Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Publisher Summary
This chapter focuses on fundamentals of poroelasticity. The presence of a freely moving fluid in a porous rock modifies its mechanical response. Two mechanisms play a key role in the interaction between the interstitial fluid and the porous rock: (i) an increase of pore pressure induces a dilation of the rock; and (ii) compression of the rock causes a rise of pore pressure, if the fluid is prevented from escaping the pore network. These coupled mechanisms bestow an apparent time-dependent character to the mechanical properties of the rock. If excess pore pressure, induced by compression of the rock, is allowed to dissipate through diffusive fluid mass transport, further deformation of the rock progressively takes place. The rock is more compliant under drained conditions than undrained ones. The chapter discusses the formulation and analysis of coupled deformation–diffusion processes, within the framework of the Biot theory of poroelasticity. The Biot model of a fluid-filled porous material is constructed on the conceptual model of a coherent solid skeleton and a freely moving pore fluid.

Fundamentals of Poroelasticity

http://www.vliz.be/imisdocs/publications/242609.pdf

Seawater intrusion processes, investigation and management: Recent advances and future challenges

A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering

article i nfo We describe a novel cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain. Special blended cubic Hermite-power-law basis functions, with arbitrary index 0b αb1, are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. The implementation of blended infinite elements to model semi-infinite crack problems is also described. Explicit formulae for the integrated kernels associated with the cubic Hermite and blended basis functions are provided. The cubic Hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions. The convergence rate of the cubic Hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the O(h 4 ) error due to the Hermite cubic interpolation. The errors due to these two approximations need to be matched in order to achieve optimal convergence. Backward Euler time-stepping yields a robust algorithm that, along with geometric increments in the time-step, can be used to explore the transition between propagation regimes over many orders of magnitude in time.

/pdf/computer-methods-in-applied-mechanics-and-engineering-53g5581srr.pdf

Computer Methods in Applied Mechanics and Engineering

This article explores the rich heritage of the boundary element method (BEM) by examining its mathematical foundation from the potential theory, boundary value problems, Green's functions, Green's identities, to Fredholm integral equations. The 18th to 20th century mathematicians, whose contributions were key to the theoretical development, are honored with short biographies. The origin of the numerical implementation of boundary integral equations can be traced to the 1960s, when the electronic computers had become available. The full emergence of the numerical technique known as the boundary element method occurred in the late 1970s. This article reviews the early history of the boundary element method up to the late 1970s.

https://www.olemiss.edu/sciencenet/benet/05eabeb.pdf

Heritage and early history of the boundary element method

Poroelastic response of a borehole in a non-hydrostatic stress field

The radial basis function (RBF) collocation method uses global shape functions to interpolate and collocatethe approximate solution of PDEs. It is a truly meshless method as compared to some of the so-calledmeshless or element-free ﬁnite element methods. For the multiquadric and Gaussian RBFs, there are twoways to make the solution converge—either by reﬁning the mesh size

Alexander H.-D. Cheng

Papers

Fundamentals of Poroelasticity

Heritage and early history of the boundary element method

Poroelastic response of a borehole in a non-hydrostatic stress field

Exponential convergence and H‐c multiquadric collocation method for partial differential equations

Material coefficients of anisotropic poroelasticity