solving stress, displacement, and plastic radius was improved by considering the effects of Biot\'s effective stress principle and the nonlinear degradation of strength and deformation parameters in plastic zone incorporating out-of-plane stress. The correctness of the proposed approach is validated by numerical simulation.

Strain-degradation analysis of tunnel structures based on twin-shear unified strength criterion

Abstract This paper focuses on the stability analysis of tunnel structures with a circular hollow section based on the with a strain-degradation model using unified strength theory. A simplified numerical approach is proposed for analyzing the elasto-plastic behavior of surrounding rocks. A constitutive equation is proposed based on the unified strength theory and the strain degradation behavior was taken into account as well with deviatoric plastic strain chosen as the degradation parameter. Meanwhile, the plastic potential function can be obtained with the associated flow rule, by which the surrounding rocks can be classified as plastic zone and elastic zone. The solution in plastic zones adopted the differential method subdividing surrounding rock into infinitesimal concentric annuli, in which the radial stresses decrease monotonically with a decrease in radial coordinate. The relationship of the increment of stresses and the increment of strains in adjacent annuli can be derived from equilibrium, constitutive and geometrical equations. The numerical solution of each annulus is calculated from the outmost annulus at the elastic-plastic interface. In addition, the impact of the intermediate principal stress factor and the critical degradation parameter on the solution, and the factors of influence on plastic radius are discussed.

Thickness effect on the mode III fracture resistance and fracture path of rock using ENDB specimens

AbstractStudy of the thickness effect in predicting the crack growth behavior and load bearing capacity of rock‐type structures is an important issue for obtaining a relation between the experimental fracture toughness of laboratory subsized samples and the real rock structures with large thickness. The fracture of rock masses or underground rock structures at deep strata may be dominantly governed by the tensile or tear crack growth mechanism. Therefore, in this research, a number of mode I and mode III fracture toughness experiments are conducted on edge notch disc bend (ENDB) specimen made of a kind of marble rock to investigate the effect of specimen thickness on the corresponding KIc and KIIIc values. It is observed that the fracture toughness of both modes I and III are increased by increasing the height of the ENDB specimen. Also, the ratio of KIIIc/KIc obtained from each thickness of the ENDB specimens is compared with those predicted by some fracture criteria, and it was shown that the minimum plastic radius (MPR) criterion is the main suitable criterion for investigating the fracture toughness ratio KIIIc/KIc. Also, the effect of ENDB height on fracture trajectory of tested samples is assessed. It is shown that the crack grows curvilinearly in thicker ENDB samples and cannot extend along the crack front in small specimens.

Anisotropic elasto-plastic solutions for cavity expansion problem in saturated soil mass

On the basis of the yield criterion of Matsuoka and Nakai (1974), novel elasto-plastic solutions are obtained for the cavity expansion problem in an anisotropic medium. The solutions are formulated by employing a small-strain analysis in an elastic zone and a large-strain analysis in a plastic zone around the cavity in a soil mass. In light of the initial condition, the plastic radius and the stress in the plastic zone are derived by incorporating the large-strain analysis, the influence of stress anisotropy (K0) and the non-associated flow rule, and two fundamental solutions for the cavity expansion problem under undrained and drained conditions are presented in an anisotropic soil mass. Finally, a comparison with the previous results is performed for verification.

Analytical Elasto-plastic Solution for Frost Force of Cold-Region Tunnels considering Anisotropic Frost Heave in the Surrounding Rock

There is an increasing trend of tunnel infrastructure construction in cold regions due to the advance of a western development strategy. To reduce the frost damage sustained by a tunnel, it is important to consider the mechanical properties of frost heave in the surrounding rock by analyzing the anisotropic frost heave of a cold-region tunnel. This paper elaborates on the behavior of anisotropic frost heave in the rock surrounding a cold-region tunnel, and the coefficient of displacement relief is adopted to indirectly quantify the influence of the tunnel construction process. The analytical elasto-plastic solutions of the stress and deformation in the surrounding rock are derived with Drucker-Prager criteria (hereinafter referred to as the D-P criteria) by modifying the constitutive equation for the frozen surrounding rock. Finally, an example is given to obtain the influence law of the distribution of stress and deformation in the surrounding rock. The results indicate that the plastic radius and frost force calculated by considering isotropic frost heave are greater than those obtained by considering anisotropic frost heave. The influence of the related parameters also analyzed. Some useful insights are provided for future numerical simulations and the design and construction of cold-region tunnels.

Effect of Gravity of the Plastic Zones on the Behavior of Supports in Very Deep Tunnels Excavated in Rock Masses

The vertical load acting on a support structure is affected by the loss of self-bearing capacity of the rock inside the plastic zone. This load can then be accounted for by analytical calculation methods capable of evaluating the stresses in the tunnel support system to proceed with the tunnel design. Generally, the effect of the rock’s own weight in the plastic zone is considered in a simplistic way by evaluating an additional vertical load given by the weight of the rock due to the thickness of the plastic zone. This approach leads to a significant increase in the vertical load with the risk of overdesigning the support structure. In this work, the effect of the rock’s own weight in the plastic zone was considered by modifying the numerical solution of the convergence-confinement method for tunnels built in rock. In this way, through the intersection of the characteristic curve of the tunnel and the intersection line of the support structure, it is possible to determine both the vertical loads (with the effect of the weight of the rock) and the horizontal load (without the effect of weight of the rock). The application of the method to a project in the Alps allowed the detection of the magnitude of the percentage increase of the vertical load and a significant increase in the thickness of the plastic zone and determination of the consequences that this may have on the designing of the radial bolting length in that zone. Increasing the plastic radius led to an increase in the length of the bolts. This is interesting because, in the area of the crown where the weight of the plasticized rock is considered, the bolts are usually installed with a greater length. In the final part of the study, a new procedure was illustrated to define the vertical and horizontal loads acting on the support structures, starting from the convergence-confinement curves obtained for the crown and for the lateral areas (sides).

Plastic zone evolution during fatigue crack growth: Digital image correlation coupled with finite elements method

Nonlinearities effects at the crack tip, due to the elastic-plastic material behavior, impact the crack growth rate and path. This paper is devoted to the study of the plastic zone evolution in the crack tip region. The methodology relies on coupling an elastic-plastic Finite Elements Method (FEM) model and experimental displacements measured by Digital Image Correlation (DIC). These latter are introduced as Dirichlet boundary conditions in the finite elements analysis. The considered FEM domain is constant, i.e. the same mesh with a centered crack is moved to each new crack tip position deduced from DIC. The new boundary conditions are updated and the residual stresses and plastic strains of the former computation are interpolated and actualized on the mesh shifted to the new crack tip position in order to incorporate them in the numerical model. The coupling method allowed applying experimental boundary conditions in order to be as close as possible to real experimental conditions and to observe the plasticity evolution from small to large scale yielding conditions. A fatigue test was conducted to validate the proposed approach. The identification residues are proved to be lower than those obtained with an experimental displacements projection onto Williams’ series basis, which is a method commonly used with DIC. The coupling results present an attractive similarity with Irwin’s model regardless of the crack length. Thus, the definition of the mask needed for the displacements fields projection on Williams’ model can be deduced with a reliable estimate of Irwin’s plastic radius.

A semi-analytical solution for shallow tunnels with radius-iterative-approach in semi-infinite space

• A radius-iterative-approach was proposed for shallow tunnels. • Plastic radius and elastic-plastic solution for shallow tunnels were proposed. • The corrected technique for plastic radius was proposed for shallow tunnels. This study presents a semi-analytical elastic-plastic solution for a shallow tunnel subjected to ground loss in the strain-softening surrounding rock. The most important contribution is the radius-iterative-approach in which the initial plastic radius is first determined by the strain continuity boundary condition on the elastic-plastic interface and then corrected to the precise one. The corrected approach follows three steps: (1) Applying the radius increment technique to semi-infinite space (2) Carrying out the plastic radius correction by using iteration method from the elastic-plastic interface to the tunnel wall. (3) If the calculated convergence value is equal to the convergence value on the tunnel wall, the accurate determination of the plastic region, stresses, and displacements, of the whole half plane, can be derived consequently. All the results compare favorably with numerical simulation results. The study completes the theoretical framework for addressing the fundamental problem of shallow tunnels excavated in the semi-infinite space and also provides a useful theoretical tool for potential application on the tunnel and underground engineering problems.

Stress Field Distribution and Deformation Law of Large Deformation Tunnel Excavation in Soft Rock Mass

The problem of large deformation is very prominent in deep-buried tunnel excavation in soft rock, which brings serious potential safety hazards and economic losses to projects. Knowledge of the stress field distribution and deformation law is the key to ensuring rational design and safe construction in large deformation tunnels of soft rock. As described in this paper, theoretical analysis, numerical simulation and field monitoring were employed to investigate the surrounding rock stress and displacement state in the Dongsong hydropower station in Sichuan Province, China. The results show that the short-bench construction method can effectively control the deformation of surrounding rock and range of the plastic zone. In order to reserve enough working space, the optimum bench length in the actual construction was 10 to 14 m. The peripheral displacement and plastic radius decreased with the increase of tunnel support strength and the advance of supporting time. The displacement can be effectively controlled by applying the second lining in time at a position about twice the diameter of the hole (16 m) from the working face. A reasonable reserved deformation should be adopted to avoid secondary expanding excavation. The values of different positions in the tunnel laterally and longitudinally may be different, and adjustments are needed according to the actual situation.

A numerical stepwise approach for cavity expansion problem in strain-softening rock or soil mass

A numerical stepwise approach for cavity expansion problem in strain-softening rock or soil mass is investigated, which is compatible with Mohr–Coulomb and generalized Hoek-Brown failure criteria. Based on finite difference method, plastic region is divided into a finite number of concentric rings whose thicknesses are determined internally to satisfy the equilibrium and compatibility equations, the material parameters of the rock or soil mass are assumed to be the same in each ring. For the strain-softening behavior, the strength parameters are assumed to be a linear function of deviatoric plastic strain (yp*) for each ring. Increments of stress and strain for each ring are calculated with the finite difference method. Assumptions of large-strain for soil mass and small-strain for rock mass are adopted, respectively. A new numerical stepwise approach for limited pressure and plastic radius are obtained. Comparisons are conducted to validate the correctness of the proposed approach with Vesic\'s solution (1972). The results show that the perfectly elasto-plastic model may underestimate the displacement and stresses in cavity expansion than strain-softening coefficient considered. The results of limit expansion pressure based on the generalised H-B failure criterion are less than those obtained based on the M-C failure criterion.

The Catastrophe Criterion for Instability of Deep Roadway based on Malignant Expansion of Plastic Zone

In order to solve the problem of discriminant criterion for deep roadway instability, the core of surrounding rock failure in deep roadway and its two types of instability mechanisms were analysed. Based on the cusp catastrophe model and taking the plastic radius function as the potential function of the roadway system, the instability discriminant of surrounding rock was derived and the catastrophic criterion for instability of deep roadway under the influence of disturbance was established. Aiming at the engineering application of this criterion, the implementation method and detailed procedure of stability discrimination for the deep roadway were proposed. The numerical results show that the malignant expansion of plastic zone often occurs in surrounding rock causing large deformation or transient instability in the ultra-high deviatoric stress environment. Under the influence of external disturbance, the plastic radius experienced two stages including slow increase and nonlinear acceleration. The discriminant conclusion of the deep roadway instability according to the calculation results of the catastrophic eigenvalues is consistent with the extension of the plastic zone during the instability process of surrounding rocks, which verifies that the established instability criterion is feasible and applicable.

Generalized Formulation for Fatigue Assessment of Laser Stake-welded T-joints Varying Thicknesses and Loading Conditions

The present work considers the evolution of the crack tip plasticity ahead of the crack-like notches with varying the loading condition and the thickness of laser stake-welded T-joints. A general procedure is described for the fatigue assessment of the joints under different loading conditions and made of different thicknesses. The new method is based on the analyses of the first order plastic radius (according to Irwin) for pure mode I, and it is generalized to mixed mode (I+II) by using the Minimum Plastic Zone Radius (MPZR) criterion. The method is validated, in the end, with a case study and explains why the fatigue slope m varies for joints loaded under bending/tension and when the thickness is reduced. The results prove that the procedure gives an excellent estimation of the fatigue life.