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 -driven. Depending on how the increment is imposed (constant, linear etc.), different strategies arise. The secant compliance is obtained by truncated expansion of the 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 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 : 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 . 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 = ψ (3) η F0 ∂σij If a time-dependent problem exhibits stationary con- where η is the 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. , 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 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). Then time is allowed to pass, and shear stress starts decreasing to 10 the limit value dictated by the yield surface (Fig. 1). 0 2 4 6 8 10 Time [s] 800 ⌧ Figure 2: Analytical and numerical results for the evolution of ⌧0 the shear stress with time for ∆t = 1s

600 In the numerical calculations, the load has been −6 400 applied in two steps: 1) ∆u = −2 · 10 m and ⌧ −6 1 ∆v = 6 · 10 m with ∆t = 0 s to generate the initial

200 stress state beyond the yield surface. 2) ∆u = ∆v = 0 and ∆t > 0. The second step has been subdivided in 0 10 increments to reach the final time of tf = 10 s (∆t = 1s). Material parameters used for this exam- 7 2 ple are: KN = KT = 10 kN/m; c = 0.0 kN/m ; −200 ⌧ tan φ = 0.577; η = 106kP a s. ⌧0 The figure shows that the strategy with θ = 1/2 −400 yields a much better approximation than θ = 0 or θ = 1. A measure of the overall error has been estab- −600 ⌧ lished as the difference of the area under the curves 1 to the one corresponding to the analytical solution, −800 −200 −100 0 100 200 300 400 500 600 700 t 800 as the total amount of viscoplastic strain will be ap- proximately proportional to that area (equation 3). The results obtained give an error lower than 1% for Figure 1: Shear stress path in σ - τ space and shear stress evolu- the strategy with θ = 1/2, and 14.1% and 14.0% for tion through time θ = 0 or θ = 1 respectively. As it would be expected, assuming constant stress for the increment equal to First, the case without cohesion is considered (c = the initial value (forward scheme with θ = 0), leads 0 in equation 6). For this particular case, it is possible to an over-prediction of the viscoplastic strain, and to find a closed-form expression for the decay of the therefore, stress decreases faster. However, assuming shear stress with time τ = f(t) : θ = 1, that is, a backward scheme taking stress at the end of the increment, normally lower, leads to an     τ − τY −KT under-prediction of the viscoplastic strain. = exp t (9) A second calculation has been run with c = τ0 − τY ηF0 10.0 kN/m2. In this case the analytical solution is not where τ0 is the value of shear stress at time t0, τY straightforward, and the numerical solution obtained is the shear yield threshold value, and KT is the for very small steps of ∆t = 0.001s which leads to practically the same results for any θ, is taken as the scheme θ = 1 (usually assumed to be more accurate) ”exact” solution. This ”exact” solution is represented leads to similar errors as the forward scheme with in Fig. 3 together with the numerical results obtained θ = 0. with θ = 0, θ = 1/2 and θ = 1, first for ∆t = 1s (left diagram), and then for ∆t = 2.0s and ∆t = 0.5s (right diagram). 6.2 Rock slope stability with zero-thickness interface elements

60 The main objective of this example is to evaluate the Num. = 1/2, t=0.01s performance of the viscoplastic relaxation iterative Num. = 0.0, t=1s procedure for a classic rock slope analysis (Fig. 4). 50 Num. = 1.0, t=1s The following geometry has been considered: L = ◦ Num. = 1/2, t=1s 100 m, H = 80 m, ∆p = 1.0 kP a/m, β1 = 55.1 and ◦ 40 β2 = 55.7 . For the inclined interfaces the strength pa- rameters used are c = 75 kP a, tan φ = 1.0 and η = 106 kP a s, while for vertical interfaces c = 1.0 kP a, 30 tan φ = 1.4 and η = 106 kP a s . Rock mass is as- sumed linear elastic with E = 2 GP a and ν = 0.2. Shear stress [kPa] First, the initial stress state due to gravity with 20 K0 = 0.8 was generated using the viscoplastic relax- ation scheme and θ = 1/2. Then, the distributed load 10 ∆p was applied incrementally using a viscoplastic re- 0 2 4 6 8 10 θ = 1/2 Time [s] laxation iterative scheme (also with ) until the failure was reached. 60 As it can be observed in Fig. 4 different compat- Num. = 1/2, t=0.01s ible failure mechanisms exist and it is not trivial to Num. = 0.0, t=0.5s determine a priori which is the most unfavorable one. 50 Num. = 1.0, t=0.5s Failure was reached after a total of 103 increments of Num. = 1/2, t=0.5s ∆p = 1.0 kP a/m , and the failure mechanism was Num. = 1.0, t=2s 40 found to be a combination of the first vertical and the Num. = 1/2, t=2s lowest inclined joints.

30

Shear stress [kPa] 7 CONCLUDING REMARKS 20 An efficient integration procedure for Perzyna-type viscoplasticity with a stress-prescribed scheme has 10 been presented, which can be used for a physical (real 0 2 4 6 8 10 Time [s] time) time-dependent problems or as a basis for a vis- coplastic relaxation procedure (fictitious time). The Figure 3: Numerical results for the evolution of the shear stress with time for ∆t = 1s (upper) and ∆t = 0.5s and ∆t = 2s latter allows to recover the elastoplastic solution as (lower). a limit inviscid case. The viscoplastic scheme is successfully imple- Fig. 3 (left) shows a similar trend as Fig. 2. Fig. 3 mented for a zero-thickness interface model origi- (right) shows some additional features of the integra- nally conceived for time-independent representations tion schemes, confirming the better performance of of rock discontinuities θ = 1/2 and showing that for larger time increments A constitutive example shows that different val- (∆t = 2s) convergence is not even obtained for θ = 0. ues of the parameter θ lead to different decay curves The overall error of the three solutions in terms of area of shear stress with time, and subsequently also under the curves for ∆t = 0.5s are: less than 0.1% for to different total values of accumulated viscoplastic θ = 1/2 in front of 4.4% and 4.3% for θ = 0 and θ = 1, strain. While for θ = 0 viscoplastic strains are under- respectively. Additionally, calculations have been also estimated, for θ = 1 an over-estimation is produced. run for ∆t = 0.2s with errors of 1.7% for both θ = 0 Best results compared to ”exact” solution are obtained and θ = 1. And one has to decrease ∆t to the value with θ = 1/2. of ∆t = 0.05s to obtain errors of less than 1% for In the context of the Finite Elements calculation those values of θ. This means that for those strategies of a rock slope stability problem, the proposed vis- (θ = 0 and θ = 1) one requires about 20 times more coplastic relaxation method is shown capable of de- time steps to obtain similar accuracy as for θ = 1/2. tecting the non-trivial failure mechanisms in fractured Also remarkable is the observation that the backward rock masses. integration procedure for viscoplastic gurson mate- rials. Computer Methods in Applied Mechanics and Engineering, 195(44-47):6146–6157, 2006.

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8 ACKNOWLEDGMENTS D. Peric. On a class of constitutive equations in viscoplasticity: Formulation and computational is- Partial support for this research has come from sues. International Journal for Numerical Methods project BIA-2012 36898 MEC (Madrid), which in- in Engineering, 36(8):1365–1393, 1993. cludes FEDER funds (European Comission). Funding P. Perzyna. Fundamental problems in viscoplastic- from 2009SGR-180 from Generalitat de Catalunya ity. Advances in Applied Mechanics, 9(C):243– (Barcelona) is also greatly appreciated. The first au- 377, 1966. thor also wishes to thank MEC (Madrid) for the doc- toral fellowship received 2009-2012 including funds J.P. Ponthot. Unified stress update algorithms for the for a 6 months stay at CU-Boulder. numerical simulation of large elasto- plastic and elasto-viscoplastic processes. Interna- REFERENCES tional Journal of Plasticity, 18(1):91–126, 2002.

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