International Journal of Geomechanics

An explicit analytical solution is developed for the problem of steady, two-dimensional seepage in the vertical plane through a fully penetrating, semipermeable cutoff wall. The flow is driven by a head drop across an overlying dam, whose width is equal to the width of the wall. Results including the discharge beneath the dam, the differential head across the wall, the hydraulic efficiency of the wall, and the exit gradient along the bed of the downstream reservoir are presented as dimensionless charts. Rigorous comparisons are made with solutions obtained by approximate methods, which are applicable to more-general problems. The analytical solutions and approximate methods are used to estimate the discharge and the differential head at the Shikwamkwa Dam in Ontario, Canada, and to assess the need for more-detailed seepage modeling.

Several complexities are inherent in numerical simulation of staged construction modeling of large-diameter steel pressure pipes. A comprehensive robust three-dimensional (3D) nonlinear finite-element analysis model was developed and verified using three experimental field tests conducted at the Rolling Hills Booster Pump Station, Fort Worth, Texas, to simulate the behavior of buried steel pipes during staged construction installation. Controlled low-strength material (CLSM) was used as the embedment material, and the depth of the CLSM embedment and the trench width varied. The developed numerical model includes three possible nonlinearities, including geometric, material, and contact nonlinear algorithms. A large deformation algorithm was considered in the finite-element model and its associated analysis using a total Lagrangian formulation. The contact between each soil layer, soil-to-trench wall, and soil to pipe was carefully implemented. The lateral effect of compaction was identified in previous studies for the soil box test (rigid trench wall), and uniform thermal loading was applied to simulate the stresses induced because of compaction forces on the pipe and trench walls. A formula was derived using mechanics of material formulations to identify the required temperature for the various embedment configurations. Finally, the vertical and lateral load deformations and strain plots obtained from the finite-element analysis results were compared with the full-scale experimental test results during the staged construction process and after the application of surcharge load. The results of the finite-element analysis accurately modeled and simulated the test results.

This paper reports the results from numerical modeling of spudcan deep penetration through three-layer clays. Two typical seabed strength profiles commonly encountered in the field, and identified as critical for potential punch-through failure, were considered: (1) uniform stiff-soft-stiff clay and (2) nonuniform clay with an interbedded stiff clay layer. Three-dimensional large deformation finite-element (LDFE) analyses were carried out with and without simulating strain softening and strain rate dependency of the shear strength. The results were compared with previously published LDFE results and centrifuge test data. A detailed parametric study was undertaken, varying the relevant range of layer thicknesses (relative to the spudcan diameter), strength ratios, normalized strength, and strength nonhomogeneity. Punch-through and rapid leg penetration (for stiff-over-soft) and squeezing (for the reverse) were demonstrated by the penetration resistance profiles and associated soil failure mechanisms. The results emphasized that the resistance profile from a spudcan continuous penetration in a three-layer sediment (e.g., stiff-soft-stiff) may be significantly different from the combined profiles resulting from two separate analyses on first-second layers and second-third layers, even if an appropriate depth factor is accounted for. For uniform stiff-soft-stiff clays, the presence of the bottom stiff clay layer within a short distance of one diameter affected spudcan bearing response in the top layer and suppressed the potential for punch-through failure (apart from rapid leg run for $t_1/D=0.75$ and $s_{u1s}/s_{u2}=3$). A trapped soil plug beneath the advancing spudcan, following penetration of the top stiff layer, caused the bottom stiff layer to be sensed earlier and led to enhancement of the limiting squeezing depth. For nonuniform clay with an interbedded stiff layer, spudcan bearing response was dominated by squeezing (with limiting squeezing depth of approximately $0.18D$) in the top soft layer and potential punch-through and rapid leg run (for $t_2/D\ge 0.5$ and $s_{u2}/s_{u3s}=2.5$) in the middle and bottom layers. In contrast to ISO recommendations, the soft soils in between the advancing spudcan base and the stronger layer did not squeeze out completely. Instead, some of the trapped material was forced into the underlying stronger layer. Punch-through distance was found to reduce significantly with increasing positive strength gradient of the bottom layer.

In the case of large-diameter monopiles, pile deflection against lateral loads must be predicted accurately to determine the direct influence on the superstructure. One-dimensional (1D) linear models of the pile-soil system, which are simple to analyze, are more commonly used in practice than the sophisticated three-dimensional (3D) simulations. This study proposes an equivalent lateral strain concept to incorporate soil nonlinearity into an equivalent linear analysis of the pile-soil system. The spring constant is obtained by iterating over soil stiffness at different strain levels and finding equivalent lateral strain from updated pile deflection at each iteration. It requires the maximum shear modulus and modulus reduction curve for different layers of the soil profile. Correlation between the equivalent lateral strain and pile deflection has been developed by comparing results from 1D and 3D simulations of long piles. The existing correlations between the lateral spring constant and Young’s modulus of soil are too approximate to help in quantifying nonlinearity of soil accurately in this comparative study. Therefore, a more accurate relationship between spring constant and Young’s modulus of soil for long piles has been proposed by incorporating relative stiffness of soil and pile.

The unprecedented increasing cost of steel coupled with the superior mechanical and anticorrosion properties of fiber-reinforced plastic (FRP) creates a huge opportunity for FRP pipes to take the market share of steel and be the pipe of choice for geotechnical engineers for a variety of applications ranging from potable-water systems to feed lines and penstock for hydroelectric power plants. Recently, very flexible large-diameter FRP pipes have been installed in eastern Canada and the United States to replace old wood-stave pipes to convey water to drive the turbines of small hydroelectric plants. The behavior of buried very flexible pipes is one of the complex soil-structure interaction (SSI) problems in geotechnical engineering and has not been studied thoroughly until now. The flexibility classification of pipes is determined based on their effective stiffness. Accordingly, flexible pipes will have an effective flexibility of $10\hspace{0.17em}\text{kPa}/\text{m}$ or less. In this paper, the results of centrifuge modeling of fully buried very flexible pipes with shallow soil cover with and without surface loading are presented to supplement the very limited data that exist for such very flexible pipes. In addition, a three-dimensional finite-element (FE) model was created as part of this study to simulate the centrifuge results obtained and to investigate the effect of surface loading on the buried pipes. In addition, the results from the model testing and the FE model were compared with the predictions of two analytical solutions to assess their suitability for such an application. Finally, the positive arching predicted by the analytical methods and its practical significance were examined.

This paper presents a novel approach to improving our understanding of complicated slope failure mechanisms that can occur in strain-softening material as well as to evaluating the accompanying postfailure deformation of slopes. Three cutting-edge technologies were integrated into one numerical analysis tool: (1) a large-deformation analysis using the updated Lagrangian formulation, (2) a stable numerical algorithm for the solution of progressive failure in strain-softening materials, and (3) an $h$-adaptive mesh refinement algorithm. The significance of this study is mainly in the adoption of a proper numerical discretization for cases of slope failure in which large deformations lead to distortion of the finite-element mesh, unreliable results, and lack of convergence. The $h$-adaptive remeshing technique is used to refine the finite-element mesh around the area of large shear strain, thereby improving the accuracy and convergence of numerical solutions. This study presents a method that can be used to simulate the failure of slopes with highly strain-softening materials that experience large deformations, so that postfailure deformations can be determined. The validity of the method is demonstrated through simulation of the failure and deformation of a highway embankment, and comparing the predicted deformations with the observed record.

A recently proposed three-dimensional (3D) linear method for examining soil-building interactions based on an input seismic wave field is, after some improvements, applied to estimate seismic building responses in the reclaimed zone of Tokyo Bay, where ground motions include a considerable amount of surface waves, thus reconfirming the effects of the method in a different situation. A seismic wave field involves seismic waves propagating in a 3D medium. The proposed method was developed with the goal of adequately treating seismic surface waves trapped by a (several-kilometers) deep underground structure in a soil-building interaction system. Two simulations were carried out. The first simulation successfully reproduced surface, downhole, and building accelerograms that were recorded at one borehole station during two medium-sized earthquakes. In the second simulation, seismic responses of a midrise RC model building and a wood model building were favorably calculated at the other borehole station for the 1923 Kanto earthquake. The building responses also were compared with those calculated by two standard response analyses, demonstrating the superiority of the proposed method.

Soil arching is a common phenomenon in pile-supported, geosynthetic-reinforced or unreinforced embankments resting on soft soil. Although the system behavior under static loading is well known (soil arching and bearing effects in geosynthetic reinforcement), the bearing behavior under dynamic loading is not yet fully understood and cannot be predicted. In this paper, the properties of soil arching under dynamic load have been investigated by performing numerical studies using the FEM and model tests. Embankments with different heights under dynamic load have been investigated. The behavior of soil arching under dynamic load can be classified into two types: for the model test without a geogrid and subsoil, the soil arching collapsed under dynamic load when the ratio of the height of the embankment to the diameter of the hole is less than 3, whereas the soil arching will not collapse under dynamic load when the ratio of the height of the embankment to the diameter of the hole is greater than 3. In numerical analysis, because of the existence of the geogrid, the value of the ratio of the height of the embankment to the center-to-center pile distance ensuring the stability of soil arching under dynamic load was 1.4, which is less than the value acquired in model tests, and it indicates that the existence of a geogrid and subsoil can improve the stability of soil arching under dynamic load.

The stability condition of the tunnel face is of great importance in both conventional and modern [tunnel-boring machine (TBM)] methods of construction. Inadequate and excessive amounts of support pressure toward the face lead to surface settlement and blowout, respectively, whereas blowout occurs only in cases of pressurized shield implementation. In this research, the stability condition of a tunnel face excavated by an earth pressure balanced shield (EPBS) in Line 3 of the Tehran subway in Iran was investigated. Considering the sensitivity of urban facilities to ground movement, and to reduce uncertainty, data simulation by Monte Carlo technique was applied. Random data generated through a simulation process were used in an analytical method to calculate the factor of safety and its probability of being less than a predetermined value for a certain amount of applied support pressure. The soil was modeled by finite-element software, and the amount of face pressure at which the face collapses was calculated. These procedures were followed for face stability evaluation in nine design sections along the tunnel path. Comparisons of the results are discussed.

The strength and failure behaviors of geomaterials, such as soils and rocks, are commonly anisotropic because of lamination and sedimentation. The main purpose of this study is to investigate the effect of anisotropy on the localization pattern and bearing capacity of geostructures by finite-element simulation. To this aim, an extended Drucker-Prager yield criterion is developed for transversely isotropic geomaterials based on the Cosserat continuum. In this criterion, the internal frictional angle is related to the material principal direction and the mixed invariant of the stress tensor and the microstructure tensor. The corresponding stress update and consistent elastoplastic tangent modulus matrix are presented. Numerical results show that the localization pattern and the bearing capacity of the geostructures are very sensitive to the material principal direction as well as the anisotropic degree.

Shallow tunneling in urban areas causes settlements whose measurement, prediction, and prevention are the most important tasks facing tunneling engineers. Researchers have proposed several analytical, semianalytical, and empirical equations for these tasks. Because of the significance of the transverse settlement trough, many correlations have been suggested for its dependency on various parameters; however, investigation into the longitudinal settlement profile is still lacking. The purpose of this paper, therefore, is to study the effect of three important parameters, namely overburden depth, round length, and cohesion, on this profile in drained conditions using a three-dimensional FEM and to propose a correlation thereto. Comparison with the monitoring results from two tunnel case histories reveals the suitability of the correlation in predicting the longitudinal settlement profile in shallow tunnels.

Bishop’s effective stress equation for unsaturated soils was abandoned for some time because of the difficulties in estimating the value of parameter $\chi$. Only recently has it been recognized that the use of Bishop’s stress equation can lead to simpler and more realistic constitutive models for unsaturated soils. Based on the analysis of the equilibrium of solid particles, it was possible to establish an analytical expression for Bishop’s parameter $\chi$ from soil pore structure. It is shown that quantitative predictions of shear strength and volume change in unsaturated soils can be made using the effective stress concept.

When the generalized plasticity model with nonassociated flow rule is utilized to simulate the mechanical behaviors of sands, the asymmetric elastoplastic matrices or coefficient matrix has to be encountered in finite-element (FE) computations, posing a great challenge to the solution of large-scale asymmetric linear systems of equations. In this study, a simple symmetrization method for the asymmetric stiffness matrix attributable to the generalized plasticity model is presented and then verified by simulating a series of triaxial tests involving the asymmetric stiffness matrices. Furthermore, the symmetric and asymmetric preconditioned iterative methods are employed to solve the asymmetric and symmetrized FE linear systems of equations, respectively, arising from a three-dimensional earth dam example. Finite-element elastoplastic dynamic analyses of the dam show that compared with the solution method based on the asymmetric FE equation, the symmetrization method can produce accurate results, but with remarkably less computational time and random-access memory (RAM) requirement.

Soil liquefaction is one of the most common geohazards that is often the root cause of damage and disruption to the civil infrastructure systems. It has been tacitly considered to only occur in loose saturated sand or low-plasticity silts. A series of strain-controlled cyclic loading tests on saturated and unsaturated Nevada sand has been conducted to reveal the fact that not only saturated soils, but also unsaturated soils, can be liquefied when certain test conditions are satisfied. In this laboratory study, different initial conditions of soil specimens are tested. These conditions include the following: relative density (30 and 70%), effective confining stress (50 and 200 kPa), and degree of saturation (90, 95, and 100%). All specimens, except the one with 70% relative density, 200 kPa confining stress, and 90% degree of saturation, reach liquefaction at the end of the tests. Although it is well established that liquefaction resistance increases with relative density and confining stress and decreases with the degree of saturation, some quantitative guidance to evaluate liquefaction of soils, especially under their unsaturated states is provided from this study. These guidelines can serve as the basis to transform current geotechnical design procedure when unsaturated soils are involved.

This paper investigates the application of high-order elements within the framework of the arbitrary Lagrangian-Eulerian method for the analysis of elastoplastic problems involving large deformations. The governing equations of the method as well as its important aspects such as the nodal stress recovery and the remapping of state variables are discussed. The efficiency and accuracy of 6-, 10-, 15-, and 21-noded triangular elements are compared for the analysis of two geotechnical engineering problems, namely, the behavior of an undrained layer of soil under a strip footing subjected to large deformations and the soil behavior in a biaxial test. The use of high-order elements is shown to increase the accuracy of the numerical results and to significantly decrease the computational time required to achieve a specific level of accuracy. For problems considered in this study, the 21-noded elements outperform other triangular elements.

Based on experimental data and the flow property analysis of the drying-wetting process of fluids under a microscope, a theoretical model is developed to consider the effect of air entrapment in the soil-water retention constitutive relationship. The effect of hysteresis on the fluid flow is considered by introducing an integrated capillary hysteretic model. There are only three conditions needed in the new model, i.e., the primary drying boundary curve, the main wetting boundary curve, and one point in the main hysteretic loop. Furthermore, as long as the previously experienced maximum matric suction in the porous medium is given, the model is capable of simulating changes of the soil-water state with the effect of air entrapment undergoing an arbitrary change of matric suction. By comparing the predictive curves with measured data from the literature, it is shown that the effects of air entrapment and capillary hysteresis are significant on the soil-water retention relationships. The model with the effects of capillary hysteresis and air entrapment should be taken into account in the soil-water relationship in order to accurately predict soil-moisture states in porous media.

In this paper, a new behavioral (plasticity) model for lignosulfonate (LS)–treated silty sand based on the disturbed state concept is proposed. Here, the relative intact (RI) responses for both the LS-treated and untreated soils are simulated using the ${\delta }_0$ version of the hierarchical single-surface plasticity models. The nonassociativeness is incorporated through the disturbance function, which shows the amount of deviation of the actual response from the reference states. In the analysis, the fully adjusted (FA) response of the untreated soil was considered similar to the shear behavior at 20% of the shear strain. Also, for the LS-treated behavior, the FA response was taken as the response of the LS bonds. In this study, the bond response was modeled separately using a linear elastic RI response and zero strength condition as the relative states. A series of laboratory direct shear tests were carried out at low effective normal stresses to determine the model parameters. The proposed model has been validated using an independent set of shear test results. The model predictions are in good agreement with the laboratory test data.

The effect of particle shape and particle crushing on the results of cone penetration testing (CPT) of granular materials in a calibration chamber has been studied using three-dimensional discrete element modeling. Simulating the whole chamber with a realistic particle size requires a large number of particles, which leads to a large computational time. Both 90° and 30° segments of a calibration chamber were used in this study to reduce computational time. The effect of particle shape was simulated by prohibiting particle rotation or using simple two-ball clumps. Prohibiting particle rotation was found to increase tip resistance significantly compared with free particle rotation, and replacing a single sphere with different shapes of simple two-ball clumps was also found to have an important effect on the tip resistance. Particle crushing was simulated during CPTs by replacing a broken particle with two new equal-sized smaller particles. The results showed that there was a considerable reduction in the tip resistance for the crushing model compared with the noncrushing model, and this reduction increased as the confining stress increased.

In this study, a three-dimensional (3D) numerical investigation of axisymmetric collapse of granular columns has been conducted using the discrete element method (DEM). The simulated granular columns have a constant initial radius of 5.68 mm and three aspect ratios: 0.55, 1.0, and 2.0. The columns consist of uniform spherical quartz particles with a diameter of 0.32 mm. In the DEM model, rotational velocities of particles are reduced by a factor at every time step to partially account for the additional rolling resistance due to the effect of particle shape and hysteretic contact behavior. The simple linear contact model is used; however, its performance is improved by using different stiffness values calculated by nonlinear Hertz–Mindlin contact model for each aspect ratio. The simulated final deposit heights, runout distances, and energy dissipation values are in good agreement with experimental observations reported in the literature. The effects of initial porosity and rotational resistance on the final deposit profile and energy dissipation at different aspect ratios are investigated through a parametric study. For different aspect ratios, a higher rotational resistance leads to higher final deposit height, shorter runout distance, and less energy dissipation. A lower value of initial porosity leads to higher final deposit height; however, the runout distance and evolution of normalized potential, kinetic, and dissipated energies versus time are insensitive to the initial porosity for the granular columns investigated.

Backfilling of underground stopes is commonly used in the mining industry. The stress state in these stopes has to be evaluated to ensure safe application of the backfill. In recent years, much work has been conducted to assess the stresses in single backfilled stopes. The stress distribution in stopes may also be affected by the excavation of multiple openings. This paper presents key results obtained from numerical simulations of two adjacent vertical stopes created in sequence. The results illustrate the influence of stope geometry (size and spacing), natural stress state, backfill properties, and excavation (and filling) sequence on the stress distribution in both stopes. The simulations indicate that the stress distribution in the first backfilled opening, following the creation of the second one, largely depends on the fill properties. These results also show the ways in which these factors may affect the stress magnitude and distribution pattern in the case of two adjacent stopes, when the creation of a second opening influences the response of the first one. It is also demonstrated that the second backfilled stope tends to behave in a manner similar to that of a single (isolated) stope, where a classical arching effect often dominates the stress distribution.

The stability of a footing may be destabilized near a retaining wall because of increased lateral displacement and decreased soil confining pressure. The stability of footing within this footing–soil–wall system is complex and not predicted using the classical solutions utilized for a semi-infinite footing–soil system. In this study, the bearing capacity of a concrete footing placed near a wall of spaced driven concrete piles using three-dimensional finite-element (FE) analysis with ANSYS 13.0 is investigated. All system materials are idealized as nonlinear, elastic–perfectly plastic, and rate independent using the upper bound function of the Drucker–Prager yield criterion. Three different soil deposits are studied: kaolin, silty clay, and kaolin–sand. A total of 144 unique FE analysis runs are executed. The footing bearing capacity is computed from the output footing pressure–settlement curve using several well-known postprocessing linearization criteria from which the lowest value is chosen. Soil yielding patterns and failure mechanisms are also studied and presented. Empirical equations capable of predicting the footing bearing capacity and the footing–pile distance without the effect of destabilizing factors are developed.

The extraction of methane gas from hydrate-bearing sediments has garnered increasing global interest in recent years. Understanding the sediment response to potential production scenarios is vital for both accurate reservoir response simulation as well as the development of field extraction methodologies. Gas hydrate has an icelike structure, and when present within sediments it will significantly alter their geomechanical behavior. Of particular interest in this paper is the observed strain-softening response of hydrate-bearing sands during drained shearing. Experimental results indicate that the strain softening during shearing is related to deviatoric behavior. A new analytical strain-softening model is proposed in which a direct relationship between volumetric expansion and reduction in hydrate saturation is developed. In the proposed model, the apparent cohesion of hydrate-bearing sediment reduces with increasing plastic shear strain. To validate the model, it was implemented into a standard numerical code, and the results were compared with those obtained from an advanced hydrate-soil numerical simulator. The model is also compared with experimental results from drained shear tests conducted on hydrate-bearing sands. The deviatoric stress results of the analytical model and numerical simulation were found to be almost identical, indicating the validity of the simple analytical formulation. In addition, the analytical model showed good agreement with the experimental results.

The penetration of steel catenary risers and other cylindrical objects, such as offshore pipelines or T-bar penetrometers, in a soft clay seabed is of practical importance in deepwater oil and gas development. Finite-element (FE) analyses of these large-deformation problems are computationally very expensive. Water can also play a significant role through development of suction behind the riser. The main objective of the present study is to develop an advanced numerical modeling technique to simulate riser–seabed–water interaction near the touchdown zone. Keeping in mind two critical issues, namely the computational cost and modeling of suction, two different numerical modeling techniques are developed. In the first one, the computational fluid dynamics (CFD) approach is used. The CFD modeling is performed using ANSYS CFX 13.0 software. Among the three different types of CFX models developed in the present study, the subdomain modeling technique is found to be the most efficient. In the second numerical modeling technique, large-deformation FE analyses are performed using the coupled Eulerian–Lagrangian (CEL) approach in Abaqus FE software. The comparison of the results of CFX and CEL shows that CFX can successfully simulate the penetration of the riser or pipeline in soft clay seabed. The main advantages of the present CFX modeling over CEL modeling are (1) the CFX can simulate suctions, and (2) the CFX modeling with a subdomain is computationally very efficient. The analyses compared in this study show that CFX simulations are computationally 10–15 times faster than CEL simulations.

This paper presents the development of next-generation prediction models for the flow number of dense asphalt–aggregate mixtures via an innovative machine learning approach. New nonlinear models were developed to predict the flow number using two robust machine learning techniques, called linear genetic programming (LGP) and artificial neural network (ANN). The flow number of Marshall specimens was formulated in terms of percentages of coarse aggregate, filler, bitumen, air voids, voids in mineral aggregate, and Marshall quotient. An experimental database containing 118 test results for Marshall specimens was used for the development of the models. Validity of the models was verified using parts of laboratory data that were not involved in the calibration process. The statistical measures of coefficient of determination, coefficient of efficiency, root-mean squared error, and mean absolute error were used to evaluate the performance of the models. Further, a multivariable least-squares regression (MLSR) analysis was carried out to benchmark the machine learning–based models against a classical approach. Sensitivity and parametric analyses were conducted and discussed. Given the results, the LGP and ANN models accurately characterize the flow number of asphalt mixtures. The LGP design equation reaches a comparable performance with the ANN model. The proposed models outperform the MLSR and other existing machine learning–based models for the flow number of asphalt mixtures.

The paper presents an analytical study of a cantilever sheet pile wall considering the effects of uncertainty of soil properties. The failure probability ($P_f$) of the sheet pile wall is combined with the sensitivity ($S$) of geotechnical random variables on the failure mode, and a new factor, the probabilistic risk factor ($R_f$), is proposed for each random variable corresponding to different variations of the random variables. The cantilever sheet pile wall that is driven into a cohesionless soil layer and backfilled by (1) cohesionless soil and (2) cohesive soil is considered separately for analysis against the rotational failure mode. $P_f$ is obtained by developing the response surface, based on finite element models. The sensitivity of each random variable is obtained by F-test analysis. It is observed that the cohesion of foundation soil ($c_2$) and water table positions are the important factors that influence the stability of the cantilever sheet pile walls to a great extent. Finally, design guidelines are provided based on the proposed risk factor–based approach for different heights of the cantilever sheet pile wall, pertaining to variation in soil properties and for varying positions of the water table.

The measurement of shear modulus at small strains in unsaturated soil is critical to evaluating the deformation behavior of soils at working loads. However, little has been done to determine if the small-strain shear modulus surface is unique during variation in either net mean stress or matric suction. In this context, uniqueness is defined as similar samples subjected to different stress paths that converge to the same stress state. Research is presented herein in which a unique constitutive shear modulus surface was established by independently controlling mean net and matric suction stress in a modified resonant column apparatus. Uniqueness was also investigated using data extracted from literature studies. Uniqueness of the shear modulus surface was shown for single-direction stress and hydraulic paths, by the convergence of modulus and by the agreement between predicted and measured modulus. The presence of a unique shear modulus surface allows for the use of a constitutive relationship based on soil type that describes the small-strain shear modulus in terms of the stress state variables.