Nonlinear Strength Criterion Research Articles

An anisotropic unified nonlinear strength criterion (AUNS) for geomaterials

Due to depositional processes and other factors, such as shear banding and the intermediate principal stress, soils and rocks are typically cross-anisotropic. Strength anisotropy should be described by the proper anisotropic strength criterion. This paper presents an anisotropic unified nonlinear strength criterion to accurately characterize the strength anisotropy of geomaterials based on a unified nonlinear strength criterion (UNS). The anisotropic state variable in terms of a joint invariant and a unit stress tensor is brought into the UNS criterion to characterize the effect of the anisotropy. The new criterion had been illustrated in both the meridian and octahedral planes, leading it the oversimplification to describe strength anisotropy behaviors of soils and rocks under different loading conditions. Moreover, only conventional triaxial test is employed to calibrate the parameters of the new criterion. The results demonstrated the ability of the anisotropic strength criterion to be applied to a wide range of soils and rocks for true triaxial tests along with the complex stress condition in torsion shear tests. By comparing the predictions to test results reported in the literature, the new criterion proposed in this paper is able to provide excellent traits of the strength anisotropy for various geomaterials.

A shear hardening plasticity model with nonlinear shear strength criterion for municipal solid waste

A shear hardening plasticity model with nonlinear strength criterion and appropriate stress-dilatancy description is presented to numerically study the bearing behavior of landfills. Simulations of drained triaxial tests are performed to illustrate the capability of the model in capturing the mechanical behavior of municipal solid waste. The bearing response and failure mode of a landfill foundation are determined using 2D and 3D finite element simulations. In contrast to the general shear failure mode obtained from rigid plasticity, the failure obtained from the hardening model shows a local mode, and prior to failure, a larger reaction force is produced for the same settlement.

Centrifuge model test and limit equilibrium analysis of the stability of municipal solid waste slopes

We have investigated the stability of a municipal solid waste (MSW) slope by the centrifuge model test and limit equilibrium analysis. The mechanical behavior of artificial MSW was studied using consolidated drained triaxial tests, and the shear strength was obtained and described by a power form failure criterion. The deformation pattern and failure mode were obtained from the movement of the monitoring point of the slope during the acceleration process in the centrifuge model test. The safety factor of the slope was calculated by the limit equilibrium method with a power form nonlinear strength criterion. Comparison of the results of the centrifuge model test and the calculated results using the Mohr–Coulomb criterion proved that the nonlinear strength criterion was a reasonable method to determine the location of the slip surface and safety factor. The Bulbul landfill failure was used as a case study to verify the applicability of the proposed method in the real-world case, and reasonable results were obtained.

An approximate nonlinear modified Mohr-Coulomb shear strength criterion with critical state for intact rocks

In this paper, the Mohr-Coulomb shear strength criterion is modified by mobilising the cohesion and internal friction angle with normal stress, in order to capture the nonlinearity and critical state concept for intact rocks reported in the literature. The mathematical expression for the strength is the same as the classical form, but the terms of cohesion and internal friction angle depend on the normal stress now, leading to a nonlinear relationship between the strength and normal stress. It covers both the tension and compression regions with different expressions for cohesion and internal friction angle. The strengths from the two regions join continuously at the transition of zero normal stress. The part in the compression region approximately satisfies the conditions of critical state, where the maximum shear strength is reached. Due to the nonlinearity, the classical simple relationship between the parameters of cohesion, internal friction angle and uniaxial compressive strength from the linear Mohr-Coulomb criterion does not hold anymore. The equation for determining one of the three parameters in terms of the other two is supplied. This equation is nonlinear and thus a nonlinear equation solver is needed. For simplicity, the classical linear relationship is used as a local approximation. The approximate modified Mohr-Coulomb criterion has been implemented in a fracture mechanics based numerical code FRACOD, and an example case of deep tunnel failure is presented to demonstrate the difference between the original and modified Mohr-Coulomb criteria. It is shown that the nonlinear modified Mohr-Coulomb criterion predicts somewhat deeper and more intensive fracturing regions in the surrounding rock mass than the original linear Mohr-Coulomb criterion. A more comprehensive piecewise nonlinear shear strength criterion is also included in Appendix B for those readers who are interested. It covers the tensile, compressive, brittle-ductile behaviour transition and the critical state, and gives smooth transitions.

Non-linear Shear Strength Criterion for a Rock Joint with Consideration of Friction Variation

Under shearing, joint asperities get sheared off and damaged. Moreover, shearing and sliding and the interaction between normal and shear stresses occur simultaneously. The nonlinearity of the shear strength envelope is closely related to the joint material, normal stress level, and morphological characteristics. This study analyzed the correlation of the friction angle with the normal stress level to overcome overestimation or underestimation at relatively low or high normal stress levels in the JRC–JCS empirical model. A hyperbolic function was adopted to describe the degradation of friction for different normal stress levels, and the modified non-linear shear criterion $$\tau = \sigma_{\text{n}} \tan (\phi_{\text{r}} + \Delta \phi /(1 + a\sigma_{\text{n}} /{\text{JCS}}))$$ was proposed. Furthermore, the proposed model avoids direct connection to the surface morphology, which is convenient for practical use. Statistical analysis of the experimental data and comparison with the JRC–JCS model verified the validity of the proposed model and present an excellent method for characterizing the non-linearity of the failure envelope for the rock joint.

An Elasto-Plastic Damage Model for Rocks Based on a New Nonlinear Strength Criterion

The strength and deformation characteristics of rocks are the most important mechanical properties for rock engineering constructions. A new nonlinear strength criterion is developed for rocks by combining the Hoek–Brown (HB) criterion and the nonlinear unified strength criterion (NUSC). The proposed criterion takes account of the intermediate principal stress effect against HB criterion, as well as being nonlinear in the meridian plane against NUSC. Only three parameters are required to be determined by experiments, including the two HB parameters σc and m i . The failure surface of the proposed criterion is continuous, smooth and convex. The proposed criterion fits the true triaxial test data well and performs better than the other three existing criteria. Then, by introducing the Geological Strength Index, the proposed criterion is extended to rock masses and predicts the test data well. Finally, based on the proposed criterion, a triaxial elasto-plastic damage model for intact rock is developed. The plastic part is based on the effective stress, whose yield function is developed by the proposed criterion. For the damage part, the evolution function is assumed to have an exponential form. The performance of the constitutive model shows good agreement with the results of experimental tests.

Improved Nonlinear Strength Criterion for Jointed Rock Masses Subject to Complex Stress States

Improved Nonlinear Strength Criterion for Jointed Rock Masses Subject to Complex Stress States

Triaxial strength behaviour of rockmass satisfying Modified Mohr-Coulomb and Generalized Hoek-Brown criteria

The triaxial strength of twenty rockmass types was predicted using two non-linear triaxial strength criteria for rockmass i.e. Modified Mohr-Coulomb (MMC) criterion and Generalized Hoek-Brown (GHB) criterion. Four different rockmass classification systems were used for the calculation of MMC criterion parameters while only GSI classification system has been used for calculation of GHB parameters. The representative value of the uniaxial compressive strength and elastic modulus of rockmass have been estimated using probabilistic approach. A hypothetical case of an unsupported tunnel has been analyzed considering both MMC and GHB criteria. The analysis was done using the convergence-confinement method with two different approaches. The first approach predicts the tunnel response using GHB criterion directly. The second approach predicts the tunnel response using equivalent Mohr-Coulomb parameters obtained by linearization of triaxial data points obtained from MMC and GHB criteria. The tunnel response has been estimated in terms of radius of plastic zone, tunnel convergence and tunnel convergence strain. For very poor rockmasses the tunnel response predicted by MMC criterion is less than that predicted by GHB criterion. For poor and fair rockmass, the tunnel response estimated considering both the criteria are comparable except for few cases. Squeezing condition in rockmass has been also evaluated.

Limit equilibrium analysis for rock slope stability using basic Hoek–Brown strength criterion

Hoek–Brown (HB) strength criterion can reflect rock’s inherent failure nature, so it is more suitable for analyzing the stability of rock slopes. However, the traditional limit equilibrium methods are at present only suitable for analyzing the rock slope stability using the linear equivalent Mohr–Coulomb (EMC) strength parameters instead of the nonlinear HB strength criterion. Therefore, a new method derived to analyze directly the rock slope stability using the nonlinear HB strength criterion for arbitrary curve slip surface was described in the limit equilibrium framework. The current method was established based on certain assumptions concerning the stresses on the slip surface through amending the initial normal stress σ 0 obtained without considering the effect of inter-slice forces, and it can satisfy all static equilibrium conditions of the sliding body, so the current method can obtain the reasonable and strict factor of safety (FOS) solutions. Compared with the results of other methods in some examples, the feasibility of the current method was verified. Meanwhile, the parametric analysis shows that the slope angle β has an important influence on the difference of the results obtained using the nonlinear HB strength criterion and its linear EMC strength parameters. For β≤45°, both of the results are similar, showing the traditional limit equilibrium methods using the linear EMC strength parameters and the current method are all suitable to analyze rock slope stability, but for β>60°, the differences of both the results are obvious, showing the actual slope stability state can not be reflected in the traditional limit equilibrium methods, and then the current method should be used.

A nonlinear dynamic uniaxial strength criterion that considers the ultimate dynamic strength of concrete

The existing test results and semi-empirical equations of the rate-dependent concrete strength have overestimated the actual dynamic strength because they do not distinguish between the actual dynamic strength related to the strain rate effects and the additional resistance caused by inertial effects. However, an ultimate strength of concrete exists at a strain rate exceeding a certain value. This paper proposes a nonlinear dynamic uniaxial strength criterion for concrete based on an analysis of the physical mechanisms governing the strain-rate-dependent behavior of concrete strength. The proposed criterion is able to describe the actual dynamic strength and to reflect the ultimate strength at a high strain rate of concrete. The results from two groups of dynamic uniaxial compressive tests and two groups of dynamic uniaxial tensile tests are used to verify the criterion. Moreover, the recommended physical parameters in the criterion are obtained by analyzing the statistical test results of dynamic uniaxial compression and tension. The recommended parameters can be used in the criterion to study the dynamic strength of concrete when dynamic tests are not feasible, and to predict the dynamic strength at high strain rates when tests are performed at lower strain rates.

Stability analysis of crack slope considering nonlinearity and water pressure

The soil slopes tend to induce the appearance of tension cracks at the top of slope under various factors, which poses an adverse impact on slope stability. Based on Power-Law nonlinear failure criterion, kinematical approach is adopted in this paper to investigate the effects of underground water and vertical tension cracks on stability of slopes. Notice that the tension cracks tend to propagate vertically, and thus a novel failure mechanism composed by the tension crack and logarithmic spiral shear slide plane is postulated. Theoretically, the worst position of crack propagation makes the stability factor minimal, and the failure surfaces can be obtained under different conditions through optimization. According to the proposed mechanism and nonlinear Mohr-Coulomb strength criterion, the stability factor and safety coefficient of slopes are derived. Subsequently, sensitivity analysis with respect to specific examples is conducted to analyze the influence of tension cracks, nonlinear strength parameters and pore water pressure on the slope stability. The results indicate that slope stability weakens gradually at the presence of tension cracks, and the changing degree is more evident with the increase of the slope angle. Meanwhile, nonlinear parameters have more significant impact on the safety factor with the decreasing value of the slope angle. However, the bigger the slope angle, the greater the height of tension cracks, and therefore the more obvious influence of pore water pressure on slope stability. When the slope angle approaches to be vertical, the height of tension cracks increases sharply. As a matter of fact, the height of tension cracks is highly related to geometrical conditions, strength parameters and water effect, which build a mutual relationship to determine slope stability.

Three-Dimensional Nonlinear Dynamic Strength Criterion for Rock

It is assumed that rocks contain three-dimensional (3D) penny-shaped microcracks. The Mode II and III dynamic stress-intensity factors at the tips of 3D penny-shaped microcracks are determined. The orientation angle of microfailure in rock materials is obtained to describe the dynamic failure of rocks. The relationship between the microfailure orientation angle and stress components is derived from the dynamic fracture criterion. Failure-characteristic parameters of penny-shaped microcracks under dynamic triaxial compressive condition are defined; these are an invariant. A 3D nonlinear dynamic strength criterion of rocks is established using the micromechanical method. The 3D nonlinear strength criterion is novel, and is not found in the existing literature. By comparison with experimental data, it is shown that the present 3D dynamic strength criterion is in good agreement with the experimental results.

Limit equilibrium slope stability analysis using the nonlinear strength failure criterion

The limit equilibrium stability analysis of two- and three-dimensional slopes with the nonlinear failure strength criterion uses a number of variables to determine the normal and shear stress on the slip surface. The equation for the nonlinear strength failure criterion is expressed using a Taylor series after analyzing the stress of an elemental slice or column. Multivariate linear equations are then derived to determine these variables based on the force and moment equilibrium conditions the sliding body is subject to. The stress on the slip surface can also be obtained to calculate the slope safety factor. The validity of the current method was verified by comparing it with established examples. Charts were produced for slope stability analysis with the nonlinear strength failure criterion under general conditions using the current method. The results of this study show that the slope safety factor decreases with an increase in the geotechnical material parameter m in the nonlinear strength failure criterion. The results of the current method are in close correspondence with other traditional limit equilibrium methods and are more reliable than the Swedish method. The charts can be used to determine slope design parameters that meet specific requirements.

Seismic bearing capacity of shallow foundations resting on rock masses subjected to seismic loads

This paper analyzes the bearing capacity of shallow foundations resting on rock masses subjected to seismic loads based on limit analysis theory. The non-linear twin shear strength criterion is used to consider the effects of intermediate principal stress on the bearing capacity of shallow foundations. The pseudo-dynamic approach is applied to account for the effects of seismic loads on the bearing capacity of shallow foundations. Both the horizontal and vertical seismic acceleration coefficients and the effects of an amplification factor on the upper bound evaluations of the foundation's bearing capacity are investigated. The ?-functions are then applied to derive the upper bound solution of the load-bearing capacity. A numerical computation is performed and indicates that the seismic acceleration coefficients, amplification factor and the intermediate principal stress parameter significantly affect the bearing capacity of shallow foundations.

A nonlinear criterion for triaxial strength of inherently anisotropic rocks

Rocks encountered at many underground construction sites are laminated and exhibit direction-dependent strength behavior. It is also a well-established fact that the strength varies in a nonlinear manner with confining pressure. There is a need of strength criterion which could capture the nonlinearity as well as the anisotropy in the triaxial strength behavior of the rocks. It is essential that the criterion should be simple and must involve minimum testing to the extent possible. Further, the parameters of the criterion should have wide acceptability among the geotechnical fraternity. In the present study, a nonlinear strength criterion for transversely isotropic rocks is presented. Critical state concept Barton (Int J Rock Mech Mining Sci Geomech Abstr 13(9):255–279, 1976) has been used to define the curvature of the criterion. With a correctly defined curvature and starting from a reference point (UCS), it is possible to accurately assess the triaxial strength for given confining pressure. An experimental study conducted on triaxial strength behavior of three types of anisotropic rocks namely phyllite, slate and orthoquartzite has been discussed. A data base comprising more than 1140 triaxial tests conducted worldwide on anisotropic rocks has been compiled. Statistical evaluation of goodness of fit of the proposed criterion to the data base has been carried out. Further, the predictive capabilities of the proposed criterion have been evaluated by determining the error in estimation of triaxial strength if only few triaxial test data are available for determining the criterion parameters. The data base has also been back analyzed to assess the critical confining pressure for anisotropic rocks. Statistically, the critical confining pressure for anisotropic rocks can be taken nearly equal to 1.25 times the maximum UCS (obtained by applying load either parallel or perpendicular to planes of anisotropy). It is concluded that reasonable estimates of the triaxial strength of anisotropic rock can be made through the proposed criterion using minimum amount of triaxial test data available.

A Numerical Approach for Evaluating the Convergence-Confinement Curve of a Rock Tunnel Considering Hoek-Brown Strength Criterion

The convergence-confinement method is frequently used for the analysis of the behavior of a tunnel and to define the support structures necessary to guarantee its stability. The reasons why it is used extensively, at least in the preliminary study phases of a tunnel are: The fundamental parameters of the problem (the diameter and depth of the tunnel, as well as the mechanical characteristics of the ground in which the tunnel is being excavated) can be considered in the calculation; the stresses and strains in the ground and the displacements of the tunnel wall can be obtained from the results of the calculation; the thickness of the plastic zone around the tunnel can be estimated by the method. The application of this method to rock masses, which are characterized by a non-linear strength criterion, has been limited to a restrictive number of works in the literature. In order to consider non-linear strength criteria, some simplifications are generally introduced in order to be able to obtain a simple expression of the stresses and strains in the rock mass and of the displacements of the tunnel wall. A numerical solution is presented in this study with which it is possible, in a simple way, to precisely describe the convergence-confinement curve of a tunnel excavated in rock masses, without the need of introducing any simplifying hypotheses for the description of the stress-strain relations in a plastic field.

Nonlinear multiaxial dynamic strength criterion for concrete material

Basic strength parameters of concrete material which are uniaxial compressive strength, uniaxial tensile strength, biaxial compressive strength and biaxial tensile strength are used to replace the four material parameters of generalized nonlinear strength theory in this study. The change rules and value ranges of four material parameters are acquired by studying the basic strength characteristics of concrete materials. Nonlinear multiaxial dynamic strength criterion for concrete material is proposed by four rate effect expressions of basic strength parameters for concrete material, and the four rate effect expressions are established on the basis of S criterion. The rate effect rules of dynamic strength parameters express that with the increase of strain rate, strength properties of concrete gradually transform to that of metal material. Dynamic strength of concrete is significantly influenced by the strain rate in the range of -3 to 3, and dynamic strength does not increase indefinitely, instead, there is a peak dynamic strength. Compared with dynamic loading test of biaxial compressive-compressive and biaxial tensile-compressive of concrete, nonlinear multiaxial dynamic strength criterion can describe the biaxial dynamic strength rule of concrete reasonably and the nonlinear strength characteristics at a certain strain rate. Strength rules are not influenced by coupling between the strain rate effect and multiaxial stress state, in other words, the dynamic strength surfaces dose not intersect at different strain rates.

Three-dimensional nonlinear strength criterion for rock-like materials based on the micromechanical method

In this paper, it is assumed the rock-like materials contain penny-shaped microcracks. The modes II and III stress intensity factors at tips of the three-dimensional penny-shaped microcracks are determined. The micro-failure orientation angle α is given out to describe the failure of rock-like materials. The formulation between the micro-failure orientation angle α and stress components is obtained from the mixed-mode fracture criterion. Failure characteristic parameters under true triaxial compression condition are defined, which are an invariant and are relative to the intermediate and minimum principal stresses. A three-dimensional nonlinear strength criterion of rock-like materials, in which the effects of the intermediate principal stress on the failure of rock-like materials is taken into account, is developed based on micromechanical methods. The theoretical nonlinear strength criterion is novel, which is not found in the previous references. By comparison with experimental data, it is shown that the present theoretical strength criterion is in good agreement with the experimental results.

A non-linear orthotropic hydrocode model for ultra-high molecular weight polyethylene in impact simulations

This paper presents detailed experimental characterization of quasi-static anisotropic directional strength properties as well as the shock behavior of ultra-high molecular weight polyethylene (UHMWPE) for the development of an advanced material model for this class of materials. Specifically, we consider Dyneema® HB26 – pressed from uni-directional (UD) tapes in a 0/90° stacking sequence. A material model based on a constitutive law with orthotropic, non-linear strength, shock response, composite failure and softening criteria is presented. A set of material parameters is derived for applications in hydrocodes (here: ANSYS AUTODYN). High- and hypervelocity impact tests with different impact velocities are used for preliminary validation and discussion of the predictive capabilities in view of future application.

Transformed Stress Method for Generalizing Soil Constitutive Models

AbstractIt is important to generalize constitutive models from the triaxial compression state to the three-dimensional stress state for soils. However, there are some deficiencies in the existing methods, e.g., the using failure criteria method, the generalized shear stress ratio method, and the shape function approach using Lode’s angle or reduced Lode’s angle, for generalizing either isotropic or anisotropic constitutive models. For example, the critical state curve may be misshapen and the yield surface will appear to be curvature discontinuous if the constitutive model is generalized with the shape function approach using reduced Lode’s angle. The transformed stress (TS) space has been proposed in the authors’ former work based on the spatially mobilized plane (SMP), the Lade’s, and the generalized nonlinear strength criterion. In this paper, the TS method based on an arbitrary failure criterion is proposed. It is a simple approach to generalize constitutive models without introducing any extra parame...