Generalised homogenisation procedures for granular materials

Engineering materials are generally non-homogeneous, yet standard continuum descriptions of such materials are admissible, provided that the size of the non-homogeneities is much smaller than the characteristic length of the deformation pattern. If this is not the case, either the individual non-homogeneities have to be described explicitly or the range of applicability of the continuum concept is extended by including additional variables or degrees of freedom. In the paper the discrete nature of granular materials is modelled in the simplest possible way by means of finite-difference equations. The difference equations may be homogenised in two ways: the simplest approach is to replace the finite differences by the corresponding Taylor expansions. This leads to a Cosserat continuum theory. A more sophisticated strategy is to homogenise the equations by means of a discrete Fourier transformation. The result is a Kunin-type non-local theory. In the following these theories are analysed by considering a model consisting of independent periodic 1D chains of solid spheres connected by shear translational and rotational springs. It is found that the Cosserat theory offers a healthy balance between accuracy and simplicity. Kunin's integral homogenisation theory leads to a non-local Cosserat continuum description that yields an exact solution, but does not offer any real simplification in the solution of the model equations as compared to the original discrete system. The rotational degree of freedom affects the phenomenology of wave propagation considerably. When the rotation is suppressed, only one type of wave, viz. a shear wave, exists. When the restriction on particle rotation is relaxed, the velocity of this wave decreases and another, high velocity wave arises

Lecture Notes (Continuum Mechanics - Basic) : Elasticity and Strength of Rock Materials

Contents 1. The Stress Tensor 2. Deformation and Strain 3. Stress-Strain Relations: Linear Elasticity 4. Mechanical Equilibrium 5. The Mohr Coulomb Yield Criterion 6. Flow Rule and Dilatancy: An Elementary Discussio

Modelling shear bands in a volcanic conduit: Implications for over-pressures and extrusion-rates

Shear bands in a volcanic conduit are modelled for crystal-rich magma flow using simplified conditions to capture the fundamental behaviour of a natural system. Our simulations begin with magma crystallinity in equilibrium with an applied pressure field and isothermal conditions. The viscosity of the magma is derived using existing empirical equations and is dependent upon temperature, water content and crystallinity. From these initial conduit conditions we utilize the Finite Element Method, using axi-symmetric coordinates, to simulate shear bands via shear localisation. We use the von Mises visco-plasticity model with constant magma shear strength for a first took into the effects of plasticity. The extent of shear bands in the conduit is explored with a numerical model parameterized with values appropriate for Soufriere Hills Volcano, Montserrat, although the model is generic in nature. Our model simulates shallow (up to approximately 700 in) shear bands that occur within the upper conduit and probably govern the lava extrusion style due to shear boundaries. We also model the change in the over-pressure field within the conduit for flow with and without shear bands. The pressure change can be as large as several MPa at shallow depths in the conduit, which generates a maximum change in the pressure gradient of 10's of kPa/m. The formation of shear bands could therefore provide an alternative or additional mechanism for the inflation/deflation of the volcano flanks as measured by tilt-metres. Shear bands are found to have a significant effect upon the magma ascent rate due to shear-induced flow reducing conduit friction and altering the over-pressure in the upper conduit. Since we do not model frictional controlled slip, only plastic flow, our model calculates the minimum change in extrusion rate due to shear bands. However, extrusion rates can almost double due to the ton nation of shear bands, which may help suppress volatile loss. Due to the paucity of data and large parameter space available for the magma shear strength our model results can only allow for a qualitative comparison to a natural system at this stage. (c) 2007 Elsevier B.V. All rights reserved

Standard equations of computational geodynamics: governing equations for mantle convection simulations

Governing equations for mantle convection, Stokes equations, heat equation, dynamic topography, computational aspects, toroidal, poloidal, thermodynamic

Continuum Mechanics A & B and Exercises

The present manuscript is an updated version of the lecture notes I had used in 1993 for lectures at Gifu University and the TU Delft. I will update this introduction as the lecture evolves. At this stage I have revised Section 1 on Vectors and Tensors. The main difference to the previous manuscript is that I have taken out the subsection on covariant differentiation, Christoffel symbols etc in acknowledgement of the fact that if curve-linear coordinates are used it is usually either is in a numerical context (iso-parametric finite elements), or one restricts oneself to cylindrical or spherical coordinates. In both cases the full differential calculus in curve-linear coordinates is not needed. However this does not mean that the ability to express geometric transformations in Lagrangian coordinates (which are curve-linear in general) is unnecessary. Quite the contrary, some of the basic geometric relationships derived in Section 1 will be very useful e.g. in the construction of constitutive relationships and the interpretation of various well-known definitions of stress and strain tensors. Accordingly we begin with a brief representation of vectors, tensors in curvilinear coordinates, whereby the curvilinear coordinates are the deformed material coordinates. The present lecture series is again intended for students who have already had their first encounter with continuum mechanics. I begin with a brief representation of vectors, tensors and equations of motion in curvilinear coordinates, whereby the curvilinear coordinates may be the deformed material coordinates

Elasticity, Yielding and Episodicity in Simple Models of Mantle Convection

We explore the implications of refinements in the mechanical description of planetary constituents on the convection modes predicted by finite element simulations. The refinements consist in the inclusion of incremental elasticity, plasticity (yielding) and multiple simultaneous creep mechanisms in addition to the usual visco-plastic models employed in the context of unified plate-mantle models. The main emphasis of this paper rests on the constitutive and computational formulation of the model. We apply a consistent incremental formulation of the non-linear governing equations avoiding the computationally expensive iterations that are otherwise necessary to handle the onset of plastic yield. In connection with episodic convection simulations, we point out the strong dependency of the results on the choice of the initial temperature distribution. Our results also indicate that the inclusion of elasticity in the constitutive relationships lowers the mechanical energy associated with subduction events

Towards A Unified Model For The Dynamics Of Planets

The way a planet deforms in response to thermal or gravitational driving forces, depends on the material properties of its constituents. The Earth's behaviour is unique in that its outermost layer consists of a small number of continuously moving plates. Venus, another planet of similar size and bulk composition to the Earth displays signs of active volcanism but there is no evidence of plate movements or plate tectonics. In this article we review Eulerian finite element (FE) schemes and a particle-in-cell (PIC) FE scheme.1 Focussing initially on models of crustal deformation at a scale of a few tens of km, we choose a Mohr-Coulomb yield criterion based upon the idea that frictional slip occurs on whichever one of many randomly oriented planes happens to be favorably oriented with respect to the stress field. As coupled crust/mantle models become more sophisticated it is important to be able to use whichever failure model is appropriate to a given part of the system. We have therefore developed a way to represent Mohr-Coulomb failure within a mantle-convection fluid dynamics code. With the modelling of lithosphere deformation we use an orthotropic viscous rheology (a different viscosity for pure shear to that for simple shear) to define a preferred plane for slip to occur given the local stress eld. The simple-shear viscosity and the deformation can then be iterated to ensure that the yield criterion is always satisfied. We again assume the Boussinesq approximation -neglecting any effect of dilatancy on the stress field. Turning to the largest planetary scale, we present an outline of the mechanics of unified models plate-mantle models and then show how computational solutions can be obtained for such models using Escript. The consequent results for different types of convection are presented and the stability of the observed flow patterns with respect to different initial conditions and computational resolutions is discussed

Thermal Effects in the Evolution of Initially Layered Mantle Material

A simplied model for anisotropic mantle convection based on a novel class of rheologies, originally developed for folding instabilities in multilayered rock, is extended through the introduction of a thermal anisotropy dependent on the local layering. To examine the eect of the thermal anisotropy on the evolution of mantle material, a parallel implementation of this model was undertaken using the Escript modelling toolkit and the Finley nite element computational kernel. For the cases studied, there appears to little if any eect. For comparative purposes, the eects of anisotropic shear viscosity and the introduced thermal anisotropy are also presented. These results contribute to the characterisation of viscous anisotropic mantle convection subject to variation in thermal conductivities and shear viscosities

Non-Newtonian Effects in Simple Models of Mantle Convection

One of the difficulties with self consistent plate-mantle models capturing multiple physical features, such as elasticity, non-Newtonian flow properties, and temperature dependence, is that the individual behaviours cannot be considered in isolation. For instance, if a viscous mantle convection model is generalized idealistically to include hypo-elasticity, then problems based on Earth-like Rayleigh numbers exhibit almost insurmountable numerical stability issues due to spurious softening associated with the co-rotational stress terms. These difficulties can be avoided if a stress limiter is introduced in the form of a power law rheology or yield criterion. A general Eulerian model is discussed and it is shown that the basic convection modes of a cooling planet are reproduced

A Theory Of Disclinations For Anisotropic Materials With Bending Stiffness

The paper considers a special type of failure in layered materials with sliding layers that develops as a progressive breakage of layers forming a narrow zone. This zone propagates as a "bending crack", i.e. a crack that can be represented as a distribution of disclinations. This situation is analysed using a 2D Cosserat continuum model. Edge dislocations (displacement discontinuities) and a disclination (the discontinuity in the derivative of layer deflection) are considered. The disclination does not create shear stresses along the axis perpendicular to the direction of layering, while the dislocation does not create a moment stress along the same axis. Semi-infinite and finite bending cracks normal to layering are considered. The moment stress concentration at the crack tip has a singularity of the power -1/4. The possibility to derive equilibrium conditions for cracks and disclinations from J-type path independent integrals is also pointed out