A continuous hyperplasticity model for sands under cyclic loading

A continuous hyperplasticity model for sands under cyclic loading G.T Houlsby Department of Engineering Science, Oxford University, UK G Mortara Department of Mechanics and Materials, University of Reggio Calabria, Italy ABSTRACT: Soils exhibit truly elastic behaviour at only very small strains, so that cycling at small to moderate strains involves hysteretic behaviour As the amplitude of cycling increases the secant modulus decreases and the damping ratio increases These facts are well established experimentally, but theories that successfully describe this behaviour are less well developed We present here a simple model for the behaviour of sand under cyclic loading, that is able to capture the main features of small-strain cycling An essential part of the model is that volume changes (or effective stress changes in the case of undrained loading) are modelled realistically The model is described using the “continuous hyperplasticity” framework Essentially this involves an infinite number of yield surfaces, thus allowing smooth transitions between elasticity and plasticity The framework allows soil models to be developed in a relatively succinct mathematical form, since the entire constitutive behaviour can be determined through the specification of two scalar functionals Dilation and compression is incorporated through the use of kinematic constraints, and dilation is accompanied by the development of anisotropy in the sand INTRODUCTION Plasticity theory is established as the most important framework for describing the behaviour of soils under monotonic loading It is capable of describing most of the salient features of soil behaviour, and many soil models based on plasticity have been developed since its first application to constitutive modelling of soils in the 1950’s However, it has been less successful in describing soil behaviour under repeated (cyclic) loading The principal problem is that, as is now well known empirically, soils exhibit elastic behaviour at only the smallest of strains, and as the amplitude of strain cycling is increased, the secant stiffness steadily reduces and the damping increases This pattern of behaviour does not match simply with plasticity theory, in which a finite elastic region is a fundamental part of the theory Plasticity has been modified in a variety of ways to cope with this problem, with the two main approaches being multi-surface plasticity and bounding surface plasticity Of these multi-surface plasticity has more justification, since bounding surface plasticity cannot describe the well-established effects of immediate stress history Multi-surface plasticity can, however, be rather cumbersome An alternative is the “continuous hyperplasticity” approach, Puzrin and Houlsby (2001a) This may be thought of as a variant of the multi-surface approach, in which the process is taken to its logical conclusion and an infinite number of surfaces are used Continuous variations of stiffness and damping can be modelled An advantage of the continuous hyperplasticity approach is that it is relatively compact mathematically The entire constitutive response is specified through just two scalar functionals, thus avoiding the plethora of ad hoc assumptions that are often encountered in complex soil models The purpose here is to develop a simple model for the behaviour of an idealised frictional material (e.g a sand) under cyclic loading Puzrin and Houlsby (2001b) present a simple model for the undrained behaviour of a cohesive material, essentially by generalising the von Mises model The task of developing an equivalent simple model for frictional behaviour is complicated by the fact that no standard model for frictional behaviour (with the same level of acceptance that the von Mises model for cohesive behaviour enjoys) is currently available In this paper we limit our attention to triaxial stress states, and so we not address the shape of the yield surface in the octahedral plane Nevertheless, even with this simplification, there is no firmly established frictional model The principal difficulty lies in the treatment of the dilation of the sand Although a fixed friction and dilation angle could be used, we set ourselves here the more ambitious task of creating a unified model for the same sand at different densities, so that variable dilation, variable friction and the approach to the critical state must all be modelled This inevitably leads to a model of some complexity, although we have attempted to minimise this here as far as possible The model developed is therefore intended as a basic model, to be used as a starting point for more sophisticated approaches In particular we not include here pressure-dependent stiffness or (as noted above) the generalisation of the model in the octahedral plane MODEL DESCRIPTION The model described here is an extension of a previous single surface model (Houlsby, 1992) within the continuous hyperplasticity framework This approach employs an infinite number of yield surfaces, which are expressed in terms of an internal coordinate η (see Puzrin and Houlsby, 2001a,b) In practice, however, the infinite number of surfaces have to be replaced by a finite number N of surfaces We label each surface n ( ≤ n ≤ N ), and the factor n N plays the same role as η In the following we present the model directly in terms of the finite number of surfaces, as this requires less sophisticated mathematics and leads more directly to the implementation It should be borne in mind, however, that the underlying model involves an infinite number of surfaces, and this can be obtained by replacing n N by η , and by replacing summations by integrals The model is formulated in terms of triaxial stress and strain variables: p' = σ'1 +2σ'3 ε p = ε1 + 2ε q = σ1 − σ (1) ε q = (ε1 − ε ) Volumetric and deviatoric plastic strains related to the nth plastic mechanism are indicated as α (np ) and ) α (n respectively The specification of two scalar q functions, a Gibbs energy function g or, alternatively, a Helmholtz free energy function f ( g = g p, q, α (p1) K α (pN ) , α (q1) K α (qN ) ) f = f (ε p , ε q , α (p1) K α (pN ) , α (q1) K α (qN ) ) d= α& (p1) K α& (pN ) , α& (q1) K α& (qN ) ) p '2 q − + K 6G 2⎞ N ⎛ + ∑ ⎜ − p ' α (pn ) − qα (qn ) + H ( n ) α (qn ) ⎟ N n =1 ⎝ ⎠ g=− n N (n ) N d = ∑d = ∑M p' α& (qn) N n =1 N n =1 N (6) for describing the behaviour of material The definition of appropriate constrains enables the introduction of dilation as well as anisotropy into the model (Houlsby, 1992): C d(n ) = α& (pn) + β c α& (qn) + β d α& (qn) a = (7) N ⎞ ⎛ N (n) A⎜ ∑ α& q − a ∑ α& (qn ) ⎟ = N ⎝ n =1 n =1 ⎠ (8) Ca = a& − The first constraint specifies that dilation is made up from isotropic (compressive) and anisotropic (dilative) parts given by functions βc and βd respectively The second constraint specifies the evolution of the anisotropy parameter a, which varies between + and − for positive and negative shearing The rate of evolution of anisotropy is determined by the constant A With dissipation specified, it is possible to obtain the yield function y ( n ) associated with each set of plastic strains through the degenerate special case of the Legendre transformation of d (n ) , which is homogeneous of degree in the rates: λ( n ) y ( n ) = χ (pn ) α& (pn ) + χ (qn )α& (qn ) − d ( n ) = where by definition: (3) χ (pn) = (i ) ∂C a (i ) ∂C d + Λd + Λa ∂α& (pn) i =1 ∂α& (pn) ∂α& (pn) ∂d (9) N ∑ (i ) ∂C a ∂d (i ) ∂C d ( n) χq = + ∑ Λd + Λa n n ( ) ( ) ∂α& q ∂α& q ∂α& (qn) i =1 N (4) (5) where H (n ) is the hardening modulus related to the nth mechanism and M is the value that the stress ratio q / p' attains at critical state conditions The normalisation term / N in (5) and (6) makes the formulation independent on the number of surfaces In the previous model (Houlsby, 1992) the energy function used was the Helmholtz free energy function, while the dissipation function was formulated in terms of strains However, the two energy functions are linked by the Legendre transformation f = g + pε p + qε q and either g and f can be used (2) and a dissipation function d d ( p, q, α (p1) K α (pN ) , α (q1) K α (qN ) , is sufficient to define completely the constitutive behaviour The following two functions are used here: (10) ) where Λ(n d and Λ a are Lagrangean multipliers From the constraints (7) and (8) it follows that Λ(dn ) = χ p and Λ a = The generalised stresses are: χ (pn) = − χ q( n) = − ∂g ∂α (pn) where the first term refers to compression and the second to dilation It is worth noting that for vλ = Γ the second term is always zero while the first one is null only when the term − a sgn(α (qn ) ) vanishes Figure shows the graphical interpretation of contractive and dilative terms in (16) =p (11) ∂g = q − 3H ( n) α (qn) ( n) ∂α q ln v ln B and Ziegler’s orthogonality condition χ leads to the yield function in terms of stress: ( ) (n) ⎛ n ⎞ q − 3H ( n ) α (qn ) = ⎜ M + βc + βd a sgn α& (qn ) ⎟ p' ⎝ N ⎠ =χ ( n) ln Γ (12) The yield surfaces exhibit kinematic hardening, given by the term 3H ( n ) α nq , where the expression for ln D the variation of the hardening modulus is: H n⎞ ⎛ = h⎜1 − ⎟ ⎝ N⎠ (n) ln p' ln p'ref b (13) ln B with h and b being parameters of the model To introduce the difference between compression and extension, the critical stress ratio M is given by: M= ( ( )) (M c + M e ) + (M c − M e )sgn α& (qn) n ln vλ − ln D N ln B − ln D n ln vλ − ln Γ βc + βd = βmax N ln D − ln Γ βc = − M (15) and densest states respectively Thus, according to constraint (7), the rate of dilation is given by α& =M n ln vλ − ln D ( − a sgn (α& (qn ) )) + N ln B − ln D − β max n ln vλ − ln Γ a sgn (α& (qn ) ) N ln D − ln Γ (1- a sgn(α (n)q )) ln Γ ln D (n) β max Nn a sgn(αq ) compression dilation Figure Density constants and graphical interpretation of compression and dilation rules (equation 16) EXAMPLE ANALYSES where βmax is the maximum rate of dilation while vλ , B , Γ and D are the specific volumes at a reference mean pressure p'ref for current, loosest, critical n q M n N (14) where M e = rec M c , and rec is the ratio between the critical stress ratios in extension M e and compression M c As in the previous model (Houlsby, 1992), the values of βc and βd depend on the state of the soil, defined by the distance between the current specific volume and the critical state line, which is assumed to be linear in a bi-logarithmic plot: α& np λ ln vλ (16) We illustrate the model by example analyses of idealised tests Although we not compare these here with specific data sets, the patterns of behaviour correspond to those that are well-established empirically The example calculations are carried out using the parameter values given in Table Figure shows a set of drained constant mean pressure tests on sands with different initial densities As the index of density vλ increases (looser samples) the strength reduces and the samples change from being strongly dilative to contractive Although not apparent in Figure 2, the denser sands show a mild peak in the stress-strain response Figure shows the results for drained cycling over a constant range of strain The upper plots are 20 MPa G Shear modulus 10 MPa H Hardening modulus constant MPa B Hardening modulus constant A Rate of anisotropy development 100 B ′ Specific volume at loosest state at p ′ = p ref 1.9 Γ ′ Specific volume at critical state at p ′ = p ref 1.8 D ′ Specific volume at densest state at p ′ = p ref 1.6 Mc Critical stress ratio in compression 1.2 βmax Maximum rate of dilation rec 120 q (kPa) Bulk modulus deviatoric stress K 80 40 -40 p ' = 100 kPa vλ = 1.85 -80 -120 -2 0.4 deviatoric strain εq (%) Ratio between critical stress ratios in extension 0.8 and compression p ' = 100 kPa vλ = 1.85 volumetric strain εp (%) Table 1: Example parameters for model deviatoric stress q (kPa) 160 120 80 vλ -2 40 deviatoric strain εq (%) 150 10 deviatoric stress deviatoric strain εq (%) q (kPa) 100 50 -50 p ' = 100 kPa vλ = 1.65 -100 -150 -2 deviatoric strain εq (%) -1 vλ -2 10 deviatoric strain εq (%) Figure Response of the model in drained compression tests on sands with different initial densities for a loose sample, which exhibits an accumulation of compressive strain The resulting densification causes a slight increase in stiffness of the response to the cycles The lower plots show the equivalent for a dense sand This time the sand dilates during the volumetric strain εp (%) volumetric strain εp (%) p ' = 100 kPa vλ = 1.65 -2 -4 -6 -2 deviatoric strain εq (%) Figure Effect of relative density on drained cyclic tests deviatoric stress q (kPa) 400 300 vλ 200 100 0 deviatoric strain εq (%) p' (kPa) 300 mean stress cycling, resulting in a slight reduction of stress in the cycles as the material loosens Stress-strain curves and effective stress paths for undrained monotonic tests on samples of different densities are shown in Figure The loose sands show a reduction in effective stress, whilst dense sands show a strong increase in effective stress Sands of medium density show a slight reduction of mean effective stress before the increase This pattern of behaviour is well known Undrained cycling over a constant stress amplitude is shown in Figure for two densities of sand A loose sand (upper plots) shows an initial reduction in effective stress, after which the p ′, q plot settles into a characteristic “butterfly” shape, and the strain amplitude also becomes constant There is a large amount of hysteresis in each cycle The dense sand shows a similar pattern, but the “butterfly” plot is narrower, and the response both stiffer and with less hysteresis Figure shows undrained cycling on a dense sand at a higher stress range Although a stable “butterfly” pattern is developed, note that this time the dilation during each cycle means that there is a net increase rather than a decrease of mean effective stress Finally, figure shows the effect of the strain amplitude in on the stress-paths of a loose sample in constant strain amplitude undrained cycling For the larger amplitude ( ∆ε q = ±2% ) stabilization of stress vλ 200 100 is achieved while for the other ( ∆ε q = ±1% ) cyclic liquefaction is obtained after just two cycles deviatoric strain εq (%) CONCLUSIONS REFERENCES Houlsby, G.T (1992) "Interpretation of Dilation as a Kinematic Constraint", Proceedings of the Workshop on Modern Approaches to Plasticity, Horton, Greece, June 1216, ISBN 0-444-89970-7, pp 19-38 Puzrin, A.M and Houlsby, G.T (2001a) "A Thermomechanical Framework for Rate-Independent Dissipative Materials with Internal Functions", Int Jour of Plasticity, Vol 17, pp 1147-1165 Puzrin, A.M and Houlsby, G.T (2001b) "Fundamentals of Kinematic Hardening Hyperplasticity", Int Jour of Solids and Structures, Vol 38, No 21, May, pp 3771-3794 q (kPa) deviatoric stress A model for the cyclic behaviour of sand under triaxial conditions has been presented The model successfully describes typical trends of behaviour for undrained and drained cycling, including typical variation of volumetric behaviour for sands of different densities 400 300 200 vλ 100 0 100 200 300 mean stress p' (kPa) Figure Response of the model in undrained compression tests on sands with different initial densities q (kPa) 120 40 deviatoric stress deviatoric stress q (kPa) 80 -40 80 40 -40 -80 -120 -80 -2 -4 -3 -2 -1 deviatoric strain εq (%) q (kPa) 40 deviatoric stress q (kPa) deviatoric stress -40 80 40 -40 -80 -120 80 -80 20 40 mean stress 60 80 100 120 mean stress 100 p' (kPa) 140 160 p' (kPa) Figure Large stress controlled undrained cyclic tests for dense sand showing increase of mean effective stress 80 120 -40 ∆εq = ± 2% q (kPa) 40 80 deviatoric stress q (kPa) 120 80 deviatoric stress -1 deviatoric strain εq (%) 40 -40 -80 -4 -2 -80 deviatoric strain εq (%) 20 40 mean stress 80 100 120 -40 q (kPa) 40 ∆εq = ± 1% 80 deviatoric stress deviatoric stress q (kPa) 80 60 p' (kPa) 40 -40 -80 -80 20 40 mean stress 60 80 100 p' (kPa) Figure Simulation of stress controlled undrained cyclic tests for loose and dense sand 20 40 mean stress 60 80 100 p' (kPa) Figure Effect of cyclic strain amplitude on the behaviour of loose samples ... more ambitious task of creating a unified model for the same sand at different densities, so that variable dilation, variable friction and the approach to the critical state must all be modelled... inevitably leads to a model of some complexity, although we have attempted to minimise this here as far as possible The model developed is therefore intended as a basic model, to be used as a starting... results for drained cycling over a constant range of strain The upper plots are 20 MPa G Shear modulus 10 MPa H Hardening modulus constant MPa B Hardening modulus constant A Rate of anisotropy