Triaxial states of stress

The contact fatigue creates independent part of the fatigue tests. As consequence of triaxial state of stress and flexible plastic state in contact area occurrence comes to very considerable scattering of experimental data. From this reason it is necessary to test statistic meaningful number of samples. 

Correlation of results from one test to another for a given material becomes difficult because of different stress states of the specimen and the associated strain rates in different tests. In the tensile-impact test, the stress state is uniaxial and it measures the tensile property at a high strain rate. In Izod and Charpy tests, the presence of notch gives a triaxial state of stress. The falling-... 

Both AS ME Code, Section 1I1, Division 2 and AS ME Code, Section III, utilize the maximum shear stress criterion. This theor) closely approximates experimental results and is also easy to use. This theory also applies to triaxial states of stress. In a triaxial stress state, this theory predicts that yielding will occur whenever one-half the algebraic difference betu een the maximum and minimum stress is equal to one-half the yield stress. Where 0 > a i> 0.3. the maximum shear stress is (oi —03)72. 

Mourad A-HI, et al. Ultra high molecular weight polyethylene deformation and fracture behaviour as a function of high strain rate and triaxial state of stress. Int J Fract 2003 120(3) 501-15. 

The state of stress in a cylinder subjected to an internal pressure has been shown to be equivalent to a simple shear stress, T, which varies across the wall thickness in accordance with equation 5 together with a superimposed uniform (triaxial) tensile stress (6). 

In order to describe completely the state of triaxial (as opposed to biaxial) stress in an anisotropic material, the compliance matrix will have 36 terms. The reader is referred to the more advanced composites texts listed in the Bibliography if these more complex states of stress are of interest. It is conventional to be consistent and use the terminology of the more general analysis even when one is considering the simpler plane stress situation. Hence, the compliance matrix [5] has the terms... 

In a recent attempt to bring an engineering approach to multiaxial failure in solid propellants, Siron and Duerr (92) tested two composite double-base formulations under nine distinct states of stress. The tests included triaxial poker chip, biaxial strip, uniaxial extension, shear, diametral compression, uniaxial compression, and pressurized uniaxial extension at several temperatures and strain rates. The data were reduced in terms of an empirically defined constraint parameter which ranged from —1.0 (hydrostatic compression) to +1.0 (hydrostatic tension). The parameter () is defined in terms of principal stresses and indicates the tensile or compressive nature of the stress field at any point in a structure —i.e.,... 

The stresses near the root of a notch are extremely complex and the stress analysis becomes exceedingly difficult when the strain is large, as is the case when yield or failure is imminent. A sharp notch causes constraints and introduces a state of triaxial tension behind the root of the notch (5). This state of stress is consistent with LeGrand s observation of the growth of a flaw behind a notch in a bar of polycarbonate (4). A blunt notch causes constraints when the thickness of the specimen is large. Such a notch can also introduce a state of triaxial tension. While it is desirable to investigate the behavior of polymers in a well-defined state of triaxial tension, it is difficult to accomplish experimentally. However, as we demonstate below, a state of plane strain is relatively easy to produce. The relationship between plane strain and brittleness of plastics is the subject of our investigation. 

A flexible boundary cubical triaxial test is another commonly used test for compression studies (Kamath et al., 1993 Li and Puri, 1996). A picture of a triaxial compression tester is shown in Figure 8. It allows not only the application of the three principal stresses independently, but also constant monitoring of the volumetric deformation and deformations in three principal directions. In a triaxial compression test, the specimen is at an initial isotropic state of stress then the three pressure lines apply the same pressure at the same rate to all six faces thus pressure is the same in all three directions (i.e., cti = 02 = 03). 

The body is uniformly cooled with the external surfaces rigidly constrained to give a well-defined triaxial tensile state of stress given by ... 

All materials tend to fracture if stressed severely enough. Some materials fracture more easily than others, and are thereby said to be brittle . Brittleness is the property of a material manifested by fracture without appreciable prior plastic deformation. In ductile fracture significant plastic flow occurs before fracture. Strain at fracture is more than a few per cent, unlike brittle fracture, and may be several hundred per cent. However, a sharp distinction cannot be made between brittle and ductile fracture since even in glassy materials some deformations take place. Further, a given material will fail in a brittle manner under some conditions and a ductile manner under other conditions. Thus, brittle fracture is favored by the low temperature, fast loading and when the state of stress approaches a uniform, i.e., triaxial or dUatational, state. Materials with low T are more... 

In any pressure vessel subjected to internal or external pressure, stresses are set up in the shell wall. The state of stress is triaxial and the three principal stresses are ... 

Division 2 stress analysis considers all stresses in a triaxial state combined in acc-ordance with the maximum shear stress theory. Division 1 and the procedures outlined in this bcx)k c-onsider a biaxial state of stress combined in accordance with the maximum stress theory. Just as you would not design a nuclear reactor to the rules of Division 1, you would not design an air receiver by the techniques of Division 2. Each has its place and applications. The following discussion on categories of stress and allowables will utilize information from Dicision 2, which can be applied in general to all vessels. 

In order to study retrofitting of beam-column coimections failing prematurely due to the collapse of joint, the ITU specimens with very low compressive strength concrete were modeled in the DIANA environment. The failure in this specimen occurred at load levels that corresponded closely to both the flexural strength of the beam and the shear capacity of the joint. This behavior was captured through the use of associated plasticity theory, with, a value for 0 that the authors recommend for stress states that have high biaxial or triaxial state of compressive stresses. 

This term is normally associated with shorter deeper beams. Failure takes place when the compression zone, after its reduetion in size due to cracking, fails under combined aetion of triaxial eompression and shear stresses. This failure type can happen at principal tensile eraeks. The limitation to this shear compression is that the average compressive stress in the concrete has only been found at failure in pure flexure. As it is based on extreme concrete fibre reaching its limiting strain it is possible that shear failure ean oecur on an inclined plane near the centre of a eompression zone. In the vessel the haunch hinge is under a multiaxial state of stress due to prestressing bonded steel, liner and penetrations. This zone will in that case resist shear compression failure. 

Eibl and others have shown that the Finite Element Method can be used with success to model pressures in silos [20]. To apply this method a constitutive model has to be used. The models of Lade [21] and Kolymbas [22] may be mentioned as examples. Each constitutive model contains parameters, which have to be identified from calibration tests. The most important demand for this calibration test is that the complete state of stress and the complete state of strain can be measured in the equivalent testers. From the mentioned testers this requirement can only be fulfilled by the true biaxial shear tester and by very special triaxial cells [2]. Lade himself and also Eibl used results from triaxial tests for calibration. Feise [5,23] could show the advantages of using the true biaxial shear tester. 

The maximum or minimum value at the normal stress at a point in a plane considered with respect to all possible orientations of the considered plane. On such principal planes the shear stress is zero. There are three principal stresses on three mutually perpendicular planes. The state of stress at a point may be (1) uniaxial, a state of stress in which two of the three principal stresses are zero, (2) biaxial, a state of stress in which only one of the three principal stresses is zero, and (3) triaxial, a state of stress in which none of the principal stresses is zero. Multi-axial stress refers to either biaxial or triaxial stress. 

It is often found that, in practice, the measured value of Gic or K c varies with the width of the specimen over a certain range of widths, and this usually arises because the state of stress near the crack tip varies from plane stress in a very thin specimen to plane strain near the centre of a wide plate. The general form of the relationship between Kic (or Gic) and specimen width, b, is shown schematically in Fig. 7.6. The value for fracture in plane strain conditions is usually less than under plane stress. This arises because the tensile stress at which a material yields is greater in a triaxial stress field (plane strain) than in a biaxial one (plane stress) and thus, in the former, a more limited degree of plasticity develops at the crack tip cf. Equations 7.26 and 7.27. The lower, conservative, plane strain value is usually the one required for engineering design and life prediction studies. The width, fe, necessary to achieve this condition is usually taken to be ... 

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