Cohesive-frictional Materials Research Articles

A B-spline based gradient-enhanced micropolar implicit material point method for large localized inelastic deformations

The quasi-brittle response of cohesive-frictional materials in numerical simulations is commonly represented by softening plasticity or continuum damage models, either individually or in combination. However, classical models, particularly when coupled with non-associated plasticity, often suffer from ill-posedness and a lack of objectivity in numerical simulations. Moreover, the performance of the finite element method significantly degrades in simulations involving finite strains when mesh distortion reaches excessive levels. This represents a challenge for modeling cohesive-frictional materials, given their tendency to experience strongly localized deformations, such as those occurring during shear band dominated failure. Hence, accurate modeling of the response of cohesive-frictional solids is a demanding task. To address these challenges, we present an extension of the material point method (MPM) for the unified gradient-enhanced micropolar continuum, aiming at the analysis of finite localized inelastic deformations in cohesive-frictional materials. The generalized gradient-enhanced micropolar continuum formulation is employed to tackle challenges related to localization and softening material behavior, while the MPM addresses issues arising from excessive deformations. The method utilizes a B-spline formulation for the rigid background mesh to mitigate the well-known cell crossing errors of the MPM. To demonstrate the performance of the method, 2D and 3D numerical studies on localized failure in sandstone in plane strain compression and triaxial extension tests are presented. A comparison with finite element results confirms the suitability of the formulation. Moreover, an efficient numerical implementation of the formulation is presented, and it is demonstrated that the additional MPM specific overhead is negligible.

Loading-unloading of spherical and cylindrical cavities in cohesive-frictional materials with arbitrary radially symmetric boundary conditions

Many scenarios in geotechnical engineering involve cavity unloading from a loaded state under different boundary conditions, especially for thick-walled cylinder tests in laboratory conditions. A novel analytical cavity expansion-contraction solution is proposed for modeling thick-walled cylinder tests during a complete loading-unloading process under arbitrary symmetric boundary conditions. The proposed solution has potential implication in investigating ambiguous boundary effects for problems involving cavity expansion-contraction. The mechanical model considers two concentric layers of materials around the cavity, where the inner layer is considered as cohesive-frictional and the circumambient infinite layer is assumed as elastic to simulate the arbitrary radially symmetric boundary condition, including two extreme scenarios with constant-pressure and zero-displacement boundaries. The derived solution provides the evolutions of cavity pressure and distributions of stresses and displacements in both layers during the loading-unloading process, which is then rigorously validated against finite element results. The effects of boundary conditions and size of the cohesive-frictional layer have been comprehensively investigated by analyzing the cavity pressure and the developments of plastic zones during expansion and contraction. The developed solution has been proved to be applicable in geotechnical engineering, with comparisons between analytical results and experimental data for pressuremeter tests in calibration chambers and pile installation tests in a rigid container. The proposed solution might also be helpful in the analysis of autofrettaged pressure vessels.

Regularization analysis of the strong discontinuity-Cosserat finite element method for modeling strain localization in cohesive-frictional materials by spectral theory

An inappropriate numerical approach adopted for modeling the strain localization of geomaterials often leads to ill-posed boundary value problems (BVP). In this article, we apply spectral analysis to study the regularization mechanism of Cosserat finite element method (CFEM) and strong discontinuity-Cosserat FEM (SD-CFEM). The 1D shear layer model and 2D plane strain tests are modeled to conduct the diagnostic spectral analysis. A 2D slope problem is simulated to demonstrate the effectiveness of SD-CFEM in simulating the characteristics of progressive deformation in geomaterials. The numerical results show that CFEM and SD-CFEM not only maintain the eigenvalues as positive, but also make the dominant eigenvectors for different mesh densities possess the identical evolution pattern which is the fundemental regularization mechanism of both methods. The deformation patterns reflected by dominant eigenvectors indicate that SD-CFEM is more robust in modeling strain localization/shear band than CFEM and SD-FEM do because the former can reflect the energy dissipation both in the shear band with a constant width and on the slip line in the shear band. It can be concluded from the displacement evolution process in the shear band that SD-CFEM inheriting the merits of CFEM and SD-FEM can physically and mathematically better simulate the whole deformation process of geomaterials from weak discontinuity (strain localization) to strong discontinuity (slip).

A unified finite strain gradient-enhanced micropolar continuum approach for modeling quasi-brittle failure of cohesive-frictional materials

In this work, a novel framework for modeling quasi-brittle crack propagation and shear band dominated failure of cohesive-frictional materials like concrete, mortar, rock, tough ceramics, energetic materials, but also granular materials like sands or powders in terms of a unified continuum approach is proposed. It is based on a combination of the gradient-enhanced continuum with gradients of internal variables for representing quasi-brittle cracking, and the micropolar continuum, accounting for the deformation of the microstructure. For developing the gradient-enhanced micropolar framework, the set of balance equations and the kinematic relations are derived, and the constitutive relations are established in a general manner. The framework is formulated in a geometrically exact setting, based on the thermodynamically sound theory of hyperelasto-plasticity, and the numerical implementation by means of the finite element method is discussed. For assessing the approach, realizations of this new approach in terms of constitutive models for particular materials are developed. They are applied to numerical benchmark examples, investigating various loading conditions, and the obtained results are validated by means of a comparison with experiments from the literature.

Determination of Rate-Dependent Properties in Cohesive Frictional Materials by Instrumented Indentation

The extraction of the elastoplastic constitutive behavior from instrumented sharp indentation is still a subject of research. Several approaches have been proposed to solve this problem, mainly based on the use of numerical techniques. This work proposes an inverse analysis approach based on dimensional analysis calibrated with finite element modeling, to extract the elastoplastic properties from instrumented sharp indentation in rate- and pressure-dependent materials, which is the typical behavior that most polymers exhibit. Pressure sensitivity was modeled with a Drucker-Prager yield criterion, while rate dependency was introduced through a power-law dependence on strain rate of the yield stress. A set of master curves is proposed that relate the experimental metrics of instrumented indentation tests with the parameters of the proposed material model. Furthermore, the analysis was experimentally validated by testing several materials, including PMMA, coarse grain copper, and ultrafine grain copper. The predictions of the inverse analysis correlated well with the known material properties.

An Extension Strain Type Mohr\u2013Coulomb Criterion

Extension fractures are typical for the deformation under low or no confining pressure. They can be explained by a phenomenological extension strain failure criterion. In the past, a simple empirical criterion for fracture initiation in brittle rock has been developed. In this article, it is shown that the simple extension strain criterion makes unrealistic strength predictions in biaxial compression and tension. To overcome this major limitation, a new extension strain criterion is proposed by adding a weighted principal shear component to the simple criterion. The shear weight is chosen, such that the enriched extension strain criterion represents the same failure surface as the Mohr–Coulomb (MC) criterion. Thus, the MC criterion has been derived as an extension strain criterion predicting extension failure modes, which are unexpected in the classical understanding of the failure of cohesive-frictional materials. In progressive damage of rock, the most likely fracture direction is orthogonal to the maximum extension strain leading to dilatancy. The enriched extension strain criterion is proposed as a threshold surface for crack initiation CI and crack damage CD and as a failure surface at peak stress CP. Different from compressive loading, tensile loading requires only a limited number of critical cracks to cause failure. Therefore, for tensile stresses, the failure criteria must be modified somehow, possibly by a cut-off corresponding to the CI stress. Examples show that the enriched extension strain criterion predicts much lower volumes of damaged rock mass compared to the simple extension strain criterion.

Strain Localization of Orthotropic Elasto-Plastic Cohesive-Frictional Materials: Analytical Results and Numerical Verification.

Strain localization analysis for orthotropic-associated plasticity in cohesive–frictional materials is addressed in this work. Specifically, the localization condition is derived from Maxwell’s kinematics, the plastic flow rule and the boundedness of stress rates. The analysis is applicable to strong and regularized discontinuity settings. Expanding on previous works, the quadratic orthotropic Hoffman and Tsai–Wu models are investigated and compared to pressure insensitive and sensitive models such as von Mises, Hill and Drucker–Prager. Analytical localization angles are obtained in uniaxial tension and compression under plane stress and plane strain conditions. These are only dependent on the plastic potential adopted; ensuing, a geometrical interpretation in the stress space is offered. The analytical results are then validated by independent numerical simulations. The B-bar finite element is used to deal with the limiting incompressibility in the purely isochoric plastic flow. For a strip under vertical stretching in plane stress and plane strain as well as Prandtl’s problem of indentation by a flat rigid die in plane strain, numerical results are presented for both isotropic and orthotropic plasticity models with or without tilting angle between the material axes and the applied loading. The influence of frictional behavior is studied. In all the investigated cases, the numerical results provide compelling support to the analytical prognosis.

A Plane Stress Failure Criterion for Inorganically-Bound Core Materials.

Inorganically-bound core materials are used in foundries in high quantities. However, there is no validated mechanical failure criterion, which allows performing finite-element calculations on the core geometries, yet. With finite-element simulations, the cores could be optimised for various production processes from robotic core handling to the decoring process after the casting. To identify a failure criterion, we propose testing methods, that enable us to investigate the fracture behaviour of inorganically-bound core materials. These novel testing methods induce multiple bi-axial stress states into the specimens and are developed for cohesive frictional materials in general and for sand cores in particular. This allows validating failure criteria in principal stress space. We found that a Mohr-Coulomb model describes the fracture of inorganic core materials in a plane stress state quite accurately and adapted it to a failure criterion, which combines the Mohr-Coulomb model with the Weakest-Link theory in one consistent mechanical material model. This novel material model has been successfully utilised to predict the fracture force of a Brazilian test. This prediction is based on the stress fields of a finite element method (FEM) calculation.

A 3D gradient-enhanced micropolar damage-plasticity approach for modeling quasi-brittle failure of cohesive-frictional materials

Continuum models based on the combination of the theories of plasticity and damage mechanics pose a powerful framework for representing the highly nonlinear material behavior of cohesive-frictional materials. However, non-associated plastic flow rules for representing the inelastic volumetric expansion of such materials may result in unstable material behavior and, accordingly, strongly mesh-dependent results in finite element simulations. Regularization techniques such as the gradient-enhanced continuum or similar nonlocal approaches, which work well for regularizing mode I failure, are often not sufficient as a remedy. In contrast to the latter, the theory of the micropolar continuum represents a suitable framework for regularizing non-associated plastic flow and shear band dominated failure properly, but it fails to do so for mode I failure. Hence, in the present contribution, a combination of the theories of the micropolar continuum and the gradient-enhanced continuum for regularizing both shear band dominated failure and mode I failure is presented. By incorporating a 3D damage-plasticity model for concrete into the proposed framework, it is demonstrated that the proposed model constitutes a physically sound and numerically stable approach for modeling the nonlinear material behavior of concrete in both the pre-peak and the post-peak regime for a broad variety of loading conditions.

Indentation hardness of the cohesive-frictional materials

An analytical model is proposed to correlate the indentation hardness with the cohesion and friction angle of a cohesive-frictional material. The model is based on the slip-line analysis of a conical (self-similar) indenter that is forced to penetrate an elastic-perfectly plastic solid of the Mohr-Coulomb type. Using Berkovich indenter as an example, we have validated the proposed model based on the numerical analyses of 50 different materials. It shows that the model predicts an accurate hardness-to-cohesion ratio for a material if its friction angle is less than or equal to 45° When the friction angle of the material is larger than 45°, the model overestimates the hardness-to-cohesion ratio. Finally, we have discussed the major reasons that cause the overestimation of the hardness-to-cohesion ratio for larger friction angles.

Shakedown analysis of cavities in cohesive-frictional materials and its application to underground energy storage caverns

In this paper, a shakedown analysis is conducted for both cylindrical and spherical cavities in cohesive-frictional materials subjected to cyclic internal pressures. Based on Melan’s lower-bound shakedown theorem, theoretical shakedown conditions for a two-layered cavity system are developed using the stress and strain solutions obtained from cavity expansion and contraction analyses. The shakedown limits are investigated in relation to the dimensions of the two-layered cavity system and the elastic/plastic material parameters. Numerical validation is conducted through a comparison of the stress path of the critical element and the failure lines of the materials. In addition, closed-form solutions for special cases, namely, the analytical shakedown conditions of cavities in an unbound cohesive-frictional material and in a bound material with different boundary conditions, are explicitly expressed. Finally, the shakedown solutions are applied to solve the thickness design problem of underground compressed air energy storage in lined and unlined caverns.

An accurate nonlocal bonded discrete element method for nonlinear analysis of solids: application to concrete fracture tests

We present a numerical procedure for elastic and nonlinear analysis (including fracture situations) of solid materials and structures using the discrete element method. It can be applied to strongly cohesive frictional materials such as concrete and rocks. The method consists on defining nonlocal constitutive equations at the contact interfaces between discrete particles using the information provided by the stress tensor over the neighbor particles. The method can be used with different yield surfaces, and in the paper, it is applied to the analysis of fracture of concrete samples. Good comparison with experimental results is obtained.

Smoothed finite element approach for kinematic limit analysis of cohesive frictional materials

By employing different smoothed finite element (S-FE) methods, the kinematic limit analysis approach has been presented by using three noded triangular elements to solve plane strain and plane stress stability problems on basis of the Mohr-Coulomb yield criterion. By using the concept of duality in conic programming, the expressions for the power dissipation function and the associated constraints have been written entirely in terms of kinematic variables. The optimization formulation is framed as a second order cone programming problem. Based on the absolute maximum magnitude of the shear strain rate, an algorithm to update the mesh has been provided. Amongst the different approaches, the node-based S-FE formulation has been found to be very accurate as well as efficient to solve large scale stability problems. The selective edge-based and the edge-based with a bubble node approaches also generate quite accurate solutions though require a little more computational time.

Application of three-dimensional shakedown solutions in railway structure under multiple Hertz loads

Shakedown analysis is a robust approach to solve the strength problem of a structure under cyclic or repeated loading, e.g. railway structure subjects to rolling and sliding loads. A three-dimensional analytical shakedown solution is developed in this paper for the strength analysis of cohesive-frictional materials under multiple Hertz loads, which is then applied into the shakedown analysis of railway structure. Different to the pavement, the railway structure subjects to multiple wheel loads acting on the surface of two parallel rails, and then transformed to the substructure, which leads to different elastic and residual stress fields. For the application of Melan's shakedown theorem, the residual stress field in railway strucuture is rationally simplified to find the most critical plane that influence the most on the shakedown limit. The elastic stresses is then analytically derived for a single layer half-space under multiple Hertz loads, and is obtained numerically by means of the finite element technique for the multi-layered railway structure. Finally, the shakedown limit can be obtained in a direct way. Parametric study indicates how wheel loads, material properties such as frictional coefficient, friction angle and Poisson's ratio, and multi-layers influence the shakedown limit and stability condition of railway structure. It is found that the shakedown load increases with the increase of cohesion ratio, ballast frictional angle, and thickness ratio. However, the increase of the shakedown load ceases once the failure location moves from ballast layers to sub-ballast layers. This indicates that there is an optimum combination of material properties and layer thickness which provides the maximum resistance to the train loads. The obtained results give a useful reference for the engineering design of the railway structure.

Comparison of a Material Point Method and a Galerkin Meshfree Method for the Simulation of Cohesive-Frictional Materials.

The simulation of large deformation problems, involving complex history-dependent constitutive laws, is of paramount importance in several engineering fields. Particular attention has to be paid to the choice of a suitable numerical technique such that reliable results can be obtained. In this paper, a Material Point Method (MPM) and a Galerkin Meshfree Method (GMM) are presented and verified against classical benchmarks in solid mechanics. The aim is to demonstrate the good behavior of the methods in the simulation of cohesive-frictional materials, both in static and dynamic regimes and in problems dealing with large deformations. The vast majority of MPM techniques in the literatrue are based on some sort of explicit time integration. The techniques proposed in the current work, on the contrary, are based on implicit approaches, which can also be easily adapted to the simulation of static cases. The two methods are presented so as to highlight the similarities to rather than the differences from “standard” Updated Lagrangian (UL) approaches commonly employed by the Finite Elements (FE) community. Although both methods are able to give a good prediction, it is observed that, under very large deformation of the medium, GMM lacks robustness due to its meshfree natrue, which makes the definition of the meshless shape functions more difficult and expensive than in MPM. On the other hand, the mesh-based MPM is demonstrated to be more robust and reliable for extremely large deformation cases.

Bonding degradation and stress–dilatancy response of weakly cemented sands

ABSTRACTThe progressive bond breakage of artificially cemented sands induced by shear straining was investigated through conventional isotropically consolidated drained triaxial compression tests. Sand specimens were prepared with a low degree of cementation by adopting a chemical grout. Test results were interpreted in terms of two stress–dilatancy theories for cohesive-frictional materials proposed in literature. The influence of debonding on the stress–dilatancy behaviour of cemented sands was analysed with particular emphasis on the ‘delayed dilatancy’ phenomenon. A bonding degradation curve was determined for each test relating the interparticle cohesion (c) to the magnitude of the total plastic strain vector (εd) and a bond degradation rate factor (Dc) was assessed from each curve. The maximum value of interparticle cohesion (c0) before the onset of bond degradation under shearing was found to correspond with a sharp decrease in the soil stiffness of the specimens. The influence of the effective confining stress (p′c) on both c0 and Dc parameters gathered from each test was also ascertained.

An Energy-Equivalent d⁺/d- Damage Model with Enhanced Microcrack Closure-Reopening Capabilities for Cohesive-Frictional Materials.

In this paper, an energy-equivalent orthotropic d+/d− damage model for cohesive-frictional materials is formulated. Two essential mechanical features are addressed, the damage-induced anisotropy and the microcrack closure-reopening (MCR) effects, in order to provide an enhancement of the original d+/d− model proposed by Faria et al. 1998, while keeping its high algorithmic efficiency unaltered. First, in order to ensure the symmetry and positive definiteness of the secant operator, the new formulation is developed in an energy-equivalence framework. This proves thermodynamic consistency and allows one to describe a fundamental feature of the orthotropic damage models, i.e., the reduction of the Poisson’s ratio throughout the damage process. Secondly, a “multidirectional” damage procedure is presented to extend the MCR capabilities of the original model. The fundamental aspects of this approach, devised for generic cyclic conditions, lie in maintaining only two scalar damage variables in the constitutive law, while preserving memory of the degradation directionality. The enhanced unilateral capabilities are explored with reference to the problem of a panel subjected to in-plane cyclic shear, with or without vertical pre-compression; depending on the ratio between shear and pre-compression, an absent, a partial or a complete stiffness recovery is simulated with the new multidirectional procedure.

FEM × DEM: a new efficient multi-scale approach for geotechnical problems with strain localization

The paper presents a multi-scale modeling of Boundary Value Problem (BVP) approach involving cohesive-frictional granular materials in the FEM × DEM multi-scale framework. On the DEM side, a 3D model is defined based on the interactions of spherical particles. This DEM model is built through a numerical homogenization process applied to a Volume Element (VE). It is then paired with a Finite Element code. Using this numerical tool that combines two scales within the same framework, we conducted simulations of biaxial and pressuremeter tests on a cohesive-frictional granular medium. In these cases, it is known that strain localization does occur at the macroscopic level, but since FEMs suffer from severe mesh dependency as soon as shear band starts to develop, the second gradient regularization technique has been used. As a consequence, the objectivity of the computation with respect to mesh dependency is restored.

THE STABILITY OF SLOPE USING CELL – BASED SMOOTHED FINITE ELEMENT METHOD AND SECOND ORDER CONE PROGRAMMING

In this paper, the numerical limit analysis procedure, associating the cell-based smoothed finite element method (CS-FEM) with the (second-order cone) primal-dual interior point algorithm, for cohesive-frictional materials problem is described. The soil is modeled as a cohesionless frictional Mohr-Coulomb material with the associated flow rule. Kinematically admissible velocity fields are established using CS-FEM. The underlying non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second order cone programming algorithm. The core purpose of this study is to evaluate collapse loads as well as failure mechanisms of footings on slope which will be obtained directly from solving the optimization problems. In this study, the properties of soil and the width of footing and distance from footing to the edge of the slope are considered. Several numerical examples of slope stability are given to show the performance of the proposed method.

Mechanical behavior of a composite interface: Calcium-silicate-hydrates

The generalized stacking fault (GSF) is a conceptual procedure historically used to assess shear behavior of defect-free crystalline structures through molecular dynamics or density functional theory simulations. We apply the GSF technique to the spatially and chemically complex quasi-layered structure of calcium-silicate-hydrates (C-S-H), the fundamental nanoscale binder within cementitious materials. A failure plane is enforced to calculate the shear traction-displacement response along a composite interface containing highly confined water molecules, hydroxyl groups, and calcium ions. GSF simulations are compared with affine (homogeneous) shear simulations, which allow strain to localize naturally in response to the local atomic environment. Comparison of strength and deformation behavior for the two loading methods shows the composite interface controls bulk shear deformation. Both models indicate the maximum shear strength of C-S-H exhibits a normal-stress dependency typical of cohesive-frictional materials. These findings suggest the applicability of GSF techniques to inhomogeneous structures and bonding environments, including other layered systems such as biological materials containing organic and inorganic interfaces.