A time dependent constitutive model for soils with isotach viscosity Teresa M. Bodas Freitas a,⇑ , David M. Potts b , Lidija Zdravkovic b a Instituto Superior Técnico, Departamento de Engenharia Civil e Arquitectura, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal (formerly at Imperial College London) b Imperial College London, Department of Civil & Environmental Engineering, Skempton Building, London SW7 2AZ, UK article info Article history: Received 5 September 2010 Received in revised form 18 May 2011 Accepted 19 May 2011 Available online 23 June 2011 Keywords: Constitutive relations Viscoplasticity Critical state line Time dependence Creep abstract This paper presents a three-dimensional elastic viscoplastic constitutive model that is able to reproduce the time dependent behaviour of soils with isotach viscosity. This constitutive law is based on the over- stress theory and incorporates some important features, namely: (i) a non-linear creep law with a limit for the amount of creep deformation under isotropic stress conditions; (ii) a ﬂexible loading surface that is capable of reproducing a wide range of shapes in p 0  J stress space and incorporates the Matsuoka– Nakai failure criterion in the deviatoric stress space and (iii) assumes that the viscoplastic scalar multi- plier is constant on a given loading surface to ensure that critical state conditions are reached. A full description of the model, its governing equations and implementation in a ﬁnite element program are presented. The model is then used to simulate the results of laboratory tests, highlighting the model’s abilities and shortcomings in reproducing the time dependent behaviour of soft clays. Furthermore, the paper investigates the importance of accounting for creep non-linearity and the consequences of iso- tach viscosity on the critical state line. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction The study of the time dependent behaviour of soils was initiated in the 1920s, prompted by the practical need to quantify the long- term settlement of soft clays. As a result, the ﬁrst mathematical formulations of the creep phenomenon were developed for one- dimensional stress conditions. In this respect the works of Buisman [1], Bjerrum [2] and Garlanger [3] are of special note. Subsequent laboratory studies have focused on the identiﬁca- tion and characterisation of the soil’s time and rate dependent behaviour under general stress conditions, namely the inﬂuence of the applied strain rate on the pre-consolidation pressure and on the size of the bounding surface [4–6], and the characterisation of undrained creep and the conditions for the occurrence of un- drained creep rupture [7,8]. These laboratory studies highlight the importance of consider- ing the time dependent behaviour of soils in the design of geotech- nical structures and in the derivation of the strength and deformability parameters appropriate for ﬁeld conditions, to en- sure that both serviceability and ultimate limit states are veriﬁed [9]. Driven by the increasing use of numerical methods in geotech- nical design, various constitutive models have been proposed to simulate the time dependent behaviour of soils that follow isotach viscosity, under general stress conditions [10]. Due to its simplicity and ﬂexibility, most of these are elastic viscoplastic (EVP) models based on Perzyna’s overstress theory [11]. Accord- ing to Perzyna’s overstress theory the size of the viscoplastic strain increment is a function of the distance between the current loading surface and the static (yield) surface. This function – referred to in the literature as the viscous nucleus or viscoplastic scalar multiplier – and thus, the size of the viscoplastic strain increment, is implicitly constant for all stress states located on a given loading surface. The individual strain components are then obtained from a plastic potential that is coincident with the current loading surface. A signiﬁcant difference between available EVP overstress-based models concerns the mathematical form of the viscoplastic scalar multiplier. In some earlier works, the viscoplastic scalar multiplier is assumed to be an exponential function of overstress and the function constants are obtained by ﬁtting laboratory test results [12,13]. An alternative is to relate the viscoplastic scalar multiplier to the observed drained creep behaviour under one-dimensional or isotropic stress conditions [14–17]. This procedure simpliﬁes con- siderably the derivation of the model parameters. But when apply- ing this latter approach some researchers [15,16,18,19] assume that the volumetric strain rate is constant on a given loading sur- face, as suggested by some laboratory test data [20,21]. However such an assumption implies that critical state conditions cannot be reached [22]. The model presented herein, assumes that the viscoplastic scalar multiplier is constant on a given loading surface [13,14,17]. 0266-352X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2011.05.008 ⇑ Corresponding author. Tel.: +351 21 8418419; mobile: +351 96 6028382; fax: +351 21 8418427. E-mail address: tmbodas@civil.ist.utl.pt (T.M. Bodas Freitas). Computers and Geotechnics 38 (2011) 809–820 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo