Built on the framework of effective interaction potentials using lattice element method, a methodology to calibrate and to validate the elasticity of solid constituents in heterogeneous porous media from experimentally measured nanoindentation moduli and imported scans from advanced imaging techniques is presented. Applied to computed tomography (CT) scans of two organic-rich shales, spatial variations of effective interaction potentials prove instrumental in capturing the effective elastic behavior of highly heterogeneous materials via the first two cumulants of experimentally measured distributions of nanoindentation moduli. After calibration and validation steps while implicitly accounting for mesoscale texture effects via CT scans, Biot poroelastic coefficients are simulated. Analysis of stress percolation suggests contrasting pathways for load transmission, a reflection of microtextural differences in the studied cases. This methodology to calibrate elastic energy content of real materials from advanced imaging techniques and experimental measurements paves the way to study other phenomena such as wave propagation and fracture while providing a platform to fine-tune effective behavior of materials given advancements in additive manufacturing and machine learning algorithms .

VL - 13 UR - http://link.springer.com/10.1007/s11440-018-0687-9 IS - 6 JO - Acta Geotech. ER - TY - JOUR T1 - Disorder-induced stiffness degradation of highly disordered porous materials JF - Journal of the Mechanics and Physics of Solids Y1 - 2017 A1 - Hadrien Laubie A1 - Monfared, Siavash A1 - Farhang Radjaï A1 - Roland Jean-Marc Pellenq A1 - Franz-Josef Ulm AB -The effective mechanical behavior of multiphase solid materials is generally modeled by means of homogenization techniques that account for phase volume fractions and elastic moduli without considering the spatial distribution of the different phases. By means of extensive numerical simulations of randomly generated porous materials using the lattice element method, the role of local textural properties on the effective elastic properties of disordered porous materials is investigated and compared with different continuum micromechanics-based models. It is found that the pronounced disorder-induced stiffness degradation originates from stress concentrations around pore clusters in highly disordered porous materials. We identify a single disorder parameter, φ*sa*, which combines a measure of the spatial disorder of pores (the clustering index, *sa*) with the pore volume fraction (the porosity, φ) to scale the disorder-induced stiffness degradation. Thus, we conclude that the classical continuum micromechanics models with one spherical pore phase, due to their underlying homogeneity assumption fall short of addressing the clustering effect, unless additional texture information is introduced, e.g. in form of the shift of the percolation threshold with disorder, or other functional relations between volume fractions and spatial disorder; as illustrated herein for a differential scheme model representative of a two-phase (solid–pore) composite model material.

VL - 106 ER - TY - JOUR T1 - Effective Potentials and Elastic Properties in the Lattice-Element Method: Isotropy and Transverse Isotropy JF - Journal of Nanomechanics and Micromechanics Y1 - 2017 A1 - Hadrien Laubie A1 - Monfared, Siavash A1 - Farhang Radjaï A1 - Roland Jean-Marc Pellenq A1 - Franz-Josef Ulm AB -

Lattice approaches have emerged as a powerful tool to capture the effective mechanical behavior of heterogeneous materials using harmonic interactions inspired from beam-type stretch and rotational interactions between a discrete number of mass points. In this paper, the lattice element method (LEM) is reformulated within the conceptual framework of empirical force fields employed at the lattice scale. Within this framework, because classical harmonic formulations are but a Taylor expansion of nonharmonic potential expressions, they can be used to model both the linear and the nonlinear response of discretized material systems. Specifically, closed-form calibration procedures for such interaction potentials are derived for both the isotropic and the transverse isotropic elastic cases on cubic lattices, in the form of linear relations between effective elasticity properties and energy parameters that define the interactions. The relevance of the approach is shown by an application to the classical Griffith crack problem. In particular, it is shown that continuum-scale quantities of linear-elastic fracture mechanics, such as stress intensity factors (SIFs), are well captured by the method, which by its very discrete nature removes geometric discontinuities that provoke stress singularities in the continuum case. With its strengths and limitations thus defined, the proposed LEM is well suited for the study of multiphase materials whose microtextural information is obtained by, e.g., X-ray micro-computed tomography. (c) 2017 American Society of Civil Engineers.

VL - 7 UR - http://ascelibrary.org/doi/10.1061/%28ASCE%29NM.2153-5477.0000125 IS - 3 JO - J. Nanomech. Micromech. ER - TY - JOUR T1 - Mesoscale Poroelasticity of Heterogeneous Media JF - Journal of Nanomechanics and Micromechanics Y1 - 2017 A1 - Monfared, Siavash A1 - Hadrien Laubie A1 - Farhang Radjaï A1 - Roland Jean-Marc Pellenq A1 - Franz-Josef Ulm AB -The poromechanics of heterogeneous media is reformulated in a discrete framework using the lattice element method (LEM) that accounts for the presence of interfaces as well as local microtextural and elastic variations. The exchange of mechanical information between pore and solid(s) is captured by means of force field potentials for these domains, which eliminate the requirement of scale separability of continuum-based poromechanics approaches. In congruence with mu VT and NPT ensembles of statistical mechanics, discrete expressions for Biot poroelastic coefficients are derived. Considering harmonic-type interaction potentials for each link, analytical expressions for both isotropic and transversely isotropic effective elasticity are presented. The theory is validated against continuum-based expressions of Biot poroelastic coefficients for porous media with isotropic and transversely isotropic elastic solid behavior. (C) 2017 American Society of Civil Engineers.

VL - 7 UR - http://ascelibrary.org/doi/10.1061/%28ASCE%29NM.2153-5477.0000136 IS - 4 JO - J. Nanomech. Micromech. ER - TY - JOUR T1 - A potential-of-mean-force approach for fracture mechanics of heterogeneous materials using the lattice element method JF - Journal of the Mechanics and Physics of Solids Y1 - 2017 A1 - Hadrien Laubie A1 - Farhang Radjaï A1 - Roland Jean-Marc Pellenq A1 - Franz-Josef Ulm AB -Fracture of heterogeneous materials has emerged as a critical issue in many engineering applications, ranging from subsurface energy to biomedical applications, and requires a rational framework that allows linking local fracture processes with global fracture descriptors such as the energy release rate, fracture energy and fracture toughness. This is achieved here by means of a local and a global potential-of-mean-force (PMF) inspired Lattice Element Method (LEM) approach. In the local approach, fracture-strength criteria derived from the effective interaction potentials between mass points are shown to exhibit a scaling commensurable with the energy dissipation of fracture processes. In the global PMF-approach, fracture is considered as a sequence of equilibrium states associated with minimum potential energy states analogous to Griffith’s approach. It is found that this global approach has much in common with a Grand Canonical Monte Carlo (GCMC) approach, in which mass points are randomly removed following a maximum dissipation criterion until the energy release rate reaches the fracture energy. The duality of the two approaches is illustrated through the application of the PMF-inspired LEM for fracture propagation in a homogeneous linear elastic solid using different means of evaluating the energy release rate. Finally, by application of the method to a textbook example of fracture propagation in a heterogeneous material, it is shown that the proposed PMF-inspired LEM approach captures some well-known toughening mechanisms related to fracture energy contrast, elasticity contrast and crack deflection in the considered two-phase layered composite material.

VL - 105 JO - Journal of the Mechanics and Physics of Solids ER - TY - JOUR T1 - Stress Transmission and Failure in Disordered Porous Media JF - PHYSICAL REVIEW LETTERS Y1 - 2017 A1 - Hadrien Laubie A1 - Farhang Radjaï A1 - Roland Jean-Marc Pellenq A1 - et al AB -By means of extensive lattice-element simulations, we investigate stress transmission and its relation with failure properties in increasingly disordered porous systems. We observe a non-Gaussian broadening of stress probability density functions under tensile loading with increasing porosity and disorder, revealing a gradual transition from a state governed by single-pore stress concentration to a state controlled by multipore interactions and metric disorder. This effect is captured by the excess kurtosis of stress distributions and shown to be nicely correlated with the second moment of local porosity fluctuations, which appears thus as a (dis)order parameter for the system. By generating statistical ensembles of porous textures with varying porosity and disorder, we derive a general expression for the fracture stress as a decreasing function of porosity and disorder. Focusing on critical sites where the local stress is above the global fracture threshold, we also analyze the transition to failure in terms of a coarse-graining length. These findings provide a general framework which can also be more generally applied to multiphase and structural heterogeneous materials.

VL - 119 IS - 7 ER - TY - JOUR T1 - Stress Transmission and Failure in Disordered Porous Media JF - Physical Review Letters Y1 - 2017 A1 - Hadrien Laubie A1 - Farhang Radjaï A1 - Roland Jean-Marc Pellenq A1 - Franz-Josef Ulm AB -

By means of extensive lattice-element simulations, we investigate stress transmission and its relation with failure properties in increasingly disordered porous systems. We observe a non-Gaussian broadening of stress probability density functions under tensile loading with increasing porosity and disorder, revealing a gradual transition from a state governed by single-pore stress concentration to a state controlled by multipore interactions and metric disorder. This effect is captured by the excess kurtosis of stress distributions and shown to be nicely correlated with the second moment of local porosity fluctuations, which appears thus as a (dis)order parameter for the system. By generating statistical ensembles of porous textures with varying porosity and disorder, we derive a general expression for the fracture stress as a decreasing function of porosity and disorder. Focusing on critical sites where the local stress is above the global fracture threshold, we also analyze the transition to failure in terms of a coarse-graining length. These findings provide a general framework which can also be more generally applied to multiphase and structural heterogeneous materials.

VL - 119 UR - https://link.aps.org/doi/10.1103/PhysRevLett.119.075501 IS - 7 JO - Phys. Rev. Lett. ER - TY - JOUR T1 - Capturing material toughness by molecular simulation: accounting for large yielding effects and limits JF - International Journal of Fracture Y1 - 2015 A1 - Brochard, Laurent A1 - György Hantal A1 - Hadrien Laubie A1 - Franz-Josef Ulm A1 - Roland Jean-Marc Pellenq A1 - Benoit A. Coasne AB -The inherent computational cost of molecular simulations limits their use to the study of nanometric systems with potentially strong size effects. In the case of fracture mechanics, size effects due to yielding at the crack tip can affect strongly the mechanical response of small systems. In this paper we consider two examples: a silica crystal for which yielding is limited to a few atoms at the crack tip, and a nanoporous polymer for which the process zone is about one order of magnitude larger. We perform molecular simulations of fracture of those materials and investigate in particular the system and crack size effects. The simulated systems are periodic with an initial crack. Quasi-static loading is achieved by increasing the system size in the direction orthogonal to the crack while maintaining a constant temperature. As expected, the behaviors of the two materials are significantly different. We show that the behavior of the silica crystal is reasonably well described by the classical framework of linear elastic fracture mechanics (LEFM). Therefore, one can easily upscale engineering fracture properties from molecular simulation results. In contrast, LEFM fails capturing the behavior of the polymer and we propose an alternative analysis based on cohesive crack zone models. We show that with a linear decreasing cohesive law, this alternative approach captures well the behavior of the polymer. Using this cohesive law, one can anticipate the mechanical behavior at larger scale and assess engineering fracture properties. Thus, despite the large yielding of the polymer at the scale of the molecular simulation, the cohesive zone analysis offers a proper upscaling methodology.

VL - 194 IS - 2 JO - Int J Fract ER - TY - JOUR T1 - Fracture toughness of calcium–silicate–hydrate from molecular dynamics simulations JF - Journal of Non-Crystalline Solids Y1 - 2015 A1 - Mathieu Bauchy A1 - Hadrien Laubie A1 - Mohammad Javad Abdolhosseini Qomi A1 - Christian G. Hoover A1 - Franz-Josef Ulm A1 - Roland Jean-Marc Pellenq AB -Concrete is the most widely manufactured material in the world. Its binding phase, calcium–silicate–hydrate (C–S–H), is responsible for its mechanical properties and has an atomic structure fairly similar to that of usual calcium silicate glasses, which makes it appealing to study this material with tools and theories traditionally used for non-crystalline solids. Here, following this idea, we use molecular dynamics simulations to evaluate the fracture toughness of C–S–H, inaccessible experimentally. This allows us to discuss the brittleness of the material at the atomic scale. We show that, at this scale, C–S–H breaks in a ductile way, which prevents one from using methods based on linear elastic fracture mechanics. Knowledge of the fracture properties of C–S–H at the atomic scale opens the way for an upscaling approach to the design of tougher cement paste, which would allow for the design of slender environment-friendly infrastructures, requiring less material.

VL - 419 JO - Journal of Non-Crystalline Solids ER - TY - JOUR T1 - Atomic-scale modelling of elastic and failure properties of clays JF - Molecular Physics Y1 - 2014 A1 - György Hantal A1 - Brochard, Laurent A1 - Hadrien Laubie A1 - Ebrahimi, Davoud A1 - Roland Jean-Marc Pellenq A1 - Franz-Josef Ulm A1 - Benoit A. Coasne KW - clay KW - elastic properties KW - fracture KW - reactive molecular simulation AB -The elastic and failure properties of a typical clay, illite, are investigated using molecular simulation. We employ a reactive (ReaxFF) and a non-reactive (ClayFF) force field to assess the elastic properties of the clay. As far as failure is concerned, ReaxFF was used throughout the study; however, some calculations were also performed with ClayFF. A crack parallel to the clay layers is found to have low fracture resistance when submitted to a tensile loading perpendicular to the crack. The mechanism of both yield and fracture failures is decohesion in the interlayer space. In contrast, under shear loading, the nanoscale failure mechanism is a stick-slip between clay layers. No fracture propagation is observed as the clay layers slide on top of each other. The low fracture resistance in mode I and the stick-slip failure in mode II are both the consequence of the lack of chemical bonds between clay layers where the cohesion is provided by non-covalent interactions. This work, which provides a description of the failure of clays at the microscopic scale, is the first step towards describing the failure of clays at a larger scale where the polycrystalline distribution of clay grains must be taken into account.

VL - 112 IS - 9-10 ER - TY - JOUR T1 - Irwin׳s conjecture: Crack shape adaptability in transversely isotropic solids JF - Journal of the Mechanics and Physics of Solids Y1 - 2014 A1 - Hadrien Laubie A1 - Franz-Josef Ulm AB -The planar crack propagation problem of a flat elliptical crack embedded in a brittle elastic anisotropic solid is investigated. We introduce the concept of crack shape adaptability: the ability of three-dimensional planar cracks to shape with the mechanical properties of a cracked body. A criterion based on the principle of maximum dissipation is suggested in order to determine the most stable elliptical shape. This criterion is applied to the specific case of vertical cracks in transversely isotropic solids. It is shown that contrary to the isotropic case, the circular shape (i.e. penny-shaped cracks) is not the most stable one. Upon propagation, the crack first grows non-self-similarly before it reaches a stable shape. This stable shape can be approximated by an ellipse of an aspect ratio that varies with the degree of elastic anisotropy. By way of example, we apply the so-derived crack shape adaptability criterion to shale materials. For this class of materials it is shown that once the stable shape is reached, the crack propagates at a higher rate in the horizontal direction than in the vertical direction. We also comment on the possible implications of these findings for hydraulic fracturing operations.

VL - 68 JO - Journal of the Mechanics and Physics of Solids ER - TY - JOUR T1 - Plane-Strain Crack Problem in Transversely Isotropic Solids for Hydraulic Fracturing Applications JF - Journal of Engineering Mechanics Y1 - 2014 A1 - Hadrien Laubie A1 - Franz-Josef Ulm AB -This paper aims at understanding and predicting how pressurized cracks propagate in anisotropic brittle solids, a situation frequently encountered in hydraulic fracturing. Special attention is paid to transverse isotropy, often used to model shale. Although the theory of linear elastic fracture mechanics of anisotropic solids is well established at present, this paper shows that the application of Muskhelishvili’s formalism to Lekhnitskii’s anisotropic complex potentials provides a powerful tool to extend the validity of the classical tools of isotropic fluid-driven crack models to the anisotropic case, provided that the appropriate elastic constants are used. These elastic constants are identified and derived in closed form for transversely isotropic solids. The constants are shown to be directly related to quantities easily measured in a laboratory at macroscopic scale through indentation tests and acoustic measurements. Moreover, several crack-kinking criteria are compared. Contrary to the isotropic case, the crack-kinking criteria are not consistent among themselves, even in the case of a pure pressure loading. The orientation at which it is easier to propagate an already existing crack is sought. A critical crack length, below which this crack orientation is the one of minimal stiffness felt by the crack, is identified.

VL - 140 IS - 12 JO - J. Eng. Mech. ER -