Principal stress, with

Linthilhac PM, Vesecky TB (1980) Mechanical stress and cell wall orientation. I. Photoelastic derivation of principal stresses. With a discussion of axillarity and the significance of the arcuate shell zone . Am J Bot 67 1477-1483 Loeb J (1918) Chemical basis of correlation. Bot Gaz 65 150-174 Lorenzi R, Horgan R, Wareing PF (1975) Cytokinins in Picea sitchensis. Biochem Physiol Pflanz 168 333-339... 

The yield surface for the von Mises yield criterion (occasionally called distor-tional strain energy criterion) is cylindrical in the space of principal stresses, with its centre coinciding with the hydrostatic axis = [Pg.90]

Fig. 8. Maximum principal stress with and without residual thermal stress.

The sample is constrained in lateral x - and y-direction by four steel plates. Vertical deformations of the sample are restricted by rigid top and bottom plates. The sample can be loaded by the four lateral plates, which are linked by guides so that the horizontal cross-section of the sample may lake different rectangular shapes. In deforming the sample, the stresses Ox and Oy can be applied independently of each other in x- and y-direction. To avoid friction between the plates and the sample the plates are covered with a thin rubber membrane. Silicone grease is applied between the steel-plates and the rubber membrane. Since there are no shear stresses on the boundary surfaces of the sample Ox and Oy are principal stresses. With the true biaxial shear tester the measurement of both stresses and strains is possible. 

The calculation was carried out using the ANSYS F.E.M. code. The pressure vessel was meshed with a 4 nodes shell element. Fig. 18 shows a view of the results of calculation of the sum of principal stresses on the vessel surface represented on the undeformed shape. For the calculation it was assumed an internal pressure equal to 5 bar and the same mechanical characteristics for the test material. 

Both Watts and sulfamate baths are used for engineering appHcation. The principal difference in the deposits is in the much lower internal stress obtained, without additives, from the sulfamate solution. Tensile stress can be reduced through zero to a high compressive stress with the addition of proprietary sulfur-bearing organic chemicals which may also contain saccharin or the sodium salt of naphthalene-1,3,6-trisulfonic acid. These materials can be very effective in small amounts, and difficult to remove if overadded, eg, about 100 mg/L of saccharin reduced stress of a Watts bath from 240 MPa (34,800 psi) tensile to about 10 MPa (1450 psi) compressive. Internal stress value vary with many factors (22,71) and numbers should only be compared when derived under the same conditions. 

Since the yield function is independent of p, the yield surface reduces to a cylinder in principal stress space with axis normal to the 11 plane. If the work assumption is made, then the normality condition (5.80) implies that the plastic strain rate is normal to the yield surface and parallel to the II plane, and must therefore be a deviator k = k , k = 0. It follows that the plastic strain is incompressible and the volume change is entirely elastic. Assuming that the plastic strain is initially zero, the spherical part of the stress relation (5.85) becomes... 

With experimental and theoretical capabilities presently in hand, materials may be studied at peak shock stresses or pressures from perhaps 100 MPa to several TPa. This work is principally concerned with pressures from this... 

That the principal stresses are not of interest in determining the strength of an orthotropic lamina is illustrated with the following example. Consider the lamina with unidirectionai fibers shown in Figure 2-16. Say that the hypotheticai strengths of the lamina in the 1-2 piane are... 

In a recent study, Saintier et al. ° investigated the multiaxial effects on fatigue crack nucleation and growth in natural mbber. They found that the same mechanisms of decohesion and cavitation of inclusions that cause crack nucleation and crack growth in uniaxial experiments were responsible for the crack behavior in multiaxial experiments. They studied crack orientations for nonproportional multiaxial fatigue loadings and found them to be related to the direction of the maximum first principal stress of a cycle when material plane rotations are taken into account. This method accounts for material rotations in the analysis due to the displacement of planes associated with large strain conditions. 

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. 

Multiaxial test geometries and test conditions may be analyzed with reference to three orthogonal principal stresses as shown in Figure 15. 

As shown in Fig. 1, a cubic body of material under consideration is deformed in the directions of orthogonal axes Xt. If this mode of deformation, the coordinate axes coincide with the principal strain axes. In the principal stresses af corresponding to the principal strains are measured as functions of stretch ratios X, in the directions of Xh W can be calculated from... 

In principle, W can be determined from Eq. (2) if principal stresses at are measured as functions of applied principal stretch ratios X,-. However, since bW/bJ/ rather than W itself are more directly connected with the stress-strain relations [see Eq. (11)], their determination from the measurements of at and X,- is more feasible than that of W. 

Consider the same unidirectional lamina with the stresses now applied perpendicular to the fiber axis as shown in Fig. 12. The local stress at the fiber matrix interface can be calculated and compared to the nominally applied stress on the whole lamina to give K, the stress concentration factor. The plot of the results of this analysis shows that the interfacial stresses at the point of maximum principal stress can range up to 2.6 times the applied stress depending on the moduli of the constituents and the volume fraction of the reinforcement. For a typical graphite-epoxy composite, with a modulus ratio of 70 and a volume fraction of 70 % the stress concentration factor at the interface is about 2.4. That is, the local stresses at the interface are a factor of 2.4 times greater than the applied stress. 

The principal strain rates are eigenvalues of the strain-rate tensor (matrix). As described more fully in Section A.21, the direction cosines that describe the orientation of the principal strain rates are the eigenvectors associated with the eigenvalues. In solving practical fluids problems, there is rarely a need to determine the principal strain rates or their orientations. Rather, these notions are used theoretically with the Stokes postulates to form general relationships between the strain-rate and stress tensors. It is perhaps worth noting that in solid mechanics, the principal stresses and strains have practical utility in understanding the behavior of materials and structures. 

The stress vector on the z plane has three components that can be determined from the projections of the principal stresses. These components, written to align with the principal axes, are... 

If the principal stresses had had shear components, which by definition they don t, then, in general, those shear components would have contributed to the stress vector on the rotated z plane. The a vector completely defines the stress state on the rotated z face. However, our objective is to determine the stress-state vector on the z face that aligns with the rotated coordinate system (z,r,G) x--, x-r, and x-e. The a vector itself has no particular value in its own right. Therefore one more transformation from cs to r is required ... 

With the normal stress in hand, turn now to the task of eliminating the direction cosines from the shear-stress expressions. Beginning with Eq. 2.163 and substituting the expressions for the principal stresses yields... 

Referring to Fig. A.2, assume that the principal coordinates align with z, r, and O. The unit vectors (direction cosines) just determined correspond with the row of the transformation matrix N. Thus, if the principal stress tensor is... 

In any layer-like arrangement of stiff crystalline and soft amorphous regions a strong influence of lamellar orientation with respect to the principal stress direction is observed. The most compliant arrangement is shown in... 

Fig. 6 Micromechanical model of a section of a semi-crystalline polymer with lamellae oriented perpendicular to the principal stress direction showing the long period L and the thicknesses of crystalline (Lc) and amorphous layers (La) the latter are composed of loose segments, entangled chains and more or less extended tie molecules. Large forces can be transferred at those points (o) where highly extended tie molecules (eTM) enter crystalline lamellae...

Fig. 10.41 The calculated distribution of the principal stress in the Lao.8Sro.2Cro.95Nio.o5C>3 3 interconnector for the standard counter-flow case 1 in Table 10.2 (a) the stress is calculated considering both the temperature and distributions in the interconnector, (b) the stress is only the thermal stress. The model geometry with 16-channels is used, and half the model is drawn in the figure.

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