Elasto-plastic Cellular Automaton Research Articles

Modelling transient unloading-triggered dynamic responses in rock mass under a non-hydrostatic geo-stress

Transient excavation unloading of in situ stress affects the stability of surrounding rock in the deep tunnel engineering. This paper focuses on the dynamic responses triggered by the transient unloading of a non-hydrostatic geo-stress field in deep-buried engineering. With resort to the integral transformation, the frequency-domain responses of stress, displacement and velocity components induced by the transient unloading are obtained under a non-hydrostatic in situ stress. Based on the numerical inversion of the Laplace transformation, the corresponding time-domain theoretical results are determined. Agreement of the current solutions with the existing results and the numerical simulations verifies the proposed scheme in this paper. The elastic analytical results indicate that the influence of unloading path on the stress redistribution is characterized by the unloading rate. The higher the unloading rate is, the larger the stress magnitude is. The smaller the unloading time is, the more remarkable vibration is. The elasto-plastic dynamic numerical responses are investigated on the basis of a self-developed code, elasto-plastic cellular automaton (EPCA), which is a module of CASRock. The numerical results demonstrate that the transient excavation unloading results in the stress redistribution and concentration of surrounding rock mass, inducing damage for the case of exceeding the capacity of rock mass. For the small lateral pressure coefficient (less than 0.25), the tensile-shear failure is the major damage mechanism of surrounding rock, while the shear failure is the major damage mechanism for the large lateral pressure coefficient. The analytical and numerical results can provide theoretical basis for the support and reinforcement of underground tunnel.

Blasting induced dynamic response analysis in a rock tunnel based on combined inversion of Laplace transform with elasto-plastic cellular automaton

SUMMARY The dynamic disturbance from explosion, earthquakes and stress impact may induce rock mass damage around the deep-buried tunnel. Here, we show the dynamic responses triggered by blasting load disturbance. In this paper, the theoretical formulae are first derived to evaluate the time-domain dynamic responses based on the wave function expansion method and the inversion of Laplace transform and then a self-developed numerical code, that is, elasto-plastic cellular automaton (EPCA), a module of CASRock, is subsequently employed to simulate the blasting induced elasto-plastic dynamic responses. It can be found from the analytical solutions that tensile stress concentration at two sidewalls occurs, while compressive counterpart at the roof and floor appears. Moreover, the radial velocity vibrations generate concentration at the two sidewalls, while the circumferential counterparts appear at the roof and floor. The numerical simulations indicate that compression-shear and tensile failure are the major mechanism for the rock mass damage around the rock tunnel. The analytical and numerical results help to understand the velocity vibrations and the major mechanism of rock mass damage and provide theoretical basis for the support of rock tunnel.

Modelling three-dimensional stress-dependent failure of hard rocks

Hard rocks exhibit three-dimensional (3D) stress-dependent failure under true triaxial compression. The deformability and strength of hard rocks under true triaxial compression differ from those under traditional loading schemes of conventional triaxial compression tests. For the purpose of characterizing these distinctive features, including 3D stress-dependent brittleness and the failure process and 3D stress-induced anisotropy, a new model suited for hard rocks was proposed in this paper. In the new model, the 3D stress-dependent brittleness under true triaxial compression tests was reflected by a determination method of the internal variable considering the influences of both $${\sigma }_{2}$$ and $${\sigma }_{3}$$ . For the failure process, different evolutions in cohesion and the friction angle were applied to realize different failure processes under different 3D stresses, and their 3D stress dependency was identified by the 3D brittleness index defined in this paper. The 3D stress-induced anisotropy involved in the deformation and failure of hard rocks under true triaxial compression was described via the deformation modulus evolution. The formulation for the stress-induced stiffness matrix and the model framework is fully thermodynamically consistent. The model was implemented in the 3D elasto-plastic cellular automaton system, and good agreement was achieved between the numerical simulation and experimental results, indicating that the new model can be applied to describe the failure behaviours of hard rocks under 3D stress.

Modeling transient excavation-induced dynamic responses in rock mass using an elasto-plastic cellular automaton

The problem of transient excavation-induced dynamic response in rock mass is discretized on both spatial and temporal scales and is solved by using the cellular automaton (CA) technique and the Newmark scheme, respectively. Using this approach, a dynamic analysis version of 3D elasto-plastic cellular automaton (EPCA3D) is developed. The advantage of this method is it avoids the solution of large-scale linear equations since only the local CA rule is used for dynamic state updating. The Den Iseger algorithm is introduced to validate the numerical method. The evolution of stress and velocity for the transient excavation in rock mass under different conditions obtained by the EPCA3D and Den Iseger methods are in good agreement. The abilities of EPCA3D in the modeling of elasto-plastic dynamics of transient excavation in rock mass are well demonstrated. By considering different in situ stresses, excavation radiuses, unloading durations and unloading paths, the factors affecting nonlinear dynamic responses are investigated. It is found that the failure extent increases with the decrease of unloading time. When lateral pressure coefficient (σx/σy) equals to 1, the failure zone is evenly distributed around the tunnel. However, with the decrease of lateral pressure coefficient, the extent of failure localization increases. The fault around the tunnel makes the dynamic failure zone asymmetric distribution. The modeling helps to understand the major mechanism of rock mass damage for the transient excavation.

Study on coupled thermo-hydro-mechanical processes in column bentonite test

An unconventional numerical scheme is developed to simulate coupled thermo-hydro-mechanical (THM) processes in partially saturated medium. The non-isothermal, unsaturated fluid flow and mechanical processes are sequentially coupled by updating all the state variables using cellular automaton technique and finite difference method on spatial and temporal scale, respectively. A new cellular automaton updating scheme is proposed by introducing a fast successive relaxation index, which greatly improves the computational efficiency in the simulation of THM coupling process. This is implemented in a self-developed numerical system, i.e., an elasto-plastic cellular automaton (EPCA3D), which was used to numerically reproduce the coupled THM behavior of bentonite pellets in a column experiment that was heated up to 140 °C firstly and then was hydrated simulating the resaturation of the backfilling. By using the cellular automaton technique in EPCA3D, the challenging courses of the changing boundary conditions over time and space during the experiment are conveniently implemented. The EPCA3D was able to reproduce the main physical processes of the in laboratory column bentonite experiment within the heating and hydration phase. The modeling results for the evolution of temperature, relative humidity, water uptake and axial pressure are consistent with the experimental data in terms of trends and magnitudes, which verifies the realistic simulation with the developed model and contributes to a deeper understanding of the observed phenomena.

An approach for simulating the THMC process in single novaculite fracture using EPCA

An approach for simulating thermal, hydraulic, mechanical and chemical (THMC) coupled processes in single rock fractures has been developed under the framework of a self-developed numerical method, i.e., an elasto-plastic cellular automaton. The balance equations of multi-physics problems to describe the THMC process in single rock fracture are solved by using cellular automaton technique on space scale and finite difference method on time scale, respectively. Using the concept of cellular automaton, a single rock fracture surface is discretized into a system composed of cell elements. Different apertures, i.e., 0 for contact and nonzero for void, are assigned to each cell element based on the fracture surface topography. The fluid flow, stress-dependent chemical reaction and solute transport are simulated by using a cellular automaton updating rule, in which only local cell balance equation is considered. The contribution of cell elements in contact to cell’s transmissivity and convection can be ignored conveniently. The Lagrangian method is used to simulate the particle transport. Special treatment for particle transport to outer boundaries and internal boundaries is adopted. As a result, the local behaviors, such as the formation of local contact, dead ends in the fracture and the local aperture change, are conveniently updated dynamically. The approach is used to simulate the coupled THMC process in a single novaculite fracture. The behaviors of pressure dissolution caused by effective stress, free-face dissolution/precipitation, thermal-dependent fluid flow and ion transport are well reproduced by using the developed approach, subject to parameter calibration. The robustness of the general approach to such complex problems is demonstrated by comparing with experimental data.

Uncertainty analysis of rock failure behaviour using an integration of the probabilistic collocation method and elasto-plastic cellular automaton

The Karhunen-Loeve (KL) expansion and probabilistic collocation method (PCM) are combined and applied to an uncertainty analysis of rock failure behavior by integrating a self-developed numerical method (i.e., the elastic-plastic cellular automaton (EPCA)). The results from the method developed are compared using the Monte Carlo Simulation (MCS) method. It is concluded that the method developed requires fewer collocations than MCS method to obtain very high accuracy and greatly reduces the computational cost. Based on the method, the elasto-plastic and elasto-brittle-plastic analyses of rocks under mechanical loadings are conducted to study the uncertainty in heterogeneous rock failure behaviour.

Development and applications of the elasto-plastic cellular automaton

The paper presents the advancement and applications of the elasto-plastic cellular automaton (EPCA), a simulator for rock mechanics and rock engineering. The most significant feature of EPCA lies in its ‘down-top’ way of dealing with nonlinear behaviors of rocks. The theory, the basic idea and associated developments, including the definition of cellular automaton, the heterogeneous material model, constitutive relations, failure criteria, the post-yield softening scheme, the thermo-hydro-mechanical coupling process, are described. The applications are presented to show the ability of EPCA to model the rock failure process, fluid flow, heat transfer, and the coupled thermo-hydro-mechanical (THM) process etc.

The influence of the intermediate principal stress on rock failure behaviour: A numerical study

The influence of the intermediate principal stress on rock strength has been studied comprehensively by previous researchers. However, the reason why rock strength firstly increases and subsequently decreases with the increase of intermediate principal stress is still unclear. In this paper, the mechanism of the intermediate principal stress effect on rock failure behaviour is revealed through a numerical method using the EPCA3D system (Elasto-Plastic Cellular Automaton). In this study, both homogeneous and heterogeneous rocks are considered. The heterogeneity of a rock specimen is modelled by introducing Weibull's statistical distribution. Two criteria, i.e. the Drucker–Prager and Mohr–Coulomb models, are used to determine whether a meso-scopic element in the rock specimen is in a failure state or not during the polyaxial stress loading process. The EPCA3D simulation reproduces the typical phenomenon of the intermediate principal stress effect that occurs in some rock experiments. By studying the EPCA3D simulated acoustic emission and complete stress–strain curves illustrating failure initiation, propagation and coalescence in the failure process of rocks, the essence of the intermediate principal stress effect is tracked. It is concluded that the heterogeneous stress distribution induced by the natural heterogeneity of rocks and the effect of the loading platen are two of the reasons producing the intermediate stress effect. Studies indicate that a moderate intermediate principal stress delays the onset of local failure, which in turn leads to an increase in the rock strength. However, once the intermediate principal stress reaches a certain value, local failure will be formed through the application of the intermediate principal stress. It is the number of failed elements in the pre-peak region that determines whether the rock strength decreases or not. The extent of rock strength reduction when the intermediate principal stress reaches a certain value is lessened with the increase in the minimum principal stress.

Effect of precrack length and inclination on tensile failure behaviour of heterogeneous rocks

The failure behaviour of heterogeneous rock specimens with prefabricated single crack under direct tensile loading is investigated numerically by using an elastoplastic cellular automaton (EPCA2D) model. The response of tensile failure behaviour of heterogeneous rocks is investigated when different crack lengths and inclinations are used. The results confirm that the tensile strength of rock specimens presents a tendency of reduction with the increment of crack length at same fixed crack inclination. However, this reduction changes non‐monotonically due to the heterogeneity of rocks. Only the precrack is longer enough can it be the main factor of rock failure. With the same crack length, but different crack inclinations, the tensile strength increases with the increment of crack inclination. With the increment of the angle between precrack and tensile loading direction, the effect of the precrack on tensile failure behaviour of rock decreases.

Elasto-Plastic Cellular Automaton Simulation of Fracturing Process in Single Pre-Fractured Rocks under Uniaxial Compression

Rock is a natural heterogeneous material and presents complicated behaviors in the fracturing process. It is prevail to study the basic failure mechanism of rocks via numerical simulation. Based on the elasto-plastic cellular automaton (EPCA) model, this paper simulates single pre-fractured rock fracturing process with consideration of rock heterogeneity on the meso-scale. In this model, the Weibull’s distribution, which characterizes heterogeneity with the homogeneous index m and the random seed parameter s, is adopted to describe the distribution of mechanical parameters of rock specimens such as cohesive strength, Young’s modulus, etc. Pre-existing crack rock specimens with different homogeneous index or the different random seed are simulated by EPCA under uniaxial compression. Numerical results show that heterogeneity has great influence on pre-fractured rock failure process, final failure modes, and the uniaxial compressive strength.

Study of failure and scale effects in rocks under uniaxial compression using 3D cellular automata

A numerical model for simulating the rock fracturing process has been developed and implemented in the three-dimensional code EPCA 3D, which is based on the earlier EPCA 2D (elasto-plastic cellular automaton). EPCA 3D has the ability to simulate the initiation, propagation and coalescence of cracks in the failure processes of rocks. Using this code, we studied the failure processes of simulated rock specimens with different sizes and shapes in three-dimensional space. It is concluded that the ‘scale effect’ is not directly related to the heterogeneity, if the rock specimens have the same homogeneity. If the platen-rock elastic mismatch effect in physical testing is not considered in the numerical simulations, there is almost no scale effect for rock specimens with different sizes and shapes. However, by considering the platen-rock interaction in the simulation of the rock failure process with the EPCA 3D code, the modeling results reproduced the well-known phenomenon of scale effects found in physical experiments.

RESEARCH ON THE EFFECT OF LOADING CONDITIONS ON THE STRENGTH AND DEFORMATION BEHAVIORS OF ROCKS

The paper aims at a numerical study of strength and deformation behaviors of rocks under different loading conditions in uniaxial compression with an elasto-plastic cellular automaton (EPCA2D) code. Two loading conditions, i.e. with and without considering frictions between loading platens and rock specimen's ends, are used in the failure processes of heterogeneous rocks. Rock specimens are assumed to be the same heterogeneity, i.e. the specimens with different sizes have the same probability of containing flaws. Under this condition, it is concluded that the strength and deformation behaviors of rocks are influenced by the heterogeneity a little. The mismatch of elastic properties of the platen and the rock in influencing the stress distribution at the ends of the specimen is the dominant cause for the so-called strength and deformation size effects, which in turn affects the final failure patterns of rocks.

Numerical simulations of Class I and Class II uniaxial compression curves using an elasto-plastic cellular automaton and a linear combination of stress and strain as the control method

Class I and Class II behavior of the rock failure process under uniaxial compression has been numerically simulated using an elasto-plastic cellular automaton (EPCA). A linear combination of stress and strain was used as the feedback signal to control the loading process in the numerical model. Uniaxial compressive simulations of heterogeneous rock failure were made using this control method and by considering different homogeneity indices and different linear combinations of parameters, as well as different yield criteria. The results indicate that Class II behavior can be numerically simulated in a complementary way to the laboratory testing method utilized in a servo-controlled testing machine. Whether the rock behaves in a Class I or Class II manner depends on the homogeneity of the rock sample, the linear combination parameters and the yield criterion selected.

Simulation of the rock microfracturing process under uniaxial compression using an elasto-plastic cellular automaton

In this paper, an elasto-plastic cellular automaton (EPCA) with the associated code was developed to simulate the non-linear fracturing process of rocks under uniaxial compression. It is a useful method for simulating the process of self-organization of the complex system using simple rules. It has the advantages of localization, parallelization, and being able to consider the heterogeneity of rocks. By this means, the fracturing processes, stress–strain curves, acoustic emissions, and compressive strength of 2-D heterogeneous rock specimens under uniaxial compression were numerically investigated using the EPCA code for different cases such as: (1) the influence of the cyclic load; (2) the influence of inhomogeneity; (3) the effect of different softening coefficients; (4) the effect of specimen sizes and height/width ratios; (5) the effect of different spatial distributions of material parameters; (6) the influence of different yield criteria; and (7) the influence of the residual strength coefficient. The results indicate that the numerical simulation reproduced some of the well-known phenomena observed by previous researchers in uniaxial compression tests.