# Notes on Geoplasticity

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Beyond the elastic limit, but still within the realm of small strain increments, a total strain increment is the sum of an elastic increment and a plastic increment. The elastic increment is computed through an incremental form of Hooke's law, isotropic or anisotropic as the case may be. Computation of the plastic part is at the core of any plasticity theory and is approached through the concept of a plastic potential. The plastic potential is a function of stresses and perhaps other material parameters such as plastic strain and temperature. Derivatives of the plastic potential with respect to stress lead to the plastic part of the total strain increment. If the yield criterion and plastic potential are the same, then the plastic stress-strain relationships are "associated rules of flow" and follow a "normality" principle. Normality is in reference to a graphical portrayal in principal stress space where the plastic strain increment is perpendicular to the yield surface. If the plastic potential and yield criterion are different, as is often the case in geoplasticity, then the rules of flow are "non-associated". Drucker's famous stability postulate implies normality at a smooth point on the yield surface, convexity of the yield function and other important features of plasticity theory in geomechanics.

However, there is no point to proceeding to theoretical analyses without physical justification. Hence, the physical foundations for application of plasticity theory to rock, jointed rock and soil are examined in Chapter 2 of this book. A brief review of continuum mechanics principles is given in Chapter 3. Chapter 4 focuses on plane plastic strain and "sliplines". 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 genuine understanding. Examples illustrate application of theory to traditional geomechanics problems such as computation of retaining wall forces in soils, foundation bearing capacity of soil and rock, wedge penetration of rock under confining pressure and others. Brief discussions of anisotropy, visco-plasticity and poro-plasticity are presented in Chapters 6, 7 and 8.

This book will be of interest to civil, geological and mining engineers, particularly those involved in reliable design of excavations and foundations beyond elasticity, especially in jointed rock.

References

2 Physical foundations of theory

2.1 Intact rock response to load

2.2 Rock joints response to load

2.3 Jointed rock response to load

2.4 Equivalent jointed rock models

2.5 Soil response to load

References

3 Elements of three-dimensional theory

3.1 Stress review

3.2 Strain review

3.3 Stress-strain relations

3.4 Principal stress space

3.5 Yield functions, failure, and loading criteria

3.6 Plastic stress-strain laws

References

4 Two-dimensional theory (plane strain)

4.1 Yield envelope in plane strain theory

4.2 The stress subsystem of equations

4.3 Regions of constant state, radial shear, and radial stress

4.4 The velocity subsystem of equations

4.5 A special plane strain velocity field (streaming flow)

4.6 Example problems

4.7 Discontinuities in velocity and stress

4.8 Envelope solutions

4.9 Numerical solution of boundary value problems

References

5 Limit theorems

5.1 Lower-bound theorem

5.2 Upper-bound theorem

5.3 Example problems

References

6 Anisotropy

6.1 Elasticity

6.2 Plasticity

References

7 Viscoplasticity

7.1 Some field data

7.2 Some laboratory data

7.3 Elastic-viscoplastic models

7.4 Finite element formulation

7.5 Example problems

7.6 Discussion

References

8 Poroplasticity

8.1 Effective stress

8.2 Poroelastic/plastic models

8.3 Finite element model

8.4 Equivalent properties

8.5 Example problems

8.6 Discussion

References

Epilogue

References

Appendix A: Axial symmetry discussion

A.1 Stress subsystem

A.2 Discussion of stress

A.3 Velocity subsystem

A.4 Discussion of velocity

Reference

Appendix B: Finite element review in brief

B.1 Finite element concept

B.2 Element equilibrium

B.3 Global equilibrium

B.4 Boundary conditions

B.5 Practical considerations

References

Currently, he is a professor emeritus and former holder of the Malcolm McKinnon endowed chair in mining engineering at the University of Utah. He joined the Department in 1971 following academic appointments at the Montana College of Science and Technology and the Pennsylvania State University. He has been a visiting academic at Brown University, Imperial College, London, and at the Commonwealth Science and Industrial Research Organization (CSIRO), Australia. He and colleagues have received a number of rock mechanics awards; he was recognized as a distinguished university research professor at the University of Utah in 1991. In 2010, he was recognized for teaching in the College of Mines and Earth Sciences with the Outstanding Faculty Teaching Award. The same year, he was honored by the Old Timers Club with their prestigious Educator Award. He was honored as a Fellow of the American Rock Mechanics Association in 2015.