Characterization of Marcellus Shale Fracture Properties through Size Effect Tests and Computations

Mechanical characterization of shale-like rocks requires understanding the scaling of the measured properties to enable the extrapolation from small scale laboratory tests to field study. In this paper, the size effect of Marcellus shale was analyzed, and the fracture properties were obtained through size effect tests. A number of fracture tests were conducted on Three-Point-Bending (TPB) specimens with increasing size. Test results show that the nominal strength decreases with increasing specimen size, and can be fitted well by Bazant's Size Effect Law (SEL). It is shown that SEL accounts for the effects of both specimen size and geometry, allowing an accurate identification of the initial fracture energy of the material, Gf, and the effective Fracture Process Zone (FPZ) length, cf. The obtained fracture properties were verified by the numerical simulations of the investigated specimens using standard Finite Element technique with cohesive model. Significant anisotropy was observed in the fracture properties determined in three principal notch orientations: arrester, divider, and short-transverse. The size effect of the measured structural strength and apparent fracture toughness was discussed. Neither strength-based criterion which neglects size effect, nor classic LEFM which does not account for the finiteness of the FPZ can predict the reported size effect data, and nonlinear fracture mechanics of the quasibrittle type is instead applicable

Asymptotic Expansion Homogenization of Discrete Fine-Scale Models with Rotational Degrees of Freedom for the Simulation of Quasi-Brittle Materials

Discrete fine-scale models, in the form of either particle or lattice models, have been formulated successfully to simulate the behavior of quasi-brittle materials whose mechanical behavior is inherently connected to fracture processes occurring in the internal heterogeneous structure. These models tend to be intensive from the computational point of view as they adopt an a priori discretization anchored to the major material heterogeneities (e.g. grains in particulate materials and aggregate pieces in cementitious composites) and this hampers their use in the numerical simulations of large systems. In this work, this problem is addressed by formulating a general multiple scale computational framework based on classical asymptotic analysis and that (1) is applicable to any discrete model with rotational degrees of freedom; and (2) gives rise to an equivalent Cosserat continuum. The developed theory is applied to the upscaling of the Lattice Discrete Particle Model (LDPM), a recently formulated discrete model for concrete and other quasi-brittle materials, and the properties of the homogenized model are analyzed thoroughly in both the elastic and inelastic regime. The analysis shows that the homogenized micropolar elastic properties are size-dependent, and they are functions of the RVE size and the size of the material heterogeneity. Furthermore, the analysis of the homogenized inelastic behavior highlights issues associated with the homogenization of fine-scale models featuring strain-softening and the related damage localization. Finally, nonlinear simulations of the RVE behavior subject to curvature components causing bending and torsional effects demonstrates, contrarily to typical Cosserat formulations, a significant coupling between the homogenized stress-strain and couple-curvature constitutive equations

Spectral Stiffness Microplane Model for Quasibrittle Textile Composites

The present contribution proposes a general constitutive model to simulate the orthotropic stiffness, pre-peak nonlinearity, failure envelopes, and the post-peak softening and fracture of textile composites. Following the microplane model framework, the constitutive laws are formulated in terms of stress and strain vectors acting on planes of several orientations within the material meso-structure. The model exploits the spectral decomposition of the orthotropic stiffness tensor to define orthogonal strain modes at the microplane level. These are associated to the various constituents at the mesoscale and to the material response to different types of deformation. Strain-dependent constitutive equations are used to relate the microplane eigenstresses and eigenstrains while a variational principle is applied to relate the microplane stresses at the mesoscale to the continuum tensor at the macroscale. Thanks to these features, the resulting spectral stiffness microplane formulation can easily capture various physical inelastic phenomena typical of fiber and textile composites such as: matrix microcracking, micro-delamination, crack bridging, pullout, and debonding. The application of the model to a twill 2$\times$2 shows that it can realistically predict its uniaxial as well as multi-axial behavior. Furthermore, the model shows excellent agreement with experiments on the axial crushing of composite tubes, this capability making it a valuable design tool for crashworthiness applications. The formulation is computationally efficient, easy to calibrate and adaptable to other kinds of composite architectures of great current interest such as 2D and 3D braids or 3D woven textiles

Lattice Discrete Particle Model (LDPM) for pressure-dependent inelasticity in granular rocks

This paper deals with the formulation, calibration, and validation of a Lattice Discrete Particle Model (LDPM) for the simulation of the pressure-dependent inelastic response of granular rocks. LDPM is formulated in the framework of discrete mechanics and it simulates the heterogeneous deformation of cemented granular systems by means of discrete compatibility/equilibrium equations defined at the grain scale. A numerical strategy is proposed to generate a realistic microstructure based on the actual grain size distribution of a sandstone and the capabilities of the method are illustrated with reference to the particular case of Bleurswiller sandstone, i.e. a granular rock that has been extensively studied at the laboratory scale. LDPM micromechanical parameters are calibrated based on evidences from triaxial experiments, such as hydrostatic compression, brittle failure at low confinement and plastic behavior at high confinement. Results show that LDPM allows exploring the effect of fine-scale heterogeneity on the inelastic response of rock cores, achieving excellent quantitative performance across a wide range of stress conditions. In addition, LDPM simulations demonstrate its capability of capturing different modes of strain localization within a unified mechanical framework, which makes this approach applicable for a wide variety of geomechanical settings. Such promising performance suggests that LDPM may constitute a viable alternative to existing discrete numerical methods for granular rocks, as well as a versatile tool for the interpretation of their complex deformation/failure patterns and for the development of continuum models capturing the effect of micro-scale heterogeneity

Experimental and Numerical Investigation of Intra-Laminar Energy Dissipation and Size Effect in Two-Dimensional Textile Composites

Design of large composite structures requires understanding the scaling of their mechanical properties, an aspect often overlooked in the literature on composites. This contribution analyzes, experimentally and numerically, the intra-laminar size effect of textile composite structures. Test results of geometrically similar Single Edge Notched specimens made of 8 layers of 0 degree epoxy/carbon twill 2 by 2 laminates are reported. Results show that the nominal strength decreases with increasing specimen size and that the experimental data can be fitted well by Bazant's size effect law, allowing an accurate identification of the intra-laminar fracture energy of the material. The importance of an accurate estimation of Gf in situations where intra-laminar fracturing is the main energy dissipation mechanism is clarified by studying numerically its effect on crashworthiness of composite tubes. Simulations demonstrate that, for the analyzed geometry, a decrease of the fracture energy to 50% of the measured value corresponds to an almost 42% decrease in plateau crushing load. Further, assuming a vertical stress drop after the peak, a typical assumption of strength-based constitutive laws implemented in most commercial Finite Element codes, results in an strength underestimation of the order of 70%. The main conclusion of this study is that measuring accurately fracture energy and modeling correctly the fracturing behavior of textile composites, including their quasi-brittleness, is key. This can be accomplished neither by strength- or strain-based approaches, which neglect size effect, nor by LEFM which does not account for the finiteness of the Fracture Process Zone

Mode I and II Interlaminar Fracture in Laminated Composites: A Size Effect Study

This work investigates the mode I and II interlaminar fracturing behavior of laminated composites and the related size effects. Fracture tests on geometrically scaled Double Cantilever Beam (DCB) and End Notch Flexure (ENF) specimens were conducted to understand the nonlinear effects of the cohesive stresses in the Fracture Process Zone (FPZ). The results show a significant difference between the mode I and mode II fracturing behaviors. It is shown that, while the strength of the DCB specimens scales according to the Linear Elastic Fracture Mechanics (LEFM), this is not the case for the ENF specimens. Small specimens exhibit a pronounced pseudo-ductility with limited size effect and a significant deviation from LEFM, whereas larger specimens behave in a more brittle way, with the size effect on nominal strength closer to that predicted by LEFM. This behavior, due to the significant size of the Fracture Process Zone (FPZ) compared to the specimen size, needs to be taken into serious consideration. It is shown that, for the specimen sizes investigated in this work, neglecting the non-linear effects of the FPZ can lead to an underestimation of the fracture energy by as much as 55%, with an error decreasing for increasing specimen sizes. Both the mode I and II test data can be captured very accurately by Ba\v{z}ant's type II Size Effect Law (SEL)

Elastic, strength, and fracture properties of Marcellus shale

Shale, a fine-grained sedimentary rock, is the key source rock for many of the world's most important oil and natural gas deposits. A deep understanding of the mechanical properties of shale is of vital importance in various geotechnical applications, including oil and gas exploitation. In this work, deformability, strength, and fracturing properties of Marcellus shale were investigated through an experimental study. Firstly, uniaxial compression, direct tension, and Brazilian tests were performed on the Marcellus shale specimens in various bedding plane orientations with respect to loading directions to measure the static mechanical properties and their anisotropy. Furthermore, the deformability of Marcellus shale was also studied through seismic velocity measurements for comparison with the static measurements. The experimental results revealed that the transversely isotropic model is applicable for describing the elastic behaviors of Marcellus shale in pure tension and compression. The elastic properties measured from these two experiments, however, were not exactly the same. Strength results showed that differences exist between splitting (Brazilian) and direct tensile strengths, both of which varied with bedding plane orientations and loading directions and were associated with different failure modes. Finally, a series of three-point-bending tests were conducted on specimens of increasing size in three different principal notch orientations to investigate the fracture properties of the material. It was found that there exists a significant size effect on the fracture properties calculated from the measured peak loads and by using the Linear Elastic Fracture Mechanics (LEFM) theory. The fracture properties can be uniquely identified, however, by using Bazant's Size Effect Law and they were found to be anisotropic

Multiphysics Lattice Discrete Particle Modeling (M-LDPM) for the Simulation of Shale Fracture Permeability

A three-dimensional Multiphysics Lattice Discrete Particle Model (M-LDPM) framework is formulated to investigate the fracture permeability behavior of shale. The framework features a dual lattice system mimicking the mesostructure of the material and simulates coupled mechanical and flow behavior. The mechanical lattice model simulates the granular internal structure of shale and describes heterogeneous deformation by means of discrete compatibility and equilibrium equations. The network of flow lattice elements constitutes a dual graph of the mechanical lattice system. A discrete formulation of mass balance for the flow elements is formulated to model fluid flow along cracks. The overall computational framework is implemented with a mixed explicit-implicit integration scheme and a staggered coupling method that makes use of the dual lattice topology enabling the seamless two-way coupling of the mechanical and flow behaviors. The proposed model is used for the computational analysis of shale fracture permeability behavior by simulating triaxial direct shear tests on Marcellus shale specimens under various confining pressures. The simulated mechanical response is calibrated against the experimental data, and the predicted permeability values are also compared with the experimental measurements. Furthermore, the paper presents the scaling analysis of both the mechanical response and permeability measurements based on simulations performed on geometrically similar specimens with increasing size. The simulated stress-strain curves show a significant size effect in the post-peak due to the presence of localized fractures. The scaling analysis of permeability measurements enables prediction of permeability for large specimens by extrapolating the numerical results of small ones

Direct Testing of Gradual PostPeak Softening of Notched Specimens of Fiber Composites Stabilized by Enhanced Stiffness and Mass

Static and dynamic analysis of the fracture tests of fiber composites in hydraulically servo-controlled testing machines currently in use shows that their grips are much too soft and light for observing the postpeak softening. Based on static and dynamic analysis of the test setup, far stiffer and heavier grips are proposed. Tests of compact-tension fracture specimens of woven carbon-epoxy laminates prove this theoretical conclusion. Sufficiently stiff grips allow observation of a stable postpeak, even under load-point displacement control. Dynamic stability analysis further indicates that stable postpeak can be observed under CMOD control provided that a large mass is rigidly attached to the current soft grips. The fracture energy deduced from the area under the measured complete load-deflection curve with stable postpeak agrees closely with the fracture energy deduced from the size effect tests of the same composite. Previous suspicions of dynamic snapback in the testing of composites are dispelled. So is the previous view that fracture mechanics was inapplicable to the fiber-polymer composites

Propagation-based x-ray phase-contrast imaging with broad focus conventional x-ray sources

A propagation-based x-ray phase-contrast imaging (PBI) setup using a conventional x-ray source (LFF Cu target) is presented. A virtual x-ray source of 40 x 50 $\mu$$m^2$ was created by using, horizontally, a $6^o$ take-off angle (with the x-ray tube working in the line focus geometry) and, vertically, a 50 $\mu$m slit . The sample was set 12 $m$ from the source. Propagation-based x-ray phase-contrast (PB) image and conventional radiography (CR) of a polypropylene tube were acquired. Edge enhanced effects and a crack, not detected in CR, were clearly seen in the PB image. Contrast, visibility of the object edges and signal to noise ratio of the acquired images were exploited. The results show that PB images can be acquired by using normal focus (macro focus) conventional x-ray sources. This apparatus can be used as an standard phase-contrast imaging setup to analyze different kind of samples with large field of view (75 x 75 $mm^2$), discarding the use of translators for sample and detector.Comment: 8 pages, 3 figure