Three-dimensional Failure Criterion Research Articles

A three-dimensional failure criterion model considering the effects of fiber misalignment on longitudinal tensile failure

This paper proposes a three-dimensional failure criterion model for unidirectional fiber-reinforced composites. In contrast to previous models that only accounted for the effect of fiber misalignment in the presence of longitudinal compressive stress, the proposed failure criterion model comprehensively considers the effect of localized misaligned regions on the failure under any stress state. Another key contribution of this study is the introduction of the effective misalignment angle. Considering effective misalignment angles, the proposed failure criterion model can reasonably reveal the effect of localized misaligned regions on the failure behavior under different stress states. The agreement between the predicted results and the experimental data proves that the proposed model has good applicability. Furthermore, the influence of initial misalignment angles on failure is analyzed under varying stress conditions. The results indicate that even under longitudinal tensile stress, the initial misalignment angle still plays an important role in the failure behavior of materials.

A conceptual three-dimensional frictional model to predict the effect of the intermediate principal stress based on the Mohr-Coulomb and Hoek-Brown failure criteria

The intermediate effective principal stress has a considerable influence on rock strength as shown by true-triaxial data on intact rock. Therefore, a failure criterion that incorporates all three principal stresses may provide an improved prediction of failure than one that considers only the major and minor effective principal stresses. This study introduces a generalisation method, which converts two-dimensional failure criteria (i.e., Mohr-Coulomb and Hoek-Brown failure criteria) to three-dimensional failure criteria, for the prediction of this true-triaxial data. This method is based on a physical mechanism, and it assumes that the potential failure surface of a material has roughness, which provides additional frictional resistance when the intermediate effective principal stress is greater than the minor effective principal stress. This roughness is introduced theoretically by assuming a repeating diamond-shaped element over the failure surface. These roughness elements have the same dip as predicted by the Mohr-Coulomb and Hoek-Brown failure criteria but are offset by the dip direction angle which is assumed similar to that calculated traditionally from the major effective principal stress direction to the failure surface. With this roughness, it is shown that the major effective principal stress at failure can be predicted using an explicit solution that uses the intermediate and minor effective principal stresses and the Mohr-Coulomb or Hoek-Brown parameters as inputs. This generalisation predicts the same strength as the Mohr-Coulomb or Hoek-Brown failure criteria under conventional triaxial loading conditions, when the intermediate and minor effective principal stresses are equal. Under general stress conditions, this generalisation produces results that are aligned closely with the true-triaxial data as well as other previous theoretical predictions. This method importantly gives a possible physical mechanism for the strengthening influence of the intermediate effective principal stress. This may lead to a more accurate prediction of the rock strength under general stress conditions.

The Generalized Mohr-Coulomb Failure Criterion

With the construction of supertall buildings such as high earth dams, the linear envelope of the Mohr-Coulomb (M-C) failure criterion fitted to lower confined pressure would significantly underestimate the loading capacity of foundations, causing a huge increase in the amount of earthwork. Given that the M-C criterion has dominated in the stability analysis of geotechnical structures, it is proposed in this study that the M-C criterion remain invariant in form but the cohesion c and the frictional factor f be related to the coefficient of intermediate principal stress b, called the Generalized Mohr-Coulomb (GMC) criterion. In other words, c and f are both functions of b, written as c(b) and f(b). In the simplest way, the GMC criterion for soils, a true three-dimensional failure criterion, can be established by using a piece of conventional triaxial apparatus. The GMC has a non-smooth strength surface like its conventional version. However, we prove from true triaxial tests and the characteristic theory of stress tensors that the failure surfaces in the stress space should be non-smooth per se for b = 0 or 1. Comparisons with other prominent failure criteria indicate that the GMC fits the test data best.

A novel three-dimensional modified Griffith failure criterion for concrete

This study proposes a three-dimensional failure criterion based on the Griffith criterion, which accounts for the nonlinear influences of hydrostatic pressure and Lode angle on the failure strength of concrete. Two approaches are utilized in the proposed criterion to describe the Lode-dependence, one is to use the biaxial compressive envelopes to derive each meridian directly, and the other is to employ a scaling function between the compressive and tensile meridians. Besides, the proposed criterion can be expressed by the principal stresses in the form of f(σ1, σ2, σ3) = 0 as well as by the stress invariants in the form of J2 = f(P, θ). To comprehensively validate the proposed criterion, it collects 37 groups of tests with nearly 1000 test data conducted on concrete under biaxial and triaxial stress states. It demonstrates that predictions of the proposed criterion, including biaxial failure envelope, compressive meridian, tensile meridian, deviatoric section, and multiaxial strength, are close to the corresponding experimental observations. The proposed criterion requires four parameters, three of which have clear physical meaning, namely the tensile, compressive, and equal-biaxial compressive strength (ft, fc, and fbc), and the rest parameter n can be determined by the triaxial compressive strength. Aimed at elaborating the physical meaning of n, it interprets the proposed criterion as a combination of the original Griffith, Mohr-Coulomb, and Rankine criteria. In addition, a simple modification based on the proposed criterion is also suggested to describe the transformation from shear failure to ductile failure of concrete with the increase of pressure. Compared with the existing Ottosen and Willam-Warnke criteria, the proposed criterion is simple in determining parameters, convenient in the application, and diverse in describing the Lode-dependence.

A generalized nonlinear three-dimensional failure criterion based on fracture mechanics

Based on fracture mechanics theory and wing crack model, a three-dimensional strength criterion for hard rock was developed in detail in this paper. Although the basic expression is derived from initiation and propagation of a single crack, it can be extended to microcrack cluster so as to reflect the macroscopic failure characteristic. Besides, it can be derived as Hoek–Brown criterion when the intermediate principal stress σ2 is equal to the minimum principal stress σ3. In addition, the opening direction of the microcrack cluster decreases with the increase of the intermediate principal stress coefficient, which could be described by an empirical function and verified by 10 kinds of hard rocks. Rock strength is influenced by the coupled effect of stress level and the opening direction of the microcrack clusters related to the stress level. As the effects of these two factors on the strength are opposite, the intermediate principal stress effect is induced.

A Non-linear Three-Dimensional Failure Criterion Based on Stress Tensor Distance

A Non-linear Three-Dimensional Failure Criterion Based on Stress Tensor Distance

Excavation-induced deep hard rock fracturing: Methodology and applications

To analyze and predict the mechanical behaviors of deep hard rocks, some key issues concerning rock fracturing mechanics for deep hard rock excavations are discussed. First, a series of apparatuses and methods have been developed to test the mechanical properties and fracturing behaviors of hard rocks under high true triaxial stress paths. Evolution mechanisms of stress-induced disasters in deep hard rock excavations, such as spalling, deep cracking, massive roof collapse, large deformation and rockbursts, have been recognized. The analytical theory for the fracturing process of hard rock masses, including the three-dimensional failure criterion, stress-induced mechanical model, fracturing degree index, energy release index and numerical method, has been established. The cracking-restraint method is developed for mitigating or controlling rock spalling, deep cracking and massive collapse of deep hard rocks. An energy-controlled method is also proposed for the prevention of rockbursts. Finally, two typical cases are used to illustrate the application of the proposed methodology in the Baihetan caverns and Bayu tunnels of China.

Multiple Impact Damage in GLARE Laminates: Experiments and Simulations.

Experiments and finite element simulations for multiple impact were performed on GLARE 5-2/1 and aluminum 2024-T3. Experiments were conducted on aluminum 2024-T3 and GLARE 5-2/1 at diverse impact energies to produce BVID (barely visible impact damage) and CVID (clearly visible impact damage). The finite element model was developed for multiple impact analysis using ABAQUS software and was confirmed by comparing the finite element analysis outcomes with experimental results. The two- and three-dimensional failure criteria model was applied to predict multiple impact behavior such as load-time history, maximum deflection-impact energy history, and damage progression. In addition, a user subroutine VUMAT was created to represent a three-dimensional progressive failure and was linked with ABAQUS. FEM results showed good agreement with experimental data.

A Simple Three-Dimensional Failure Criterion for Jointed Rock Masses under True Triaxial Compression

The joint configuration and the intermediate principal stress have a significant influence on the strength of rock masses in underground engineering. A simple three-dimensional failure criterion is developed in this study to predict the true triaxial strength of jointed rock masses. The proposed failure criterion in the deviatoric and meridian planes adopts the elliptic and hyperbolic forms to approximate the Willam–Warnke and Mohr–Coulomb failure criterion, respectively. The four parameters in the proposed failure criterion have close relationships with the cohesion and the internal friction angle and can be linked with the joint inclination angle using a cosine function. Two suits of true triaxial strength data are collected to validate the correctness of the proposed failure criterion. Compared with other failure criteria, the proposed failure criterion is more reasonable and acceptable to describe the strength of jointed rock masses.

Investigation on three-dimensional failure criterion of asphalt mixture considering the effect of stiffness

In order to reveal the effect of complex stress state on the characteristics of strength and stiffness of asphalt mixture, the self-developed double confining pressure triaxial test method was used in this study. The one-time failure and resilient modulus tests in complex stress state were carried out on the SMA-13 and AC-13 asphalt mixtures widely used in China's asphalt pavement, especially in the combined tensile-compressive stress state. The failure principal stress of material and its stress-strain data under the step loading condition were obtained. The calculation model of resilient modulus in complex stress state and the failure criterion I1-J2 in the principal stress space were established. On this basis, based on the generalized Hooke's law, a three-dimensional strain failure criterion model for asphalt mixture considering the effect of stiffness was also established by the conversion method. The research results can provide experimental and theoretical support for the reasonable selection and resistance calculation of material parameters of asphalt mixture in complex stress state.

Development of a poly-axial platen for testing true-triaxial behavior of rocks

Mining at greater depths can lead to stress-induced failure, especially in areas of high horizontal in situ stress. The induced stresses around the opening are known to be in a poly-axial stress state where, σ1 ≠ σ2 ≠ σ3 with special case of σ3 = 0 and σ1, σ2 ≠ 0 at its boundary, where σ1, σ2, σ3 are major, intermediate, minor principal stress, respectively. The conventional triaxial testing does not represent the actual in situ strength of the rock in regions of high horizontal stress, as it ignores the influence of intermediate principal stress (σ2). The typical poly-axial testing (biaxial and true-triaxial tests) of intact rock mostly requires sophisticated and expensive loading systems. This study investigated the mechanical behavior of intact rock under a poly-axial stress state using a simple and cost-effective design. The apparatus consists of a biaxial frame and a confining device. The biaxial frame has two platens that apply equal stress in both directions (σ1 = σ2) on a 50.8 mm cubical specimen when placed inside the uniaxial loading device. The confining device performed separate biaxial tests under constant intermediate principal stress (σ2 = constant) and true-triaxial tests when used along with the biaxial frame. This study then compared the failure modes and peak strength of Berea sandstone specimens with other biaxial–triaxial devices to validate the design of the poly-axial apparatus. We also performed uniaxial tests on both standard cylindrical samples and prismatic specimens of different slenderness ratios. These tests provided a complete understanding of the failure mode transition from standard uniaxial compressive tests to triaxial stress conditions on cubical specimens. Additionally, this study determined best-fitted strength envelopes for biaxial and triaxial stress state. Based on regression analysis, we found a quadratic polynomial to be a good fit to biaxial strength envelope. For the true-triaxial strength envelope, we found the three-dimensional (3D) failure criterion to be a good fit with R2 of 0.964.

A general failure criterion for soil considering three-dimensional anisotropy

Abstract This paper proposes a generalized three-dimensional (3D) failure criterion for cross-anisotropic soils based on the spatially mobilized plane (SMP) criterion. In the proposed criterion, the anisotropic property of soil is considered by introducing a novel modification stress tensor, in which the angle between the principal stress and the soil depositional direction serves as a basic variable. Taking advantage of the SMP criterion, the proposed anisotropic criterion is subtly extended into a 3D version. The proposed criterion only involves two friction angles, φ v and φ h , which are determined by triaxial compression test along and perpendicular to the depositional direction, respectively. To account for the effects of the depositional direction, two special modification stress tensors regarding different depositional conditions (vertical and inclined) are discussed in this study. Comparisons between the predictions from the proposed strength criterion and the observations of the true triaxial tests on four cross-anisotropic soil samples show that the proposed generalized failure criterion can well capture both the cross-anisotropic behaviors and the 3D strength of cross-anisotropic soils.

Mixed mode testing of a multi-directional woven laminate

Mixed-mode fracture toughness tests were carried out on a multi-directional laminate making use of a Brazilian disk specimen. The composite laminate consisted of plain, balanced woven plies with tows in the $$0^{\circ }/90^{\circ }$$ -directions and the $$+45^{\circ }/-45^{\circ }$$ -directions. The delamination was placed between two of these plies by means of a polytetrafluoroethylene (PTFE) film. Tests at various loading angles were performed in order to obtain a wide range of mode mixities. Each ply was homogenized by means of a micro-mechanical model to obtain its effective properties. The specimens were analyzed by means of the finite element method. An interaction energy or M-integral was extended for this interface to obtain the stress intensity factors. The latter were used to determine the critical interface energy release rate and two phase angles. Employing this information, a three-dimensional failure criterion proposed elsewhere was presented. The criterion may be used to predict failure of a composite structure containing a delamination along the interface and material studied here.

Two- and Three-Dimensional Failure Criteria for Laminate Composites

Abstract Several two- and three-dimensional mixed-mode interface failure criteria are proposed for predicting delamination failure in multidirectional, laminate composites. The proposed criteria, based on the stress intensity factors K1, K2, and KIII, as well as the critical interface energy release rate Gic and phase angles ψ and ϕ, are examined using results obtained from Brazilian disk mixed-mode fracture toughness tests. Two material systems are considered. The first contains a delamination along an interface between a unidirectional fabric and a plain woven fabric. The second is composed of a plain woven fabric with fibers oriented in different directions in succeeding plies. The former was manufactured by means of a wet-layup and the latter is a prepreg. Finally, a statistical analysis is carried out to obtain a failure curve or surface with a 10% probability of unexpected failure and a 95% confidence. These curves or surfaces may be used to predict failure of structures containing these laminates and to assist in composite design.

Simple approach for solution of the quasi-plane-strain problem in a circular tunnel in a strain-softening rock mass considering the out-of-plane stress effect

The out-of-plane stress is sometimes the major or intermediate principal stress in a circular tunnel opening. The influences of the out-of-plane stress and axial strain are often neglected in the stability analyses of tunnel excavation, which can induce significant errors in the determination of surrounding rock deformations. In this paper, the use of a simple approach is proposed to solve the quasi-plane-strain problem of circular tunneling considering the effect of the out-of-plane stress, which is deformation-dependent and influenced by the in situ stress. As the intermediate principal stress is deformation-dependent, to obtain the numerical solution of the intermediate principal stress, the quasi-plane-strain problem is defined based on assumptions that the initial axial total strain is a nonzero constant (ε0) and that the axial plastic strain is nonzero. With the numerical solution for the plastic strain, obtained using the plastic potential functions based on the three-dimensional failure criteria, the formula for the intermediate principal stress can be derived using Hooke’s law. The proposed approach can be utilized to obtain the numerical solution for the intermediate principal stress, which is deformation-dependent, and the numerical results can be simplified as the solution presented by Pan and Brown. The proposed approach can also be used to obtain the solution for the strain softening of the surrounding rock. To verify its validity and accuracy, the results obtained using the proposed approach are compared with the solution of Pan and Brown. In addition, parametric studies are performed to address the influences of the out-of-plane stress on the stress and displacement in the circular tunnel.

Mixed mode interface fracture toughness of a multi-directional composite – UD/woven pair

Twenty-seven mixed mode Brazilian disk specimens were tested to obtain the fracture toughness of a delamination between two fiber reinforced composite plies in a multi-directional laminate, manufactured by means of a wet-layup. Using finite element analyses and conservative integrals, mechanical and residual curing stress intensity factors were found and used to calculate the interface energy release rate and phase angles. Based on these results, a three-dimensional failure criterion is proposed to obtain a failure surface. This criterion may be used to predict failure of a structure containing this material and interface.

Progressive Failure Simulation of Notched Tensile Specimen for Triaxially-Braided Composites.

The mechanical characterization of textile composites is a challenging task, due to their nonuniform deformation and complicated failure phenomena. This article introduces a three-dimensional mesoscale finite element model to investigate the progressive damage behavior of a notched single-layer triaxially-braided composite subjected to axial tension. The damage initiation and propagation in fiber bundles are simulated using three-dimensional failure criteria and damage evolution law. A traction–separation law has been applied to predict the interfacial damage of fiber bundles. The proposed model is correlated and validated by the experimentally measured full field strain distributions and effective strength of the notched specimen. The progressive damage behavior of the fiber bundles is studied by examining the damage and stress contours at different loading stages. Parametric numerical studies are conducted to explore the role of modeling parameters and geometric characteristics on the internal damage behavior and global measured properties of the notched specimen. Moreover, the correlations of damage behavior, global stress–strain response, and the efficiency of the notched specimen are discussed in detail. The results of this paper deliver a throughout understanding of the damage behavior of braided composites and can help the specimen design of textile composites.

Progressive damage modelling and experimental investigation of three-dimensional orthogonal woven composites with tilted binder

This paper investigates the tensile properties of 3D orthogonal woven carbon fiber composites with tilted binder by experiment and simulation. The tensile failure strain and fracture mode of this composite show distinguished discrepancy with idealized 3D orthogonal woven composites experimentally. In order to explain this specific failure mechanism, a unit cell finite element model incorporated with damage models of constituents is established to reproduce the damage initiation and propagation of 3D orthogonal woven composites with tilted binder during tensile test. A three-dimensional failure criterion based on Hashin's criterion and Pinho's criterion is utilized to describe the progressive damage of yarns, while the non-linear behavior of the matrix is predicted by Drucker-Prager yield criterion. Besides, a traction-separation law is applied to predict the damage of yarn-matrix interface. The proposed unit cell model is correlated and validated by global stress–strain curves, DIC full-field strain distributions and modulus history curve. The damage evolution process of 3D orthogonal woven carbon fiber composites with tilted binder, including fiber tow failure, matrix cracking, and interfacial debonding, is recorded and investigated by the modulus history curve from simulation.

Simple Progression Law in Predicting the Damage Onset and Propagation in Composite Notched Laminates

The aim of this article is to simulate the damage initiation and progression in unidirectional (UD) laminates. A three-dimensional (3D) failure criteria of Puck incorporated with degradation scheme is developed. Two types of degradation law known as sudden degradation are used to predict the damage progression in UD laminates. The establishment of constitutive law in progressive damage model (PDM) is achieved through implementation of user subroutines in Abaqus. The failure analysis is applied to various composite stacking sequences and geometries, as well as different fiber reinforced polymer (FRP) composite materials. The comparative studies revealed that the predicted ultimate failure load agree well with those available in the literature.

Simple Progression Law in Predicting the Damage Onset and Propagation in Composite Notched Laminates

The aim of this article is to simulate the damage initiation and progression in unidirectional (UD) laminates. A three-dimensional (3D) failure criteria of Puck incorporated with degradation scheme is developed. Two types of degradation law known as sudden degradation are used to predict the damage progression in UD laminates. The establishment of constitutive law in progressive damage model (PDM) is achieved through implementation of user subroutines in Abaqus. The failure analysis is applied to various composite stacking sequences and geometries, as well as different fiber reinforced polymer (FRP) composite materials. The comparative studies revealed that the predicted ultimate failure load agree well with those available in the literature.