Constitutive modeling for sand with emphasis on the evolution of bounding and phase transformation lines

A recently proposed model for drained and undrained behavior of sand under monotonic and cyclic loading conditions is briefly presented. The model is formulated in the framework of classical elastoplasticity, and combines features of: (a) the bounding surface plasticity, (b) the critical state concept, and (c) a hardening evolution law and unloading-reloading rule of the modified Bouc-Wen type. Emphasis is given on the evolution of the bounding, Ms, and phase transformation, Mpt, lines. A new set of functions is proposed for the variation of Ms and Mpt based on Bolton’s relative dilatancy index. It is shown that the new set of Ms and Mpt combined with the appropriate calibration according to Bolton (1986), offers a certain degree of flexibility and accuracy that can provide a high level of predictability for both loose and dense states of sand. The ability of the model to realistically reproduce complex patterns of monotonic and cyclic b haviour such us densification and associated strength hardening in drained cyclic loading, loss of strengt and cyclic mobility in undrain onotonic and cyclic loading, respectively, without readjustme t of its parameters, is highlighted through a series of nu erical examples in p-q and 3D stress space. strength) in compression, e M in extension and s M in simple shear, while   3 3 2 2 3 3 3 2 / J cos θ J  . The yield function takes into account both the initial Ko and reversal loading conditions through the stress ratio tensor rmax and stress ratio value nmax. The stress ratio tensor rmax is defined as: max max max max max max p p p    I s σ r (7) where max σ and max p are equal to the initial stress values at the beginning of loading. In the following, they obtain the stress values at the pivot points once reversal of loading occurs. It is evident that the modified deviatoric stress   1 2 m / max max q ( p) : ( p)    s r s r always obtains zero value at the beginning of loading, for all Ko initial conditions, and at each load reversal. The stress ratio value max n is defined as the inner product of two tensors, such as max max n : n r , where n is a normalized stress ratio tensor showing the loading direction and it is equal to the derivative of the m q with respect to σ; thus, normal to f: max / max max p ( p) : ( p)         s r n s r s r 1 2 (8) The properties of the tensor n are: tr 0  n and 2 tr 1 :   n n n . In either case, nmax, incorporates the effects of the deviatoric stress ratio due to initial and pivot-point stress conditions (rmax) on the current loading direction/path (through current tensor n). Therefore, nmax, compensates for the return of the deviatoric stress m q to the hydrostatic axis at each load reversal. The yield surface can be rewritten as: 2 0 3 s θ M p   s n , : (9) 2.4 Hardening Parameter, ζ Following Eqn. (9), the hardening parameter, ζ, is defined as: 2 3 s ,θ : ζ M p  s n (10) The hardening parameter, ζ, is bounded, strictly obtaining values within the range [0 1]. In particular, ζ obtains zero value at the beginning of the loading and at reversal points, which leads to zero values of the hardening matrix H according to Eqn. (4). Consequently, the elastoplastic matrix, ep h E becomes equal to the elastic matrix, e E , following Eqn. (3). The gradient to the yield surface [Eqn. (6)] is obtained as: , f s θ ζ f M      Φ σ n I 1 2 3 3 (11) 2.5 Plastic Flow Rule The stress-dilatancy relationship, adopted by the model, is based on Rowe’s dilatancy theory (Rowe 1962). The ratio of the plastic volumetric strain increment, dεp , over the plastic deviatoric strain increment, dεq p depends on the distance of the current stress ratio, q/p in conventional p-q space from the phase transformation line, Mpt, as follows: p p pt p q dε q M p dε         (12) The dilatancy in 3D formulation remains a scalar quantity, calculated by: , , : : pt θ pt θ d M M p p                   s s n n n 2 2 3 3 (13) The gradient to the potential function g, considering non-associative plasticity, is given by: g g d      Φ σ n I 1