Non-associated Flow Laws Research Articles

A New Numerical Procedure for the Excavation Response in Mohr–Coulomb Rock Mass Exhibiting Strain-Softening Behavior

A new numerical procedure for calculating the excavation response of the Mohr–Coulomb rock mass considering strain-softening behavior is proposed in this article. In this method, the plastic zone of the strain-softening rock mass is divided into the sufficient small plastic concentric annulus with constant radial stress increment, where the stress and strain distributions are characterized based on the existing analytical solutions of the brittle–plastic rock in the plastic zone. According to the equilibrium equation, geometric equation, and Mohr–Coulomb yield criterion, the stresses of each annulus can be calculated, and the explicit form of the displacement can also be determined by invoking the non-associated flow law and Hooke’s law. On this basis, the excavation disturbance-induced response and the ground reaction curve (GRC) in the strain-softening rock mass can be calculated by iterative computation. The proposed method is verified by comparing both the numerical simulation results and the existing theoretical solutions. Extensive computations are then carried out to clarify some practical questions, including the effect of ground condition, the computation efficiency, and the engineering applicability. It is found that the proposed numerical procedure behaves more efficiently and accurately than the previous one for the strain-softening rock mass. This might, therefore, provide convenience and benefits from a computation standpoint for the preliminary design of underground openings in rock masses with slight deformation.

Theoretical model for the shear strength of rock discontinuities with non-associated flow laws

In an earlier publication (Serrano et al. 2014), the theoretical basis for evaluating the shear strength in rock joints was presented and used to derive an equation that governs the relationship between tangential and normal stresses on the joint during slippage between the joint faces. In this paper, the theoretical equation is applied to two non-linear failure criteria by using non-associated flow laws, including the modified Hoek and Brown and modified Mohr-Coulomb equations. The theoretical model considers the geometric dilatancy, the instantaneous friction angle, and a parameter that considers joint surface roughness as dependent variables. This model uses a similar equation structure to the empirical law that was proposed by Barton in 1973. However, a good correlation with the empirical values and, therefore, Barton's equation is necessary to incorporate a non-associated flow law that governs breakage processes in rock masses and becomes more significant in highly fractured media, which can be induced in a rock joint. A linear law of dilatancy is used to assess the importance of the non-associated flow to obtain very close values for different roughness states, so the best results are obtained for null material dilatancy, which considers significant changes that correspond to soft rock masses or altered zones of weakness.

Mechanical behavior of low-rank bituminous coal under compression: An experimental and numerical study

The mechanical behavior of coal has an important influence on coal reservoir stimulation and the dynamic change of coal reservoir permeability during coalbed methane (CBM) production. Generally, coal is regarded as an elastic-brittle material, which is reasonable when the surrounding stress is small. However, with the increasing depth of CBM exploitation, the in-situ stress increases continuously, so the coal will inevitably show some plastic behavior. In this work, the typical low-rank bituminous coal from the Baode CBM block on the eastern margin of the Ordos Basin was selected to carry out uniaxial and triaxial compression tests under various confining pressures (Pc) for further investigation of mechanical behavior. The results show that, as a complex porous material, the coal samples have a series of deformation responses during the loading process, which can be classified into four phases. Both the yield stress (σy) and the peak stress (σpeak) increase nonlinearly as Pc increases. Additionally, with increasing Pc, the elastic modulus (E) increases at a decreasing rate, while the Poisson's ratio (ν) decreases at a decreasing rate. Moreover, a change from volumetric compressibility to dilatancy in the coal was clearly observed in the tests, and the stress at the onset of dilation (σd) is also significantly influenced by Pc. Finally, based on irreversible thermodynamics and empirical support, a coupled elastoplastic damage constitutive model for low-rank bituminous coal was established by introducing the non-associated plastic flow laws and the damage evolution rules; the numerical simulation result of the model is in good agreement with the experimental measurements. This model can be used for further investigation of the mechanical properties of coal reservoirs under different in-situ stress conditions during the exploitation of CBM.

Closed system of coupling effects in generalized thermo-elastoplasticity

In this paper, the field equations of the generalized coupled thermoplasticity theory are derived using the postulates of classical thermodynamics of irreversible processses. Using the Legendre transformations two new thermodynamics potentials <i>P</i> and <i>S</i> depending upon internal thermodynamic forces Π are introduced. The most general form for all the thermodynamics potentials are assumed instead of the usually used additive form. Due to this assumption, it is possible to describe all the effects of thermomechanical couples and also the elastic-plastic coupling effects observed in such materials as rocks, soils, concretes and in some metalic materials. In this paper not only the usual postulate of existence of a dissipation qupotential (the Gyarmati postulate) is used to derive the velocity equation. The plastic flow constitutive equations have the character of non-associated flow laws even when the Gyarmati postulate is assumed. In general formulation, the plastic strain rate tensor is normal to the surface of the generalized function of plastic flow defined in the the space of internal thermodynamic forces Π but is not normal to the yield surface. However, in general formulation and after the use the Gyarmati postulate, the direction of the sum of the plastic strain rate tensor and the coupled elastic strain rate tensor is normal to the yield surface.

Multiscale Numerical Study of Fracture and Strength Characteristics of Zirconium Alumina Concrete with Use of the Particle-based MCA Method

In the present paper the multiscale approach to the construction of multiscale rheological models of brittle materials with heterogeneous internal structure is implemented within the numerical method of movable cellular automata (MCA). Nikolaevsky's plasticity model with non-associated flow law was chosen as a rheological model for zirconium alumina concrete at the different structural scales. The mechanical properties (including parameters of the rheological model and strength values) of zirconium alumina concrete were obtained for both macroscopic and mesoscopic scales. The effect of strain rate on material behavior was analyzed in case of uniaxial compression and tension.

Convergence of circular tunnels in elastoplastic rock masses with non-linear failure criteria and non-associated flow laws

This work provides new findings on the theory of non-associated plasticity applied to the study of convergence in tunnels. It focuses on elasto-plastic rock masses with linear and non-linear strength criteria and associated and non-associated flow laws. It seeks to obtain the “Characteristic Rock mass Curve” based on the “Convergence–Confinement” method. The results that are obtained are compared to other calculation procedures. Detailed mathematical expressions are proposed for implementation in a computer spreadsheet.

The Local Conditions of Uniqueness and Plastic Strain Localization Part I: Determination of the Critical Hardening Moduli for Some Elastoplastic Materials

The classical, most frequently cited constitutive equations of elastic—plastic materials with non-associated and associated plastic flow laws are specified. Local sufficient uniqueness conditions for the solution of the boundary-value problem and local necessary conditions of initiation of deformation localization in the form of a Rice—Rudnicki localization plane are derived. We show that sufficient local uniqueness conditions for these constitutive equations with sign (≤) on the right-hand side become local necessary conditions of non-uniqueness of the solution of the boundary-value problem.

Generalized Coupled Thermoplasticity Taking into Account Large Strains: Part II. A Model of Non-compressible Elastic–Plastic Solid with Non-associated Plastic Flow Law

This paper is the second part of our work on generalized coupled thermoplasticity taking into consideration the large strains. In the first part of the work global uniqueness conditions for solution of fundamental incremental boundary-value problem were derived. In this part a model of non-compressible elasto-plastic body with non-associated plastic flow law is analyzed. The assumptions and constitutive equations for this model are presented and suitable comparison bodies dependent on kinematically admissible fields of velocity strain and statically admissible fields of velocity stress are derived. An attempt to derive local sufficiwent uniqueness condition for comparison bodies is also presented. This attempt was unsuccessful since it lead to contradictory results, which means that the derivation of local sufficient uniqueness condition for comparison bodies of non-compressible elasto-plastic body is still unsolved.

Plastic collapse in non-associated hardening materials with application to Cam-clay

This paper presents a variational formulation for the analysis of plastic collapse conditions for a class of hardening materials that accounts for some non-associated flow laws such as the modified Cam-clay model of soils. In this framework, classical statical and kinematical principles of limit analysis do not hold. The variational principle is formulated for the general class of materials whose flow equations are derived from a kind of generalized potentials named bipotentials by de Saxcé. The plastic collapse phenomenon for hardening materials is considered first and formulated as a system of equations. In particular, the case of the usual modified Cam-clay model is analyzed. The paper follows with the proposal of a minimization principle whose solution is then related to the solution of the plastic collapse problem. We demonstrate the use of this minimum principle in a simple example of triaxial compression of a modified Cam-clay material. Finally, we discuss the particular form of the proposed variational formulation for the case of associated plasticity.

Stability of highly fractured infinite rock slopes with nonlinear failure criteria and nonassociated flow laws

The effect of the dilatancy phenomenon on rock masses is not usually considered when calculating slope stability. A theoretical analysis has been conducted on the stability of infinite rock slopes, with varied hypotheses of simplified seepage flow nets. The calculations were made in the most general way possible, for any type of failure criterion and for linear-type flow laws. When the calculations were completed they were applied to the original and modified Hoek–Brown failure criteria and also to the associated flow and constant-dilatancy conditions. The study reveals the great importance of the dilatancy value and the "non-conservative" nature of an associated flow law hypothesis, which is the hypothesis that is generally used. The important effect of groundwater flow in the slope has also been enhanced. A quantitative estimation of the stability is provided for homogeneous and isotropic rock media under different angles of dilatancy and water flow nets. Lastly, the equilibrium is dependent on the following parameters: slope inclination angle (α), specific weight of the rock mass (γR*), type of rock (m0), unconfined compression strength (σc*), and geological strength index (GSI).Key words: infinite rock slope, Hoek Brown, dilatancy.

Numerical modelling of thrust structures in unconsolidated sediments: implications for glaciotectonic deformation

The mechanics of thrust faulting induced by a differential vertical load has been studied by application of a dynamic finite element model. The modelled scenario is analogous to the glaciostatic load imposed onto an unconsolidated sedimentary substrate by a stationary ice sheet. The objective is to evaluate whether large-scale glaciotectonic deformation may be induced by gravity spreading in front of a stationary glacier or if a push from the rear is required. The model is capable of handling large strains and rotations combined with visco-elastic-plastic rheologies. Model parameters are derived from a Pleistocene large-scale glaciotectonic thrust complex in the eastern Danish North Sea. The plastic behaviour of the unconsolidated sediments is described by a time and pressure dependent Drücker–Prager yield criterion combined with a non-associated flow law. The modelling results show that gravity spreading in front of a stationary ice-sheet can form strain localisations and possibly cause thrust faults to nucleate in unconsolidated sediments several kilometres beyond the snout of the ice-sheet, provided a décollement surface is situated at a suitable depth relative to the effective load of the ice-sheet. Hence it is concluded that large-scale glaciotectonic thrusting may be formed by gravity spreading alone, and a push from the rear is not generally required to explain the observed deformation.

Comparisons between analogue and numerical models of thrust wedge development

Analogue and finite element numerical models with frictional and viscous properties are used to model thrust wedge development. Comparison between model types yields valuable information about analogue model evolution, scaling laws and the relative strengths and limitations of the techniques. Both model types show a marked contrast in structural style between ‘frictional-viscous domains’ underlain by a thin viscous layer and purely ‘frictional domains’. Closely spaced thrusts form a narrow and highly asymmetric fold-and-thrust belt in the frictional domain, characterized by in-sequence propagation of forward thrusts. In contrast, the frictional-viscous domain shows a wide and low taper wedge and a thrust belt with a more symmetrical vergence, with both forward and back thrusts. The frictional-viscous domain numerical models show that the viscous layer initially simple shears as deformation propagates along it, while localized deformation resulting in the formation of a pop-up structure occurs in the overlying frictional layers. In both domains, thrust shear zones in the numerical model are generally steeper than the equivalent faults in the analogue model, because the finite element code uses a non-associated plasticity flow law. Nevertheless, the qualitative agreement between analogue and numerical models is encouraging. It shows that the continuum approximation used in numerical models can be used to model frictional materials, such as sand, provided caution is taken to properly scale the experiments, and some of the limitations are taken into account.

A numerical model for masonry implemented in the framework of a discrete formulation

A direct discrete formulation suitable for the nonlinear analysis of masonry structures is presented. The numerical approach requires a pair of dual meshes, one for describing displacement fields, one for imposing equilibrium. Forces and displacements are directly used (instead of having to resort to a model derived from a set of differential equations). Associated and nonassociated flow laws are dealt with within a complementarity framework. The main features of the method and of the relevant computer code are discussed. Numerical examples are presented, showing that the numerical approach is able to describe plastic strains, damage effects and crack patterns in masonry structures.

On smooth bifurcations in non-associative elastoplasticity

The second-order incremental constitutive equations proposed by Petryk and Thermann [(1985) Second-order bifurcation in elastic-plastic solids. J. Mech. Phys. Solids 33, 577–593] are generalized to include non-associativity of the plastic flow rule. It is shown that the exclusion principle of Raniecki [(1979) Uniqueness criteria in solids with non-associated plastic flow laws at finite deformations. Bull. Acad. Polon. Ser. Sci. Tech. XXVII(8–9), 391–399] for first-order bifurcations is sufficient to exclude second-order bifurcations. The result holds true under specific regularity conditions and, accepting stronger regularity conditions, is extended to the case of nth-order bifurcations.

Finite deformation plasticity with strain induced anisotropy and shear banding

Theoretical description of large plastic deformations accounting for the effects of kinematic hardening and shear banding is proposed. Macroscopic measure of the rate of plastic deformation produced by micro-shear banding is derived and the idea of multiple potential surfaces forming a vertex on the smooth extremal surface is applied to display the connection with the geometric pattern of micro-shear bands. New form of non-associated flow law is studied and the perspectives of possible aplications of the theory are discussed.

Anisotropic Hardening Plasticity Model for Sands

In this study, a bounding surface-based anisotropic hardening softening elasto-plasticity model has been formulated for the behavior of sand. The model employs both joint invariants of stress tensor and rotational tensor, and satisfies the objectivity requirements within the form-invariance principle. The second order rotational tensor represents the rotation of the symmetric axis of the bounding surface, and is postulated to reflect the evolution of microstructures of particulate sand. An evolution law for the rotational hardening is forwarded, along with a nonassociated flow law. The plastic moduli are derived in terms of stress increment and strain increment to facilitate numerical simulation of stress- or strain-controlled tests. The performance of the model is demonstrated by comparisons of numerical simulations with the results of tests with a hollow cylinder and a cube.

Bifurcations in finite element models with a non-associated flow law : De Borst, R Int J Num Anal Meth GeomechV12, N1, Jan–Feb 1988, P99–116

Bifurcations in finite element models with a non-associated flow law : De Borst, R Int J Num Anal Meth GeomechV12, N1, Jan–Feb 1988, P99–116

Damage evolution in mild steel

Macroscopic attributes of damage resulting from development of microvoids in a cold-strained mild steel are examined. One of them is the relative density change, the evolution of which has been found experimentally in torsion and in a process close to uniaxial tension. In the latter case, softening of the material understood as a reduction of yield stress has also been measured. The theoretical description is based on the appropriate yield condition taking into account pressure sensitivity as well as hardening and softening (together with decohesion) of a ductile porous material. Respecting the change in density during torsion, the non-associated flow laws were adopted. In the relative density change evolution equation both nucleation and void growth were incorporated. Emphasis has been put on confirming reliability of the model by obtaining quantitative agreement between the experimental and theoretical data.

Bounding approach to the bifurcation point of annular plates with nonassociated flow law subjected to uniform tension at their outer edges

A bounding approach based on numerical procedure is developed to investigate the bifurcation behavior of nonuniformly stressed solids obeying the nonassociated flow law. An improved lower bound to the bifurcation point is determined by optimizing a positive parameter r in the linear comparison solid. Accuracy of the obtained lower bound is examined by the upper bound analysis. The nonaxisymmetric bifurcation behavior of pressure-sensitive and dilatant annular plates subjected to uniform tension at their outer edges has been analyzed. The results indicate that the optimization of the parameter r is essential in achieving considerable improvement of the lower bound to the bifurcation point. Nonassociateness always destabilizes the deformation. Furthermore, the effect of work hardening rate and yield stress on the bifurcation behaviors has been clarified.

Bifurcation behaviour of non-uniformly stressed solids with the non-associated flow law.

In order to establish a numerical procedure for predicting the bifurcation behaviour of a non-uniformly stressed body, the generalized version of Hill's theory of uniqueness and bifurcation to materials obeying the non-associated flow rule, proposed recently by Raniecki et al., is employed. To obtain an improved lower bound to the bifurcation point, the steepest descent method is introduced to optimize a parameter r^-, which is introduced into the constitutive equation to establish a variational principle governing the bifurcation behaviour. The obtained results for pressure-sensitive and dilatant annular plates, subjected to uniform tension at the outer edge, show that the bifurcation is notably accelerated compared with that for plates obeying the associated flow rule. The effect of the pressure-sensititve yielding on the bifurcation behaviour is more remarkable than that of the dilatant plasticity. It is clarified that optimization of the parameter r^- is quite important in achieving considerable improvement of the lower bound of the bifurcation point.