Dilation Equation Research Articles

Elastoplastic constitutive model of gravelly soil and its validation through numerical analysis of the Kobe Port Island liquefaction

The liquefaction of gravelly soil has been demonstrated in many earthquakes, but the constitutive model to predict its liquefaction behavior is limited. The dilatancy equation plays an important role in establishing elastoplastic models of geotechnical materials. Recognizing the unique dilatancy characteristics of gravelly soil, this study introduces an improved nonlinear dilatancy equation into the generalized plasticity model. Consequently, a unified elastoplastic constitutive model is proposed, aimed at accurately predicting both the monotonic and cyclic behaviors of saturated gravelly soil under diverse drainage conditions. A series of drained and undrained triaxial tests of gravelly soil under monotonic and cyclic loadings are carried out to validate the model performance. In addition, the model performance is also examined through monotonic and liquefaction tests of gravelly soil obtained from prior publications. Furthermore, the proposed model is implemented into the nonlinear finite element program GEODYNA and applied to the numerical simulation of the Kobe Port Island liquefaction to better validate the capabilities of the proposed model. From the comprehensive analysis of triaxial tests and numerical simulations, the proposed model performs well in predicting mechanical behavior of gravelly soil and can provide a powerful tool for seismic liquefaction analysis of gravelly soil sites.

Constitutive models of ultra-high performance concrete (UHPC) under ture tri-axial compression and an analysis-oriented model for FRP-confined UHPC

The confinement mechanism of ultra-high performance concrete (UHPC) has not been fully explored yet, and there is no analysis-oriented model available for FRP-confined UHPC. This study aims to fill this research gap by implementing new constitutive models of tri-axial compression behavior of UHPC in an existing analysis-oriented FRP-confined concrete model. In the first part, the test findings of tri-axial compression tests on 22 UHPC cubes with different fiber lengths and confining stresses were briefly summarized. In the second part, assessments were performed for four relevant axial stress-strain models using the test results. Generally, the four models yielded overestimations for the axial strains at the peak stresses by at least 30%, and the predicted post-peak decline curves misaligned with the test curves. Inspired by the assessment findings, a new axial stress-strain model was proposed using Liao et al.’s model as the basic form, and replacing the post-peak decline curve with a power-law function to capture the steep descending trend. The new axial stress-strain model showed excellent performance in predicting the peak axial stresses and corresponding strains (average error at around 5% and 9%, respectively), and depicting the tri-axial compressive stress-strain curves. With a newly calibrated dilation equation, the proposed axial stress-strain model was applied for the development of an analysis-oriented model for FRP-confined UHPC, which showed corroborated results with the verification tests in axial stress-strain curves, as well as the ultimate axial strengths and corresponding axial strains (average error at 6.5% and 13.3%, respectively).

Dilatancy Equation Based on the Property-Dependent Plastic Potential Theory for Geomaterials

The dilatancy equation ignores the noncoaxiality of granular soil for the coaxial assumption of the direction of the stress and strain rate in conventional plastic potential theory, which is inconsistent with extensive laboratory tests. To reasonably describe the noncoaxial effects on dilatancy, the energy dissipation of plastic flow is derived based on the property-dependent plastic potential theory for geomaterials and integrates the noncoaxiality, the potential theory links the plastic strain of granular materials with its fabric, and the noncoaxiality is naturally related to the mesoscopic properties of materials. When the fabric is isotropic, the dilatancy equation degenerates into the form of the critical state theory, and when the fabric is anisotropic, it naturally describes the effects of noncoaxiality. In the plane stress state, a comparison between a simple shear test and prediction of the dilatancy equation shows that the equation can reasonably describe the effect of noncoaxiality on dilatancy with the introduction of microscopic fabric parameters, and its physical significance is clear. This paper can provide a reference for the theoretical description of the macro and micro mechanical properties of geomaterials.

Crushing and mechanical characteristics of coral sands under various stress paths

Coral sand is known to have stress path dependence and fragility. In this research, a series of drained triaxial tests were conducted to explore stress path effects on the mechanical and particle breakage characteristics of coral sands. Several mechanical characteristics, such as critical state, strength, and dilatancy characteristics were studied. The results revealed that the dilatancy behaviors, stress-strain responses, and crushing of coral sands were strongly affected by consolidation stress and stress path. The critical state lines of coral sands were straight in p‘-q space and e-(p‘/P a)0.44 plane regardless of stress path. Based on these characteristics, a dilatancy equation was derived and validated through test results. The developed dilatancy equation could predict the dilatancy relationships of coral sands under various stress paths.

Breakage critical state of gravels with different gradings. Part II: Constitutive modelling

A unified critical state surface for gravels considering the effect of initial grading and particle breakage was proposed in Part I based on the test results of large-scale triaxial compression tests. In this study, we proposed a state-dependent elastoplastic constitutive model to predict the shear behaviors of gravels in the framework of bounding surface plasticity and critical state theory. A simple state parameter with a reference to the unified critical state surface was applied to the dilatancy equation and the bounding stress ratio in the proposed model. The proposed constitutive model can well capture the grading state index, the relative breakage index, the deviatoric stress and the void ratio at the end of tests, and the stress–strain relationships during the shearing process. The model can still be improved by considering the influence of the confining pressure on the critical state strength.

Stress–dilatancy behaviour of remoulded Fujinomori clay

Abstract The effect of the degree of consolidation and the stress path on the behaviour of remoulded Fujinomori clay for drained triaxial compression and extension was analysed using the Frictional State Concept. It is shown that the stress–dilatancy behaviour can be approximated by a linear general dilatancy equation given by the critical frictional state angle and two soil parameters. The newly formulated dilatant failure state is represented on the stress ratio plastic dilatancy plane by points lying on the friction state line defined by the friction state angle and the Friction State Concept parameters α=0 and β=1. It has been shown that the stress ratio–plastic dilatancy relationship, which is very rarely used in the interpretation of test results, is important for a complete description of the behaviour of soils during shearing.

Modification of a Constitutive Model for Gassy Clay

Gassy clays containing large, discrete gas bubbles are widely spread in deposits in shallow waters. The existence of gas bubbles may impair the original soil structure, resulting in the instability of offshore foundations and the occurrence of submarine landslides. Although gassy clay was found to exhibit different undrained shear behaviors with variations of the initial pore water pressures and initial gas volume fractions, more experimental studies and theoretical interpretations of the relationship between the effective confining pressure and undrained responses are required. A series of undrained triaxial compression tests is conducted to compare the responses of reconstituted gassy and saturated specimens at an effective confining pressure of 200, 400, or 600 kPa. An existing elastoplastic constitutive model based on the critical state is improved by updating its stress-dilatancy function and yield surface. The dilatancy equation proposed has the potential to quantify the effect of gas bubbles on the dilatancy of the soil matrix. The yield surfaces are reproduced reasonably well by optimizing the expressions of shape parameters under a variety of effective confining pressures. The model developed can describe the stress-dilatancy and stress-strain responses of both the gassy and saturated specimens.

Experimental Study on the Effect of Coarse Grain Content on the Dilatancy and Particle Breakage Characteristics of Coarse-Grained Soils

The dilatancy deformation and particle breakage effects of coarse-grained soil have a significant influence on the safety and stability of high earth and rock-fill dams. To study the shear deformation and particle breakage characteristics of coarse-grained soil, four groups of soil sample with different coarse grain contents were to conduct consolidated drained triaxial compression tests by DJSZ-150 large-scale triaxial compression test machine. The results showed that the volume of coarse-grained soil changes significantly under deviation stress. The soil samples underwent positive dilatancy under the confining pressures of 200 kPa and 400 kPa but negative dilatancy under the confining pressures of 600 kPa and 800 kPa. When increasing the coarse grain contents in the soil sample under the same confining pressure, the positive dilatancy was more significant. A three-parameter dilatancy equation of coarse-grained soil considering confining pressure was established, in which the parameter L = 4 could serve as a criterion for whether the coarse-grained soil underwent positive dilatancy. Under the load, the coarse-grained soil showed three particle breakage modes: abrasion, attrition, and fracture. The coarse-grained soil particle size distribution before and after breakage conformed to the fractal characteristics, and the particle breakage rate increased significantly with larger soil particle size fractal dimension differences before and after the test. The coarse grain content and the confining pressure are two important factors affecting the mechanical properties of coarse-grained soil, and they control the shear deformation of soil.

Stress-Dilatancy Behaviourof Fibre-Reinforced Sand.

This paper analyses the stress-strain behaviour of fibre-reinforced sand using the results obtained by drained triaxial compression tests presented in the literature. The general stress-plastic dilatancy equation of the Frictional State Concept has been used to describe the behaviour of fibre-reinforced sand for different shear phases. The behaviour of pure sand is taken as a reference for the behaviour of sand with added fibres. It is shown that the characteristic shear phases can only be determined when the η-Dp relationships are used, which are very rarely demonstrated in the results of shear tests presented in the literature. It has been shown that tensile strains must occur in order to achieve the strengthening effect of fibre reinforcement. A reduction in the stiffness of the fibre-sand composite is observed in the absence of tensile strains below certain threshold values.

The constitutive behavior and dissociation effect of hydrate-bearing sediment within a granular thermodynamic framework

Considering the energy dissipation caused by hydrate dissociation and granular rearrangement at the micro level, a multiphase constitutive model of hydrate-bearing sediment is established within a granular thermodynamic framework. This model incorporates the nonlinear expression of bonding stress into the elastic energy density function, considers the relative velocities of the gas and liquid phases in terms of the solid phase in the dissipative force system, and introduces the coupling effect of heat conduction into the migration coefficient matrix to capture the complex behavior of the sediment. Defining the migration coefficient matrix and elastic energy function, the effective stress is obtained and is suitable for constant hydrate saturation and hydrate dissociation conditions. Dilatancy equations are improved by considering the effect of compactness, bonding stress, and hydrate saturation. Meanwhile, the calculation method of bonding stress is suitable for both strong and weak cementation conditions. The deduced model is validated against the test results conducted on natural and synthetic samples under different hydrate saturations, sediment porosities, and hydrate habits and can effectively capture the strain hardening and softening as well as the dilatancy properties of the sediments and the loading path effect.

The Stress-Dilatancy Behaviour of Artificially Bonded Soils.

In this study, the results of triaxial compression tests of some naturally and artificially bonded soils presented in the literature were analysed. It was shown that the three characteristic stages of plastic flow during shear can be identified. In all stages, the stress–dilatancy behaviour could be approximated by the general linear stress–dilatancy equation of the Frictional State Concept. For many shear tests, the failure states and newly defined dilatant failure states are not identical. The points representing dilatant failure states lie on a straight line, for which the position and slope in the η-D plane depend on the soil type and the amount of cement admixture. This line defines the critical frictional state angle, and its slope for bonded soils is greater than for unbonded soils.

A general plastic model for rockfill material developed by using Caputo fractional derivative

The gradient direction and fractional derivative direction of one function can be coaxial or non-coaxial, and this depends on the order value of fractional derivative. A fractional plasticity model was developed by using Caputo fractional derivative and a dilatancy equation pondering particle breakage for rockfill material in this study. In this fractional plasticity model, the gradient direction and fractional derivative direction of the plastic potential function derived from a dilatancy equation were deduced and further used as the plastic flow direction and loading direction respectively, and the order controls the non-associated feature between the plastic flow direction and loading direction. For validation, drained triaxial test results under different stress paths of five rockfill materials are simulated by the proposed fractional plasticity model. It can be found from comparisons between the drained triaxial compression test results of rockfill materials and corresponding model predictions that the proposed model can well capture the stress-strain characteristics of rockfill material, including strain hardening and volume change behaviors etc.

Experimental study on critical state behaviour of unsaturated silty sand under constant matric suctions

There are relatively few experimental studies on the hydromechanical and critical state behaviours of unsaturated silty sand. In this paper, suction-controlled, consolidated, drained triaxial shear tests were conducted on silty sand under saturated (net confining pressures of 100, 200, 300 and 400 kPa) and unsaturated (suctions of 30, 100 and 200 kPa and net confining pressures of 100, 200 and 300 kPa) conditions. For unsaturated silty sand, the effects of dry density, matric suction and net confining pressure on the hydromechanical and critical state behaviour were investigated, and the critical state parameters were obtained. The main findings for unsaturated silty sand included shear shrinkage at low dry density (1·6 g/cm3) and dilatancy at higher density (1·96 g/cm3). The peak shear strength was positively correlated with dry density, matric suction and net confining pressure. The critical state volumetric strain and critical state degree of saturation were negatively correlated with dry density and matric suction but positively correlated with net confining pressure. A new dilatancy equation that includes the state parameter for unsaturated silty sand was established. A unique critical state line passes through the origin in the effective stress [Formula: see text]–deviator [Formula: see text] plane, and its slope is the saturated critical state stress ratio [Formula: see text]. In the lnp*–v plane ([Formula: see text] is the specific volume), the critical state lines corresponding to different matric suctions are parallel.

Dilatancy Equation and Yield Function for Granular Materials Based on Energy Accumulation

Abstract The dilatancy equation and yield function are the two important components of the elastoplastic constitutive model for soils. Because the incremental energy equations of the Cam-Clay and Rowe models do not consider the particle breakage energy, the predicted dilatancy for the granular materials is not convincing. Accordingly, the total input energy during shearing, Es, is taken as the equivalent of the energy equation, and Es is found to be directly proportional to the shearing strain, εs. The slope, K, of the Es∼εs line is theoretically induced and experimentally demonstrated to be equal to the critical shear stress, qc. Thus, a new dilatancy equation and yield function are derived and preliminarily verified using different kinds of granular materials, e.g., calcareous sand, gravel, and rockfill. The shape of the proposed yield surface can be adjusted automatically from bullet-shaped to drop-shaped with an increasing Mc, which is better for expressing the yielding behavior of granular materials. Additionally, another significant advantage of the proposed dilatancy equation and yield function is that no additional fitting or nonphysical parameters except Mc (critical stress ratio) is included, whereas the predictions for the granular materials are still satisfactory.

Unified modeling behavior of rockfill materials along different loading stress paths

Rockfill materials used for dam construction show obvious stress path dependency during construction process. The constitutive model for rockfill materials are mainly established based on conventional triaxial test. This brings some deviation to the numerical calculation for earth rockfill dam engineering, and it is necessary to develop a constitutive model considering stress paths effect. In the framework of generalized plasticity, plastic modulus can be expressed as the function of dilatancy equation and tangent modulus, hence the influence of stress paths on model can be transfer to the dilatancy equation and tangent modulus. This leads to a new method to describe stress-strain response of rockfill under various loading stress paths. Based on the analysis of large-scale triaxial experiments of rockfill materials with various stress paths, we proposed new dilatancy equation and tangent modulus to better describe stress paths effect and corresponding modified generalized plasticity model is established. Through the prediction and comparison of three groups of rockfill materials triaxial tests, the model established in this paper can simulate behaviors of rockfill materials under various stress paths, using only a set of parameters.

Experimental Study on the Dilatancy Characteristics and Equation of Saturated Sand–Gravel Composites during the Whole Shearing Process

Experimental Study on the Dilatancy Characteristics and Equation of Saturated Sand–Gravel Composites during the Whole Shearing Process

Dilatancy of the foundation filling material of island-reefs in the South China Sea

Calcareous gravelly sand (CGS) is the primary building material used in the foundation filling of island-reefs in the South China Sea. In this study, multiple consolidation-drained triaxial shear tests were conducted on CGS, in order to reveal the effects of relative density (Dr) and effective confining pressure (σ3′) on the CGS dilatancy. The applicability of classical dilatancy equations to CGS was tested. An empirical dilatancy equation applicable to calcareous sand was proposed and the rationality of this dilatancy equation was validated. According to the results: (I) Phase-transition stress ratio (ηp) and failure dilatancy ratio (df) are important indices for assessing the CGS dilatancy. Physical models were built to predict the ηp and df of CGS based on its Dr and σ3′, respectively. (II) The dilatancy equation of the modified Cam-clay model could not reflect the CGS dilatancy. Rowe’s dilatancy equation could reflect the CGS dilatancy to some extent, but it underestimated the CGS dilatancy when σ3' ≤ 50 kPa and overestimated the CGS dilatancy when σ3' ≥ 400 kPa. (III) A good linear relationship was identified between dilatancy ratio and stress ratio-related parameter (ηf2-η2η0.5). Therefore, the empirical dilatancy equation built on this basis is highly applicable to calcareous sand.

Modified anisotropic generalized plasticity model for rockfills based on spatial mobilized plane

A modified anisotropic generalized plasticity model for rockfills is developed by improving the previous version. The improvements include a new anisotropic variable ω', a new anisotropic state parameter ξ', and a modified dilatancy equation. The ω' is constructed on the basis of the angle between the spatial mobilized plane and compaction plane and can degenerate to zero in the case of vertical compaction. The anisotropic state parameter ξ', which includes the effects of ω' and mean principal stress, can simulate the initial moduli changing with the direction angles of major principal stress under different confining pressures. The modification of the dilatancy equation including ξ' and correction term can reduce the overestimation of the volumetric contraction in the initial shear stage. To verify the modified model, triaxial tests for rockfill materials with different confining pressures and different direction angles are simulated and analyzed. The simulation results are in good agreement with the test values. Compared with the original model, the modified model is more effective in describing the mechanical properties of rockfill materials, especially in the initial shear stage, and more practical.

Elasto-plastic constitutive modelling of compacted rockfill materials: a physically based approach

An elasto-plastic constitutive model for rockfill materials considering particle breakage is proposed. Although numerous constitutive models have been proposed for rockfill materials, most of them are mainly based on empirical results, without consideration of the physical meaning of their parameters. In the present work, a yield function is derived based on the evolution of microstructures formed by rockfill particles. The dilatancy equation is derived based on both the theory of breakage mechanics and the energy dissipation equations. The breakage-induced hardening is also formulated to account for both the volumetric and shearing hardening of rockfill materials. It is demonstrated that, without compromising the simplicity of the model, most of the parameters in the proposed constitutive model have concrete physical meanings. Comparison of the model simulation with experimental data shows that the proposed model can describe reasonably the static loading characteristics of rockfill materials within a wide range of confining stresses.

Large-Scale Triaxial Tests on Dilatancy Characteristics of Lean Cemented Sand and Gravel

The dilatancy equation, which describes the plastic strain increment ratio and its dependence on the stress state, is an important component of the elastoplastic constitutive model of geotechnical materials. In order to reveal their differences of the dilatancy value determined by the total volume strain increment ratio and the real value of lean cemented sand and gravel (LCSG) materials, in this study, a series of triaxial compression tests, equiaxial loading and unloading tests, and triaxial loading and unloading tests are conducted under different cement contents and confining pressures. The results reveal that hysteretic loops appear in the stress–strain curves of equiaxial loading and unloading tests, and triaxial loading and unloading tests and that the elastic strain is an important component of the total strain. The hysteretic loop size increases with an increase in the stress level or consolidation stress, whereas the shape remains unchanged. Furthermore, with an increase in the cement content, the dilatancy value determined by the total volume strain increment ratio becomes smaller than that determined by the plastic strain increment ratio, and the influence of the elastic deformation cannot be ignored. Thus, in practical engineering scenarios, especially in the calculation of LCSG dam structures, the dilatancy equation of LCSG materials should be expressed by the plastic strain increment ratio, rather than the total volume strain increment rati.