Elastic shear stress 57 equations transfer

The most commonly used theory used to model the stiffness of this type of composite was developed by Cox [1] and further improved by Krenchel [2, 3]. Cox s shear lag model was developed for aligned discontinuous elastic fibers in an elastic matrix. The applied load is transferred from the matrix to the fiber via interfacial shear stresses, with the maximum shear at the fiber ends decreasing to zero at the centre. Thus, the tensile stress in the fiber is zero at the ends and maximum in the middle. Thus, although the efficiency of stress transfer increases with fiber length, it can never reach 100%. In order to accommodate this dependence of reinforcement efficiency on fiber length, Cox introduced a fiber length efficiency factor -rij into the rule-of-mixtures equation for the composite modulus Eg. 

The model used to arrive at equation (5.102) is based on a constant shear stress approximation for transfer of stress across the interface between matrix and ceramic fiber, and this is often interpreted as the flow stress of the matrix or as the friction stress at the interface. Such a model leads to the view that the average fiber stress varies linearly as a function of distance along the fiber which is proportional to shear stress. Thus the shear stress at the interface is expected to equal the frictional stress only over a specific part of the fiber and then to decrease steadily with distance along the interface. A more detailed model concentrating on the transfer of load across a frictional interface between elastic solids is reported by Dollar and Steif, which suggests that the results obtained by using equation (5.102) are approximations because the constant shear stress approximation overestimates the extent of slip. Furthermore, the error increases as the coefficient of friction in the interface increases and as the load increases. Nevertheless this hardness indentation procedure does lend itself to obtaining much useful comparative data for one type of ceramic fiber say, in a series of matrices. Thus the technique is in line with how hardness indentation methods are most commonly used. 

In this case the only mechanism of stress transfer is an elastic one, governed by shear lag equations, such as those of Gresczczuk [7] and Lawrence [8]. 

For composite PHE-Gr, it was shown [23] that the dependence of the elasticity modulus (E) on (pf is well described by the Kerner equation which assumes strong interactions at the interface. Because of it we can suppose that by fracture of a composite there exists the possibility of transfer of the applied stress through the interfacial border. The tensile strength of the composite in the given case should be the function of the shear strength of the... 

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