RICE UNIVERSITY Plasticity, Localization, and Friction in Porous Materials by Daniel Christopher Badders A T h e s is S u b m i t t e d in P a r t ia l F u l f il l m e n t o f t h e R equirements f o r t h e D e g r e e Doctor of Philosophy A p p r o v e d , T h e s is C o m m i t t e e : Michael M. Carroll, Chairman Dean of the George R. Brown School of Engineering and Burton J. and Ann M. McMurtry Professor of Engineering Mary F. Wbjeeler and Applied Mathematics Yves C. Angel Associate Professor of Mechanical Engineering and Materials Science Houston, Texas May, 1995 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted Ihus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. 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Abstract Plasticity, Localization, and Friction in Porous Materials by Daniel Christopher Badders Original variations of two-invariant rate-independent plasticity models capture many features of the homogeneous behavior of porous materials. All of the variations proposed are based on a simple 7 parameter critical state plasticity model proposed by Carroll (1991). The models 'Utilize an associative flow rule along with a yield surface dependent on Terzaghi effective pressure and shear stress with plastic volume strain as the only hardening variable. In Carroll’s original model, a parabolic yield surface accommodates hardening by translating along the pressure axis while the top of the yield surface moves along a critical state line. The base of the parabolic yield surface is of constant width, and hardening is linear in plastic volume strain. Even in this simple form with only 5 plastic constants the model can predict dilation, shear enhanced compaction, critical state behavior, hardening, softening, and yielding during unloading. To test Carroll’s simplifying assumptions and to extend the constitutive formu­ lation for more complex behavior, variations of the model are compared to test data published by Terra Tek. The large groups of test data for sandstone and di- atomite include varied stress paths for axisymmetric loading, unloading and reload­ ing. Modifications proposed to Carroll’s model address nonlinear hardening, initial elastic transverse isotropy, variation of the shape of the yield surface, variation of elastic moduli due to pore closure, anisotropic hardening, and pore fluid pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. interaction with matrix compressibility. Evaluation of these modifications indicates they are secondary effects and points to the strength of Carroll’s original model, while providing refinements for treating these aspects of poro-behavior. Plastic softening creates potential for localization and and subsequent frictional behavior. While this behavior is not represented in the constitutive model or test data, a framework is outlined for treating formation and growth of localization bands based on a bifurcation analysis of the plasticity model. The most practical application of this would be using finite elements modified to include an embedded localization. Extreme localization leading to fracture can switch the dominant form of inelastic deformation from plasticity to frictional slip. It should be possible to accommodate this behavior within an embedded localization finite element. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments I thank members of my thesis committee, Dr. Carroll, Dr. Angel, and Dr. Wheeler for being sources of inspiration and knowledge. I thank Kirk Hansen, John Schatz, and Don Schroeder for their special assistance. Finally, I thank my wife, Anne-Lancaster Emery Badders, for her steadfast sup­ port along with my family and friends. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Abstract ii Acknowledgments iv List of Figures viii List of Tables x Glossary of Notation xi 1 Porous Behavior, Applications, and Background 1 1 . 1 Behavior of Rocks and Other PorousMaterials ....................................... 1 1.2 The Motivation for Better Porous Material Models .............................. 3 1.2.1 Subsidence at Japan’s New Kansai A irport ............................... 3 1.2.2 Compaction and Subsidence at Ekofisk Oil Field ...................... 4 1.2.3 Testing Core Samples from Oil Reservoirs ............................... 6 1.2.4 Cup-and-Cone Failure of Tensile Specimens ............................ 7 1.3 Continuum M echanics............................................................................ 8 1.3.1 Kinematic and Constitutive Variables ........................................ 8 1.3.2 Invariance Under Rigid Body M otion ........................................ 11 1.4 General Plasticity Issu e s ......................................................................... 15 1.4.1 Stress and Strain Space Yield Criteria ...................................... 15 1.4.2 Defining Finite Plastic Deformation ............................................ 17 1.4.3 Associated flo w ............................................................................. 19 1.4.4 Basis of Smooth Yield Behavior .................................................. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.4.5 Generalization to Multisurface P lasticity .................................. 21 1.5 Effective S tre s s .......................................................................................... 23 2 Experimental Studies of Porous Materials 26 2.1 Boise Sandstone .......................................................................................... 26 2.2 Monterey D iato m ite ................................................................................ 26 3 Plasticity of Porous Media 37 3.1 Micromodel Based Volumetric P la stic ity .............................................. 37 3.1.1 G re e n ........................................................................................... 37 3.1.2 Gurson-Tvergaard ...................................................................... 38 3.2 Cap M odels ............................................................................................... 40 3.3 Cam-clay Critical State M odels .............................................................. 43 3.4 Carroll’s Critical State Plasticity Model ................................................... 46 3.4.1 Extracting the Strain-Stress Relation ....................................... 48 3.4.2 Extracting the Stress-Strain Relation ....................................... 50 4 Localization and Friction 53 4.1 Localization in One Dimension ................................................................ 53 4.2 Extending Localization Ideas to Three Dimensions .................................. 54 4.3 Localization Band Analysis ................................................................... 55 4.4 Localization Leading to Frictional Sl i p ................................................... 58 4.4.1 Simple Frictional L aw s ................................................................ 59 4.4.2 Mohr Envelopes ............................................................................ 59 4.4.3 Frictional Slip Accompanied by D ila tio n .................................. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Combined Plasticity, Localization, and Friction 63 5.1 Towards a Finite Element Model .........................................................