Spherical Cavity Expansion Research Articles

Similarity Solutions for Spherical Cavity Expansion in Strain-Softening Soils and its Application in Cone Penetration Test

The cavity expansion theory is commonly employed to predict the grouting pressure and cone penetration resistance. However, existing theoretical solutions for cohesive-frictional soils with dilation fail to simultaneously consider both the elastic deformation in the plastic zone and strain-softening behaviour of soils. To address this limitation, a similarity solution was derived for the expansion of spherical cavities from a zero initial radius with an initial confining pressure in strain-softening Mohr–Coulomb materials. The results demonstrate good agreement with previous theories, predicting an ultimate pressure higher than that obtained by neglecting elastic deformation in the plastic zone but lower than that calculated by neglecting softening characteristics. Moreover, several influential factors affecting the radius of the plastic zone and ultimate pressure were meticulously examined. On-site cone penetration test data and the theoretical results showed notable agreement. This indicates that the proposed solution exhibits a better conformance with the field data than do the previous solutions.

Research On Soil Displacement Prediction Method Caused By High-pressure Jet Grouting Pile Group Construction Under Adjacent Pile Response

This paper analyzes the soil displacement effect during high-pressure jet grouting pile construction by simplifying the construction process as a series of pressure-controlled spherical cavity expansions in a semi-infinite soil medium. Based on the elastic-plastic solution of pressure-controlled spherical cavity expansion theory, a calculation method is proposed to estimate the soil displacement caused by high-pressure jet grouting pile construction. Non-linear contact theory between the pile and soil is introduced, and the finite difference method is employed to determine the internal forces and deformations of the pile. The proposed method is applied to field construction cases, and its validity is verified by comparing the results with on-site monitoring data. The research findings reveal that the soil displacement effect decreases with depth and primarily occurs in shallow soil layers during the construction of individual high-pressure jet grouting piles. The lateral displacement at the ground surface increases first and then decreases with the increase in horizontal distance, while the ground uplift exhibits an exponential decay trend.

Cavity expansion-based interpretation of piezocone penetration test (CPTu) data in clays

A cavity expansion-based method is proposed in this paper to correlate the relevant findings with the piezocone penetration test (CPTu) data in clays. The rigorous and explicit solution for both spherical and cylindrical undrained cavity expansion is adopted based on a unified critical state constitutive model for clays and sands. The analogous model developed is comprehensively validated against two series of advanced numerical simulations and calibration chamber tests of the CPTu in fine-grained soils, along with bias analyses. The effects of the overconsolidation ratio and initial state parameter are extensively investigated for various clay types. Both theoretical and empirical correlations are then proposed for back-calculations of cone factor, undrained shear strength, overconsolidation ratio and initial state parameter of clays. Two case studies of CPTu tests are conducted in addition to examine correlations and contours of the in situ state parameter, demonstrating the applicability of the proposed theoretical approach for interpretation of the CPTu data.

Oblique penetration of tungsten spheres against steel targets based on compressible and incompressible cavity expansion theory

In penetration problems, empirical formulas from different sources may easily lead to calculation conflicts. In the present work, based on the spherical cavity expansion theory of compressible and incompressible models, the resistance decay model and the kinematics analysis of projectiles, a 3-Danalytical model for the oblique penetration of tungsten spheres against low-carbon steel targets was built. The modeling method was described in detail. Subsequently, calculation results were compared with experimental data of tungsten spheres penetrating 100 mm thick Q235 steel targets and 8 mm thick Q345 steel targets to check computational performance of the established model. It is proved that resistance models can be used to calculate penetration problems after considering the resistance correction coefficient. The 3-D model is suitable for penetration problems of plastic cratering of targets.

Analytical analysis of spherical cavity expansion considering particle breakage effect of sand and its application for CPT tests

Particle breakage in sand significantly influences the stress and deformation predictions of the spherical cavity expansion theories. However, existing theories often overlook or lack a quantitative description of particle breakage, leading to significant errors in the actual results. Based on the Simple Critical State Sand (SIMSAND) model, considering the particle breakage effect, the traditional Euler’s problem of the drainage spherical cavity expansion is converted into a set of first-order ordinary differential equations described by Lagrangian. Fontainebleau sand is taken as an example, and the influence of the initial stress of the sand on the stress state and void ratio is studied. An analysis of the relative breakage surrounding the cavity reveals that the primary impact zone extends to a distance thrice the expansion radius. An axisymmetric model of the cone penetration test is established, the analytical and numerical solutions of expansion stress and cone tip resistance are analysed and the correctness of the spherical cavity expansion theory is verified. Furthermore, the cone tip resistance results, derived from the theory of spherical cavity expansion, are consistent with the results of numerical calculations. Notably, when the particle breakage effect is not considered, the calculated results of cone tip resistance significantly increase.

Spherical cavity expansion method dependent on strain, strain rate, and temperature

This paper presents a spherical cavity expansion method based on the self-similar field considering dynamic hardening, which includes plastic strain, plastic strain rate, and temperature for an infinite compressible medium under J2 plasticity. This study proposes an approach based on the gradients of plastic strain and temperature under self-similarity transformation to deal with dynamic hardening models such as the Johnson-Cook (JC) and the modified Johnson-Cook (mJC) models. A theoretical model shows that the effective and radial stresses are reduced near the cavity surface and increased away from the cavity under the mutual influence of hardening and thermal softening. The JC hardening model is used to analyze the behavior of the cavity expansion for AISI 4340 steel. The proposed theoretical model was validated by comparing the result of the explicit finite element method (FEM). This study discusses the behavior of the expanding cavity problem for thermal softening and strain rate sensitivities. In addition, rigid penetration problems are simulated to analyze the application of the presented method. FEM and the proposed cavity expansion method were compared with the penetration depth of AA6061-T651 target using the mJC hardening model for a hemispherical-nosed penetrator. A comparison is made for the performance of the penetration depth with an ogive-nose projectile for AA7075-T651 and AA6061-T651 materials using the mJC hardening model. This study concludes that the proposed method can be applied without modifying the hardening models to spherical cavity expansion problems that involve dynamic hardening models dependent on strain, strain rate, and temperature under adiabatic heat.

Interpretation of piezocone tests in normally- and over-consolidated clay using a spherical cavity expansion solution

This paper presents the derivation of closed-form expressions for the interpretation of piezocone tests in clay on the basis of an undrained cavity expansion solution. The cavity expansion solution employs the modified Unified Hardening model to describe the undrained behaviour of normally- and over-consolidated clays upon shearing. The derived expressions correlate the measured tip resistance and pore pressure with the undrained shear strength and the over-consolidation ratio, using a cone factor that, unlike empirical formulas, is calculated directly from the constitutive model parameters and the cone geometry. The cavity expansion solution is validated via comparison of its results against published studies, and accordingly the accuracy of the proposed expressions is benchmarked against field data. We show that, despite the simplifications introduced in the derivation, the calibrated expressions are capable of reproducing the field data with reasonable accuracy.

Rigorous solution for drained analysis of spherical cavity expansion in soils of finite radial extent

This paper revisits the expansion problem of a spherical cavity in an elastic–plastic soil mass with a finite radial extent under drained conditions, which proves to be non-self-similar. Rigorous semi-analytical solutions for the analysis of this problem are developed adopting several well-known soil models. With the small strain assumption, analytical expressions for both elastic stresses and displacements are obtained considering the stress dependency of soil moduli and influences of the outer boundary. In the plastic zone, the non-self-similar expansion process is formulated into a set of first-order partial differential equations (PDEs) based on the innovative combination use of the Lagrangian and Eulerian descriptions, and an efficient algorithm for solving these equations is developed. The solution method is quite general to be extended to many other constitutive models and is validated by comparing with existing solutions in some special cases. Then the cavity expansion behaviours in both clay and sand are discussed with a particular focus on the boundary effect. It is shown that, when the ratio of the outer to inner radius is small, the boundary effect may significantly influence the cavity expansion response. The present solution may provide a useful theoretical method for the validation of relevant finite element calculations and interpretation of some typical geotechnical problems such as cone penetration tests in small-size chambers, considering the boundary effect.

Modeling of temporary cavity in ballistic gelatin based on spherical cavity expansion model

To investigate the expansion mechanism of the temporary cavity produced by high-speed penetrators into ballistic gelatin, a dynamic spherical-cavity-expansion model for gelatin targets was proposed with consideration of the outer boundary size. The theoretical solution of the cavity radial stress was obtained and the energy conversion and conservation relation during the cavity expansion process was established. The influence of the finite boundary to the solutions of the cavity radial stress and the deformation energy in the medium was analyzed. The energy conversion and conservation relation was applied to penetration process of high-speed penetrators into soft targets and regarded as the temporary cavity expansion model. This model is a nonlinear first order differential equation about the cavity radius, which can be solved numerically to obtain the variation of temporary cavity volume with time. Experimental results of high-speed penetrators (bullets and fragment simulating projectile with different shapes) penetrating into ballistic gelatin were analyzed. Temporary cavity results of rhombic fragment simulating projectiles were used to estimate the parameters in the model. Then, model predictions were compared with the temporary cavity results of bullets, spheres and cylinders to verify the reliability of the model. It was found that: (1) if the ratio of the cavity radius to the initial target boundary radius exceeds 0.62, the error caused by ignoring the boundary size effect will exceed 10%; (2) Model predictions for temporary cavity volume of penetration produced by various penetrators are shown to be in good agreement with the corresponding experimental results.

An Application of Spherical Cavity Expansion Theory in Soft Rock Pile-Base Resistance

The failure mode should be taken into account in the analytical solution for pile base resistance. In order to determine the pile’s failure pattern in soft rock, the graphical model test method is used. That the compression zone below the pile base in soft rock is compatible with the spherical cavity expansion is shown. An improved calculation model for the ultimate base resistance may be developed by considering the force balance of the rigid cone under the pile-end in the ultimate stress state and the limit pressure required to expand a spherical cavity. To account for the influence of socket length, the limiting pressure for the spherical cavity expansion in infinite space is adjusted to that in semi-infinite space. The ultimate resistance of a base is computed by factoring in the length effect. The normalization of the base resistance-displacement relation is provided based on the findings of pressure chamber experiments on pile load-settlement, which are independent of socket length and overburden pressure. This semi-empirical technique is verified via the use of pressure chamber experiments and field testing of piles in soft rock.

Modelling and analysis of penetration resistance of probes in cultivated soils.

At present, the measurement of tillage depth is mainly based on manual measurement, but the manual raking method results in low measurement accuracy and high labor intensity. Due to the complexity of soil, theoretical research on tillage depth is relatively scarce. In order to provide a new research direction and research idea for soil stratification, topsoil was taken as the research object of this paper. The correlation between penetration resistance and penetration depth of a probe in a cultivated soil was studied, and a mathematical model was established. There is a certain similarity between the process of spherical cavity expansion and the process of probe penetration, so we introduced the theory of spherical cavity expansion into the modeling of penetration resistance of the cultivated soil. In this paper, the spherical cavity expansion theory of unsaturated soil was used as the basis for solving the penetration resistance. And the unified strength criterion was employed as a yield condition of the soil to set a stress solution and a displacement solutionin into of the probe penetrating into the elasto-plastic zone of the cultivated soil to determine the model of expansion force. We have carried out indoor tests to revise the expansion force model. Firstly, according to the range of soil density and water content in the field, the soil densities were classified into 1.1×103kg/m3, 1.2×103kg/m3 and 1.3×103kg/m3, and the water contents were divided into 10%, 15% and 20%. In addition, the orthogonal tests were performed at different levels. The soil was put into the barrel, and the probe was inserted into the soil in the barrel at the speed of 8mm/s to determine the test values of the change of the probe penetration resistance with depth. Finally, the expansion force model was fitted with the results of the indoor test, and coefficient B was introduced to express the influence degree of density and water content on the resistance. Coefficient B was substituted into the expansion force model to obtain the penetration resistance model of the cultivated soil. Through the goodness of fit analysis of the penetration resistance model, the results show that the overall average goodness of fit of the penetration resistance modelat was up to 0.871 at different water contents and densities, which was a good fit and could present novel insights into the study relating to soil stratification theory.

Dynamic Penetration Process of Torpedo Anchors into Sand Foundation

Torpedo anchors are a new type of anchoring system in deepwater that is much more economical than conventional anchoring methods. The dynamic penetration process is vitally important to the installation of torpedo anchors. Based on the spherical cavity expansion theory, the dynamic response characteristics of pressure-locked soils are analyzed using the Mohr-Coulomb criterion. The equations for the penetration of torpedo anchors with bullet-shaped heads are established considering rod friction. Subsequently, the analytical solutions for velocity, acceleration, and final penetration depth and the approximate analytical solution for penetration depth vs. time are obtained. The established penetration equation is solved using MATLAB software to obtain a semi-analytical solution, and the model tests on the penetration of a torpedo anchor with different initial velocities into saturated sand with different densities were conducted. A comparison of the test results shows that the analytical solution and the semi-analytical solution can well predict the model test results, indicating that the established analytical method can be used to analyze the penetration process of torpedo anchors. The research results can provide a guideline to the installation of torpedo anchors into the seabed in actual engineering.

Revisiting undrained cavity expansion problem in critical state soils: A simple graph‐based approach

AbstractThis paper presents analytical solutions for the finite expansion problems of a spherical or cylindrical cavity, using a simple yet novel graphical approach recently proposed by Chen & Abousleiman in 2022, in both original Cam Clay (OCC) and modified Cam Clay (MCC) soils under undrained conditions. It is shown that, for a soil mass subjected to isotropic in situ stress conditions, the stress paths in the deviatoric plane for the spherical and cylindrical cavity expansions turn out to be two straight lines, which correspond to Lode angles equal to and , respectively. The desired limiting cavity pressure therefore can be directly and accurately evaluated through simple numerical integration with respect to the mean effective stress, while the relationship between the internal cavity pressure and the cavity radius, the cavity expansion curve, may be equally conveniently determined. Numerical results obtained from the current graphical method, for a range of the values of over consolidation ratio considered, compare extremely well with those from the conventional semianalytical formulations of the undrained cavity problem that involve solving a system of coupled governing differential equations. It is interesting to note that the representative and approximate solution developed by Collins & Yu in 1996 indeed is a correct one for the spherical cavity expansion problem, and, with minor modifications, will be applicable for the accurate calculation of the responses of the cylindrical cavity as well.

Improved cone penetration test predictions of the state parameter of loose mine tailings

The most widely used in situ testing instrument for tailings storage facility (TSF) is the cone penetration test (CPT), which uses calibration chamber (CC) test data as the preferred method to correlate the soil's state with the CPT acquired data. Many, if not most, mine tailings are different from the soils historically used in CC studies (mainly sands). Moreover, these tests were conducted at denser states than typically found in TSFs. Additionally, CC tests cannot practically be performed for every new soil-specific project. For these reasons, practitioners usually perform numerical simulations to approximate these correlations. One of the most popular techniques for this is the spherical cavity expansion approach. This study compares novel small CC testing of two different silty mine tailings, and their respective spherical cavity expansion simulations. Since these simulations use the NorSand constitutive model, this study also presents the calibration process that includes the triaxial laboratory testing to calibrate the NorSand properties, and the corresponding iterative process to infer the additional plastic and elastic parameters of the model. Finally, an adjustment equation is introduced for the contractive state of these silty materials.

Evaluation of stress history and undrained shear strength of three marine clays using semi-empirical methods based on Piezocone Test

The paper presents a comparative study between semi-empirical methods for the estimation of pre-consolidation pressure and undrained shear strength from Piezocone (CPTu) data. The first method, proposed by Massad, was developed from observing the variation of these parameters with depth; the second method, proposed by Mayne, was developed from simplifications and relationships between the Spherical Cavity Expansion Theory (SCET) and the Critical State Theory; the third method was proposed by Mayne, which considers the variations due to soil type from the CPT Index to estimate the pre-consolidation pressure. The methods were validated based on their applications to the marine clay from Santos Coastal Plain, Brazil, Bothkennar clay from Scotland, and Torp Clay from Sweden. It is intended to verify if the results are consistent with each other, with the stress history of these soils and with the available test results. The application of the Massad’s method led to results close to the available reference values. The results of the Mayne’s method based on SCET showed great variability in behavior comparing to the test data depending on the case study. By the Mayne’s method based on CPT Index values, the calculated pre-consolidation pressures were slightly higher than the values of the available test data. The variations in the results highlighted the importance of validating estimates based on semi-empirical methods through specific tests and the knowledge of geological history contributes to predicting the behavior of clays, since they showed good agreement with the available data from oedometer tests.

The optimal nose shape of a rigid projectile deeply penetrating into a solid target considering friction

The paper deals with the optimal nose shape of a perfectly rigid axi-symmetric projectile which deeply penetrates into a semi-infinite target accounting for the dry friction between the nose surface and the target. Analytical expressions of the contact stresses at the “projectile-target” interface and of the total resistance force as well as of the final penetration depth are developed for targets described by a locked hydrostat and a linear shear failure-pressure relationship. These analytical expressions are developed using the cylindrical cavity expansion (CCE) and the spherical cavity expansion (SCE) approaches. A comparison of these approaches on the optimization problem solution is performed. The effect of dry friction on the optimal shape of the projectile nose is investigated and two main approaches to solving the problem of shape optimization are studied. Two optimization approaches are compared: the first optimization approach is based on minimizing the resistive force (MRF) acting on the projectile, and the second approach is based on maximizing the penetration depth (MPD). The optimal shapes are compared for a fixed shank radius and evaluations are carried out for different nose lengths and different values of the coefficient of friction. It is shown that friction may have a visible effect on the final penetration depth, although in all cases, the friction effect on the optimal shape of the projectile nose is negligible. The obtained optimal nose shapes are implemented in computational analysis in an own developed code that had been verified with numerous test data to examine whether these shapes actually lead to a minimum resistive force and to a maximal penetration depth. Significant effectiveness of the optimal shape performance is shown for larger radius projectiles having a defined nose length.

Modelling the penetration of subsonic rigid projectile probes into granular materials using the cavity expansion theory

Typical subsurface investigation tools used on Earth are not applicable for subsurface exploration of the Moon and other extraterrestrial bodies due to payload limitations in space missions. Instead, light and compact subsonic projectile probes can be considered as an alternative subsurface investigation tool in such exploratory missions to overcome a host of challenges. Such probes can be launched from a lunar orbiter or lander to the surface of the Moon to provide the initial effective penetration from the impact. Here, we develop a model based on the spherical cavity expansion theory to predict the deceleration rate and final penetration depth of a rigid projectile probe into geological targets under the perpendicular subsonic impact. Two stress fields are assumed to propagate in the medium, plastic (near field) and elastic (far field), upon the impact. The stresses at the cavity wall are obtained by combining the Mohr–Coulomb failure criterion for the target failure (plastic region) considering two different assumptions for plastic wave propagation. Two field experiments are used to compare and assess the robustness of the proposed solutions in the subsonic range. Base on the simulation results and the experiments, it is concluded that the cavity expansion model considering the locked hydrostat assumption, with the modification here introduced for the volumetric strain, can provide us with a reasonable prediction of the projectile penetration, final penetration depth and the stresses on the probe. Thus, our proposed solution can be used as a benchmark for sophisticated and computationally-expensive numerical calculations.

Research progress of target resistance model of cavity expansion theory and its application

The cavity expansion theory is one of the main basic theories for the theoretical analysis of penetration problems. It is mainly used to analyze the failure response characteristics of typical target materials under impact load, and then to determine the penetration resistance of the target. It is widely used in the analysis of high-speed impact penetration and failure problems. Domestic and foreign scholars have made abundant research achievements on plastic and (quasi) brittle materials based on the theory of cylindrical and spherical cavity expansion. Starting from the theoretical system of the static/dynamic cavity expansion model, the results of the cavity expansion model in different directions are introduced, mainly involving the cavity expansion pressure theoretical calculation model and numerical simulation method under ideal penetration conditions, and the application of cavity expansion model to typical penetration problems and complex missile target conditions. The theoretical calculation model under ideal penetration conditions based on cavity expansion theory mainly discusses the influence aspects of target material, yield criterion and equation of state on target resistance and the applicability of the cavity expansion model. According to the different initial conditions in the numerical simulation, two numerical simulation methods of cavity surface constant velocity/constant pressure are introduced, and the reliability of the numerical simulation method is proved. The basic assumptions, application scope and engineering application characteristics of the cavity expansion model are summarized, and its applications in typical penetration problems and complex missile targets such as multilayer composite target plate, constrained target, projectile grooves and projectile body with special cross-section are listed. Based on the current status of the cavity expansion model, we summarized the current cavity expansion model application direction in the field of impact dynamics, and the problems existing in the application of the cavity expansion model, as well as the key development direction in the cavity expansion model.

Theoretical analysis of long-term setup of driven piles considering soil anisotropy, destructuration and creep effects

The soil adjacent to the pile suffers a severe destructuration during pile installation and then regains its strength during the setup stage. This paper presents a theoretical approach to evaluate the long-term setup of driven piles in soft sensitive clays. An advanced elastic-viscoplastic model that accounts for fabric anisotropy, particle bonding and viscosity, is used in both stages of pile installation and setup. A semi-analytical solution for the undrained cylindrical and spherical cavity expansion in soft sensitive clays is developed to capture the pile installation effects. The changes in the stress state and fabric anisotropy of the soil adjacent to the pile caused by pore pressure dissipation and sustained service loads are analyzed. The evolution of soil mechanical behavior is incorporated into the load-transfer models of shaft friction and base resistance. Comparisons between measured and predicted results of field tests at three different sites demonstrate the applicability of the proposed approach. The calculated results illustrate that the ultimate bearing capacity increases steadily with time during the long-term setup stage. An extensive study is carried out to evaluate the effects of soil's initial anisotropy and bonding on the load-settlement behavior of driven piles.

Spherical cavity expansion in porous rock considering plasticity and damage

AbstractIn rock engineering, damage evolution upon loading imposes significant impacts on the stiffness of rock mass and its deformation characteristics. In order to investigate the influence of both damage and plasticity on cavity expansion, a plastic damage solution is derived for undrained spherical cavity expansion in rock medium. For the consideration of plasticity‐damage, Modified Cam‐Clay (MCC) model is selected as the plasticity driver, a damage evolution criterion is adopted and coupled with MCC. The coupled damage MCC model is validated against experimental data in the literature. The proposed cavity expansion solution with the consideration of plasticity damage is verified through a classic solution in literature. The role of damage in undrained spherical cavity expansion is investigated by studying the spatial variations of effective stress, pore pressure and damage for cases with different stress ratios. Distribution of cavity expansion induced plastic and damage zones for cases with different stress ratios are also reproduced and discussed. Cavity expansion results show that, the damage zone should be considered for engineering application as the plastic zone is affected due to the damage evolution. In addition, the stability (e.g., radial stress) of rock mass is overestimated in classical solution without the consideration of damage zone.