An analytic approach for the evolution of the static/flowing interface in viscoplastic granular flows

International audienceObserved avalanche flows of dense granular material have the property to present two possible behaviours: static (solid) or flowing (fluid). In such situation, an important challenge is to describe mathematically the evolution of the physical interface between the two phases. In this work we derive analytically a set of equations that is able to manage the dynamics of such interface, in the thin-layer regime where the flow is supposed to be thin compared to its downslope extension. It is obtained via an asymptotics starting from an incompressible viscoplastic model with Drucker-Prager yield stress, in which we have to make several assumptions. Additionally to the classical ones that are that the curvature of the topography, the width of the layer, and the viscosity are small, we assume that the internal friction angle is close to the slope angle (meaning that the friction and gravity forces compensate at leading order), the velocity is small (which is possible because of the previous assumption), and the pressure is convex with respect to the normal variable. This last assumption is for the stability of the double layer static/flowing configuration. A new higher-order non-hydrostatic nonlinear coupling term in the pressure allows us to close the asymptotic system. The resulting model takes the form of a formally overdetermined initial-boundary problem in the variable normal to the topography, set in the flowing region only. The extra boundary condition gives the information on how to evolve the static/flowing interface, and comes out from the continuity of the velocity and shear stress across it. The model handles arbitrary velocity profiles, and is therefore more general than depth-averaged models

Accounting for localized deformation: a simple computation of true stress in micropillar compression experiments

Compression experiments are widely used to study the mechanical properties of materials at micro- and nanoscale. However, the conventional engineering stress measurement method used in these experiments neglects to account for the alterations in the material's shape during loading. This can lead to inaccurate stress values and potentially misleading conclusions about the material's mechanical behavior especially in the case of localized deformation. To address this issue, we present a method for calculating true stress in cases of localized plastic deformation commonly encountered in experimental settings: (i) a single band and (ii) two bands oriented in arbitrary directions with respect to the vertical axis of the pillar (either in the same or opposite directions). Our simple analytic formulas can be applied to homogeneous and isotropic materials and crystals, requiring only standard data (displacement-force curve, aspect ratio, shear band angle and elastic strain limit) obtained from experimental results and eliminating the need for finite element computations. Our approach provides a more precise interpretation of experimental results and can serve as a valuable and simple tool in material design and characterization.Comment: arXiv admin note: text overlap with arXiv:2012.1278

Shell design from planar pre-stressed structures

The interplay between mechanics and geometry is used to construct a theoretical framework able to describe the class of three-dimensional objects that can be fabricated from suitable planar designs by using relaxation of pre-strains/stress in ultra-thin films. Small deformations and large rotations are used here to model the elastic relaxation into various three-dimensional shapes. Over the kinematics associated with the designed mid-surface, a small perturbation of Love–Kirchhoff type is considered in order to deduce the design plate-to-shell equations for orthotropic materials with an important pre-stress/strain heterogeneity. The resulting equations for the efforts average and efforts moments provide the supplementary equations to compute the in-plane pre-strain/stress. In particular, for materials with a weak material transversal heterogeneity the moments equations involve only the thickness, the curvature tensor, and the pre-strain/stress moments. Special attention is devoted to materials that can be obtained by layer-by-layer crystal growth (molecular beam epitaxy), which posses an in-plane isotropic pre-strain. We have found that a rectangular plate could relax both into a cylindrical surface or on a part of a sphere in which case it should have a small diameter with respect to the sphere radius. In both cases, the theoretical estimates have been compared with the experimental realizations and finite-element numerical computations and we found very good agreement among all of them. In addition, for the cone geometry we found that the design is not possible from an isotropic pre-stress with an in-plane homogeneity. However, the 3D finite-element computation of the relaxed surface with a (necessary) non-isotropic pre-stress obtained from the theoretical estimates matches remarkably well the designed conical surface. © The Author(s) 2020

Design of pre-stressed plate-strips to cover non-developable shells

In this paper we address the following design problem: what is the shape of a plate and the associated pre-stress that relaxes toward a given three-dimensional shell? As isometric transformations conserve the gaussian curvature, three-dimensional non-developable shells cannot be obtained from the relaxation of pre-strained plates by using isometric transformations only. Overcoming this geometric restriction, including small-strains and large rotations, solves the problem for small areas only. This paper dispenses with the small-area restriction to cover three-dimensional shells fully by using shell-strips. Since shell-strips have an additional geometric parameter, we show that under suitable assumptions that relate the width of the strip to the curvature of the shell, we are able to design arbitrary shell surfaces by covering them with shell-strips. As an illustration, we provide optimized covers of the sphere in a variety of different surface-strips relaxed from plate-strips with homogeneous and isotropic pre-stress. Moveover, we propose the design of the torus, of the helicoid and of the non-developable M\"obius band, which requires inhomogeneous and anisotropic pre-stress.Comment: Minor changes in Introduction, Sections 3 and 4, Conclusions and Reference

Instabilities in the antiplane problem with slip dependent friction

Quasi-static loading process and dynamic process modelling the interaction with slip displacement friction between an elastic body and a rigid body are considered. In these processes, stick-slip motions are related to the earthquake instabilities. On the contact interface we use the friction Coulomb law with a slip dependent friction coefficient in the case of a prescribed normal pressure. In quasi-static loading, the qualitative behaviour of the solution is decided by the competition between two parameters which involve the geometry, the normal stress, the elasticity properties and the dependence on the slip displacement of the friction coefficient. We explain how slow loads generate time discontinuities and space nonhomogeneities on the contact interface at a large time scale. In the dynamic process we provide a linear stability analysis: rapidly after the initial perturbation, the dominant exponential part governs the time and space evolution of the slip during the initiation phase. For large times, the slip velocity behaves qualitatively as it would be in the case of a propagating crack. Between these two phases a transition phase exists caracterised by an extremely high apparent velocity (supersonic) of the rupture front

The blocking of an inhomogeneous Bingham fluid. Applications to landslides

This work is concerned with the flow of a viscous plastic fluid. We choose a model of Bingham type taking into account inhomogeneous yield limit of the fluid, which is well-adapted in the description of landslides. After setting the general threedimensional problem, the blocking property is introduced. We then focus on necessary and sufficient conditions such that blocking of the fluid occurs. The anti-plane flow in twodimensional and onedimensional cases is considered. A variational formulation in terms of stresses is deduced. More fine properties dealing with local stagnant regions as well as local regions where the fluid behaves like a rigid body are obtained in dimension one

A Lagrangian DG-Method for Wave Propagation in a Cracked Solid with Frictional Contact Interfaces

We developed a discontinuous Galerkin (DG) numerical scheme for wave propagation in elastic solids with frictional contact interfaces. This type of numerical scheme is useful in investigations of wave propagation in elastic solids with micro-cracks (cracked solid) that involve modeling the damage in brittle materials or architected meta-materials. Only processes with mild loading that do not trigger crack fracture extension or the nucleation of new fractures are considered. The main focus lies on the contact conditions at crack surfaces, including provisions for crack opening and closure and stick-and-slip with Coulomb friction. The proposed numerical algorithm uses the leapfrog scheme for the time discretization and an augmented Lagrangian algorithm to solve the associated non-linear problems. For efficient parallelization, a Galerkin discontinuous method was chosen for the space discretization. The frictional interfaces (micro-cracks), where the numerical flux is obtained by solving non-linear and non-smooth variational problems, concerns only a limited number the degrees of freedom, implying a small additional computational cost compared to classical bulk DG schemes. The numerical method was tested through two model problems with analytical solutions. The proposed Lagrangian approach of the nonlinear interfaces had excellent results (stability and high accuracy) and only required a reasonable additional amount of computational effort. To illustrate the method, we conclude with some numerical simulations on the blast propagation in a cracked material and in a meta-material designed for shock dissipation

Domain decomposition method for dynamic faulting under slip-dependent friction

International audienceThe anti-plane shearing problem on a system of finite faults under a slip-dependent friction in a linear elastic domain is considered. Using a Newmark method for the time discretization of the problem, we have obtained an elliptic variational inequality at each time step. An upper bound for the time step size, which is not a CFL condition, is deduced from the solution uniqueness criterion using the first eigenvalue of the tangent problem. Finite element form of the variational inequality is solved by a Schwarz method assuming that the inner nodes of the domain lie in one subdomain and the nodes on the fault lie in other subdomains. Two decompositions of the domain are analyzed, one made up of two subdomains and another one with three subdomains. Numerical experiments are performed to illustrate convergence for a single time step (convergence of the Schwarz algorithm, influence of the mesh size, influence of the time step), convergence in time (instability capturing, energy dissipation, optimal time step) and an application to a relevant physical problem (interacting parallel fault segments)

A Lagrangian DG-Method for Wave Propagation in a Cracked Solid with Frictional Contact Interfaces

We developed a discontinuous Galerkin (DG) numerical scheme for wave propagation in elastic solids with frictional contact interfaces. This type of numerical scheme is useful in investigations of wave propagation in elastic solids with micro-cracks (cracked solid) that involve modeling the damage in brittle materials or architected meta-materials. Only processes with mild loading that do not trigger crack fracture extension or the nucleation of new fractures are considered. The main focus lies on the contact conditions at crack surfaces, including provisions for crack opening and closure and stick-and-slip with Coulomb friction. The proposed numerical algorithm uses the leapfrog scheme for the time discretization and an augmented Lagrangian algorithm to solve the associated non-linear problems. For efficient parallelization, a Galerkin discontinuous method was chosen for the space discretization. The frictional interfaces (micro-cracks), where the numerical flux is obtained by solving non-linear and non-smooth variational problems, concerns only a limited number the degrees of freedom, implying a small additional computational cost compared to classical bulk DG schemes. The numerical method was tested through two model problems with analytical solutions. The proposed Lagrangian approach of the nonlinear interfaces had excellent results (stability and high accuracy) and only required a reasonable additional amount of computational effort. To illustrate the method, we conclude with some numerical simulations on the blast propagation in a cracked material and in a meta-material designed for shock dissipation