A thermodynamic framework to model thixotropic materials

Thixotropic materials are widely used in a variety of industrial applications. The constitutive relations to describe these materials are based on one-dimensional experiments in which the material is subjected to a shear motion and there is no unique methodology to obtain proper three-dimensional models. The path towards generalization to a three-dimensional framework is invariably carried out in a ad hoc manner. Here we propose a three-dimensional model that stems from a general thermodynamic framework that has proved to be quite robust in the development of constitutive relations, namely the application of the second law of thermodynamics together with the maximization of the entropy production. This leads to a constitutive equation that has the same form of a generalized Upper Convected Maxwell equation, if we require that changes of microstructure due to the deformation of each Maxwell element that comprises the model are reversible. Changes in microstructure are governed by a potential that is a measure of the difference between the current structure and the equilibrium structure associated with it. The equilibrium structure associated with the current structure is determined by the current value of stress, considered the main break up agent. We assume that the state of equilibrium would be achieved in a Motion With Constant Stress History, starting from the current stress state, until a steady state where the kinematics is not changing.     

Stress intensity factors for a moving cylindrical crack in a nonhomogeneous cylindrical layer in composite materials

Stresses are determined for a finite cylindrical crack that is propagating with a constant velocity in a nonhomogeneous cylindrical elastic layer, sandwiched between an infinite elastic medium and a circular elastic cylinder made from another material. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. An internal gas pressure is then applied to the crack surfaces. The solution is derived by dividing the nonhomogeneous interfacial layer into several homogeneous cylindrical layers with different material properties. The boundary conditions are reduced to two pairs of dual integral equations. These equations are solved by expanding the differences in the crack surface displacements into a series of functions that are equal to zero outside the crack. The Schmidt method is then used to solve for the unknown coefficients in the series. Numerical calculations for the stress intensity factors were performed for speeds and composite material combinations.     

Toward a Turbulence Constitutive Relation for Geophysical Flows

Rapidly rotating turbulent flows are frequently in approximate geostrophic balance. Single-point turbulence closures, in general, are not consistent with a geostrophic balance. This article addresses and resolves the possibility of a constitutive relation for single-point second-order closures for classes of rotating and stratified flows relevant to geophysics. Physical situations in which a geostrophic balance is attained are described. Closely related issues of frame-indifference, horizontal divergence, and the Taylor–Proudman theorem are discussed. It is shown that, in the absence of vortex stretching along the axis of rotation, turbulence is frame-indifferent. Unfortunately, no turbulence closures are consistent with this frame-indifference that is frequently an important feature of rotating or quasi-geostrophic flows. A derivation and discussion of the geostrophic constraint which ensures that the modeled second-moment equations are frame-invariant, in the appropriate limit, is given. It is shown that rotating, stratified, and shallow water flows are situations in which such a constitutive relation procedure is useful. A nonlinear nonconstant coefficient representation for the rapid-pressure strain covariance appearing in the Reynolds stress and heat flux equations, consistent with the geostrophic balance, is described. The rapid-pressure strain closure features coefficients that are not constants determined by numerical optimization but are functions of the state of turbulence as parametrized by the Reynolds stresses and the turbulent heat fluxes as is required by tensor representation theory. These issues are relevant to baroclinic and barotropic atmospheric and oceanic flows. The planetary boundary layers in which there is a transition, with height or depth, from a thermally or shear driven turbulence to a geostrophic turbulence is a classic geophysical example to which the considerations in this article are relevant. Received 14 October 1996 and accepted 9 June 1997     

Mixing in Isotropic Turbulence with Scalar Injection and Applications to Subgrid Modeling

A new technique for injecting scalar fluctuations in a DNS of isotropic turbulence is presented. It is used to study statistically steady states associated with different levels of mixing. The results are analysed in terms of spectra and PDF, and they are used as a database to investigate the effect of the filtering operation that is performed in LES. It is shown that the PDF of the scalar is substantially affected by the filtering operation. It is also shown that the Cook and Riley [1] subgrid model allows reconstruction of a PDF which is in fairly good agreement with the unfiltered DNS results. The consequences of estimating the scalar subgrid variance by scale similarity assumptions are investigated. It is found that the results are improved by a local determination of the model constant. This revised version was published online in July 2006 with corrections to the Cover Date.     

Implicit boundary method for determination of effective properties of composite microstructures

A method that uses a structured grid to perform micromechanical analysis for determining effective properties of a composite microstructure is presented. This approach eliminates the need for constructing a mesh that has nodes along the interfaces between constituent materials of the composite. Implicit boundary method is used to ensure that interface conditions are satisfied at the material boundaries. In this method, solution structures for test and trial functions are constructed using approximate step functions such that the interface conditions are satisfied, even if there are no nodes on the material interface boundary. Since a structured grid does not conform to the geometry of the analysis domain, the geometry of the microstructure is defined independently using equations of the interface boundary curves/surfaces. Structured grids that overlap the geometry are easy to generate, and the elements in the grid are regular shaped and undistorted. A numerical example is presented to demonstrate that the proposed solution structure accurately models the solution across material interface, and convergence analysis is performed to show that the method converges as the grid density is increased. Fiber reinforced microstructures are analyzed to compute the effective elastic properties using both 2D and 3D models to show that the results match closely with the ones available in the literature.     

Rapid conical flows of viscoelastic solutions

Several examples of conical, stretching flows of viscoelastic solutions are described. Two cases are then examined in more detail, the rapidly stretching free jet and an internally pressurised sheet of liquid in which extension takes place in a circumferential direction.It is shown that both the stress and strain rate may readily be calculated at different positions, provided certain assumptions are made. The changes of extensional viscosity necessary to produce the specified flow geometries are then shown to be anomalous and inconsistent. If, however, a solid-like model based on the Green measure of strain is used, a more satisfactory interpretation of the behaviour is achieved.It is emphasised that these are high-speed, high Deborah number flows and that such a flow pattern is not a general one.An example is also given in which the stretching of rubber is shown to be consistent with the same solid-like model, and values of the extensional moduli of elasticity are quoted for both liquids and rubber.     

An energy localization principle and its application to fast kinetic Monte Carlo simulation of heteroepitaxial growth

Simulation of heteroepitaxial growth using kinetic Monte Carlo (KMC) is often based on rates determined by differences in elastic energy between two configurations. This is computationally challenging due to the long range nature of elastic interactions. A new method is introduced in which the elastic field is updated using a local approximation technique. This involves an iterative method that is applied in a sequence of nested domains until a convergence criteria is satisfied. These localized calculations yield energy differences that are highly accurate despite the fact that the energies themselves are far less accurate: an effect referred to as the principle of energy localization. This is explained using the continuum analogue of the discrete model and error estimates are found. In addition, a rejection algorithm that relies on a computationally inexpensive estimate of hopping rates is used to avoid a substantial fraction of the elastic updates. These techniques are applied to 1+1-dimensional KMC simulations in physically interesting regimes.     

Flow Characteristics of a “Multiconfigurational”, Shear Thinning Viscoelastic Fluid with Particular Reference to the Orthogonal Rheometer

This paper deals with the flow characteristics of a class of nonsimple viscoelastic fluid models developed by Rajagopal and Srinivasa (1999). The central feature of these models is that the stress response is lastic from a changing natural configuration with the viscous dissipation occurring due to changes in the natural state. The class of models considered are characterized by three independent parameters that represent respectively the elasticity, the viscosity and the shear thinning index. The stress relaxation response of the material is compared with experimental data reported by Bower et al. (1987) for polyisobutelene in cetane, and parameters that fit the data are calculated. The flow of such a fluid between parallel disks rotating about noncoincident axes (the orthogonal rheometer) is then studied. It is shown that the assumed velocity field leads to a system of second-order nonlinear ordinary differential equations (Rajagopal, 1982). A parametric study is then undertaken to see the effect of the various material, geometrical, and flow parameters on the flow characteristics. It is observed that inertial effects and shear thinning effects are roughly complementary in the range of parameters considered. While it is well known that boundary layers occur in these flows due to inertial effects, it is demonstrated that these boundary effects are insensitive to the Reynolds number but rather are determined by the absorption number. Finally, in the range of parameters that are commonly observed in such rheometers, it is shown that neglect of inertia causes significant discrepancies in the calculation of the boundary shear rates. Received 3 June 1999 and accepted 2 October 1999     

Control of a Nonlinear System Using the Saturation Phenomenon

In this paper, a theoretical investigation of nonlinear vibrations of a 2 degrees of freedom system when subjected to saturation is studied. The method has been especially applied to a system that consists of a DC motor with a nonlinear controller and a harmonic forcing voltage. Approximate solutions are sought using the method of multiple scales. It is shown that the closed-loop system exhibits different response regimes. The nature and stability of these regimes are studied and the stability boundaries are obtained. The effects of the initial conditions on the response of the system have also been investigated. Furthermore, the second-order solution is presented and the corresponding results are compared with those of the first-order solution. It is shown that by increasing the amplitude of the excitation voltage, the higher-order term in the solution becomes significant and causes a drift in the response. In order to verify the obtained theoretical results, they are compared with the corresponding numerical results. Good agreement between the two sets of results is observed.     

A principle of virtual work for combined electrostatic and mechanical loading of materials

The equations governing mechanics and electrostatics are formulated for a system in which the material deformations and electrostatic polarizations are arbitrary. A mechanical/electrostatic energy balance is formulated for this situation in terms of the electric enthalpy, in which the electric potential and the electric field are the independent variables, and charge and electric displacement, respectively, are the conjugate thermodynamic forces. This energy statement is presented in the form of a principle of virtual work (PVW), in which external virtual work is equated to internal virtual work. The resulting expression involves an internal material virtual work in which (1) material polarization is work-conjugate to increments of electric field, and (2) a combination of Cauchy stress, Maxwell stress and a product of polarization and electric field is work-conjugate to increments of strain. This PVW is valid for all material types, including those that are conservative and those that are dissipative. Such a virtual work expression is the basis for a rigorous formulation of a finite element method for problems involving the deformation and electrostatic charging of materials, including electroactive polymers and switchable ferroelectrics. The internal virtual work expression is used to develop the structure of conservative constitutive laws governing, for example, electroactive elastomers and piezoelectric materials, thereby determining the form of the Maxwell or electrostatic stress. It is shown that the Maxwell or electrostatic stress has a form fully constrained by the constitutive law and cannot be chosen independently of it. The structure of constitutive laws for dissipative materials, such as viscoelastic electroactive polymers and switchable ferroelectrics, is similarly determined, and it is shown that the Maxwell or electrostatic stress for these materials is identical to that for a material having the same conservative response when the dissipative processes in the material are shut off. The form of the internal virtual work is used further to develop the structure of dissipative constitutive laws controlled by rearrangement of material internal variables.     

Experiment and modelling of birefringent flows using commercial CFD code

It is well-known that certain fluids are birefringent and when flows are viewed in polarised light interference fringes are observed. The fringes are caused by a phase shift in the light passing through the fluid and are proportional to the integral of the maximum shear strains in the fluid. In order to understand what is happening within the three dimensional flow and overcome the difficulties due to this integration, additional computational or experimental information is needed.

In this work, a commercially available computer code (Fluent) is used for the first time to model the flows. The flow data are then exported to a spreadsheet where the shear rates are integrated across the field and then banded for graphical output. The results from this are then compared to results generated from birefringent flow experiments and the agreement is found to be good since the modelled fringes show the same patterns as those in the experiment. This novel use of computational and experimental techniques together will allow quantitative analysis of three-dimensional flows in the future.

Currently, there are still a lot of empirical variables involved in fitting the computational fringes to the experiment, but the results of this preliminary study show that this is a promising approach to this type of problem.     

In-plane waves in an elastic solid containing a cracked slab region

Propagation of longitudinal and transverse waves in an elastic solid that contains a cracked slab region is investigated. The cracks have a uniform probability density in the slab region, are parallel to the boundaries of the slab, and the solid is uncracked on either side of the slab. The waves are normally incident on the cracks. It is shown that the resulting average total motion in the solid is governed by a pair of coupled integral equations. These equations are solved under the special assumption that the average exciting motion near a fixed crack is equal to the average total motion. In this case, one finds that in the cracked region, where multiple scattering occurs, there is a forward motion and a backward motion. The two motions have identical frequency-dependent velocity and attenuation, for which simple closed-form formulae are obtained. Simple formulae are also obtained for the wave amplitudes outside the slab. Numerical results corresponding to the velocity, attenuation, reflection amplitude, and transmission amplitude are presented for several values of crack density and slab thickness.     

Deployable lenticular stiffeners for origami-inspired mechanisms

Light-weight origami-inspired mechanisms can provide advantages in deployable space systems and other applications. However, a significant challenge in their design is ensuring that they are su?ciently stiff. Compliant, deployable stiffeners utilizing a profile that approximates the Euler Spiral are proposed as one possible solution. It is shown that a structure with this specific profile, called a lenticular stiffener, permits stiffeners to be flattened using a force applied at their edge. Formulas for calculating the increase in stiffness are developed. Relations needed to design the deployment behavior of the stiffeners are also derived. Finally, advantages of different configurations of stiffeners are evaluated. These results are verified through simulation and experiments.     

On a Class of Deformations of Compressible, Isotropic, Nonlinearly Elastic Solids

We consider deformations of unconstrained, isotropic hyperelastic solids which satisfy the condition that the determinant of the deformation gradient is constant. In the absence of body forces, it is shown (i) that a certain deformation in this class (which describes the bending of rectangular blocks into annular cylindrical sectors) is not possible in any of the considered materials, (ii) that in the case when the body fills the whole space, it is composed of a compressible neo-Hookean material and it is subjected to relatively moderate loads, these deformations are necessarily homogeneous and (iii) that for boundary conditions of place and relative to a certain sub-class of the class of considered materials, these deformations are globally stable, in the sense that they are minimizers for the total energy with respect to smooth variations that are compatible with the boundary conditions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Quasi-static response of linear viscoelastic cantilever beams subject to a concentrated harmonic end load

This study evaluates the response of a uniform cantilever beam with a symmetric cross-section fixed at one end, and submitted to a lateral concentrated sinusoidal load at the free extremity. The beam material is assumed to be homogeneous, isotropic and linear viscoelastic. Due to the nature of the loading and the beam slenderness, large displacements are developed but the strains are considered small. Consequently, the mathematical formulation only involves geometrical non-linearity. It is also assumed that the beam is inextensible (neutral axis length is constant) and that inertial forces are negligible, i.e., dynamic effects are insignificant and the system can thus be modeled quasi-statically. The beam is therefore subject to oscillations caused by the sinusoidal time-dependent load, leading to a transient response until the material stabilizes and the system exhibits a periodic response, which can be conveniently described in the frequency domain. The time domain solution of this problem is elaborated by considering the quasi-static response for each time interval. The mathematical equations are presented in dimensional and dimensionless forms, and for the latter case, a numerical solution is generated and several case studies are presented. The problem is governed by a set of non-linear ordinary differential equations encompassing functions of space and time that relate the curvature, rotation angle, bending moment and geometrical coordinates. In this study, an elegant solution is deduced using perturbation theory, yielding a precise steady-state solution in the frequency domain with considerable computational economy. The solutions for both time and frequency domain methods are developed and compared using a case study for a series of dimensionless parameters that influence the response of the system.     

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

Consider a cylinder (not necessarily of circular cross-section) that is composed of a hyperelastic material and which is stretched parallel to its axis of symmetry. Suppose that the elastic material that constitutes the cylinder is homogeneous, transversely isotropic, and incompressible and that the deformed length of the cylinder is prescribed, the ends of the cylinder are free of shear, and the sides are left completely free. In this paper it is shown that mild additional constitutive hypotheses on the stored-energy function imply that the unique absolute minimizer of the elastic energy for this problem is a homogeneous, isoaxial deformation. This extends recent results that show the same result is valid in 2-dimensions. Prior work on this problem had been restricted to a local analysis: in particular, it was previously known that homogeneous deformations are strict (weak) relative minimizers of the elastic energy as long as the underlying linearized equations are strongly elliptic and provided that the load/displacement curve in this class of deformations does not possess a maximum.     

A finite element approach to study cavitation instabilities in non-linear elastic solids under general loading conditions

This paper proposes an effective numerical method to study cavitation instabilities in non-linear elastic solids. The basic idea is to examine—by means of a 3D finite element model—the mechanical response under affine boundary conditions of a block of non-linear elastic material that contains a single infinitesimal defect at its center. The occurrence of cavitation is identified as the event when the initially small defect suddenly grows to a much larger size in response to sufficiently large applied loads. While the method is valid more generally, the emphasis here is on solids that are isotropic and defects that are vacuous and initially spherical in shape. As a first application, the proposed approach is utilized to compute the entire onset-of-cavitation surfaces (namely, the set of all critical Cauchy stress states at which cavitation ensues) for a variety of incompressible materials with different convexity properties and growth conditions. For strictly polyconvex materials, it is found that cavitation occurs only for stress states where the three principal Cauchy stresses are tensile and that the required hydrostatic stress component at cavitation increases with increasing shear components. For a class of materials that are not polyconvex, on the other hand and rather counterintuitively, the hydrostatic stress component at cavitation is found to decrease for a range of increasing shear components. The theoretical and practical implications of these results are discussed.     

On the sensitivity of non-generic bifurcation of non-linear normal modes

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r=1,3,5,… . Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears, and generic bifurcation appears in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.     

Modelling framework for mass-growth III: Isochoric growth

Mass-growth is usually perceived as a non-isochoric process, but classes of soft tissue that exhibit incompressible or nearly incompressible in vitro behaviour may have gone through growth stages which are isochoric or nearly isochoric. The present paper aims thus to complement and complete the non-isochoric mass-growth modelling framework presented in [1], [2] by presenting a relevant formulation for isochoric deformation processes that exhibit features of simultaneous elastic and plastic mass-growth. The refined modelling route that is followed is slightly different, and more general to that followed in [2], to which, however, is also applicable. Because mass density and stress levels are expected to increase faster than they would in analogous non-isochoric mass-growth situations, purely pseudo-elastic or purely pseudo-plastic stages of isochoric mass-growth are rather unlikely to alternate in the manner implied in [1] for their non-isochoric counterparts. Purely pseudo-elastic and purely pseudo-plastic isochoric mass-growth models can however still be obtained as particular cases of the present formulation. These issues as well as additional features that characterise the present model are detailed and clarified further through the complete, closed form solution of a particular, example problem application in which the mass density and the shape of the growing continuum are subjected to continuous time change.