On Soil Yielding and Suitable Choices for Yield and Bounding Surfaces

Transcription

1 On Soil Yielding and Suitable Choices for Yield and Bounding Surfaces Andrés Nieto Leal 1 and Victor N. Kaliakin 2 Research Report Newark, Delaware, U.S.A. December 213 1,, Newark, DE 19716, U.S.A. and Department of Civil Engineering, Universidad Militar Nueva Granada, Bogotá, Colombia. 2,, Newark, DE 19716, U.S.A.

2 Contents 1 Introductory Remarks Yielding of Soils Experimental Approaches Mathematical Expressions for Yield and Bounding Surfaces Simple Geomechanical Functions Mohr-Coulomb Surface Drucker-Prager and Related Surfaces Original Cam Clay Yield Surface Basic Geometric Functions Modified Functions Modified Elliptical Functions Modified Lemniscate of Bernoulli Functions Eight-curve functions Analytical Forms Proposed for Bounding Surfaces New Functional Form for a Yield or Bounding Surface Description in Multiaxial Stress Space Description of Isotropic Case in Terms of Stress Invariants Application of Isotropic Case in the Role of a Bounding Surface Concluding Remarks

3 1 Introductory Remarks The earliest scientific investigations of yielding of soils were carried out in the late 193 s by Rendulic [42] and Hvorslev [25]. It was not until the 195 s, however, that a mathematical description of this behavior was realized through the application of rate-independent elastoplasticity to geomaterials. This evolution was strongly influenced by the well-established mathematical theory of metal plasticity. As a result, since the 195 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 = that is taken as the yield criterion. If f <, the stress state lies inside the yield surface and corresponds to purely elastic response. Finally, the condition f > represents inaccessible states. A hypothetical yield surface in biaxial principal stress space is shown in Figure 1. 2 inaccessible states, f > yield surface, f = elastic domain, f < uniaxial case 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 (σ 1, σ 2, σ 3 ) such as the three-dimensional Haigh-Westergaard space 2 A. Nieto-Leal and V. N. Kaliakin

4 (Figure 2) or a two- or three-dimensional space spanned by stress invariants. Two invariant constitutive models are often formulated in terms of the mean normal effective stress p = (σ v + 2σ h )/3 and the deviatoric stress (or principal stress difference) q = σ v σ h, where σ v is the vertical (axial) stress and σ h is the lateral stress. space diagonal (hydrostatic axis) octahedral plane 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 modulus (K p ). 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 loading, 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

5 Figure 3. In this figure the magnitude of the uniaxial plastic modulus (i.e., the slope of the σ - ε p curve at any point) depends upon the distance δ between the true state of stress A and its image point Ā on the corresponding bound. Figure 3: The bounding state concept in uniaxial stress space Figure 4 shows a general bounding surface in biaxial stress space. In this particular representation, 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 (σ ij ) and an associated image stress ( σ ij ) 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

6 σ mn σ ij F δ σ ij F r/s p r σ ij β ij o β ij a ij o a ij Elastic Nucleus Bounding Surface F( σ ij, q n )= Figure 4: Schematic illustration of the bounding surface and radial mapping rule in multiaxial 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

7 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, desiccation, and freeze-thaw effects. Field tests have shown that these clays are often stiffer and more linear in behavior than is commonly assumed [22]. Natural cementation increases the resistance of a soil to deformation. When breakdown of the cementation occurs, the response changes from rather stiff, small strain to a more flexible response in which the magnitude and rate of subsequent deformation are large. Consequently, lightly overconsolidated, cemented clays are particularly useful in determining yield points. Thus, a rather large number of investigations of yielding in clays used such soils. Mitchell [37] reported one of the earliest investigations of yielding in clays 1. He performed fully drained triaxial tests on lightly overconsolidated samples of a strongly cemented sensitive natural (Leda) clay. Yield points were identified from an abrupt change in slope on a curve of volumetric strain (ε v ) versus vertical effective stress (σ 1). At stresses less than the yield stress the specimen compression was relatively small and mostly recoverable; it was approximately a linear function of the applied stress. At stresses exceeding the yield stress, the compression was relatively large and primarily irrecoverable. Figure 5 shows similar segments in the volumetric strain (ε v ) - p plane, where p = (σ 1 + 2σ 3)/3. Mitchell [37] next performed fully drained triaxial tests at various constant stress ratios for various orientations (i.e., horizontal, vertical and inclined 45 o to the horizontal) of the bedding planes. He concluded that a yield curve could be established for natural Leda clay. However, due to apparent anisotropy, such curves were not the same for specimens oriented in different directions during testing. Sangrey [45] performed drained and undrained triaxial tests on samples of three different natural cemented clays (Leda, Mattagami, and Labrador). He determined yield envelopes that were similar to those reported by Mitchell [37]. Sangrey [45] suggested that the yield 1 It is timely to note that Graham [19] appears to have performed an earlier investigation of yielding in a soft, sensitive clay. This was subsequently followed by a similar investigation [2]. 6 A. Nieto-Leal and V. N. Kaliakin

8 .2 Elastic behavior ε v [%].4.6 Yield p [MPa] Figure 5: Yield point determination involving change in stiffness [47] envelope changed abruptly from horizontal to vertical in the vicinity of the effective preconsolidation pressure (p c). To investigate the yielding in sands, Tatsuoka and Ishihara [49] performed a series of drained triaxial compression tests on Fuji River Sand at three different densities. They adopted the approach suggested by Poorooshasb et al. [41] in which drained triaxial tests are performed with different stress paths involving loading, unloading, and reloading. The deviator stress (q = σ 1 σ 3) is then plotted versus the shear strain (ε 1 ε 3 ) for the reloading portion. Two approximately linear portions are extended near the sudden increase in strains, and a yield point is located at the intersection. The points at which yielding occurs are then marked in two-dimensional stress space. By connecting these points, it is possible to determine the shape of the yield curve. Figure 6 shows the different types of stress paths used in this series of tests. In these figures p = (σ 1 + 2σ 3)/(3p a ), where p a is the atmospheric pressure. In Type A tests the sample is first isotropically consolidated to point 1. It is next sheared to Point 2, unloaded to point 3, and then isotropically loaded to point 4. Yielding is then noted while the sample is sheared from up to point 6. In Type B tests the sample is sheared up to point 2, is then unloaded to point 3, is isotropically swelled to point 4, and then is sheared up to point 6. Type C tests combined Tests A and B. 7 A. Nieto-Leal and V. N. Kaliakin

9 Figure 6: Stress Paths used in yield point determination by Tatsuoka and Ishihara [49] Wong and Mitchell [52] performed drained triaxial and plane strain tests on a typical Ottawa area Champlain Sea clay. They determined yield points from curves of q versus distortional strain ε s = 2(ε 1 ε 3 )/3. Since the yield points were defined by triaxial tests with vastly different effective stress paths, it was concluded that the position of this yield curve was a function of p and q only, and was independent of the imposed stress path. The experimental yield curve is not symmetrical about the p axis, indicating that yielding of this clay is anisotropic. Crooks and Graham [3] performed drained triaxial analyses on intact, overconsolidated samples of a soft natural Belfast estuarine deposits of organic silty clay from two sites (Holywood and Kinnegas). They concluded that in standard odometer tests the fabric of natural sensitive clays breaks down at p c, whereas for samples reconstituted anisotropically to the field overburden stress and tested under undrained triaxial conditions, yielding occurs at maximum value of q. In commenting on Crooks and Graham s yield curves, Leroueil and Tavenas [32] showed that the shapes of yield curves determined for several natural clays reflected the stress anisotropy prevailing during deposition and consolidation. In particular, these curves tended to be centered on the K line, whereas Crooks and Graham s results indicated that the yield surface was not symmetrical about the K line. Parry and Nadarajah [39] presented the results of constant stress ratio (η = q/p ) strain controlled triaxial compression and extension tests on normally consolidated and lightly overconsolidated samples of laboratory prepared Spestone kaolin (LL = 72, P L = 4). In particular, the following tests were performed: (a) drained triaxial compression tests, keeping the axial effective stress constant and reducing the lateral effective stress, (b) drained triaxial compression test, keeping lateral effective stress constant and increasing the axial effective stress, (c) reconsolidation along a K swelling line (K =.64 for this soil), (d) deviator stress removed and sample isotropically consolidated, and (e) undrained triaxial compression test. Both anisotropically and isotropically consolidated samples were tested. 8 A. Nieto-Leal and V. N. Kaliakin

10 The volume change v (for drained tests) or pore pressure u (for undrained tests) were plotted against q or p. The yield points below the strength envelope were once again determined from the intersection of the two linear segments that best show the change of the soil stiffness. To determine the yield curve in the overconsolidated range, isotropic consolidation tests, in conventional axisymmetric triaxial devices, are performed on overconsolidated samples. In this case the yield point is often taken as the maximum deviator stress, or the residual deviator stress in the stress-strain curve. Figure 7 shows some typical results obtained in the aforementioned tests (a) to (e). Figure 8 shows the volumetric yield surface determined by Parry and Nadarajah [39], which has a rather sharp nose at the K line. Parry and Nadarajah [39] noted that the shape of their surface was similar to that determined for natural Leda clay by Mitchell [37]; they concluded that the shape of the yield surface at its intersection with the K line was influenced by the specific undrained stress paths followed in the various triaxial compression and extension tests. 9 A. Nieto-Leal and V. N. Kaliakin

11 Figure 7: Determination of yield points for Spestone kaolin [39] 1 A. Nieto-Leal and V. N. Kaliakin

12 Figure 8: Volumetric yield surface determined for Spestone kaolin specimen with OCR = 1.6 [39] Baracos et al. [1] investigated the yielding of a highly plastic natural inorganic clay from Winnipeg, Canada. Yield envelopes were determined from the results of drained triaxial tests on anisotropically consolidated specimens that were sheared along various stress paths. The yield envelopes were well-defined for shear failure of the soil structure, but were ill defined or conditions of increasing octahedral stress. Graham et al. [22] appear to have continued the work of Baracos et al. [1] on Winnipeg clay. They performed oedometer, K triaxial and drained axisymmetric triaxial tests at different stress ratios on normally and lightly overconsolidated samples of the clay. In addition to the resulting yield surface, Graham et al. [22] also showed that the plastic strain vectors are not always normal to the yield surface. Figure 9 summarizes both the yield surface and the plastic strain vectors. Although yield stresses have been determined by plotting different quantities (e.g., σ 1 versus ε 1, p versus specific volume v, q versus ε s, σ 3 versus ε 3 ), Graham et al. [22] found that the yield values obtained from various graphs were usually quite similar. Consequently, they concluded that the yield stresses indeed constitute an inherent component of soil behavior. 11 A. Nieto-Leal and V. N. Kaliakin

13 1.75 Winnipeg, Natural Clay q/σ p p/σ p Figure 9: Yield points and plastic strain increment vectors for Agassiz Lake clay from Winnipeg, Canada [22] Graham et al. [21] examined the interpretive procedures used for identifying yielding in clay soils. They considered implications of plotting techniques for defining yielding, features related to asymmetry of yield envelopes about the p axis, and factors that influence the three-dimensional shape of yield envelopes in various clays. They concluded that yielding is an inherent feature of most lightly overconsolidated clays. The shape of yield envelopes was thought to depend on the composition, anisotropy and stress history of a clay. Typically the envelopes were not symmetrical about the K line or, for moderately over consolidated samples, the p axis. Diaz-Rodriguez [15] performed anisotropic consolidation tests on Mexico City clay at different constant stress ratios. Yield points below the strength envelope were thus determined. Conventional triaxial tests on Isotropically overconsolidated samples were also performed in order to determine the yield points in the overconsolidated range. Figure 1 shows the yield curves for Mexico City clay. 12 A. Nieto-Leal and V. N. Kaliakin

14 6 q [kpa] 4 2 Mexico City Clay p [kpa] Figure 1: Yield points from experimental results for Mexico City Clay [15] Using a similar methodology, Mendoza [36] performed tests in order to determine the yield curve for Bogota Clay, a natural clay from Bogota, Colombia. Figure 11 shows some typical yield points for Bogota Clay. q [kpa] 18 Bogota Clay p [kpa] Figure 11: Yield points from experimental results for Bogota Clay [36] Diaz-Rodriguez [15] also presented yield curves that were determined for other different natural soils. Figure 12 shows some typical yield curves for these soils. Jiang and Ling [26] also presented a compilation of such curves for natural soils. 13 A. Nieto-Leal and V. N. Kaliakin

15 .8.6 φ=17.5 Winnipeg.8.6 φ=23 Alchafalaya Perno q/σ p.4.2 q/σ p.4.2 q/σ p q/σ p p/σ p 1.5 φ=25 27 Otaniemi Riihimakj Saint Louis Ottawa Osaka p/σ p 1.5 Saint Jean Vianney Bogota Favren φ= p/σ p q/σ p q/σ p p/σ p φ=28 3 Backebol Drammem Pomic p/σ p φ=32 35 Mexico p/σ p Figure 12: Yield points for several clays [15] Newson [38] tested both normally and overconsolidated samples of Speswhite kaolin. Yield points were then located from the experimental results. Figure 13 shows these yield points, as well as related results obtained by Bondok 2. 2 Bondok, A.R. (1989), Constitutive relations for anisotropic soils, Ph.D. Thesis, University of Wales. 14 A. Nieto-Leal and V. N. Kaliakin

16 .8.6 After Bondok 1989 After Newson 1992 q/p p/p Figure 13: Yield points from experimental results of Speswhite kaolin [38] Futai et al. [18] investigated the shape of the yield curve for a tropical clayey-sand and silty-sand soil, with a plasticity index between 16 and 29. Figure 14 shows the yield points for samples located at 1 m and 5 m depth and for samples at different depths, respectively. From these figures it is evident that the yield surface tends to be oriented towards a non-zero η, indicating that the soil is anisotropic. However, for samples located at 1 m and 2 m depths, the response was found to be isotropic, perhaps because these samples are not consolidated at K conditions due to their more shallow depth. Figure 14: Yield points corresponding to a natural tropical soil at (a) 1 and 5 m depth, and (b) Varying depths [18] 15 A. Nieto-Leal and V. N. Kaliakin

17 Sultan et al. [47] performed triaxial tests on intact Boom clay, a relatively stiff natural clay with plasticity index of approximately 5%. Two series of triaxial tests were performed. The first series of tests was intended to investigate the yield locus of the clay in its initial natural state. Consequently, the first series of tests involved normally consolidated samples consolidated under relatively low isotropic stresses (e.g., p max =.365 MP a). In the second series of tests the samples were isotropically consolidated at a very high mean stress (e.g., p = 9 MP a) and then unloaded to different degrees of overconsolidation. The main conclusions reached in the work of Sultan et al. [47] were that the yield curve in the first series of tests is inclined along the K line, while the yield curve of specimens overconsolidated from high stress level is oriented along the p axis (Figure 15). This was explained by the fact that overconsolidation erases all the soil memory about anisotropy consolidation (K ). q [kpa] Natural clay k q x1 2 [kpa] Isotropically consolidated to 9 MPa q/p Natural clay Isotropically consolidated k p [kpa] p x 1 2 [kpa] p/p Figure 15: Yield points from experimental results of Boom clay [47] Temperature also has an effect on the yield curve. Marques et al. [35] performed special oedometer and both isotropic and anisotropic triaxial tests on Sea Clay from Quebec, Canada. In these tests the temperature was maintained constant at values between 1 and 5 o C. Figure 16 shows the resulting yield curves, which are seen to decrease in size as the temperature increases. 16 A. Nieto-Leal and V. N. Kaliakin

18 q [kpa] T=1 o C T=2 o C T=5 o C k p [kpa] Figure 16: Canada [35] Influence of temperature on yield surface size for Sea clay from Quebec, Instead of relying on changes in slope on a response curve, Tavenas et al. [5] used strain energy to define the yield curves for four natural clays from eastern Canada. They performed stress controlled, drained triaxial tests on overconsolidated samples of these clays. Along a K stress path, the threshold energy at the yield state was found to be a linear function of the preconsolidation pressure. For other stress paths, at a given void ratio, the threshold energy was was found to be a function of the stress state at the yield point. Citing difficulties associated with the aforementioned experimental studies involving stress paths and sudden changes in a soil s stiffness, Lade and Kim [29] examined contours of plastic work and compared them to experimental data for yield surfaces. Such an approach does avoids tests with complicated stress paths and avoids the difficulties associated with the determination of yield points. The computation of plastic work contours is relatively straightforward and seemed to capture yielding in terms of shear strains as well as volumetric strains [29]. Consequently, a yield function was proposed such that yield surfaces are equivalent to plastic work contours. The associated yield surface is shown in Figure 17. It resembles an axisymmetric teardrop that is centered on the hydrostatic axis in principal stress space. 17 A. Nieto-Leal and V. N. Kaliakin

19 Figure 17: Yield surface proposed by Lade and Kim [29] As evident from the previous discussion, experimentally determined yield points tend to form yield curves that have relatively similar shapes for most of the natural and remolded clays considered herein. The size of the yield curve is controlled by the preconsolidation pressure. In addition, the yield curve is oriented towards the K line, which implies anisotropy. For moderately and heavily over consolidated soils, and for soils overconsolidated under high isotropic stresses, the resulting yield curves tend to resemble those associated with isotropically consolidated soils. This is explained by the fact that high isotropic consolidation pressures affect the soil structure, making it more isotropic. 18 A. Nieto-Leal and V. N. Kaliakin

20 3 Mathematical Expressions for Yield and Bounding Surfaces The experimentally determined yield curves presented in the previous section are typical for many natural and remolded soils. In three-dimensional stress space the yield curve generalizes to a yield surface. To mathematically simulate the yielding of soils using constitutive relations, suitable expressions for yield surfaces must be developed that are in reasonable agreement with the experimental data. The functional form of such expressions should be as simple as possible; it should involve a minimum number of parameters to define its shape and orientation in stress space. Since yield curves are geometric entities, many potential functions are available in fundamental mathematical textbooks (e.g., [31]). Such functions are typically written in the Cartesian (x-y) plane. They can, however, be easily written in q p stress space, and then suitably generalized to three-invariant space. 3.1 Simple Geomechanical Functions The simplest yield surfaces for geomaterials are typically defined from existing failure surfaces. Mohr-Coulomb Surface Perhaps the most fundamental functional form for a yield surface is obtained by adopting the Mohr-Coulomb failure criterion. This surface is an irregular hexagonal pyramid in threedimensional stress space. The axis of the surface coincide with the hydrostatic axis. The apex of the pyramid is located at or near the origin in stress space. The failure surface has an irregular hexahedral cross-section in the octahedral plane. This irregularity is due to the fact that according to the Mohr-Coulomb criterion, the strength in compression is higher than in extension. This is a consequence of assuming the same friction angle in triaxial compression as in extension. Drucker-Prager and Related Surfaces In 1952, Drucker and Prager [17] presented the first major extension of metal plasticity to geomaterials. The so-called Drucker-Prager failure criterion represents an attempt to create a smooth, three-dimensional approximation to the Mohr-Coulomb failure surface in the same manner as the von Mises criterion approximates the Tresca criterion for metals. Mathematically the Drucker-Prager surface is expressed as f (I 1, J 2 ) = J 2 αi 1 k = where I 1σ is the first invariant of the stress tensor, J 2 is the second invariant of the deviatoric stress tensor, and α and k are positive model parameters. The parameter α is related to the opening angle of the pyramid. The parameter k is related to the tensile strength of the 19 A. Nieto-Leal and V. N. Kaliakin

21 material. For α >, the surface is a circular cone with its axis equally inclined to the coordinate axes (and thus coincident with the hydrostatic axis), and its apex in the tension octant (Figure 18). In three-dimensional stress space, the section of this cone through the octahedral plane is a circle. When plotted in the hydrostatic pressure-deviatoric stress plane, the cone consists of two lines with identical slope. This implies that the friction angles in triaxial compression and extension are different. J 2 tan -1 k I 1 Figure 18: Drucker-Prager failure criterion plotted in I 1 J 2 space A major shortcoming associated with constitutive relations based on the Mohr-Coulomb and Drucker-Prager yield criteria is the inability to predict yielding for hydrostatic states of stress. This is attributed to the absence of a cap on the end of the surface. Drucker, Gibson and Henkel [16] eliminated this shortcoming by introducing a work-hardening cap to the perfectly plastic yield surface such as the Mohr-Coulomb or Drucker-Prager type. Original Cam Clay Yield Surface The original Cam Clay model, Roscoe et al. [44] used a bullet shaped yield surface defined by ( ) q p Mp + ln = p where M is the slope of the critical state line q p stress space (i.e., at critical state, η = M), and p is the preconsolidation pressure, which characterizes the size of the yield surface (Figure 19). The major drawback of this surface was that it possessed a corner at p = p. Consequently, the predictions generated for isotropic consolidation (i.e., for q = ) were problematic since non-zero deviator shear strains were predicted. 2 A. Nieto-Leal and V. N. Kaliakin

22 Original Cam Clay model Modified Cam Clay model CSL M = 1.25 assumed 8. q p Figure 19: Yield surfaces associated with cam clay models for clays 3.2 Basic Geometric Functions Historically, elliptical functions have been one of the most popular yield shapes because of their simplicity. Figure 2 shows a general elliptical yield surface in q p stress space. Beginning with the general equation defining an ellipse, it follows that or, in homogeneous form, ( p p f) 2 ( p f ) 2 + q 2 ( Mp f ) 2 = 1 f (p, q) = ( p p f) 2 ( p f ) 2 + q 2 ( Mp f ) 2 1 = (1) 21 A. Nieto-Leal and V. N. Kaliakin

23 Figure 2: General elliptical yield surface in q p stress space For the particular case of the Modified Cam Clay model [43], p f = p /2. Equation (1) thus reduces to ] f = M [(p 2 ) 2 p p + q 2 = or f = q 2 M 2 p (p p ) = (2) When p = p, q =. Differentiating equation (2) with respect to p and q gives f p = M 2 (2p p ) f q = 2q At critical state, p = p f = p /2, giving f/ p = as required. Also, for isotropic states of stress, q =, implying that f/ q =. Another function that has been used to describe yield surfaces is the Lemniscate of Bernoulli; viz., ] f = [(p [ 2 ] ) 2 + q 2 (p ) 2 (p ) 2 q 2 = (3) Pestana and Whittle [4] modified this basic function in the manner described in the next section. Finally, the eight curve function defined by f = (p ) 4 (p ) 2 [ (p ) 2 q 2 ] = (4) or ( ) ] f = q 2 (p ) [1 2 p 2 = p 22 A. Nieto-Leal and V. N. Kaliakin

24 has been also used as a yield surface. Manzari and Dafalias [34] modified this basic function, leading to a new yield surface very useful for sands. Figure 21 compares curves in the (q p ) plane generated using the three aforementioned basic geometric functions..5 Elliptical Functions.5 Lemniscate of Bernoulli.5 Eight Curve q/p q/p q/p p/p p/p p/p Figure 21: Comparison of basic geometric functions for yield surfaces in q p stress space 3.3 Modified Functions Having considered the basic geometric functions, attention is now turned to modified forms of these functions. The development of such forms is motivated by the observation that basic functions often do not accurately capture some aspects of a soil s response. Modified Elliptical Functions As a first example of a modified elliptical yield function, consider the basic function given by equation (2). To better account for anisotropic soil response, q is replaced by the term (q pα), where α controls the rotation of the yield surface towards the K line. In addition, the slope of the critical state line (M) is replaced by the N. Since N is not a fundamental material parameter it can be changed to suitably modify the shape of the yield surface. Equation (2) thus becomes f = (q p α) 2 N 2 p (p p ) = (5) When p = p, q = q = αp, which becomes q = for isotropically consolidated soils (i.e., for α = ). When q =, equation (5) gives two roots p = and ( ) M p 2 = p α 2 + M 2 which correctly reduces to p = p for α =. Figure 22 shows the basic elliptical yield surface, which is used in the modified Cam-clay model, and the modified elliptical yield surface given by equation (5). 23 A. Nieto-Leal and V. N. Kaliakin

25 .6.4 q 2 M 2 pp (1 p/p ) 1 (q pα) 2 N 2 pp (1 p/p ) Mc=1.2 q/p.2 q/p.5 α= Me= p/p p/p Figure 22: Basic and modified elliptical yield surfaces in q p stress space Dafalias [7] presented an anisotropic critical state model. The yield surface associated with this model has the following function form: f = (p p ) p + 1 [ ] (q p α) 2 + α 2 p (p p M ) = (6) 2 When p = p, q = q = αp (point A in Figure 23), which becomes q = for isotropically consolidated soils (i.e., for α = ). When q =, equation (6) gives two roots p = and ) p = (1 α2 p (point D in Figure 23) which correctly reduces to p = p for α =. M 2 Figure 23: Anisotropic yield surface proposed by Dafalias [7] 24 A. Nieto-Leal and V. N. Kaliakin

26 Collins [2] proposed a two-parameter family of critical state models that was similar to the earlier work of Lagioia et al. [3]. The yield surface associated with such models is given by the following general expression: or ( q f = M ( ) p p 2 ( ) 2 s q + = 1 (7) (1 γ)p + p s M [(1 α)p + α p s] ) 2 [(1 γ)p + p s] 2 + (p p s) 2 [(1 α)p + αp s] 2 [(1 γ)p + p s] 2 [(1 α)p + αp s] 2 = where γ, α 1 are surface configuration parameters, and p s = γ p /2 is a shift pressure. If α = γ = 1, equation (7) reduces to the elliptical yield surface associated with the Modified Cam Clay model (equation 2). Figures 24 and 25 show the effect that variations in α and γ have on the shape of the yield surface. Decreases in α tend to make the yield surface more tear drop shaped. Figure 24: Effect of varying α on yield surface proposed by Collins [2] 25 A. Nieto-Leal and V. N. Kaliakin

27 Figure 25: Effect of varying γ on yield surface proposed by Collins [2]. Remark: At critical state, p = p s. From equation (7) it follows that implying that q = Mp s. q 2 = M 2 [(1 α)p s + α p s] 2 Remark: For isotropic states of stress, q = and p = p. From equation (7) it follows that (p p s) 2 = [(1 γ)p + p s] 2 or p = 2 p s γ p s = γp 2 Lower values of γ thus give more elongated surfaces (see Figure 25).. 26 A. Nieto-Leal and V. N. Kaliakin

28 Modified Lemniscate of Bernoulli Functions In order to modify the basic lemniscate function given by equation (3), q is replaced by the term (q p α)/m. This leads to the following functional form: ( ) [ p f = (q p α) 2 m 2 p 2 2 ( ) ] q p α 1 + = (8) mp p where α controls the orientation of the yield surface and m controls its slenderness [48]. If only isotropic hardening is desired, α = and the function is controlled just by the parameter m. Even though this function has only one parameter, the mathematical expression is not simple and the derivatives of f with respect to p and q are not trivial. Figure 26 shows the effect that varying α and m has on the shape of the yield surface given by equation (8). q/p 1.5 m=1.45 α= q/p α=.45.6 m= p/p p/p Figure 26: Modified lemniscate yield surface: effect of varying α and m In their unified model for clays and sands, Pestana and Whittle [4] proposed a distorted lemniscate yield surface that is described by the following functional form: f = (q p α) 2 (m 2 + α 2 2 qp ) [ ( ) p α p 2 n ] 1 = (9) where m and n are parameters that describe the shape of the surface, and α is a state variable that controls the rotation of the surface. Figure 27 shows the effect that varying α, m, and n has on the shape of the yield surface given by equation (9). p 27 A. Nieto-Leal and V. N. Kaliakin

29 q/p m=.8 n= p/p.45 α= q/p α=.45 n=.5.7 m= p/p q/p 1.5 α=.45 m=1.5 n= p/p Figure 27: Distorted lemniscate yield surface proposed by Pestana and Whittle [4]: effect of varying α, m, and n Eight-curve functions The eight-curve is modified by replacing q by the rotational hardening term (q αp )/m, where α and m again control the rotation and the shape of the yield surface, respectively. Introducing this new term, equation (4) thus becomes ( ) ] f = (q αp ) 2 m 2 (p ) [1 2 p 2 = (1) Figure 28 shows the effect that varying α and m has on the shape of the yield surface given by equation (1). p 1 m=1 1 α= q/p α= q/p.5 m= p/p p/p Figure 28: Modified eight-curve yield surface: effect of varying α and m Manzari and Dafalias [34] further enhanced the modified eight-curve given by equation (1) so as to better capture the response of sands. They introduced a new parameter n that controls the curvature of the surface cup; the parameter m now controls the opening at the origin [48]. Thus, the equation of this new modified surface is ( ) f = (q αp ) 2 m 2 (p ) [1 2 p n ] = (11) p 28 A. Nieto-Leal and V. N. Kaliakin

30 In addition to the above changes, Manzari and Dafalias [34] specified the numerical range of values for these parameters in order to get the desirable surface shape. If a very small value of m (i.e. m << 1) and large values of n 2 are used, the surface will have a narrow wedge-type open shape. This shape ensures plastic strain increments when small changes in the stress ratio η occur. For stress path along η =constant and inside the yield surface, only elastic strain increments will be predicted. Figure 29 shows the effect that varying α has on the shape of the yield surface given by equation (11)..6.4 m=.1 n=2.45 q/p.2 α= p/p Figure 29: Modified eight-curve yield surface proposed by Manzari and Dafalias [34]: effect of varying α 3.4 Analytical Forms Proposed for Bounding Surfaces Early forms of the bounding surface model for cohesive soils [8 11], as well as subsequent unified versions [4, 5] employed a composite surface consisting of a compression ellipse, a hyperbola and a tensile ellipse. In their time-independent anisotropic bounding surface model, Liang and Ma [33] also employed a composite form of the bounding surface, only this form consisted of two ellipses and a sinusoid instead of the hyperbola. Although continuity of the surface and its slope across ellipse/hyperbola and ellipse/sinusoid interfaces was maintained, the discontinuities in the second derivatives precluded the use of this form of the surface in conjunction with certain stress point integration algorithms [46]. In order to avoid the complexity and shortcomings of composite versions, Kaliakin and Dafalias [27] used a single elliptical function as the bounding surface. In order to generalize the shape of the surface with respect to equation (2), they introduced a new parameter R that allows the surface to move along the p -axis (this had been done earlier in conjunction with the compression ellipse in the composite form of the surface). Since the surface intersects this axis at p, this movement is directed in the negative p direction. Consequently, the slenderness of the surface increases. f = (p p ) [ p + ( R 2 R ) ] ( q ) 2 p + (R 1) 2 = (12) M 29 A. Nieto-Leal and V. N. Kaliakin

31 or ( p f = p ) 2 2 ( ) p + R p ( ) 2 R R ( ) 2 q + (R 1) 2 = Mp where R is a shape configuration parameter (R 2.). If R = 2., the surface reduces to that associated with the Modified Cam Clay model (equation 2). Figure 3 shows the effect of R on the shape of the surface. Figure 3: Modified elliptical bounding surface proposed by Kaliakin and Dafalias [27] 3 A. Nieto-Leal and V. N. Kaliakin

32 4 New Functional Form for a Yield or Bounding Surface Based on the previous experimental results related to yielding and in light of the previous functional forms investigated above, a new simple functional form for a yield surface is proposed in order to capture the clay response under monotonic loading. This shape is defined purely based on the aforementioned experimental results. The basic equation is called the Right Strophoid [31], and is given by the following function: q 2 (p + p ) = (p ) 2 (p p ) (13) Replacing q by the quantity (q p α)/m in equation (13) and multiplying through the resulting expression by m 2 gives the following new simple yield function: f = (q p α) 2 (p + p ) m 2 (p ) 2 (p p ) (14) where m is a parameter defining the shape of the surface and α is again a state variable that controls the rotation of the yield surface. Figure 31 shows the effect that varying m and α has on the shape of the yield surface given by equation (14). Figure 32 shows the effect that variations in the parameter m have on the the yield surface when α =. q/p m= p/p α=.45 q/p m= p/p.45 α= Figure 31: Effect of varying m and α on the Right Strophoid yield function 31 A. Nieto-Leal and V. N. Kaliakin

33 .5 m=1. m=.8 q/p p/p Figure 32: Effect of varying m on the Right Strophoid yield function for α = In Figure 33 the yield surfaces obtained using the Right Strophoid function given by equation (14) are compared with both experimental results and with the modified eightcurve function (equation 1). From this figure it is evident that both functions fit the experimental results quite well. However, in all cases the top point of the Strophoid shape is lower than the eight-curve, showing a better agreement with the experimental data. The same value of α was used in both yield functions, so the yield surfaces were fitted only by changing m. In all cases α was approximately equal to the value of (1 sin φ), indicating the natural anisotropic soil consolidation. That is, for natural clays the yield surface should be rotated towards the K line. However, if the natural soil is isotropically consolidated under high stresses, the yield surface will tend to be oriented towards the horizontal p -axis, as shown by [47] (See Figure 15). 32 A. Nieto-Leal and V. N. Kaliakin

34 1 Strophoid m=1.45 Eight curve m=.9 φ=17 1 Strophoid m=2.2 Eight curve m=1.4 φ= q/σ p q/σ p p/σ p 1 Strophoid m=2.7 Eight curve m=1.7 φ= p/σ p.8.6 Strophoid m=1. Eight curve m=.55 Newson, q/σ p.5 q/p.4.2 q/p p/σ p Strophoid m=1.6 Eight curve m=1. Futai, 24. q/p p/p Strophoid m=1.21 Eight curve m=.8 Mendoza, 29 Bogota clay p/p.5 1 p/p Figure 33: Comparison of the Right Strophoid and eight-curve function with experimental results 4.1 Description in Multiaxial Stress Space Having shown the new yield surface equation in p q space, it is timely to describe this expression in multiaxial stress space. The relation between the deviatoric stress tensor s = σ p I in the multiaxial generalization and the deviator stress q is [48] 3 2 s : s = q2 33 A. Nieto-Leal and V. N. Kaliakin

35 where the symbol : represents the trace of the product of two tensors (i.e., s : s = tr s 2 ). In addition, the tensor for the deviatoric stress ratio is defined as η = s/p, which in triaxial space corresponds to η = q/p. Also the effective deviatoric stress tensor is written as s pα, where α is now a second-order tensor defining the rotation of the yield surface. The multiaxial expression of the Right Strophoid given by equation (14) is thus f = 3 2 (s p α) : (s p α)(p + p ) m 2 (p ) 2 (p p ) = or f = 3 ) ) 2 (s p α) : (s p α) (1 + p m 2 (p ) (1 2 p = p p For the special case of isotropically consolidated soils, α =. 4.2 Description of Isotropic Case in Terms of Stress Invariants It is next useful to write equation (14) in terms of the direct stress invariants. For isotropically consolidated soils (α = ), the first invariant of effective stress tensor (σ ) is denoted by I = trσ = σ kk. The second invariant of (s) is J = 1 2 tr (s)2 = 1 2 s ijs ij = q/ 3. As a result, the Right Strophoid is written as F = J 2 (I + I ) m2 I 2 (I I ) (15) where the shape parameter m is as previously defined, and I (= 3p ) is the value of the first effective stress invariant at the preconsolidation state (or the intersection of the yield surface with the positive I axis in I J space). If the parameter m is related to the critical state line through M, it should be manipulated in order to get the correct relation between M and N, where M is the critical state line slope in the p q plane and N is the critical state line in the I J plane; this relation is given by M = 3 3N. In order to ensure that ε p v = at at the critical state, m has to be a function of the slope M of the critical state line in p q space. As a result m = 2.586M, implying that m 2 = M 2 3 2M 2. Equation (15) is thus re-written as F = J 2 (I + I ) N 2 I 2 (I I ) = (16) where It follows that L ij = F = F σ ij I I + F σ ij J J + F σ ij θ θ σ ij F I = J N 2 I(3I 2I ) I σ ij = δ ij F J = 2J(I + I ) 34 A. Nieto-Leal and V. N. Kaliakin

36 J σ ij = 1 2J s ij Since the third invariant (the Lode angle θ) is included in the formulation, its derivatives are also required. Noting that N = 2k 1 + k (1 k) sin(3θ) N c where k = N e /N c, it follows that the last two terms in equation (16) become F θ = F N N θ θ = θ J + θ σ ij J σ ij S S σ ij where and Consequently, In addition, N θ = θ σ ij = F N = 6 2 NI 2 (I I ) 6k(1 k) N c [1 + k (1 k) sin(3θ)] 3(1 k) 2 = N 2 cos(3θ) 2N e F θ = 9 2N 3 I 2 (1 k) (I I ) cos(3θ) N e [ ( ) ] 3 3 s ik s kj S s ij 3 2Jcos(3θ) J 2 2J J 2 3 δ ij 4.3 Application of Isotropic Case in the Role of a Bounding Surface If equation (16) is used to describe a bounding surface, then it must be re-written as F = J 2 (Ī + I ) N 2 Ī 2 (Ī I ) = (17) where barred quantities refer to the image point on the bounding surface, and J = bj, Ī = b(i CI ) + CI, and θ = θ. In the version of the bounding surface model that employs a radial mapping rule [11], an explicit expression is required for b. Substituting the above expressions for Ī and J into equation (17) gives, after some algebraic manipulation, the following cubic equation for b: where A b 3 + B b 2 + C b + D = 35 A. Nieto-Leal and V. N. Kaliakin

37 A = [J ] 2N 2 (I CI ) 2 (I CI ) [ B = J 2 (C + 1) + 3 ] 2N 2 (I CI ) 2 (3C 1) C = 3 2N 2 C(I CI )(3C 2)(I ) 2 D = 3 2N 2 C 2 (C 1)(I ) 3 The solution for b is then given by b = + [ ] 1 ω 4 + (ω 4 ) 2 + [ω 5 (ω 3 ) 2 ] 3 3 [ ] 1 ω 4 (ω 4 ) 2 + [ω 5 (ω 3 ) 2 ] ω3 I where ω 3 = B 3A ω 4 = (ω 3 ) 3 + B C 3A D 6(A ) 2 ω 5 = C 3A Since I can equal to CI, it follows that A can equal zero, thus making ω 3, ω 4, and ω 5 undefined. This is a problem that plagues cubic (and higher-order) functional forms of the bounding surface. 36 A. Nieto-Leal and V. N. Kaliakin

38 5 Concluding Remarks Motivated by the results of the aforementioned experimental studies of yielding in soils, several specific functional forms, possessing varying degrees of complexity, have been proposed for yield surfaces. Several general functional forms have also been proposed for such surfaces [14, 23, 51]; many of these can be reduced to some of the more simple functions presented in this report. Taiebat and Dafalias [48] recently reviewed many yield functions appropriate for soil plasticity and made general recommendations for ones suitable for sands and clays. When considering potential functional forms for bounding surfaces, the discussion is complicated by the requirement to explicitly solve for the variable b that is associated with the radial mapping version of the model. In particular, it is possible that seemingly suitable cubic and higher-order functional forms will be undefined at certain points in stress invariant space, thus precluding a successful solution. 37 A. Nieto-Leal and V. N. Kaliakin

39 Bibliography [1] A. Baracos, J. Graham, and L. Domaschuk. Yielding and rupture in a lacustrine clay. Canadian Geotechnical Journal, 17(4): , 198. [2] I. F. Collins. Elastic/plastic models for soils and sands. International Journal of Mechanical Sciences, 47(4-5):493 58, 25. [3] J. H. A. Crooks and J. Graham. Geotechnical properties of the belfast estuarine deposits. Géotechnique, 26(2): , [4] R. S. Crouch and J. P. Wolf. Unified 3-d critical state bounding-surface plasticity model for soils incorporating continuos plastic loading under cyclic paths i: Constitutive relations. International Journal for Numerical and Analytical Methods in Geomechanics, 18(11): , [5] R. S. Crouch, J. P. Wolf, and Y. F. Dafalias. Unified critical state bounding surface plasticity models for soil. Journal of Engineering Mechanics, ASCE, 12(11): , [6] Y. F. Dafalias. On Cyclic and Anisotropic Plasticity: i). A General Model Including Material Behavior under Stress Reversals, ii). Anisotropic Hardening for Initially Orthotropic Materials. PhD thesis, University of California, Berkeley, Department of Civil Engineering, [7] Y. F. Dafalias. An anisotropic critical state soil plasticity model. Mechanics Research Communications, 13(6):313 33, [8] Y. F. Dafalias and L. R. Herrmann. A bounding surface soil plasticity model. In G. N. Pande and O. C. Zienkiewicz, editors, Proceedings of the International Symposium on Soils Under Cyclic and Transient Loading, pages , Swansea, UK, 198. A. A. Balkema, Rotterdam. [9] Y. F. Dafalias and L. R. Herrmann. Bounding surface formulation of soil plasticity. In G. N. Pande and O. C. Zienkiewicz, editors, Soil Mechanics Transient and Cyclic Loads, chapter 1, pages John Wiley and Sons, Chichester, UK, [1] Y. F. Dafalias and L. R. Herrmann. A generalized bounding surface constitutive model for clays. In R. N. Yong and E. T. Selig, editors, Application of Plasticity and Generalized Stress-Strain in Geotechnical Engineering, pages 78 95, Hollywood, FL, ASCE, New York. 38