Notes on Geoplasticity

243 Pages·2019·13.489 MB·\243

by  William G. Pariseau (Author)

Notes on Geoplasticity

Notes on Geoplasticity
William G. Pariseau
Malcolm McKinnon Endowed Chair – Emeritus
Department of Mining Engineering
University of Utah, Salt Lake City, UT, USA
Cover photo: a portion of a 45 deg slope in an open pit copper mine in
the western USA where mining depth is approaching 1000 m. The benign
appearance of the slope masks complex geology replete with joints, faults,
dikes, underground workings and groundwater. Poro-elastic/plastic analysis
details are given by Schmelter and Pariseau, 1997 (see reference in Chapter 8).
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DOI: https://doi.org/10.1201/9780429428272
Contents
Prologue viii
Acknowledgements x
1 Introduction 1
References 5
2 Physical foundations of theory 6
2.1 Intact rock response to load 9
2.2 Rock joints response to load 29
2.3 Jointed rock response to load 32
2.4 Equivalent jointed rock models 34
2.5 Soil response to load 37
References 38
3 Elements of three-dimensional theory 39
3.1 Stress review 39
3.2 Strain review 47
3.3 Stress-strain relations 50
3.4 Principal stress space 51
3.5 Yield functions, failure, and loading criteria 52
3.6 Plastic stress-strain laws 57
References 60
4 Two-dimensional theory (plane strain) 62
4.1 Yield envelope in plane strain theory 63
4.2 The stress subsystem of equations 66
4.3 Regions of constant state, radial shear, and radial stress 70
4.4 The velocity subsystem of equations 74
4.5 A special plane strain velocity field (streaming flow) 78
4.6 Example problems 79
4.7 Discontinuities in velocity and stress 92
4.8 Envelope solutions 98
4.9 Numerical solution of boundary value problems 107
References 113
vi Contents
5 Limit theorems 114
5.1 Lower-bound theorem 114
5.2 Upper-bound theorem 115
5.3 Example problems 116
References 119
6 Anisotropy 120
6.1 Elasticity 120
6.2 Plasticity 129
References 152
7 Viscoplasticity 153
7.1 Some field data 154
7.2 Some laboratory data 157
7.3 Elastic-viscoplastic models 162
7.4 Finite element formulation 168
7.5 Example problems 169
7.6 Discussion 178
References 179
8 Poroplasticity 182
8.1 Effective stress 183
8.2 Poroelastic/plastic models 185
8.3 Finite element model 186
8.4 Equivalent properties 187
8.5 Example problems 190
8.6 Discussion 202
References 203
Epilogue 205
References 211
Appendix A: Axial symmetry discussion 212
A.1 Stress subsystem 214
A.2 Discussion of stress 215
A.3 Velocity subsystem 216
A.4 Discussion of velocity 218
Reference 219
Appendix B: Finite element review in brief 220
B.1 Finite element concept 220
B.2 Element equilibrium 223
Contents vii
B.3 Global equilibrium 224
B.4 Boundary conditions 226
B.5 Practical considerations 226
References 227
Index 228
Prologue
My interest in rock mechanics in general and soil plasticity in particular – geoplasticity –
began some time ago in graduate school at the University of Minnesota. My research topic
focused on ore passes. Ore passes are long, steeply inclined “tunnels” in underground mines.
They are conduits used to transport broken ore and waste rock (“muck” in miner’s language)
under the action of gravity from one mine level to a level below. Backfill may also be trans-
ported underground in this manner. Ore passes malfunction on occasion because of “hang-
ups,” stoppages caused by interlocking blocks and cohesive action of “fines.” Piping is also
a malfunction when flow only occurs in the central portion of an ore pass. Flow segregation
can be problematic. Soil plasticity appeared to be a logical choice, if not the only choice,
for a rational analysis of the “muck” mechanics involved, although my interest was in flow
rather than stability.
The timing was fortunate because of the resurgent interest and development of theoretical
soil plasticity, especially at Brown University in the Department of Applied Mathematics
and Mechanics, where oft-cited publications were produced by such luminaries as W. Prager,
P. G. Hodge, D. C. Drucker, A. H. Greenberg, R.T. Shield, and others.
Design of bins, hoppers, chutes, and silos for handling bulk materials in numerous indus-
trial operations pose problems closely related to ore pass design. Although less well known,
the development and application of practical approaches to design of bins and hoppers were
advanced at the University of Utah Engineering Experiment Station. Not glamorous but of
considerable importance in many industries, the technology was intensively studied under
the guidance of Professor A. Jenike and his student J. R. Johanson, subsequent principals in
the highly respected firm of Jenike and Johanson with specialization in bulk materials han-
dling solutions for industrial processes.
An initial interest and exploration of soil and metal plasticity, where extrusion and finite
strain have some similarity to flow in ore passes, quite naturally led to possible applications
to rock. Although, at the time, common knowledge considered rock a brittle material that
failed catastrophically by fast fracture with any attempt to continue loading beyond the elas-
tic limit. The elastic limit was thus the “end of the story.” Much laboratory test experience
confirmed this view. However, casual observations of rock at the engineering scale – for
example, in road cuts – revealed numerous fractures, cracks, faults, and joints. Certainly,
fracture was not the end of the story at the engineering scale. Noticeable “squeeze” in deep
hardrock mines also indicated inelasticity. Plasticity theory based on first principles offered
a way forward.
Prologue ix
An important objective of this book is to present a clear but compact account of “sli-
plines.” The technical literature is replete with numerous diagrams of sliplines, especially in
discussions of foundations on soils, but the relevant mathematics is often lacking and with
it a genuine understanding. A short introduction to this book is presented in Chapter 1. But
there is no point to proceeding to theoretical analyses without physical justification. Hence,
the physical foundations for application of plasticity theory to rock are examined in Chap-
ter 2. No justification for application to soils is needed. A brief review of continuum mechan-
ics principles is given in Chapter 3. Chapter 4 focuses on plane plastic strain and goes to the
origins of this book. Examples illustrate application of theory to traditional geomechanics
problems such as computation of retaining wall forces in soils, foundation-bearing capac-
ity of soil and rock, wedge penetration of rock under confining pressure, and others. Axial
symmetry is relegated to an appendix despite superficial similarities to plane strain analysis.
Chapter 4 also contains a brief exposition of boundary value problem types and numerical
solutions. Limit theorems that were of great interest before the digital era but are now largely
of academic interest are discussed in Chapter 5. Discussions of anisotropy, viscoplastic-
ity, and poroplasticity are presented in Chapters 6, 7, and 8. Anisotropy is as much the rule
as the exception in geomechanics. Consideration of a viscous component of deformation
introduces a rate and therefore a time dependency. Slipline analyses do not do justice to the
importance of fluid flow in porous rock and soil. The chapter on poroplasticity is intended to
open the door to this important phenomenon in geomechanics.
The advent of the computer and concurrent development of numerical methods such as
the discrete element method (DEM), discontinuous deformation analysis (DDA), and vari-
ants has since revolutionized design analysis involving bulk materials handling. Indeed, the
popular finite element method (FEM) has made routine complex elastic-plastic analyses of
soil and rock mechanics problems. Mechanics of ice and snow could be added to the mix of
geo-materials. Indeed, geoplasticity is now well-known and widely accepted technology in
the realm of geotechnical engineering.
These “notes” were begun shortly after graduation and formed a basis for graduate courses
in geoplasticity and post-graduate research as well. They have evolved in time. The digital
age prompted development of a second course, one in numerical modeling, with focus on
the finite element method. A brief appendix outlines the fundamentals of the method. Exam-
ple problems are found in Chapters 7 and 8. Much has been accomplished by the technical
community since these notes were begun, especially in the realm of numerical methods that
allow for site-specific design analyses of a great variety of geotechnical problems, which
potentially involve deformation beyond the purely elastic range, that is, geoplasticity.
William G. Pariseau
Salt Lake City, Utah, USA
October 2018