Issue 41

A. Mardaliazad et alii, Frattura ed Integrità Strutturale, 41 (2017) 504-523; DOI: 10.3221/IGF-ESIS.41.62 505 assumes that the mechanical characteristics, i.e. the stress-strain behaviour, are linear all the way through the critical cross- sectional area of the beam [5]. Due to the sensitivity of this test to the boundary conditions, it may be inaccurate in large displacement (which is similar to the direct tensile test) [6]. In addition to these difficulties, the strength of quasi-brittle materials, e.g. rock, is considered to be dependent on their size and scale [7-9]. However, the Flexural test is considered as the simplest method to investigate the creep (time-dependent behaviour) of rock [10], and accordingly, represents an interesting alternative for the investigation of the stress-strain relationships [11, 12]. The experimental campaign designed for this research study follows the protocols of the ASTM standard and the results can therefore be easily compared with similar studies, i.e. on other materials. The mechanical properties of several types of rock materials, including sandstones have been studied in a large variety of research. Among them, the Berea sandstone which is deposited sub-aqueously as offshore beds in a considerable number of well drilling applications, is widely investigated by the oil and gas industries [13, 14]. It is a sedimentary rock with high porosity and mostly composed of sand-sized grains [15], however, it is not a highly accessible material resulting in a relatively high expense for testing purposes. An extensive literature review [16-18] exposed the presence of another rock material, which is called Pietra Serena sandstone, with similar mechanical characteristics to Berea sandstone. These similar properties led the authors to perform the experimental tests on the Pietra Serena sandstone. It is anticipated that the experimental data of the flexural test on the Pietra Serena can be used to have an outlook of the behaviour of Berea sandstone. The development of a reliable numerical method in conjunction with an accurate constitutive material model has been considered an essential tool in the stress analyses. Several numerical modelling techniques have been recently implemented [19-23] to simulate the rock materials. One of the most common and accurate numerical simulation techniques, which is implemented in research fields as well as in industrial applications, is the Lagrangian Finite Element Method (FEM). However, this method cannot appropriately deal with large deformations and tearing, which are often present in the numerical modelling of fractured rock. The Smooth Particles Hydrodynamics (SPH), on the other hand, is a mesh-free Lagrangian method that discretizes a system into a number of “mesh-points” (or particles) carrying the field variables. Due to the not fixed nodal connectivity of this method, the SPH is able to cope with highly distorted elements [21, 24, 25]. The FEM is however more efficient in terms of accuracy and computation time when SPH particles are used. Therefore, an advanced technique is exploited, inspired by the research study of the same authors of the present paper, Bresciani et al., [25] to take advantage of both the Lagrangian FEM (before the occurrence of high distortion) and the SPH methods to deal with large deformation, mesh distortion, etc. In this numerical model, which is called FEM-coupled to-SPH, the specimen is initially modelled by Lagrangian 3D elements and subsequently by means of an eroding algorithm. The elements which reach a specific failure level are eroded, and subsequently these eroded elements are transformed to SPH particles with the same mechanical properties. Two commercial numerical solvers, LS-DYNA and ABAQUS, are used to replicate the experimental tests in conjunction with two constitutive material models: the Karagozian and Case Concrete (KCC) model and the Extended (Linear) Drucker- Prager (LDP) model. The KCC is an advanced material model, developed by Malvar et al. [26-29], available in LS-DYNA, that decouples the volumetric and deviatoric responses. This material model consists of three-independent failure surfaces to determine the accumulated damage. The LDP, on the other hand, is based on the conventional Drucker-Prager model. This material model which is available in ABAQUS, takes advantage of some improvements, i.e. the flow rule plasticity [30] and is particularly interesting due to the potential use of different shapes of the yield function (linear, hyperbolic and exponential), and additional cap yield function and the possibility to be used in conjunction with an equation of state. However, due to the lack of a tension (or pressure) cut-off level, the numerical results have to be critically evaluated. The numerical results of the KCC and LDP are thus compared in detail to show the reliability of the models. The article is divided into the following sections. The methods and the results obtained during the experimental tests are reported in section 2. In section 3, the theories of the KCC and the LDP material models are discussed in detail. Then the experimental configuration suggested by the ASTM is replicated in LS-DYNA and ABAQUS by the FEM-coupled to-SPH method in conjunction with the KCC and the LDP material models, respectively. The numerical results of these models are then compared with each other and discussed in section 4 by the further comparison of the numerical modelling results with the experimental data. E XPERIMENTAL TEST he experimental configuration for the Flexural test is designed based on the protocols of the ASTM standard [31]. This configuration consists of a rectangular cubic specimen which is supported by two fixed rollers near the end of its length span (see Fig. 1a). Thus, the specimen is loaded vertically by means of two compressive rollers at a certain T