Cox models

There are two well-accepted models for stress transfer. In the Cox model [94] the composite is considered as a pair of concentric cylinders (Fig. 19). The central cylinder represents the fiber and the outer region as the matrix. The ratio of diameters r/R) is adjusted to the required Vf. Both fiber and matrix are assumed to be elastic and the cylindrical bond between them is considered to be perfect. It is also assumed that there is no stress transfer across the ends of the fiber. If the fiber is much stiffer than the matrix, an axial load applied to the system will tend to induce more strain in the matrix than in the fiber and leads to the development of shear stresses along the cylindrical interface. Cox used the following expression for the tensile stress in the fiber (cT/ ) and shear stress at the interface (t) ... 

State Full Box-Cox Model InQ on left hand side... 

These results can be used to great advantage in deriving the actual second derivatives of the log likelihood function for the Box-Cox model. Hint See Example 10.11.)... 

The two common types of model, one widi an intercept and one without an intercept term, are related. Models excluding the intercept are often referred to as Sheffe models and those with die intercept as Cox models. Normally a full Sheffe model includes all higher order interaction terms, and for diis design is given by... 

Table 14.2. A Cox model for the five-year risk of ipsilateral ischemic stroke on medical treatment in patients with recently symptomatic carotid stenosis ...

Tibshirani, R. (1997). The Lasso method for variable selection in the Cox model. Statist. Med, 16 ... 

The solid line in Figure 8 is a least squares fit of the data. It has a slope of 0.92 and an intercept of 0.28 o strain. This line reflects the average behaviour of the fibres during deformation. The dashed line in Figure 8 has a slope of unity and corresponds to e = e which would be the behaviour expected from the Cox model (r8) r an ideal uniaxially-aligned continuous fibre composite with no complication due to thermal stresses. The experimental line confirms that the surface fibres are on average subjected to a thermal compressive strain of 0.287. if it is assumed that the linear dependence of peak position upon e shown in... 

Fig re 3 can be extrapolated for compressive strains. However, since the slope of the line is close to unity (0.92) it shows that when the prepreg composite is deformed the fibres are strained in tension as predicted by the Cox model. We feel that these are two important observations relevant to the deformation of both carbon-fibre/PEEK matrix composites and composites in general. 

Micromechanical models such as Cox shear-lag and Halpin-Tsai are often used to predict the stiffness and strength of discontinuously short-fiber reinforced composites. Experimental results of tensile measurements are then compared or correlated with such theoretical models. The shear-lag analysis originally proposed by Cox considered a discontinuous fiber embedded in an elastic matrix with a perfectly bonded interface and loaded in tension along the fiber direction [25]. The analysis tabes into account the difference in strain displacements of the fiber and matrix along the interface. The stress transfer depends on the interfacial shear stress between the fiber and the matrix. The stress transfer from fiber ends is neglected in the analysis. The Cox model incorporates the aspect ratio (a = l/d where I is the fiber length and d the diameter) of the fiber into... 

Using the Cox model for a fibre composite, already discussed [9], and... 

On the other hand, Hofiman experimentally studied the dynamic contact angle—interface velocity relation using different liquids and alarge range ofvelocities. He obtained results compatible with Cox model, seen in Fig. 14. ° ... 

This agreement between Cox model and experimental data obtained with systems with 2ero or close to zero static contact angle, is due to the fact that viscous eflFects are well determined in that case and the factor Qe (molecular dissipation in the very vicinity of the contact line) is negligible compared to the factor Ln K/ls) 14. 

Fig. 14. Variation of the contact angle with the capillary number for silicon oil displacing air. Experimental data were obtained by Hoffman and the solid lines correspond to Cox model. From Ref 101.

Fig. 15. Variation of the contact angle with the velocity for glycerol/water displacing air on a PET surface. Solid line corresponds to molecular-kinetic theory and dotted line to Cox model. " ...

Using the Cox model for a fibre composite, already discussed [12], and neglecting the very small contribution E" V" arising from the tensile modulus of the comphant matrix, the extensional modulus E of the highly oriented polymer becomes... 

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