A mathematical model for an upper convected Maxwell fluid with an elastic core: Study of a limiting case

In this paper we formulate a mathematical model for a continuum which behaves like an upper convected visco-elastic Maxwell fluid if the stress is above a certain threshold and like a neo-Hookean elastic solid if the stress is below that threshold. The constitutive equations for each phase are derived within the context of the theory of natural configurations and by means of the criterion of the maximization of the rate of dissipation [11]. We then focus on a limiting case in which the continuum becomes an elastic-rigid body. In this limiting case the constitutive relation of the material becomes implicit and, although there is no energy dissipation, it cannot be included in the class of hyperelastic (or Green) bodies. The stress indeed cannot be expressed as a function of the strain. This class of materials was first introduced by Rajagopal in [15] and is the subject of the forthcoming papers [3] and [4].     

Full Eulerian simulations of biconcave neo-Hookean particles in a Poiseuille flow

For a given initial configuration of a multi-component geometry represented by voxel-based data on a fixed Cartesian mesh, a full Eulerian finite difference method facilitates solution of dynamic interaction problems between Newtonian fluid and hyperelastic material. The solid volume fraction, and the left Cauchy–Green deformation tensor are temporally updated on the Eulerian frame, respectively, to distinguish the fluid and solid phases, and to describe the solid deformation. The simulation method is applied to two- and three-dimensional motions of two biconcave neo-Hookean particles in a Poiseuille flow. Similar to the numerical study on the red blood cell motion in a circular pipe (Gong et al. in J Biomech Eng 131:074504, 2009), in which Skalak’s constitutive laws of the membrane are considered, the deformation, the relative position and orientation of a pair of particles are strongly dependent upon the initial configuration. The increase in the apparent viscosity is dependent upon the developed arrangement of the particles. The present Eulerian approach is demonstrated that it has the potential to be easily extended to larger system problems involving a large number of particles of complicated geometries.     

A framework for coupled deformation–diffusion analysis with application to degradation/healing

This paper deals with the formulation and numerical implementation of a fully coupled continuum model for deformation–diffusion in linearized elastic solids. The mathematical model takes into account the effect of the deformation on the diffusion process, and the affect of the transport of an inert chemical species on the deformation of the solid. We then present a robust computational framework for solving the proposed mathematical model, which consists of coupled non‐linear partial differential equations. It should be noted that many popular numerical formulations may produce unphysical negative values for the concentration, particularly, when the diffusion process is anisotropic. The violation of the non‐negative constraint by these numerical formulations is not mere numerical noise. In the proposed computational framework, we employ a novel numerical formulation that will ensure that the concentration of the diffusant be always snon‐negative, which is one of the main contributions of this paper. Representative numerical examples are presented to show the robustness, convergence, and performance of the proposed computational framework. Another contribution of this paper is to systematically study the affect of transport of the diffusant on the deformation of the solid and vice versa, and their implication in modeling degradation/healing of materials. We show that the coupled response is both qualitatively and quantitatively different from the uncoupled response. Copyright © 2011 John Wiley & Sons, Ltd.     

Mechanical modeling of incompressible particle-reinforced neo-Hookean composites based on numerical homogenization

In this paper, the mechanical response of incompressible particle-reinforced neo-Hookean composites (IPRNC) under general finite deformations is investigated numerically. Three-dimensional Representative Volume Element (RVE) models containing 27 non-overlapping identical randomly distributed spheres are created to represent neo-Hookean composites consisting of incompressible neo-Hookean elastomeric spheres embedded within another incompressible neo-Hookean elastomeric matrix. Four types of finite deformation (i.e., uniaxial tension, uniaxial compression, simple shear and general biaxial deformation) are simulated using the finite element method (FEM) and the RVE models with periodic boundary condition (PBC) enforced. The simulation results show that the overall mechanical response of the IPRNC can be well-predicted by another simple incompressible neo-Hookean model up to the deformation the FEM simulation can reach. It is also shown that the effective shear modulus of the IPRNC can be well-predicted as a function of both particle volume fraction and particle/matrix stiffness ratio, using the classical linear elastic estimation within the limit of current FEM software.     

A coupled discrete element‐finite difference approach for modeling mechanical response of fluid‐saturated porous materials

A model of fluid‐saturated poroelastic medium was developed based on a combination of the discrete element method and grid method. The developed model adequately accounts for the deformation, fracture, and multiscale internal structure of a porous solid skeleton. The multiscale porous structure is taken into account implicitly by assigning the porosity and permeability values for the enclosing skeleton, which determine the rate of filtration of a fluid. Macroscopic pores and voids are taken into account explicitly by specifying the computational domain geometry. The relationship between the stress–strain state of the solid skeleton and pore fluid pressure is described in the approximations of simply deformable discrete element and Biot's model of poroelasticity. The developed model was applied to study the mechanical response of fluid‐saturated samples of brittle material. Based on simulation results, we constructed a generalized logistic dependence of uniaxial compressive strength on loading rate, mechanical properties of fluid and enclosing skeleton, and on sample dimensions. The logistic form of the generalized dependence of strength of fluid‐saturated elastic–brittle porous materials is due to the competition of two interrelated processes, such as pore fluid pressure increase under solid skeleton compression and fluid outflow from the enclosing skeleton to the environment. Copyright © 2015 John Wiley & Sons, Ltd.     

Oscillatory torsional flow of a viscoelastic solid

 We show that a viscoelastic solid, modelled by a three-dimensional analogue of the Kelvin-Meyer-Voigt equation (the neo-Hookean rubber-like body and the Oldroyd-B element in parallel), has an exact similarity solution in the torsional flow geometry, including inertia. Numerical results obtained by a finite difference method for the oscillatory torsional flow indicate that the flow loses its stability at high amplitudes of oscillations. This is partly explained by a linear stability analysis of a simpler flow generated by a constant angular displacement. Received 10 December 2001 / Accepted 12 March 2002     

Thermomechanics of the joint–synovial fluid system under periodic and impulse actions

The thermomechanical reaction of the joint–synovial fluid system to periodic and impulse actions has been investigated. The behavior of the body of the joint near the articular surface — resistance of the bone material of the joint to periodic mechanical actions — has been established. It has been shown that the most destructive actions for the joint are instantaneous impulse shock mechanical actions.     

Propagation of waves in a fluid-saturated porous elastic solid

A theory is developed for the propagation of waves in a porous elastic solid containing a compressible viscous fluid using a homogenization process. The matrix is a lattice of periodically distributed gaps of arbitrary shape, the period of the lattice being small compared with the wave length. The present treatment is concerned with materials where fluid and solid are of comparable densities. Two cases are considered: the situation in which the pores are connected and that in which they are not. When pores are closed, the bulk medium behaves like an elastic medium; when they are connected, the fluid filtration and the bulk deformation are coupled. Boundary conditions, for macroscopic variables, at the interface between such a porous medium and the adjacent free flow are given.     

Structure and properties of nylon 6–clay nanocomposites: effect of temperature and reprocessing

Melt-compounding is a technique which has been commonly used for producing polymer–clay nanocomposites with enhanced mechanical, thermal, and physical properties. Twin-screw extruders have been found to effectively exfoliate the clay platelets due to their high shear intensity. However, concerns about polymer and organoclay degradation have been raised in some studies. In this investigation, a composite of nylon 6–Cloisite 30B with fully exfoliated and well-dispersed clay particles was produced using a single-screw extruder and hence with limited polymer degradation. We show that processing temperature plays an important role in enhancing dispersion and that reprocessing at a higher temperature can enhance both dispersion and exfoliation and thus can result in composites with superior properties. We attempt to elucidate how the change in melt viscosity—coupled with the change in processing temperature—affects clay exfoliation and dispersion.     

Extraction from a porous body in the presence of periodic fluid flow on it

This paper considers the nonstationary process of extraction from a solid body modeled by a system of semiinfinite capillaries connected with a group of no-flow channels when the mass transfer velocity in the flow is composed of two components — a constant velocity component and a time-periodic addition to the first one that is assumed to be small relative to the amplitude. We have obtained analytical dependences for the masstransfer characteristics that are of practical interest: the concentration and the diffusion flow for both the main approximation and with correction for the periodic action on the system.     

Plug flow formation and growth in Da Vinci fluids

A model is discussed for flow of dense granular matter—a da Vinci fluid. The local properties of the fluid are generically different from ordinary fluids in that energy is dissipated by solid friction. We discuss the equation of flow of such a fluid and show that it gives rise to formation and growth of plug regions—a phenomenon observed frequently in flow of granular matter. Simple explicit examples are presented to illustrate the evolution of plug flow regions.     

Generalized Micromorphic solids and fluids

Micromorphic theory (MMT) envisions a material body as a continuous collection of deformable particles; each possesses finite size and inner structure. It is considered as the most successful top-down formulation of a two-level continuum model, in which the deformation is expressed as a sum of macroscopic continuous deformation and internal microscopic deformation of the inner structure. In this work, the kinematics including the objective Eringen tensors is introduced. Balance laws are derived by requiring the energy equation to be form-invariant under the generalized Galilean transformation. The constitutive theories for generalized Micromorphic solid and fluid are formulated. Applications of MMT in micro/nanoscale are discussed.     

Large deformation response of a hyperelastic fibre reinforced composite: Theoretical model and numerical validation

The mechanical behaviour of an incompressible neo-Hookean material directionally reinforced with a generalised neo-Hookean fibre is examined in the finite deformation regime. To consider the interaction between the fibre and the matrix, we use a composite model for this transversely isotropic material based on a multiplicative decomposition of deformation, which decouples the uniaxial deformation along the fibre direction from the remaining shear deformation. The model is then verified numerically by a unit cell model with periodic boundary conditions. The strain energy stored in the unit cell is compared with the energy predicted by the proposed theoretical model and excellent agreement is reported.     

Probing material properties with sharp indenters: a retrospective

A retrospective on the use of sharp, fixed-profile indenters as materials probes is presented. Indentation is proposed as a simple but powerful methodology for evaluating basic mechanical properties—elastic modulus, hardness, toughness—in all classes of materials. Indentation also provides unique insight into fundamental deformation and fracture processes. Of particular interest is the existence of intrinsic size effects as characteristic contact dimensions pass from macro- to micro- to nano-scale dimensions. The utility of indentations as ‘controlled flaws’ in the context of strength of materials is outlined. The roles of two other important material factors—rate effects and microstructure—are considered. Examples of technological and biological applications are presented as illustrations of the widespread power of the technique. Strengths and limitations of the methodology as a routine testing protocol are discussed.     

Poroviscoelastic analysis of borehole and cylinder problems

Summary This paper addresses the phenomena of mechanical creep and deformation in rock formations, coupled with the hydraulic effects of fluid flow. The theory is based on Biot's poroelasticity, generalized to encompass viscoelastic effects through the correspondence principle. Based on the resultant poroviscoelastic theory, stress and deformation analyses are performed. The interactions between the fluid pore pressure diffusion and the elastic/viscoelastic rock matrix deformation are illustrated via two important examples. First, the problem of a borehole subject to a non-hydrostatic stress state, but deforming under plane strain condition, is examined. Second, a cylinder under generalized plane strain conditions is solved. Three rocks, Berea Sandstone, Danian Chalk, and a deep water Gulf of Mexico Shale, covering a wide range of permeabilities, are considered. The significance of poro-and viscoelastic time-dependent effects is discussed.     

Fiber-reinforced hyperelastic solids: a realizable homogenization constitutive theory

A new homogenization theory to model the mechanical response of hyperelastic solids reinforced by a random distribution of aligned cylindrical fibers is proposed. The central idea is to devise a special class of microstructures—by means of an iterated homogenization procedure in finite elasticity together with an exact dilute result for sequential laminates—that allows to compute exactly the macroscopic response of the resulting fiber-reinforced materials. The proposed framework incorporates direct microstructural information up to the two-point correlation functions, and requires the solution to a Hamilton–Jacobi equation with the fiber concentration and the macroscopic deformation gradient playing the role of “time” and “spatial” variables, respectively. In addition to providing constitutive models for the macroscopic response of fiber-reinforced materials, the proposed theory also gives information about the local fields in the matrix and fibers, which can be used to study the evolution of microstructure and the development of instabilities. As a first application of the theory, closed-form results for the case of Neo-Hookean solids reinforced by a transversely isotropic distribution of anisotropic fibers are worked out. These include a novel explicit criterion for the onset of instabilities under general finite-strain loading conditions.     

Partially crystallized elastomer as a nanocomposite model

Data on the influence of crystallization on the mechanical properties of elastomers — the elastic modulus, the relaxation properties, in particular, restorability in compression, and the tensile strength — have been generalized. These data have been compared to those on the influence of active fillers and a much higher crystallization efficiency has been shown. The size of single crystals has been evaluated for most crystallizable rubbers. It has been inferred that the nanosize of single crystals of elastomers and their direct bond with the elastomer matrix influence the mechanical properties of elastomer materials. In considering a partially crystallized elastomer as a nanocomposite model, one can formulate requirements imposed on efficient nanofillers for elastomer materials. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 78, No. 5, pp. 19–23, September–October, 2005.     

Skew shock wave on a gas-solid body interface

The interaction between a skew shock wave (SSW) and a plane surface of a solid body is considered. Break decay (BD) on the gas—solid body interface at an arbitrary angle of impact is calculated within the limits of applicability of the hydrodynamic theory of shock waves.     

Simulation of the supersonic turbulent flow around a cylinder with coaxial disks

The turbulent axisymmetric flow around a stepped body — a cylinder with coaxial front and rear disks — has been calculated with the aid of a VP2/3 package based on multiblock computational technologies and the generalized procedure of pressure correction. The computational model has been tested with the example of a supersonic flow around a sphere. The numerical forecasts made with the use of Spalart–Allmares shear stress transfer and eddy viscosity transfer models have been compared with the data of the aeroballistic experiment, wind tunnel tests, and the results of the calculation of the flow around the disk–cylinder arrangement by a simplified zonal model in a wide range of variation of the incident flow Mach number (from 1.5 to 4). We have obtained a good agreement between the calculated transverse flow density distributions in the front stalling zone and those determined from the interferograms for the wave-drag-rational disk–cylinder arrangement. The influence of the rear disk on the drag of the disk–cylinder–disk arrangement has been estimated.