On Soil Yielding and Suitable Choices for Yield and Bounding Surfaces
Andr´esNieto Leal1
and
Victor N. Kaliakin2
Research Report
Department of Civil and Environmental Engineering University of Delaware Newark, Delaware, U.S.A.
December 2013
1Department of Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, U.S.A. and Department of Civil Engineering, Universidad Militar Nueva Granada, Bogot´a, Colombia. 2Department of Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, U.S.A. Contents
1 Introductory Remarks...... 2 2 Yielding of Soils...... 6 2.1 Experimental Approaches...... 6 3 Mathematical Expressions for Yield and Bounding Surfaces...... 19 3.1 Simple Geomechanical Functions...... 19 Mohr-Coulomb Surface...... 19 Drucker-Prager and Related Surfaces...... 19 Original Cam Clay Yield Surface...... 20 3.2 Basic Geometric Functions...... 21 3.3 Modified Functions...... 23 Modified Elliptical Functions...... 23 Modified Lemniscate of Bernoulli Functions...... 27 Eight-curve functions...... 28 3.4 Analytical Forms Proposed for Bounding Surfaces...... 29 4 New Functional Form for a Yield or Bounding Surface...... 31 4.1 Description in Multiaxial Stress Space...... 33 4.2 Description of Isotropic Case in Terms of Stress Invariants...... 34 4.3 Application of Isotropic Case in the Role of a Bounding Surface... 35 5 Concluding Remarks...... 37
1 University of Delaware Research Report Department of Civil and Environmental Engineering
1 Introductory Remarks
The earliest scientific investigations of yielding of soils were carried out in the late 1930’s by Rendulic [42] and Hvorslev [25]. It was not until the 1950’s, however, that a mathematical description of this behavior was realized through the application of rate-independent elasto- plasticity to geomaterials. This evolution was strongly influenced by the well-established mathematical theory of metal plasticity. As a result, since the 1950’s, elastoplasticity theory has been rather extensively used to simulate the complex behavior of geomaterials. General elastoplasticity theory has four fundamental ingredients [24], namely: • A suitable elastic idealization • A yield criterion • An associative or non-associative flow rule • Suitable hardening and possibly softening laws In stress space the boundary of the yield criterion defines a surface, the so-called yield surface. A yield surface is generally a convex, smooth, closed surface in stress space that bounds stress states that can be reached without initiating plastic strains. As a matter of convenience, the yield surface is mathematically represented by a scalar yield function f = 0 that is taken as the yield criterion. If f < 0, the stress state lies inside the yield surface and corresponds to purely elastic response. Finally, the condition f > 0 represents inacces- sible states. A hypothetical yield surface in biaxial principal stress space is shown in Figure1.
inaccessible states, f > 0 s 2 yield surface, f = 0
elastic domain, f <