Multilaminate Framework Research Articles

Probabilistic Evaluation of Geomechanical Risks in CO2 Storage: An Exploration of Caprock Integrity Metrics Using a Multilaminate Model

The probabilistic uncertainty assessment of geomechanical risk—specifically, caprock failure—attributable to CO2 injection, as presented in a simplified hypothetical geological model, was the focus of this study. Our approach amalgamates the implementation of a multilaminate model, the creation of a response surface model in conjunction with the Box–Behnken sampling design, the execution of associated numerical modeling experiments, and the utilization of Monte Carlo simulations. Probability distributions to encapsulate the inherent variability (elastic and mechanical properties of the caprock and reservoir) and uncertainty in prediction estimates (vertical displacement, total strain, and F value) were employed. Our findings reveal that the Young modulus of the caprock is a key factor controlling equivalent total strain but is insufficient as a stand-alone indicator of caprock integrity. It is confirmed that the caprock can accommodate significant deformation without failure, if it possesses a low Young’s modulus and high mechanical strength properties, such as the friction angle and uniaxial compressive strength. Similarly, vertical displacement was found to be an unreliable indicator for caprock integrity, as caprock failure can occur across a broad spectrum of vertical displacements, particularly when both the Young modulus and mechanical strength properties have wide ranges. This study introduces the F value as the most dependable indicator for caprock failure, although it is a theoretical attribute (the shortest distance between the Mohr circle and the nearest failure envelope used to measure the sensitivity to failure) and not physically measurable in the field. Deviatoric stress levels were found to vary based on stress regimes, with the maximum levels observed under extensive and compressive stress regimes. In conjunction with the use of the response surface method, this study demonstrates the efficacy of the multilaminate framework and the Mohr–Coulomb constitutive model in providing a simplified, yet effective, probabilistic model of the mechanical behavior of caprock failure, reducing mathematical and computational complexities.

A state-dependent multilaminate constitutive model for anisotropic sands

Experimental studies show that initial fabric and its evolution under different stress paths greatly influences soil behaviour. Even though different sample preparation methods create different inherent anisotropies and cause different material responses, the same initial fabric structure under different stress paths also results in different material behaviours. In this paper, a simple state-dependent, bounding surface-based elastoplastic constitutive model, which can simulate the anisotropic nature of sands including the effect of principal stress rotation, is described. The model is developed based on a semi-micromechanical concept within the multilaminate framework and, to include the inherent anisotropy of sand, a deviatoric fabric tensor describing the initial microstructure is introduced. In addition, a fabric evolution rule compatible with anisotropic critical state theory is employed to describe the evolving fabric structure and induced anisotropy towards the critical state. In contrast to the classical strain-driven formulation for fabric evolution, a micro-level evolution rule is proposed. This paper presents concise theoretical aspects of the multilaminate framework and the anisotropic elastoplastic constitutive formulation. The model's capability under drained and undrained monotonic loading conditions at different stress states, relative densities and principal stress orientations is demonstrated by simulating experimental data for Toyoura sand.

Cyclic stress–dilatancy relations and plastic flow potentials for soils based on hypothesis of complementarity of stress–dilatancy conjugates

Soil, rocks and rock masses dilate or compact when sheared, i.e., distortion necessitates volume change. This coupling between distortional strains and volumetric strains, described by stress–dilatancy theories, endows soils with manifestation of peculiar characteristics when they are subjected to shear. Stress–dilatancy theories have become central in describing the mechanical energy dissipation mechanism and further establishing flow rules in constitutive modelling of soils. The classical stress–dilatancy theories, such as Taylor's and Rowe's, are endowed with simplicity and descriptive power, but they were developed for describing the dilatancy behaviour of soils subjected to loading in shear (mobilizing away from isotropic stress state) and needed to be extended for describing plastic dissipation and shear-induced volumetric changes when soils are subjected to cyclic shear. In this paper, hypothesis of complementarity of stress–dilatancy conjugates is proposed as a unifying hypothesis for deriving stress–dilatancy relations for both loading in shear and unloading in shear. Then, plastic potential functions are derived based on the resulting stress–dilatancy relations. In so doing, the resulting stress–dilatancy relations and plastic potential functions are rendered with a quality to be used for the modelling of deformation behavior of soils subjected to both monotonic and cyclic shearing. The theoretical framework is applied first for plane strain and axisymmetric stress–strain conditions; and then extended for the general stress condition considering the Lode angle dependency of the shear strength of soils, using the multilaminate framework and applying the Matsuoka–Nakai spatial mobilized plane.

Numerical analysis of cyclic loading effect on progressive failure of an earth dam upon a multi-laminate framework

Numerical analysis of cyclic loading effect on progressive failure of an earth dam upon a multi-laminate framework

Behaviour of ballast under principal stress rotation: Multi-laminate approach for moving loads

Abstract Railway tracks are subjected to millions of loading cycles over time and at high speeds, these moving trains induce dynamic amplification of vertical stresses and rotation of principal stress axes in the track layers. It is important to predict and analyse the behaviour of ballast under these loads with complex stress paths involving principal stress rotation. In this paper, a constitutive model based on a multi-laminate framework is used to predict the deformation and degradation of ballast under complex stress paths. The yield and plastic potential surfaces are developed based on a non-linear critical state and bounding surface plasticity concepts. The proposed model is validated with independent test data to capture the influence of confining stress, loading frequency, Cyclic Stress Ratio ( CSR ) and Shear Stress Ratio ( η τ ) on the permanent strain response. Furthermore, the response of ballast under traffic loading stress paths with different CSR and η τ is analysed. These model predictions show that higher CSR and η τ values lead to exacerbated particle breakage of ballast, large and unstable axial strains and dilatant volumetric strains. Furthermore, a stability surface is proposed based on model predictions, to estimate the allowable CSR and η τ for a stable response.

A novel semi-micro multilaminate elasto-plastic model for the liquefaction of sand

After the liquefaction of sand, the prediction of anisotropy and heterogeneity is one of the complexities of constitutive law. This study aimed to develop a method to more effectively assess anisotropy and strain and stress distributions, and determine their history in cohesionless soil. To achieve this objective, instead of defining all the direction-dependent events on the three orthogonal planes of the Cartesian coordinate system, numerical integration was utilized to make use of 17 planes with pre-defined directions. This leads to a more accurate and powerful assessment of anisotropy and its effects. The constitutive equations of the proposed model were adjusted with a multilaminate framework, and its result for different monotonic and cyclic loading, drained and undrained conditions, and different pressures and void ratios were verified using the experimental data. Finally, the model's performance in predicting induced anisotropy is demonstrated under cyclic mobility conditions on the 17 planes.

Soil–Pile Interaction Analysis Using Multi-laminate Elasto-Plastic Modelling

The safe and economical design of pile foundations are particularly important due to their usage in sensitive structures. This research aims to investigate soil–pile interaction using the numerical method and will provide greater familiarity with force distribution in deformation mechanisms in the soil as well as soil–pile interaction. Mohr–Coulomb model was used as the behavioral pattern. This model poorly applies the effects of soil anisotropy on mathematical relations. Therefore, this constitutive model was formulated within the multi-laminate framework which is of the great importance and value among different methods for material behavior modeling. This model can predict complex behavioral features, the effect of the principal stress axes rotation during plastic deformations, and prediction capability of the failure mechanism direction. The model is capable of predicting softening behavior of the soil in a reasonable manner due to using unconventional plasticity. The influences of induced anisotropy are included in a rational way without any additional hypotheses owing to natural properties of the multi-laminate framework. Results of this model were compared with test data and reasonable agreement was found. In this research, a pile sample was considered with its characteristics examined using a computer program within the framework of non-linear finite element method and multi-laminate theory. Calibrating the appropriate parameters of contact surfaces, the soil–pile interaction was estimated. Finally using the presented curves for planes, soil deformation, strain and tension controlling mechanism were identified to determine the shear failure route and critical areas with the risk of slip occurring at the pile shaft–soil interface.

In-plane Failure Analysis of URM Structures Based on Strain Hardening and Softening in the Multilaminate Framework

In-plane Failure Analysis of URM Structures Based on Strain Hardening and Softening in the Multilaminate Framework

Numerical evaluation of geomaterials behavior upon multiplane damage model

Among the various numerical simulating constitutive models for geomaterials, the constitutive models which formulated within multilaminate framework have an excellent position. The major advantage of these models is because of its simplicity of working instead of complicated microscopic models such as discrete particles models and do not have the shortcomings of macroscopic models based on the stress or strain invariants.The object of this study is to deal with the problem exists in the present multilaminate models built on a numerical integration upon 13 sampling points, which both compatibility and equilibrium conditions are not simultaneously satisfied. To overcome this problem, a logical 17 sampling points of numerical integration is presented. The obtained results show a good match with experimental tests.

Influence of Anisotropic Small Strain Stiffness on the Deformation Behavior of Geotechnical Structures

AbstractConstitutive models based on the multilaminate framework naturally account for strain-induced anisotropy, but inherent anisotropy needs additional considerations. A recently introduced feature is the capability of the model to take into account anisotropic elasticity in the small strain range. The implications of taking into account anisotropy of small strain stiffness for the analysis of geotechnical problems is addressed in this paper by solving two relatively simple benchmark problems: a deep excavation and a strip footing. It follows from these studies that, if displacements and thus strains remain moderate (which is often the case for these types of problems), the influence of small strain stiffness anisotropy is more pronounced compared with problems where relatively large strains occur. However, the influence of stiffness anisotropy can be approximately accounted for by using the average of the axial stiffnesses.

A multilaminate constitutive model accounting for anisotropic small strain stiffness

SUMMARYIn this paper an extension of existing multilaminate soil models is presented, which can account for inherent and stress‐induced cross‐anisotropic elasticity in the small strain range and its dependency on the load history. In the multilaminate framework, material behaviour is formulated on a number of local planes in each stress point, and the macroscopic response of the material is obtained by integration of the local contributions. Strain‐induced anisotropy, which adds to the stiffness anisotropy inherently present in the material, is therefore intrinsically taken into account. Micro–macro relations between local parameters on plane level and global parameters on macroscopic level are obtained by the spectral decomposition of the global elastic compliance matrix. The model is implemented into a finite‐element code, and model predictions are compared with experimental data of triaxial tests on different soils involving small and large load cycles. The importance of cross‐anisotropic elasticity within the small strain range for predicting ground deformations in geotechnical boundary value problems is discussed at the example of an excavation problem. Copyright © 2012 John Wiley & Sons, Ltd.

Nonlocal Multilaminate Model for Strain Softening Analysis

A constitutive model formulated within the multilaminate framework is described in this paper. The basic model, incorporating deviatoric and volumetric hardening, was developed for normally consolidated or slightly overconsolidated clay. This model has been extended to account for shear softening by using a nonlocal formulation. The nonlocal plasticity is based on weighted average of plastic strains taken from neighboring material points within a certain distance which is a function of an internal length parameter. This parameter is related to the mean grain size that restricts the element size. By using a softening scaling this restriction can be eliminated and the model is therefore capable of simulating geotechnical boundary value problems. The capability of the model for simulating strain softening behavior is shown by numerical simulation of biaxial tests. Finally, the ground response curve obtained from the proposed model for excavation of a tunnel in soil is discussed.

A Multilaminate Model with Destructuration Considering Anisotropic Strength and Anisotropic Bonding

Many constitutive models have been developed to describe the behaviour of soils, but only more recently a limited number of them have paid attention to include effects of structure and consequently destructuration due to volumetric and/or deviatoric straining. Most of them consider isotropic behaviour for strength and structure of soil. In this paper a multilaminate type of constitutive model is presented in which anisotropic strength and anisotropic structure are taken into account. According to the mathematical formulation of the multilaminate framework, this kind of model is capable of taking into account induced anisotropy intrinsically. Inherent anisotropy can be introduced to the model by varying the model properties over sampling planes using a so-called microstructure tensor. Only mechanical destructuration, namely degradation of structure with deformation is considered in this paper. The proposed model is capable to consider anisotropy in bonding and preconsolidation pressure as well as inherent anisotropy in strength thus the hardening of the yield surface and strength parameters can be considered direction dependent. In this paper the drained and undrained behaviour of natural clays is simulated and the effect of different assumptions with respect to structure degradation and anisotropy is evaluated.

A multilaminate framework for modelling induced and inherent anisotropy of soils

In this paper the basic features of a constitutive model for normally and lightly overconsolidated soils, based on the multilaminate framework, are discussed. Multilami-nate models simulate the stress–strain behaviour of a material by considering the response on so-called integration or sampling planes. Yield and plastic potential functions are expressed in terms of normal and shear stresses on these integration planes, and thus the mathematical formulations remain relatively simple even for complex strain-hardening/softening models. The model includes a deviatoric yield surface with a non-associated flow rule and a volumetric yield surface with an associated flow rule. Induced anisotropy and the effect of rotation of principal stress axes are intrinsically taken into account in multilaminate models without requiring additional material parameters. Inherent anisotropy can be modelled by introducing a structural tensor. A slight disadvantage of the multilaminate approach is that no function for a yield surface in three-dimensional stress space exists, and therefore comparison with other constitutive models is difficult on a visual basis. However, it is shown that the model produces approximately a Mohr–Coulomb failure surface in the deviatoric plane, which can be easily modified to incorporate anisotropic behaviour with respect to strength, providing a significant extension of the model. The influence of the integration rule on the obtained failure surface is discussed. Comparison with experimental data from a comprehensive series of stress-path-controlled triaxial tests on Poko clay shows the capability of the approach for modelling the mechanical behaviour of soft clays. The significant importance of the formulation of the flow rule on the model's performance for undrained triaxial stress paths is discussed by comparison with experimental data. Finally some results from a slope stability analysis are presented.

On a multilaminate model for soil incorporating small strain stiffness

AbstractIn this paper a constitutive model for soils incorporating small strain stiffness formulated in the multilaminate framework is presented. In the multilaminate framework, the stress–strain behaviour of a material is obtained by integrating the mechanical response of an infinite number of randomly oriented planes passing through a material point. Such a procedure leads to a number of advantages in describing soil behaviour, the most significant being capture of initial and induced anisotropy due to plastic flow in a physically meaningful manner. In the past, many soil models of varying degree of refinement in the multilaminate framework have been presented by various authors. However, the issue of high initial soil stiffness in the range of very small strains and its degradation with straining, commonly referred to as ‘small strain stiffness’, has not been addressed within the multilaminate framework. In this paper, we adopt a micromechanics‐based approach to derive small strain elastic stiffness of the soil mass. Comparison of laboratory test data with results obtained from numerical simulations based on the proposed constitutive model incorporating small strain stiffness is performed to demonstrate its predictive capabilities. The model is implemented in a finite element code and numerical simulations of a deep excavation are presented with and without incorporation of small strain stiffness to demonstrate its importance in predicting profiles of deformation. Copyright © 2008 John Wiley & Sons, Ltd.

Application of a multilaminate model to simulate the undrained response of structured clay to shield tunnelling

A constitutive model based on the multilaminate framework has been implemented into a finite element program to investigate the effect of soil structure on the ground response to tunnelling. The model takes into account the elastic unloading–reloading, inherent and induced anisotropy, destructuration, and bonding effects. The model is successfully calibrated and used to investigate the undrained response of structured sensitive clay in the construction of the Gatineau tunnel in Gatineau, Quebec. Numerical results were compared to the field measurements taken during tunnel construction. To improve the performance of the numerical model, an implicit integration algorithm is implemented and proven to be very effective when coupled with the multilaminate framework as compared to the conventional explicit integration methods. The effect of different soil parameters including bonding and anisotropy on the tunnelling induced displacements and lining stresses is also examined using a comprehensive parametric study. The results indicated that soil bonding and anisotropy have significant effects on the shape of the settlement trough as well as the magnitudes of surface displacements and lining stresses induced by tunnelling.

Modelling the behaviour of an embankment on soft clay with different constitutive models

The paper investigates the effect of constitutive models on the predicted response of a simplified benchmark problem, an embankment on soft soil. The soft soil is assumed to have the properties of POKO clay from Finland and five different constitutive models are used to model the deposit. Two of the models are isotropic models, i.e. the Modified Cam Clay model and the Soft-Soil model. The other models are recently proposed constitutive models that account for plastic anisotropy. The S-CLAY1 and S-CLAY1S models are embedded in a standard elasto-plastic framework and account for anisotropy via a rotational hardening law. In addition, the S-CLAY1S model accounts for bonding and destructuration. In contrast, the Multilaminate Model for Clay (MMC) accounts for plastic anisotropy by utilizing so-called multilaminate framework. The results of numerical simulations show that accounting for anisotropy results in notable differences in the predicted settlements and horizontal movements compared to the predictions using the isotropic models. There are also significant differences in the K0 predictions by the different constitutive models and this has a significant impact on the results. Copyright © 2006 John Wiley & Sons, Ltd.

On the modelling of anisotropy and destructuration of soft clays within the multi-laminate framework

The possibilities and limitations of modelling the behaviour of soft natural clays within the multi-laminate framework are discussed in this paper. The presented version of the constitutive model was developed to describe two important characteristics of the soft clay behaviour: strength anisotropy and destructuration. Viscous effects are not taken into account. Strength anisotropy of the soft clay is achieved using the multi-laminate model with controlled spatial distribution of the preconsolidation pressure over the sampling planes. This spatial distribution is related to the initial stress state and is additionally adjusted by the distribution of bonding. Bonding or structural anisotropy is not permanent in the model and may be reduced during the process of destructuration. Interrelation between stress-induced and bonding anisotropy is shown by investigating the shape changes of a macro yield surface resulting from the multi-laminate modelling. Performance of the new model is demonstrated by standard element tests and also by the boundary value problem of an embankment on soft ground.

Description of soil anisotropy based on multi‐laminate framework

AbstractIn this paper, a description of inherent anisotropy of soil, based on multi‐laminate framework, is presented. The problem is formulated by assuming that the conditions at failure are governed by Coulomb criterion in which both the cohesion and the angle of internal friction are functions of the orientation of the sample. The distribution of these parameters is described by employing a set of symmetric second‐order tensors. The formulation of the problem is illustrated by some numerical examples. In particular, the directional dependence of the strength characteristics is examined for samples subjected to both plane strain and triaxial compression. Copyright © 2001 John Wiley & Sons, Ltd.

Multi-laminate framework of soil models — plasticity formulation : Pietruszckak, S; Pande, G N Int J Num Anal Meth GeomechV11, N6, Nov–Dec 1987, P651–658

Multi-laminate framework of soil models — plasticity formulation : Pietruszckak, S; Pande, G N Int J Num Anal Meth GeomechV11, N6, Nov–Dec 1987, P651–658