New types of reinforced composite materials

NEW TYPES OF REINFORCED COMPOSITE MATERIALS

J.G.MORLEY Wolfson Institute of In terfacial Technology, University of Nottingham, England

~P~C

NORTH-HOLLAND PUBLISHING COMPANY

—

AMSTERDAM

PHYSICS REPORTS (Section C of Physics Letters) 28, No. 3(1976) 245—302. NORTH-HOLLAND PUBLISHING COMPANY

NEW TYPES OF REINFORCED COMPOSITE MATERIALS J.G. MORLEY Wolfson Institute of Interfacial Technology, University of Nottingham, England Received June 1976 Contents: 1. Introduction 2. Preliminary considerations 3. Stiffness and strength of conventional composites contaming continuous reinforcing members 3.1. Reinforcement by elastic modulus difference 3.2. Reinforcement by differential stressing 3.3. Reinforcement of an elastic/plastic matrix 4. Stress transfer in conventional composites 4.1. Stress transfer by elastic shear displacement (theoretical analysis) 4.2. Stress transfer in real composites 4.3. Strengthening and stiffening by discontinuous fibres 5. Stress transfer in composites with non-fracturing reinforcing elements 5.1. Theoretical considerations 5.2. Design of experimental systems (duplex fibres) 5.3. Decoupling load Lmax for helical core systems 6. Reinforcement by core/sheath reinforcing members 6.1. General considerations 6.2. Longitudinal stress distributions between core and sheath 7. Fracture of conventional unidirectional composites under tensile loading 7.1. Continuous fibres —~ single and multiple fracture 7.2. Continuous fibres of uniform strength —work of fracture

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7.3. Discontinuous fibres 7.4. General summary of fracture processes occurring in conventional composites under tensile loading 8. Fracture in composites containing non-fracturing reinforcing members 8.1. General considerations 8.2. Simple tensile loading of a unidirectional composite structure with non-fracturing components 8.3. Propagation of an edge crack in a duplex fibre reinforced sheet by a monotonic crack opening force 8.4. Crack stability in a composite sheet reinforced by non-fracturing reinforcing members 9. Behaviour of reinforced composites under cyclic loading 9.1. Conventional composites 9.2. Fatigue crack growth in composites containing non-fracturing reinforcing members 10. Possible future developments and alternative nonfracturing reinforcing members 10.1. Hybrid systems based on non-fracturing reinforcing members 10.2. Hybrid conventional reinforced composites 10.3. Stress controlled decoupling by plastic deformation 11. Conclusions References

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J.G. Morley, New types of reinforced composite materials

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Abstract: The physical properties of solids determine their usefulness as structural materials. Metals have some disadvantageous characteristics which reduce their effectiveness in critical engineering applications. These limitations can be overcome by the use of certain types of fibrous reinforced composites which have become available over the last few years. However, these materials in turn have their own inherent limitations, particularly in their mode of fracture under overload conditions. In this review the basic properties of conventional fibrous composites are discussed, particular emphasis being given to physics of these failure processes. In an attempt to overcome some of these limitations a new type of fibre reinforced composite has been designed and preliminary research data on the physical properties of these systems has now been obtained. The primary reinforcing elements extend throughout the whole length of the composite but differ from others in that they do not fracture when the composite is subjected to a wide range of loading and deformation conditions. These characteristics are achieved because the interface between the primary reinforcing members and the rest of the composite structure is responsive to the local stress carried by the reinforcing members. This stress controlled decoupling/recoupling process is very broadly analbgous to the transition between elastic and plastic deformation in metals and the physical principles underlying the design of the reinforcing members are outlined. Because the reinforcing members do not fracture any failure process is confined to the rest of the composite structure. Various interactions occur between the non-fracturing and fracturable parts of the system and these suppress crack growth in the latter, thus producing a structure possessing very considerable damage tolerance. A preliminary analysis of the basic physics of these interactions is given. Possible future developments of these materials are outlined. Also ways are discussed in which the simple analytical models, developed in the study of the fracture processes occurring in these materials, may be applied to more conventional fibrous composites.

1. Introduction Until recently the efficiency of metals has been unchallenged as structural materials and there are good reasons why this should be so. Any elastic solid supports an externally applied load as a result of forces generated by displacing the atoms, or molecules, of which it is formed from their equilibrium positions. This holds for crystalline materials and amorphous solids such as glass. In contrast with elastic brittle crystalline materials, a metal can deform in a ductile manner at some fairly clearly defined yield stress and can absorb a large amount of energy in doing so. This makes it very difficult for cracks to propagate under monatonic loading because of the large amount of energy absorbed in rupturing the material. Even more importantly, the yield stress can be arranged to increase as deformation proceeds by a process known as work hardening or strain hardening. This feed back mechanism stabilises the failure process and prevents premature fracture of the material. Metals have two basic disadvantages as structural materials. The first limitation is that their Young’s modulus and strength values are, in general, almost directly proportional to their densities. Since it is the Young’s modulus to density ratio and strength to density ratio which determines structural efficiency, there are no major advantages to be gained in this respect by using one metal alloy rather than another. The second difficulty is that, under cyclic loading conditions, a single fatigue crack can grow until it becomes of a sufficient size to cause catastrophic failure of an engineering component. Because of these inherent limitations a serious interest in the development of new types of fibrous composite materials began about 1960. In these systems various types of reinforcing fibres having high strengths, high Young’s moduli and low densities, are embedded in various types of matrix substances. Over the last ten years or so a considerable effort has been devoted to the development of these materials and to the scientific study of their properties. A list of some of the more recent books, conference reports and review articles is given [1—7]. As a result of these efforts new fibrous composites of remarkable strength, Young’s

248

J. G. Morley, New types of reinforced composite materials

modulus and resistance to fatigue failure have been produced and these are now finding applications in demanding engineering situations, particularly in the aerospace field. Chronologically the first of the new fibres was boron, produced by depositing boron on to a hot tungsten wire from boron halides in the presence of hydrogen [3, 51. The tungsten core is converted into tungsten borides during the deposition process. The complex microstructure of boron fibres leads, very conveniently, to residual stress distributions which leave the surface in compression. This makes the fibre less sensitive to surface damage which, in general, produces stress concentrating flaws and would, in other circumstances, drastically reduce the strength of the material. Residual internal stresses also reduce the effect of internal defects which are associated with the transformation of the tungsten wire core. Boron fibres are produced in two sizes of diameter 0. 10 mm and 0. 1 5 mm in diameter. They have elastic modulus to density ratios an order of magnitude greater than structural metals and show similar improvements in specific strength values. They are also perfectly elastic having failing strains of rather less than 1%. Boron fibres are being used to reinforce polymers and aluminium alloy matrix materials. The second technologically attractive ceramic reinforcing fibre to be produced during the 1960’s was carbon. These fibres are produced by the pyrolysis of a textile precursor fibre, usually rayon or polyacrylonitrile (PAN). The polymer is converted to an imperfect graphitic structure which is preferentially aligned with the axis of the fibre [8]. The presence of weak planes aligned with the fibre axis makes carbon fibres relatively insensitive to surface flaws and they have been used very effectively to reinforce polymeric matrix materials. The small diameters of the fibres (an order of magnitude less than boron fibres) make carbon fibre composites more susceptible to failure in compression by the buckling of individual fibres. This form of failure is strongly dependent on the mechanical properties of the matrix and the presence of small defects in it. Again the failing strain of carbon fibres is in the region of 1% so that relatively small amounts of energy can be absorbed elastically by the composite before fibre fracture occurs. The third advanced reinforcing fibre to become available is an organic fibre of very low density having a Young’s modulus about one third that of boron fibres and a diameter similar to carbon fibres. Because of its very low density, its stiffness to weight ratio is comparable with most types of carbon fibres. Its tensile failing strain is something like 2% and composites utilising these fibres are therefore capable of absorbing appreciably more energy through tensile deformation before fracture than is the case with carbon fibre composites. The main disadvantages of these high stiffness polymer fibres, which have the trade name “Kevlar” (Du Pont trade mark), lies in their relatively poor performance in compression although specific compression strengths comparable with high duty metal alloys are observed in composites utilising these fibres. Progress has been made towards the development of high temperature composite systems by the controlled solidification of metal alloys to give an aligned microstructure of either fibres or lamellae within a metal matrix. The reinforcement consists of intermetallic compounds or metal carbides. It has proved possible to control the composition of the metallic matrix and to modify its metallurgical properties by subsequent heat treatment. All of the fibrous composites mentioned above have superior characteristics under cyclic loading over conventional metal alloys. The composites do suffer from fatigue damage but this is gradual and progressive and does not lead to catastrophic failure as can be the case with metal alloys. Also the important structural ratios of Young’s modulus/density and the strength/density of the new reinforcing fibres are about an order of magnitude greater than those of metal alloys and this is reflected in the properties of the composite systems in which they are used.

J.G. Morley, New types of reinforced composite materials

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However, despite these undoubted advantages, existing advanced fibrous composites have some characteristics which place them at a disadvantage when compared with metals. Their behaviour is elastic up to failure, and their failing strains are low 1%), so consequently, little energy is absorbed before fracture. As a result, although strong, they can suffer severe damage under impact conditions, particularly when in the form of relatively thin sheets. In similar circumstances, a metal would absorb the impact energy by plastic deformation and suffer little change in its load bearing ability. The most widely used advanced fibrous composites make use of a polymeric matrix. The main advantage in using a polymer is that fabrication is relatively easy for simple engineering structures. However, large expensive multi-axis filament winding machines are necessary for the fabrication of large engineering structures having complex shapes and designed to meet complex stress distributions. This is necessary because the relatively poor mechanical properties of polymers show to disadvantage when stresses are applied at an appreciable angle to that of the reinforcing fibres so that they have to be aligned in various directions to support the stresses carried by the composite structure. Thus the problem of dealing with complex stresses can be alleviated to some extent, by producing laminated structures having fibres aligned in various directions. However, it should be noted that, in order to produce a sheet structure having properties substantially isotropic in the plane of the sheet, a reduction in mechanical properties to less than one third of the corresponding unidirectional structure has to be accepted. For this condition, and remembering that the fibre volume fraction is usually not greater than 60%, the margin of improvement in terms of strength/density and Young’s modulus/density of advanced composites over isotropic metals is largely eliminated. The mechanical properties of fibrous composites with reinforcement oriented in three dimensions are much worse than those of metals. A further disadvantage of polymeric matrix systems is their sensitivity to errosion and these materials often require surface protection. Because of the limitations of polymers as matrix materials a considerable effort has been made towards the development of ceramic fibre reinforced metals. The advantages expected of this combination follow from the improved mechanical properties of metals, which reduce the need for multidirectional reinforcement in many engineering situations and also give improved surface durability and temperature capability in the composite system. The primary difficulties in this area stem from the physical and chemical problems encountered in the fabrication of these materials. This topic is outside the scope of this review. So far the fabrication processes used are slow and expensive and not readily applicable to the manufacture of large components. Only one ceramic fibre metal matrix system (boron—aluminium) has been produced on an engineering scale and it now seems very unlikely that metals with attractive engineering properties, such as stainless steel, can be utilised as matrix materials by these fabrication procedures. As indicated above, the fundamental limitation of existing fibrous composites is associated with the physical processes occurring during fracture under monatonic loading conditions. The failing strains of the fibres govern the failing strain of the composite and are at least an order of magnitude less than those of ductile metals. The primary energy absorbing process is due to broken fibres being pulled out of the matrix, across a crack, as the fracture faces separate. This proceeds at a diminishing stress level so that the failure process is confined to a localised region which forms the “weakest link” of the composite system. It is true that some types of failure process can proceed at an increasing stress level notably multiple matrix cracking discussed in section 7 but these are again limited to the failing strain of the reinforcing fibres. The “unforgiving” nature of fibrous composites also presents problems in engineering design, manufacture and (—j

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—

250

J.G. Morley, New types of reinforced composite materials

quality control. These materials are much more limited in their ability to deform (and so relieve unforseen local high stresses) than is the case with metal structures. This limitation requires improved standards in the design and manufacture of engineering components using composite materials and can require the use of non destructive testing techniques to detect the presence of flaws. Joining also presents difficulties and much higher engineering efficiency is achieved with composites if the complete structure is fabricated in one piece. It is clear that the properties of fibrous composites would be considerably improved if it could be arranged for these materials to accept a tensile deformation much greater than the failing strain of the fibres whilst still retaining at least a major portion of their tensile load bearing ability. Since the physical properties of the fibres and matrix cannot be changed significantly, attention has to be focussed on the properties of the interface between the reinforcing members and the rest of the composite structure. It is via this interface that excessive stresses in the matrix are transferred to the fibres and conversely where excessive stresses in individual fibres are transferred back to the composite structure. Following this line of argument a new type of fibre reinforced composite stnicture has been investigated. The new composites differ from conventional fibrous composites in that the reinforcing members are prevented from fracturing by a stress controlled decoupling mechanism which decouples the reinforcing members from the rest of the composite structure in regions where they are experiencing excessive stress. This decoupling process is reversible and the physical principles upon which it depends are outlined in section 2. Under normal engineering design loads the reinforcing members stiffen and strengthen the composite structure in a manner similar to that of conventional reinforcing elements. Attention has been focussed on the reinforcement of sheet metal and the reinforcing elements studied have had diameters in the region of 1 mm, i.e. an order of magnitude greater than boron fibres. The relatively large size of the reinforcing elements has made it practicable to manufacture composites by the use of straightforward fabrication techniques. The basic characteristics of the new composites are discussed in sections 5, 6, 8 and 9 of this review and, in order to provide a basis for comparison, the properties of conventional composites, with particular reference to the physics of their fracture processes, are discussed in sections 3, 4 and 7.

2. Preliminary considerations In conventional materials of all types, the application of a concentrated load results in a relatively localised region of high stress concentration and this can lead to local failure. The concentrated load can be produced, for example, by the local application of an external force, by a geometrical discontinuity in an engineering component or by the presence of some form of local damage such as a flaw or crack. In the case of fibre reinforced materials, there are clearly major advantages to be gained through the use of a powerful stress diffusing mechanism which would limit the development of very localised high stresses in the reinforcing fibres. This process already occurs to some extent in conventional fibrous composites but is not completely effective because of the constant value of the shear strength of the fibre matrix interface. Thus these materials are insensitive to the very localised high stresses occurring around the tip of a small crack or notch, but fibre fracture and composite fracture occurs if the high stresses are applied over a more extended region. The situation can be improved

J.G. Morley, New types of reinforced composite materials

251

if it is arranged for the reinforcing members to become decoupled from the rest of the composite structure in any region of high stress, whatever its extent. Thus the excessive tensile loads carried by the reinforcing members would be transferred back to the composite system only in those regions which are not experiencing excessive stress. If it is arranged for the decoupling process to proceed progressively as the stress carried by the reinforcing member increases, the concentrated loads can be accommodated by the elastic extension of increasing lengths of decoupled reinforcing members. Such a stress diffusing process would proceed at an increasing stress level and thus stabilise any fracture process. Also fracture of the reinforcing members would be prevented if complete decoupling is achieved at a stress level less than the breaking stress of the reinforcing members. In the limit a continuous reinforcing member would become everywhere decoupled from the composite structure except near its free ends which must be carrying zero stress. If the interfacial coupling is generated by frictional effects any further deformation would result in the reinforcing member being drawn through the composite structure against frictional losses. Experimental systems possessing these characteristics have been constructed from thin walled steel hypodermic tubes of about 1 mm in diameter containing a steel wire core. The core is the primary reinforcing member and the tube is attached directly to a metal sheet or embedded in a matrix. A frictional bond is generated between the core and the tube and this is made responsive to the tensile load carried by the core by arranging for the core to be slightly convoluted, usually helically. In the systems studied, the diameter of the steel core wire is only slightly less than the internal diameter of the tube and the cross sectional areas of wire and tube are similar. The geometrical form of a typical core/tube reinforcing member is illustrated in fig. 1. When the helical core is subjected neither to longitudinal tensile loads nor lateral restraint, its overall diameter is greater than that of the internal diameter of the tube. The reinforcing member is fabricated by drawing the core into the tube and as this occurs the deformations experienced by core and tube are purely elastic and reversible. Under normal engineering loads the core exerts a considerable pressure on the walls of the tube thus developing a strong frictional bond and generating a store of strain energy in the system. Due to the very large effective Poisson’s ratio of the core, the pressure exerted by the core on the tube and hence the frictional shear strength of the core/tube interface, falls rapidly as the tensile strain of the core is increased. At the same time the initial store of strain energy in the system is correspondingly reduced and this has the effect of increasing the longitudinal elastic modulus of the core (see section 6). It is arranged for the core to become completely decoupled from the tube while carrying a load less than its tensile breaking load. Hence fracture of the core is prevented and the composite system has residual “fail safe” characteristics. Both core and tube act as a single reinforcing member, providing the core is carrying a load less than its decoupling load and the load carried by the tube is less than its breaking load. Outside these conditions the tube elements behave as conventional reinforcing members and can fracture under appropriate circumstances. However, any fracture process is confined to the tubes and matrix and a crack in-this part of the composite system is bridged by the core reinforcing members. They can interact very effectively with the stress fields around a crack and in certain circumstances can prevent the occurrence of unstable crack growth. This feature is discussed in section 8. Preliminary studies of the fatigue behaviour of these systems have been carried out and it has been shown that the experimental core reinforcing members can accept cyclic stresses which are many times greater than the design loads of highly stressed engineering components subjected to cyclic loading. This feature coupled with the ability to prestress a sheet metal “matrix” in corn-

J.G. Morley, New types of reinforced composite materials

252

~

~

0.805mm

0.72 mm din.

din.

1072mm din. ~ \~\

____

LONGITUDINAL

\

~

SECTION

U

CROSS

SECTION

Fig. 1. Illustrating the geometrical form of a typical core/tube reinforcing member.

pression is of considerable value in suppressing the growth of fatigue cracks in sheet metal structures. This aspect is discussed in section 9. All of the experimental work to date has been carried out using hypodermic tubes as the outer part of the reinforcing member. This material is intrinsically much cheaper than, for example, boron fibres and seems to be quite realistic for use in advanced engineering applications. However, engineering structures possessing the above mentioned features but fabricated from sheet metal can be envisaged and hybrid systems containing quantitie~of stiff ceramic reinforcing fibres can also be postulated. These are discussed in section 10. This approach offers the possibility of utilising, in some measure, the engineering advantages of stiff, lightweight materials in combination with the further advantages of the new composite properties outlined above. Also, since conventional fabrication techniques are suitable, a wide range of material types can be incorporated as components in the composite structure.

3. Stiffness and strength of conventional composites containing continuous reinforcing members Fibrous composites are designed so that the reinforcing elements should carry the maximum practical proportion of an applied tensile load. This will be governed by a combination of economic and technical factors. When the reinforcing members are uniformly distributed and are continuous or have very high aspect ratios it can be assumed that the longitudinal tensile strain in both fibres and matrix will be equal. Hence, if the Poisson’s ratios of the two phases are equal, we have the relationship, Ec=EfVf+EmVm

(1)

J.G. Morley, New types of reinforced composite materials

253

where E is the Young’s modulus, V the component volume fractions and the subscripts c, f and m refer to composite, fibres and matrix respectively. If the Poisson’s ratios of fibres and matrix (i’~and Vm) are different, the mutual constraints which one phase imposes on the other lead to increases in the distortional energy at any particular strain and hence to an increase in the stiffness of the composite structure as was pointed out by Hill [9]. Eq. (1) therefore represents the lower bound for the longitudinal stiffness of an aligned fibrous composite but, in most cases, the difference between v~and Vm is too small to have a significant effect. If both fibres and matrix are perfectly elastic up to fracture the tensile stress carried by the composite, ~ during elastic tensile deformations will be given by, UcfVf+UmVm_UfVf+(l_Vf)fjm

(2)

where Cf and Cm are the stresses carried by fibres and matrix. 3.1. Reinforcement by elastic modulus difference When both fibres and matrix are elastic and are carrying a tensile strain e, the load carried by the fibres and the matrix is Ef V~eand Em VmC respectively. In the case of fibre reinforced polymers Ef is usually between one and two orders of magnitude greater than Em and Vf is usually between 0.1 and 0.7 so that the reinforcing fibres carry most of the load applied to the composite structure. 3.2. Reinforcement by differential stressing The fibres can be preferentially stressed in tension so that in the absence of an external load, the matrix carries a corresponding longitudinal compressive load. This technique is used in reinforced concrete structures and is particularly valuable when the fibres can carry a relatively high elastic strain before failing, but the matrix can support a very small tensile strain before cracking. In these circumstances it is desirable to limit the tensile strain carried by the matrix when the composite structure is under load. If Cf and ~ are the initial tensile stress in the fibres and corresponding compressive stress in the matrix we have for the initial equilibrium condition, VfCf1

—

VmUmi

=

0

(3)

and by combining both mechanisms in one composite structure we have, =

Ec6

=

(Uf~ + cEf) Vf +

(CErn

—

Umi)Vm

where e is the longitudinal tensile strain and a~the average stress developed in the composite structure under the application of an external tensile load. 3.3. Reinforcement of an elastic/plastic matrix When the matrix is a ductile metal, which has a yield point lower than the failing strain of the fibres and also has a work hardening capability, the composite characteristics are more complex. For strains up to the elastic limit of the metal matrix the relationships described above are applicable. Beyond this point a simple approach would suggest that the effective elastic modulus of the

254

J. G. Morley, New types of reinforced composite materials

composite E~at a strain e will be given by, E~=EfVf+ [dCm/dIeVm

(5)

where [dUm/d] ~ is the tangent modulus of the matrix at the strain e. The rate of work hardening of a metal matrix however can be apparently increased by the presence of reinforcing fibres [10]. Since the maximum strength (u.t.s.) of a work hardening metal matrix will be developed at strains very much greater than the failing strain of most reinforcing fibres (which is 1%) it is possible for the ultimate tensile strength of a metallic matrix to be reduced by the presence of a small volume fraction of strong brittle fibres. Providing the fibre volume fraction Vf is sufficiently large, the tensile strength of a unidirectionally reinforced ductile matrix is given by, 0cult

0f

=

ult

Vf

+

o~n(l Vf)

(6)

—

where the subscript ult refers to the ultimate tensile strength of composite and fibres and a~is the stress carried by the matrix at the ultimate failing strain of the fibres. If the composite strength is to be greater than the strength which can be developed by the work hardened matrix. 0mult’ then, UccTfVf+U~fl(l

—

Vf)>fJmult.

Hence, the critical volume fraction of fibres ~ is given by, Vent

(Cmuit

1J~fl)/(CfuIt

which must be exceeded for fibre-strengthening (7)

U~).

At the breaking strain of the fibres the average stress carried by the matrix is U~. The maximum stress which the matrix can support is ~~muit so that failure of the fibres results in immediate failure of the composite if, C ‘1 V~
V

f.ult

f

0m~

f

This determines a minimum volume fraction Vmjn which must be exceeded if the strength of the composite is to be given by eq. (6). Hence, V —(a ~u’~11u ~ 1 —

mm

‘

m.ult

m11~ f.ult

m.ult

The arguments outlined above were first given by Kelly and Davies [11]. In practical systems containing fibres which are very much stronger than the matrix Vmin Vcrit u~/u~ ult where ~ is the work hardening capacity of the metal matrix. 4. Stress transfer in conventional composities Engineering loads are applied to the matrix and are then transferred to the reinforcing fibres. The mechanism of stress transfer is therefore of paramount importance in controlling the characteristics of composite systems.

J.G. Morley, New types ofreinforced composite materials

255

4.1. Stress transfer by elastic shear displacements (theoretical analysis) The all elastic situation has been examined analytically by Cox [12] Dow [13] and Rosen [141. The treatment given by Cox is outlined here. All three approaches give similar results but the numerical values obtained are only very approximate because of the assumptions made in constructing the analytical model. In the model used by Cox an element of the composite structure is considered to be a cylinder of matrix material containing a reinforcing member along its axis. The ends of the fibre are considered to be unloaded and stress concentrations associated with the fibre ends are ignored. The matrix cylinder is considered to be subjected to a uniform longitudinal strain , fig. 2. The rate of transfer of load from the matrix to the fibre at a distancex from one end of the fibre is proportional to the difference between the displacements of the fibre u and the displacement v which would have occurred in the matrix in the absence of the fibre. Hence, if P is the load carried by the fibre a distance x from the end we have, ,

dP/dxH(u—v)

(9)

where H is a constant which depends on the geometrical arrangement of the fibre and matrix and their elastic constants. Since P = EfAf (du/dx), whereAf is the cross sectional area of the fibre, and dv/dx = e, eq. (9) can be differentiated to give, C). (10) d2P/dx2 = H(P/EfAf —

This equation is solved by the substitution P = EfAfC + R sinh f3x + S cosh 13x. Where = (H/EfAf)112 and R and S are constants. Since the load on the fibre must be zero at each end it follows that

r

P=EA elI— ‘ L ~‘

coshl3(l/2—x)1 I cosh6(l/2) J

(11)

so that P approaches EfAf e at the centre of a long fibre. For a cylindrical fibre of radius rf embedded axially in a cylinder of radius rm Cox obtained, H = 2ir Gm/ln(rm/rf)

(12)

where Gm is the shear modulus of the matrix. Hence, 3= [27rGm/EfAfln(rm/rf)]1~’2.

(13)

xi.:

po Fig. 2. The

Af

Irm

D

composite element used to derive the theory of stress transfer (after Cox [12]).

256

J.G. Morley, New types of reinforced composite materials

~‘~:x: ~

Fig. 3. The change in fibre tensile stress o and the shear stress at the interface r along the length of a fibre within an elastic matrix.

The analyses of Dow and Rosen contain the factor Gm/Eç in the constant f3 and only differ in the parts which involve the geometrical arrangement of the fibres. If r(r) is the shear stress in the direction of the fibre axis on planes parallel to this axis then at the surface of the fibre r = rf and dP/dx

=

—-2lrrfr(rf).

(14)

Also P= ufAf whereof is the tensile stress carried by the fibre. By differentiating eq. (11), and rearranging, an expression for r can be obtained. This has its largest value at the fibre ends (fig. 3). For long fibres the ratio of the maximum shear stress at the interface to the maximum tensile stress in the fibre is found to be comparable with Gm/Ef for ordinary fibre volume fractions. This analysis therefore implies that the stresses in the matrix at the fibre ends will be similar to the maximum tensile stresses developed in the fibre. If the stress concentrating effects of the geometrical form of the fibre ends is taken into account the shear stresses in the matrix near the fibre ends will be even higher. Photo elastic studies confirming the magnitude of the stress concentrations near the fibres ends have been made by Schuster and Scala [151 and Tyson and Davies [16]. It follows that when the matrix is a ductile metal, yield will take place in the matrix near the fibre ends and where the matrix is an elastic polymer, or ceramic, shear failure of the matrix or the fibre matrix interface will take place in this region. 4.2. Stress transfer in real composites Since some sort of yield must occur near the fibre ends the actual stress distribution there cannot be given by fig. 3 which is based on elastic behaviour. Piggott [171 has calculated the stress profile in a fibre in a part elastic—-part plastic matrix. Outwater [181 has considered the case of a glass fibre reinforced polymer composite where the high shear stresses near the fibre ends cause failure of the fibre matrix bond and shear stress transfer in this region takes place by frictional interactions. The development of various possible stress distributions along a fibre is shown in fig. 4. As a load is applied to the composite, the stresses are everywhere below the elastic limit of the matrix and also the strength in shear of the interface (providing the effect of any severe stress concentrations at the fibre ends are neglected). Thus for this condition the elastic analysis applies (fig. 4a). As the applied load is increased the elastic limit of the matrix is reached, if it is an elastic/ plastic material, or the strength in shear of the interface (or the matrix near the interface) is reached if the matrix is wholly elastic. If we assume a constant interfacial shear strength r, gener-

J. G. Morley, New types of reinforced composite materials

257

(b)

(a)

Cc) Cd) Fig. 4. Illustrating the possible tensile stress distributions along a fibre and the development of interfacial debonding or plastic shear in the matrix.

ated either by ductile shear in a metal or by a sliding friction shear stress (if the interface has debonded), we have for the stress

C

at a distance x from the end of a circular fibre of diameter 2r,

C2rx/r.

(15)

This condition holds for values of x up to the point beyond which the elastic analysis holds (fig. 4b). As the tensile load increases eq. (15) holds for an increasing proportion of the length of the fibre. If the fibre is short enough the constant shear strength interface zones meet at the centre of the fibre before it fractures (fig. 4c). The limiting critical length l~for fibre non fracture is therefore given by,

l~=

Cf

~15r/r.

(16)

If the fibre length is greater than this value it will fracture as the load on the composite is increased. A fibre of perfectly uniform strength would fracture at itscentral point. In practical systems fibre strengths are not uniform so that the fibre fractures at some weak point or flaw somewhere in the highly stressed region of the fibre (fig. 4d). To a first approximation it is convenient to consider the stress distribution along a fibre to increase linearly from the free ends according to eq. (15). The central regions of a long fibre carry an almost constant load and in this region the tensile strain in the fibre is the same as that in the matrix. Since there is no relative displacement between fibre and matrix there is no stress transfer between fibre and matrix in this region. 4.3. Strengthening and stiffening by discontinuous fibres If the reinforcing fibres are considered to behave independently, i.e. to be unaffected by the presence of nearby fibre ends, the strengthening and stiffening effect of the fibres is reduced to the calculation of the average stress carried by the fibres up to the point of failure of the composite. This approach has been considered by Kelly [19],Kelly and Tyson [20] ,Spencer [211 and Kelly [22]. If the fibre length 1 is less than l~composite failure will be governed by the properties of the

258

J. G. Morley, New types of reinforced composite materials

matrix. The average stress on the fibres when the composite fails will be lufUIt/21C and the strength of the composite is then given by, (17) = lUf~~fl Vf/2lC + 0m ult V. If the fibres are longer than l~composite failure will be governed by the properties of the fibres and the average stress & carried by a discontinuous fibre at its failing stress 0f~.uItwill be given by. U =

~

—

l~/2l)

(18)

so that the strength of the composite becomes, CC

~t.uitVfU

~lc/2l)+U~Vm

(19)

where u~is the stress carried by the matrix at the failing strain of the fibres. It follows that discontinuous fibres strengthen and stiffen a matrix to a lesser extent than continuous fibres but that when the fibre length is large compared with l~this reduction will be negligible. However, it should be noted that when the fibre length equals l~the effectiveness of the fibre is reduced to one half that of the corresponding continuous fibre. Consideration has been given to the local increase in the stress carried by fibres adjacent to the ends of discontinuous fibres, or to fractured nominally continuous fibres. Work in this area has been carried out by Riley [23] and Hedgepeth and Van Dyke [241. A measure of uncertainty exists in the interpretation of the above mentioned, and also other studies, of this problem. However, it does not appear to be a serious factor in controlling the behaviour of technological fibrous composites.

5.

Stress transfer in composites with non-fracturing reinforcing elements

5. 1. Theoretical considerations As described in section 4 stress transfer in conventional composites takes place primarily either by frictional interactions at the fibre matrix interface or, in the case of a ductile metal matrix, by yield in shear in the matrix near to the surface of the fibre. Both mechanisms produce an approximately uniform rate of stress transfer so that for a fibre of circular c~rosssection, daf/dx = 2r/r where ~ is the tensile stress carried by the fibre at a distance x from the end, 2r is the fibre diameter and r is either the sliding friction of the fibre matrix interface or the yield strength in shear of a metal matrix. We can modify the situation so that r, instead of being constant, becomes a function of a so that T 0fTOKU

(20)

where i~0f is the local shear strength of the interface when the local tensile stress carried by the fibre is a~and r0 is the shear strength of the interface when the local tensile stress carried by the

J.G. Morley, New types fo reinforced composite materials

259

full

/

/~ r

Fig. 5. Illustrating tensile stress distributions along fibres with constant interfacial shear strengths and stress controlled interfacial shear strengths.

fibre is zero. K is a constant. If T0f becomes zero when the tensile stress carried by the fibre reaches some maximum limiting value Umax we have,

K

(21)

To/Cf max

and the fibre will not fracture [25] as a result of shear stresses acting at the fibre matrix interface providing that Cf max < Cf ult It follows that the limiting stress distribution along the fibre as it is being drawn through the matrix is given by, 2r0x/Cfmaxr)} (22) CfX = Cfmax{l exp(— where Cf ~ is the stress carried by the fibre at a distance x from a free end. Since the fibre does not fracture, this condition corresponds to that in a conventional composite when the fibre length 1 ~ l~.Thus, by controlling the shear strength of the interface by the tensile stress carried by the fibre,.the fibre critical length and stress transfer length can be made infinitely long. At lower stress levels the stress transfer length is reduced considerably and in fig. 5, the stress distribution along such a reinforcing member is shown for different stress levels and compared with that of a conventional fibre having a constant shear strength interface. —

5.2. Design of experimental systems (duplex fibres) Various experimental systems containing stress controlled frictional interfaces have been studied. These have consisted of two part core sheath reinforcing members in which the core is in frictional contact with the sheath. The shear strength of the frictional interface between the core and the sheath has been made strongly dependent on the tensile stress carried by the core member by arranging for the pressure exerted by the core on the wall of the sheath to be similarly dependent. This can be achieved by arranging for the core to possess a very high effective Poisson’s

J. G. Morley, New types of reinforced composite materials

260

ratio. In any practial system, variations in dimensions will exist so that it is necessary to ensure that the core member undergoes a very large lateral contraction for a given tensile extension in order to ensure that stress controlled decoupling of the core tube interface will occur. This condition is obtained if the core member is slightly convoluted. Various geometries have been considered including a sawtooth waveform, a sinusoidal waveform or a helical core [261. However, the bulk of the research carried out to date has been concerned with tubular sheath/helical core systems. Stainless steel hypodermic tubes and steel piano wire cores have been mainly used in their construction. In these systems the stress controlled interface is the core/tube interface and the geometry of the system ensures that fracture of the core member is prevented. The tubular portion of the member functions in the same way as a conventional reinforcing member, strengthening and stiffening a matrix in which it is embedded via the tube/matrix interface. A frictional interface between a helical core and an outer tube will be developed if the diameter of the helix, when under zero external load, is greater than the internal diameter of the tube. Thus when the helix is pulled into the tube a pressure is generated at the helix tube interface and a corresponding frictional shear strength is developed. If only geometrical factors are considered, it is clear that the helix will become decoupled from the tube when it carries a tensile load of sufficient magnitude to extend the helix longitudinally and reduce its diameter to that of the internal diameter of the tube. The helical carbon steel wire cores used in the experiments described below have been prepared in the following way. A portion of a length of steel wire was closely wound on to a suitably sized mandrel which was rotated slowly in a lathe. The closely wound helix produced in this way was then formed into an elongated helix, having an overall diameter greater than the internal diameter of the tube to be used, by loading it in tension to a predetermined level. A suitable pre-stress load was determined for each wire/tube combination used, by first inserting the straight portion of the core through the tube and then repeatedly pulling the partially extended helix through a long length of tube, using a mechanical test machine, until the loads developed remained constant. This value was then used as a prestress load for further samples of the same wire/tube combination. This procedure ensured that the diameter of the unloaded helix was as large as possible consistent with the requirement that the deformation caused by pulling the core into the tube should remain within the elastic range of the material. The loads required to decouple a core helix of a given geometry has been examined by Chappell, Morley and Martin [27]. 5.3. Decoupling load (Lmax) for helical core systems A longitudinal tensile load L applied to a helix of radius r and pitch angle a can be resolved into a flexural couple, M = Lr sin a which tends to decrease the curvature of the wire, and a torsional couple given by, 1’ = Lr cos a which acts about the central axis of the wire forming the helix. In cases where the helix angle a is large the torsional couple can be neglected. This is the case with the systems studied since a is in the region of 850. The flexural couple Lr sin a tends to straighten the wire changing the helix

J. G. Morley, New types of reinforced composite materials

261

angle a and reducing the diameter of the helix. At the same time the tensile load L tends to cause the wire forming the helix to extend longitudinally in accordance with Hooks law. In this analysis it is assumed that this extension will increase the overall length of the helix and possibly the number of convolutions, but will not change the geometry of the individual convolutions, except by reducing the helix diameter, by an insignificant amount, through the Poisson contraction of the wire itself. This assumption is justified because, as is shown below, the effective Poisson’s ratio of the helix for the values of a used in these studies is an order of magnitude greater than the Poisson contraction of a conventional elastic solid. If a short length of helix formed from a wire of circular cross section is considered as a beam having an initial radius of curvature Pi then the application of a tensile load L along the axis of the helix produces a bending moment in the beam tending to straighten it and increasing its radius of curvature to P2. The bending moment M required to produce such a change in the radius of curvature of the segment is given by, MEI(l/p

1

l/p2)

—

(23)

where E is the Young’s modulus of the core and I is the geometrical moment of inertia of its cross section. Thus if torsional and shear effects are neglected we can write, Lmax

r2sina2EI(l/p1

—

(24)

l/p2)

where r2 is the radius of the axis of the wire forming the helix when the external diameter of the helix equals the internal radius of the tubular sheath and a2 is the corresponding helix angle. If we assume that the radius of curvature of the segment of helix under consideration is the same as that of the maximum radius of curvature of an ellipse inclined at an angle a to the longitudinal axis of the helix we have from the properties of the ellipse, 2a p = r/cos where r is the radius of the cylinder upon the surface of which the axis of the helical core lies. Hence, LmaxT2 sin a

2 a 2

EI[(cos

=

2a 1)/r1

—

(cos

2)/r2]

(25)

where r1 is the radius of the axis of the helix when unrestrained and under zero tensile load and a1 is the corresponding helix angle. Since for a core of circular cross section and diameter D, 4/64 1 irD we have, EirD4

rcos2a

2a

1

Lmax=

I—~-

.

64r2slna2L

r1

cos —

r2

2l

I

.

(26)

j

Now sin a2 = ?‘~2/l,cos a1 = 27rr1/l and cos a2 = 2rrr2/l where X2 is the wavelength of the helix when inside the tube and 1 is the length of core material in one wavelength of the helix. Therefore, T

~max

—L’ 3n4( — i.~iT ‘.

\/1~

S

~r1 r2~1iur2,~2 —

262

J.G. Morley, New types of reinforced composite materials

:t~°°

-

Li V

r~ ~

-~

V

/ 0

‘1~

V

C-

~

1030

V

V ~

V

•

A /

/

,/:

3 2

0)

C.~C

4 S

~—---~

/0

C

~~~TI0N OCVOF/ING

00

1000 CALL3T0~SLUE0

1500 3MAXiMVMP3LL-T~p0L33~/

2300

Fig. 6. Comparison of experimental and calculated values of pull through stress for various core/tube combinations (from ref. [27]).

and Umax =

E~r2D2(r1 r2)/4r2X2l. —

(28)

The general validity of eq. (28) has been confirmed [271, for various tube wire combinations and the correlation between theoretical values and observed values is shown in fig. 6. The limits of applicability of eq. (28) due to gross plastic deformation of the hypodermic tube as a result of large contact pressures is also indicated in fig. 6. A further test on the validity of eq. (28) has been carried out by observing the change in diameter of a helical wire with increasing applied tensile load [271. In this experiment the tube is not present. The helix is considered to be represented by a line drawn through the locus of the mid point of the wire cross section so the helix diameter a under a given load L is obtained by rearranging eq. (27) and substituting, 3D4)} +D. (29) a 2r1/{l +(16LX21/Eir In figs. 7 and 8 values of a calculated from eq. (29) are compared with experimental values. Fig. 7 shows the change in diameter of a helix initially fairly closely coiled, as the applied load is increased. This deformation causes the wire to deform plastically and the load was increased by steps of SON and cycled several times at each step interval in order to develop a reproducible elastic condition at each point. Fig. 8 illustrates the elastic behaviour of a helix after being prestressed to a by the expected debonding load Lmax, and its behaviour after being pulled through the tube several times in order to arrive at a reproducible condition. As discussed above it is desirable that the effective Poisson’s ratio of the helical core should be as large as possible in order that the effect of manufacturing tolerances on the dimensions of the component parts should be reduced to a negligible level. Direct measurements of the effective Poisson’s ratio of a helical core have been made and the results are illustrated in fig. 9. In this par-

J.G. Morley, New types of reinforced composite materials

500

I

263

500 Wire diameter

~——

400

400

300

300

V\

\

~

r Pullint\

~ \\Initial

tree wire

Wire afte

200

200

through l9tube

~

50

V

Os

Os

100

0

________________________________________________

_________________________________

External diameter of helix (mm)

______________________________________________________________________________________ 0 0.72 0,76 Ui 80 • 0.84 088 External helix diameter (mm)

Fig. 8. Experimental i~v and calculated 0 a values of helix diameters under load before and after being pulled through tube (from ref. [271).

Fig. 7. Experimental o, and calculated ~ values of helix diameter under load (from ref. [271).

Helix length~1030mm

16

084

83

14 S 12

0.82

10

0.81

-E

I 080

~

0.79

-

8 0 6

V

0

100

___0.76 200 Load (N)

300

N0

078 0.77

400

Fig. 9. Longitudinal extension versus tensile load ~ with the corresponding helix diameter 0 (from ref. [271).

264

J. G. Morley, New types of reinforced composite materials

pu))- ri _______________________________

_______

wire reinforced by bonded tube tenste loud

_______________

push-out

Ax

-~I _______________

—______________________

sliding

fit

/ support sleeve

compressive load

Fig. 10. General experimental arrangement used to measure compressive displacement loads for various lengths of core member (from ref. [26]).

ticular system the numerical value of the effective Poisson’s ratio is found to be 7.7 which is about 25 times greater than that of a normal solid. Measurements have been made of the compressive loads which can be supported by various lengths of helical core within a tube [261. In fig. 11 experimental values of pull-out and push-in loads are shown. The compressive loads which can be supported increase very rapidly with increasing core length. There is reasonable agreement between the experimental points and the smooth curve which corresponds to the equation, L =L el-- -Kx x max’ e The constant K in this equation was taken as the average of the values computed for each of the experimental points with Lmax being obtained from the extrapolated experimental data. In order

°

V

dup[eo

0.25

2 L.eogiflo.Lcore 2

I

fibre

300

lube JfliViiaciVifli.1Y1L

0~

/ 05

Fig. 11. Compressive and tensile loads required to cause the longitudinal displacement of various lengths of core element (from

ref. [261).

J.G. Morley, New types ofreinforced composite materials

265

to delay the onset of buckling in the core member when compressive loads were applied the experimental arrangement shown in fig. 10 was used.

6. Reinforcement by core/sheath reinforcing members 6.1. General considerations The reinforcing mechanisms available with core/sheath duplex fibres differ from those of conventional fibrous composites in many important respects. The primary objectives, in conventional fibrous composites, are that the fibres should carry a disproportionately large share of the applied load and that they should stiffen the matrix. In the case of polymeric matrix systems both these objectives are achieved because the fibre stiffness is usually two orders of magnitude greater than that of the matrix and, since the principle of equal tensile strains in both fibres and matrix is generally applicable, the fibres carry almost all the applied load even when present in relatively small volume fractions. In the case of metal matrix systems the Young’s modulus of the reinforcing fibre and the matrix are more nearly the same but here load transfer between the ends of the fibres and the matrix can take place very efficiently as a result of ductile shear in the matrix. It so happens that the only ceramic fibre metal matrix system under development on an appreciable scale is the aluminium boron system in which the elastic modulus of the boron fibres is about six times that of the aluminium matrix. For this system therefore the fibres, at a typical volume fraction of 0.5, carry about 85% of the applied axial tensile load (see section 3). Duplex fibres differ from other reinforcing fibres in that the core elements can be placed in residual tension with the rest of the composite structure under residual compression. Thus it is not mandatory for the core reinforcing fibres to be stiffer than the matrix in order that they may attract a disproportionate share of the load carried by the composite. However, if the core member is also stiffer than the matrix it will, in addition, attract a larger share of any stress applied to the composite just as is the case with orthodox composites (section 3). Two part core/sheath fibres can be considered to provide two separate reinforcing components. The tubes reinforce the matrix in just the same way as do conventional reinforcing fibres. At engineering design strains the cores and tubes can be considered as acting together as a single reinforcing member, except for any initial differential stress condition as indicated previously. The core and tube also interact to produce differential lateral elastic strains and, as a result of this, the longitudinal stiffness of the core is modified. As mentioned in section 5 the helical core has an overall diameter when unstressed, which is larger than the bore of the tube. It follows that, when the core is contained within the tube, hoop tensile stresses are developed within the tube and corresponding radial compressive stresses are developed in the core. Besides developing a strong frictional bond at the core—tube interface these stresses provide a store of strain energy which can modify very considerably the longitudinal stiffness of the core. Thus, through a choice of the geometrical form of the duplex reinforcing element, it is possible to modify its longitudinal stiffness independently of the intrinsic elastic properties of the material used in its construction. Some preliminary observations of this effect are discussed below.

J.G. Morley, New types of reinforced composite materials

266

(a) .

L max

(c)

L max

I~ Ix

core

zero load

S___-

zero load on

-~.~--~--r-~~

~

~

tube

•

r—~-~

L max (b)

(d)

t _

4

xbe

_

.~

~

—~

Fig. 12. Illustrating various possible longitudinal stress distributions between core and tube (from ref. [28]).

6.2. Longitudinal stress distributions between core and sheath The experimentally observed relationship between the loads applied to the ends of core members of various lengths in order to withdraw them from or insert them into a tube have been discussed in the previous section. It would be expected that the tensile load distribution along a long length of core member as it was being withdrawn from a tube would follow the same load/distance relationship as indicated in fig. 11 and this has been confirmed recently by direct experiment [28]. Measurements have been made of the tensile strain experienced at various points along the length of a tube as a core member was being withdrawn. (This situation corresponds to that indicated in fig. 1 2a.) From the strain measurements, the load experienced by the tube at the position of the strain gauge could be obtained since the tensile elastic modulus of the tube was known. The loads applied to the end of the core wire were also known and, from the directly observed relationship between the pull through load and the length of core member within the tube, the assumed load distribution along the core member could be plotted. Now the total load carried by the tube/core assembly must be constant at any cross section and equal to the total tensile load applied to the system. Hence, from the experimental relationship between the pull-out load and the length of core in the tube, the assumed load carried by the tube at any position of the strain gauge can be calculated. In fig. 1 3 the loads calculated in this way are shown at one position along the length of the tube as various lengths of core element are being withdrawn. These are compared with the corresponding values calculated from the straih gauge readings. Various possible stress distributions between core and sheath are shown diagramatically in figs. 1 2a, 1 2b, 1 2c and 1 2d. If, after reaching the situation indicated in fig. 1 2a, the load applied to the

J.G. Morley, New types ofreinforced composite materials

267

400 / / “0 /0

o ~

/

300

/ /

I

/

0

/ / / 200

/

0

/ 0/

100,

/

0

/

100

200

300

400

Load on tube calculated from stress transfer IN I

Fig. 13. Loads carried by tube calculated from core pull-out data compared with values ~btained from observed values of tube strain (from ref. [28]).

end of the core member is reduced the helix expands laterally and develops frictional contact with the wall of the tube. The frictional bond resists differential movement at the core tube interface so that, in the central region of the assembly, the tube is placed under compression by the helical core which is undergoing longitudinal elastic relaxation. Clearly, at all positions, in the absence of any external forces the tensile load carried by the core must balance the compressive load carried by the tube. Also the strain energy in the system must tend to a minimum value so that on the basis of these considerations the final stress distribution is calculable from a knowledge of the elastic characteristics of the core sheath assembly. The situation is complicated however by the presence of frictional effects, which dissipate energy by differential movements at the core sheath interface, during decoupling and recoupling. Hence, in the situation indicated in fig. I 2b some of the strain energy originally contained in the system will be dissipated by frictional losses. If an increasing tensile load is now applied to both ends of the core until a value of Lmax is reached, the core will become debonded over its entire length and the tube will carry zero stress. The stress distribution corresponding to this situation is indicated in fig. 1 2c. If the tensile load carried by the core is now reduced to zero the load distribution will again return to the situation indicated in fig. l2b. If a tensile load is applied to the ends of the tube, initially in the condition described in fig. 1 2b, load is transferred from the tube to the core. In contrast with the condition indicated in fig. 1 2c it is not possible, in this situation, to cause the tensile load carried by the core to increase above Lmax, although this load level can be approached closely. The situation when the load distribution along the core approaches its limiting condition is indicated diagramatically in fig. 1 2d. If additional load is now applied to the tube, and it does not fracture, this must be supported by an additional elastic extension of the tube. Since the load carried by the core cannot be increased above its limitingvalue and, assuming that the characteristics of the stress transfer mechanism are unchanged, the additional load would cause the upper curve of fig. 1 2d (indicating the load carried

268

J. G. Morley, New types of rein!breed composite materials

/7~>\\

4O0~

/

/

300

/

7

A’,,

e N,

Pull in load from load cell reoding

0

0

~. .-

0-

N,, 0 Local tensile load in wire calculated tram - o. s/rain in tube

A’

-

//

200~

.1

/

~

/1

3

-

1/JO

200

300

COO

500

600

700

LeL/gih. 3f.0050_io.0s.onlerface I m ml Fig. 14. Longitudinal distribution of residual tensile load carried by core member (from ref. 128]).

by the tube) to be translated upwards. The shape of both stress distribution curves however would remain the same. One feasible engineering objective would be to arrange for the core reinforcing member to be placed in residual tension with the tube member, and any matrix to which it is attached, in residual compression. Such a condition is useful in suppressing crack growth in the matrix. This is very important under conditions of cyclic loading when the matrix is a metal, particularly since it has been observed that laboratory duplex elements are capable of withstanding high stress levels for large numbers of load cycles. This issue is discussed in section 9. An experimental load distribution curve corresponding to that indicated diagrammatically in fig. I 2b is shown in fig. 1 4. For this particular system the tensile load carried by the core in its central regions is almost 300N and this corresponds with a similar compressive load in the tube. Details of this particular system are given in table 1. Also shown in fig. 14 is the limiting load distribution carried by the core member as it is being pulled through the tube in each direction. This load line therefore indicates the maxmium tensile load which the core can carry at any position as a result of frictional stress transfer generated by differential displacements at the core tube interface. This particular system was designed for the convenience of the experimental investigation and no attempt was made to maximise its mechanical properties with a view to possible engineering applications. It can be seen from fig. 14 that the effective stress transfer length for the core member in this system under these loading conditions is about 250 mm. Since the central region of the element has reached a limiting stress distribution condition this will be the case whatever the absolute length of the core/tube system. If the tube member is now caused to undergo a uniform tensile extension the load carried by the central region of the core member will also increase because it too will undergo an elastic extension. in the central regions the elastic extension of the core and tube will be the same so that.

J.G. Morley, New types of reinforced composite materials

269

Table 1 Material

Tube stainless steel

Straight wire piano wire

Outside diameter Inside diameter Metal cross sectional area Longitudinal elastic modulus Mandrel diameter Tensile load to form helical core Helix diameter Helix wavelength Extrapolated value of maximum limiting debonding load (Lmax)

2.347 mm 1.794 mm 2 1.8004 mm 1.56 X i05 Nmm2 3.05 mm

1.175 mm 1.084 mm2 2.0 X i05 Nmm2

Helix piano wire

0.79 X 1O~Nmm2 1420 N 1.835 mm 15.4.2 mm 480 N

if the average increased stress, over the initial equilibrium condition, carried by both parts of the reinforcing element is a, we have, (31)

o=E~V~e+E~V~e where e is the increase in longitudinal strain, E~and E

6 refer to the effective Young’s modulus of core and tube and V~and V~their respective volume fractions. As the load carried by the core member increases the core stress transfer length will also increase from its initial length of about 250 mm and will follow the limiting condition in4icated in fig. 14. Eventually when the core is carrying a load approaching 480 N its stress transfer length becomes too great to be accommodated within the composite structure, whatever its length, and the core can support no further increase in load. For the particular example shown in fig. 14, the load carried by the core in the central region of a very long reinforcing member could be increased from its initial value of about 300 N to its maximum decoupling value of about 480 N. For this condition the change in stress of the individual components, and the two part element as a whole, is given by eq. (31) with the appropriate value for the strain e inserted. The stress carried by the core and sheath in the central regions of a long duplex reinforcing element, perturbed from its initial equilibrium state by a general tensile strain e, therefore depends on the magnitude of the strain, the volume fraction and elastic modulus of the core and the sheath, and also on the initial differential stress situation. Further, in a composite engineering structure an additional term Em Vm would have to be added to eq. (31) to account for the contribution of the matrix. Hence, for a general applied tensile strain e we have, =

V~(E~ + ~

+ V~(E9e—

c~~) + Vm(EmC

—

Gmi)

(32)

where o~is the average effective tensile stress carried by the composite, V~,V~and Vm are the volume fractions of core tube and matrix and E~,E8 and Em their corresponding effective Young’s moduli. The initial residual differential stresses carried by core, tube and matrix are given by ~ and ~Jmirespectively. These are shown, as appropriate, with negative signs in eq. (32) to indicate the initial compressive condition. If the system is assembled by loading up the core elements in tension to their decoupling load, ~

270

J. G. Morley, New types of reinforced composite materials

and then allowing the core members to relax, the magnitude of the residual tensile load carried by the core members will depend among other things on the compliance of the tube and matrix assembly. The residual stress situation can be modified by arranging for the tube and matrix assembly to be loaded in tension, or compression, independently of the core members while the latter are stressed to their decoupling load. In this way the numerical values of Ui~ Oti and 0mj in eq. (32) can be controlled. This experiment has been carried out on an individual reinforcing member by Morley, Millman and Martin [28] using a simple load splitting lever arrangement. By this means the relative magnitude of the tensile loads applied to the core and tube could be varied over wide limits. At one end of the specimen the core and tube were bonded together and the load splitting lever arrangement was fixed to the other. The tensile load applied to the system was then increased until the load carried by the core element exceeded Lmax. The load was then reduced until core recoupling occurred. At this stage the core member was catrying a load Lmax and the tube was also carrying a tensile load, the magnitude of which is set by the lever arm ratio of the load splitting device. As the total applied load was reduced to zero, the ratio between the loads carried by core and tube remained the same. When the load applied to the system has been reduced to zero, the residual differential longitudinal stress carried by the core and tube will only be zero if they both relax by the same amount e. Therefore for this condition,

4.0/1 load ratio tube / wire

77

3.011

..-•

.0/

1 std deviation

1 Compressive strain on lube

234

residual

567 tensile s/rain

0101112131415 on

tube lE~10’l

Fig. 15. Residual elastic strain in the tube for various ratios of core/tube load (from ref. [281).

J.G. Morley, New types of reinforced composite materials

271

CE~A~ = a~A~ = and eE~A~ =

=

L~,

(33)

where E~and Ec are the longitudinal stiffness values of the tube and core when frictionally locked together, and A, a and L are the corresponding cross sectional areas, tensile stresses at decoupling, and applied loads. For equilibrium (zero residual longitudinal strain), A~E~/ACEC =L~/L~.

(34)

The residual strain at the centre of the tube was noted for various lever arm ratios and these results are shown plotted in fig. 15. From this data it can be seen that a lever arm ratio of 1.85/1 gives almost zero residual longitudinal strain for this particular system. The ratio of the effective elastic moduli of tube and core-wire is therefore given by, 85A~/A~ = 1.11. (35) E~/EC= l. This ratio is very different from the ratio of effective elastic moduli of the free core wire and tube measured separately, i.e. 1.97 (from table 1). Observations were made of the longitudinal stiffness of the core/tube assembly after it was initially placed under zero differential longitudinal stress. These values were obtained from a strain gauge placed at various positions along the length of the tube. The tensile loads were applied directly to the ends of the tube and for the first part of the load extension curve both tpbe and core extended equally, fig. 16. For this condition we have therefore, A 5E5 =A~E~ +A1E~

(36)

whereA~and E5 are respectively the combined cross sectional areas and effective longitudinal elastic modulus of the combined core/tube system. After an initial linear relationship a change in the slope of the load extension curve is observed and this new slope corresponds to the elastic modulus of the tube alone. It follows from these simple experiments that the effective longitudinal stiffness of the tube is not detectably changed by the presence of the core wire, but the effective stiffness of the helical core, when inside the tube, has been increased by about 1.8 times its independent value for the particular combination examined. It seems probable that the observed increased effective longitudinal stiffness of the core, when inside the tube, is due to the large hoop tensile stresses generated in the tube, the corresponding radial compressive stresses generated in the core and the very high effective Poisson’s ratio values obtained for the helical core. The combined system contains a large store of strain energy when the core and tube are under zero differential longitudinal stress. When a tensile load is applied to the combined system it elongates and as a consequence of the large effective Poisson’s ratio of the core, the lateral pressure exerted by the helix on the tube is reduced. Thus, the initial store of strain energy in the system associated with the tensile hoop strains in the tube is correspondingly diminished. Eventually, when the elastic strain in the system is sufficient to decouple the core, this initial store of strain energy is reduced to zero and the system extends as two separate springs loaded together in parallel. Up to the decoupling condition, the effective Young’s modulus of the core is enhanced. The magnitude of this effect depends on

272

J.G. Morley, New types of reinforced composite materials

12/JO

‘A

1100

100/1

//

7/ 0 N,’

900~

7 ‘.1

800

/7

,,1~

boor 400/

/

7

~/

i

001

.002

.003

Strain

Fig. 16. Load/strain relationship for core, tube and con3bined system.

the rate at which the initial store of hoop strain energy is released during longitudinal extension of the system, i.e. on the effective Poisson’s ratio of the core. This mechanism can be considered in the following way. When core and tube are under zero differential longitudinal stress the core element is restrained from relaxing laterally by the presence of the tube. Hence, under these conditions the core is longer, by an amount 1~,than would be the case were it freely suspended under zero longitudinal load, fig. 1 7. Decoupling occurs when the load carried by the core reaches Lmax and the core has extended under load by a further distance l~.The core stiffness at higher loads becomes that of the helix loaded independently of the rest of the structure. The enhanced stiffening of the core is maximised when 1~has as large a value as possible and when 1~has a minimum value. The strain energy associated with a core extension of lo must be balanced by the strain energy associated with the hoop tensile stresses in the tube, and 11 is controlled by the effective Poisson’s ratio of the core. Clearly it is desirable for the tube hoop stiffness to be as large as possible. There will be an optimum value for the wall thickness and this will depend on the mechanical properties of the materials used. As the tube wall thickness increases its hoop stiffness increases, but the relative volume fraction of the core member diminishes. Also the loads are applied to the tube along an interfacial strip, over which the contact pressure is not uniform, and the situation is complicated further by the possibility of elastic/ plastic effects over the area of contact and by frictional effects associated with the decoupling mechanism. This interaction phenomenon has not yet been studied in detail.

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273

/ /

~

4

I~

~ Ex tens ton

Fig. 17. Illustrating change in initial core length when inside the tube and the increase in the effective longitudinal stiffness of the core.

7. Fracture of conventional unidirectional composites under tensile loading In principle, fracture can occur through a material reaching some general overall limit of stress or extension. In practical systems constructed from components having non uniform properties this general strength will not be approached unless the material possesses some stabilising mechanism which prevents premature failure by the growth of some type of flaw. Conventional fibrous composites possess this feature in some measure but it is much less powerful than the corresponding mechanism of work hardening available in tough metal alloys. Thus, the strengths of conventional composites are less consistent than those of metal alloys. 7.1. Continuous fibres

—

single and multiple fracture

The fracture properties of a composite stressed in tension in the direction of the fibre alignment has been discussed in general terms by Cooper El]. Cooper considered an idealised composite shown in fig. 18. This consists of two phases of constant cross section. At a low strain value e below the failing strain of either phase we have, =

V1E1e+ V2E2e

(37)

where V1E1 and V2E2 refer to the volume fractions and Young’s moduli of phases 1 and 2, and a~is the stress carried by the composite. If the failing strain e1 of phase I is greater than the failing strain e2 of phase 2 we have just as phase 2 fractures = V1E1e2 + V2E2e2 =

V1E1e2

+

V2a2

and the load V2a2 is redistributed to phase 1. This will fail if the failing stress a1 .ult of phase 1 is exceeded by the addition of the extra load, i.e. V1a1~1~
+

a~V2

(38)

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274

____

-~

~—

~

~/// E,V

16~ ti,

~tuIt

V~
//

/A

~

E2V2O’2E2

o~uitv ) E1V~E~. ~ V1

p7Jr7T~vJrtv)r Hr/i vir~~r Fig. 18. Single and multiple fracture: the more brittle phase is shaded (from ref. [I]).

and if this is the case the composite will separate into two pieces by the propagation of a planar crack. This process is defined as “single fracture”. If inequality (38) is not satisfied the composite remains unbroken although phase 2 is cracked. More cracks are produced in phase 2 as the load on the composite is increased until failure of phase I occurs at a stress, =

V1 u1~1~.

(39)

This type of behaviour is termed multiple fracture and is characterised by a stress strain curve which has a “yield point”, fig. I 9b. If both phases are completely elastic, a material showing single fracture, fig. 1 9a, has a stress strain gradient E5 V1 + E2 V2 until the strain reaches e2 and catastrophic failure occurs. The falling portion of the stress strain curve may be affected to some extent by portions of phase 2 being pulled out of phase 1. The energy absorbed by fibre pull-out is discussed in more detail below. The initial portion of the stress strain curve for the composite showing multiple fracture is the same as before having a gradient F1 V1 + E2 V2. However, after initial failure of phase 2 at strain 2 progressive fracture of phase 2 proceeds. This behaviour depends upon the existence of a distribution of flaws of varying severity within phase 2 and generates a regime in

ult

d

~

~V2~E,V,E~

I

Fig. 19. Stress—strain curves characteristic of single and multiple fracture (a) single fracture, (b) multiple fracture (from ref. [1]).

J. G. Morley, New types ofreinforced composite materials

Multiple trocture

Single fracture

~t

~i~ri<~~0)

It

~

I

Multiple tracture

275

ult

im:~~~dt~

Fig. 20. Illustrating the conditions under which single and multiple fracture occur (from ref. [11).

which the stress carried by the composite rises slowly in an irregular manner as phase 2 breaks into smaller and smaller pieces at flaws of reducing intensity. This process ends when phase I can no longer support the load which has been transferred to it as defined in eq. (39). Final separation of the broken pieces occurs on a falling load-extension curve as for single fracture. In the case of conventional fibre reinforced composites the fibres invariably have a greater failing stress than the matrix but may or may not have agreater failing strain. The relative volume fractions of the phases is also a factor which controls whether or not multiple or single fracture occurs as is indicated by inequality (38). There are four combinations of these variables two of which lead to single fracture and two to multiple fracture. Figs. 20a and 20b describe the situation at the point of initial failure. In these diagrams a~represents the stress carried by the fibres at the initial failing stress of the matrix. Fig. 20b refers to strong fibres reinforcing a brittle matrix which has the lower failing strain. As multiple fracture occurs a~increases by an amount determined by the nature of the flaw distribution in the matrix. Fig. 20a refers to strong brittle fibres in a ductile metal matrix. It should be noted that technically interesting composites of this type containing a large volume fraction of strong fibres will fail by single fracture. This is a reasonable approximation to the observed behaviour of say, boron fibre reinforced aluminium, but the variability in fibre strengths causes some fibres to fracture at points some distance away from the primary fracture plane. Fig. 20b approximates to the situation found with glass fibre reinforced plaster. It should be noted that in both types of system the ultimate failing strain of the composite is limited by the failing strain of the fibres. The volume fractions of fibre necessary to generate multiple fibre fracture with appreciable plastic deformation of a metallic matrix are too low to be technically interesting. Also, although a measure of “strain hardening” by multiple fracture can be obtained with a brittle ceramic matrix, the ultimate failing strain is less than the failing strain of the fibres which is 1%. ‘~

7.2. Continuous fibres of uniform strength

—

work of fracture

7.2.1. Brittle fibres brittle matrix The fracture processes occurring during multiple fracture of a unidirectionally aligned fibre composite having a brittle matrix with a substantially uniform strength Gm ult’ have been discussed by Aveston, Cooper and Kelly [4] and Hale and Kelly [2]. In the situation considered the matrix —

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fractures at a much lower elongation than the fibres. As discussed in the previous section, so long as the fibres are capable of bearing the applied load the matrix becomes traversed by an increasing number of parallel cracks. The limiting separation of the cracks is determined by the rate of stress transfer between fibres and matrix and for cylindrical fibres will lie between x and 2x where, x

(40)

= (‘Vm/Vf)Umultr/2T

where the fibre diameter equals 2r, and r is the strength in shear of the fibre matrix interface. The failing strain of the matrix can be increased above its normal value when the fibres are sufficiently small. The reason for this is outlined below. As the matrix fractures, the load carried by the matrix per unit area of composite, i.e. 0m.ult Vm~will be thrown on to the fibres bridging the crack. The fibres will elongate further elastically under this additional load and the matrix will relax elastically in the vicinity of the crack since it is unloaded at the crack face. Frictional energy is dissipated by this differential movement over a length of fibre equal to the stress transfer length (x in eq. (40)). The additional stress carried by the fibres will vary linearly between zero at a distance x from the crack to a maximum of Umult (Vm/Vf) at the crack. The mean additional strain over the distance 2x is thus aem uit/2 where muIt is the failure strain of the matrix and a = EmVm/EfVf. If the strength of the matrix is substantially constant the cracking will continue at a stress until the matrix is broken down into a set of blocks each of length between x and 2x. This occurs because matrix blocks of length greater than 2x develop a sufficiently high stress in their central regions to fracture while those that are just less than 2x cannot fracture in this way. For the lower bound of crack spacing x, the maximum additional stress on the fibre at the crack is still Umuit( Vm/Vf) but the distance over which this additional stress is transferred back to the matrix is reduced to x/2. The mean additional strain over the length x is therefore now given by 3ctem uit/~~ The upper and lower limits for the total strain Eme on the composite when multiple matrix cracking has ceased are therefore, Fe Cm ult

Cmuit(l +

a/2) <

6mc

<

mult(l

+

344).

Further increases to the load carried by the composite will cause further elastic extension of the fibres with frictional losses occurring due to differential movement with respect to the matrix blocks. The matrix blocks can carry no further increase in load so that the Young’s modulus of the composite now becomes E~ V~.It will fail eventually at a stress ~f uit 17f and a strain ~, given by aemuit/2} <

< ~f~

aemuit/4}.

The stress strain curve for such a composite is shown schematically in fig. l9b. The system shows a measure of strain hardening in that immediate fracture does not occur when fracture of the matrix first takes place and the system can absorb energy as the applied load increases. During cyclic loading hysteresis loops are generated [4]. This is due to frictional losses occurring during differential movements between the continuous fibres and the blocks of matrix. Despite the stabilising effect of multiple fracture, it should be noted that the failing strain of the composite cannot exceed the failing strain of the fibres and will generally be less than this value.

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277

It has been observed by Cooper and Sillwood [29] that multiple matrix fracture as described above can be suppressed if the fibres are small enough. This occurs because the amount of energy released by the presence of the matrix crack depends upon the size of the fibres. When insufficient energy is released to balance the energy absorbed in rupturing the matrix, fracture does not occur at the normal matrix failing strain. Instead the strain on the composite has to be increased before multiple matrix fracture occurs. That is Cmult is increased to a new value Cmuc given by, (41) [l2r7mEfV~IEcE~grVm]113 where ‘y~is the work of fracture of the matrix. Crack suppression in the matrix increases the amount of energy absorbed during fracture until Ecmuc = ~ V~when there is a transition to single fracture, and, as a consequence, a sudden decrease in the work of fracture. However, as pointed out by Aveston, Cooper and Kelly [4], the failing strain of the system and the additional load which can be supported after initial cracking are both reduced so that suppression of matrix cracking by this means seems to be of little practical value in increasing the apparent fracture toughness of brittle matrix composites. ~muc

7.2.2. Brittle fibres ductile matrix An example of this type of system comprising brittle tungsten wires in a soft copper matrix was examined by Cooper and Kelly [30]. When the fibres have a uniform strength, composite fracture is substantially planar and fibre fracture occurs before the matrix fails. The composite structure is held together by matrix bridges, after the fibres fracture, and work is done by the plastic deformation of the bridges until the composite separates into two halves. It is during this stage of the fracture process that most of the energy is absorbed and the work of fracture depends on the volume of the matrix material which is taking part in the failure process. The extent x of the deformed zone of matrix on either side of the fracture plane of the fibres is given by, —

Vf

2r

(42)

and is thus directly proportional to the fibre diameter (2r). The yield strength in shear of the matrix is given by r. The volume of material in the matrix bridges is 2x Vm, per unit area of fracture surface. If it is assumed that the work done in deforming a unit volume of this material to fracture, is the same as that required to deform a unit volume of matrix to fracture in the absence of fibres, Um~we have for the work of fracture of the composite G 0, V~amr G~=-~---——Um~

(43)

Since r is taken as the strength of the matrix in shear, we can assume Gm = 2r and from eq. (42) we see that, for normal fibre volume fractions of about 0.5, the distance x is comparable with the fibre diameter. Hence, the volume of matrix taking part in the failure process is very small as is the corresponding work of fracture of the composite. It will be noted from eq. (43) that G0 is directly dependent on the diameter of the fibres. This prediction is in broad agreement with a wide range of experimental observations.

J.G. Morley, New types of reinforced composite materials

278

Crack growth and toughness in composite systems comprising a ductile matrix and unidirectional ductile fibres has been studied in detail by McGuire and Harris [31] . The behaviour is similar to that of brittle fibre ductile matrix systems except that an additional term has to be included to account for the work of fracture during localised failure of the fibres. The contribution of the fibres to the work of fracture was observed to be important at high volume fractions. 7.2.3. Continuous fibres of non uniform strength in a ductile matrix

This is the condition which is normally encountered in practical systems such as boron fibre reinforcing aluminium. Cooper [32] has considered an idealised situation in which the fibre strengths are everywhere a~except at uniformly spaced weak points where the strength is reduced to a volume a”. When fibres of this type are traversed by an orthogonal matrix crack the fibres will break at the weak points and pull out of the matrix if the spacing between the weak points is less than d~where, (a~ u*) —

l~.

=

(44)

Ut

If the matrix is brittle the fibres may debond (see section 7.3) and pull out against residual frictional forces. If the matrix is ductile and is strongly bonded to the fibres the pull out process involves intense local plastic shear of the matrix. This is the mode of failure encountered in boron aluminium composites and occurs because the boron fibres contain a distribution of flaws of varying intensity. The detailed analysis of this type of fracture process is complicated because the fracture of one fibre can generate static and dynamic stress concentrations which can influence the failure of neighbouring fibres. 7.3. Discontinuous fibres In section 4 the mechanism of stress transfer between matrix and fibres was discussed. It was shown that when a conventional unidirectional composite is stressed in the direction of the fibre alignment, plastic zones (or regions of interface failure) develop at the ends of the fibre and grow inwards towards the centre of the fibre. These zones are defined as the fibre stress transfer length. In the central region of the fibre the longitudinal strains in fibre and matrix are equal and since there is no transfer of load in this region the fibre stress is approximately constant. As the tensile load applied to the composite increases, the length of the stress transfer zones increase and the stress carried by the central region of the fibre increases. Fibre fracture occurs when this reaches the fibre failing stress. The average stress ~ carried by the fibres of length 1 at failure is therefore, =

a

1~15(l —

l~/2l).

(45)

It has been shown by Kelly [19, 221 that when the fibre length 1> l~,and the fracture of one fibre does not influence preferentially the behaviour of its neighbour, a fraction l~/lwill be pulled out of the matrix and the rest of the fibres will fracture in the vicinity of the plane of fracture within the composite system. The work We done in pulling a fibre of length 1 out of a matrix against a constant interfacial shear stress r is given by

J. G. Morley, New types ofreinforced composite materials

279

(46)

We=(~T7rd)P.

If 1 < 1~all the fibres will be pulled out from each side of the matrix crack but when the fibres are randomly arranged the length of fibre which will be extracted from the matrix will vary from 0 to 1/2. The average work done in extracting a fibre will be given by, (l~l~) W~ j~ irrrl2 and if the fibres are longer than the critical fibre length, Wp=(l~/l)~irrrl~ (l>~la).

(47)

(48)

Thus, as has been discussed by Hale and Kelly [2] and Cottrell [33] the maximum work of fracture occurs when 1 = In general toughness, by fibre pull out, is only obtained at the expense of the reinforcing effect of the fibres. Although the maximum work of fracture by fibre pull out occurs when 1= l~the strengthening and stiffening effect of such fibres is only half that of continuous fibres. However, not all composites of the same strength have the same toughness. This depends on the flaw distribution [32]. When the matrix is brittle and the fibres are left bridging a matrix crack, work can be done by debonding the fibres before extracting them from the matrix against frictional forces. This has been discussed by Outwater and co-workers [34]. They calculated that the maximum length of a fibre xmax of uniform strength ~fult which can be debonded before fracture is given by, Xmax

= ~

~

—

4EG r

1/2 ]

(49)

where r 5 is the sliding friction shear stress after debonding, G11 is the fracture energy in shear of the fibre/matrix interface, and for a composite containing many fibres the total work of debonding becomes, Wtotai

=

VfUf 1~r 2~ [~fuIt

(4EfG11)’~].

(50)

As with the equations governing the work of fracture by fibre pull out, or by deforming matrix bridges after fibre fracture, the work done increases with increasing fibre diameter and as the degree of coupling between the fibre and matrix decreases (as both r~and G11). However, as discussed by Kelly [35], and Cooper [1], the energy absorbed by this process is less by about two orders of magnitude than that absorbed by fibre pull out. The maximum length of fibre which can be involved in either process is half the critical length l~.Equating the energy absorbed in debonding the fibre over this distance with the strain energy contained in the fibre after debonding, the work done during pull out is related to the work done during debonding by [2] 3(E/a~). We/Wd =

Since E/af is usually at least 50, Wd is small compared with the maximum achievable work of fracture by fibre pull out, W 5. However, it should be remembered that in practical systems containing continuous fibres, the actual work done during pull out will depend on the flaw distribution in the fibres and can be very much less than We.

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280

It will be noted from eq. (49) that there is a minimum size of fibre below which a crack in the composite will fracture both fibres and matrix alike. The suppression of matrix cracks by a reduction in the size of the fibres, discussed in section 7.2.1 (eq. (41)), can therefore lead eventually to extreme brittleness in the composite system if the fibre size is reduced to this critical value. 7.4. General summary of fracture processes occurring in conventional composites under tensile

loading A very important aspect of composite fracture by fibre pull out, preceeded or not by interfacial debonding, is that the failure process proceeds at a diminishing stress level. Hence, the zone failure is localised and the maximum deformation which can occur is given by the relationships discussed above. This may be very small compared with the total dimensions of a composite structure. For structural stability it is necessary for a deformation process to take place at an increasing stress level. The arguments outlined above have related to unidirectional systems loaded in the direction of fibre alignment. Some deformation is possible, without fibre fracture occurring, when metallic matrix composites are loaded at an angle to the direction of fibre alignment. Longitudinal shear can take place in the matrix with the angle of fibre alignment rotating towards the direction of the applied load [361, and thus the composite structure can in effect strain harden. However, this process involves co-operative deformation throughout the composite structure and will be inhibited, for example, when localised impact loads are applied. Further constraints on this type of deformation are imposed when the fibres are crossplied in layers. In general the maximum deformation of conventional fibrous composites is limited by the failing strain of the fibres to a value of about 1%. The most effective toughening mechanism (fibre pull out) proceeds at a diminishing stress level and results in locahised failure. A measure of work hardening is achieved by the multiple cracking of a brittle matrix but the magnitude of the energy absorbed is small and the maximum deformation is again limited to the failing strain of the fibres. 8.

Fracture in composites containing non-fracturing reinforcing members

8. 1. General considerations

In the case of composites reinforced with core/sheath reinforcing members, it can be arranged that the core reinforcing members do not fracture under any circumstances. Their integrity can be maintained under conditions of tensile deformation, cyclic loading or combinations of both effects. The core members provide a residual fail safe characteristic and any fracture process is confined to the rest of the composite structure. The stress controlled decoupling of the core members provides two further features of interest. Firstly, it provides a stress diffusing mechanism, making it impossible for the stress carried by the core members to be confined to a localised region. This feature can be used to provide the composite structure with a strain hardening and tensile deformation mechanism. The second feature of interest arises from the interactions between the non fracturing and fracturable parts of the composite structure, since these can be used to suppress crack growth in the latter. This can take place as a result of differential stressing with the core members placed in residual tension and the

J.G. Morley, New types of reinforced composite materials

281

tubes and matrix in residual compression, as described in section 6. It can also occur through the core members limiting the extent of the elastic relaxation of the rest of the composite structure, in the presence of a crack, and by absorbing energy through frictional losses during crack growth. These occur as a result of displacements at the interfaces between the fracturing and non fracturing parts of the composite structure. 8.2. Simple tensile loading of a unidirectional composite structure with non-fracturing compo-

nents Up to the point of initial fracture, a composite system utilising two part core/sheath reinforcing elements behayes as a conventional composite. If the matrix has a lower failing strain than the sheath elements it will fail by the propagation of a transverse crack. This situation is similar to that described in section 7.1, except that there are now three phases present in the composite structure. The sheath and matrix will have fairly clearly defined failing strains and corresponding failing stresses but the core elements, of course, do not fracture and have a maximum load bearing capacity of Lmax~Also, again in contrast with conventional composites, account has to be taken of any initial differential stress which might have been introduced into the system. One issue of particular significance is whether, after fracture of the tubes and matrix, the load carried by the core elements is less than Lmax~If this is the case the crack bridging core elements can carry an increasing load as an increasing tensile deformation is applied to the composite structure. As this occurs the core members become progressively decoupled from the rest of the composite structure in the vicinity of the matrix crack so that further matrix cracking will not occur in this region. The load applied to the ends of the composite structure is supported by an ever increasing length of crack bridging core element as decoupling proceeds. Thus the composite extends elastically under increasing load conditions and energy is absorbed during this process by frictional losses due to differential movement at the core/tube interface. Eventually, the core element reaches its maximum limiting load condition, at the position of the crack, and further tensile deformation causes the core elements to be drawn bodily through the composite structure against frictional losses. Clearly very considerable tensile extensions are possible during this stage of the deformation process. Preliminary studies have been made of this behaviour at high rates of deformation [37]. Flat plate specimens, consisting of a single layer of resin bonded steel tube/steel core reinforcing members, were held in grips at each end. The grips were free to rotate but not to approach each other and the specimen was impacted by a missile travelling at speeds of up to 200 m s in a direction normal to the plane of the specimen. Fracture of the resin and tubes occurred followed by considerable deformation as the core elements were drawn out of the composite structure. On a weight for weight basis the energy absorbed by these specimens was greater than that of existing metal alloys. 8.3.

Propagation of an edge crack in a duplex fibre reinforced sheet by a monotonic crack opening force

Studies of crack propagation under these loading conditions have been carried out using a modified double-cantilever beam arrangement [38]. The specimens used consisted of large thin brass sheets. The reinforcing elements were attached directly to one face of the brass sheet and opposite

J. G. Morley, New types of reinforced composite materials

282

L~

13

Crock lengths

Crosshead movement

Fig. 21. Schematic load/crosshead movement curve during crack growth (from ref. [381).

edges of the sheet were attached to the cantilever beams. As the crack extended, the beams were observed to deflect in a manner very similar to that which would be obtained by loading them at three points at each end and at the position of the crack tip. Hence, F~bL is effectively constant, where F is the load applied at the mouth of the crack, L is the crack length and b is the overall length of the beams. In order that the crack should propagate it is necessary to increase continuously the distance of separation of the ends of the beams, although the load being applied as the crack extends is falling. In this arrangement the crack is very stable and is observed to propagate in a series of short steps. In fig. 21 the relationship between load applied and the movement of the crosshead of the testing machine asthe crack length increases is illustrated diagramatically for an unreinforced sheet of metal. After the crack has propagated a convenient distance the applied load is reduced to zero, and the work done in propagating the crack is calculated from the area under the load extension curve. In these studies the reinforcing elements were aligned perpendicularly to the direction of crack propagation and the tubular portion of the two part reinforcing elements were attached directly to one face of the sheet. If the core members had been drawn into continuous tubes a residual tensile stress would have been developed in the core with a corresponding compressive stress in the tubular member and the sheet metal. In these experiments this condition was avoided by arranging for the tubular members to be discontinuous and quite short (20 mm). They were threaded on to a long length of core which was stressed in tension above its decoupling stress. Short gaps (— 1.0 mm) were left between adjacent elements. On releasing the load applied to the core member frictional contact was regained between the core and the tubes but the residual stresses, which had maximum values at the centre of each tube segment, were negligible. The core/tube assembly was then attached to the sheet metal by soft soldering the centre point of each tube to the sheet. In this arrangement therefore the contribution by the tube elements to the strength of the composite structure was negligible. Observed values of the crack opening force versus the crosshead movement of the testing machine obtained for both unreinforced and duplex reinforced sheets are shown in fig. 22. Also indicated are the associated crack lengths. It is seen that, unlike the unreinforced case, the load to continue crack propagation for the duplex reinforced sheet does not tend towards zero with increasing crack length but remains relatively constant. As an edge crack propagates through a duplex reinforced sheet under the action of a crack opening force there are three distinct phases of energy absorption by the crack bridging reinforc—

J.G. Morley, New types of reinforced composite materials

1.5

283

crack lengths (mm) 310

225

bOO

1.0

580 Reintorced sheet

600

293 Urireiritorced sheet

0.5

520

620

0

1

2 Crossheod

750 I 3 1. mouemer,t (mm I

5

6

Fig. 22. Observed load/crosshead movement curves for unreinforced and duplex reinforced brass sheet (from ref. [38]).

ing elements. Initially relative movement occurs between the reinforcing elements and the sheet in the vicinity of the crack as the elements are loaded up to their debond load. The energy absorbed during this stage is relatively small. Secondly, the debonded zone of the reinforcing elements extends out towards the extremities of the sheet. In principle there are two ways in which this can occur. If the crack opening force is applied locally in the region of the crack, compressive loads can be supported by the sheath elements and tensile loads by the core elements. If the load is applied to the reinforced sheet at a remote position the core members will still carry a tensile load, but the sheath elements (and the sheet to which they are attached), will now relax elastically as the decoupled zone spreads outwards. During both these processes the loads carried by the crack bridging core elements remain substantially constant and the separation of the crack faces is controlled by elastic deformations occurring over an increasing volume of the composite structure. The third phase occurs when the debonded zone extends sufficiently to allow the core elements to pull through the composite structure. If the structure is large compared with the stress transfer length this will again occur at an almost constant load and work will be done against frictional losses as the crack faces separate. Thus, it is clear that for edge crack propagation under the action of a crack opening force in a reinforced structure of this type, the observed work of fracture, with increasing crack length, is dependent on: (a) The volume fraction, load to debond and load/pull out characteristic of the core reinforcing elements. (b) The area between the crack faces. This is so because the crack ar@a corresponds approximately to the integrated pull through distance of the crack bridging core elements. (c) The size of the composite structure. The work of fracture will be maximised when the dimensions of the structure are large enough to enable the core pull through load to approach Lmax. The core elements can be regarded as a toughening component in the composite structure whose effectiveness increases with increasing crack length. The specific work of fracture (G5~)for

J.G. Morley, New types of reinforced composite materials

284

10’

Deboad stress

/

H.

....~.

Dalu~o

L 13 10

~_~_.i_ 10~

.~ Crack lengthI kmmj

100

Fig. 23. Calculated values of the specific work of fracture of two types of core reinforcing elements (from ref.

1381).

the reinforcing elements is given by: =

W/Vfp

where W is the work done on the core elements during crack extension in the sheet, Vf is the cone element volume fraction and p is the core element density. In fig. 23 calculated values of ~ are given, for increasing crack lengths, for the reinforcing elements used in this study. Calculated values of ~ are also shown for similar reinforcing elements studied elsewhere, [26, 371. The latter elements consisted of sinusoidally crimped steel wire within a stainless steel hypodermic tube and the debonding process involved some plastic deformation of the core wire. However, the bulk of the energy absorbed during pull through with this system is generated by frictional losses. In both cases the crack is assumed to be triangular, the crack length being fifty times greater than its width at the mouth. The core elements are assumed to be uniformly distributed and, for cracks of an appreciable length, they develop specific work of fracture values greater than those of existing metal alloys. In principle, G5~values several times greater than those in fig. 23 would be attainable were the core elements to consist of strong light-weight materials such as resin bonded carbon fibres. These and similar possibilities are discussed in section 1 0. Higher work of fracture values would also be observed for greater separations of the crack faces.

J.G. Morley, New types of reinforced composite materials

285

For the purpose of these calculations it is assumed that the tube elements have properties identical with those of the matrix and can be considered part of it. In a practical engineering situation, however, the reinforcing elements might well be spaced appreciable distances apart. In such circumstances the tube elements could act as crack inhibitors in their own right and their contribution to the effective work of fracture of the sheet would need to be considered in addition to those of the core elements given in fig. 23. 8.4.

Crack stability in a composite sheet reinforced by non-fracturing reinforcing members

The fracture mechanics of a thin sheet, containing a crack bridged by duplex reinforcing elements, when a uniform tensile load is applied at the remote edges of the sheet, has been considered by Morley and McColl [38]. It is necessary to evaluate the form of the strain field around the crack, and the way in which it is modified by the progressive reduction in the shear strength of the interface between the composite structure and the crack bridging reinforcing elements, as the stress they carry increases. An approximate solution to the problem was obtained by the use of a simplified analytical model of the situation. As a first step the model assumes a linear change of strain in the unreinforced matrix in the vicinity of the crack (fig. 24). The partially relaxed zone around the crack takes the form of an ellipse and is considered to be split into a number of independent strips each parallel with the major axis of the ellipse and of unit cross-sectional area. Each strip is considered to be subjected only to longitudinal strain. The strain gradient for each strip therefore increases from a minimum value, for the strip positioned at the major axis of the ellipse, to infinity for the strip at the crack tip. By neglecting interactions between adjacent strips, the strain energy contained in the zone can be obtained, simply by summing the tensile strain energy contained by each strip in the zone. The size of the partially relaxed zone is chosen so that the total amount of strain energy released by the presence of the crack (on the basis of these simplifying assumptions) is the same as that released in the classical case of a crack in an infinite elastic sheet. This is equivalent to the complete relaxation of an elliptical zone around the crack twice the area of a circle having the crack as its diameter. The major axis of the partially relaxed elliptical zone shown in fig. 24 is therefore three times the crack length. The approximate effect of the reinforcing elements on the stress distribution around the crack can now be obtained by considering the elements to be distributed uniformly over the sheet and Strain

Centre

3a

of crack

Fig. 24. Idealised strain distribution around a crack in an unreinforced sheet under longitudinal stress (from ref. [38]).

J.G. Morley, New types of reinforced composite materials

286

/

__ Ciston.cv..Jroffijree end of core

Fig. 25. Idealised stress distribution along the core of a duplex reinforcing element (from ref. [38j).

all the elements contained within any particular strip to behave identically. Since a core element cannot fail in tension it bridges the crack in the sheet and carries a load across the crack faces. If the maximum load carried by the reinforcing element (in the presence of a crack in the matrix sheet), is less than the debonding load, the element will remain in frictional contact with the matrix and an increasing load will be transferred from the fibre to the matrix with increasing distance from the edge of the crack. The load transfer will be governed by the strength in shear of the interface between the reinforcing element and the rest of the composite structure. In order to compute the approximate effect of the reinforcing elements from simple geometric considerations, a constant shear strength interface is assumed up to a particular stress level at which complete debonding occurs. This idealised stress distribution is indicated in fig. 25. In the first condition considered it is assumed that the maximum strain carried by the reinforcing element is always less than the debonding strain, ~d in fig. 25. The longitudinal strain distribution within one strip can now be computed and this is showi~diagrammatically in fig. 26 for a strip at the major axis of the ellipse. The line OP shows the strain distribution for the unreinforced material, the matrix strain increasing with increasing distance from the crack face, rising to e~the bulk strain in the matrix material at the edge of the elliptical zone. The strain gradient is therefore e~/3afor this condition. In the presence of the reinforcing core elements the matrix strain is increased, compared with the unreinforced material, due to stress transfer from the cores. The elastic strain carried by the core members decreases correspondingly with increasing distance from the crack face, the rate of change of core tensile stress being given by, da~/dx= —2r/r where 2r is the effective diameter of a circular sectioned core member and r is the effective shear strength of the interface. (In these calculations the area over which stress transfer occurs is taken as the surface area of the core.) The rate of change of core strain is therefore given by de~/dx= ~2r/Er

(51)

where Ef is Young’s modulus of the core and e~is its elastic strain. Now, over a short distance dx, the change in load carried by the cores per unit cross sectional area of the composite structure is

J.G. Morley, New types ofreinforced composite materials

287

V

~

su

LV///////////////~ i/~~

r

I

y.

0

/ 0

/

/

/

/~I /i I II

I

._—~

I

,—.-.

I

.-~ .~

~l. ~l

I I

L

0L~

3a ~istarice from crock face

Fig. 26. Strain distribution along a strip perpendicular to the crack

~


(from ref.

[38]).

given by, —2r

dcj~V~=— V~dx where V~is the volume fraction of the core elements. This change in load is linked with a corresponding change in the load carried by the matrix. If this results in a change in matrix strain of dCm we have, dEm

=

2r Vfdx/Emr Vm

where Em is the Young’s modulus of the matrix and Vm is its volume fraction. (Here the tubular portion of the reinforcing element is considered to be part of the matrix sheet.) The rate of change of strain in the matrix, due to stress transfer from the crack bridging core elements, has to be added to the rate of change of strain already existing round the crack in the unreinforced sheet so that, near the crack, the rate of change of matrix strain in a strip lying along the major axis of the ellipse shown in fig. 24 is given by dEm —=-—

dx

2r

Vf

Ca

—+—-

EmrVm

3a

(52)

where a = half crack length. Eqs. (51) and (52) define the slopes of the lines VQ and OQ in fig. 26. The tensile strain carried by the reinforcing element decreases with increasing distance from the edge of the crack and the corresponding tensile strain carried by the matrix sheet increases with increasing distance from the edge of the crack. The distance OL1 is effectively the core element stress transfer length. At the position L1 the tensile elastic strain in both core and matrix is the same and the differential movement between the components is zero, so that no further stress transfer takes place between core and matrix. In constructing the model it is therefore assumed that the elastic strain in both fibre and matrix remains the same between L1 and the point L2

J. G. Morley, New types of reinforced composite materials

288

which lies on the strain distribution line for the unreinforced matrix. From L2 to a distance 3a from the edge of the crack, the strain distribution for both core elements and matrix is assumed to he the same and to correspond to the strain distribution for the unreinforced sheet in this region. Since the strain redistributions are assumed to be confined within the elliptical zone around the crack (shown in fig. 24) the total integrated strain of the core reinforcing element from the centre of the crack to the position P, must equal the original integrated 3aea. Over the distancestrain 0L in the absence of the crack. The original integrated strain is simply 0 (fig. 26), the core carries an additional strain and thus an increase in integrated strain corresponding to the area VRS. From L0 to 3a the core carries a reduced strain with a corresponding decrease in integrated strain proportional to the area SQTP. Therefore, in order that there should be no change in the total strain carried by the core reinforcing element over the length 3a, the area VRS must equal the area SQTP. Thus it is possible to define the point V and hence the numerical of e~.~. 6r value approaches An inspection of fig. 26 shows that, as L1 becomes small compared with 3a, This is because, so far, it is assumed that e~has a maximum value less than Cd, fig. 25. It follows that, since the slope VQ is fixed, the area VRS has a maximum value when the core elements are able to carry the total load applied to the composite without debonding. Thus the area SQTP also l1as a maximum value and (Ca Cr) will become approximately inversely proportional to 3a when 3a becomes large compared with L 1. Under these conditions, therefore, the amount of strain energy released by the reinforced strip of matrix sheet becomes a very small fraction of the amount which would be released in the corresponding unreinforced condition. From the position L1 to the position 3a both the matrix sheet and the non fracturing core members are experiencing the same strain, and the Young’s modulus of the composite structure is given by E~= VmEm + I/flff. If the strain energy contained within the partially relaxed reinforced strip illustrated in fig. 26 is given by W~.We have, 2 2 E L L1 ex 2 E 3a x 2 =~j~ [VmEm(~) + V~E~{e~_~ Cr)) ]dx+~f ~dx+—~f [~ e~ —

—

The total strain energy initially contained within one reinforced strip when uncracked and subjected to a uniform strain is E~e~3a/2 so that if the amount of energy released from the strip by the crack is W~ we have: 3E~C~a/2 V~E~(e~ + ~p~r + e~)L1/6 VmEmC~Li/6 E~e~(L2 3— L 3}/54a2. (53) —-E~e~.f(3a) 2 The procedure discussed above can be applied to other strips which are parallel to the major axis of the ellipse (shown in fig. 24) so that a representative value of the amount of strain energy available to propagate a matrix crack is given by summing the energy released by each strip over the whole of the elliptical zone. It will be noted that such a calculation does not take into account energy losses due to frictional displacements between the reinforcing member and the metal sheet between the edge of the crack and the point L ~

=

—

—

—

—

1.

So far it has been assumed that the non-fracturing reinforcing members are capable of carrying the total load applied to the composite structure without reaching their debonding stress value. The mathematical model discussed above has been extended to permit initial estimates to be made

J.G. Morley, New types of reinforced composite materials

0

VL

0

L1

S

289

12 30 Distance trom crac1~ face

Fig. 27. Strain distribution along a strip perpendicular to the crack (e~ ed). (from ref. [38]).

of the amount of strain energy locked in the composite structure by the reinforcing members after they have reached their decoupling loads as defined in fig. 25. On the basis of this simple model the reinforcing member is assumed to be suddenly decoupled from the matrix when it reaches a critical stress ad and corresponding critical strain Cd. The strain distribution along a strip of matrix sheet, parallel to the major axis of the ellipse in fig. 24, containing such reinforcing members, is shown in fig. 27. The reinforcing members are completely decoupled from the composite structure from the edge of the crack up to the point L0. Between L0 and L1 stress is transferred between the reinforcing members and the matrix through a constant shear strength interface and the rest of the strain distribution pattern is as defined for the non-decoupled reinforcing members, illustrated in fig. 26. Again the integrated strain of the reinforcing members is unchanged so that the areas shown shaded in fig. 27 are equal. The amount of strain energy available from a single strip W~now becomes, E

E ic \2 c1a1

3 2 C 2 Wxc=~Eceaa——(L2—Li)Cr

V E —

rC

m m

3/3

—

—

[L1

r

V E /~ \2 mm1 a~L3 6 ~3a)

{(3a) ~ —L2 }

L~/3 V(L 2 1 —

—

L~)+ J”2(L 1

L0)]

2

—

L1—L0 2L V~E~e 0 V~E~e~ 2 [(S—L 3—(S—L 3] 2 6S 0) 1)

—

(54)

—

where rLo

—

e 0L1

edLl S=

Cr~O

LoCr

—

~,

Cd ~Cr

—

Lied

L0—L1

290

J.G. Morley, New types of reinforced composite materials

Eq. (54) therefore defines the amount of strain energy released by a strip of reinforced sheet and available for the propagation of a matrix crack. Energy losses due to frictional displacements are again neglected. The overall effect of the crack bridging core fibres is to reduce the amount of strain energy released by the reinforced sheet compared with the unreinforced situation. An approximate numerical estimate of this effect can be obtained by summing the effect of the strain energy locking mechanism over a number of parallel strips extending over the whole of the elliptical zone illustrated in fig. 24. The analysis assumes that the parallel strips behave independently of each other and are subjected only to longitudinal stresses, i.e. shear effects between adjacent strips are ignored. The bulk of the effect of the crack bridging core members occurs in the central region of the crack. The amounts of strain energy released, or not released, in the regions around the crack tip are relatively small. In the central region of the crack around the major diameter of the elliptical regions, depicted in fig. 24, the value of Cr does not change rapidly with distance along the length of the crack. Morley and McColl [381 estimated the rates of release of strain energy, with increasing crack length, by numerical differentiation of the values obtained by summing the energy released by each strip using eqs. (53) and (54). They suggest that this gives a reasonable approximation to the actual situation, particularly so when the crack lengths are long compared with the core element stress transfer length. Using this technique they calculated the expected behaviour of a duralumin sheet reinforced with steel core/sheath elements having properties similar to laboratory produced systems of this type. The volume fraction of core elements in the composite structure was assumed to be 0.2. The sheath elements were assumed to have mechanical properties identical with those of the aluminium alloy sheet. This assumption is justified on the grounds that the specific strength and specific stiffness values of the tubes are similar to those of duralumin. Also, since the sheath elements would be expected to be relatively massive compared with the sheet thickness and to be separated by appreciable distances, it is to be expected that they would temporarily bridge the tip of a crack in the sheet and hinder its propagation. The effective specific work of fracture of the tube elements would therefore be expected to be at least comparable with that of the duralumin sheet. In fig. 28 the rate of release of strain energy with increasing crack length is shown for both a reinforced and unreinforced sheet at various strain values. Equal strain values approximate to equal tensile loading on a weight for weight basis. The values of rate of release of strain energy with increasing crack lengths for the unreinforced sheet are shown as straight lines having different slopes for different tensile stress values (and corresponding strain values). This is regarded as a reasonable approximation for the situation considered because the crack is much longer than any plastic zone around the tip. Thus the Griffith—Irwin—Orowan relationship equating the rate of release of strain energy at the semi critical crack length acrjt with the work of fracture G~and the Young’s modulus of the material can be assumed to hold, i.e. G~= 7racritEe~

(55)

where C~is the general tensile strain applied to the sheet. The rate of release of strain energy, for this situation, increases linearly with increasing crack length and G~becomes the work of fracture of the sheet metal “matrix” Gm~In fig. 28 values of 2Gm and 2 VmGm are plotted because the crack

J.G. Morley, New types of reinforced composite materials

I

750

!

-

/

0.OOb

/ C 0

f

Reintorced Unroinforced

.____....

I

/

0;Ô03

~

500

/ /

~

/

291

/

/

2Gm--—

0oo~ 2VmGm

I/

//

250 0.0015 0.5

I

1.0 Crock length ml

I 1.5

Fig. 28. Calculated values of strain energy release rates for reinforced and unreinforced duralumin sheet (from ref. [381).

lengths are expressed as the total crack length instead of the half crack length value included in eq. (55). For strain values up to e~,which corresponds to the composite stress at which the core elements are just capable of carrying all of the applied load (i.e. e~is just less than Cd), the rate of release of strain energy tends to a limiting value. Providing the work of fracture of the sheet is sufficient, catastrophic crack growth is not possible irrespective of crack length. This contrasts with the fracture mechanics of the unreinforced sheet where catastrophic crack growth occurs at the point at which the rate of release of strain energy with increasing crack length, equals the amount of energy which can be absorbed in rupturing the material. As the load applied to the reinforced structure is increased, so that the general strain also increases, the strain energy locking effect becomes less powerful because a portion of the crack bridging core fibres become decoupled from the matrix sheet and therefore no longer restrict its elastic relaxation in these regions. In the limit the crack bridging core elements become decoupled from the sheet, except at the remote edges, and only influence the growth of a crack in the sheet through the portion of the total applied load which they carry. It will be noted, however, that the composite structure has a fail safe characteristic in that the tensile load it can carry cannot be reduced below the load bearing capability of the crack bridging core elements. Recently calculations have been made of the frictional energy lost by differential movement at the core/tube interface (McColl and Morley, unpublished work). These calculations show that this frictional term is increased with increasing crack length and becomes more important at increasing general strain values, since under these conditions, the amount of differential movement at the interface increases. It is convenient to regard these frictional effects as an additional component to the work of fracture of the matrix sheet. Its effect is to increase the crack stability and is complementary to the strain energy locking term described above. —

292

J. G. Morley, New types of reinforced composite materials

1500r

1o0o~

Specific Lood IN / Specific Gravity)

/

530~—

Deflection 1mm)

Fig. 29. Illustrating energy losses occurring during cyclic loading in bending (from ref. [39]).

Frictional energy losses have been observed in polymers reinforced with core/sheath elements, when subjected to cyclic loading [39]. The specimens used were rectangular in section and severely notched. They were loaded in three point bending to cause a crack to extend in the notched region and this crack was, of course, bridged by the non fracturing core elements. As the deflection increased the crack extended and the specimens were permanently deformed by core fibre pull through. Cyclic loading, within the elastic range of the material, was carried out at various stages as the permanent deformation of the sample was increased (fig. 29). Pronounced hysteresis effects were observed with these samples.

9. Behaviour of reinforced composites under cyclic loading 9.1.

Conventional composites

Failure processes occurring during cyclic loading in conventional fibrous composites are fairly complex and will be given only in outline here. They have been discussed fairly recently by Cooper [11 and Hale and Kelly [21 and at greater length by Owen and Hancock [401. Strong brittle ceramic reinforcing fibres are inherently resistant to failure as a result of cyclic loading. They can, however, suffer stress corrosion effects, in which preferential chemical attack on the highly stressed region at a crack tip causes a progressive weakening of the fibre. This effect, known as static fatigue, can take place under the application of a constant tensile stress and occurs particularly in the case of silica and silicate systems where the weakening mechanism is associated with the presence of water vapour. The same mechanism produces progressive weakening under the application of any cyclic load pattern which has a tensile component. Advanced rein-

J.G. Morley, New types of reinforced composite materials

293

forcing fibres such as boron or carbon tend to fracture as a result of internal flaws of one sort or another and consequently show no significant delayed fracture effects. If there is no way in which damage in the matrix can cause failure of the reinforcing fibres and if the fibres are capable by themselves, of carrying the applied load, the fatigue effect will be very small. When a system of this type is loaded cyclically in tension, parallel to the fibre direction, the presence of the matrix is largely irrelevant and there is very little degradation in the strength of the composite after very large numbers of applied load cycles. This is particularly so when the fibres are very stiff since the elastic strains experienced by the matrix are then very small. Composites of this type loaded in this way are also insensitive to notches and other stress concentrators providing the fibre/matrix interfacial characteristics are properly designed. The mechanical properties of a fibrous composite are degraded at an increased rate, during cyclic loading, if the matrix is subjected to local high stresses. This can occur when the composite is put into compression during any part of its load cycle because the matrix is then required to provide local support for the fibres to prevent them buckling. This tendency is greater at the surface of a specimen and is made worse by the presence of flaws, regions having non uniform properties, voids, etc. Debonding of interfaces in conventional fibre reinforced polymers occurs when there is a stress component transverse to the fibres. The debonded regions extend and form matrix cracks which continue to develop but their progress is hindered when they have to cross other fibres. The failure processes occurring during fatigue in composites are not localised as is the normal situation with metal alloys. Metal matrix composites similarly tend to fail in fatigue by multiple cracking. The cracks can be generated internally for example at the ends of broken fibres where local matrix stresses are very high. Conversely fibre reinforced metals are not susceptible to fatigue failure from surface damage as is the case with metal alloys. Fibre fracture can occur during cyclic loading, and the number of matrix cracks tend to multiply. Again, as with polymeric matrix composites, the fibres hinder crack propagation by deflecting cracks and by maintaining loads across crack faces. Fatigue failure is gradual and progressive and has been described as “wear out”. The progressive, non catastrophic failure processes occurring in conventional fibrous composites is one of their main attractions from the point of view of structural engineering. 9.2. Fatigue crack growth in composites containing non-fracturing reinforcing members Preliminary studies have been made of the effect of crack bridging non fracturing reinforcing elements on crack growth in a thin brass sheet containing an edge crack and subjected to a cyclic crack opening force [41]. In these experiments the edges of the sheet were attached to a pair of cantilever beams in an arrangement similar to that described in section 8.3. For simplicity the reinforcing elements were attached directly to the beams in a manner used earlier by Morley [42] to study the propagation of an edge crack under monotonic loading. The non fracturing core reinforcing members bridged the edge crack in the metal sheet The load applied to the ends of the cantilever beams was cycled between nominally zero and an upper fixed limit, with a triangular waveform and at frequencies of between 300 and 500 cycles per minute. For the particular samples used, the upper limits of load was 500 N for the unreinforced sheets and 550 N for the reinforced sheets. This additional load compensated for the additional cross sectional area of the core elements which was approximately 10%. Again the presence

294

J.G. Morley, New types of reinforced composite materials

t

-

la

Llnreinforced specimens

0

E -~

~2

10

-

01

a C

Reinforced specimens

id~-

I

I

100

__

2OO~

300

Crack length 1 mm I

Fig. 30. Observed crack propagation rates for reinforced and unreinforced sheets (from ref. [41]).

of the tube elements was neglected, this being justified by the assumption that their mechanical properties, including their resistance to fatigue crack propagation, would not be worse than that of the brass sheet. Fig. 30 shows the observed crack propagation rates with increasing crack length for both reinforced and unreinforced sheets. In the case of the unreinforced sheets the incremental crack growth per cycle increases progressively with increasing crack length. Eventually the crack length becomes critical, for the particular loading condition, and catastrophic failure ensues. In the case of the reinforced system studied, the incremental crack growth in the metal sheet still occurred but proceeded at an ever decreasing rate and the applied load was supported by the crack bridging core fibres. The core elements were therefore subjected to the applied cyclic load and were also drawn through the composite structure as the crack in the metal plate extended and as the separation of the crack faces increased. For the particular i~einforcingelements used in the investigation the decoupling stress was about 0.53 GNm2 and this was chosen to be less than the fatigue limit of the core elements which was observed to be not less than 0.54 GNm2. At the decoupling stress level used in these experiments core element lifetimes of at least 3 X 106 cycles were observed. Preliminary tests were carried out under cyclic loading with the peak core stress values just less

J. G. Morley, New types of reinforced composite materials

295

than the decoupling load. Under these conditions no observable displacement of the core took place. More recently similar core elements having a slightly different helical geometry have been produced which can accept peak stress values in the region of 1.0 GNm2 for greater than 106 cycles (McColl and Morley, unpublished work). These core elements also have longitudinal stiffness values when encased within the tube (see section 6.2) which are not significantly less than that of the corresponding straight piece of core wire. These stress values, and the corresponding elastic strain values, are many times greater than those commonly used in lightweight highly stressed engineering structures. Hence, a relatively small volume fraction of such reinforcing elements, perhaps bonded directly to sheet metal would be capable of supporting all of the applied load. Such a structure would have fail-safe characteristics and catastrophic crack growth in the metal plate would not be possible at engineering design stresses (section 8.4). Thus, the tensile load bearing capability of the core members is assured and the additional load bearing ability of the plate is preserved. It will be noted that the core members, in a reinforced sheet metal system of this type can be placed initially in tension with the tubular members (and the metal sheet to which they are directly attached) in residual compression. This principle has great promise for two reasons. Firstly, at current engineering design stress values and under cyclic loading, it is possible to arrange for the core elements to be operating at peak stress levels which are within their endurance limits, while the rest of the composite structure is experiencing very small or even zero applied tensile stresses. This condition would be expected to exert a profound influence on fatigue crack growth in the metal plate and preliminary observations (McColl and Morley, unpublished work) indicate that this is so. It will be noted that the prestressed core members are fully load bearing and in no way parasitic in the sense of contributing to the weight, but not the load bearing ability, of the structure. Secondly, the prestressing principle enables the full tensile load bearing ability of the core elements to be used without them necessarily having to be particularly stiff. In conventional composites, in which the tensile elastic strains in both the reinforcing members and the rest of the composite structure are assumed to be the same (section 3), it is essential for the reinforcing members to be much stiffer than the matrix if they are to carry the major portion of the applied load at a particular strain value. The prestressing principle allows a stiff matrix to contribute to the composite stiffness while a strong, relatively compliant, reinforcing phase provides most of the tensile strength and resistance to crack growth in the composite structure.

10. Possible future developments and alternative non-fracturing reinforcing members 10.1. Hybrid systems based on non-fracturing reinforcing members It is apparent from fig. 28 that at elastic strains corresponding to engineering design strains (0.003 or less), catastrophic crack growth cannot occur in an aluminium sheet reinforced with a relatively small volume fraction (0.2) of non fracturing steel core elements. indeed the work of fracture Gm of the aluminium sheet is much greater than is necessary in order to prevent catastrophic crack extension. It is therefore possible to consider modifications to the aluminium alloy which reduce its work of fracture but which also improve other desirable properties, or alternatively, to substitute other materials for part of the aluminium sheet. One obvious application of

296

.1. G. Morley, New types of reinforced composite materials

the latter principle would be the addition of resin bonded boron, or carbon fibres to the aluminium sheet in order to increase its stiffness for an acceptable reduction in its work of fracture. It will be noted that an increase in the effective stiffness of the metal sheet, in comparison with the crack bridging core members, will modify the strain energy locking effect, described above, by changing the values ofE~and Em in eqs. (53) and (54), and thus modifying the effect shown in fig. 28. It is also feasible to consider making use of the prestressing principle, discussed in sections 6.2 and 9.2, in sheet metal hybrid structures. The carbon fibre, or boron fibre polymer bonded laminate, could be attached to the metal sheet before or after the core members are inserted. If the latter, it is clearly possible to arrange for the ceramic fibre component to be carrying zero tensile stress when no external forces are being applied to the composite system. If the former, then the ceramic fibre component is also placed in initial compression. In many engineering situations, particularly in structures having high rotational speeds, fatigue failures are often caused by resonance effects. In these situations it is desirable to use materials with high damping capacities. It is thus not clear at first sight whether, in such a situation, it would be more beneficial to increase the degree of prestressing in order to reduce the growth of a fatigue crack. By increasing the degree of prestressing, the amount of frictional interaction between the core members and the rest of the composite structure, and hence the damping capacity of the system, would be reduced. So far the discussion has been concerned with sheet metal systems to which convoluted steel wire core/steel tube two part reinforcing elements have been attached. The diameters of the core element wires have been in the region of 1.0 mm and are thus an order of magnitude greater than boron fibres and two orders of magnitude greater than glass and carbon fibres. Stainless steel hypodermic tubes have been used for the sheath elements. These are relatively expensive compared with conventional materials having costs, for a given weight, comparable with aero engines and subsonic aircraft structures. However, small diameter metal tubes are clearly intrinsically cheaper to produce than, for example, boron fibres. Alternative manufacturing routes to the construction of metal composite systems are clearly possible. One possible approach based on sheet metal technology is shown in fig. 3 1. Because of the relatively large size of the core members it becomes feasible to consider using them to feed concentrated loads into an engineering structure. The stress controlled interface prevents the occurrence of local excessive stresses. Also, when centrifugal loads are applied to the composite structure, it seems possible to adjust the core/tube stress transfer rate and the overall length of the component, so that the matrix will be generally subjected to longitudinal compressive forces with the core members supporting tensile loads. This is an alternative approach to the prestressing mode, previously described, for placing the matrix structure in compression. Since the core reinforcing members are only subjected, in operation, to small elastic deformations it is feasible to consider manufacturing them from strong stiff lightweight materials. The most attractive possibility would be to use polymer bonded carbon fibre bundles as core members. These could be fabricated by pultrusion techniques. If bonded with thermoplastic polymers a two stage manufacturing process would be possible. The first stage of the process would be to produce continuous straight “wire”, the second stage to produce suitable convolutions in the core by a combination of local heating and deformation. It should also be remembered that techniques for producing individually crimped carbon fibres have been reported through the use of bromine and iodine monochioride, in both the liquid and vapour states, as a “plasticising” agent [431.

J.G. Morley, New types of reinforced composite materials

297

Fig. 31. Schematic arrangement for a duplex reinforced structure manufactured from sheet metal (from ref. [281).

10.2. Hybrid conventional reinforced composites The theoretical model discussed in section 8.4 assumes for simplicity a constant shear strength interface between the non-fracturing core reinforcing members and the rest of the composite structure, up to the condition at which decoupling of the core members suddenly occurs. Thus, provided the applied loads, and corresponding strains, are less than the decoupling strain, 6d’ the model is applicable to a conventional fibre reinforced material having a constant shear strength fibre matrix interface. This condition will apply when the reinforcing fibres are capable of carrying all of the load applied to the composite. Fig. 28 can therefore be considered to apply to this situation. It is apparent from fig. 28 that if the work of fracture of the matrix Gm is sufficiently small, transverse crack propagation in the matrix is possible even when the reinforcing fibres are capable of carrying all of the applied load. This condition is, of course, observed experimentally as multiple fracture in brittle matrix composites (section 7). Thus, from fig. 28, matrix cracking would be expected to be inhibited by increasing the work of fracture of a brittle matrix. This seems to be achievable by incorporating within the matrix a second set of reinforcing fibres so as to increase the matrix work of fracture. Following this line of argument, therefore, we can postulate a hybrid fibrous composite system containing two types of reinforcing fibres. One set is designed to stiffen and strengthen the matrix and to limit the amount of strain energy released by a matrix crack through the strain energy locking mechanism described in section 8.4. The second set of reinforcing fibres is provided simply to enhance the work of fracture of a brittle matrix. Some further factors need to be born in mind. The second, matrix toughening, fibre phase would need to operate effectively in the very localised regions of high stress, and corresponding strain, in the vicinity of the matrix crack tip. These fibres could also make a contribution to the matrix work of fracture as the crack faces separate. Here again the fibres would need to be loaded to a high stress level in a localised region since the separation of the crack faces is small in absolute terms. One possible approach therefore would be to arrange for the second matrix toughening phase to consist of small fibres. These could absorb energy by pull out, by interfacial debonding or by both mechanisms. For pull out to occur, continuous fibres would require a suitable flaw distribution. This line of argument, therefore, leads to the design of hybrid composites containing two sets of reinforcing fibres of widely differing sizes and perhaps different interfacial characteristics.

J.G. Morley, New types of reinforced composite materials

298

The second factor requiring consideration is the desirability of minimising the Young’s modulus of the matrix so that the strain energy released by a matrix crack of given geometry is minimised. However, this factor cannot be treated in isolation because strain energy is also absorbed by frictional losses generated by displacements occurring at the interface between the large crack bridging fibres and the matrix. The total situation is therefore quite complex but seems to offer a possible route towards the suppression of cracking in the brittle matrix of a fibre reinforced composite structure. 10.3. Stress controlled decoupling by plastic deformation So far this review has been concerned with stress controlled decoupling mechanisms based on a core member in the form of an extended helix. Other ~convo1utedforms, in which the decoupling of the core element is achieved by elastic deformation, can be used and some examples have been studied by Morley and Millman [26]. An alternative decoupling mechanism, based on the controlled plastic deformation of a high strength metal tube, has also been investigated [44]. In this arrangement a high strength, and hence relatively brittle metal tube acts as the primary reinforcing member. It is bonded strongly to the matrix which, in the experimental samples so far studied, has been a relatively brittle epoxy resin. A loosely fitting core element which may be in the form of a tube or a solid rod is carried inside the outer tube. Under normal loading conditions the core member does not carry any load and the stiffening and strengthening of the composite structure is generated solely by the tube elements. The behaviour of a reinforcing member of this type under tensile loading can be described with reference to fig. 32 which illustrates the load/extension curve, characteristic of a typical tube and core element, in a sharply notched brittle resin matrix. The first part of the load extension curve p—1.OSmm

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Fig. 32. Load extension characteristics of reinforcing member with controlled plastic decoupling (from ref. [44]).

J.G. Morley, New types ofreinforced composite material:

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represents the tensile elastic extension of the composite structure. This extends to the point B where the resin at the notch starts to crack. Several cracks may be generated and this is accompanied by some short-range debonding of the load carrying tube on each side of the notch. At this point the outer tube alone is carrying the load in this region, and the curve follows the transition between elastic and plastic deformation as the tube approaches its failing stress. When the tube starts to fail by local necking (point C), it collapses inwards and the wall thickness is reduced. This condition is indicated by the slight fall in the load carried. In the absence of the core member the outer tube would continue to deform plastically in the necked down region and would eventually fracture. The presence of a core member of a suitable size prevents the occurrence of the final stages of the deformation process and, because the outer tube grips the central core, the core member supports part of the applied load in the necked down region. The mechanism can be considered as analogous to strain hardening in metals. Here the additional load bearing ability, resulting from tensile extension, is generated by a change in the geometrical form of the composite system. (This contrasts with the strain hardening process in metals which results from the progressive entanglement of dislocation arrays within the crystal structure of the material.) The further course of the deformation process is illustrated in fig. 33. Since the initially deformed region is no longer the weakest part of the system, subsequent deformation has to take place in the intermediate regions, not yet in contact with the core, on each side of the collapsed region. These regions of local deformation now propagate along the outer tube. During this process the load bearing outer tube is progressively debonded from the matrix by the deformation process, and each portion of this tube is subjected sequentially to deformation. The tensile load path extends from the still bonded matrix, through the intermediate collapsing region of the tube to the collapsed core. As the collapsed zone approaches the end of the structure a point is reached at which there is insufficient tube/matrix interface to carry the deformation load. At this point (D, fig. 32) the remaining interface fails in shear and the load which can be supported drops to a level E (fig. 32), which is generated by residual frictional effects as the end portions of the reinforcing element are drawn through the cylindrical hole in the polymer matrix. It is clear that the successful operation of this process will depend on the relative dimensions of the inner and outer member and on the way in which the outer member deforms. Millman and Morley attempted to establish a relationship between the geometrical parameters and the magnitude of the deformation process but could find no clear pattern of behaviour from the particular systems studied.

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300

J.G. Morley, New types of reinforced composite materials

It is possible to draw some conclusions from these experiments and to consider engineering applications of these systems. The feature of central importance is that, although some ductility is necessary in the outer load bearing tube, it is only used as a trigger mechanism for the deformation process which is stabilised by the ensuing geometrical changes. Hence a significant degree of strain hardening is not required in the outer member and indeed, in the samples examined, failure would have taken place at overall elongations of around 3%. This was increased by about an order of magnitude by the presence of the core member and these elongations are comparable with those which are encountered in an orthodox strain hardening metal. In the case of conventional metals, yield strength and ductility are basically incompatible requirements. The first depends upon hindering the movement of dislocations within the crystal structure from the outset and the second requires that initially their movement should be relatively easy and become progressively more difficult with increasing deformation. By the use of mechanism described above, it is possible to design composite structures, using relatively brittle metal components as reinforcing members, which show bulk deformation and failure stabilisation characteristics similar to those of ductile strain hardening metals. This feature can be utilised in order to combine high yield strengths with large amounts of tensile deformation.

11. Conclusions The bulk of the research carried out to date on fibre reinforced composites has been concerned almost exclusively with systems consisting of strong brittle reinforcing members embedded in, and separated from each other by some sort of matrix. In almost all cases the reinforcing members have been approximately circular in cross section, although a small portion of the work carried out has been concerned with laminated composites. The reinforcing component has been strong but brittle and the interface between the reinforcing members and the matrix has fixed characteristics. Under load both matrix and fibres elongate by approximately equal amounts. Prestressing has been limited to effects produced by differential contractions in the components after high temperature fabrication and this usually places the matrix in slight longitudinal tension. In all cases the reinforcing members fracture as an increasing tensile load is applied to the composite structure and this usually occurs at a strain of about 1%. In this review the properties of conventional composites are compared with systems in which the fibre/composite interfacial characteristics are active in the sense that they are modified by the stress carried by the reinforcing members. By this means, fracture of the primary reinforcing members can be prevented thus providing a fail safe characteristic in the composite structure. In addition control can be exercised over the amount of initial differential stress between the primary reinforcing members and the rest of the composite structure. Crack stopping and energy absorbing deformation mechanisms provide enhanced toughness compared with conventional composites. Much of the emphasis has been placed on reinforced sheet materials but applications are not necessarily limited to this type of composite structure. Metals have so far been used in the construction of experimental versions of these systems, but ductility is not required in the convoluted core reinforcing members (and only to a limited degree in the arrangement discussed in section 10.3). The approach may make it feasible to use metal alloys which, although having many desirable characteristics, are too brittle to be used satisfactorily in monolithic form. It is possible that new high temperature engineering composites might be produced in this way.

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High strength, light weight strong ceramic fibres can also be used in the construction of these systems so that fail safe characteristics can be added to the engineering advantages already offered by the use of these materials. Finally, the construction processes considered so far are based on existing sheet metal technology thus avoiding the difficult and expensive fabrication procedures so far necessary with existing high performance composites.

References [1] G.A. Cooper, The structure and mechanical properties of composite materials, Rev. Phys. Technology, The Institute of Physics 2, Number 2 (1971) 49. [2] D.K. Hale andA. Kelly, Strength of fibrous composite materials, Ann. Materials Science 2 (1972) 405. [3] L.J. Broutman and R.H. Koock (eds.), Modern CompositeMaterials (Addison-Wesley, 1967). [4] The Properties of Fibre Composites (Conf. Proc. National Physical Laboratory), (I.P.C. Science and Technology Press, 1971). [5] L.J. Broutman and R.H. Krock (Eds.), Composite Materials (8 volumes), (Academic Press, New York and London, 1974). [6] E. Scala, Composite Materials for Combined Functions (Hayden Book Company Inc., Rochelle Park, N.J., U.S.A., 1973). [7] J.G. Morley, Fibre Reinforcement of Metals and Alloys, International Metals Reviews (Institute of Metals and American Society for Metals) in press. [8] J.G. Morley, Carbon Fibers, Kirk-Othmer Encyclopedia of Chemical Technology, Supplement Volume, 2nd Ed. (John Wiley and Sons Inc., 1971). [9] R. Hill, I. Mech. Phys. Solids 12(1964)199. [10] A. Kelly and H. Lilholt, Phil. Mag. 20(1969) 311. [11] A. Kelly and G.J. Davies, Metall. Rev. 10 (1965) 1. [12] H.L. Cox, Br. J. AppI. Ploys. 3 (1952) 72. [13] N.F. Dow, General Electric Report (1963), R. 63, SD, 61. [14] B.W. Rosen, Fiber Composite Materials (American Society for Metals, Metals Park,Ohio, and Chapman and Hall, London, 1964) p. 37. [15] D.M. Schuster and F. Scala, J. Metals 15 (1963) 697, also, Trans. Metall. Soc. A.I.M.E. 230 (1964) 1635. [16] W.R. Tyson and G.J. Davies, Br. J. Appl. Ploys. 16 (1965) 199. [11] MR. Piggott, Acta Metall. 14 (1966) 1429. [18] J.O. Outwater, Mod. Plastics 33 (1956) 156. [19] A. Kelly, Proc. Roy. Soc. A 282 (1964) 63. [20] A. Kelly and W.R. Tyson, J. Mech. Phys. Solids 13 (1965) 329. [21] A.J.M. Spencer, Int. J. Mech. Sci. 7 (1965) 197. [22] A. Kelly, Strong Solids (Clarendon Press, Oxford, 1966 and 1973). [23] V.R. Riley, J. Composite Mater. 2 (1968) 436. [24] J.M. Hedgepeth and P. Van Dyke, J. Composite Mater. 1 (1967) 294. [25] J.G. Morley, Proc. Roy. Soc. A 319 (1970) ill. [26] J.G. Morley and R.S. Millman, J. Mater. Sci. 9 (1974) 1171. [21] M.J. Chappell, J.G. Morley and A. Martin, J. Phys. D, Appl. Phys. 8(1975)1071. [28] J.G. Morley, R.S. Millman and A. Martin, J. Ploys. D, Appl. Ploys. 9 (1976) 1031—1047. [29] G.A. Cooper and J.M. Sifiwood, J. Mater. Sci. 7 (1972) 325. [30] G.A. Cooper and A. Kelly, J. Mech. Phys. Solids 15 (1967) 279. [31] M.A. McGuire and B. Harris, J. Phys. D, Appl. Phys. 1 (1974) 1788. [32] G.A. Cooper, J. Mater. Sci. 5 (1970) 645. [33] A.H. Cottrell, Proc. Roy. Soc. A 282 (1964) 2. [34] J.O. Outwater and W.O. Carnes, Final Report Contract DAA-21-67-C-0041 (AD-659363) (1967); J.O. Outwater and M.C. Murphy, Final Report Contract DAA-21-67-C-0041 (extended) (1968); J.O. Outwater and M.C. Murphy, Paper 1 1-C, 26th Annual Conf. Reinforced Plastics and Composites Division of Society of Plastics Industry (1969). (See also G.A. Cooper [1].) [35] A. Kelly, Proc. Roy. Soc. A 319 (1970) 95. [36] KM. Prewo, J. Composite Mater. 6 (1972) 442. [37] R.S. Millman and J.G. Morley, Materials Science and Engineering 23 (1976) 1—10. [38] J.G. Morley and I.R. McColl, J. Phys. D, Appl. Phys. 8 (1975) 15.

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[39] M.J. Chappell and J.G. Morley, J. Mater. Sci. 11(1976) 57. [40] M.J. Owen (Chapters 7 and 8) and J.R. Hancock (Chapter 9), Composite Materials, Vol. 5, Fracture and Fatigue, eds. L.J. Broutman and R.H. Krock (Academic Press, New York and London, 1974). [41] I.R. McColl and J.G. Morley, J. Phys. D, Appl. Phys. 8 (1975) Ll00. [421 J.G. Morley, Farathy Special Discussions of the Chemical Society, No. 2 (1972) 109. [431S.B. Warner, L.H. Peebles and D.R. Uhlman, Carbon Fibres, Their Place in Modern Technology, Proc. Second Intern. Carbon Fibres Conference, London (1974) p. 16. [441R.S. Millman and J.G. Morley, J. Ploys. D, AppI. Phys. 8 (1975) 1065.