Combined stress states

For the analysis of combined stress in the two-dimensional situation the Mohr circle diagram (see Appendix A1.8) is of value. Normal stresses are represented along the x axis and shear stresses along the y axis, so the Mohr circle thus 

For the analysis of combined stress in the two-dimensional situation the Mohr circle diagram (see standard texts [13]) is of value. Normal stresses are represented along the 1 axis and shear stresses along the 2 axis, so that the Mohr circle thus represents a state of stress, with each point representing the stresses on a particular plane. The direction of the plane normal is given relative to the directions of the principal stresses by the rule that a rotation in real space of 0 in a clockwise direction, corresponds to a rotation in Mohr circle space of 20 

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The second special case is an orthotropic lamina loaded at angle a to the fiber direction. Such a situation is effectively an anisotropic lamina under load. Stress concentration factors for boron-epoxy were obtained by Greszczuk [6-11] in Figure 6-7. There, the circumferential stress around the edge of the circular hole is plotted versus angular position around the hole. The circumferential stress is normalized by a , the applied stress. The results for a = 0° are, of course, identical to those in Figure 6-6. As a approaches 90°, the peak stress concentration factor decreases and shifts location around the hole. However, as shown, the combined stress state at failure, upon application of a failure criterion, always occurs near 0 = 90°. Thus, the analysis of failure due to stress concentrations around holes in a lamina is quite involved. 

With increase of the roadway depth, the left side and the right side of the roadway are subjected to tensile, shear, or combined stress state. Thus, tension crack, shear crack or mixed mode crack are existed in these locations. 

Although relatively few studies have been made of the fracture of elastomers under complex stress conditions, some general conclusions can be drawn regarding fracture under specific combined-stress states, as follows. 

In a continuum, combined stress states are usual, and there are six components in each stress and strain. Strictly speaking, strength criteria should be functions of all the components. The simplest criterion for combined stress conditions are the maximum stress function shown in the following equation. 

SCC has been defined as failure by cracking under the combined action of corrosion and stress (Fig. 9.1). The stress and corrosion components interact S3mergistically to produce cracks, which initiate on the surface exposed to the corrodent and propagate in response to the stress state. They may run in any direction but are always perpendicular to the principal stress. Longitudinal or transverse crack orientations in tubes are common (Figs. 9.2 and 9.3). Occasionally, both longitudinal and transverse cracks are present on the same tube (Fig. 9.4). Less frequently, SCC is a secondary result of another primary corrosion mode. In such cases, the cracking, rather than the primary corrosion, may be the actual cause of failure (Fig. 9.5). 

Now that the basic stiffnesses and strengths have been defined for the principal material coordinates, we can proceed to determine how an orthotropic lamina behaves under biaxial stress states in Section 2.9. There, we must combine the information in principal material coordinates in order to define the stiffness and strength of a lamina at arbitrary orientations under arbitrary biaxial stress states. 

Structural analysis of the solid rocket case-grain system using experimentally determined propellant response properties may permit a complete description of the combined stresses and resultant deformations, but a statement expressing the ability of the propellant to withstand these stresses is also required. Such a statement, which relates the physical state at which failure occurs to some material parameters, is called a failure criterion. The criterion for failure permits a prediction of safety margins expected under motor operation and handling and defines the loading regimes where abnormal operations will occur with intolerable frequency. 

A combination of an energy criterion and the failure envelope has been proposed by Darwell, Parker, and Leeming (22) for various doublebase propellants. Total work to failure was taken from the area beneath the stress-strain curve, but the biaxial failure envelope deviated from uniaxial behavior depending on the particular propellant formulation. Jones and Knauss (46) have similarly shown the dependence of failure properties on the stress state of composite rubber-based propellants. 

This operation can be seen through combinations of the foregoing operations. Assume that a given (stress) state is represented by T in one coordinate system and T in a rotated... 

This brought a bout a keen interest in other methods of intensification in processing. Lately, the directed effect of physical (mechanical) fields on molten polymers has become one such area. These effects, as demonstrated in many works published in the 1970s and in the 1980s, (see for examples [6-9]) result in altered parameters of micro- and macrostress of the system. Molding under conditions of directed physical fields, in particular, in the case of mechanical and acoustic vibration effects upon melts, is performed so that an additional stress superimposed on the polymer s main shear flow and the state of material is characterized by combined stress. 

As discussed in section 7.1.6.4, semidilute solutions of rodlike polymers can be expected to follow the stress-optical rule as long as the concentration is sufficiently below the onset of the isotropic to nematic transition. Certainly, once such a system becomes nematic and anisotropic, the stress-optical rule cannot be expected to apply. This problem was studied in detail using an instrument capable of combined stress and birefringence measurements by Mead and Larson [109] on solutions of poly(y benzyl L-glutamate) in m-cresol. A pretransitional increase in the stress-optical coefficient was observed as the concentration approached the transition to a nematic state, in agreement of calculations based on the Doi model of polymer liquid crystals [63]. In addition to a dependence on concentration, the stress-optical coefficient was also seen to be dependent on shear rate, and on time for transient shear flows. 

In the model derived by McClintock and Irwin the shape and size of the plastic zone were calculated by a combination of the stress field at the crack tip (e.g. Eq. (2)) with a yield criterion (e.g. von Mises, Tresca). This leads to the well known dog-bone type of plastic zone showing the influence of stress state. Its form is often approximated to by a circle of radius Tp, where... 

There now exists some evidence for a tendency for fault sealing capacity to relate to the azimuth of fault strike in part of the North Sea. It has previously been suggested that this relates to the azimuth of the maximum and minimum horizontal stresses. However, this relationship can be inferred to relate to some yet unspecified combination of the present day stress state and the overall geological history, including previous stress states. We suspect that minor shear displacements can potentially greatly increase the stress-sensitivity of seals composed of mudrock. 

In actuallity, both factors occur together and it is difficult to separate them. As shown later, this separation is possible by calculations based on the theory introduced here. However, it is not so simple and adds an extra factor to the analysis of the failure mechanism. In order to avoid confusion caused by the combined factors, it is advantageous to cariy out the failure experiment under a simple stress state such as constant stress and so on. 

In uniaxial extension, for example, the cross-sectional area of the body becomes thiner and thiner with increasing creep strain and then the body is broken like the break down of a thread. This class is also a kind of the combined type of. the first and the second factors. Another class of creep failure, which will be considered in this article is that in which the creep strain is not so large and the stress state is kept nearly constant and failure occurs suddenly at some time. The creep process of vulcanized rubbers is one of the processes closest to this second class. That is, both the strain rate and the creep strain are not so high in this creep process except at the moment just after... 

Bulk-micromachined membranes are usually formed from dielectric materials like silicon oxide or silicon nitride combined with additional materials for example, in pressure sensors, silicon is used to increase the membrane thickness to the required values and in thermal sensors, platinum or other metals are needed for the sensing elements. The overall stress state of the membranes has to be controlled well to prevent buckhng (under high compressive stress) or fracture (under high tensile stress). With proper processing control, silicon oxide and silicon nitride thin films meet this requirement, making them ideal candidates for membrane-type devices. 

The best way to determine the stresses in the scale would be a direct measurement. However, X-ray methods have usually a limited spatial resolution which makes it difficult to measure nonuniform stress fields. The application of OFS for scales consisting of a-Al203 provides a sufficient spatial resolution and permits, in principle, to examine stress variations in the scale. However, only the trace of the stress tensor can be measured for an untextured polycrystalline scale. Thus, anisotropic stress states have to be analysed in combination with a mechanical modelling of the scale loading in order to deduce the stress components from the trace of the stress tensor. 

Thermomechanlcal Behavior. Requirements for optical performance Impose unprecedented requirements for dimensional stability of polymers used In hlgh-concentratlon reflectors. Requirements for mechanical compatibility are also strict for photovoltaic systems subjected to moisture and thermal stresses. Moisture, temperature, and UV, separately and In combination, can change the volume and thus the stress state of polymers. For example, temperature and humidity cycles alone do not cause surface micro-cracks In polycarbonate. However, In the presence of UV radiation, such cycles cause microcracks, while UV alone does not [32]. An understanding of these relationships Is essential to permit reliable design of equipment that uses polymers. 

As stated previously, stresses are meaningless until compared to some stress/failure theory. The significance of a given stress must be related to its location in the vessel and its bearing on the ultimate failure of that vessel. Historically, various tlieories have been derived to combine and measure stres.ses against the potential failure mode. A number of stress theories, also called yield criteria, are available for describing the effects of combined stresses. For purposes of this book, as these failure theories apply to pressure vessels, only two theories will be discussed. 

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