Implementation and use of constitutive models in geomechanics: analysis of shallow footings on sand

Abstract

This thesis work deals with the resolution of boundary value problems via Finite Element (FE) analyses. Since powerful computers have come to be accessible to the common users, the use of FE methods to solve engineering problems has been stimulated. Many FE codes have been developed, both for professional and for research purposes. The standard FE method has been widely developed starting from the pioneering work carried out in the last century by many researcher in both mathematics and engineering fields. A geometrical representation of the boundary value problem, object of the study allowing the decomposition of the geometry for a complex domain in simpler sub-domains, has to be provided by the user; the sub-domains are the elements, which are defined as finite since they are bounded by polygons so that each one occupies a finite area, and the entire set of elements forms a mesh of finite elements. Each element is characterized by a finite number of integration points (Gauss points) in which the governing equations are formulated. The overall solution, in terms of nodal displacements, is achieved in a variational form by enforcing the solution of an energy function typically formulated in an integral form, the so called “weak form” of the Virtual Work Theorem. Because of the discrete nature of the formulation, the integration operation becomes a finite sum over the Gauss points of the whole domain. Once the general solution is achieved, enforcing the fulfilment of the boundary conditions, the state of stress is determined for each Gauss point from the stress-strain relations, and the local solution is extended to the vertices of the elements (nodes) and then to the whole domain. The resolution of a boundary value problem via FE analyses requires the assumption of a constitutive model in order to simulate the stress-strain behaviour at Gauss point level for each material involved in the analysis. For some of the materials commonly used in civil and industrial engineering, relatively simple constitutive laws are able to describe and to model the real mechanical behavior: for example, as for steel structures, a linear elastic relation is able to well capture the material behaviour for loading levels far from the yielding load. If we complete the former relation with a yield criterium aiming to predict the overall behaviour when the yielding load is approached, we obtain a more complex law but now able to predict the mechanical behaviour for a wider loading range. For more complex materials, such as concrete or granular materials, the need of constitutive laws more realistic than a simple Young elastic relation is nowadays commonly acknowledged.Moreover, the availability of higher-performance CPUs has encouraged an intense development of complex constitutive laws starting first within the framework of the elastoplasticity, and then following even different mathematical structures. As for the geomaterials, the work of the Cambridge group, and mainly by Schofield and Wroth, has brought to the development of the Cam-Clay model (Schofield and Wroth, 1967), that is a milestone in the world of elastoplasticity. Most of the constitutive models nowadays available for the resolution of BVP involving geomaterials, such as the extensions or modifications proposed by many researchers (see e.g.(Nova, 1977),(Nova, 1988),(Borja and Tamagnini, 1998),(Lade, 1977)) stem from Cam-Clay pioneer model. Besides, many other constitutive models have been developed outside the framework of elastoplasticity: in the recent past a new family of constitutive laws has been proposed coming from the studies of two separate research groups, the former in Karslruhe, which brought to the K-hypoplasticity, the latter in Grenoble, with the CLoE hypoplastic model. The predictive capabilities of the hypoplastic models for the resolution of BVPs have been investigated in many papers (e.g.(Viggiani and Tamagnini, 2000),(Leoni and Montrasio, 2003),(Leoni et al., 2003)) for different classes of geotechnical engineering problems involving rather different stress paths. Whatever constitutive model is chosen for the soil, the best confirmation of its predictive capacity comes from the comparison with measurements on real structures, since the more complex a constitutive law is, the larger is the set of parameters involved in its definition. A higher number of parameters to be calibrated brings to an higher level of uncertainty on the final results: one can therefore understand the importance of a reliable set of experimental measurements allowing to perform a comparison between FE predictions and real data, in order to assess the reliability of the FE model for a particlar class of BVP. Nevertheless, it must be pointed out how hard is to have a good set of experimental measurements available due to the intrinsic difficulties for which special equipments are required, and to their consequent high cost. In this thesis work, two different constitutive models have been implemented: an elastoplastic model (SSC) which is a simplified version of the original “Sinfonietta Classica” (Nova, 1988), and the hypoplastic CLoE model (Chambon et al., 1994). We have chosen to make some simplifications in the original SC mainly because of its formulation: the original SC has a complicated definition for the constitutive functions, so that their mathematical formulation has been restricted to the first and second invariant of stress tensor, thus neglecting the dependence on the Lode’s angle. Moreover, while the hardening law, which relates the evolution of the hardening parameter to the plastic deformations, has been implemented in differential form, in the original SC an integrated formula is given. The commercial FE code chosen for the implementation is Abaqus, by Hibbitt, Karlssonn and Sorensen, due to its wide diffusion in both research and applied engineering domains and the possibility given to the user to implement its own constitutive relations.A large set of experimental measurements was available coming from the work by Montrasio and Nova (Montrasio and Nova, 1997), thus the laboratory tests have been simulated via FE analyses in order to compare the numerical results with the experimental ones and to evaluate the predictive capabilities of the two models. In the next paragraphs of Chapter 1, an overview on the resolution techniques for the problem of shallow foundations followed by some details about the FE method will be given, together with a description of the experimental set-up and the laboratory tests. In Chapter 2 a wide description of the two models will be given, while in Chapter 3 full details of the implementation process are depicted, along with the description of the validation tests performed in order to assess the reliability of the two implementations. In Chapter 4 the BVP of a rigid shallow foundation on sand is studied, and the results are compared to those obtained in the laboratory tests, aiming to assess the capability of the models to capture the real behaviour of the footing during loading tests. In addition to the complex constitutive models, an extended version of a more simple well-known Mohr-Coulomb model, already implemented in the FE code, is used. Its description is given in the next introductory pages; moreover, after a discussion about the obtained results, some conclusions are drawn in Chapter 5.