Application of a State Variable Description of Inelastic Deformation to Geological Materials
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APPLICATION OF A STATE VARIABLE DESCRIPTION OF INELASTIC DEFORMATION TO GEOLOGICAL MATERIALS A thesis submitted for the degree of Doctor of Philosophy (University of London) and for the Diploma of University College by Stephen John Covey-Crump Department of Geological Sciences, University College London, Gower Street, London, WC1E 6BT ProQuest Number: 10630768 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10630768 Published by ProQuest LLC(2017). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 2 ABSTRACT The work reported in this thesis is an investigation of the potential for employing non steady-state constitutive relations to describe the inelastic deformation properties of rock forming minerals. Attempts to determine generally applicable constitutive equations for inelastic deformation are complicated by the severe path dependence of the deformation response and by the wide range of mechanisms by which that deformation is accomplished. In the geological literature it has been normal practice to circumvent these problems by approximating the deformation as occurring at steady-state. This presents difficulties for the description of deformation by inherently transient mechanisms, for the description of small strain deformation, and for the reliability of extrapolating laboratory mechanical properties to geological deformation conditions. In contrast, in the materials science literature several systems of inelastic constitutive equations which do not make the steady-state approximation have been proposed. One of the oldest and most widely applied of these i.e. that due to Hart and coworkers, was chosen for investigation here to determine its potential for geological applications. To be successful any deformation constitutive equation must satisfy three criteria. Firstly it must provide an adequate description of the material behaviour, secondly it must be analytically / numerically integrable for deformation modelling purposes, and thirdly it must contain material parameters which can be evaluated from the results of deformation experiments. In the first half of this thesis the descriptive capacity and integrability of Hart’s equations are investigated by attempting to provide a comprehensive description of inelastic deformation from the perspective offered by Hart’s analysis. Hart bracketed from consideration several aspects of inelastic deformation which are of considerable geological importance i.e. in particular the influence of nominally deformation independent recovery processes (active at high temperatures), of solute impurities, of finely dispersed inclusions and of grain-size. Procedures for identifying the effect of these factors on the results of deformation experiments, and possible strategies for extending the analysis to accommodate them are outlined. The second half of the thesis describes an experimental programme designed to apply Hart’s description to the inelastic deformation of Carrara marble at 200 MPa confining pressure in the temperature range 120 to 700°C. The primary aim of die experimental programme was to determine whether die material parameters in Hart’s description can be determined with sufficient accuracy at elevated confining pressures (given the technical limitations on the quality of the data obtained from such tests) for the approach to be of interest for the characterization of geological materials. From die results of several hundred experiments it is found that those material parameters can be evaluated sufficiendy accurately. Furthermore, the proposed strategy for extending Hart’s analysis to high temperatures is shown to have considerable potential. In so doing the first description of the inelastic deformation properties of calcite at temperatures below 400°C is presented, and an experimental programme is outlined which can be applied to materials of even greater geological significance. 3 ACKNOWLEDGEMENTS The work reported in this thesis and presently ongoing, has had a long gestation period. The aim has always been to work towards a description of inelastic deformation which is above all comprehensive, but yet which is also practical in the sense that the material parameters in it can be readily evaluated through experiments conducted at high confining pressures. The fact that it has been possible to work towards such a general goal, is to a great measure due to the outstanding nature of the courses that I have been able to take in my university career. Of particular importance has been the advanced igneous / metamorphic and structural geology third year option in my undergraduate degree at Imperial College, London. It was the awesome content of this course combined with the possibility of doing varied projects all of sufficient length to allow some quality thinking time, which has proved so valuable. My thanks extend to all involved but particularly to Ernie Rutter, Kate Brodie, Paul Suddaby, Bob Thompson and Jack Nolan. A further two years at Harvard University under the auspices of Rick O’Connell, provided die mathematical literacy and a certain rigour of thinking without which much of the work presented here would never have been imagined. The experimental work reported here has been carried out in the Rock Deformation Laboratory which was at Imperial College, London but which moved to Manchester University in July 1989. The work has involved considerable attention to experimental detail and much of this has been made possible by the lab experimental officer, Rob Holloway. It is not just his intimate knowledge of the apparatus that has been important, but also the extremely high quality of his work, for it is this more than anything which has raised the quality of the mechanical data obtained beyond even my most optimistic expectations and allowed me to draw the conclusions that I have. This combined with the fertile source of experimental and interpretative possibilities provided by Ernie Rutter has made the experimental work at times positively exciting. Thanks are also due to my contemporary John Bloomfield. The excellence of his own experimental results on the two-phase flow problem, has been matched only by his generosity in involving me in their interpretation. The paper that we have written together has served as an inspiration for the long task of writing up the PhD. Above all I thank my supervisors Ernie Rutter and Stan Murrell. I acknowledge with a great deal of gratitude their patience and faith in allowing me to pursue my own enquiries for what must have seemed interminable lengths of time. In particular this must have involved considerable sacrifice to Emie as I worked on material which was only indirectly relevant to my role as his research assistant on his project on the deformation properties of ultra-fine grained •materials. Without this freedom, this PhD would certainly never have been completed. 4 CONTENTS Abstract 2 Acknowledgements 3 Contents 4 1. Introduction 11 1.1 The nature of the task 11 1.1.1 Inelastic constitutive equations employed for geological materials 12 1.1.2 Inelastic constitutive equations employed in the materials’ sciences 13 1.2 Scope of the present study 14 PART I : THE THEORETICAL FRAMEWORK 2. Theoretical framework for Hart’s analysis of inelastic deformation 16 2.1 Definitions and assumptions in the initial analysis 16 2.2 Derivation of the existence conditions : the isothermal case 17 2.3 Derivation of the existence conditions : the non-isothermal case 18 2.4 Definition of a mechanical state variable 19 2.5 Comparison of Hart’s theoretical analysis with Caratheodory’s treatment of the first and second laws of thermodynamics 19 3. Experimental verification of the existence conditions for a state variable description of inelastic deformation 22 3.1 Verifying that y and v are unique functions of a and dr 22 3.1.1 Verifying that y is a unique function of a and dr 22 3.1.2 Verifying that v is a unique function of a and dr 23 3.1.3 The approach of MacEwen 23 3.2 Verifying that curves of constant <r* are unique and form a one parameter family 24 3.2.1 The observed form of constant <j * curves 24 3.2.2 Verifying that constant a* curves form a one parameter family 25 3.2.3 Verifying that constant cr* curves are unique 25 3.3 The effect of temperature 28 4. Derivation of the constitutive relations 29 4.1 Derivation of the equation of state 29 4.1.1 Constraints on the form of the plastic equation of state 29 4.1.2 Special forms adopted for the equation of state 32 4.2 Derivation of the mechanical state evolution equation 35 4.2.1 General form of the integrating factor 35 4.2.2 Special forms of the integrating factor 35 5. A general model for inelastic deformation 40 5.1 Description of the general model 40 5.1.1 General descriptions of anelastic and of plastic deformation 40 5.1.2 Presentation of the model 43 5.1.3 Deformation of the model 44 5.2 Extension of Hart’s model to the deformation of anisotropic materials in the presence of multiaxial differential stresses and/or large strains 48 5 5.2.1 Multiaxial formulation 48