This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.

The Mechanics of Earthquakes and Faulting

Basic error estimates for elliptic problems

Three-dimensional Problems of the Theory of Elasticity. By A. I. Lur’e.1964. (Interscience Publishers)

We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume ∼hβ, where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼hα. Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α=β=0, nonlinear bending theory to α=β=2, von Karman theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [29] which states that for maps v:(0,1)3→ℝ3, the L2 distance of ∇v from a single rotation is bounded by a multiple of the L2 distance from the set SO(3) of all rotations.

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A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence

https://repositorium.sdum.uminho.pt/bitstream/1822/19623/1/6.7.32%202012.pdf

Compliant contact force models in multibody dynamics : evolution of the Hertz contact theory

Elastic structures, conceived as slender bodies able to transmit loads, have been studied by scientists and engineers for centuries. By the seventeenth century several useful theories of elastic structures had emerged, with applications to civil and mechanical engineering problems. In recent years improved mathematical tools have extended applications into new areas such as geomechanics and biomechanics. This book, first published in 1998, offers a critically filtered collection of the most significant theories dealing with elastic slender bodies. It includes mathematical models involving elastic structures, which are used to solve practical problems with particular emphasis on nonlinear problems. This collection of interesting and important problems in elastic structures will appeal to a broad range of scientists, engineers and graduate students working in the area of structural mechanics.

Mathematical Models for Elastic Structures

A model for an elastic-plastic body

Qualitative Methods in Elasticity

The normal impact of an elastic sphere against a rigid plane is described using an elastic solution for a heavy sphere on a rigid plane found by Bondareva (1970). The solution is valid only in the case of equilibrium, but may be extended also to the dynamical case, provided that the time of duration of the impact is much larger that the time employed by the elastic waves in traversing the whole sphere after the first impact. The fundamental result is that, according to the new solution, the time of adherence between the first contact and the detachment is slightly higher than the same time evaluated according to the classical Hertz's theory.

Piero Villaggio

Papers

Mathematical Models for Elastic Structures

A model for an elastic-plastic body

Qualitative Methods in Elasticity

The Rebound of an Elastic Sphere Against a Rigid Wall

A Free Boundary Value Problem in Plate Theory