The relationship between viscoelasticity and elasticity

Soft materials that are subjected to large deformations exhibit an extremely rich phenomenology, with properties lying in between those of simple fluids and those of elastic solids. In the continuum description of these systems, one typically follows either the route of solid mechanics (Lagrangian description) or the route of fluid mechanics (Eulerian description). The purpose of this review is to highlight the relationship between the theories of viscoelasticity and of elasticity, and to leverage this connection in contemporary soft matter problems. We review the principles governing models for viscoelastic liquids, for example solutions of flexible polymers. Such materials are characterized by a relaxation time λ, over which stresses relax. We recall the kinematics and elastic response of large deformations, and show which polymer models do (and which do not) correspond to a nonlinear elastic solid in the limit λ → ∞. With this insight, we split the work done by elastic stresses into reversible and dissipative parts, and establish the general form of the conservation law for the total energy. The elastic correspondence can offer an insightful tool for a broad class of problems; as an illustration, we show how the presence or absence of an elastic limit determines the fate of an elastic thread during capillary instability

Mini-Workshop: The Mathematics of Electro-Active Smart Materials

[no abstract available

A GPU-accelerated package for simulation of flow in nanoporous source rocks with many-body dissipative particle dynamics

Mesoscopic simulations of hydrocarbon flow in source shales are challenging, in part due to the heterogeneous shale pores with sizes ranging from a few nanometers to a few micrometers. Additionally, the sub-continuum fluid-fluid and fluid-solid interactions in nano- to micro-scale shale pores, which are physically and chemically sophisticated, must be captured. To address those challenges, we present a GPU-accelerated package for simulation of flow in nano- to micro-pore networks with a many-body dissipative particle dynamics (mDPD) mesoscale model. Based on a fully distributed parallel paradigm, the code offloads all intensive workloads on GPUs. Other advancements, such as smart particle packing and no-slip boundary condition in complex pore geometries, are also implemented for the construction and the simulation of the realistic shale pores from 3D nanometer-resolution stack images. Our code is validated for accuracy and compared against the CPU counterpart for speedup. In our benchmark tests, the code delivers nearly perfect strong scaling and weak scaling (with up to 512 million particles) on up to 512 K20X GPUs on Oak Ridge National Laboratory's (ORNL) Titan supercomputer. Moreover, a single-GPU benchmark on ORNL's SummitDev and IBM's AC922 suggests that the host-to-device NVLink can boost performance over PCIe by a remarkable 40\%. Lastly, we demonstrate, through a flow simulation in realistic shale pores, that the CPU counterpart requires 840 Power9 cores to rival the performance delivered by our package with four V100 GPUs on ORNL's Summit architecture. This simulation package enables quick-turnaround and high-throughput mesoscopic numerical simulations for investigating complex flow phenomena in nano- to micro-porous rocks with realistic pore geometries

Reference map technique for finite-strain elasticity and fluid-solid interaction

The reference map, defined as the inverse motion function, is utilized in an Eulerian-frame representation of continuum solid mechanics, leading to a simple, explicit finite-difference method for solids undergoing finite deformations. We investigate the accuracy and applicability of the technique for a range of finite-strain elasticity laws under various geometries and loadings. Capacity to model dynamic, static, and quasi-static conditions is shown. Specifications of the approach are demonstrated for handling irregularly shaped and/or moving boundaries, as well as shock solutions. The technique is also integrated within a fluid–solid framework using a level-set to discern phases and using a standard explicit fluid solver for the fluid phases. We employ a sharp-interface method to institute the interfacial conditions, and the resulting scheme is shown to efficiently capture fluid–solid interaction solutions in several examples.United States. Dept. of Energy. Office of Science (Computational and Technology Research, contract number DE-AC02-05CH11231)National Science Foundation (U.S.) (Grant DMS-0813648)National Science Foundation (U.S.) (Mathematical Sciences Postdoctoral Research Fellowship)Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Program)

Multiscale modeling of segregation in granular flows

Modeling and simulation of segregation phenomena in granular flows are investigated. Computational models at different scales ranging from particle level (microscale) to continuum level (macroscale) are employed in order to determine the important microscale physics relevant to macroscale modeling. The capability of a multi-fluid model to capture segregation caused by density difference is demonstrated by simulating grain-chaff biomass flows in a laboratory-scale air column and in a combine harvester. The multi-fluid model treats gas and solid phases as interpenetrating continua in an Eulerian frame. This model is further improved by incorporating particle rotation using kinetic theory for rapid granular flow of slightly frictional spheres. A simplified model is implemented without changing the current kinetic theory framework by introducing an effective coefficient of restitution to account for additional energy dissipation due to frictional collisions. The accuracy of predicting segregation rate in a gas-fluidized bed is improved by the implementation. This result indicates that particle rotation is important microscopic physics to be incorporated into the hydrodynamic model. Segregation of a large particle in a dense granular bed of small particles under vertical vibration is studied using molecular dynamics simulations. Wall friction is identified as a necessary condition for the segregation. Large-scale force networks bearing larger-than-average forces are found with the presence of wall friction. The role of force networks in assisting rising of the large particle is analyzed. Single-point force distribution and two-point spatial force correlation are computed. The results show the heterogeneity of forces and a short-range correlation. The short correlation length implies that even dense granular flows may admit local constitutive relations. A modified minimum spanning tree (MST) algorithm is developed to asymptotically recover the force statistics in the force networks. This algorithm provides a possible route to constructing a continuum model with microstructural information supplied from it. Microstructures in gas fluidized beds are also analyzed using a hybrid method, which couples the discrete element method (DEM) for particle dynamics with the averaged two-fluid (TF) equations for the gas phase. Multi-particle contacts are found in defluidized regions away from bubbles in fluidized beds. The multi-particle contacts invalidate the binary-collision assumption made in the kinetic theory of granular flows for the defluidized regions. Large ratios of contact forces to drag forces are found in the same regions, which confirms the relative importance of contact forces in determining particle dynamics in the defluidized regions

Numerical Simulations of Bubble Dynamics Near Viscoelastic Media

Cavitation-induced damage occurs in a wide range of applications, including in naval hydrodynamics, medicine, and the Spallation Neutron Source. Local transient pressure decreases in liquid flow may give rise to explosive bubble growth and violent collapse, with shock waves produced at collapse interacting with neighboring solids. Although the mechanisms of erosion to hard, metallic solids can be predicted in relatively simple geometries, damage to soft materials (e.g., elastomeric coatings, soft tissue) or in confined geometries is less well understood. In such problems, the constitutive models describing the medium are non-trivial and include effects such as (nonlinear) elasticity, history (relaxation effects) and viscosity. As a result, the influence of the shock on the material and the response of the material to the shock are poorly understood. To gain fundamental insights into cavitation-induced damage to both soft objects and rigid materials, we develop a novel Eulerian approach for numerical simulations of wave propagation in heterogeneous viscoelastic media. We extend the five-equations multiphase, interface-capturing model, based on the idea that all the materials (gases, liquids, solids) obey the same equation of state with spatially varying properties, to incorporate the desired constitutive relation. We consider problems in which the deformations are small, such that the substances can be described by linear viscoelastic constitutive relations. One difficulty is the calculation of strains in an Eulerian framework, which we address by using a hypoelastic model in which an objective time derivative (Lie derivative) of the constitutive relation is taken to evolve strain rates. The resulting numerical framework is a solution-adaptive, high-order interface-capturing approach for compressible, multiphase flows involving linear viscoelasticity at all speeds. We then utilize this numerical framework to gain fundamental insights into cavitation damage (i) in a confined geometry, (ii) in shock wave lithotripsy, and (iii) to rigid objects covered by an elastomeric coating. We examine the maximum stresses, pressures, and temperatures along/in rigid/compliant objects. We quantify the effect of confinement on an inertially collapsing bubble and determine the appropriate scaling governing the maximum pressures can predicted on the surfaces. We investigate the stresses produced to a model kidney stone due to a shock wave by examining the amplification of tensile stresses in the stone when a gas bubble is present. The impact loads on a polymeric coating relevant to naval engineering applications by shock-induced bubble collapse indicate how pitting and coating material ejection may take place by repeated cavitation events near the surface.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147536/1/mrdz_1.pd

An Eulerian projection method for quasi-static elastoplasticity

A well-established numerical approach to solve the Navier--Stokes equations for incompressible fluids is Chorin's projection method, whereby the fluid velocity is explicitly updated, and then an elliptic problem for the pressure is solved, which is used to orthogonally project the velocity field to maintain the incompressibility constraint. In this paper, we develop a mathematical correspondence between Newtonian fluids in the incompressible limit and hypo-elastoplastic solids in the slow, quasi-static limit. Using this correspondence, we formulate a new fixed-grid, Eulerian numerical method for simulating quasi-static hypo-elastoplastic solids, whereby the stress is explicitly updated, and then an elliptic problem for the velocity is solved, which is used to orthogonally project the stress to maintain the quasi-staticity constraint. We develop a finite-difference implementation of the method and apply it to an elasto-viscoplastic model of a bulk metallic glass based on the shear transformation zone theory. We show that in a two-dimensional plane strain simple shear simulation, the method is in quantitative agreement with an explicit method. Like the fluid projection method, it is efficient and numerically robust, making it practical for a wide variety of applications. We also demonstrate that the method can be extended to simulate objects with evolving boundaries. We highlight a number of correspondences between incompressible fluid mechanics and quasi-static elastoplasticity, creating possibilities for translating other numerical methods between the two classes of physical problems.Comment: 49 pages, 20 figure

Soft body impact modeling and development of a suitable meshless approach

Cette thèse présente des travaux récents relatifs à la modélisation d'impact de projectiles mous et le développement d'une méthode numérique sans maillage. En premier lieu, la théorie rattachée aux impacts d'oiseaux ainsi que les méthodes numériques et résultats expérimentaux disponibles sont donnés afin d'établir des normes pour les simulations d'impact d'oiseaux. Les connaissances générales concernant les impacts d'oiseaux sont ensuite améliorées par des tests récents qui utilisent un substitut pour l' oiseau. Une recette pour le substitut est donnée afin de servir de référence dans les procédures de certification pour les impacts d'oiseaux. Les résultats sont également fournis afin de valider les modèles numériques et promouvoir l'utilisation des outils numériques dans le design de structures aéronautiques ainsi que dans le processus de certification. Les détails du montage expérimental sont donnés ainsi qu'une analyse de la précision des résultats obtenus et quelques sources d'erreurs à éviter dans l'éventualité où d'autres tests auraient lieu. Finalement, la méthode sans maillage smoothed particles hydrodynamics (SPH) est modifiée afin qu'un algorithme maison puisse traiter le problème d'impact d'oiseaux. Afin de rencontrer cet objectif, des améliorations sont apportées à la formulation mathématique afin de traiter les problèmes d'instabilités numériques rapportés dans la littérature. Ensuite, des lois de comportement et des équations d'état ont été ajoutées. L'algorithme résultant peut être utilisé pour plusieurs types de problèmes, ce qui rend la méthode SPH très attrayantes pour les simulations numériques avec grandes déformations

Fluid-fluid phase separation in a soft porous medium

Various biological and chemical processes lead to the nucleation and growth of non-wetting fluid bubbles within the pore space of a granular medium, such as the formation of gas bubbles in liquid-saturated lake-bed sediments. In sufficiently soft porous materials, the non-wetting nature of these bubbles can result in the formation of open cavities within the granular solid skeleton. Here, we consider this process through the lens of phase separation, where thermomechanics govern the separation of the non-wetting phase from a fluid-fluid-solid mixture. We construct a phase-field model informed by large-deformation poromechanics, in which two immiscible fluids interact with a poroelastic solid skeleton. Our model captures the competing effects of elasticity and fluid-fluid-solid interactions. We use a phase-field damage model to capture the mechanics of the granular solid. As a model problem, we consider an initial distribution of non-wetting fluid in the pore space that separates into multiple cavities. We use simulations and linear-stability analysis to identify the key parameters that control phase separation, the conditions that favour the formation of cavities, and the characteristic size of the resulting cavities