Alan S. Wineman

On boundary conditions for a certain class of problems in mixture theory

Abstract An additional boundary condition is proposed for solid-fluid mixtures for the situation in which a mixture boundary is in a saturated state. This condition on the boundary is derived from a thermodynamic characterization of the state and takes the form of a relationship between the total stress tensor, the stretch tensor and the volume fraction of the solid. This additional condition is sufficient to make several boundary value problems involving mixtures, determinate.

Applications of the theory of interacting continua to the diffusion of a fluid through a non-linear elastic media

The theory of interacting continua is applied to the problem of diffusiion of a fluid through a non-linear elastic layer and a hollow sphere. Using methods which are by now standard in continuum mechanics expressions and restrictions are derived from a thermodynamic standpoint for the partial stresses for the fluid and solid and the diffusive body force. In order to obtain detailed solutions to specific boundary value problems a choice of a particular form for the free energy function for the mixture is made based on statistical theory. To simplify the problem, we assume that the fluid in question is ideal. The difficulties inherent to a clear definition of the boundary conditions for the partial stresses are overcome by the use of the Flory-Huggins equation. Two specific examples are considered. The first is the problem of diffusion through a stretched layer and second is diffusion through a spherical shell. Results of the numerical solution enable the construction of the pressure difference-flux relations, which have been shown to be in good agreement with experimental data.

On a boundary layer theory for non-Newtonian fluids

Abstract Theoretical support for the existence of a boundary layer theory, similar + o the classical boundary layer theory for linearly viscous fluids, is offered in the case of the non-Newtonian fluid of second grade. It is also pointed out that unless certain assumptions are made regarding the flow, assumptions which are not alluded to in earlier work in this area, in addition to the assumptions usually made in the case of the linearly viscous fluid, the theory so developed might have inherent flaws.

Some nonlinear diffusion problems within the context of the theory of interacting continua

Abstract Recently, a method for generating additional boundary conditions which are required for solving boundary value problems within the context of the theory of interacting continua, has been developed by Rajagopal, Wineman and Gandhi (Int. J. Engng Sci. 24, 1453, 1986). In this paper, using these boundary conditions the problems of radial diffusion of a fluid through a hollow non-linear elastic cylinder, and the diffusion through a sheared non-linear elastic layer, are studied in detail. It is found that shearing and stretching have qualitatively different effects on the diffusion process.

Material identification of soft tissue using membrane inflation

The constitutive equation for large elastic deformations is often used to model the mechanical response of soft tissue. This paper is concerned with the applications of the method of material identification to the determination of the strain energy density functions ( W) in such a mode, under the assumption that the tissue is incompressible and isotropic. It is shown that an identification experiment based on inflation by lateral pressure ofan initially flat circular membraneous specimen has a number of advantages. These are: the method of clamping the specimen, the ease of labelling material particles and measuring current coordinates, the easily determined domain of identification of I% and a means of systematically determining Wover a large deformation range. An example in the form of a hypothetical experiment is presented. 1. lNTRODUCDON The constitutive equation for large elastic defor- mations is often used to model the mechanical re- sponse of soft biological tissues as well as rubbery materials. Its application depends on the knowledge of the strain energy density function (W) for the material under consideration. Forms for W for incompressible rubbery materials are reasonably well established. On the other hand, a significant amount of activity is concerned with the determination of strain energy density functions for soft tissues. The procedures which have been used to establish forms for W for rubber appear to be the guide for some current approaches for the determination of W for soft tissue. It will be useful, for present purposes, to briefly review the determination of W for rubber. The initial forms for W were developed from an experimental program based on subjecting specimens to unequal biaxial homogeneous deformattons. In order to assess the Mooney form of W in predicting non- homogeneous deformations, Adkins and Rivlin (1952) used it in the calculation of the deformed profiles of a clamped, initially flat, circular rubber membrane which had been pressurized on one side. These were then compared with actual profiles which had been previously measured by Treloar (1944). Agreement between the measured and calculated profiles was satisfactory at low inflation levels but less so at higher levels. Later, Klingbeil and Shield (1964), and then Hart-Smith and Crisp (1967) introduced improved forms for W for which the match between measured and calculated profiles was very good. Recently, a method has been suggested which combines these stages of construction from homo- geneous deformations, verification against non- homogeneous deformations and subsequent improve-

Large Axisymmetric Inflation of a Nonlinear Viscoelastic Membrane by Lateral Pressure

The large axisymmetric deformation of a plane circular membrane into surfaces of revolution by a lateral pressure is considered. The material is taken to be a styrene‐butadiene rubber for which a nonlinear integral type constitutive equation incorporating measured properties has been presented by McGuirt and Lianis [Trans. Soc. Rheol., 14, 117, (1970)]. The formulation is reduced to a two‐point boundary value problem governed by a system of nonlinear partial differential‐integral equations of Volterra type for principal stretch ratios and a related kinematic variable. A numerical procedure is outlined which reduces at each time step to solving a system of equations having the same general structure as that for the corresponding problem assuming the membrane to be elastic. Stretch ratio and stress variation and deformed profile histories are computed for prescribed pressure histories, the latter being most useful for comparison of predictions and experiment.

Flow of electro-rheological materials

SummaryAn electro-rheological fluid is a material in which a particulate solid is suspended in an electrically non-conducting fluid such as oil. On the application of an electric field, the viscosity and other material properties undergo dramatic and significant changes. In this paper, the particulate imbedded fluid is considered as a homogeneous continuum. It is assumed that the Cauchy stress depends on the velocity gradient and the electric field vector. A representation for the constitutive equation is developed using standard methods of continuum mechanics. The stress components are calculated for a shear flow in which the electric field vector, is normal to the velocity vector. The model predicts (i) a viscosity which depends on the shear rate and electric field and (ii) normal stresses due to the interaction between the shear flow and the electric field. These expressions are used to study several fundamental shear flows: the flow between parallel plates, Couette flow, and flow in an eccentric rotating disc device. Detailed solutions are presented when the shear response is that of a Bingham fluid whose yield stress and viscosity depends on the electric field.

New universal relations for nonlinear isotropic elastic materials

A nonlinear isotropic elastic block is subjected to a homogeneous deformation consisting of simple shear superposed on triaxial extension. Two new relations are established for this deformation which are valid for all nonlinear elastic isotropic materials, and hence are universal relations. The first is a relation between the stretch ratios in the plane of shear and the amount of shear when the deformation is supported only by shear tractions. The second relation is established for a thin-walled cylinder under combined extension, inflation and torsion. Each material element of the cylinder undergoes the same local homogeneous deformation of shear superposed on triaxial extension. The properties of this deformation are used to establish a relation between pressure, twisting moment, angle of twist and current dimensions when no axial force is applied to the cylinder. It is shown that these relations also apply for a mixture of a nonlinear isotropic solid and a fluid.

A mathematical model for the determination of viscoelastic behavior of brain in vivo—I Oscillatory response

Abstract In a recent experiment for determining the mechanical response of brain in vivo , a probe, inserted through scalp, skull and dura, is placed in contact with and normal to the brain, given a prescribed motion, and the time variation of corresponding force is measured. In the corresponding continuum mechanical model, brain is idealized as a linear isotropic viscoelastic solid constrained by a rigid skull. At the mating surface, the shear stress and normal displacement vanish everywhere except under the probe which exerts a local radial displacement. This model introduces effective viscoelastic moduli in shear, which is unknown, and in dilation, which is considered known from other sources. Part I of this study considers steady oscillations of the probe. A transcendental equation for the complex shear modulus is established in terms of probe displacement and force amplitude ratio and phase lag and is solved for specific test data. The corresponding stress and displacement fields are evaluated so that the probe influence may be assessed.

Chemorheological Relaxation, Residual Stress, and Permanent Set Arising in Radial Deformation of Elastomeric Hollow Spheres:

Recently, a constitutive theory for rubber-like materials has been developed by which stress arises from different micromechanisms at different levels of deformation. For small deformations, the stress is given by the usual theory of rubber elasticity. As the deformation increases, there is scission of some junctions of the macromolecular microstructure. Junctions then reform to generate a new microstructure. The constitutive equation allows for continuous scission of the original junctions and formation of new ones as deformation increases. The macromolecular scission causes stress reduction, termed chemorheological relaxation. The new macromolecular structure results in permanent set on release of external load. The present work considers a hollow sphere composed of such a material, also assumed to be incom-pressible and isotropic, which undergoes axisymmetric deformation under radial traction. There develops an outer zone of material with the original microstructure and an inner zone of material having undergone macromolecular scission, separated by a spherical interface whose radius increases with the deformation. The stress distribution, radial load-expansion response, residual stress distribution, and permanent set on release of traction are determined. It is found that a residual state of high compressive stress can arise in a thin layer of material at the inner boundary of the sphere.