W.M. Coombs

Observations on Mohr-Coulomb Plasticity under Plane Strain

Linear elastic-perfect plasticity using the Mohr-Coulomb yield surface is one of the most widely used pressure-sensitive constitutive models in engineering practice. In the area of geotechnical engineering, a number of problems, such as cavity expansion, embankment stability, and footing bearing capacity, can be examined using this model together with the simplifying assumption of plane strain. This paper clarifies the situation regarding the direction of the intermediate principal stress in such an analysis and reveals a unique relationship between hydrostatic pressure and the principal stress ratio for Mohr-Coulomb and Tresca perfect plasticity under those plane-strain conditions. The rational relationship and direction of the intermediate principal stress are illustrated through both material-point and finite-element simulations. The latter involves the analysis of a rigid strip footing bearing onto a weightless soil and the finite-deformation expansion of a cylindrical cavity.

Numerical Simulation of Fracking in Shale Rocks: Current State and Future Approaches

Extracting gas from shale rocks is one of the current engineering challenges but offers the prospect of cheap gas. Part of the development of an effective engineering solution for shale gas extraction in the future will be the availability of reliable and efficient methods of modelling the development of a fracture system, and the use of these models to guide operators in locating, drilling and pressurising wells. Numerous research papers have been dedicated to this problem, but the information is still incomplete, since a number of simplifications have been adopted such as the assumption of shale as an isotropic material. Recent works on shale characterisation have proved this assumption to be wrong. The anisotropy of shale depends significantly on the scale at which the problem is tackled (nano, micro or macroscale), suggesting that a multiscale model would be appropriate. Moreover, propagation of hydraulic fractures in such a complex medium can be difficult to model with current numerical discretisation methods. The crack propagation may not be unique, and crack branching can occur during the fracture extension. A number of natural fractures could exist in a shale deposit, so we are dealing with several cracks propagating at once over a considerable range of length scales. For all these reasons, the modelling of the fracking problem deserves considerable attention. The objective of this work is to present an overview of the hydraulic fracture of shale, introducing the most recent investigations concerning the anisotropy of shale rocks, then presenting some of the possible numerical methods that could be used to model the real fracking problem.

Parallel computations in nonlinear solid mechanics using adaptive finite element and meshless methods

Purpose – A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities. Design/methodology/approach – A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Bely...

A posteriori discontinuous Galerkin error estimator for linear elasticity

Abstract This paper presents for the first time the derivation of an hp a posteriori error estimator for the symmetric interior penalty discontinuous Galerkin finite element method for linear elastic analysis. Any combination of Neumann and Dirichlet boundary conditions are admissible in the formulation, including applying Neumann and Dirichlet on different components on the same region of the boundary. Therefore, the error estimator is applicable to a variety of physical problems. The error estimator is incorporated into an hp-adaptive finite element solver and verified against smooth and non-smooth problems with closed-form analytical solutions, as well as, being demonstrated on a non-smooth problem with complex boundary conditions. The hp-adaptive finite element analyses achieve exponential rates of convergence. The performances of the hp-adaptive scheme are contrasted against uniform and adaptive h refinement. This paper provides a complete framework for adaptivity in the symmetric interior penalty discontinuous Galerkin finite element method for linear elastic analysis.

A modified CPT based installation torque prediction for large screw piles in sand.

The intermediate soil (e.g. silt, sandy silt, clayey silt) response at the standard cone penetration (CPT) velocity of 20 mm/s is generally partially drained, falling between that of sand and clay. As a result, a proper interpretation of CPT (or CPTU) in such mixed soils is not always straightforward. In order to properly analyse the in situ soil response and avoid incorrect estimates of soil parameters, the preliminary assessment of drainage conditions is essential. In this paper, changes in normalized CPTU measurements caused by changes in cone velocity are analysed. Penetration rate effects are assessed by means of No. 8 piezocone tests, with penetration rates ranging from about 0.9 to 61.7 mm/s. Tests were performed at a site located at the southern margin of the Po river valley (Northern Italy), where the subsoil mainly consists in a clayey silt deposit. Limitations on the applicability of some widely used empirical correlations, proposed for sands, are investigated and some preliminary results are shown. tests carried out at variable rates are nowadays recognized as an effective way to identify the transition point from undrained to partially drained and drained responses. A dimensionless velocity V (Finnie & Randolph 1994), given by:

Fast native-MATLAB stiffness assembly for SIPG linear elasticity

When written in MATLAB the finite element method (FEM) can be implemented quickly and with significantly fewer lines, when compared to compiled code. MATLAB is also an attractive environment for generating bespoke routines for scientific computation as it contains a library of easily accessible inbuilt functions, effective debugging tools and a simple syntax for generating scripts. However, there is a general view that MATLAB is too inefficient for the analysis of large problems. Here this preconception is challenged by detailing a vectorised and blocked algorithm for the global stiffness matrix computation of the symmetric interior penalty discontinuous Galerkin (SIPG) FEM. The major difference between the computation of the global stiffness matrix for SIPG and conventional continuous Galerkin approximations is the requirement to evaluate inter-element face terms, this significantly increases the computational effort. This paper focuses on the face integrals as they dominate the computation time and have not been addressed in the existing literature. Unlike existing optimised finite element algorithms available in the literature the paper makes use of only native MATLAB functionality and is compatible with GNU Octave. The algorithm is primarily described for 2D analysis for meshes with homogeneous element type and polynomial order. The same structure is also applied to, and results presented for, a 3D analysis. For problem sizes of 106 degrees of freedom (DOF), 2D computations of the local stiffness matrices were at least ≈24 times faster, with 13.7 times improvement from vectorisation and a further 1.8 times improvement from blocking. The speed up from blocking and vectorisation is dependent on the computer architecture, with the range of potential improvements shown for two architectures in this paper.

Improving seabed cable plough performance for offshore renewable energy.

Seabed ploughing plays an important part in providing protection to subsea cables that connect offshore renewables, and the process represents a significant portion of the overall installation cost. Current models for predicting the required tow forces for seabed ploughs are based on semi-empirical methods as conventional geotechnical finite element analysis is not suited to modelling this process due to the large de-formations involved. The project described in this paper aims to develop a new material point method numerical modelling software to predict seabed plough response, which will enable new designs to be optimised. This will be validated against physical modelling, a part of which is the focus of this paper. This includes both 1g and centrifuge testing as well as new techniques such as 3D soil surface scanning. The testing also provides insights into share geometry influences on plough behaviour, and will allow improvements to existing empirical models.

Reuleaux plasticity : improving Mohr-Coulomb and Drucker-Prager.

The yielding of soil exhibits both a Lode angle dependency and a dependency on the intermediate principal stress. Ignoring these leads to a loss of realism in geotechnical analysis, yet neither of the widely used Mohr-Coulomb (M-C) or Drucker-Prager (D-P) models include both. This paper presents a simple pressure-dependent plasticity model based on a modified Reuleaux (mR) triangle which overcomes these limitations and yet (like the M-C and D-P formulations) allows for an analytical backward-Euler stress integration solution scheme. This latter feature is not found in more sophisticated (and computationally expensive) models. The mR deviatoric function is shown to provide a significantly improved fit to experimental data when compared with the M-C and D-P functions. Finite deformation finite-element analysis of the expansion of a cylindrical cavity is presented, verifying the use of the mR constitutive model for practical analyses.