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Efficient Numerical Integration of Perzyna Viscoplasticity, With Efficient Numerical Integration of Perzyna Viscoplasticity, with Application to Rock Slope Stability using Zero-Thickness Interface Elements I. Aliguer & I. Carol Geotechnical Engineering and Geosciences Department, ETSECCPB Universitat Politecnica` de Catalunya, Barcelona, Spain ABSTRACT: In this paper, Perzyna-type viscoplastic rate equations are integrated for a time step by con- sidering the step as stress-driven. Depending on how the increment is imposed (constant, linear etc.), different strategies arise. The secant compliance is obtained by truncated expansion of the yield function. The viscoplas- tic model can be applied to materials exhibiting rate-dependent behavior, but it can also be used to recover an inviscid elastoplasticity solution when stationary conditions are reached. Within this framework, a viscoplastic relaxation iterative strategy is developed, relating the iterations with the fictitious time steps. Some examples of application are presented in the context of the Finite Element Method with zero-thickness interface elements for slope and stability problems with discontinuities. 1 INTRODUCTION amount of iterations are required. From the viewpoint of the type of viscoplastic for- Viscoplasticity has been widely used for engineer- mulation there are also differences. While for Duvaut- ing materials with (physical) time-dependent be- Lions formulations a good compromise between com- haviour over a treshold stress level (Cormeau, 1975, plexity and cost has been reached via quasi-linear ex- Zienkiewicz and Cormeau, 1974), or in the context ponential algorithms and the formulation of a consis- of viscoplastic relaxation strategies to obtain the sta- tent viscoplastic tangent operator (Simo and Hughes, tionary solution of an inviscid problem via a ficti- 1998, Simo et al., 1988), for Perzyna-type viscoplas- tious (non-physical) pseudo-time (Underwood, 1983, ticty there seems to be no equivalent approach. Zhang and Yu, 1989). In either case, the rate-type In Rock Mechanics, time-dependency was incorpo- infinitesimal viscoplastic formulation requires a time rated in the analysis procedures relatively early, for integration strategy to a) discretize time in increments the overall homogenized behavior of rock masses, and b) evaluate a linearized relation between stress and also associated to creep phenomena or salt formations strain increments for each time step and, possibly, (Winkel et al., 1972, Zienkiewicz and Pande, 1977). some residual force calculation and iterative strategy. In this paper, the continuum is considered as linear- Typically, the algorithms are based on the initial stress elastic, and the non-linearities are restricted to vis- scheme used in FEs, in which the strain increments coplastic behavior of the rock discontinuities, which are prescribed to the constitutive equations. A variety are explicitly modeled using zero-thickness interface of such algorithms has been proposed since the orig- elements. inal constant stiffness and constant stress procedures (Hughes and Taylor, 1978, Peirce et al., 1984, Peric, 1993, Simo, 1991) to more recent and sophisticated 2 CLASSICAL PERZYNA-TYPE contributions (Alfano et al., 2001, Betegon et al., VISCOPLASTICITY 2006, Lorefice et al., 2008, Ponthot, 2002). In con- trast, stress driven schemes are not that common (Ca- ballero et al., 2009, Zienkiewicz and Cormeau, 1974). Within the classical framework of small strain Their implementation can be numerically advanta- Perzyna viscoplasticity Perzyna (1966), total strain geous (explicit integration of the constitutive equa- can be additively split into the elastic and the vis- tions and simple coding) but their stability is strongly coplastic: related to the size of the time step and, as a conse- el vp quence, if a small time step is required an enormous ij = ij + ij (1) The elastic strain tensor is obtained by considering Zienkiewicz and Cormeau (1974) with constant stiff- isotropic linear elasticity: ness is recovered. For other values of θ greater than zero, the calculation of ∆vp involves ∆σ , and el −1 n+1 n+1 ij = C0 ijklσkl ;C0 ijkl = D0 ijkl (2) therefore it is required to iterate within the time step until the viscoplastic strain estimates satisfy the pre- where D0 is a elasticity stiffness matrix, symmetric −1 scribed stress increment. and positive definite. D0 denotes its inverse, i.e. the In the implementation developed, a first order ex- elasticity compliance matrix, which will be referred pansion of the yield function and the plastic poten- to as C0, and σij is the Cauchy stress tensor. tial is proposed to evaluate the expression above, The stress loading function F (σij) is defined to which leads to a linearized form of ∆ in terms of distinguish between elastic states (F ≤ 0) and vis- n+1 ∆σn+1, as described in more detail in Aliguer et al. coplastic states (F > 0). In the latter case, the clas- (2013). sical Perzyna viscoplastic strain rate is considered: vp 1 F (σ) @Q 4 VISCOPLASTIC RELAXATION _ij = (3) η F0 @σij If a time-dependent problem exhibits stationary con- where η is the viscosity of the material, F0 is a refer- ditions, those are characterized by no change in ence value of the yield function for normalization, and strains. In terms of stress and for the problem at hand, Q the viscoplastic potential typical of non-associated this can be interpreted as stress states evolving from formulations. the viscoplastic region (F (σ ) > 0) to the yield sur- vp ij Finally, the accumulated viscoplastic strain ij in face/elastic region (F (σij) ≤ 0 or stationary state). (1) can be obtained by integrating in time the vis- The viscoplastic relaxation technique takes advantage coplastic strain rate: of this concept, in order to reach the solution of the Z t Z t inviscid problem as the stationary solution of a ficti- vp vp 1 F (σ) @Q tious viscoplastic problem. ij = _ij dt = dt (4) 0 0 η F0 @σij The resulting scheme can be summarized as fol- lows: in the iteration, the viscoplastic strain increment is calculated using only the value of the stress at the 3 STRESS-DRIVEN NUMERICAL SCHEMES i FOR VISCOPLASTIC TIME-INTEGRATION beginning of the increment, σn and the secant stiff- ness is calculated if the value of θ is different than i As the main difference to traditional elastoplasticity, 0. Once the system of equations is solved for δun+1, i in viscoplasticity stress states are allowed outside the the increment of stress for the iteration (∆σn+1) can loading surface. The proposed scheme takes advan- be obtained. The iterative procedure reaches conver- tage of this fact to obtain a simple but effective al- gence when the stress state is sufficiently close to the gorithm for the integration of Perzyna viscoplasticity. yield surface, so that the value of residual stress is in- The main distinctive feature of the proposed scheme ferior than a tolerance value. is that it is stress-driven, in contrast to traditional strain-prescribed procedures for elastoplasticity. 5 CONSTITUTIVE MODEL FOR INTERFACES The main advantage of a stress-prescribed scheme is that the integration of the constitutive relation can The above schemes for viscoplasticity and viscoplas- be reduced to the numerical calculation of the inte- tic relaxation have been applied to the simplified ver- gral expression (4), which becomes relatively simple. sion of an existing zero-thickness interface constitu- In this expression, all the terms are computed in ad- tive model, which was previously developed within vance, in order to update the current value of the vis- the context of quasi-static elastoplasticity Caballero coplastic strain, instead of the usual implicit proce- et al. (2008) . dures resulting from strain-prescribed schemes. The frictional hyperbolic loading function for the The integration of the expression (4) can be done interface is defined in terms of normal and shear via the trapezoid rule in the interval ∆t = t − n+1 n+1 stresses (σ,τ1,τ2) and the strength parameters cohe- tn. Depending on the assumption of the stress incre- sion and friction angle (c,φ) ment within the interval the following expression for the viscoplastic strain increment can be obtained: q 2 2 2 2 F = − tan φ(a − σ) + τ1 + τ2 + a tan φ (6) vp ∆tn+1 ∆n+1 = ((1 − θ)F (σ) + θF (σ + ∆σ)) m (5) ηF0 where a = c= tan φ. The flow rule considered in this model is non- where θ is a fixed parameter with value between 0 associative and depends on whether normal stress and 1 and m is the flow rule. For θ = 0 the original state is tension or compression. In the case of the ten- viscoplastic algorithm of Zienkiewicz and Cormeau sion, the flow rule is radial towards a point near the origin, while in compression, only shear viscoplastic transversal stiffness modulus of the interface. Expres- strains are generated : sion (9) indicates that, at the limit of infinite time, the value τ − τY tends to zero, that is, the shear stress tends to the shear stress yield limit value τ , and @Q=@σ ! 2(σ − σcut) ! Y the elastoplastic solution is recovered. The resulting tens @Q=@τ 2τ m = 1 = 1 (7) curve has been represented in Fig. 2, together with @Q=@τ 2τ 2 2 the numerical results for the same case obtained with a ∆t = 1s, and three different values of parameter θ 0 ! equal to 0, 1 and 1/2. compr m = 2τ1 (8) 2τ2 60 where σcut is a value of small compressive stress Analytical which acts as the center for the radial flow rule in ten- Num. θ = 0.0 sion. 50 Num. θ = 1.0 Num. θ = 1/2 6 NUMERICAL EXAMPLES 40 6.1 Relaxation test at constitutive level 30 The first example of application consists of a relax- ation test at constitutive level. Normal stresses are ap- Shear stress [kPa] plied first, followed by prescribed shear relative dis- 20 placement (at constant normal stress).
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