Tryfan Rogers memorial lecture

Composirrs Manujbcturing 6 (1995) 119-I22 c, 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 09567143/95/$10.00

3rd International

Tryfan

Conference on Flow Processes in Composite Galway, Ireland, 7-9 July 1994

Rogers Memorial

Materials

94

Lecture

A. J. M. Spencer Faculty of Engineering, University of Nottingham, Nottingham NG7 ZRD, UK

I know there are many in thins audience who knew Tryfan Rogers, but there must be many more who did not, and will wonder why a memorial lecture for a Welsh applied mathematician should be presented at a conference on Flow Processes in Composite Materials in Galway. I will try to explain. About eight years ago Neil Cogswell brought together a group which included Ken Walters, Spencer Jones, Tryfan Rogers and myself, with a view to undertaking

COMPOSITES

University

Park,

some fundamental research on the deformation processes that occur in forming processes of sheets reinforced by continuous unidirectional fibres. More importantly, he persuaded ICI to give us some financial support through their Joint Research scheme. Tryfan and I had, for many years before then, been interested in the mechanics of fibre-reinforced solids and we found that our interests and those of the group at Aberystwyth, together with the practical input from ICI and other industrial

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collaborators, complemented each other very well, and I think we had some success. Additional financial support was obtained from the Science and Engineering Research Council, the Ministry of Defence, British Aerospace and Westlands. This group gradually widened, and we began to share our experience with Pat Mallon, Conchur 0 Bradaigh and others. We held, and still hold, progress meetings two or three times a year. These meetings we found to be scientifically stimulating and also socially enjoyable. Tryfan was a very sociable person and contributed enormously to the success of the group, not only by his scientific contributions (which were numerous) but by his humour and goodwill that played a large part in keeping us working together. It was during one of these meetings just over a year ago that he became ill with what eventually turned out to be a fatal illness. This is the third of three conferences that have been constructed around our research. The first was organized by Neil Cogswell at Brunel University in 1988 under the auspices of the British Society of Rheology, the second by Pat Mallon at Limerick in 1991. Tryfan participated actively in the previous two meetings, and I know he was looking forward to this one, especially as he and I have long-standing connections with, and many friends in, Galway. It is therefore fitting that we should remember him at this meeting. Tryfan was a wonderful person and had many friends all over the world. He was a very close friend to me, and I was only one of many. He was a delightful companion, with a great sense of humour, and one who would always go out of his way to help others. He was a man of very wide interests and achievements. He was a local magistrate in Nottingham, a prominent figure in the local Labour Party, and he served Nottingham University in many ways. Next to mathematics, his greatest love was sport any sort of sport. The Welsh Rugby team, Nottingham Forest football and Boston Redsox baseball, were just a few of the teams he followed with great enthusiasm. However, at this time it is most appropriate to recall his scientific and engineering achievements. Tryfan was an applied mathematician in the British tradition. He graduated from Manchester University, which has always been a stronghold of fluid mechanics, but his main interest (there were many) was in solid mechanics. After Manchester he spent three years in the USA at Brown and Stanford Universities, before coming to Nottingham in 1963 where he remained until his death, eventually becoming Professor and Head of the Department of Theoretical Mechanics. I thus had the good fortune to be his colleague as well as friend for some 30 years, for most of which we worked closely together; in fact I share some 30 papers with him and a succession of research students and postdoctoral researchers whom we jointly supervised. In the course of this we had a lot of fun - the great thing about work with Tryfan was that work was fun - and we both had tremendous pleasure out of working together. So did others with whom he collaborated, notably Ras Lee and Jack Pipkin. He was a

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gregarious person, and always liked to share his ideas rather than pursue solitary research. Tryfan’s interests in the mechanics of polymers began early. One of his earliest papers’ dates from 1963 and was entitled ‘Solution of viscoelastic stress analysis problems using measured creep or relaxation functions’. This title illustrates his approach to research in applied mathematics. He loved mathematics and its rigour, but he also liked it to be useful and directed towards practical problems. Other early papers related to the polymer processing field, which have become near classics, are refs 2 and 3. In 1965 Tryfan and I took on as our first joint research student a graduate from Galway. One of the results was a theory4 to describe the behaviour of fibre-reinforced plastic materials. This was one of our earliest efforts in the field of composite materials, and one of the first to exploit mathematically the anisotropic nature of fibrereinforced composites and show that it can be used to advantage, rather than as a complicating factor. Sadly Sean Mulhern, the research student from Galway, also died tragically young. A few years later Tryfan spent a sabbatical year at Brown University, and in collaboration with Jack Pipkin, realized that the work we had done was much more general than we had recognised. They developed5 a comprehensive theory of ideal fibre-reinforced materials, the full significance of which has only recently begun to be recognised. It would be inappropriate to present a catalogue of Tryfan’s scientific papers, but I will pick out a few more highlights that are relevant to this meeting - I emphasise that he had many other scientific interests besides composite mechanics. In ref. 6 we gave a simple explanation of the springback effect. Papers 7, 8 and 9 are contributions to previous conferences in this series. Paper 10 is an important contribution to the rheological characterization of fibre composites. My collaboration with Tryfan continued to his death, and as my tribute I should like in this lecture to present some extracts from the last article we wrote together, in collaboration with a former research student, Dr Brian Hull”. This is a review article that includes much of the work we did jointly in the composite processing area. One of the things that gave Tryfan (and indeed gives any applied mathematician) great pleasure was to use a simple bit of mathematics to obtain a significant and useful result. The subject of composite processing is a complex one, but it does provide opportunities for the bright idea that can produce meaningful results with little apparent effort. The opening section of ref. 11 contains examples of the sort of mathematics that Tryfan enjoyed. It is easy to share the enjoyment, and in his memory I should like to present a summary of this section. The objective is to construct a mathematical model for a system comprising liquid resin reinforced by continuous fibres. Most of the theory is equally applicable to thermoplastic or to thermosetting composites while they

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Tryfan Rogers memorial

are in a liquid state. Typically, such systems contain about 40-60% volume fraction of aligned fibres. The point of view adopted is a continuum approach, so that macroscopically the resin and fibres are regarded as forming a single homogeneous material and, as far as the overall behaviour is concerned, no distinction is made between the fibre and the resin. At this level the principal mechanical effect of the fibres is to introduce a preferred direction (the fibre direction) in the fluid. Since the fibres are densely distributed and aligned with one another, in the continuum model we attribute a fibre direction to each particle of fluid; this fibre direction is characterized by a unit vector field a which may vary in space and in time as the flow proceeds. Then, in the mathematical description, the fibres are the trajectories defined by the vector a. The material is thus, locally, a transversely isotropic fluid with respect to the fibre direction a. The flow is described in terms of a fixed rectangular Cartesian coordinate system 0x1x2x3 and all vector and tensor components are components in this system. The velocity vector u has components ui(i = 1,2,3), and the components dv of the rate-of-deformation tensor d and R, of the vorticity tensor fl are

The bulk density of the composite is denoted by p, and the continuity equation, representing conservation of mass, has the form

g+pg=o where we employ the repeated suffix summation convention, and D/Dt is the material or convected derivative, so that, for example, DP

-3+g

Dt - at

x,

lecture: A. J. M. Spencer

expression of this assumption is aiajd2i = 0

or, equivalently,

We make the further assumption that the resin and fibres move as a single continuum, i.e. that adjacent fibres and liquid particles remain adjacent during flow and no percolation of liquid through the fibres takes place. The justification for this is that, in the case of densely packed aligned fibres, the characteristic time for percolation greatly exceeds the time required for the flow processes of interest to us. The mathematical form of the condition that the fibres convect with the resin is

where 6, represents the Kronecker delta (zero when i # j, unity when i = j). If the fibres are inextensible, then using (2) the condition (3) reduces to au; Da, - ak--Dt axk

(4)

Multiplying this by Lli,and remembering that a is a unit vector, leads us back to (2). Hence (4) implies (2), and (2) becomes redundant if (4) is adopted. This completes the description of the flow kinematics. The essential equations are the scalar equation (l), and the vector equation (4), which are restrictions on the allowable flows of our idealized fibre-reinforced fluids. To consider the stress response, we take as our basic constitutive assumption that the stress depends on the rate-of-deformation d, the fibre direction u, temperature 8 and density p. Thus, if u denotes the Cauchy stress tensor, with components crij we have u = a(d, u, 8, p)

(5)

The sense of the fibres has no significance, and hence dependence on the vector a can be replaced by dependence on the symmetric second-order tensor A = a @ u, with components A, = UiU,

where dp/dt represents time differentiation at a fixed x. For the materials of interest flow often takes nlace with negligible volume change (this of course may not apply in the case of consolidation flows) and to a good approximation the fluid may be regarded as incompressible. In such a case Dp/Dt = 0, and we have

dlri-0 ax; -

The purpose of introducing fibres into the composite is to provide stiffness and strength in the fibre direction in the finished product. In the fluid state, with aligned continuous fibres, the fibre-resin system shows much greater resistance to deformation by stretching in the fibre direction than to deformation by shear. This mechanical behaviour is idealized by assuming that the fibre is inextensible in the fibre direction. The mathematical

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Thus, suppressing explicit dependence on 0 and p, c=a(d,A)

This is the generalization to transversely isotropic fluids of the constitutive equation U = b(d) of an isotropic non-linear viscous fluid. From (5) it follows, using standard results in the theory of tensor representations, that the stress can be expressed explicitly in the form g = o,Z+ n2A + tr3d+ qd* + a5(Ad + dA) + a6(Ad’ + d*A)

(6)

where cyl, cx2,. . , a6 depend on p, 8 and the invariants trd,

trd*,

trd”,

trAd:

trAd2

and ‘tr’ denotes the trace operation.

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The above applies in the case of an unconstrained transversely isotropic material. If the material is subject to kinematic constraints, such as incompressibility or inextensibility in a specified direction, the stress can be decomposed into two parts a=an+r where on is a reaction stress, that is a reaction to the constraints, and r is the extra-stress, which requires a constitutive equation for its determination. For a material which is incompressible and inextensible in the direction a, the reaction stress is a hydrostatic pressure p and a fibre tension T, so that ba = -pZ+

TA

wherep and Tare arbitrary in the sense that they are not related to the deformation and eventually are determined by equations of motion and boundary conditions. For a material that is incompressible and inextensible in direction a, the terms ail and cqA in (6) can be absorbed in ba, and we obtain

REFERENCES 1

2

3

a=aa+T = -pZ+

geometries; a number of solutions are given in ref. 10. The equations are also amenable to numerical solution11”2. I would not like to leave the impression that Tryfan was the sort of mathematician who considers his task complete when he has formulated some equations. He was essentially a problem solver, and in ref. 10 and elsewhere in his papers there can be found an abundance of solved problems. Time does not allow me to give any illustrations. It has been a privilege to give this memorial lecture, but Tryfan’s true memorial is the body of work that he has left us. He detested any form of secrecy in science, and his work is openly published and speaks for itself.

TA+qd+~d~

4

+ fi3(Ad+ dA) + K4(Ad2 + d2A)

(7) where kl, ICY,63 and k4 are functions of 8, trd2, trd3 and trAd2. For some fibre-resin systems, it has been observed that to a good approximation the relation between 7 and d is linear. In this case (7) reduces to 0 = -pZ+

TA + 2md+

2(qL - m)(Ad+

dA)

(8)

and only two viscosity constants, nr and x, are required to characterize the material. Here nr is the viscosity for shear flow transverse to the fibres, and nL the viscosity for shear flow parallel to the fibres. This completes the formulation of a simple but realistic theory. The system of governing equations consists of (I), (4), (8) and the equations of motion (or, for slow flows, equations of equilibrium). Only two material constants, m and nL, are required for material characterization. The equations are sufficiently simple to admit a variety of analytical solutions for simple

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5 6

I 8

Lee, E.H. and Rogers, T.G., Solution of viscoelastic stress analysis problems using measured creep or relaxation function, J. Appl. Mech. 1963, 30, 127 Lee, E.H., Rogers, T.G. and Woo, T.C., Residual stresses in a glass plate cooled symmetrically from both surfaces, J. Am. Cerum. Sot. 1965,48,480 Lee, E.H. and Rogers, T.G., On the generation of residual stresses in thermoviscoelastic bodies, J. Appl. Mech. 1965, 32, 814 Mulhem, J.F., Rogers, T.G. and Spencer, A.J.M., A continuum Proc. Roy. Sot. model for fibre-reinforced plastic materials, 1967, A301,413 Pipkin, A.C. and Rogers, T.G., Plane deformations of incompressible fiber-reinforced materials, J. Appl. Mech. 1971,38, 634 O’Neill, J.M., Rogers, T.G. and Spencer, A.J.M., Thermally induced distortions in the moulding of laminated channel sections, Math. Eng. Ind. 1988, 2, 65 Rogers, T.G., Rheological characterisation of anisotropic materials, Composites 1989, 20, 21 Golden, K., Rogers, T.G. and Spencer, A.J.M., Forming kinematics of continuous fibre reinforced laminates, Composites Manufacturing 1991, 2, 261 Rogers, T.G. and O’Neill, J.M., Theoretical analysis of forming flows of fibre-reinforced composites, Composites Manufacturing 1991,2, 153 Rogers, T.G., Shear characterisation and inelastic torsion of fibre-reinforced materials, in ‘Inelastic Deformation of Composite Materials’ (Ed. G.J. Dvorak), Springer Verlag, New York, 1991, pp. 653-674 Hull, B.D., Rogers, T.G. and Spencer, A.J.M., Theoretical analysis of forming flows of continuous fibre-resin systems, in ‘Flow and Rheology in Polymer Composites Manufacturing’ (Ed. S.G. Advani), Elsevier, 1994, pp. 203-256 0 Bradaigh, C.M., Sheet forming of composite materials, in ‘Flow and Rheology in Polymer Composites Manufacturing’ (Ed. S.G. Advani) Elsevier, 1994, pp. 517-569

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