Some aspects of linear and nonlinear viscoelastic behaviour of polymer melts in shear

In linear viscoelastic investigations the frequency dependence of the phase shift between stress and strain appears to be very characteristic of the molecular structure of the material. This function is also a good approximation of the slope of the double logarithmic plot of the absolute value of the shear modulusG _d vs. the angular frequency. The product (G _d /) sin 2 comes very close to the relaxation spectrumH(), with = 1/, in all physical states of the material.The experimentally observed separability of time and strain effects in nonlinear viscoelasticity of highly viscous isotropic polymer fluids imposes restraints to the form of the constitutive equation. A single integral superposition equation of the Boltzmann type containing the product of a time function and a nonlinear strain function gives good results in describing experimental data in shear as well as in elongation. The molecular structure affects both functions in a different way. A universal definition of the nonlinear tensorial strain measure has not yet been developed. There are some indications that a definition on the basis of the principal stretch ratios may be fruitful.Invited paper, presented at the First Conference of European Rheologists at Graz (Austria), April 14–16, 1982.     

Secondary cells and separation in developing laminar curved-pipe flows

Laminar flows through 180° curved bends of circular cross section are investigated numerically. For small curvature ratio, , defined as pipe radius over mean bend radius, the governing equations could be parabolized. The equations are solved for an range of from 0.04 to 0.143, a Dean number (De) range of from 277.5 to 1360, and for a uniform flow, a potential vortex, and a parabolic flow inlet condition. In all these studies a zero cross-stream flow at the inlet is assumed. A detailed study of the effects of , De, and inlet condition on the secondary flow pattern is carried out. Within the range of parameters investigated, up to three secondary cells are found in the cross-stream half-plane of a curved pipe. They are the Dean-type secondary cell, a secondary separation cell near the inner bend (closest to the center of curvature of the bend), and a third cell near the pipe center. The number of secondary cells in the cross-stream half-plane is greatly influenced by the inlet flow, and to a much lesser extent by and De. For example, only the Dean cell is found in a curved-pipe flow where and De are small and the inlet flow is either uniform or a potential vortex. When the inlet condition of the same case is changed to a parabolic flow, a three-cell structure results. Furthermore, as De increases to 1180, incipient axial flow separation begins at around 23° downstream of the curved-pipe entrance. The formation and extent of the separation and third cells are investigated together with their dependence on the parameters studied. This investigation further shows that, within the range of parameters examined, there is no secondary cell occurring near the outer bend, contrary to some earlier findings on fully developed curved-pipe flows.This work was supported by the Office of Naval Research under Grant No. N0014-81-K-0428 and by DTRC, Annapolis, Maryland, under Contract No. N00167-86-K-0075. Also, support in the form of an IPA awarded to RMCS during his sabbatical leave at DTRC, Annapolis, Maryland, in the spring of 1990 is gratefully acknowledged.     

Reduced‐order modelling of an adaptive mesh ocean model

A novel proper orthogonal decomposition (POD) model has been developed for use with an advanced unstructured mesh finite‐element ocean model, the Imperial College Ocean Model (ICOM, described in detail below), which includes many recent developments in ocean modelling and numerical analysis. The advantages of the POD model developed here over existing POD approaches are the ability:

• 1. To increase accuracy when representing geostrophic balance (the balance between the Coriolis terms and the pressure gradient). This is achieved through the use of two sets of geostrophic basis functions where each one is calculated by basis functions for velocities u and v.
• 2. To speed up the POD simulation. To achieve this a new numerical technique is introduced, whereby a time‐dependent matrix in the discretized equation is rapidly constructed from a series of time‐independent matrices. This development imparts considerable efficiency gains over the often‐used alternative of calculating each finite element over the computational domain at each time level.
• 3. To use dynamically adaptive meshes in the above POD model.

Copyright © 2008 John Wiley & Sons, Ltd.     

A highly redundant coordinate formulation for constrained rigid bodies

The paper presents a formulation for the dynamic analysis of rigid multibodies. An introductory part carries out the kinematic analysis and the definition of the highly redundant differential framework along with the choice of unknowns and equations. From the differential formulation the variational principles, either in Lagrangian or Hamiltonian form, are developed. The Hamiltonian formulation is then used to develop the numerical approximation by applying the finite element method in time. The application of the method in its multifield form is discussed and a solution algorithm is proposed. Some examples are finally presented in order to verify the effectiveness of the formulation.

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Sommario Il lavoro presenta una formulazione per lo studio della dinamica dei sistemi multicorpo rigidi. Nella parte introduttiva viene svolta l'analisi cinematica e si definisce il quadro differenziale con la scelta delle incognite e delle equazioni. Dalla formulazione differenziale vengono poi sviluppati dei principi variazionali nella forma Lagrangiana ed Hamiltoniana. La formulazione Hamiltoniana é quindi utilizzata per sviluppare l'approssimazione numerica col metodo degli elementi finiti di tempo. Viene discussa l'applicazione del metodo nella forma multi-campo e viene proposto un algoritmo di soluzione. Da ultimo, vengono discussi alcuni esempi per verificare la correttezza della formulazione.

     

Compatibility conditions for the Cauchy-Green strain fields: Solutions for the plane case

This paper considers the issues related to uniqueness and existence of a finite deformation generated by prescribed right or left Cauchy-Green strain tensor field in the plane. First, the questions of uniqueness and existence to a pre-assigned right strain field C are discussed. It is shown that the existence condition, in the context of continuum mechanics, are naturally posed using the field corresponding to the square root of C instead of C, the latter a classical approach. Then, the corresponding questions for the left strain field are considered, which is more involved. The analysis of uniqueness gives rise to an appropriate classification of the deformation fields. The question of existence is discussed and a complete solution is presented. In both the right and left cases, we stress the techniques for obtaining the corresponding deformation fields.     

A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void

In this paper, we carry out an explicit analysis of a bifurcation problem for a solid circular cylinder composed of a particularcompressible nonlinearly elastic material. This problem is concerned with the bifurcation of a solid body into a configuration involving an internal cavity. A discussion of its physical interpretation is then carried out. In particular, it is shown that this model may be used to describe the nucleation of a void from apre-existing micro-void.     

The elastic modulus of particulate composites using the concept of a mesophase

A theoretical model for the evaluation of the elastic modulus in particulate composites has been developed. The method takes into account the existence of a mesophase between main phases, which constitutes an important parameter influencing the behaviour of a composite material. This layer between the matrix and filler develops different physico-chemical properties from those of the constituent phases and variable ones along its thickness. The effect of the progressive variation of the elastic modulus of the mesophase on the modulus of the composite was estimated by applying various simple laws of variation. Convenient laws of variation were introduced, varying from a simple one, assuming a linear law, to a more refined one using a parabolic law. Experimental results with particulates, based on iron-filled epoxy composites, compared satisfactorily with other models. However, the model based on a parabolic law was superior to all others on physical grounds.List of symbols E elastic modulus - v Poisson's ratio - volume fraction - r radius - V volume - K bulk modulus - r thickness of mesophase - parameter which depends on the sudden changes of the heat capacity for the filled and unfilled polymer C _p ^f and C _p ⁰      

Quasiconvexity and partial regularity in the calculus of variations

We prove partial regularity of minimizers of certain functionals in the calculus of variations, under the principal assumption that the integrands be uniformly strictly quasiconvex. This is of interest since quasiconvexity is known in many circumstances to be necessary and sufficient for the weak sequential lower semicontinuity of these functionals on appropriate Sobolev spaces. Examples covered by the regularity theory include functionals with integrands which are convex in the determinants of various submatrices of the gradient matrix.     

Kinking of a crack emanating from the free surface of a beam under pure bending

A perturbation technique developed by Karihaloo et al. is employed to obtain the stress intensity factors at the tip of a kinking crack that emanates from the free surface of a beam under pure bending. Under the condition that the kink extends in the direction of vanishing K₁₁ the crack path is obtained as well as a path stability condition. From conditions on K₁ a material parameter r^* akin to that of Ramulu and Kobayashi's r_c is obtained. By analysis of the slope of the kinking crack a stability condition is obtained corroborating the stability condition from consideration of vanishing K₁₁. It is shown that for a beam in pure bending the nonsingular remote stress term T must be greater than some positive critical value for kinking to occur confirming the results of Sayir and Schindler.