Fibre tensile stress, equations

For example, on the side face of a bending bar the tensile stress decreases with distance from the specimen edge. With increasing curvature of the bending bar, the cracks are driven inward to the neutral fibre. The position of the crack tip,yljp, at maximum bending deformation has to be inserted into (7) in order to calculate the tensile stress at the crack tip olip = oxx(ylip) (Fig. 11).The fracture toughness follows then from the equation Kk = K,([Pg.153]

One further topic merits discussion in this section in view of its success in dealing with the mechanical properties of oriented fibres, which are after all anisotropic polymers. That is the theory of kinetic fracture, developed mainly by Zhurkov and co-workers. Evidence has been presented from electron spin resonance (e.s.r.X " mass spectrometry, and infra-red spectroscopy that when highly oriented fibres or heavily cross-linked rubbers experience a tensile stress (along the axis for fibres) an appreciable fraction of main-chain bonds are broken by the applied stress. These scission events are observed to occur more or less homogeneously throughout the fibre and are not localised in the fracture plane. Many sets of data show that the lifetime tb of a fibre under stress is described approximately by the following equation... 

Fii>. 9. Stress distributions associated with a reinforcing fibre in a more ductile matrix (according to equations 6.40 (tensile stress) and 6.49 (shear stress) in Kelly and Macmillan, 1986),... 

The prepreg fibre bed is typically assumed to be an elastic porous medium with incompressible and inextensible fibres and fully saturated with the resin. The resin is assumed to flow in the pores between the fibres, and the fibre mass in the laminate remains constant during cure. The governing equations of the system must describe the behaviour of the composite constituents the fibre bed and the resin. Firstly, the equilibrium of forces on the representative element is considered. Secondly, the mass conservation for the representative element must be satisfied. For a porous medium saturated with a single phase fluid, the total stress tensor a,) is separated into two parts as (tensile stresses are considered positive) ... 

The reduction in tensile stress towards the ends of each fibre inevitably leads to a decrease in the tensile modulus compared with the continuous filament case. Consider a plane drawn perpendicular to the stress direction in an aligned discontinuous fibre composite (Figure 8.4), which must intercept individual filaments at random positions along their length. Hence the stress carried by the composite must be lower than that for the continuous filament case, and is dependent on the length of each fibre. Cox [9] predicted a correction factor rji for the tensile modulus in the axial direction that takes into account the finite length of the fibres so that Equation (8.3) is modified to... 

E and A are Young s modulus and cross-sectional area of the fibre, respectively, is the uniform matrix strain, G is the shear modulus of the matrix. Corresponding diagrams of bond stress t and tensile stress a in the fibre are shown in Figure 8.11. Values of the tensile stress a (x) in the fibre are obtained directly from equation (8.9). 

Although Galileo s stress equation was later shown to have a slight numerical error of 6, because the tensile stress across the outer fibres of a cantilevered elastic beam is 6FL/bd, the overall basis of deflection and strength of beams was established... 

Based on these assumptions, Cox [2] derived the following equations for the tensile stress, a(x), in the fibre, and the elastic shear stress at the interface, t(x),... 

The tensile stress in thefibre can be calculated from this equation, and its maximum is approximately at the exit point of the fibre. To determine the total stress of the fibre at this point, the stress induced by the bending moment, M, should be added ... 

In order to simplify the discussion and keep the derivation of the formulae tractable, a fibre with a single orientation angle is considered. In a creep experiment the tensile deformation of the fibre is composed of an immediate elastic and a time-dependent elastic extension of the chain by the normal stress ocos20(f), represented by the first term in the equation, and of an immediate elastic, viscoelastic and plastic shear deformation of the domain by the shear stress, r =osin0(f)cos0(f), represented by the second term in Eq. 106. 

Failure starts when the stress in matrix or fibre reaches the yield or tensile strength. Similar to the elastic case, we can apply an isostrain rule of mixtures (equation (9.2)) to calculate the stress in the composite ... 

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