Anomalous proximity effects have been observed in adhesive systems ranging from proteins, bacteria, and gecko feet suspended over semiconductor surfaces to interfaces between graphene and different ...substrate materials. In the latter case, long-range forces are evidenced by measurements of non-vanishing stress that extends up to micrometer separations between graphene and the substrate. State-of-the-art models to describe adhesive properties are unable to explain these experimental observations, instead underestimating the measured stress distance range by 2-3 orders of magnitude. Here, we develop an analytical and numerical variational approach that combines continuum mechanics and elasticity with quantum many-body treatment of van der Waals dispersion interactions. A full relaxation of the coupled adsorbate/substrate geometry leads us to conclude that wavelike atomic deformation is largely responsible for the observed long-range proximity effect. The correct description of this seemingly general phenomenon for thin deformable membranes requires a direct coupling between quantum and continuum mechanics.
The Malliavin calculus is an extension of the classical calculus of variations from deterministic functions to stochastic processes. In this paper we aim to show in a practical and didactic way how ...to calculate the Malliavin derivative, the derivative of the expectation of a quantity of interest of a model with respect to its underlying stochastic parameters, for four problems found in mechanics. The non-intrusive approach uses the Malliavin Weight Sampling (MWS) method in conjunction with a standard Monte Carlo method. The models are expressed as ODEs or PDEs and discretised using the finite difference or finite element methods. Specifically, we consider stochastic extensions of; a 1D Kelvin-Voigt viscoelastic model discretised with finite differences, a 1D linear elastic bar, a hyperelastic bar undergoing buckling, and incompressible Navier-Stokes flow around a cylinder, all discretised with finite elements. A further contribution of this paper is an extension of the MWS method to the more difficult case of non-Gaussian random variables and the calculation of second-order derivatives. We provide open-source code for the numerical examples in this paper.
In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of ...derivative information amounts to a correction to the standard Monte Carlo estimation procedure that reduces the variance under certain conditions. We express the finite element models in variational form using the high-level Unified Form Language (UFL). We derive the tangent linear model automatically from this high-level description and use it to efficiently calculate the required derivative information. To study the effectiveness of the derivative-driven method we consider two stochastic PDEs; a one-dimensional Burgers equation with stochastic viscosity and a three-dimensional geometrically non-linear Mooney–Rivlin hyperelastic equation with stochastic density and volumetric material parameter. Our results show that for these problems the first-order derivative-driven Monte Carlo method is around one order of magnitude faster than the standard Monte Carlo method and at the cost of only one extra tangent linear solution per estimation problem. We find similar trends when comparing with a modern non-intrusive multi-level polynomial chaos expansion method. We parallelise the task of the repeated forward model evaluations across a cluster using the ipyparallel and mpi4py software tools. A complete working example showing the solution of the stochastic viscous Burgers equation is included as supplementary material.
•Derivative-driven Monte Carlo for fast propagation of uncertainty through models.•Automatic computation of the sensitivity derivative with FEniCS.•Comparison of the approach with multi-level polynomial chaos.•Superior performance of estimation for first and second order quantities.
•A general approach for the stochastic finite element analysis of soft tissue deformation is presented.•Sensitivity derivative Monte Carlo method to propagate uncertainty through hyperelastic models ...with stochastic parameters.•The proposed method is up to two orders of magnitude faster than the standard Monte Carlo approach.•The method is able to provide the user with statistical results on quantities of practical interest.•We applied the approach to a simple academic example and to the stochastic deformation of a brain.
We present a simple open-source semi-intrusive computational method to propagate uncertainties through hyperelastic models of soft tissues. The proposed method is up to two orders of magnitude faster than the standard Monte Carlo method. The material model of interest can be altered by adjusting few lines of (FEniCS) code. The method is able to (1) provide the user with statistical confidence intervals on quantities of practical interest, such as the displacement of a tumour or target site in an organ; (2) quantify the sensitivity of the response of the organ to the associated parameters of the material model. We exercise the approach on the determination of a confidence interval on the motion of a target in the brain. We also show that for the boundary conditions under consideration five parameters of the Ogden–Holzapfel-like model have negligible influence on the displacement of the target zone compared to the three most influential parameters. The benchmark problems and all associated data are made available as supplementary material.
Understanding complex materials at different length scales requires reliably accounting for van der Waals (vdW) interactions, which stem from long-range electronic correlations. While the important ...role of many-body vdW interactions has been extensively documented for the stability of materials, much less is known about the coupling between vdW interactions and atomic forces. Here we analyze the Hessian force response matrix for a single and two vdW-coupled atomic chains to show that a many-body description of vdW interactions yields atomic force response magnitudes that exceed the expected pairwise decay by 3-5 orders of magnitude for a wide range of separations between perturbed and observed atoms. Similar findings are confirmed for carbon nanotubes, graphene, and delamination of graphene from a silicon substrate previously studied experimentally. This colossal force enhancement suggests implications for phonon spectra, free energies, interfacial adhesion, and collective dynamics in materials with many interacting atoms.
The aim of this work is to characterize the mechanical parameters governing the in-plane behavior of human skin and, in particular, of a keloid-scar. We consider 2D hyperelastic bi-material model of ...a keloid and the surrounding healthy skin. The problem of finding the optimal model parameters that minimize the misfit between the model observations and the in vivo experimental measurements is solved using our in-house developed inverse solver that is based on the FEniCS finite element computational platform. The paper focuses on the model parameter sensitivity quantification with respect to the experimental measurements, such as the displacement field and reaction force measurements. The developed tools quantify the significance of different measurements on different model parameters and, in turn, give insight into a given model's ability to capture experimental measurements. Finally, an a priori estimate for the model parameter sensitivity is proposed that is independent of the actual measurements and that is defined in the whole computational domain. This estimate is primarily useful for the design of experiments, specifically, in localizing the optimal displacement field measurement sites for the maximum impact on model parameter inference.
•Framework for identifying the mechanical parameters of heterogeneous soft tissue.•Model parameter sensitivity quantification with respect to measurement uncertainty.•Prior estimate of model parameter sensitivity and localization of measurement sites.
A 3D numerical model with strong discontinuities implemented within the Enhanced Finite Element Method (E-FEM) is developed to address multi-cracking problems. Two failure criteria are proposed for ...two (tensile and shear) fracture kinematics; namely an anisotropic Mohr–Coulomb criterion with sliding and an anisotropic principal strain criterion with pure opening. This model is used to reproduce induced fracture networks around drifts after an underground excavation. A transversely isotropic behaviour is considered for the host rock. The influence of the anisotropy of rock properties and the in situ stress field on the induced fractures and the convergence of drifts are also studied.
Le stockage des déchets radioactifs dans des formations géologiques profondes nécessite d'excaver de la roche en grande profondeur pour accueillir les différentes installations. L'argilite du ...Callovo-Oxfordian a été choisie comme une potentielle formation hôte grâce à sa capacité de rétention et ses propriétés hydromécaniques. L'objectif de la thèse est d'étudier numériquement la fissuration induite lors du creusement des galeries souterraines à -490 m de profondeur. Un modèle Éléments Finis 3D (méthode E-FEM) est développé pour représenter la fissuration. Plusieurs critères sont proposés pour caractériser les fissures avec une ouverture en mode I ou un glissement en mode II. L'influence de l'anisotropie des propriétés de l'argilite et du champ de contrainte in situ sur les réseaux de fractures et la convergence des galeries est principalement étudiée. L'origine géologique de l'argilite, la complexité de sa microstructure et les grandes dimensions des galeries amènent à des incertitudes sur les propriétés hydromécaniques de ce matériau. La prise en compte des variabilités spatiales des paramètres mécaniques de la roche se fait au travers de champs aléatoires corrélés. Des Méthodes Éléments Finis probabilistes avec des formulations de Galerkin (méthodes d’intégration stochastique indirecte) sont ensuite développées avec une approche non intrusive pour propager ces incertitudes pour des systèmes linéaires et non linéaires (avec et sans fissuration). Ces méthodes sont ensuite appliquées à des problèmes d’excavation pour propager les incertitudes paramétriques associées au comportement de l'argilite. Les résultats sont comparés par rapport à ceux obtenus par des méthodes d’intégration stochastique directe (famille des méthodes dites de Monte-Carlo).
A 3D numerical modelling using the Enhanced Finite Element Method (E-FEM) is developped to address induced fracture networks around drifts after an excavation in Callovo-Oxfordian claystone (COx). A transversely isotropic behaviour is considered for the host rock and two failure criteria are proposed and implemented to characterize shear and tensile fractures: an anisotropic/isotropic Mohr Coulomb criterion with sliding (mode II) and an anisotropic principal strain criterion with mode I opening. Numerical simulations of underground excavations are presented and the results are discussed compared to in situ observations. The influence of the anisotropy of rock properties and in situ stress field on the induced fractures and the convergence of drifts are also studied. The geological origin of the COx as well as the large size of drifts lead to a large uncertainty related to hydro-mechanical properties of this material. Stochastic problems for linear and non linear systems are more and more of interest because it is an important issue to quantify uncertainties when parameters (loading, material properties) are modelled by correlated random fields. Despite a slow convergence, Monte Carlo methods are the most often used to solve these problems thanks to its easy implementation (non intrusive computation). Probabilistic Finite Element methods like Galerkin formulations which produce a coupled system are often viewed as intrusive. Here, we want to show that is possible to compute them in a non intrusive way and with the same accuracy as Monte Carlo methods which are considered as a reference. These methods are applied to take into account and to propagate parametric uncertainties related to claystone's behavior. 3D Numerical results are presented and discussed concerning the induced fracture networks around drifts and the anisotropic convergence of walls.
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