Finite Element Theory And Application Of Strain Localization Of Rock And Soil

There is a band region caused by inconsistent deformation in materials, a narrow zone of intense deformation compared with outside the zone, in the process of deformation. This phenomenon is called localized deformation bifurcation. Generally, localized deformation in the form of shear bands and slip surface ubiquitously exists in a variety of inhomogeneous solid materials, such as metals, composite materials and geomaterials.In the current, the finite deformation theory non-related flow theory hyperelastic theory and gradient plastic theory are adopted to analyze strain localization in geomaterials. This paper mainly discusses mechanism of onset of bifurcation behaviors through analyzing localized deformation in the form of slip and shear bands occurs naturally in geomaterials such as soil and rock. As a result, the limitation that the above theories can't describe the whole process for shear bands in loaded geomaterial to deform from toughness to brittleness, and then lead to fracture is presented in the paper.For hydraulic engineering, many geomechanical problems require a mathematical description of failure and post-failure condition of geotechnical structures such as dams, embankments, excavations, foundations, and slopes. Failures of these structures are often accompanied by the appearance of highly localized patterns of deformation. As well know, strain localization is the precursor of failure. Therefore, in order to know about causes of failure, the process of failure and estimate of safety, it's necessary to simulate the strain localization in the hydraulic structures for the geotechnical engineer or engineering geologist to make informed analysis and decisions for geomechanical structures.In the recent, there are two methods for failure mechanism of structures in hydraulic engineering. Firstly, it's the limit equilibrium methods, which are based on defining a failure mechanism, orientation of failure surface, taking into account simplifying assumptions to make the problem statically determinate and doing a stability analysis. Advantages to this method include ease of use and relatively good results for simple problems, such as geometry and loading conditions. However, the method becomes unwieldy for problems with complex geometry and soil constitutive model. Also, itdoes not account for a softening response along the discontinuity and cannot be used for deformation type analysis.Secondly, it's the finite element method, which has been used widely for addressing issues of consistent deformations. As for conventional FEM, the issue of localization can't be solved well. In the absence of a material 'length scale', the conventional FEM do not satisfy two necessary criteria for a finite element solution to be meaningful (i.e., mesh-independent): objectivity with respect to mesh refinement and insensitivity to mesh alignment. Therefore, a more sophisticated numerical model needs to be implemented.Because of the onset of localized deformations in geomaterial mediums, there is the loss of ellipticity and hyperbolicity in the governing equation under static and dynamic loadings respectively. Consequently, just mentioned above, the results of finite element analysis suffer from pathological mesh-dependence. For decades, many attempts have been made to numerically model localized deformation using regularization techniques in framework of FEM. These attempts have been proposed to circumvent the problem of mesh dependence in finite element modeling of strain localization, and also to describe the mechanism of pre-localization and post-localization. This paper firstly classes these methods, which have different advantages and disadvantages respectively based on different assumptions and conceptions, and compares their applications on engineering projects, respectively. The regularization techniques are improved governing equations, adaptive remeshing and discontinuous shape functions.The technique of improved governing equations keeps the governing equeation ellipticity and hyperbolicity by adopting a material length scale in order to make the result of finite element method objective. But the criterion of bifurcation is based on the loss of ellipticity. As a consequence, there is no information for onset of bifurcation. So that the orientation of localized bands can not be decided since the bifurcation criteria depends on classic bifurcation theory. This technique needs many new parameters, including material parameters length scale and so on. Therefore, the main work of this method is to define these parameters. However, the work can't be done because the current apparatus conditions. And some of them have to be assumed, so that these parameters have no exact physical meaning. Consequently, the results are hardly to beestimated for these new parameters. Because of above reasons, this method is only limited to research theoretically.Through refining mesh in the region of intense deformation, adaptive remeshing can simulate the strain localization. And so the technique needs a strong mesh generator. But the geomaterials is special, in which joints, faults and soft stratums ubiquitously exist. Therefore this characteristic brings some difficulties for meshing, such as too large calculation and hard to pre-process. Although the technique is regarded as a prospective method to address this issue, recently it is hardly to be implemented for reasons above.In order to consider the effect of localized bands, the technique of discontinuous shape functions modifies interpolation function. It's easy to be implemented for finite element method, so that this technique gets to be researched widely and deeply. The model presented in this paper pertains to this technique.This paper presents a simplified finite element method using visco-plastic strain softening models to capture the strain localization points in hydraulic structures, in which the classic discontinuous bifurcation theory is adopted as the condition of onset of strain localization. The trait of this method is that the continuum elastic-plastic tangent modulus needed in the bifurcation theory is just used in the judgment of localization in every time step, and makes no difference to the process of visco-plasticity return mapping algorithm proposed in the paper. Compared with other methods, this model is not only based on simple theory, but also is easy to program. And numerical results show that this method is reasonable and feasible through the analysis of slope stability.Then, this paper presents a model with embedded localized softening bands incorporating into the idea of strong discontinuities. Based on the virtual work principle, the equilibrium equations of FEM considering the effect of localization deformation are established. And the bifurcation theory is adopted as the criterion of onset of strain localization. The model can describe the mechanism of pre-localization and post-localization continuously by regarding localization as a visco-plastic flow procedure. The band is considered as a virtual sub-element, in which Mohr-Coulomb's yield criterion is employed together with cohesion softening and cutoff for the post-localization behavior. Compared with the intense plastic deformation inside the band,the plastic deformation outside is neglected. Assumption that the plastic deformation is focus in the band, elastic unloading outside. The deformations of the band are regarded as the rigid deformations, which are distributed to nodes of the localized element in the proportion. Consequently, a discontinuous displacement band is formed in the initial continuous displacement field.The FE model is implemented into a FORTRAN code FEA-LOC using constant strain triangular (CST) elements and four node elements. The model is formulated using standard Galerkin finite element method. A band-tracing algorithm is implemented to track the propagation of the shear band. The formulation does not need extra parameters and require static condensation to be performed on the element level. So the main advantages of this algorithm are that the model needs little amount of computation, and it has exact physical meaning compared with other methods. Furthermore, it can be incorporated into the conventional finite element analysis procedure. Numerical results have been carried out to show the validity of the model, and demonstrate near mesh-independence.As mentioned above, when analyzing geomechanical structures with complex geometry and geomaterial behavior, the limit equilibrium method is unwieldy. So for the analysis of slop stability, the model of embedded localized softening band is implemented. Numerical results have been carried out to show the validity of the proposed model through simulating the propagation of strain localization band in the slope. This model not only can find the slip surface in the slope, but also simulate the whole process of deformation and provide the information of strain and stress. Meanwhile, however, the real factors are neglected, such as temperature, seepage and so on.