K1 of Chevalley groups are nilpotent

Abstract Let F be a reduced irreducible root system and R be a commutative ring. Further, let G(F,R) be a Chevalley group of type F over R and E(F,R) be its elementary subgroup. We prove that if the rank of F is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(F,R)/E(F,R) is nilpotent by abelian. In particular, when G(F,R) is simply connected the quotient K1(F,R)=G(F,R)/E(F,R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.

Coupled consolidation of a solid, infinite cylinder using a Terzaghi formulation

Axisymmetric consolidation is a classical boundary value problem for geotechnical engineers. Under some circumstances an analysis in which the changes in pore pressure, effective stress and displacement can be uncoupled from each other is sufficient, leading to a Terzaghi formulation of the axisymmetric consolidation equation in terms of the pore pressure. However, representation of the Mandel-Cryer effect usually requires more complex, coupled, Biot formulations. A new coupled formulation for the plane strain, axisymmetric consolidation problem is presented for small, linear elastic deformations. A single, easily evaluated parameter couples changes in pore pressure to changes in effective stress, and the resulting differential equation for pore pressure dissipation is very similar to Terzaghi’s classic formulation. The governing equations are then solved using finite differences and the consolidation of a solid infinite cylinder analysed, calculating the variation with time and with radius of the excess pore pressure and the radial displacement. Comparison with a previously published semi-analytical solution indicates that the formulation successfully embodies the Mandel-Cryer effect.