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. inaccessiblestates, f>0 s 2 yieldsurface, f=0 elasticdomain, f<0 uniaxialcase s 1 Figure 1: Hypothetical yield surface in biaxial stress space The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space (σ1, σ2, σ3) such as the three-dimensional Haigh-Westergaard space 2 A. Nieto-Leal and V. N. Kaliakin University of Delaware Research Report Department of Civil and Environmental Engineering (Figure2) or a two- or three-dimensional space spanned by stress invariants. Two invari- ant constitutive models are often formulated in terms of the mean normal effective stress 0 0 0 0 0 p = (σv + 2σh)=3 and the deviatoric stress (or principal stress difference) q = σv − σh, where 0 0 σv is the vertical (axial) stress and σh is the lateral stress. s III spacediagonal (hydrostaticaxis) q s s II I octahedralplane Figure 2: Haigh-Westergaard stress space with an octahedral plane shown When the stress state lies on the yield surface the material is said to have reached its yield state and the material is said to have become plastic. Further deformation of the material causes the stress state to remain on the yield surface, even though the surface itself may change size and possibly shape as the plastic deformation evolves due to material hardening. The yield surface, in conjunction with the consistency condition, defines the plastic mod- ulus (Kp). If an associative flow rule is used in the mathematical theory of plasticity, then the normal to the yield surface defines the direction of the plastic strain increment. If, on the other hand, a non-associative flow rule is used, the direction of the plastic strain increment is then given by the normal to a suitably defined plastic potential [24]. Experimental evidence indicates that geomaterials deform inelastically within the yield surface [43, 52]. Consequently, soils do not exhibit the sharp change between elastic and inelastic response assumed in standard elastoplasticity. Consequently, geomaterials are also simulated using constitutive models based on the concept of a bounding surface in stress space. The bounding surface concept was originally introduced by Dafalias [6] and Dafalias and Popov [12, 13] and independently by Krieg [28] in conjunction with an enclosed yield surface for the description of monotonic and cyclic behavior of metals. This concept and the name were motivated by the observation that any stress-strain curve for monotonic load- ing, or for monotonic loading followed by reverse loading, eventually converges to certain well-defined \bounds" in the stress-strain space. These bounds cannot be crossed but may change position in the process of loading. In addition, the rate of convergence, expressed by means of the plastic modulus, depends upon the \norm" or \distance" (in a proper metric space) between the current state and a corresponding \bounding state". These concepts are better illustrated by considering the typical uniaxial stress-plastic strain response shown in 3 A. Nieto-Leal and V. N. Kaliakin University of Delaware Research Report Department of Civil and Environmental Engineering Figure3. In this figure the magnitude of the uniaxial plastic modulus (i.e., the slope of the σ - "p curve at any point) depends upon the distance δ between the true state of stress A and its \image" point A¯ on the corresponding bound. Figure 3: The bounding state concept in uniaxial stress space Figure4 shows a general bounding surface in biaxial stress space. In this particular rep- resentation, which is appropriate for geomaterials, the bounding surface always encloses the origin and is origin-convex; i.e., any radius emanating from the origin intersects the surface at only one point. The essence of the bounding surface concept is the hypothesis that plastic deformations can occur for stress states either within or on the bounding surface depending on the distance δ between the actual stress state (σij) and an associated \image" stress (¯σij) that is defined through a suitable \mapping rule". Thus, unlike classical yield surface elastoplasticity, the plastic states are not restricted only to those lying on a surface. This fact has proven to be a great advantage of the bounding surface concept. 4 A. Nieto-Leal and V. N. Kaliakin University of Delaware Research Report Department of Civil and Environmental Engineering σ mn σ ∇F ij δ r/s σ ∇F p ij r σ ij β ij a ij o β a o ij ij Elastic Nucleus 0 Bounding Surface F( σ , q )=0 ij n Figure 4: Schematic illustration of the bounding surface and radial mapping rule in multi- axial stress space This report first investigates yielding in soils. This consists of a review of experimental findings aimed at defining yielding and the shape of yield surfaces for soils. This is followed by a review of mathematical expressions used in defining yield and bounding surfaces for soils. A new yield/bounding surface is next proposed that simulates the response of normally consolidated and overconsolidated soils more accurately than elliptical shapes, yet is not overly complex analytically. The final section discusses limitations associated with surfaces used in conjunction with the radial mapping version of the bounding surface model for cohesive soils. 5 A. Nieto-Leal and V. N. Kaliakin University of Delaware Research Report Department of Civil and Environmental Engineering 2 Yielding of Soils The definition of yielding in geomaterials such as soils is typically not as straightforward as in the case of other materials such as metals. This is largely due to the fact that as a soil is loaded, it continuously develops both elastic and plastic strains without a distinct yield state (e.g., a yield point on a stress-strain curve) that delineates elastic from inelastic material states. Since it is not generally easy to determine if strains in soils are elastic or elastoplastic, researchers have typically employed different experimental techniques in order to determine the shapes of yield curves (a two-dimensional section of the yield surface) for various types of soils. Of particular interest to this report are cohesive soils. 2.1 Experimental Approaches Most naturally occurring soft clays are lightly overconsolidated due to a variety of processes such as erosional unloading, groundwater level changes, cementation, porewater chemistry changes, delayed compression,