Civil Engineering April 2022 | Vol 30 No 3

Civil Engineering April 2022 11 the principles and behaviours of soils described by these complex models are buried deep in memory or in dusty bookshelves, and a brief refresher may go some way towards re-enlightenment. So, without intending to read like a textbook, the following sections may give the reader renewed perspective. CSSM states that a soil will reach a failure state at combinations of mean ef- fective stress (p’), deviatoric stress (q) and void ratio (e) defined by a line in p’, q, e space termed the critical state line (CSL). This line is unique for a specific material. 2D projections of the CSL are shown in p’:q and p’:e space in Figure 1. In addition to the mathematical formulae defining the critical state line, the NorSand model comprises various relationships which regulate a material’s elastic (recoverable) and plastic (non- recoverable) behaviour before failure occurs. It is therefore a relatively complex constitutive model when compared to elasto-plastic models. A total of 10 parameters are needed to fully define the NorSand constitutive model. The boundary between elastic and plastic strain occurs on a yield surface. NorSand yield surfaces are curves in p’:q space, each with constant void ratio do- mains. All states within the yield surface are elastic, however, should a stress state contact the yield curve (1) plastic strain results and (2) the yield surface changes in size. With increasing strain, the soil state moves towards the CSL. The yield surface applicable to dilative soils expands until the stress state contacts the CSL and failure occurs. Conversely, contractive soils are controlled by a shrinking yield surface enroute to the CSL. The two projections in Figure 1 can be combined to form a 3D space defining the NorSand framework as shown in Figure 2. The critical state line (green line) forms a non-linear 3D curve. The red line shows the stress path for a monotonically loaded tailings element as post-processed from a finite element model. The purple lines show the yield curves plotted intermit- tently along the stress path. CSSM MATERIAL BEHAVIOUR AND LIQUEFACTION Material behaviour, as alluded to above, can be categorised as dilatant or contrac- tive. In a practical sense, contractive materials are prone to strain softening and liquefaction and are therefore a cause for concern, requiring extra care, particu- larly when considering the stability of an upstream tailing’s storage facility. The state parameter is the numeric difference between the actual void ratio and the corresponding critical state void ratio for a given stress state. Positive state parameters are synonymous with soils that are “loose” of critical or contractive and negative state parameters suggest soils that are dilative or “dense” of critical. Theoretically, a material is contractive Figure 2 NorSand framework in 3D space q p’ CSL Stress path Yield curves CSL e p’ Yield curves Stress path CSL q q Stress path Yield curves p’ e q e p’ Yield curves Stress path CSL