Drucker-Prager Failure Criterion Research Articles

A length-free phase field model for epoxy adhesive materials based on modified Drucker–Prager criterion

Characterizing the cracking behavior of epoxy adhesive material is of great importance for composite structures. Recently, the phase field model has emerged as the preferred method for predicting the crack propagation. This paper presents a length-free phase field model considering the characteristics of epoxy adhesive materials. The proposed model is formulated within a framework that decomposes the phase field fracture driving force, ensuring that it adequately considers the tension/compression asymmetry of the epoxy adhesive materials. The effects of hydrostatic pressure, deviational stress, and friction are incorporated into the model by employing a modified Drucker-Prager failure criterion. The failure criterion is included in the phase field model using the effective stress invariants. Furthermore, the internal length scale, previously considered to lack real physical meaning, is reformulated based on material strength. This model addresses the challenge of measuring the internal length of epoxy materials. It serves as a valuable reference for applying phase field models to epoxy materials. The accuracy of the model is validated through an L-shaped plate simulation, and an attempt using the model to predict the crack development in thick epoxy adhesive joints is undertaken. Several representative samples are simulated, including three double cantilever beams with different configurations. The numerical results demonstrate a good correlation with experimental data, highlighting the potential of the model in simulating the crack propagation in epoxy adhesive joints.

Numerical analysis of the indentation of a poro-elastoplastic half space by a rigid cylinder

Poro-elasto-plastic cylindrical indentation as a means for material characterization is explored numerically in this work. When a rigid cylinder is being pressed into a semi-infinite, fully saturated poroelastic medium via step displacement loading, undrained and drained elastic constants are reflected in the responses at the early and late times, while hydraulic diffusivity is manifested in the transient behavior. Such an indentation test can therefore be used in principle to identify material parameters. For geomaterials, plastic deformation may occur due to the indentation action. How the hydromechanically coupled indentation response is affected by plastic deformation is investigated here using finite element analysis. We consider the problem to be plane strain with frictionless contact. Drucker-Prager failure criteria with and without a cap are considered. We show that if there is only limited plastic deformation, the concept of using the transient response for determining hydraulic diffusivity may be extended from poroelasticity to poro-elasto-plasticity.

Numerical modelling approach for estimation of a yield zone in the face of a deep longwall panel

In the Longwall mining method, the coal face is being supported with two-legged or four-legged shield supports which improve better roof control between the face and canopy tip. The development of load on the face increases with the face retreating rate or the increase of the overhung length behind the shield support and as a result, the coal face tends to yield/fail. The yield/failure zone in the face extends depending on intensity of developed load due to the depth of mining, panel size, height of excavation and geometry of the overhung. The broken coal from the yielded face may fall on the shearer, armoured flexible conveyor (AFC), shield supports or may hit the deployed manpower in the face and causes the face stoppage which in turn results in to the loss of the project. It is therefore, a study is conducted to understand the face yield/failure behaviour considering various geo-mining conditions based on input parameters such as main roof thickness, overhung length, and material type and setting load. This paper develops a unique statistical model to predict the yield/failure zone in the longwall face. For this purpose, a total of fifty-four (54) three-dimensional finite element models in ANSYS software are developed and analyzed considering Drucker-Prager failure criterion.

Phase field study on fracture behavior of crushable polymer foam

Polymer foam is a widely used material across various industries for its lightweight and exceptional energy absorption capability. Nevertheless, its porous structure and compaction properties present challenges in simulating crack initiation and propagation. The phase field method has attracted much attention for its advantages in predicting cracks. In this paper, we combine the phase field method with the “crushable foam” model and introduce the modified Drucker-Prager failure criterion to establish the phase field model of polymer foam materials. Material parameters are calibrated using experimental data from uniaxial compression, uniaxial tensile and shear punching experiments. The typical tensile, shear and compression instability fracture processes of polyurethane are then simulated using the proposed model, and the numerical results agree well with the experimental results. This indicates that the coupled phase-field crushable foam model not only exhibits the compaction properties of polymer foam materials but also accurately predicts its crack propagation under complex loading. Studying the fracture behavior of polymer foams under complex loads can enhance their engineering applications.

A general framework for decomposing the phase field fracture driving force, particularised to a Drucker–Prager failure surface

Due to its computational robustness and versatility, the phase field fracture model has become the preferred tool for predicting a wide range of cracking phenomena. However, in its conventional form, its intrinsic tension–compression symmetry in damage evolution prevents its application to the modelling of compressive failures in brittle and quasi-brittle solids, such as concrete or rock materials. In this work, we present a general methodology for decomposing the phase field fracture driving force, the strain energy density, so as to reproduce asymmetrical tension–compression fracture behaviour. The generalised approach presented is particularised to the case of linear elastic solids and the Drucker–Prager failure criterion. The ability of the presented model to capture the compressive failure of brittle materials is showcased by numerically implementing the resulting strain energy split formulation and addressing four case studies of particular interest. Firstly, insight is gained into the capabilities of the model in predicting friction and dilatancy effects under shear loading. Secondly, virtual direct shear tests are conducted to assess fracture predictions under different pressure levels. Thirdly, a concrete cylinder is subjected to uniaxial and triaxial compression to investigate the influence of confinement. Finally, the localised failure of a soil slope is predicted and the results are compared with other formulations for the strain energy decomposition proposed in the literature. The results provide a good qualitative agreement with experimental observations and demonstrate the capabilities of phase field fracture methods to predict crack nucleation and growth under multi-axial loading in materials exhibiting asymmetric tension–compression fracture behaviour.

A modified damage constitutive model of gas hydrate-bearing sediments based on discrete element method

Based on a series of discrete element method (DEM) simulations, a modified damage constitutive model that can predict the mechanical behavior of gas hydrate-bearing sediments (GHBS) has been developed. A new method for preparing the GHBS sample with different gas hydrate saturations is proposed. According to the triaxial compression simulation tests, the stress–strain curves, Mohr’s circles, and their envelopes of GHBS samples were obtained. Based on the results of simulation tests, material parameter equations for GHBS were established. Using the Drucker–Prager failure criterion, a modified damage constitutive model was constructed that can predict the mechanical behavior of the GHBS under high confining pressure more accurately than a typical damage constitutive model. The effects of confining pressure and gas hydrate saturation on the mechanical behavior of GHBS were investigated using the modified damage constitutive model. The model’s correctness is demonstrated by a comparison with simulation test results. This article can be used as a reference by researchers working on gas hydrate drilling design and analysis.

Influence of non-uniformity in inter-fibre distance on strength distribution of unidirectional fibre-reinforced polymer composites

Unidirectional fibre-reinforced polymer (UDFRP) composites exhibit statistical variations in the material properties. It is essential to acknowledge these variations and use a statistics-based strength property for designing composite structures. In this work, the effect of non-uniformity in inter-fibre distance on the statistics of strength distribution of UDFRP composite is studied through computational micromechanics. Micromechanics-based realisations of the UDFRP composite are developed with a quantified measure of non-uniformity (MoN) in fibre distribution through an algorithm. Several realisations of these models are created for each non-uniform fibre distribution. External stress is applied to each realisation in the transverse direction till failure initiates in the matrix. The modified Drucker–Prager failure criteria are implemented for matrix failure. This externally applied stress is termed as the failure initiation strength. Statistical distribution of failure initiation strength is obtained for each fibre distribution. Variations of mean value, standard deviation and B-basis of failure initiation strength with the measure of non-uniformity in fibre distribution are studied for the fibre volume fractions of 0.5 and 0.6. In addition to the mean value, non-uniformity in fibre distribution also significantly influences the standard deviation and B-basis of strength distribution.

Three-dimensional Nonlinear Finite Element Analysis for Load-Carrying Capacity Prediction of a Railway Arch Bridge

Calculation of load-carrying capacity of old railway masonry arch bridges as crucial infrastructures is one of the vital issues in the railway industry. The field test of the 2PL20 bridge reveals important properties such as the initial stiffness and cracking pattern, but the ultimate capacity of the bridge under static loading was not achieved due to the field limitations. Therefore, the present study aims to predict the total nonlinear response of the 2PL20 bridge up to the failure status. All the geometrical characteristics of the 2PL20 bridge are modeled precisely and the failure state is predicted using the three-dimensional nonlinear finite element analysis. Moreover, to determine the load-carrying capacity of the 2PL20 bridge, the behavior of materials is regarded nonlinearly in which the Williams and Warnke and the Drucker-Prager failure criteria are taken into account for concrete and soil materials, respectively. To obtain precise results of ultimate capacity of the 2PL20 bridge under field test, the vertical and horizontal displacements of the crown on the northern span are chosen as the calibration criterion and, consequently, the vertical and horizontal load-deformation curves are verified based on the field test results. The obtained results indicate that the load-carrying capacity of the 2PL20 bridge equals 8880 kN under the static field test. Finally, the calculated safety factor is equal to 3.2 and increasing axial load up to 35-ton is admissible based on UIC 776-1 for a 25-ton axial load.

Investigation of blasting impact on limestone of varying quality using FEA

Large deformation and rapid pressure propagation take place inside the rock mass under the dynamic loads caused by the explosives, on quarry faces in order to extract aggregate material. The complexity of the science of rock blasting is due to a number of factors that affect the phenomenon. However, blasting engineering computations could be facilitated by innovative software algorithms in order to determine the results of the violent explosion, since field experiments are particularly difficult to be conducted. The present research focuses on the design of a Finite Element Analysis (FEA) code, for investigating in detail the behavior of limestone under the blasting effect of Ammonium Nitrate & Fuel Oil (ANFO). Specifically, the manuscript presents the FEA models and the relevant transient analysis results, simulating the blasting process for three types of limestone, ranging from poor to very good quality. The Finite Element code was developed by applying the Jones-Wilkins-Lee (JWL) equation of state to describe the thermodynamic state of ANFO and the pressure dependent Drucker-Prager failure criterion to define the limestone plasticity behavior, under blasting induced, high rate stress. A progressive damage model was also used in order to define the stiffness degradation and destruction of the material. This paper performs a comparative analysis and quantifies the phenomena regarding pressure, stress distribution and energy balance, for three types of limestone. The ultimate goal of this research is to provide an answer for a number of scientific questions, considering various phenomena taking place during the explosion event, using advanced computational tools.

Experimental Investigation and Finite Element Modelling of the Influence of Hydrostatic Pressure on Adhesive Joint Failure

Structural adhesives have been increasingly used in automotive industries in conjunction with lightweight structural components to maximize weight reduction in modern car design. They are used to bond various kind of lightweight materials such as FRP composites or metallic alloys and provide a good distribution of force across joining area without damaging the substrates. For automotive applications, bonded components have been often used in complex loading environments which brings failure prediction of adhesive joints a key of success for this technology. As polymer-based materials, hydrostatic pressure can have a major impact on adhesive joint failure in addition to temperature and strain rate. Its influence can be investigated using a specific test fixture known as a modified-Arcan fixture. This article presents the characterization of epoxy-based adhesive using a series of modified-Arcan tests to apply different loading taking in account effect of hydrostatic pressure onto adhesive joints. A high-order exponential Drucker-Prager failure criterion is chosen to interpret the yield and failure surface. Finally, the non-linear behaviour of adhesive is identified via 0° and 90° Arcan tests which correspond respectively to tri-axial and shear modes and the validation of this behaviour is carried out by mix-mode 60° and 120° Arcan tests. The simulation shows a good agreement only with the result in direction 60° while in direction 120° the divergence is obvious due to the different in damage mechanism of this mode.

An Approach for Stability Analysis of Gas/Fluid-Filled Cylindrical Hole in Laminated Materials

An explicit analytical workflow for cylindrical hole stability analyses in general laminated materials that possess transversely isotropic (TI) anisotropy is presented. In this approach, the calculation of the distribution of the stresses around a cylindrical hole and the failure evaluation at the hole wall consider the effects of both material elasticity anisotropy and strength anisotropy caused by material laminated structures. Material strength anisotropy is assumed to be caused by the sliding of preexisting weakness planes oriented parallel to the isotropic plane of the material. The effect of anisotropy on strength is modeled by combining a shear failure criterion for the intact matrix and a weak plane failure criterion for the planes of weakness. We derive critical pressure solutions for the stability of the intact matrix around a hole filled with gas or fluid based on the Mohr–Coulomb failure criterion and Drucker–Prager failure criterion; either one of them can be combined with the weak plane failure criterion to give the solution for hole wall shear failure pressure. The solution for hole wall fracture initiation pressure is derived based on the tensile failure criterion. This approach can be applied to holes of arbitrary orientation in general laminated materials.

Geotechnical localization analysis based on Cosserat continuum theory and second-order cone programming optimized finite element method

Based on the Hellinger-Reissner mixed variational principle, a second-order cone programming optimized finite element method for elastic and elastoplastic analyses of Cosserat continuum (named CosFEM-SOCP) is proposed. To examine the correctness and effectiveness of CosFEM-SOCP, three well-known numerical examples are revisited. In the plane-strain elastic problem of stress concentration around a circular hole, the stress concentration factors calculated by CosFEM-SOCP are in good agreement with the existing solutions, and the correctness of CosFEM-SOCP can be verified. In the analysis of load-displacement response of a flexible strip footing resting on the soil ground, it is found that load-displacement response may be affected by the internal characteristic length lc, and generally a larger lc may result in a wider and deeper slip surface as well as more elements being included in the shear bands. From the plane-strain soil strip with the Drucker-Prager (DP) failure criterion subjected to the prescribed incremental displacement, it can also be observed that the pathologically mesh-dependent problem can be partially or even completely removed by employing CosFEM-SOCP, as long as the mesh is fine enough so that the average element size in the mesh discretization is smaller than lc.

A Closed-Form Analytical Solution for Circular Opening in Rocks Using Drucker–Prager Criterion

Wellbore and tunnel problems are of true triaxial stress state, even if the ground is under axisymmetric loading condition. A closed-form analytical solution is proposed using Drucker–Prager failure criterion. The solutions are obtained for rock mass exhibiting elastic–perfectly plastic or elastic–brittle–plastic behaviour. The proposed solution is then compared with the finite element analysis (FE-analysis) results. Parametric studies are also carried out. The results of the proposed analytical solution are found to be in good agreement with the FE-analysis results. The proposed analytical solution can thus be used for predicting the stresses and deformation of underground circular openings considering true triaxial stress state.

Study of adhesive bonded joint failure in composite-metal joining

Currently, composites play an important role in the automotive industry where the weight reduction is a key of success due to its high strength to weight ratio. However, the replacement of metals with composites cannot be entirely archived, the joining between composite-metal is required. Bonding with adhesive is one of the most important methods in composite-metal joining. However, the accurate failure prediction of adhesive layers is still lacking due to the complicated state of stress at free-edges of specimens. The aim of this work is to investigate the failure behavior and analysis of acrylic adhesive which is particularly designed to provide superior bonding between composites and metals. Since an adhesive is generally a polymer, its failure depends on the hydro-static pressure. The chosen criterion needs to take into account this dependency. In this study, the Drucker-Prager failure criterion has been used. The parameters were characterized by tensile, compression and double lap shear tests. Then the FEM simulation was performed to predict a failure of single lap shear tests as a validation of model.

Investigation of the tool-rock interaction using Drucker-Prager failure criterion

Mechanical rock breaking in different forms have found wide applications in the mining and civil engineering industry such as drilling, tunnelling and grinding. A detailed understanding of the tool-rock interaction is essential to achieve high efficiency in rock breaking and to optimize the cutting parameters. The finite element simulation of the tool-rock interaction remains a challenging task due to the complexity caused by the physical rock properties, the crack initiation and propagation, as well as the chip generation phenomenon. This study has implemented the Drucker-Prager constitutive model with an isotropic damage model in a finite element analysis to examine the rock failure modes, considering the interaction of cutting tool and the rock. The material parameters for the rock derive from the uniaxial compressive strength (UCS) and Brazilian tensile strength (BTS) tests performed on rock specimens. This study includes an experimental program to examine the crack initiation, crack propagation and the chip generation in rock specimens. The results show the Drucker-Prager failure criterion with the calibrated material parameters and element sizes simulate closely the tool-rock interaction phenomenon. The mesh size imposes a great effect on the parameter selection of the rock model, particularly for the fully damaged plastic strain value. This study leads to an enhanced understanding of rock breaking mechanisms in the mechanical excavation, and provides the basis to improve the rock excavation machine design.

Numerical Studies on the Effectiveness of Dynamic Compaction in Loose Granular Deposits Using Shear Wave Velocity Profiling

Over the past few decades, Dynamic compaction (DC) has gained popularity as an effective improvement technique for geomaterials in view of its simplicity, low-environmental impact and cost effectiveness. The low carbon footprint associated with this ground remediation method addresses the adverse ecological threats imposed on the environment and society due to unsustainable geotechnical engineering practices encountered in the face of rapid urbanization. In the literature, studies related to numerical modelling of DC are limited, and the existing databases are founded on field trials, past experience and empirical equations. Further, till date, numerical evaluation of improvement in soil strength post DC is restricted primarily to change in relative density of soil samples before and after impact, whereas, in the field, shear wave velocity (Vs) profiling is frequently adopted as a monitoring technique for measuring the degree and depth of improvement. This necessitates a quantitative correlation between the DC design parameters (tamper radius, energy and momentum) and the available shear wave profile data measured in the field for effective design and execution of DC methodology. In order to overcome the above mentioned research gaps, an elasto-plastic soil model with Drucker–Prager failure criteria is incorporated in the present study using FE software ABAQUS. The response of the soil model to large strains developed during multiple tamper drops on dry sand is investigated numerically, and validated with the results of a centrifuge model test, and numerical analyses published in literature. Further, the shear wave velocity of soil samples is assessed numerically based on the value of shear modulus, and subsequent improvement in model soil due to impact (66% in the present case) is studied to arrive at a better practical application. The results are compared with physically observed field data, and are found to corroborate well. Subsequent parametric studies are carried out by varying the design parameters related to DC, which indicates that the degree and depth of improvement of soil in terms of Vs increases substantially (about 40%) with an increase in momentum and decreasing tamper radius (about 60%), whereas, energy imparted has comparatively lesser impact on improvement. A method is eventually proposed with design equations to calculate the improvement after DC in field based on Vs profiling, depending on momentum and radius of tamper. Further, structural requirements coupled with Vs profile data computed in the ground remediated by DC can help in avoiding construction of expensive deep foundations in sites exhibiting poor subsoil profiles, thereby economizing the project. In addition, the above concept ensures sustainability in engineering practices by enabling land-reclamation and utilization of sites exhibiting locally available compressible soils for infrastructure construction and foundation support.

A coupled thermo-mechanical damage model for granite

With the increasing demand for mineral and energy resources, deep underground operations in mining, oil and gas industries become more frequent. This also resulted in problems associated with high temperature and pressure (HTP) that rock is exposed to in deep underground conditions. It is therefore imperative to understand and correctly model the coupled thermo-mechanical (TM) behavior of rocks for better and safer designs. For this purpose, based on the Weibull distribution and Lemaitre's strain-equivalent principle, a framework for the coupled thermo-mechanical (TM) damage model for granite is developed to simulate the deformation and failure process of rock under the high temperature and pressure (HTP) conditions. The thermal damage caused by the temperature change and the mechanical damage caused by high pressure were analyzed, and a nonlinear coupled total damage parameter is proposed in this study. The Weibull distribution parameters involved in the damage model are derived by combining the geometric conditions obtained from the conventional triaxial compression tests. In order to compare the performance of the proposed damage model, Drucker-Prager failure criterion is used as an example in the nonlinear TM damage model. For the validation of the proposed model, test results for granite samples at various temperature and confining pressures were compared in which the theoretical results are found to be in a good agreement with the experimental results. The proposed coupled TM damage model can be used in conjunction with any other failure criteria to account the damage evolution under HTP conditions.

Modelling tensile/compressive strength ratio of artificially cemented clean sand

The present work proposes a new theoretical model for predicting both the splitting tensile strength (qt) and the compressive strength (qu) of artificially cemented sand and assesses their ratio for a given material. The proposed model is based on the concept of the superposition of the failure strength contributions of the sand and cement phases. The sand matrix obeys the concept of critical state soil mechanics, while the strength of the cemented phase can be described using the Drucker-Prager failure criterion. The analytical solutions are compared against the results of tests on three different types of cemented clean sand cured for different time periods. While the analytical relation fits the experimental data well, it also provides a theoretical basis for the explanation of some features related to the experimentally derived strength relationships for cemented clean sand. The value of the power relationship between the strength and the porosity/cement ratio index seems to be governed by the soil matrix properties, while the interdependency of the strength and the curing time can also be captured. For a given cemented sand, the model equally confirms the existence of a unique tensile/compressive strength ratio (qt/qu), independent of the curing time and primarily governed by the compressive to tensile strength ratio (or the friction properties) of the cement. It is also confirmed that the qt/qu ratio changes within a narrow range for different frictional properties of the cementing phase.

Modelling tensile/compressive strength ratio of fibre reinforced cemented soils

The present work proposes a theoretical model for predicting the splitting tensile strength (qt) - unconfined compressive strength (qu) ratio of artificially cemented fibre reinforced soils. The proposed developments are based on the concept of superposition of failure strength contributions of the soil, cement and fibres phases. The soil matrix obeys the critical state soil mechanics concept, while the strength of the cemented phase can be described using the Drucker-Prager failure criterion and fibres contribution to strength is related to the composite deformation. The proposed developments are challenged to simulate the experimental results for fibre reinforced cemented Botucatu residual soil, for 7 days of cure. While the proposed analytical relation fits well the experimental data for this material, it also provides a theoretical explanation for some features of the experimentally derived strength relationships for artificially fibre reinforced cemented clean sands. A parametric study to analyse the effect of adding different fibre contents and fibre properties is provided. The proposed modelling developments also confirm the existence of a rather constant qt/qu ratio with moulding density, cement and fibre contents .

Hydrate failure in ITZ governs concrete strength: A micro-to-macro validated engineering mechanics model

Ever since the early days of Féret (1892) and Abrams (1919), concrete research has targeted at relating concrete composition to uniaxial compressive strength. While these activities were mainly characterized by empirical fitting functions, we here take a more fundamental approach based on continuum micromechanics. The loading applied at the concrete level, is first concentrated (“downscaled”) to maximum stresses related to cement paste volumes which are directly adjacent to the aggregates, i.e. to the interfacial transition zones (ITZ). These maximum stresses are further “downscaled” to the micron-sized hydrates, in terms of higher-order stress averages. The latter enter a Drucker-Prager failure criterion with material constants derived from nanoindentation tests. The model is successfully validated across the hydrate-to-concrete scales. Strength magnitude is governed by ITZ stress concentrations, and the water-to-cement ratio is its dominant mixture design parameter.