Triaxial Stress State Research Articles

Intragranular three-dimensional stress tensor fields in plastically deformed polycrystals.

The failure of polycrystalline materials used in infrastructure and transportation can be catastrophic. Multiscale modeling, which requires multiscale measurements of internal stress fields, is the key to predicting the deformation and failure of alloys. We determined the three-dimensional intragranular stress tensor fields in plastically deformed bulk steel using a high-energy x-ray microbeam. We observed intragranular local stresses that deviated greatly from the grain-averaged stresses and exceeded the macroscopic tensile strength. Even under deformation smaller than the uniform elongation, the intragranular stress fields were in highly triaxial stress states, which cannot be determined from the grain-averaged stresses. The ability to determine intragranular stress tensor fields can facilitate the understanding and prediction of the deformation and failure of materials through multiscale modeling.

Evaluation of Brittle Failure Characteristics using the Finite Method and Discrete Element Method

The relationship between brittle failure and applied stress conditions using finite difference method and discrete element method was evaluated. The characteristics of brittle failure were classified into the initiation stress, extent and depth. FLAC3D and PFC3D were adopted as FDM and DEM, respectively. Numerical experiments were performed under triaxial stress conditions and compared with the previous experiments. The CWFS model in FDM could simulate the V-shaped failure well. The magnitude of the differential stress perpendicular to the opening axis and the extent/depth of failure were linearly proportional. Spalling could be simulated as particle separating in DEM. The crack initiation stress in DEM can be a good indicator of when failure starts in the experiment or field. PFC3D can reflect the effects of 3D stress conditions, but there were difficulties in distinguishing the depth and extent of failure. Therefore, it is necessary to select appropriate numerical methods in assessing brittle failure.

Version of the deformation theory of plastic ductility of concrete in a plane stress state

Reinforced concrete as one of the main materials for a wide class of building structures for civil, industrial and transport purposes has a number of specific properties: physical nonlinearity, anisotropy, and crack formation. The behavior of reinforced concrete in the elastoplastic stage before its destruction is more characterized by deformation of concrete. It is shown that the physical nonlinearity of concrete is due to plastic deformations, which are characteristic of various types of stress state. For a triaxial stress state, the system of equations in the mechanics of a deformable solid, it includes two groups of formulas that combine nine equations that include 15 unknowns (three displacements, six strain components, and six stress components). In order for the system to be closed, it is necessary to supplement it with six equations. Such equations are the basic physical relationships that relate the six stress components to the six strain components. The use of linear relationships between stresses and strains introduces the greatest error in the assessment of the stress-strain state (NDS) of structures made of materials with the properties of nonlinearly deformable bodies. In this regard, the more correctly the physical law defining the correlation reflects the material, according to which the material resists various types of deformations, the less error will be allowed in the assessment of the NDS of structures. The article proposes a new approach to the construction of basic physical relationships based on an invariant solution to the problems of mechanics of a deformable solid for concrete in a plane stress state. The correspondence of the proposed dependences to the real stress and deformable state of the material is shown.

Effect Of Stress Triaxiality On The Strain Concentration Factor

Axisymmetric machine element with irregularities such as notches encountered with effects of stress triaxiality on the strain concentration factor (SNCF) at the reduced section. The effect of notch geometries on the triaxial stress state development in the critical section of a notched cylindrical bar is studied here using FEM. In addition, the effect of triaxial stress state (TSS) on the SNCF is evaluated. To this end, a notched cylindrical bars with notch depths from extremely deep notch (do/Do = 0.2) shallow notch (do/Do = 0.95) has been employed. The results show that the notches introduce a TSS at the critical section, which strongly affected by the notch depth as well as the notch radii. In this paper, a new concentration factor is introduced as the ratio of the stress triaxiality factor at the notch root (TFNR) to the average triaxiality on the critical section (), i.e. the triaxiality concentration factor KTF. The numerical results reveal that the variation of the average triaxiality factor with total strain shows the same trend as that of the SNCF. The variation of the elastic values of TFCN, , , and SNCF with do/Do and show that the minimum TFNR leads to the maximum elastic SNCF. It is prominent that elastic TFNR is less that elastic TFCN for 0.2 ≤ do/Do ≤ 0.85, while it is greater for shallow notches. The current results indicate a strong compatibility between the newly defined triaxiality concentration factor and the SNCF up to general yielding.

X-ray diffraction assessment of complex sample states on the basis of “combined analysis”

The multi-reflection grazing-incidence X-ray diffraction technique and the analysis of the thereby recorded datasets are extended in this paper on hexagonal lattice types with distinct elastic anisotropy. The evaluation of the complete stress tensor is based on X-ray diffraction pattern decomposition and considers for texture by the spherical harmonics approach. This procedure, following the directives of “combined analysis”, enables the extraction of the mean macroscopic strain/stress tensor weighted by texture assuming a quasi-isotropic state with respect to sample elasticity. The procedure has been applied to study plastic deformation and the resulting evolution of triaxial stress states in zinc green-compacts after uniaxial compression. Due to the application of asymmetric diffraction and associated with the chosen incidence angle the method focuses on the near-surface region.

Experimental and Numerical Investigation on Ductile Fracture of Steel Pipelines

Abstract Rupture of steel pipelines leads to the loss-of-containment that may be accompanied with loss of life or damage to property and environment. Therefore, the understanding of the fracture characteristics of steel grades used in the pipelines is essential for a safe and reliable design. In this study, a set of small-scale fracture tests was designed and conducted in order to characterize the fracture of X65 steel grade. The experimental results show that not only is the fracture strain dependent on the triaxial stress condition but also the three-dimensional nature of the stress field considerably affects the ductile fracture toughness. Moreover, parallel finite element (FE) simulation of experiments were conducted and a hybrid experimental–numerical approach was used to calibrate the Mohr–Coulomb fracture criterion and obtain the equivalent plastic strain to fracture of X65 steel as a three-dimensional function of stress triaxiality and Lode angle. An engineering application friendly ductile fracture model is proposed for X65 steel pipelines.

Field experiments on multi-stage chemical diversion in low-permeability HPHT reservoirs

Efficiently producing gas from low-permeability reservoirs relies on a complex and dense fracture network. Since mechanical plugs or isolation tools become less reliable at high pressure and high temperature, chemical diversion is likely one of the most promising stimulation methods for the multi-stage hydraulic fracturing. To meet field operation needs, a series of diverters has been developed; they have different shapes and sizes, and can be applied in reservoirs with temperatures up to 200 °C and pressures up to 140 MPa. After the stimulation, diverters can degrade at the reservoir condition, leaving no residue that causes formation damage. Laboratory evaluation methods were introduced that could obtain the plugging process and plugging efficiency of diverters with different shapes and sizes, as well as the initiation of secondary fractures under tri-axial stress conditions. However, due to the limitation of sample dimensions, these methods still cannot capture the plug formation and fracture initiation in the middle section of previously generated fractures, and thus unable to evaluate the intra-stage diversion that is more crucial to the generation of the complex fracture network. Therefore, different diversion treatments with different diverters were tested and compared in 43 vertical wells in the Tarim Basin with reservoir temperatures ranging from 120 °C to 181 °C. Results showed the successful operations of both intra-stage and inter-stage diversions during the multi-stage diversion treatment, where the former one seemed more effective on production enhancement. Moreover, the fiber-shaped diverters could enhance the formation of complete plugs in the previously generated fractures, and allow more secondary fractures to branch off from these fractures to achieve a better stimulation. Findings of this paper could provide crucial guidance on optimizing the diverters for stimulating the low-permeability reservoirs.

Non-orthogonal elastoplastic constitutive model for sand with dilatancy

A non-orthogonal elastoplastic constitutive model for sand with dilatancy is presented in the characteristic stress space. Dilatancy of sand is represented by the direction of plastic flow, which can be directly determined by applying the non-orthogonal plastic flow rule to an improved elliptic yield function. A new hardening parameter is developed to describe the contractive and dilative volume change during the shear process, which is coordinated with the non-orthogonal plastic flow direction. The combination of the non-orthogonal plastic flow rule and the proposed hardening parameter renders the proposed model with the ability to reasonably describe the stress-strain relationship of sand with dilatancy. The model performance is evaluated by comparing with the experimental data of sand under triaxial stress conditions.

Nonlinear Elastic Effects on High-Strength Concrete Stress Invariants

Abstract Computational modeling of concrete behavior and most failure criteria for concrete strongly rely on the use of invariants of stress. These invariants allow a straightforward geometric interpretation of stress states and concrete failure surfaces in the three-dimensional Haigh-Westergaard stress space. Although second-order linear elastic relations have been traditionally applied to evaluate stresses in concrete, the magnitudes of concrete third-order parameters based on finite deformation theory—Murnaghan parameters l, m, n—are such that, if nonlinear stress-strain relations are considered, the stress invariants’ magnitudes are directly affected. In this work, acoustoelastic stress-strain relations for a general triaxial stress state and ultrasonically calculated second- and third-order elastic parameters are used in the evaluation of three high-strength concrete mixtures with compressive strengths: f′c = 43.0, 44.8, and 56.0 MPa (6.2, 6.5, and 8.1 ksi, respectively). The results demonstrate that nonlinear elastic effects introduce anisotropy when uniaxial stresses are applied and the magnitudes of concrete elastic and shear moduli change because of the applied stress field. When triaxial loads are considered, all stress invariants—first (I1), second (I2), and third (I3)—also undergo variations in their magnitudes. The stress-induced anisotropy due to uniaxial loading application and also the changes in the stress invariants’ magnitudes calls attention to the importance of considering nonlinear third-order effects in cement-based materials’ stress-strain relations and failure criteria.

Mechanical response of three semi crystalline polymers under different stress states: Experimental investigation and modelling

The mechanical responses of high‐density polyethylene (HDPE), polypropylene (PP) and polyamide 6 (PA 6) were experimentally investigated for a wide range of stress states and strain rates. This was accomplished by testing numerous specimens with different geometries. The uniaxial compression of cylindrical unnotched specimens and the uniaxial tensile behaviour of dumbbell specimens at different strain rates, was determined. A series of biaxial loading tests (combined shear and tension/compression, pure shear, pure tension/compression) using a designed Arcan testing apparatus were also performed. Flat and cylindrical notched specimens with different curvature radii were additionally tested in order to explore a wider range of stress states. The Drucker‐Prager yield criterion was calibrated with a set of experimental data, for which analytical formulae for stresses are available, and then applied to predict the deformation behaviour under different stress states, prior to strain localization. The results of the numerical simulations show that the Drucker‐Prager model can capture the initial elastic range and the post‐elastic response very satisfactorily. For triaxial and biaxial stress states there is a good agreement, however some load‐displacement responses are only satisfactorily described. Deviations observed in the predicted and experimental results are very likely attributed to the third invariant stress tensor, which was not explored in the model calibration. The evolution of stress triaxiality and Lode angle parameters with equivalent plastic strain were extracted and analysed for several specimens. The results show a plastic yielding behaviour sensitive to the stress state, which can be attributed to different combinations of stress triaxialities and Lode angle parameters.

Influence of gas migration on permeability of soft coalbed methane reservoirs under true triaxial stress conditions.

The permeability of the coal body is the key parameter restricting the efficient extraction of coalbed methane, and scholars have analysed it from two angles of the change of stress state and porosity of the coal body. However, there is still a lack of study on the mechanism of gas migration and movement in soft coalbed methane reservoir under the coupling between the true triaxial stress field (maximum principal stress σ1 > intermediate principal stress σ2 > minimum principal stress σ3) and the gas pressure field. In this paper, the coal gas adsorption and seepage experiments are conducted through the self-developed true triaxial ‘gas–solid’ coupled coal mass seepage system with gas as the adsorption and seepage medium and coal briquette taking the place of soft coalbed methane reservoirs. Furthermore, the coal gas adsorption deformation model and the permeability evolution model taking gas adsorption into account are developed. Through analysis of both experimental and theoretic results, the main conclusions are drawn as follows: (i) With the increase in gas pressure, the adsorption deformation variation of coal mass is divided into a slow growth zone, a stable growth zone and a rapid growth zone. (ii) The gas adsorption deformation model developed can predict the variation trend of coal mass adsorption volumetric strains for different types of soft coalbeds, and the fitting variance of experimental and theoretical volumetric strains is above 98%. (iii) With the increase in maximum principal stress difference, the coal permeability variation curve shows two obvious turning points, which can be divided into a slow reduction zone, a rapid reduction zone and a steady reduction zone. (iv) The permeability model of coal mass considering the gas adsorption effect can reflect the variation characteristics of permeability in the rapid reduction zone, and the overall fitting variance of experimental and theoretical permeabilities is above 91%. The above results could provide a reliable experimental and theoretical basis for improving coalbed methane extraction rates.

A More Realistic Estimation of Ductility in Reinforced Concrete Beams Through Three-Dimensional Finite Elements

Ductility in reinforced concrete (RC) beams subjected to bending is generally assessed on the basis of the one-dimensional (1D) beam theory through the moment–curvature relationship of its cross-section. The inhomogeneity of concrete, however, inevitably imposes a three-dimensional (3D) stress state in any RC structural member. The importance of this work is its different approach to assessing the ductility in RC beams, taking into account this triaxial stress state. A concrete model which has been well documented experimentally in the literature is used, in conjunction with an in-house 3D smeared fixed-crack finite element code. Simply supported beams under three-point monotonic loading are used as numerical examples. After validating the results with experiments, three important parameters which affect the ductility and strength of RC beams are investigated. These are the tension reinforcement ratio, the yield strength of tension reinforcement steel and the compression reinforcement ratio. It appears that, for the first two, the general trend remains the same as with the 1D approach of the Codes, whereas the third does not affect the ductility of the under-reinforced beams, contrary to the Codes.

Modeling the mechanical response of gas hydrate reservoirs in triaxial stress space

Exploitation of gas from deep-sea methane hydrate-bearing layers might lead to some geological disasters, including marine landslides and excessive settlement of marine ground. The first offshore gas production tests for methane hydrate-bearing sediments were carried out in eastern Nankai Trough. However, knowledge on mechanical behavior of gas hydrate reservoirs with similar gradation and minerology component to the marine sediment is still insufficient. Consequently, proper modeling of geomechanical properties of methane hydrate-bearing sediments is crucial for reservoir simulation and deep ocean ground stability analysis for long-term gas production in the future. This study conducted a series of triaxial shear tests to examine the shear response of methane hydrate-bearing sediments with a similar grading curve and minerology components to the hydrate-rich sediments in Nankai Trough. The test results demonstrated that the presence of hydrate mass between sand grains altered the stress-strain pattern from strain-hardening to postpeak strain-softening. A simple constitutive model based on several empirical relationships of granular materials is proposed to describe the stress-strain relationship of methane hydrate-bearing sediments under triaxial stress condition. This model can reproduce the enhancement of shear strength, initial stiffness, and dilation behavior of methane hydrate-bearing sediments containing different amounts of fines content with a rise in the methane hydrate saturation at a wide range of effective confining pressures. The numerical results indicate that the parameter A associated with initial stiffness of stress-strain curve and the parameter α related with dilation properties are jointly governed by the confining pressure, fines content, and hydrate saturation.

Simulation of shear fracture in sheet metal forming of thick plates under triaxial stress states

In industrial sheet forming of high strength steel sheets, shear fracture has been reported to occur in narrow die shoulders where the sheet is subject to excessive bending and stretching. For high strength steel plates, which generally show low ductility, the die curvature becomes relatively large for thin sheets, increasing the risk of fracture. Under these circumstances, failure is caused by the shear stresses induced by the high through-thickness stress gradient. Since shear fracture cannot be predicted by the strain-based forming limit diagram, efforts have been made to overcome this limitation and eventually led to the triaxiality failure diagram. In this study, we investigate shear fracture occurring in the forming of thick high strength steel sheets by means of finite element analysis. To do so, we implement the TFD in the commercial finite element analysis software AFDEX through a subroutine. We compare and analyze the predicted damage values obtained for diverse damage models with those obtained via the subroutine so to gain better insight in the theory and background of shear fracture. Finally, the influence of the mesh quality on the predicted stress triaxiality-dependent damage parameter is investigated.

Fracture initiation and propagation under different perforation orientation angles in supercritical CO2 fracturing

Supercritical carbon dioxide (SC-CO2) fracturing has attracted considerable attention on exploiting shale gas and tight gas. This study investigated the influence of the perforation orientation on fracture initiation and propagation induced by SC-CO2 fracturing through a series of laboratory fracturing experiments under tri-axial stress states and water was also used as a fracturing fluid to compare with SC-CO2. It was found that breakdown pressure increases with the increase of perforation angle according to the pressure curves in SC-CO2 fracturing and hydraulic fracturing. AE energy characteristic has shown that AE energy release rate of SC-CO2 fracturing is larger than that of hydraulic fracturing, indicating that fractures induced by SC-CO2 fracturing are more complex. Fracture geometry and energy surges verified that there were more fractures induced by SC-CO2 fracturing with increase of the perforation orientation angle. It was observed that hydraulic fracturing mainly induced a bi-wing fracture while SC-CO2 fracturing induced bi-wing and tri-wing fracture. At low perforation angle (α ≤ 45°), perforation orientation has little influence on fracture propagation direction and fracture extends along the direction of the maximum principle direction in SC-CO2 fracturing. At high perforation angle (α ≥ 60°), fracture propagation shows an independence of loading stress, fractures may extend in three directions. The findings are beneficial for further understanding the mechanism of SC-CO2 fracturing and optimizing perforation parameters in SC-CO2 field fracturing.

Combined crystal plasticity and grain boundary modeling of creep in ferritic-martensitic steels: I. Theory and implementation

This paper presents a physically-based microstructural model for creep rupture at 600 °C for Grade 91 steel. The model includes constitutive equations that reflect various observed phenomena in Grade 91, and it is incorporated into a mesoscale finite element model with explicit geometry for the prior austenite grains and grain boundaries. Creep within the grains is represented using crystal plasticity for dislocation motion and recovery along with linear viscous diffusional creep for point defect diffusion. The grain boundary models include physics-based models for cavity growth and nucleation that accurately capture tertiary creep and creep rupture. Simulations of creep at 100 MPa are performed, and the contribution of each mechanism is analyzed. The overarching goal is to gain a mechanistic understanding of the material to improve the prediction of creep rupture for long service lives in elevated temperature operating conditions. The creep response of the material at different stress levels, stress states, and temperatures is studied in Part 2 of this paper in order to determine the implications of the simulations on high temperature design practice. Furthermore, the second part explores the effect of triaxial stress states on the creep response and finds a transition from notch-strengthening behavior at high stress to notch-weakening behavior at lower stresses.

Brittleness Index: From Conventional to Hydraulic Fracturing Energy Model

The success of a hydraulic fracturing (HF) operation is strongly dependent on brittleness of the formation. Several models based on mechanical testing, mineral composition, and sonic log data have been proposed to quantify the brittleness of formations known as brittleness index (BI). The limitations of these conventional BI models, in particular, the consistency and applicability at field scale with the complex in situ conditions are poorly understood. We therefore developed a novel BI model based on the hydraulic fracture propagation energy (HFPE) criterion. A set of hydraulic fracturing (HF) experiments was conducted on samples with different mineralogy at true tri-axial stress conditions for model definition, i.e., the scaling law was employed to ensure that hydraulic fracture propagation resembles the field conditions. To assess the performance of the proposed BI model in predicting the rock fracability, the obtained results of the model on different samples were compared with conventional models. It was shown that the predictions of the conventional BI models are predominantly related to the failure characteristics of the rock rather than its fracability. We showed that such BI models cannot assess the hydraulic fracturing feasibility where complex failure mechanisms (i.e., splitting, shear, friction, etc.), geological condition (i.e., in situ stress) and operational factors (i.e., injection rate, fluid viscosity, etc.) are involved. The new proposed BI model, however, successfully predicted the fracability of different rock types.

A unified elasto-plastic constitutive model for sandy and clayey soils accounting for at-rest condition

ABSTRACT Taking advantages of some existing two-surface soil constitutive models in a critical state framework, a unified two-surface constitutive model was developed for rigorous simulation of elasto-plastic deformation of different soils such as loose and dense sands, normally and over-consolidated clayey soils subjected to any type of monotonic loadings. Employing the dilatancy parameter, the strain-softening and stress-dilatancy were described using non-associated flow rule to generalise the application of the model to different soils under various stress states. The applicability of the proposed model was validated comparing the numerical simulations with the existing experimental data for variety of soils subjected to different phases of monotonic loadings. The model was found to be capable of simulating drained and undrained behaviour of clayey and sandy soils under triaxial stress conditions as accurate as the constant stress-ratio loading state. The results also demonstrated the ability of the model to determine the at-rest earth pressure coefficient (K0 ) for variety of soils suitable to be applied to the geotechnical problems affected by this parameter such as piles groups and retaining earth structures. While the formulation of hardening parameter in the integration scheme is based on volumetric and deviatoric plastic strains, the simulation of dilatancy response especially in dense sands showed an excellent agreement with the experimental data.

The effect of gas migration on the deformation and permeability of coal under the condition of true triaxial stress

The dynamic characteristics of gas flow are often not considered in a true triaxial stress environment, making the simulation test hard to meet the requirement for coal and gas outburst in reality. To solve this issue, we investigated the deformation and gas flow characteristics of coal-like briquette under true triaxial stress conditions using self-developed triaxial servo-controlled seepage equipment for fluid-solid coupling of coal containing methane. Analysis of the experimental data revealed the relationship of coal body’s stress to strain as well as the relationship of their permeability. The results showed that in the conventional triaxial and true triaxial compression seepage tests, the relationship of stress-strain-permeability varies at different stages. The stress-strain curves in the whole stress-strain process of coal show a trend of decrease after increase, while the permeability-strain curves show a trend of increase after decrease. There is a “stress-sensitive zone” in the influence of the intermediate principal stress on the minimum permeability of the coal. The gas pressure promotes the “fluctuation” strain of the coal body, resulting in an instantaneous increase in the strain rate. The research provides the experimental basis for studying the permeability of coal body in the true triaxial stress environment.

A reduced Yld2004 function for modeling of anisotropic plastic deformation of metals under triaxial loading

Solid elements are increasingly utilized in finite element analysis of sheet metal stamping processes with the rising application of advanced high strength steels (AHSSs) and aluminum alloys in auto industries as these advanced materials fail in a form of rupture with negligible necking during sheet metal stamping. However, there are rare orthotropic yield functions for the description of directional dependence of plastic deformation of body-centered cubic (BCC) materials and face-centered cubic (FCC) metals under spatial loading, especially for metals with moderate anisotropy. For this purpose, the Yld2004-18p function is revisited and modified to provide satisfactory predictability of orthotropic behavior of BCC and FCC materials under spatial loading with reduced experimental costs for the calibration of orthotropic coefficients. The reduced Yld2004 equation is utilized to characterize the directional dependence of plastic behavior of AHSSs and aluminum alloys to verify its promising accuracy and cost efficiency. The reduced Yld2004 function is also implemented into finite element analysis to assess its accuracy and computation efficiency compared with the Hill48, Yld2004-18p and the anisotropic Drucker equations. The application in both analytical prediction and numerical analysis proves that the reduced Yld2004 function provides competitive predictability of anisotropic plastic deformation for BCC and FCC metals with moderate directionality under triaxial stress states, but the cost for the parameter calibration is cut down by about half compared to the Yld2004-18p function. Considering the satisfactory accuracy, reduced cost for the parameter calibration as well as the fact that most metals are of gentle dependence on loading direction, the reduced Yld2004 function is expected to reliably reproduce actual stamping processes of the majority of AHSSs and aluminum alloys with virtual models thereby improving the credibility of both numerical design of tools and process optimization.