Technical Communication On the modeling of the state dependency of granular soils Ali Lashkari School of Civil Engineering, Islamic Azad University of Shiraz, Shiraz, Iran article info Article history: Received 18 December 2008 Received in revised form 24 May 2009 Accepted 17 June 2009 Available online 17 July 2009 Keywords: Sand State Critical state Constitutive equation Bounding surface Stress ratio abstract Experimental studies have revealed that principal elements of the mechanical behavior of granular soils like the angles of internal peak friction and dilatancy are highly influenced by the combined effect of soil density and mean principal effective stress. In the literature, various empirical correlations between these elements and some parameters indicating soil state have been suggested. Herein, by using two well- known empirical expressions for state dependent peak friction and dilatancy angles, proper constitutive equations are derived and implemented in a stress ratio-based bounding surface plasticity framework. It is shown that the modified model is capable of simulating sand response in either loose or dense states using a unique set of parameters. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction The mechanical response of granular soils subjected to shear is of great interest in soil mechanics and geotechnical engineering. In general, sand behavior is contractive when sand is in loose state, and is dilative in dense state [1–6]. Density is the first important factor which has influence on sand response. The second, but not the less important factor is mean principal effective stress. It has been observed that increase in mean principal effective stress leads to rise in contractiveness of behavior [1–6]. To describe overall soil behavior, some state parameters are required to define the state of sand uniquely. From the historical viewpoint, relative density, I D =(e max  e)/(e max  e min ), is one of the first parameters of soil state used in soil mechanics and engineering practice, where e max , e min , and e indicate the maximum, minimum, and current values of void ratio, respectively. However, it must be noticed that state parameters like relative density which does not consider the effect of normal effective stress are not capable of accounting for sand behavior over wide ranges of normal stress. To improve this limi- tation, Been and Jefferies [3] introduced w = e  e c as state param- eter, where e c is the critical void ratio corresponding to the current amount of mean principal effective stress. Ishihara [5] suggested state index, I s =(e u  e)/(e u  e Q ), as another sand state parameter, where e u and e Q are the upper limit and quasi-steady state void ra- tios, respectively. Some researchers suggest that the ratio (e/e c ) can play the role of a state parameter (e.g., [7]). On the other hand, a number of researchers prefer to work with the current and critical mean principal effective stresses (e.g., [8,9]). For example, Wang et al. [8] introduced the concept of state pressure index, I p = p/p c , as another parameter of sand state. Using extensive sets of exper- imental data on various sands, Bolton [10] proposed index of rela- tive dilatancy, I R = I D (Q  ln p c )  R (where Q and R depend on material type), and suggested a number of correlations between maximum dilatancy, and peak internal friction angle and I R . Critical state soil mechanics has been successfully applied to constitutive modeling of clays (e.g., [11]). However, this concept has been applied to granular soil models in recent years. Using the initial value of w, Jefferies [12] introduced a Cam-Clay type state dependent sand model so-called Nor-sand. Wood et al. [13] suggested the concept of virtual peak stress for modeling peak strength of granular soils subjected to drained shear. Extending the latter theory in order to include dilatancy, Manzari and Dafalias [14] proposed a versatile two-surface plasticity model. In a same path, Gajo and Wood [15] pioneered their first proposition [13] and developed a framework so-called Severn-Trent sand model. A parallel approach in multi-yield surface framework was sug- gested by Latifi [16]. Using the ratio (e/e cr ), Wan and Guo [7] pro- posed a simple sand model which is accurately capable of simulating sand behavior in triaxial space. In the context of hypo- plasticity, a similar approach has been proposed by Herle and Gudehus [17]. Cubrinovski and Ishihara [18] considered sand state dependency by the use of I s in an improved hyperbolic model. Re- cently, Wang et al. [8] proposed a state dependent bounding sur- face hypoplasticity model by means of state pressure index, I p . In the literature, a number of correlations between some princi- pal elements of soil behavior and sand state parameters can be found. For example, using experimental data on several sands, 0266-352X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2009.06.003 E-mail address: lashkari_ali@hamyar.net Computers and Geotechnics 36 (2009) 1237–1245 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo