Cavity Expansion in Soil of Finite Radial Extent

2014 ◽  
Vol 553 ◽  
pp. 464-469
Author(s):  
Mohammad Pournaghiazar ◽  
Adrian R. Russell ◽  
Nasser Khalili
Keyword(s):  
Analytical Solution ◽  
Quartz Sand ◽  
Stress Condition ◽  
Constitutive Models ◽  
Solution Procedure ◽  
Cavity Expansion ◽  
Finite Boundary ◽  
Spherical Cavities ◽  
Radial Extent ◽  
Zero Radius

The problem of drained cavity expansion in a soil of finite radial extent is investigated. Spherical cavities expanded from zero radius subjected to a constant stress condition at the finite boundary are considered. The new analytical solution procedure presented enables more advanced constitutive models to be implemented than possible than when using other solution procedures. Cavity expansion results generated for a Sydney quartz sand highlight substantial differences between cavity limit pressures for boundaries of finite and infinite radial extent.

Cavity Expansion in Cohesive-Frictional Soils with Limited Dilation

Géotechnique ◽  
2021 ◽  
pp. 1-20
Author(s):  
John P. Carter ◽  
Hai-Sui Yu
Keyword(s):  
Analytical Solution ◽  
Characteristic Length ◽  
Cavity Expansion ◽  
Cavity Pressure ◽  
Initial Radius ◽  
Limit Pressure ◽  
Zero Radius ◽  
Self Similar ◽  
Limiting Pressure ◽  
Frictional Soils

The problem of cavity expansion from zero radius has no characteristic length and therefore possesses a similarity solution, in which the cavity pressure remains constant and the continuing deformation is geometrically self-similar. In this case, the incremental velocity approach first used by Hill (1950) to analyze cavity expansion in Tresca materials can be extended to derive a solution for limiting pressure of cavity expansion in other types of material. In this article, a rigorous semi-analytical solution is derived, following Hill's incremental velocity method, for the expansion of cavities from zero initial radius in cohesive-frictional soils with limited dilation. In particular, the radius of the elastic-plastic interface c is used in this article as the time scale and the solution for the limit pressure has been presented. Solutions are evaluated for a number of cases representative of a range of cohesive-frictional and dilatant soils. A comparison is also made between the solutions presented here and previous solutions for cohesive-frictional soils that have unlimited (on-going) plastic dilation. In particular, the influence of limited plastic dilation on the cavity limit pressure is identified and discussed.

Cavity expansion in unsaturated soils of finite radial extent

2018 ◽  
Vol 102 ◽  
pp. 216-228 ◽  
Author(s):  
Yan Cheng ◽  
Hong-Wei Yang ◽  
De'an Sun
Keyword(s):  
Unsaturated Soils ◽  
Cavity Expansion ◽  
Radial Extent

scholarly journals A Full-Fledged Analytical Solution to the Quantum Harmonic Oscillator for Undergraduate Students of Science and Engineering

Physics ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 541-570
Author(s):  
Arturo Rodríguez-Gómez ◽  
Ana Laura Pérez-Martínez
Keyword(s):  
Analytical Solution ◽  
Harmonic Oscillator ◽  
Undergraduate Students ◽  
Solution Procedure ◽  
Quantum Harmonic Oscillator ◽  
Web Based ◽  
Chemistry Students ◽  
Physics Students ◽  
Step Procedure ◽  
And Mathematics

The quantum harmonic oscillator is a fundamental piece of physics. In this paper, we present a self-contained full-fledged analytical solution to the quantum harmonic oscillator. To this end, we use an eight-step procedure that only uses standard mathematical tools available in natural science, technology, engineering and mathematics disciplines. This solution is accessible not only for physics students but also for undergraduate engineering and chemistry students. We provide interactive web-based graphs for the reader to observe the shape of the wave functions for an electron and a proton when both are subject to the same potential. Each of the eight steps in our solution procedure is treated as a separate problem in order to allow the reader to quickly consult any step without the need to review the entire article.

An Analytical Solution of the Generalized Phase-Lagging Equation for Ultra-Fast Heat Transfer in One-Dimensional Semi-Infinite Domain

2012 ◽  
Author(s):  
Vladimir Kulish ◽  
Kirill V. Poletkin
Keyword(s):  
Analytical Solution ◽  
Domain Boundary ◽  
Integral Form ◽  
Solution Procedure ◽  
Generalized Derivatives ◽  
Infinite Domain ◽  
One Dimensional ◽  
Confluent Hypergeometric Functions ◽  
The Laplace Transform ◽  
Derivatives Of

The paper presents an integral solution of the generalized one-dimensional phase-lagging heat equation with the convective term. The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of non-integer orders). Confluent hypergeometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the temperature and heat flux. The solution is valid everywhere within the domain, including the domain boundary.

scholarly journals Drained cavity expansion analysis with a unified state parameter model for clay and sand

2018 ◽  
Vol 55 (7) ◽  
pp. 1029-1040 ◽  
Author(s):  
Pin-Qiang Mo ◽  
Hai-Sui Yu
Keyword(s):  
Analytical Solution ◽  
Plastic Region ◽  
State Parameter ◽  
Auxiliary Variable ◽  
Cavity Expansion ◽  
Large Strains ◽  
Initial State ◽  
Model Soil ◽  
Spacing Ratio ◽  
Soil Behaviour

This paper presents an analytical solution for drained expansion in both spherical and cylindrical cavities with a unified state parameter model for clay and sand (CASM). The solution developed here provides the stress and strain fields during the expansion of a cavity from an initial to an arbitrary final radius. Small strains are assumed for the elastic region and large strains are applied to soil in the plastic region by using logarithmic strain definitions. Since its development, the unified CASM model has been demonstrated by many researchers to be able to capture the overall soil behaviour for both clay and sand under both drained and undrained loading conditions. In this study, the CASM model is used to model soil behaviour whilst a drained cavity expansion solution is developed with the aid of an auxiliary variable. This is an extension of the undrained solution presented by the authors in 2017. The parametric study investigates the effects of various model constants including the stress-state coefficient and the spacing ratio on soil stress paths and cavity expansion curves. Both London clay and Ticino sand are modelled under various initial stress conditions and initial state parameters. The newly developed analytical solution highlights the potential applications in geotechnical practice (e.g., for the interpretation of cone penetration test data) and also provides useful benchmarks for numerical simulations of cavity expansion problems in critical state soils.

The Non-Stationary Dynamic Analytical Solution of a Spherical/Cylindrical Cavity Expansion

2020 ◽  
Vol 87 (11) ◽  
Author(s):  
V. R. Feldgun ◽  
D. Z. Yankelevsky
Keyword(s):  
Analytical Solution ◽  
General Solution ◽  
Cylindrical Cavity ◽  
Constant Velocity ◽  
Present Model ◽  
Elastic Region ◽  
Cavity Expansion ◽  
Special Cases ◽  
Significant Difference ◽  
Cavity Boundary

Abstract A review of the pertinent literature related to the dynamic expansion of a spherical/cylindrical cavity shows that all the solutions with kinematic boundary conditions deal with a constant velocity at the cavity boundary. This paper develops a new general solution of the nonstationary dynamic problem of cavity expansion, which allows the application of time-dependent motion conditions at the cavity boundary. This solution can be used, for example, in the development of approximate approaches for projectiles penetrating with a non-constant velocity into different targets. Due to the complexity of the nonlinear nonstationary problem, an analytical solution of the problem may be developed if simplified constitutive relationships are used. In the present model, a simplified material model with a locked equation of state and a linear shear failure relationship is implemented. This solution may be applied to different materials such as concrete, soil, and rock. Special cases of the newly developed nonstationary solution are compared with different spherical and cylindrical cavity expansions solutions reported in the literature, and a good agreement is obtained. The capability of the present model is demonstrated in a following investigation of representative cases of cavity expansion with zero, constant, and variable acceleration of the cavity boundary. A significant difference in the stress variation for the different cases is shown. Along with the general solution which deals with an elastic–plastic region, a simplified solution which disregards the contribution of the elastic region is presented and the evaluation of the elastic region effect may be assessed.

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