Stress Space Research Articles

On stress ratio equations in three-dimensional stress space for modelling soil behaviour

Constitutive models and failure criteria of soils, rocks, and other materials often need to be extended beyond the triaxial state where they are usually defined, for plane strain, axisymmetric, or three-dimensional analyses. This extension is commonly done by making the stress ratio a function of the Lode angle. This process turns two-dimensional yield, plastic potential, bounding, dilatancy and similar surfaces into three-dimensional shapes such as cones and bullets. The equations used have a range of shapes on the octahedral or π-plane between Mohr-Coulomb’s irregular hexagon and Drucker-Prager’s circle. Nine stress ratio generalization equations popular in soil mechanics are evaluated based on numerical stability, agreement with available data, and ease of implementation. The computed limits on their convexity, and the flexibility they offer in the calibration process are discussed. At the end, a new equation that satisfies all these criteria while remaining simple and easy to calibrate is proposed, implemented in a finite element model, and demonstrated to improve numerical stability and efficiency.

Pseudo-dynamic stability analysis of 3D rock slopes considering tensile strength-modified Hoek–Brown failure criterion: Seismic UBLA implementations

The Hoek-Brown (HB) criterion has found extensive application in the stability analysis of intact or jointed rock slopes, in both two-dimensional (2D) and three-dimensional (3D) contexts. In the early days, the assessment of Hoek-Brown slopes rarely incorporated tensile strength until, recently, limited studies examined the tension-controlled toppling failure at the sliding head. Nevertheless, the adapted tensile strength, which is extrapolated from the HB envelope the tensile regime, has been found overestimate experimental results. Thus, for more realistically representing the influence of rock tensile strength, the present study introduces a novel procedure for assessing the seismic stability of 3D HB slopes. In this study, the tensile strength-modified HB envelopes considering cut-off effects are derived in the principal stress space and Mohr space. To assess the slope stability, the obtained HB envelope is employed in a multi-cone mechanism based on upper bound limit analysis (UBLA). Specifically, the counter of this mechanism is determined by 10 optimized dependent rupture angles, highly considering the nonlinearity of the Hoek-Brown envelope compared to the traditional linearization strategies. As an improved seismic stability analysis, the seismic load is characterized by a modified pseudo-dynamic method. Analytical expressions for two stability indicators, i.e., the stability number and safety factor, are derived. Comparisons with numerical simulations demonstrate the rationality of the proposed procedure. Parametric analysis indicates the significant effect of the tension cut-off on the HB rock slope stability and the contour of the sliding surface.

Nucleation of Fracture: The First-Octant Evidence Against Classical Variational Phase-Field Models

Abstract As a companion work to [1], this article presents a series of simple formulae and explicit results that illustrate and highlight why classical variational phase-field models cannot possibly predict fracture nucleation in elastic brittle materials. The focus is on “tension-dominated” problems where all principal stresses are nonnegative, that is, problems taking place entirely within the first octant in the space of principal stresses.

Optimal mud pressure design using nonlinear failure criteria for wellbores in shaley sedimentary reservoir

AbstractWellbore drilling disturbs the equilibrium stress state in the rock mass, resulting in stress redistribution around the opening. Wellbore stability in the altered stress state is vital for engineering applications, as the wellbore instability results in cost overrun. Accurate estimation of rock mechanical properties, in‐situ stresses, and required mud pressure is crucial for safe drilling. If the mud pressure is lower than required, the shear failure of the rock takes place, and conversely, if the mud pressure is higher than the upper limit, tensile failure occurs. A strength criterion that can accurately predict the mud pressure may help significantly reduce non‐productive time and cost in well drilling. The commonly used Mogi‐Coulomb (MGC) failure criterion for estimating critical mud pressure neglects the few fundamental aspects of rock failure characteristics observed in the laboratory, such as nonlinear strength response in major‐minor principal stress space and prediction of multiple failure stress values near the triaxial axial extension boundary. The present study uses the Modified Mohr‐Coulomb true‐triaxial failure criterion (MMC_TT), which predicts the strength of rock better than the MGC in laboratory true‐triaxial tests to overcome the limitations. Moreover, based on the data from previously published five vertical wells in the Krishna‐Godavari basin (K‐G basin), an empirical relationship is proposed to obtain the strength parameters for the MMC_TT criterion for shaley sedimentary rocks as the existing correlations do not cater for the parameters required for MMC_TT criterion. The comparative study of the MMC_TT with MGC, Modified Mohr‐Coulomb triaxial, and Mohr‐Coulomb failure criteria, showed that for most of the K‐G basin wells, the MMC_TT criterion predicted close to the MGC. However, for well‐13, the MMC_TT criterion results are closer to the mud pressure used in actual drilling than the MGC.

Neuroinflammation modifies the relationship between stress and perivascular spaces in an elderly population with different levels of cognitive impairment.

Perivascular spaces (PVS) are fluid-filled spaces surrounding the brain parenchymal vasculature. Literature suggests that PVS may play a significant role in aging and neurological disorders, including Alzheimer's disease (AD). The aim of this study is to investigate whether the relationship between MRI-visible PVS and stress is influenced by neuroinflammation in an elderly population with different levels of cognitive impairment. Using brain MRI scans acquired at 1.5 T, PVS were quantified in a cohort of 461 individuals, consisting of cognitively healthy controls (n = 48), people with mild cognitive impairment (MCI, n = 322) and Alzheimer's disease (AD, n = 91). PVS volume fraction was calculated in the basal ganglia and centrum semiovale using a semi-automated segmentation approach. Stress was quantified with levels of salivary cortisol. Inflammatory biomarkers measured from plasma included cytokines, matrix metalloproteinases and C-reactive protein. General linear models were used to test the relationship between PVS and cortisol, when interacting with inflammatory markers. This was done on the whole cohort and within each clinical cognitive group. In the centrum semiovale, higher inflammation levels reduced the relationship of cortisol with PVS. In basal ganglia, higher levels of C-reactive protein reduced the negative relationship of cortisol with PVS. All analyses were accounted for age, sex, body mass index (BMI) and total hippocampal volume. There was a significant interaction effect between cortisol and C-reactive protein on PVS volume fraction in the MCI group. These findings suggest an influence of neuroinflammation on the PVS structure in Alzheimer's disease spectrum, and offer insight for better understanding physiological processes of cognitive impairment onset.

Anisotropy cyclic plasticity constitutive modelling for Ni-based single-crystal superalloys based on Kelvin decomposition

Nickel-based single-crystal (SC) superalloys exhibited excellent exceptional mechanical properties at high temperatures due to the elimination of internal grain boundaries, contributed a strong orientation-dependent material response. The anisotropy of SC superalloys was modeled viscoplastically from a macroscopic viewpoint based on the Kelvin decomposition theory [1] which was a decomposition of the stress space according to the elastic matrix eigen-directions to control the viscoplastic flow by Kelvin stress decoupled from each other. Compared to the classical phenomenological macro model, the proposed model effectively captures the slip deformation mechanism of SC superalloys with the inherent ability to simulate anisotropic because of the two criterions framework controlled by Kelvin stress. Compared with others, the proposed model was able to simulate time-dependent inelastic deformation and cyclic deformation behavior under complex loading. The kinematic hardening and isotropic hardening models incorporated microscopic quantities, such as dislocation density and channel phase width, connecting the macroscopic mechanical response with the microscopic state to achieve multiscale constitutive modelling. The parameter identification and finite element implementation were conducted on a SC superalloy [2]. Simulation results demonstrated the accuracy of the proposed model in predicting deformation behavior under various orientations, rate-dependent effects, isothermal and non-isothermal cyclic deformation. Comparison with the classical anisotropic matrix macroscopic phenomenological approaches highlights the superior capability of the proposed model to simulate the orientation-dependent mechanical properties of single-crystal alloys.

Influence of Strain Rate and Temperature on the Multiaxial Failure Stress Locus of a Polyamide Syntactic Foam

This study introduces a comprehensive experimental methodology allowing for the direct measurement of the rate dependent multiaxial response of polymer syntactic foams under combined direct-shear loading. The combined tension-torsion behaviour of a syntactic foam and its rate dependence are investigated for the first time.Dynamic tension-torsion experiments were conducted using a newly developed Tension-Torsion Hopkinson Bar (TTHB) enabling the measurement of the combined tensile-shear response of engineering materials at high rates of strain.The response and multiaxial failure envelope of a polyamide syntactic foam were experimentally measured and analysed to determine the combined influences of stress state, strain rate, and temperature. The multiaxial failure stress locus was defined in both the normal versus shear stress space and the principal stress space, providing a comprehensive characterisation of the behaviour of the material under various loading and environmental conditions.The suitability of existing pressure dependent failure criteria to represent the measured experimental data was also assessed. The Drucker-Prager pressure dependent criterion proved to be effective in capturing the measured quasi-static and dynamic multi-axial stress loci at different temperatures.The effects of temperature, loading rate and stress state on the deformation and failure modes were analysed by means of SEM micrographs of the tested samples.

An exact implementation of the generalized Zhang-Zhu criterion for elasto-plastic finite element calculations

The generalized Zhang-Zhu (GZZ) strength criterion was proposed as an extension to the Hoek-Brown criterion and the Mogi criterion. The introduction to mean normal stress results in a non-smooth and non-convex yield surface, which presents a challenge for updating plastic stress. Current research primarily focuses on modified smooth GZZ criteria or approximate solutions, which inevitably lead to increased computational costs or inaccuracies. In this paper, an accurate stress updating algorithm is proposed based on the original GZZ criterion. The algorithm operates entirely in the principal stress space, where numerical singularities at the intersection of yield surfaces are avoided by defining four different types of stress updating. This approach simplifies the GZZ criterion compared to its formulation in general stress space. The return mapping is employed to compute the updated stress and consistent stiffness matrix, facilitating calculations using both finite element implicit and explicit algorithms. Finally, the accuracy of the proposed method is validated using rock true triaxial test data and semi-analytical solutions for stresses and displacement around a circular opening under the GZZ criterion.

A comparative study of three types of strength criteria for rocks

Currently, the unified strength criterion (USC), the three-dimensional Hoek-Brown (3D H-B) criterion and the generalized unified strength theory (GUST) are the three types of typical rock strength criteria. In this paper, a comparative study of the three types of strength criteria is performed for rocks. Based on the nonlinear characteristics of rock strength on meridian and deviatoric planes, the USC can predict rock strength under triaxial stress state. The USC is composed of two failure functions on meridian and deviatoric planes. The failure surface of the USC in principal stress space satisfies smoothness and convexity. The predicted strength for five types of rock under the true triaxial tests were compared among the USC, the GUST, and the 3D H-B criterion. The results indicate that the USC can effectively reflect the influence of the intermediate principal stress on rock strength and accurately predict rock strength under both triaxial tension (σ1=σ2>σ3) and triaxial compression (σ1>σ2=σ3). Additionally, the conventional triaxial tests were conducted on other eight types of rock to measure the strength on meridian plane. The predicted strengths for the eight types of rock on meridian plane were compared between the USC and the original H-B, which suggests that the USC is suitable for various types of rock and provides higher accuracy.

A new three-dimensional strength criterion considering the transverse isotropy based on the α-Spatial Mobilized Plane

Natural geotechnical materials are affected by sedimentation and exhibit significant anisotropy. To study the transverse isotropy characteristics of soil, the influence of intermediate principal stress and loading direction must be considered. Currently, research on transverse isotropy primarily focuses on the modified stress space, which is cumbersome to apply in multi-yield surface constitutive models. To describe the three-dimensional mechanical properties of geomaterials in real stress space, the α-Spatial Mobilized Plane strength criterion is introduced. Then, combined with the structure tensor, the transverse isotropic three-dimensional strength criterion can account for the effect of the loading angle. Finally, the three-dimensional strengths of Fukakusa clay, unsaturated SP-SC soils, uncemented Monterey sands, Yamaguchi marble, San Francisco Bay mud, Toyoura sand, and Santa Monica Beach sand are predicted on the π-plane. The results show that the αmn-SMP criterion, in the context of transverse isotropy, can describe the three-dimensional mechanical properties reasonably, and it can provide an accurate strength criterion for geotechnical engineering practice.

Development of a Novel Beam-Based Finite-Element Approach for the Computationally Efficient Prediction of Residual Stresses and Displacements in Large 3D-Printed Polymer Parts

Recently, 3D printing of large, structural polymer parts has received increasing interest, especially for the creation of recyclable structural parts and tooling. However, the complexity of large-scale 3D polymeric printing often dictates resource-intensive trial and error processes to achieve acceptable parts. Existing computational models used to assess the impact of fabrication conditions typically treat the 3D-printed part as a continuum, incorporate oversimplified boundary conditions and take hours to days to run, making design space exploration infeasible. The purpose of this study is to create a structural model that is computationally efficient compared with traditional continuum models yet retains sufficient accuracy to enable exploration of the design space and prediction of part residual stresses and deformations. To this end, a beam-based finite element methodology was created where beads are represented as beams, vertical springs represent inter-bead transverse force transfer and multi-point, linear constraints enforce strain compatibility between adjacent beads. To test this framework, the fabrication of a large Polyethylene terephthalate glycol (PETG) wall was simulated. The PETG was modeled as linearly elastic with an experimentally derived temperature-dependent coefficient of thermal expansion and elastic modulus using temperature history imported from an ABAQUS thermal model. The results of the simulation were compared to those from a continuum model with an identical material definition, showing reasonable agreement of stresses and displacements. Further, the beam-based model required an order of magnitude less run time. Subsequently, the beam-based model was extended to allow separation of the part from the printing bed and the inclusion of part self-weight during fabrication to assess the significance of these effects that pose challenges for existing continuum models.

Ratcheting assessment of additively manufactured SS316 alloys at elevated temperatures within the dynamic strain aging domain

The present study has evaluated the accumulation of plastic strain in additively manufactured (AM) stainless steel 316 samples undergoing asymmetric loading cycles at elevated temperature range where the dynamic strain aging (DSA) phenomenon occurred. The ratcheting response of AM steel samples were assessed through the use of the combined isotropic-kinematic hardening framework. The dynamic strain aging was introduced into the dynamic recovery term of the Ahmadzadeh-Varvani (A-V) kinematic hardening rule by an exponential function ψ(p, T). This function enabled the hardening framework to evaluate the ratcheting of AM steel samples at operating temperatures within the DSA domain. The DSA temperature range for AM SS316 samples fell between 700 K to 1140 K. Within this temperature domain, the interaction between solute atoms and dislocations led to an increase in material strength and suppressed the magnitude of ratcheting over the loading cycles. The influence of DSA on the backstress evolution and the yield surfaces translated in the deviatoric stress space and their subsequent influence on the ratcheting of samples using the A-V model were further discussed. The predicted ratcheting curves at the DSA temperature domain were found in close agreement with those of measured values.

Delamination free forming of novel high interface strength metal-polymer laminates

A new class of metal-polymer-laminates, fabricated using wire mesh and “steel-chips” interlayer, is shown to have significantly high interface strength. A finite element model, calibrated using tensile, lap shear, and peel tests, is used to predict the deformation and failure of the laminates. Formability of the developed laminates is studied using the limiting-dome-height tests. The metal-polymer-laminates do not delaminate in any of the tests. A complete shift of forming-limit-curve towards biaxial-tension region is observed in strain-based forming limit diagram for the polymer side. Strain-path independent forming-limits are plotted in stress space, to negative the effect of non-proportional strain histories.

To justification of concrete strength criterion at biaxial compression

Introduction. Numerous experimental data from Russian and foreign researchers indicate that the classical plasticity hypotheses do not take into account the different resistance to uniaxial tension and compression, the influence of the spherical tensor. At the same time, experiments show that the ultimate resistance depends on the type of stress state, and hydrostatic pressure increases the strength and plasticity of solids. Aim. To establish the dependence of the influence of the second component of stresses during biaxial compression of concrete on the parameters of the complete diagrams of deformation of the material σbR and εbR it is necessary to describe these diagrams, and to construct a closed curve on the plane of the main stresses (concrete strength criterion). Materials and methods. Based on the experimental materials of foreign and domestic researchers, including the experiments of the authors of the article, methods of mechanics of deformed solids, limit curves and a closed curve on the plane of the main stresses in the form of a chain line forming the smallest area strength surface in the form of a catenoid are proposed. Results. The article provides an analysis of the known strength criteria from the point of view of their geometric interpretation in the stress space. It is shown that these studies relate mainly to metals and metal structures, and for the design of reinforced concrete and steel-concrete structures in a complex stress state, it is necessary to develop an appropriate criterion for the strength of concrete. Conclusions. As a result, the proposed limit curves and the surface (material strength criterion) on the plane of main stresses in the form of a catenoid accurately reflect the behavior of concrete under conditions of uniform and uneven flat stress state, and the equation of the surface in the form of a catenoid is a generalization of the equations of limit curves for each of the three types of flat stress state. At the same time, there is no overestimation of strength in the "compression – compression" area in this case.

Polyaxial failure criteria for in situ stress analysis using borehole breakouts: Review of existing methods and development of an empirical alternative

Analysis of compressive wellbore failure, or breakouts, is one of the primary methods of constraining the maximum horizontal stress in deep boreholes. To estimate stress using the observation of breakouts, one needs to measure the breakout width from image logs and use a failure theory to predict the stress that led to the development of the measured breakout. Most commonly, Mohr–Coulomb failure criterion has been used which disregards the influence of intermediate stress on strength. Hence, various polyaxial criteria have been proposed to include this effect. Here, we first review some selected polyaxial criteria: Drucker–Prager, Mogi, Modified Wiebols–Cook, and Modified Lade, and we conclude that their application in breakout analysis may be cumbersome and often unreliable. One reason for these problems is that the criteria are defined using stress invariants, while the stress estimation is most easily performed and analyzed in the principal stress space. Therefore, an alternative is to define the polyaxial criterion as a simple relation between maximum and intermediate stresses. We propose to define such an empirical criterion as a second order polynomial which fits trends observed in polyaxial laboratory strength data. Such approach allows to limit strength overestimation, often associated with the use of previous polyaxial criteria, and to easily relate uncertainties in strength estimation to uncertainty in maximum horizontal stress prediction.

Nonlinear seismic response analysis of liquefiable sites based on effective stress method

In this study, the one-dimensional nonlinear Matasovic model was extended to the two- and three- dimensional stress space by replacing the shear strain with the generalized shear strain. The extended Matasovic model was subsequently combined with Dobry’s excess pore water pressure generation model, and the reliability of the proposed loosely coupled effective stress analysis model was verified through undrained cyclic triaxial tests and a one-dimensional site response analysis. Finally, this method was applied to an engineering site in China to evaluate the nonlinear seismic response of liquefiable sites. The results show that the proposed effective stress analysis method can capture the dynamic behavior of the soil and the generation of pore water pressure during strong shaking. Moreover, there are differences between the proposed effective stress analysis method and the total stress analysis method for liquefiable sites, which indicates the need for careful selection in practical applications.

True-triaxial strength characteristics of cement stone subjected to sulfuric acid corrosion: An experimental and theoretical study

True triaxial tests were conducted to investigate the effect of sulfuric acid corrosion on the true triaxial strength of cement stone under different acid immersion times, including the common loading path and novel loading path with constant Lode angle. The chemical composition and microstructure of cement stone under different acid immersion times (0, 7, 14, and 28 d) were examined via X-ray diffraction and scanning electron microscopy. As the duration of acid immersion increases, the concentrations of portlandite (Ca(OH)2) and calcium silicate hydrate (C-S-H) within the cement stone exhibit a gradual decline, while the presence of calcium sulfate (CaSO4) progressively rises. The microstructure of the cement stone was primarily affected by two factors: the damaging and strengthening effects of sulfuric acid and CaSO4, respectively. The damaging effects of the acid were dominant for days 0–14 of immersion, with continuous increase in microstructure damage. Subsequently, the strengthening effect of CaSO4 became apparent for days 14–28 of immersion, with continuous improvement in microstructure integrity. Consequently, the strength of cement stone exhibited a diminishing trend during the initial immersion period of 0–14 days, followed by an enhancement in strength from 14 to 28 days of immersion. Furthermore, a three-dimensional strength criterion was proposed by combining the Hoek–Brown and Matsuoka–Nakai criteria. The failure envelope first shrank and then expanded with increasing immersion time, accurately describing the variation in strength in the three-dimensional principal stress space.

A method to represent the strain hardening effect in the hyper-elastic model within a fully Eulerian framework

We propose a method to treat the strain hardening effect in the hyperelastic model. Coupled with the level set interface capturing method and a real ghost fluid method, it is formulated in a fully Eulerian framework. The HLLD approximate Riemann solver is used to obtain the interface state. We use the elastic inverse deformation gradient tensor to track the material deformation state. Once the solid starts to yield, along the normal direction in stress space, an iterative algorithm is used to relax the elastic inverse deformation gradient tensor to target value where the equivalent stress meets the requirement of consistency condition in plasticity. This approach is intuitive and can avoid the use of the physical delay time, which is difficult to calibrate from experiments. We use the Wilkins flying plate impact, cylindrical shell collapse, Taylor impact and the conical-nosed projectile penetration test cases to validate the method. The results show good agreement with the theoretical analysis and experiments. This approach can also apply without further extension when thermal softening effect or damage models are considered.

The mesoscale mechanics of compacted ductile powders under shear and tensile loads

A discrete numerical analysis of the yield and damage properties associated with a cohesive granular system composed of ductile particles is hereby presented. Such a modelling approach aims at better understanding damage mechanisms which are often encountered during the powder compaction process, widely used in the metallurgical and pharmaceutical fields. The analysis was based on the micromechanical modelling of an idealised granular system in the framework of the multi-particle finite element method, in which particle deformation was fully taken into account. An adhesive interaction law, presented in Audry et al. (2024), was used in the purpose of estimating the averaged mechanical properties associated with the modelled elementary volume. The focus was put on tensile and highly deviatoric loadings, which are usually related to the failure of powder compacts. The specific contact area developed through inter-particles contacts was used as an indicator of the mechanical strength of the elementary volume. Threshold surfaces corresponding to yielding and contact decohesion mechanisms were plotted in the stress space.

Strength characteristics of different stress spaces of red sandstone under loading and unloading tests

Red sandstone tests were conducted on the rock mechanics test system to clarify the strength and failure characteristics of red sandstone under triaxial loading and different unloading confining pressure rates. Strength change rules of the red sandstone in various stress spaces were analyzed. Results show that a large initial confining pressure indicates a long yield section and large axial strain. A small unloading confining pressure rate indicates a long yield section and large axial strain at failure under the identical initial confining pressure. The confining pressure reduction increases with the unloading confining pressure rate increasing when the red sandstone fails. The stress path of unloading confining pressure weakens the red sandstone cohesion and strengthens its internal friction angle. The general octahedral shear stress formula for judging whether the red sandstone fails under loading and different unloading rates was acquired on the basis of the octahedral shear stress characteristics of the red sandstone. In addition, a 3D empirical criterion for evaluating the red sandstone strength was constructed through the deviator plane function of Eekelen and the strength envelope curve on the meridian plane. A great initial confining pressure indicates serious internal damage of the red sandstone at failure under the identical unloading rate of confining pressure. A low unloading rate of confining pressure indicates internal serious damage of the red sandstone at failure when the initial confining pressure is the same. The analysis of macroscopic failure characteristics revealed that the stress path significantly influences the failure mechanism of the red sandstone. The results are instructive for determining the stability of underground engineering of the red sandstone.