Stress radial

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

The nylon ring may be considered as a thick wall cylinder subjected to this internal pressure (see Appendix D). At the inner surface of the ring there will be a hoop stress, <7, and a radial stress, Cr. Benham et al. shows these to be... 

If the acetal ring is considered as a thick wall cylinder, then at the inner surface there will be hoop stresses and radial stresses if it is constrained in a uniform manner ... 

Assuming the appropriate boundary conditions between the internal sphere and any number of spherical layers, surrounding it, in the RVE of the composite, which assure continuity of radial stresses and displacements, according to the externally applied load, we can establish a relation interconnecting the moduli of the phases and the composite. For a hydrostatic pressure pm applied on the outer boundary of the matrix... 

The principal stresses (see Section 13.3.1) acting at a point in the wall of a vessel, due to a pressure load, are shown in Figure 13.1. If the wall is thin, the radial stress comparison with the other stresses, and the longitudinal and circumferential stresses o and <72 can be taken as constant over the wall thickness. In a thick wall, the magnitude of the radial stress will be significant, and the circumferential stress will vary across the wall. The majority of the vessels used in the chemical and allied industries are classified as thin-walled vessels. Thick-walled vessels are used for high pressures, and are discussed in Section 13.15. 

Thick walls are required to contain high pressures, and the assumptions made in the earlier sections of this chapter to develop the design equations for thin-walled vessels will not be valid. The radial stress will not be negligible and the tangential (hoop) stress will vary across the wall. 

When the probe makes contact with the film, it generates a radial stress field around the point of contact. If the film is isotropic, it deforms in a uniform ring around the probe, as shown in Fig. 8.11 a). If the film is oriented, it deforms in a non-uniform manner. When the film is mildly oriented, the deformation area becomes ellipsoidal, as we see in Fig. 8.11 b), with its long axis... 

The annealing experiments on spin cast films near Tg show a drop in Ie/Im, although after 24 hours of annealing Ie/Im is still higher than for a solvent cast film that has undergone no radial stress. This implies that non-equilibrium chain structures still exists in the spin cast films even after long annealing times. 

An additional radial stress, i(a,z), acts at the interface that arises from the differential Poisson contraction between the fiber and the matrix when the matrix is subjected to an axial tension at remote ends. q (a,z) is obtained from the continuity of tangential strain at the interface (i.e. e (a,z) = e (a,z)) (Gao et al., 1988)... 

The radial (compressive) stress, qo, is caused by the matrix shrinkage and differential thermal contraction of the constituents upon cooling from the processing temperature. It should be noted that q a, z) is compressive (i.e. negative) when the fiber has a lower Poisson ratio than the matrix (vf < Vm) as is the normal case for most fiber composites. It follows that q (a,z) acts in synergy with the compressive radial stress, 0, as opposed to the case of the fiber pull-out test where the two radial stresses counterbalance, to be demonstrated in Section 4.3. Combining Eqs. (4.11), (4.12), (4,18) and (4.29), and for the boundary conditions at the debonded region... 

There are many features in the analysis of the fiber push-out test which are similar to fiber pull-out. Typically, the conditions for interfacial debonding are formulated based on the two distinct approaches, i.e., the shear strength criterion and the fracture mechanics approach. The fiber push-out test can be analyzed in exactly the same way as the fiber pull-out test using the shear lag model with some modifications. These include the change in the sign of the IFSS and the increase in the interfacial radial stress, (o,z), which is positive in fiber push-out due to expansion of the fiber. These modifications are required as a result of the change in the direction of the external stress from tension in fiber pull-out to compression in fiber push-out. 

For the cylindrical coordinates of the fiber push-out model shown in Fig. 4.36 where the external (compressive) stress is conveniently regarded as positive, the basic governing equations and the equilibrium equations are essentially the same as the fiber pull-out test. The only exceptions are the equilibrium condition of Eq. (4.15) and the relation between the IFSS and the resultant interfacial radial stress given by Eq. (4.29), which are now replaced by ... 

Also, from the general relations between strains and stresses given by Eqs. (4.8) and (4.9), and the additional radial stress 9i(u,z) of Eq. (4.18), the strains in the fiber and matrix at the interface for fiber pull-out are obtained as ... 

Fig. 7.15, Normalized residual radial stress as a function of Young s modulus ratio, Ej/Em, for varying coating thickness, t/a = 0.05, 0.1, 0.2. Coefficients of thermal expansion (CTE) of the coating (a) Kc = 100 X 10-V°C (b) = 20 X 10-V°C. After Kim and Mai (1996a, b).

The implementation of mandrel removal is of particular interest in winding model formulation. When the mandrel is removed, there is no radial stress [Pg.407]

This requirement is imposed by adding a radial stress at the cylinder inner diameter that is equal but opposite in magnitude to the contact stress between the mandrel and cylinder. The contact stress corresponds to the radial pressure at the interface at the time the mandrel is removed. 

The calculation of the strength resp. the admissible internal pressure varies with the wall-thickness thick-walled hollow cylinders are calculated by neglecting the radial stress (equal to the pressure) which is small compared to the tangential. On the other side the thick-walled hollow cylinders are calculated with the Lame equations (1833). 

D/D0 inner / outer diameter, a, tangential stress, ot residual tangential stress ar, a radial stress, a/stress ratio R, Ri radius, inner (J0.2 yield stress... 

Up wind Total stress = Axial stress - Radial stress... 

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