An improved strain-softening constitutive model of granite considering the effect of crack deformation

This paper presents an improved strain-softening constitutive model considering the effect of crack deformation based on the triaxial cyclic loading and unloading test results.The improved model assumes that total strain is a combination of plastic,elastic,and crack strains.The constitutive relationship between the crack strain and the stress was further derived.The evolutions of mechanical parameters,i.e.strength parameters,dilation angle,unloading elastic modulus,and deformation parameters of crack,with the plastic strain and confining pressure were studied.With the increase in plastic strain,the cohesion,friction angle,dilation angle,and crack Poisson's ratio initially increase and subsequently decrease,and the unloading elastic modulus and the crack elastic modulus nonlinearly decrease.The increasing confining pressure enhances the strength and unloading elastic modulus,and decreases the dilation angle and Poisson's ratio of the crack.The theoretical triaxial compressive stress-strain curves were compared with the experimental results,and they present a good agreement with each other.The improved constitutive model can well reflect the nonlinear mechanical behavior of granite.

Ground reaction curves for circular excavations in non-homogeneous,axisymmetric strain-softening rock masses

Fast methods to solve the unloading problem of a cylindrical cavity or tunnel excavated in elasto-perfectly plastic, elasto-brittle or strain-softening materials under a hydrostatic stress feld can be derived based on the self-similarity of the solution. As a consequence, they only apply when the rock mass is homogeneous and so exclude many cases of practical interest. We describe a robust and fast numerical technique that solves the tunnel unloading problem and estimates the ground reaction curve for a cylindrical cavity excavated in a rock mass with properties depending on the radial coordinate, where the solution is no longer self-similar. The solution is based on a continuation-like approach(associated with the unloading and with the incremental formulation of the elasto-plastic behavior), fnite element spatial discretization and a combination of explicit sub-stepping schemes and implicit techniques to integrate the constitutive law, so as to tackle the diffculties associated with both strong strain-softening and elasto-brittle behaviors. The developed algorithm is used for two practical ground reaction curve computation applications. The frst application refers to a tunnel surrounded by an aureole of material damaged by blasting and the second to a tunnel surrounded by a ring-like zone of reinforced(rock-bolted) material.

Stability analysis of slope in strain-softening soils using local arc-length solution scheme

Soils with strain-softening behavior — manifesting as a reduction of strength with increasing plastic strain — are commonly found in the natural environment. For slopes in these soils,a progressive failure mechanism can occur due to a reduction of strength with increasing strain. Finite element method based numerical approaches have been widely performed for simulating such failure mechanism,owning to their ability for tracing the formation and development of the localized shear strain. However,the reliability of the currently used approaches are often affected by poor convergence or significant mesh-dependency,and their applicability is limited by the use of complicated soil models. This paper aims to overcome these limitations by developing a finite element approach using a local arc-length controlled iterative algorithm as the solution strategy. In the proposed finite element approach,the soils are simulated with an elastoplastic constitutive model in conjunction with the Mohr-Coulomb yield function. The strain-softening behavior is represented by a piece-wise linearrelationship between the Mohr-Coulomb strength parameters and the deviatoric plastic strain. To assess the reliability of the proposed finite element approach,comparisons of the numerical solutions obtained by different finite element methods and meshes with various qualities are presented. Moreover,a landslide triggered by excavation in a real expressway construction project is analyzed by the presented finite element approach to demonstrate its applicability for practical engineering problems.

Instability criterion for the system composed of elastic beam and strain-softening pillar based on gradient-dependent plasticity

A mechanical model is proposed for the system of elastic beam and strain-softening pillar where strain localization is initiated at peak shear stress. To obtain the plastic deformation of the pillar due to the shear slips of multiple shear bands, the pillar is divided into several narrow slices where compressive deformation is treated as uniformity. In the light of the compatibility condition of deformation, the total compressive displacement of the pillar is equal to the displacement of the beam in the middle span. An instability criterion is derived analytically based on the energy principle using a known size of localization band according to gradient dependent plasticity. The main advantage of the present model is that the effects of the constitutive parameters of rock and the geometrical size of structure are reflected in the criterion. The condition that the derivative of distributed load with respect to the deflection of the beam in the middle span is less than zero is not only equivalent to, but also even more concise in form than the instability criterion. To study the influences of constitutive parameters and geometrical size on stability, some examples are presented.

A Method for Evaluating the Maximum Bending Degree of Flexural Toppling Rock Masses Based on the Rock Tensile Strain-Softening Model

Flexural toppling occurs when a series of layered rock masses bend towards their free face.It is important to evaluate the maximum bending degree and the requirement of supports of flexural toppling rock mass to prevent rock mass cracking and even failure leading to a landslide.Based on the rock tensile strain-softening model,this study proposes a method for calculating the maximum curvature(C_(ppmax))of flexural toppling rock masses.By applying this method to calculate Cppmax of 9 types of rock masses with different hardness and rock layer thickness,some conclusions are drawn:(1)the internal key factors affecting C_(ppmax)are E^(⋆)(E^(⋆)=E_(ss)/E_(0),where E_(0)and E_(ss)are the mean deformation moduli of the rock before and after reaching its peak tensile strength,respectively),the strainεt corresponding to the tensile strength of rock,and the thickness(h)of rock layers;(2)hard rock layers are more likely to develop into block toppling than soft rock layers;and(3)thin rock layers are more likely to remain in flexural toppling state than thick rock layers.In addition,it is found that C_(ppmax)for flexural toppling rock masses composed of bedded rocks such as gneiss is related to the tensile direction.

Progressive failure analysis of slope with strain-softening behaviour based on strength reduction method

Based on the strength reduction method and strain-softening model,a method for progressive failure analysis of strain-softening slopes was presented in this paper.The mutation is more pronounced in strain-softening analysis,and the mutation of displacement at slope crest was taken as critical failure criterion.An engineering example was provided to demonstrate the validity of the present method.This method was applied to a cut slope in an industry site.The results are as follows:(1) The factor of safety and the critical slip surface obtained by the present method are between those by peak and residual strength.The analysis with peak strength would lead to non-conservative results,but that with residual strength tends to be overly conservative.(2) The thickness of the shear zone considering strain-softening behaviour is narrower than that with non-softening analysis.(3) The failure of slope is the process of the initiation,propagation and connection of potential failure surface.The strength parameters are mobilized to a non-uniform degree while progressive failure occurs in the slope.(4) The factor of safety increases with the increase of residual shear strain threshold and elastic modulus.The failure mode of slope changes from shallow slip to deep slip.Poisson's ratio and dilation angle have little effect on the results.

Modified ground response curve(GRC)in strain-softening rock mass based on the generalized Zhang-Zhu strength criterion considering over-excavation

The ground response curve(GRC)depicts the relationship between support reaction force and ground displacement,which improves the understanding of ground-support interaction and provides important references to the tunnel design.However,it is difficult to anticipate the tunneling-induced large deformation with sufficient reliability in soft rock with high geostress since the small strain theory is not applicable.When large deformation occurs,the tunnel needs to be over-excavated.Thus,the GRC should be modified considering the enlarged excavation radius since the actual excavation radius is usually greater than the designed one.To overcome the shortcomings of small strain theory in recognizing ground-support interaction under large deformation circumstances,a new large strain numerical approach for modifying the GRC was proposed considering over-excavation in strain-softening rock masses based on the generalized Zhang-Zhu strength criterion.A case study was conducted based on the Lianchengshan tunnel in China.The modified GRC was employed to investigate the ground-support behavior for different support schemes and to explore the applicability of the stress release measures.Combined with field tests,the proposed approach was validated.By comparing with GRCs proposed by previous work,the present modified GRC was proved to be superior to others.Parametric studies were conducted and it is found that over-excavation,for example,reserving a very large clearance between the surrounding rock and the support,is necessary to reduce ground pressure to a large extent.The yielding supports which can provide high support pressure during the process of deformation are highly recommended when tunneling in high geostress environment.However,if the initial geostress is not very high,it is not necessary to pursue unwarranted overexcavation since the ground pressure applied on the support is mainly the loosening stress when the deformation is large.Ample support stiffness should be provided in the process of deformation to prevent uncontrolled large deformation of surrounding rock.

Analytical solutions for deep tunnels in strain-softening rocks modeled by different elastic strain definitions with the unified strength theory

This paper presents the analytical solutions for the responses of tunnels excavated in rock masses exhibiting strain-softening behavior. Since previous analyses give little consideration to the effect of the intermediate principal stress on the strain-softening rock behavior, the unified strength theory was introduced to analyze the tunnel response. Four cases of different definitions of the elastic strain in the softening and residual regions, used in the existing solutions, were considered. The tunnel displacements,stresses, radii of the softening and residual zones and critical stresses were deduced. The proposed solutions were verified by comparing with numerical simulations, model tests and existing solutions. Furthermore, the solutions of the four cases were compared with each other to investigate the influence of the elastic strain expressions on the tunnel responses. The results showed that the intermediate principal stress coefficient b has a significant effect on the tunnel displacements, stress fields, and plastic radii. Parametric studies were performed to analyze the influences of the softening and residual dilatancy coefficients,softening modulus and residual strength on the tunnel responses. The parametric analysis indicated that the existing models should be carefully evaluated in the analysis of tunnels constructed through average-quality rocks;the proposed solutions outperformed the existing models in solving the mentioned problem.

Analytical prediction for time-dependent interaction of a circular tunnel excavated in strain-softening rock mass

Viscoelastic plastic solutions for tunnel excavation in strain-softening rock mass and tunnel-rock interaction are proposed based on the Mohr-Coulomb and the Generalized Zhang-Zhu(GZZ)strength criterion considering stress path.The solutions are verified by numerical simulations,results show that the theoretical solutions are close to the simulated data.The evolutions of rock stresses,strains,displacements and support pressure were investigated and the influences of residual strength parameter,support stiffness,support timing,initial support pressure and viscosity coefficient on the rock deformation and the support pressure are discussed by proposed solution.It is found that strain-softening results in large deformation and high support pressure,with stiffer support and a larger viscosity coefficient contributing to even greater support pressure.Ductile support is recommended at the first stage to release the energy and reduce the support pressure by allowing a relatively large deformation.The support pressure,especially the additional support pressure at the second stage will be much smaller if a higher initial support pressure is applied at the first stage.This can not only control the displacement rate of surrounding rock and improve the tunnel stability at the first stage by exerting sufficient support pressure immediately after tunnel excavation,but also greatly reduce the pressure acted on permanent support and improve the structure stability at the second stage.Therefore,to avoid the instability of support structure,ductile support,which could not only deform continuously but also provide sufficient high support pressure,is recommended at the first stage.

Viscoelastic plastic interaction of tunnel support and strain-softening rock mass considering longitudinal effect

A simplified two-stage method was employed to provide an explicit solution for the time-dependent tunnel-rock interaction,considering the generalized Zhang-Zhu strength criterion.Additionally,a simplified mechanical model of the yielding support structure was established.The tunnel excavation is simplified to a two-stage process:the first stage is affected by the longitudinal effect,while the second stage is affected by rheological behavior.Two cases are considered:one is that the rigid support is constructed during the first stage,and the other is that constructed at the second stage.Distinguished by the support timing at the seconde stage,different kinds of the“yield-resist combination”support method are divided into three categories:“yield before resist”support,“yield-resist”support,and“control-yield-resist”support.Results show that the support reaction of“control-yield-resist”is much higher than that of“yield before resist”if the initial geostress is not very high,but the effect is not obvious on controlling the surrounding rock deformation.So,the“yield before resist”support is much more economical and practical when the ground stress is not very high.However,under high geostress condition,through applying relatively high support reaction actively to surrounding rock at the first stage,the“control-yield-resist”support is superior in controlling the deformation rate of surrounding rock.Therefore,in the high geostress environment,it is recommended to construct prestressed yielding anchor immediately after excavation,and then construct rigid support after the surrounding rock deformation reaches the predetermined deformation.

Design of the yielding support used highly deformable elements for a tunnel excavated in squeezing rock

Yielding support is often used in the squeezing tunnel to prevent damage to the lining induced by large deformation of the surrounding rock.Highly Deformable Elements(HDE)which is often installed along the circumferential direction of the shotcrete lining is a common type of yielding support.To determine the yield parameters of HDE,the support characteristic of the lining using HDE and the ground pressure considering strain-softening of soft rock were analyzed by an analytical method.The analytical solution showed that when considering the strain-softening of squeezing ground,the ground pressure has a non-zero minimum value.The minimum value of ground stress can be used to determine the constant yield stress of the HDE,and the corresponding deformation of the minimum ground pressure can be used to determine the deformation capacity of the HDE.Based on the variation in the design constant yield stress and yield displacement of HDE with the in-situ stress and the mechanical parameters of the soft rock,equations were proposed for determining of the yield parameters of the HDE.

Failure characteristics and its influencing factors of rock-like material with multi-fissures under uniaxial compression

The compression test on rock-like specimens with prefabricated closed multi-fissures made by pulling out the embedded metal inserts in the precured period was done on the servo control uniaxial loading instrument. The influence of fissure inclination angle and distribution density on the failure characteristics of fissure bodies was researched. It was found that, the fissure inclination angle was the major influencing factor on the failure modes of fissure bodies. The different developmental states of micro-cracks would appear on specimens under different fissure inclination angles. However, the influence of fissure distribution density on the failure mode of fissure bodies was achieved by influencing the transfixion pattern of fissures. It was shown by the sliding crack model that, the effective shear, which drove the relative sliding of the fissure, was a function of fissure inclination angle and friction coefficient of the fissure surface. The strain-softening model of fissure bodies was established based on the mechanical parameters that were obtained by the test of rock-like materials under the same experimental condition. And the reliability of experimental results was identified by using this model.

Softening response under undrained compression following anisotropic consolidation

Experimental evidence has indicated that clay exhibits strain-softening response under undrained compression following anisotropic consolidation.The purpose of this work was to propose a modeling method under critical state theory of soil mechanics.Based on experimental data on different types of clay,a simple double-surface model was developed considering explicitly the location of critical state by incorporating the density state into constitutive equations.The model was then used to simulate undrained triaxial compression tests performed on isotropically and anisotropically consolidated samples with different stress ratios.The predictions were compared with experimental results.All simulations demonstrate that the proposed approach is capable of describing the drained and undrained compression behaviors following isotropic and anisotropic consolidations.

多级开挖边坡强度折减法本构模型分析

运用FLAC3D软件,在Drucker-Prager模型和Strain-Softening模型中引入强度折减概念,实现了Drucker-Prager模型和Strain-Softening模型的强度折减法,并通过简单算例讨论了其在边坡工程数值模拟中的适用性。以湖北省宜昌市白洋长江公路大桥左侧路堑高边坡开挖工程为背景,采用不同本构模型强度折减法对比分析了边坡开挖稳定性。结果表明:边坡开挖中应充分考虑岩土体特性,选用恰当本构模型,才能得到更合理的分析结果。

Numerical Analysis of Cylindrical Cavity Expansion in Sand Considering Particle Crushing and Intermediate Principal Stress

Considering the effects of particle crushing and intermediate principal stress on material yielding strength, the spatial mobilization plane(SMP) yielding criterion and state parameter model including a general critical state line are selected in the analysis of cylindrical cavity expansion.Meanwhile, combining Rowe s flow rule and Bolton s simplification to stress-dilatancy relationship to reflect soil shear dilatancy and softening behavior, this paper analyzes the problem of cylindrical cavity expansion i...

Effect of intermediate principal stress on strength of soft rock under complex stress states

A series of numerical simulations of conventional and true triaxial tests for soft rock materials using the three-dimensional finite difference code FLAC3D were presented. A hexahedral element and a strain hardening/softening constitutive model based on the unified strength theory(UST) were used to simulate both the consolidated-undrained(CU) triaxial and the consolidated-drained(CD) true triaxial tests. Based on the results of the true triaxial tests simulation, the effect of the intermediate principal stress on the strength of soft rock was investigated. Finally, an example of an axial compression test for a hard rock pillar with a soft rock interlayer was analyzed using the two-dimensional finite difference code FLAC. The CD true triaxial test simulations for diatomaceous soft rock suggest the peak and residual strengths increase by 30% when the effect of the intermediate principal stress is taken into account. The axial compression for a rock pillar indicated the peak and residual strengths increase six-fold when the soft rock interlayer approached the vertical and the effect of the intermediate principal stress is taken into account.

Effect of pore pressure on deformation and unstable snap-back for shear band and elastic rock system

Fast Lagrangian analysis of continua(FLAC) was used to study the influence of pore pressure on the mechanical behavior of rock specimen in plane strain direct shear, the distribution of yielded elements, the distribution of displacement and velocity across shear band as well as the snap-back (elastic rebound) instability. The effective stress law was used to represent the weakening of rock containing pore fluid under pressure. Numerical results show that rock specimen becomes soft (lower strength and hardening modulus) as pore pressure increases, leading to higher displacement skip across shear band. Higher pore pressure results in larger area of plastic zone, higher concentration of shear strain, more apparent precursor to snap-back (unstable failure) and slower snap-back. For higher pore pressure, the formation of shear band-elastic body system and the snap-back are earlier; the distance of snap-back decreases; the capacity of snap-back decreases, leading to lower elastic strain energy liberated beyond the instability and lower earthquake or rockburst magnitude. In the process of snap-back, the velocity skip across shear band is lower for rock specimen at higher pore pressure, showing the slower velocity of snap-back.

ON DILATATIONAL PLASTIC CONSTITUTIVE EQUATION OF DUCTILE MATERIALS AND PLASTIC LOADING PATHS AT BIFURCATION

The dilatational plastic constitutive equation presented in this paper is proved to be in aform of generality. Based on this equation, the constitutive behaviour of materials at themoment of bifurcation is demonstrated to follow a loading path with the response as "soft"as possible.