3D Stress State Research Articles

A quantitative study on the thermomechanical coupling effect of elasticity and plasticity of polymethyl methacrylate

Quantitative study on the elastic and plastic thermomechanical coupling effect of solid materials has important theoretical significance for monitoring the deformation state and stability evaluation of engineering components. In this study, the sample is made of polymethyl methacrylate (PMMA), and uniaxial compression experiments are carried out. The sample experiences the elastic state and then the plastic state until the failure of the sample. The thermoelastic effect and thermoplastic effect in the deformation process of PMMA materials are monitored by infrared thermography and digital speckle system (DIC). That is, the relationship between the reversible elastic temperature change ΔTe and average stress change Δσm, and the relationship between the irreversible plastic temperature rise ΔTp and the dissipated energy ΔUp during the failure deformation process. The results show that when the sample is in the elastic regime, there is only elastic temperature change ΔTe inside the sample. When the sample enters the plastic deformation regime, the total temperature change ΔT includes elastic temperature change ΔTe and plastic temperature change ΔTp, and the temperature change has an obvious memory effect on the previous maximum load. When the previous maximum load exceeded, ΔTp increases rapidly. When the load P is removed, ΔTe disappears, and ΔTp remains. Combined with the theory of thermodynamics and solid mechanics, starting from the 3D stress state, the approximate linear incremental relationship between Te and σm, and the accurate exponential total quantity relationship are derived. The incremental relationship and the full relationship combined with thermodynamic theory and the energy conservation principle, the relationship between the overall plastic temperature rise ΔTp and local plastic temperature rise ΔTpmax of the sample and the energy dissipation value ΔUp experienced by the sample and the local dissipation specific energy Δup is obtained. The theoretical calculation results of ΔTe and ΔTp are in good agreement with the experimental results. The research results can provide a new method for monitoring and identifying the stress and deformation state of engineering components through the temperature field.

Enhanced Fractional Model for Soil–Structure Interface Considering 3D Stress State and Fabric Effect

Enhanced Fractional Model for Soil–Structure Interface Considering 3D Stress State and Fabric Effect

Characteristics of Seismogenic Structures and 3D Stress State of the Delaware Basin of West Texas as Constrained by Earthquake Source Mechanisms

Abstract Seismicity in the Delaware basin is suggested to be associated with oilfield operations. To better characterize the seismogenic structures revealed by the induced seismicity, in this study we determined source mechanisms for the Delaware basin of Texas and leveraged the obtained source mechanisms to perform stress inversion for evaluating the region’s stress state. Based on the seismogenic patterns and seismicity distribution, we identified seven distinctive seismogenic zones. Within each zone, earthquakes form several parallel-trending linear clusters. Most notably, there is an observable change in the seismicity trend on either side of the basin-bisecting Grisham fault zone. In addition, the extent of hypocentral depths vary drastically across the fault zone, with events located below the basin–basement interface north of the fault zone and more shallowly to the south. We also see spatial variations of source mechanism patterns and the direction of the maximum horizontal stress across the Delaware basin. A vast majority of seismic moment release can be attributed to the basement-rooted tectonic faults in the Culberson–Mentone seismogenic zone. A statistic deficit present in the magnitude–frequency distribution suggests that earthquakes with Mw 3.8–4.7 are needed to fulfill this scaling law. It indicates a strong possibility for future occurrence of earthquakes within this magnitude range.

3D numerical simulations and microstructural modeling of anisotropic and tension–compression asymmetric ductile materials

In the present work, we have analyzed the effect of anisotropy on void growth and stress–strain behavior for materials that exhibit remarkable tension–compression asymmetry (i.e., zirconium alloys). For that purpose, we have performed finite element simulations using a cubic 3D cell with a spherical void inside and subjected to periodic boundary conditions. Nonlinear kinematic constraints are also imposed as boundary conditions in order to maintain the values of macroscopic ratios constant during the whole loading history of the cell and account for a general (3D) stress state. The behavior of the matrix material is described by the CPB06 anisotropic criterion developed by Cazacu et al. (2006). The numerical results are compared to those considering 3D homogeneous (without void) cell with the same initial porosity as the voided one and governed by the anisotropic porous yield criterion developed by Stewart and Cazacu (2011). To investigate the influence of prescribed stress, strength differential parameter and strain hardening exponent on stress–strain behavior and void growth in the non-homogeneous (with void) and the homogeneous (without void) cells, we have used several stress ratios, three strength differential parameters and three strain hardening exponents. Finite element results obtained from different stress ratios show the strength differential parameter significantly affect void growth in both homogeneous and non-homogeneous cells. Moreover, comparison of two cells proves that both stress–strain behavior and porosity evolution are in good qualitative agreement for all three values of strength differential parameter. In contrast, as the value of strain hardening exponent increases, the agreement between results obtained from homogeneous and non-homogeneous cells is worse. An heuristic extension of the Stewart and Cazacu (2011)’s model is proposed in this work in an attempt to improve the accuracy of the model.

Pile-soil interaction under cyclic lateral load in loose sand: Experimental and numerical evaluations

Pile foundations in loose sand are occasionally subjected to cyclic loading initiated by the influence of wind, wave, traffic loads, etc. Such load reversals alter the strength and stiffness of surrounding loose sand affecting the ultimate capacity and serviceability of the pile foundation. Although such cyclic loading may be under vertical, lateral or torsional modes or a combination, the lateral cyclic load dominates the other modes. To carry out an in-depth study on pile-soil interaction under lateral cyclic load in loose sand, a series of laboratory model tests were performed with 2 × 2 pile group, followed by developing two alternative numerical models, i.e., boundary element and finite element models (i.e., BEM and FEM). The BEM involved a p-multiplier technique to incorporate the group effect, while the FEM was developed by ABAQUS software incorporating 3D stress conditions. As observed, the BEM slightly over-predicts while the FEM marginally under-predicts the experimental observations. The lateral cyclic loading was found to produce stiffening effect on loose sand which increased the pile capacity and reduced the pile head displacement. Sand relative density is also found to affect the test and numerical results significantly. A set of important conclusions are drawn from the entire study.

A study on the effects of internal architecture on the mechanical properties and mixed-mode fracture behavior of 3D printed CaCO3/ABS nanocomposite samples

PurposeThe purpose of this study was to investigate the effects of CaCO3 nanoparticles on the mechanical properties, and mixed-mode fracture behavior of acrylonitrile butadiene styrene 3D printed samples with different internal architectures.Design/methodology/approachThe nanocomposite filaments have been fabricated by a melt-blending technique. The standard tensile, compact tension and special fracture test samples, named Arcan specimens, have been printed at constant extrusion parameters and at four different internal patterns. A special fixture was used to carry out the mixed-mode fracture tests of Arcan samples. Finite element analyses using the J-integral method were performed to calculate the fracture toughness of such samples. The fractographic observations were used to evaluate the mechanism of fracture at different concentrations of nanoparticles.FindingsThe addition of CaCO3 nanoparticles has resulted in a significant increase in the fracture loading of the samples, although this increase was not consistent for all the filling patterns, being more significant for samples with linear and triangular structures. According to the fractographic observations, the creation of uniformly distributed microvoids due to the blunting effect of nanoparticles and 3D stress state at the crack tip in the samples with linear and triangular structures justify the enhancement in the fracture loading by the addition of CaCO3 nanoparticles in the matrix.Originality/valueThere is a significant gap in the knowledge of the effects of different nanoparticles in the polymer samples produced by the fused filament fabrication process. One of such nanoparticles is an inorganic CaCO3 nanoparticle that has been frequently used as nanofillers to improve the thermomechanical properties of thermoplastic polymers. Here, experimental and numerical studies have been conducted to investigate the effects of such nanoadditives on the mechanical and fracture behavior of 3D printed samples.

New method for the mechanical characteristics of self-weight collapsibility testing of loess foundation based on centrifugal model test

The mechanical characteristics of self-weight collapsibility of loess is mainly studied through laboratory tests, numerical simulations, and field immersion tests. However, a laboratory test under the lateral restraint condition cannot reflect the 3D stress state of soil and the influence of lateral extrusion deformation. The numerical simulation method is complex, and the calculation results depend on the accuracy of the constitutive model of loess deformation. Although the field immersion test produces results consistent with the stress and deformation conditions of actual foundations, the test cycle is long, and the cost is high. To overcome these issues, based on the strong self-weight collapsible loess in Ili, Xinjiang, and the variation laws of its physical and mechanical properties along the buried depth, a site is generalized and layered, and typical soil layers are selected to prepare a centrifugal model. A centrifugal model test is then conducted on the self-weight collapsible loess foundation. Taking the regional correction coefficient as the index, the rationality of the new method is verified by comparing the results of the centrifugal model test with those of the laboratory test and field immersion test.

A UMAP-based clustering method for multi-scale damage analysis of laminates

In this study, a novel multi-scale damage model based on the parametric finite-volume direct averaging micromechanics theory (parametric FVDAM), extended finite element method (XFEM), and a clustering algorithm, is built, to describe the progressive damage evolution of laminates, where a uniform manifold approximation and projection (UMAP)-based clustering method is specially designed and incorporated with the FVDAM and XFEM for the first time. The subvolumes in the unit cell generated by FVDAM with similar mechanical responses were included in the same cluster using the UMAP-based clustering method. The semi-analytical framework of FVDAM allows the efficient generation of unit cells. The built-in advantage in the visualization of the proposed UMAP-based clustering method allows researchers in the field of mechanics to have a clear and intuitive understanding of the clustering results. The clustered unit cell is then conjugated to the XFEM, which allows the damage information to be transmitted between different scales. Three damage mechanisms of laminates under a three-dimensional (3D) stress state are considered, which include matrix damage, fiber breakage, and delamination. Furthermore, a shear nonlinear matrix Ds for a 3D element is derived and introduced in the model, which enables the description of the unique elastic nonlinear behavior of laminates. The model is implemented using UDMGINI and USDFLD subroutines in ABAQUS, which not only allows the calculation of the load–displacement curve, but also the simulation of clear and smooth crack propagation paths. To verify the effectiveness of the model, the experiments of [±60]2s and [90]8 notched glass-fiber/epoxy laminates under tension were performed by digital image correlation (DIC) monitoring, and the results compared were in good agreement with the simulations, thus proving the effectiveness of the proposed model. The code of the UMAP-based clustering method developed in this study can be downloaded from https://github.com/Danhui-Yang/MicroMechanicsUmapClustering.

An analytical approach to the ultimate bearing capacity of smooth and rough strip foundations on rock mass considering three-dimensional (3D) strength

An analytical approach to the ultimate bearing capacity qu of strip foundations on rock mass under both smooth and rough base conditions is presented. To consider the three-dimensional (3D) stress state, a 3D version of the Hoek-Brown (HB) criterion is combined with equilibrium equations to derive the governing equations by using the characteristics method. Then, a finite difference-based approach is used to solve the stresses below the foundation. Finally, integration of the vertical stress on the foundation base is performed to determine the qu of a strip foundation with different base conditions. Validation of the proposed approach is performed through comparisons with model test results and the effect of 3D strength is investigated by comparing the proposed approach with a two-dimensional HB criterion-based solution. Finally, parametric analyses are conducted to study the effects of rock constant mi, unconfined compression strength of intact rock, geological strength index, Poisson’s ratio of rock mass, and foundation width on the qu, failure zone, and vertical stress on the base. The results show that ignoring 3D strength and unit weight of rock leads to underestimations of qu and it is important to consider the different factors when designing a strip foundation on rock mass.

Performance Check of Asphalt Mixture Layer Based on Yield Criterion and Normalized Fatigue Equation

The fatigue resistance of an asphalt mixture is crucial to the pavement’s durability and serviceability of pavement. The pavement structure is in a three-dimensional (3D) stress state. However, the conventional fatigue-resistance design index mostly considers a one-dimensional stress state, which is different from the field conditions. In this paper, the unconfined compression test (UCT), direct tension test (DTT), and splitting test (SPT) were selected to conduct the strength and fatigue tests. Based on the Ayyagari Vural yield criterion, the strength damage surface under various loading rates was established. Furthermore, a normalized fatigue equation was developed in a 3D stress state by determining the fatigue-damaged path based on the yield criterion. Finally, the fatigue-resistance models for the structural design were proposed by combining the strength models (including the octahedral shear stress model and the Ayyagari Vural model) and the normalized fatigue equation. The results indicate that the strength damage surface model is in an elliptic form, and the surface is abducent as the loading rates increase, and the Ayyagari Vural yield criterion can describe the fatigue failure process. Both fatigue-resistance design methods for pavement resistance take the interaction among three principal stresses into consideration and comply with the failure characteristics of actual pavement structures; the equivalent stress as a design index is more feasible in its actual application. This study could provide a reference for the fatigue-resistance design of asphalt mixture layers in a complex stress state and improve the reliability in the pavement structure.

Effect of natural aging time on anisotropic plasticity and fracture limit of Al7075 alloy

Numerical approaches for fracture modeling have been proposed for various metallic materials, including high-strength steels and aluminum alloys. However, the quantitative relationship between the Portevin–Le Chatelier (PLC) effect and the fracture initiation has not been yet fully explored, considering the influence of the natural aging time (NA) on the material deformation behavior. To address this gap, this study investigated the PLC effect on the anisotropic plasticity and fracture limit of Al7075 alloy according to the natural aging time. Aging behavior and PLC effects are experimentally evaluated through uniaxial tensile and fracture tests using the 3D digital image correlation method. The natural aging-dependent PLC effect is tracked in the visualization of the strain rate map and thickness strain distributions with plastic deformation. Variations in the anisotropic plasticity are captured using Barlat’s YLD2000‐2d anisotropic yield function at different natural aging times, and fracture envelopes that cover general 3D stress states are identified based on the Lou–Huh ductile fracture criterion. A natural aging-dependent fracture modeling approach is proposed to predict the influence of natural aging on the fracture limit transition. Finally, the strain- and stress-based fracture forming limit diagrams, including their corresponding magnitude of the stress vector, are represented. The results reveal that no general trend for the natural aging-dependent fracture limit based on the strain metric is observed, whereas the fracture limit based on the stress metric tends to increase with natural aging in various loading conditions.

Pre-peak acoustic emission characteristics of tight sandstone failure under true triaxial stress

In this study, experimental investigations were conducted on low-porosity sandstone specimens to reveal their pre-peak cracking characteristics under constant mean stress (σm, 100 MPa) and varying stress Lode angles (θσ, −30° to +30°). The acoustic emission (AE) characteristics 50 s before specimen failure, as well as the strengths and fracturing surface networks, are presented and discussed. The results show that the values of the three principal stresses when the specimen failure occurs change linearly with increasing θσ, in which the maximum principal stress linearly decreases. The AE energy distribution exhibits three ranges, which can be speculated as the different stages of rock cracking propagation. The AE peak frequency is distributed unevenly in seven bands, in which over 90% of the AE signals are located in the 75–95 kHz and 96–130 kHz ranges, which correspond to the tensile and shear cracks, respectively. The RA-AF dynamic evolution curves reveal that the ultimate failure is driven by shear fractures under 3D stress states (for all θσ). The pattern of macrocrack networks shows the “double-X″ through-going mode, and clear tensile and shear areas were directly observed on the fracturing surfaces. These findings are deemed essential for determining the failure mechanism of deep engineering rock masses.

Developing a Three Dimensional (3D) Elastoplastic Constitutive Model for Soils Based on Unified Nonlinear Strength (UNS) Criterion

To achieve versatility, a unified nonlinear strength (UNS) criterion is put forward for capturing the complicated strength behaviors exhibited by geomaterials under three dimensional (3D) stress paths. The UNS criterion, widely covering meridian planes and octahedral planes, can serve for describing the nonlinear strength behaviors exhibited by soils, as well as confirm how the intermediate principal stress affects the strength of different materials. Based on UNS strength criterion, an elastoplastic constitutive model is presented, with the purpose of predicting the strength as well as deformation behavior exhibited by soils under 3D stress conditions. Besides, although the proposed model is extremely simple, it is fit for predicting the results of true triaxial tests in related literature with the help of the UNS criterion, and meanwhile can confirm how intermediate principal stress affects material strength and material deformation when the stresses are different.

2D closed form solution for bending of edge bonded dissimilar beams: An application of EKM

Dissimilar material like composite laminated materials combined with metals like aluminium/steel have been extensively used to reduce the weight in automotive industries and aerospace industries. Moreover, prosthetic limbs are also designed to have varying material along the length for better suitability. At these interfaces, 3D state of stress will exist. Therefore, accurate estimation of deflection and bending stresses is very important and will lead to better design of structures. The objective of this paper is to develop a closed-form bending solution for edge-bonded beams in two dimensions (2D). Beams can be segmented along x-axis and can have any materials. Using Extended Kantorovich method, governing equations are formulated in mixed form and two sets of ordinary differential equations are obtained. Continuity of displacement and stresses are satisfied at interface of each segment. Two segmented beam having aluminium and Gr/Ep equal and unequal segment are considered. A Four segmented Al/SiC beam having gradual material variation is also analysed. The deflection and stresses are compared with the finite element solution and found in good agreement. The formulation is thoroughly verified by comparing the results with 2D FE. The present development will lead to developed solution for dissimilar plates and more complex cases.

An extended invariant approach to laminate failure of fibre-reinforced polymer structures

AbstractThis paper presents the extension and validation of omni-failure envelopes for first-ply failure (FPF) and last-ply failure (LPF) analysis of advanced composite materials under general three-dimensional (3D) stress states. Phenomenological failure criteria based on invariant structural tensors are implemented to address failure events in multidirectional laminates using the “omni strain failure envelope” concept. This concept enables the generation of safe predictions of FPF and LPF of composite laminates, providing reliable and fast laminate failure indications that can be particularly useful as a design tool for conceptual and preliminary design of composite structures. The proposed extended omni strain failure envelopes allow not only identification of the controlling plies for FPF and LPF, but also of the controlling failure modes. FPF/LPF surfaces for general 3D stress states can be obtained using only the material properties extracted from the unidirectional (UD) material, and can predict membrane FPF or LPF of any laminate independently of lay-up, while considering the effect of out-of-plane stresses. The predictions of the LPF envelopes and surfaces are compared with experimental data on multidirectional laminates from the first and second World-Wide Failure Exercise (WWFE), showing a satisfactory agreement and validating the conservative character of omni-failure envelopes also in the presence of high levels of triaxiality.

ГИПОТЕЗЫ О ДЕПЛАНАЦИИ СЕЧЕНИЯ ОТ ДЕФОРМАЦИЙ СДВИГА В ЖЕЛЕЗОБЕТОННЫХ КОНСТРУКЦИЯХ, ИСПЫТЫВАЮЩИХ КРУЧЕНИЕ С ИЗГИБОМ

The article analyzes the hypothesis of deplanation of angular deformations in reinforced concrete structures during bending with torsion. A simple method from the field of the grid methods for approximating the deformations of complex functions under the 3D stress state is considered. Diagrams of angular and linear deformations in such structures were built and analyzed to find these deformations, bending and torques perceived by the concrete of the compressed area using the accepted hypotheses and coefficients for projecting normal and shear stresses (strains) through diagrams of compressed concrete and working reinforcement. Approximating the calculated section by small squares, a spatial surface of deformations was constructed. This is with the corresponding gradients of these deformations in the section with a crack, and on this basis, expressions were written for the total longitudinal deformations in the section and shear deformations in the section of the element during bending with torsion. Using the membrane analogy of the Timoshenko-Goodyear stress functions and the proposed version of new complex functions from the field of the grid method, an analysis was made of its error, when we find the value of complex functions at the considered characteristic points (2%) and at any points of the cross section (7%). The use of the proposed hypotheses and the given version of complex functions makes it possible considering the warping of the 3D stressed section of a reinforced concrete element with cracks undergoing bending with torsion.

On strength prediction of laminated composites

Great challenge exists in prediction of a laminate strength, due to too many tough issues involved. The constitutive relation of each lamina in the laminate is described by Bridging Model. Both matrix plasticity and interface debonding induced slippages have been taken into account. Significant differences do sometimes exist between predicted laminate strengths with and without matrix plasticity. Whereas stress calculation and failure detection for the fiber is relatively easy, the homogenized matrix stresses obtained micromechanically must be converted into true quantities prior to a failure assessment. The conversion for the bi-axial transverse stresses is re-established in this paper. A matrix tensile failure subjected to a 2D (two-dimensional) stress state is detected by a recently developed physics based failure criterion, while that under a 3D stress state is estimated using Tsai-Wu's criterion. Matrix compressive failures under both 2D and 3D stress states are assessed using physics based slippage failure criterion in which the normal stress influencing coefficient for the different stress state should be different. A matrix failure induced lamina failure can correspond to an ultimate failure only when an additionally critical strain condition is fulfilled. These conditions are established in the paper. All of the test cases in the first and second world wide failure exercises (WWFEs) have been analyzed using the original input data provided by the exercise organizers. All of the predictions agree well or reasonably with the available experiments. All the analyzing formulae involved are analytical and in closed-form, without any iteration for a solution.

Three-dimensional finite element simulations of milling carbon/epoxy composites

Mechanical milling is an extensively used finishing process for fiber-reinforced polymer (FRP) composites. To better understand the characteristics in machining FRPs for higher quality and efficiency, various studies have been conducted based on the finite element (FE) method as it can give insight on the mechanisms of material removal during cutting. Yet there is still a lack of three-dimensional FE models suitable for studying the mechanisms involved in the milling of FRPs. This paper developed a 3D meso-scale FE model for simulating CFRP milling by considering the damage behavior of the composites in 3D stress state. A progressive damage model was proposed and implemented using the VUSDFLD subroutine to predict the continuous damage evolution and stiffness degradation of the laminae. The simulated cutting forces and sub-surface damage agreed with the experimental observations. Numerical studies also showed that the failure criterion affected the simulated cutting forces and sub-surface damage. Upon increasing the tool edge radius, the simulated sub-surface damage affected a larger zone beneath the machined surface; however, the simulated cutting force and the sub-surface post-failure-initiation damage area experienced no obvious changes for the studied edge radius range.

An Open-Access Stress Magnitude Database for Germany

Abstract. The stress field in the Earth's crust plays a central role in the site-selection process for a deep geological repository for high-level nuclear waste. Site selection and construction planning must take into account several factors that are influenced by the stress state. These include the excavation damage zone, the hydraulic permeability of the host rock, the self-sealing capacity, the effects of seismic events and the possible reactivation of faults as migration pathways for fluids and radionuclides. Likewise, the initial stress state is of central importance for the long-term studies to prove site safety over 1 Ma. To obtain a continuous description of the current 3D stress state, 3D geomechanical numerical models are used. These models have to be calibrated with data on stress magnitudes to obtain robust predictions. One of the central goals of the SpannEnD project (Spannungsmodell Endlagerung Deutschland, http://www.spannend-projekt.de, last access: 31 October 2021) was to build the first comprehensive and publicly accessible stress magnitude database for Germany, including a quality ranking of the data compiled from different methods. This database is the logical extension of the database of the World Stress Map project, in which so far only information on stress orientations and the stress regime has been compiled systematically. We present this first compilation of stress magnitude data published and made available by Morawietz et al. (2020). The stress data density is generally low and heterogeneous, so that a model calibration at the scale of a site model is not possible. Therefore, the main objective of the SpannEnD project is to develop a 3D geomechanical numerical model for the whole of Germany. The resulting 3D stress field will provide the basis for regional and local models in a later phase of the site selection process. Details on this are presented in three complementary contributions in this symposium by Reiter et al., Röckel et al. and Ahlers et al. The new Geology Data Act (Geologie-Datengesetz) now allows access to considerably more data, which will be incorporated into an update of the database after assessment according to the defined quality criteria. This database extension will improve the reliability of the predictions of the geomechanical models on different spatial scales.

Accurate Effective Stress Measures: Predicting Creep Life for 3D Stresses Using 2D and 1D Creep Rupture Simulations and Data

Operating structural components experience complex loading conditions resulting in 3D stress states. Current design practice estimates multiaxial creep rupture life by mapping a general state of stress to a uniaxial creep rupture correlation using effective stress measures. The data supporting the development of effective stress measures are nearly always only uniaxial and biaxial, as 3D creep rupture tests are not widely available. This limitation means current effective stress measures must extrapolate from 2D to 3D stress states, potentially introducing extrapolation error. In this work, we use a physics-based, crystal plasticity finite element model to simulate uniaxial, biaxial, and triaxial creep rupture. We use the virtual dataset to assess the accuracy of current and novel effective stress measures in extrapolating from 2D to 3D stresses and also explore how the predictive accuracy of the effective stress measures might change if experimental 3D rupture data was available. We confirm these conclusions, based on simulation data, against multiaxial creep rupture experimental data for several materials, drawn from the literature. The results of the virtual experiments show that calibrating effective stress measures using triaxial test data would significantly improve accuracy and that some effective stress measures are more accurate than others, particularly for highly triaxial stress states. Results obtained using experimental data confirm the numerical findings and suggest that a unified effective stress measure should include an explicit dependence on the first stress invariant, the maximum tensile principal stress, and the von Mises stress.