Loading perpendicular to the fibres

If the fibres are oriented perpendicular to the loading axis, this is called serial connection in analogy to springs connected this way. To simplify matters, we assume that the fibres are plates extending over the whole cross section of the material and oriented perpendicularly to the applied load (figure 9.2(a)). 

In the serial connection, the balance of forces must hold at each fibre-matrix interface  

Again in complete analogy to springs, both materials can deform in different 

If we use Hooke s law and the condition (9.4), we find, after some rearrangement, Young s modulus 

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Under compressive loads perpendicular to the fibre direction, the matrix may shear on planes parallel to the fibres. In this case, the fibres are irrelevant for the compressive strength. Shearing on planes cut by the fibres is not possible because the fibres impede this. If shear occurs in the direction of the fibres, either the matrix itself can shear between the fibres or there may be shearing along the interface. The strengthening effect of the fibres is small in the latter case as well. If the interface is weak, the strength of the composite may even be smaller than that of the pure matrix material [122]. 

Make the simplifying assumption in the case of loading perpendicular to the fibres that the fibres are plates extending throughout the volume (see figures 9.1(a) and 9.2(a)) Neglect the transversal contraction ... 

Example 3.8 A thin unidirectional carbon fibre composite is loaded as shown in Fig. 3.14 and has the properties listed below. If the fibres are aligned at 35° to the x-axis, calculate the stresses parallel and perpendicular to the fibres. 

In tensile loading of a specimen, the maximum shear stresses occur on planes with their normals at 45° to the tensile stress direction. In tensile loading of a 45° specimen we are interested therefore in laminar shear on planes with their normals parallel to the sheet surface which either contain or are perpendicular to the fibre axis i.e. are at 45° to the tensile stress direction). Since easy shear on either set of planes is consistent with the observed high value of S44, additional measurements (such as X-ray diffraction) must be made during deformation in order to determine the relative importance of the possible molecular deformation modes. Such measurements were not attempted in the above study. 

The filament s transverse isotropy implies that under compressive loading normal to the fibre axis the stresses in the transverse plane are identical in form to those for the compression of an isotropic cylinder and, provided that the length of the filament under compression is long compared with the width of the contact strip 2b, friction ensures that compression occurs under plane strain conditions. As = 0 only a normal stress acts along the filament axis, which can be found in terms of the normal stresses and oy, in the perpendicular plane, i.e. 

The spirally wound E-glass/epoxy tube of Problem 6.10 is loaded to failure in axial tension. Predict the failure stress and describe the mode of failure. (The wall of the tube may be taken to have a tensile strength perpendicular to the fibres af = 40 MPa and a shear strength rf2 = 60 MPa.)... 

If a composite with unidirectional fibres is loaded in tension or compression perpendicular to the fibre direction or in axial shear in fibre direction, it can fail without failure of the fibres by fracture, buckling, or kinking. These cases are therefore called matrix-dominated failure. 

We want to calculate Young s modulus of a fibre composite loaded in parallel and perpendicular to the fibre direction (see sections 9.2.1 and 9.2.2). We start by considering a polymer matrix composite with perfectly aligned, infinitely long, uniaxial fibres. 

A simple analysis of this problem follows from the condition that the contact zone can be arranged to be small compared with the radius of the monofilament. To calculate the deformations in the diametral plane, it is then adequate to consider the problem as the compression of a cylinder under concentrated loads (Figure 8.11(b)). For an isotropic cylinder, this is a well-known problem to be found in textbooks on elasticity (see Reference 20, p. 122). It is necessary to satisfy the boundary conditions on the surface of the cylinder, and this is done by addition of an isotropic tension in the plane perpendicular to the fibre axis. 

The qualitative observations can be characterized in the following way In the specimen loaded to a maximum strain of 0.001 a distributed microcracking pattern perpendicular to the fibres can be observed. No cracks penetrate the specimen from one side to the other and many cracks are limited to a length corresponding to the average distance between adjacent fibres. 

At UTS an arbitrary fibre intersecting the macrocrack under an angle (tt/2 0) is transmitting a load perpendicular to the crack plane P (0)cos0, where P (0) is the load acting in the fibre at the crack plane. A 3-D UR... 

X >1 when o6 60 . Specimens with the oriented fibres /ID orientation/ subjected to impact loads perpendicular to the direction of fibres = 90V exhibited from 1.19 to 1.59 the impact resistance of the specimens with 2D orientation. When... 

The fibre sanples (ca 0.5g), consisting of tows of aligned fibres with a volume fraction of ca 10%, were loaded into lOmm id fused silica tubes, heated under vacuum (0.01 iribar) at 200°C and the tubes then sealed under vacuum. In this way absorbed water was eliminated from the fibres. The neutron transmission was ca 70-80% for these amounts of fibres. The scattering data from the two-dimensional area detector were found to be anisometric and were therefore converted to one-dimensional data by using sector masks. Scattering in the fibre direction was derived frcm a 75° wide mask parallel to the fibres and the scattering normal to the fibres was derived from a 15° wide mask perpendicular to the fibre direction. 

The Z-direction is perpendicular to the page. For simplicity the material is assumed to be isotropic, ie same properties in all directions. However, in some cases for plastics and almost always for fibre composites, the properties will be anisotropic. Thus E and v will have different values in the x, y and z direction. Also, it should also be remembered that only at short times can E and v be assumed to be constants. They will both change with time and so for long-term loading, appropriate values should be used. 

When a load is perpendicular to the direction of fibres, the components of the composite react independently ... 

At higher concentrations of fibres or at intermediate concentrations when a few fibres around the crack tip are orientated perpendicular to the notch plane, the loading curve increases linearly up to a maximum load Pi as the load is transferred onto the fibres at the crack front and a process zone develops. Fracture of the fibres lying normal to the notch plane results in unstable crack propagation until it is arrested by a packet of fibres favourably orientated then the applied load must be increased to create a new frontal process zone. Tlierefore the successive unstable crack extensions result in a saw-tooth like loading curve behaviour (types 3 and 3 loading curves in Table II, associated with Figures I OB and lOE, I OF respectively). 

This approximation is called the Voigt model, and the value of the elastic modulus is often known as the Voigt bound. The expression is identical to that for a continuous aligned fibre composite under a longitudinal load, and gives the elastic modulus when the load is applied parallel to the sheets. Similarly, if the stress is applied perpendicular to the layers, and an iso-stress condition applies (the Reuss model), the elastic modulus is ... 

The ends of the microfibrils create about Itf m point vacancies in the microfibrillar superlattice (Fig. 11). Under applied tensile load they may fail first, eventually by microcrack formation so that the adjacent microfibrils have to carry a heavier load than the rest of the sample. Hence they are first candidates for rupture detectable by the rascals formed at the rupture of tie molecules in at least one amorphous layer of the microfibril affected. Depending on the ratio of axial strength to lateral adhesion of the microfibrils the microcracks will grow parallel (high ratio) or perpendicular (low ratio) to the fibre axis yielding a large number of broken, chains and radicals in the former and a small one in the latter case. Nylon is an example of the former and linear polyethylene of the latter type. 

When 4" 0 the method models continuous systems of fibre and matrix materials stacked perpendicular to the load direction. 

The loading direction is parallel (0°) to the fibres The loading direction is perpendicular (90°) to the fibres... 

In this section, we start by discussing the behaviour of fibre composites under tensile loads, at first for the simplest case of continuous fibres. Subsequently, we will discuss the load transfer between the matrix and non-continuous fibres and see how this determines the failure properties and the fracture toughness of the material. For this, we also have to consider that fibre properties are statistically distributed. Finally, we will discuss the behaviour under compressive loads, loads perpendicular to fibre direction, and arbitrarily oriented loads. 

Fig. 9.16. Hierarchical structure of adult human bone. Tropocollagen molecules are arranged in a so-called quarter-stagger structure, with platelets of hydroxy apatite in between. The fibres formed by this structure unite to fibre bundles which in turn form lamellae. The major part of the bone consists of osteons made of ring-shaped lamellae. Near the bone s surface, the lamellae are parallel to the surface. The orientation of the fibre bundles within the lamellae depends on the mechanical loads on the bone in tensile regions, they are ahgned in the loading direction as shown in the figure, in compressive regions, the fibre bundles of some lamellae are perpendicular to the loading direction...

The fibre has birefringence in the absence of an external load, of which several factors, such as geometric, UV-induced and stress-induced, may be at the origin. However, for mathematical convenience, it is assumed that the initial birefringence in the FBG is due to a residual strain state in the fibre core that is described by the principal strains e o and 820 (Sio > 820)- These principal strains are in directions perpendicular to each other and to the fibre axis, and direction 1 makes an angle f with the x axis. According to Eq. (10.18), the wavelength separation can be expressed as ... 

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