View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Granular Matter (2010) 12:469–476 DOI 10.1007/s10035-010-0206-7 Stress-induced anisotropy in sand under cyclic loading Minyun Hu · Catherine O’Sullivan · Richard R. Jardine · Mingjing Jiang Received: 17 December 2009 / Published online: 7 September 2010 © The Author(s) 2010. This article is published with open access at Springerlink.com Abstract The anisotropy of a granular material’s structure ratio of the normal to shear spring stiffness at the particle con- will influence its response to applied loads and deformations. tacts in the DEM model affects the evolution of anisotropy. Anisotropy can be either inherent (e.g. due to depositional Higher ratios reduce the degree of anisotropy induced by process) or induced as a consequence of the applied stresses anisotropic consolidation. The anisotropy induced by cyclic or strains. Discrete element simulations allow the interac- loading depends on the amplitude of the loading cycles and tions between individual particles to be explicitly simulated the initial anisotropy. and the fabric can be quantified using a fabric tensor. The eigenvalues of this fabric tensor then give a measure of the Keywords Sand · Anisotropy · Discrete element method · anisotropy of the fabric. The coordination number is a particle Fabric tensor scale scalar measure of the packing density of the material. The current study examines the evolution of the fabric of a granular material subject to cyclic loading, using two-dimen- 1 Introduction sional discrete element method (DEM) simulations. Isotropic consolidation modifies and reduces the inherent anisotropy, There are many situations where soil is subject to extended but anisotropic consolidation can accentuate anisotropy. The periods of drained cyclic loading. Practical foundation sce- narios involving drained cyclic loading include road pave- M. Hu ments (e.g. Lekarp et al. [1]), reciprocating machines and Key Laboratory of Geotechnical and Underground Engineering storage tanks (Sweeney and Lambson [2]) as well as offshore of Ministry of Education, Tongji University, Shanghai, and wind turbine foundations. Other examples are offered by China the soil backfill adjacent to integral bridge abutments (e.g. M. Hu (B) Clayton et al. [3]) and earth dams subject to reservoir level College of Civil Engineering and Architecture, Zhejiang University fluctuations (Tedd et al. [4]). Janbu and Seneset [5] proposed of Technology, Hangzhou 310014, China that observed time-dependant settlements of structures on e-mail: [email protected] high permeability soils may be due to combined effects of C. O’Sullivan · R. R. Jardine cumulative displacements under cyclic loading and creep. Department of Civil & Environmental Engineering, Imperial Jardine et al. [6] noted important interactions between age- College London, Skempton Building, South Kensington Campus, ing, creep and cyclic loading in field tests on piles driven London SW7 2AZ, UK e-mail: [email protected] in sand. Piles that could be brought to failure by high level drained cyclic loading could also show a benign or even ben- R. R. Jardine e-mail: [email protected] eficial response to low level drained cycling. A fundamental understanding of the processes that define cyclic stiffness, M. Jiang the growth of permanent deflections, the potential effects on Department of Geotechnical Engineering, Tongji University, creep and implications for failure loads would offer signifi- 1239 Siping Road, Shanghai 200092, China cant practical benefits. This paper reports a contribution that e-mail: [email protected] focuses on drained load cycling in high permeability granular 123 470 M. Hu et al. media. A simplified and idealized system is considered that strain accumulation with increasing number of cycles. O’Sul- offers insight into evolution of fabric and stress anisotropy livan et al. [17] simulated drained strain controlled cyclic under extended cyclic loading. triaxial tests. Good agreement was found between the DEM Anisotropy of soils can be classified into two components: simulations and physical tests on an ideal granular material. inherent anisotropy and stress induced anisotropy. Inher- The material’s internal structure continued to evolve during ent anisotropy is attributed to preferred particle and contact cyclic loading and the variation in fabric was found to depend orientations that develop during deposition. Stress-induced on the applied strain level. anisotropy, on the other hand, results from the deformation The current study investigated the evolution of anisotropy associated with different changes in applied stress. Oda et al. of sand during stress controlled drained, cyclic loading. 2D [7] listed three approaches to quantify anisotropy in a gran- circular disk particles were used allowing the stress induced ular material: (1) distribution of contact normal orientations, anisotropy given by the contact normal orientations to be (2) the orientation of void spaces; and (3) the orientation of isolated. A parametric study considered different interac- nonspherical particles. Moreover, Oda et al. observed that the tion parameters, different consolidation states, and different inherent anisotropy due to (1) and (2) can be altered during cyclic stress amplitudes. the early stage of inelastic deformation, while the inherent anisotropy due to (3) persists even after shearing to large strains. 2 Fabric tensor Motivated by the need to advance understanding of lique- faction, a significant number of research studies have exam- A convenient means to quantify the overall orientation of ined the response of soil to undrained cyclic loading. Fewer the contact normals in a granular material is the fabric ten- studies have examined the response to many cycles of drained sor. Key references describing the use of the fabric tensor to cyclic loading. Recent experimental studies considering the characterize the fabric of granular materials include Satake response of sand to repeated loading in drained conditions [18], Rothernberg and Bathurst [19], Oda [20], and Oda et al. include the work of Cosgrove et al. [8], AnhDan and Loseiki [7]. There are subtle differences in the equations presented [9], Wichtmann et al. [10], Wichtmann and Triantafyllidis in the literature for the fabric tensor. The expression for the [11] and Festag and Katzenbach [12]. While these experi- 2nd order (2nd rank) fabric tensor adopted here follows that mental studies have provided insight into the overall mate- used by Thornton [21] and is given by: rial response, the issue of soil fabric has not been considered in detail. Other studies on the mechanical response of soil 1 ij = ni n j (1) indicate that fabric must be carefully considered. For exam- Nc Nc ple, Hyde et al. [13] studied the behavior of silt following = = = undrained cyclic loading and found that initial anisotropy where ij fabric tensor; Nc number of contacts; ni unit can affect postcyclic recompression, stiffness, and consoli- vector normal to contact. The matrix form of fabric tensor in dation strength of soil. A better understanding of the evolu- 2D is ⎡ ⎤ tion of soil micro-structure under cyclic loading is required n n n n ⎢ 1 1 1 2 ⎥ to improve fundamental understanding of soil response and Nc 1 ⎢ Nc ⎥ hence our ability to predict deformations. The development = ⎣ ⎦ (2) Nc n n n n of constitutive models that include fabric parameters is ongo- 2 1 2 2 N Nc ing. For example, Papadimitriou and Bouckovalas [14]devel- c oped an analytical approach to account for fabric changes Similar to the stress tensor, σ , the eigenvalues of this during cyclic loading in a continuum elasto-plastic constitu- matrix give the principal values of fabric: 1 and 2.The tive model. difference between the maximum and minimum eigenvalues One approach that can be adopted to gain data on fabric 1 − 2, is defined as the deviator fabric, and this param- evolution is to simulate the response of assemblies of parti- eter quantifies the anisotropy of the material. An isotropic cles using the distinct element method (DEM) as proposed material will have 1 = 2 = 0.5. by Cundall and Strack [15]. In DEM, the individual particles A second (scalar) measure of fabric is the coordination are simulated and simple contact laws are used to describe number, this gives the density of contacts in the system and their interactions. Then, the system evolution is obtained by is calculated as integrating the equations of motion for each particle. There 2Nc have been limited applications of DEM to study the cyclic Z = (3) N response of granular materials. Using DEM, David et al. [16] p studied the effect of cyclic loading on the development of where Nc is the total number of contacts and N p is the number strain within a granular material, demonstrating significant of particles. 123 Stress-induced anisotropy in sand under cyclic loading 471 3 Materials and methods 3.2 Model and parameters 3.1 Sample preparation The simple linear–elastic contact law provided in PFC2D and commonly used in DEM analysis was adopted in this A sample of circular disks, with diameters between 0.20 and study. While more sophisticated models are available, the 0.50mm was created within the two-dimensional DEM code linear model was considered adequate for the 2D simula- PFC 2D. The initial particle coordinates were determined by tions considered here. In the simulations the particles inter- digitizing photographs of physical biaxial compression tests act only if they are in contact, and the force between them on “Schneebeli” rod specimens described by Thomas [22]. can be decomposed into a normal and a tangential part. A uniform scaling was then applied to the particle sizes and The normal and tangential spring stiffness between parti- 7 positions to reduce the particle radii to sizes closer to real cles were selected to be either kn = ks = 2 × 10 N/m or 7 soil. A monotonic simulation using this input configuration kn = 3×10 N/m.