Stress biaxial

During the optical coat work stress examination method the upper plate of the head of some of the bolts was covered with an optical coat work (Fig. 4). On the head of some other bolts strain gauges were stuck which measured the plain biaxial stress state in the middle of the top surface of the head of the bolt (3.5 x 3 mm). The magnetic probe detected average stresses up to 0.1 mm depth in an area of 14 mm diameter in the middle of the head of the bolt. 

Using von Mises Theory from equation 4.58, the probabilistic requirement, P, to avoid yield in a ductile material, but under a biaxial stress system, is used to determine the reliability, R, as ... 

The reason for the activity of the above named classes of liquids is not fully understood but it has been noted that the most active liquids are those which reduce the molecular cohesion to the greatest extent. It is also noticed that the effect is far more serious where biaxial stresses are involved (a condition which invariably causes a greater tendency to brittleness). Such stresses may be frozen in as a result of molecular orientation during processing or may be due to distortion during use. 

Note that the ratio of the ratio of the hoop stress (pR/h) to the axial stress (pR/lh) is only 2. From the data in this question the hoop stress will be 8.12 MN/m. A plastic cylinder or pipe is an interesting situation in that it is an example of creep under biaxial stresses. The material is being stretched in the hoop direction by a stress of 8.12 MN/m but the strain in this direction is restricted by the perpendicular axial stress of 0.5(8.12) MN/m. Reference to any solid mechanics text will show that this situation is normally dealt with by calculating an equivalent stress, Og. For a cylinder under pressure Og is given by 0.5hoop stress. This would permit the above question to be solved using the method outlined earlier. 

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... 

If the material is subjected to biaxial stresses in both the x and y directions then the strains will be... 

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. 

For each of the failure criteria, we will generate biaxial stresses by off-axis loading of a unidirectionally reinforced lamina. That is, the uniaxial off-axis stress at 0 to the fibers is transformed into biaxial stresses in the principal material coordinates as shown in Figure 2-35. From the stress-transformation equations in Figure 2-35, a uniaxial loading obviously cannot produce a state of mixed tension and compression in principal material coordinates. Thus, some other loading state must be applied to test any failure criterion against a condition of mixed tension and compression. 

Figure 2-35 Biaxial Stresses from Off-Axis Uniaxial Loading...

In applications of the maximum stress criterion, the stresses in the body under consideration must be transformed to stresses in the principal material coordinates. For example, Tsai [2-21] considered a unidirec-tionally reinforced composite lamina subjected to uniaxial load at angle 6 to the fibers as shown in Figure 2-35. The biaxial stresses in the principal material coordinates are obtained by transformation of the uniaxial stress, a, as... 

Results for this criterion are plotted in Figure 2-40 along with the experimental data for E-glass-epoxy. The agreement between the Tsai-Hill failure criterion and experiment is quite good. Thus, a suitable failure criterion has apparently been found for E-glass-epoxy laminae at various orientations in biaxial stress fields. 

Figure 2. Thermal strain vs temperature curves for VsSi measured along [001] on heating (4.2-60K) and cooling (4.2-1.5K). Curve (a) is for an uniaxial stress (s 0.03o doo)) along [001] (b) and (c) are for biaxial stress applied along [100] and [010] with 0.5o (ioo> and o (ioo>, respectively. The x-ray data of Batterman and Barrett (reference 15) are also plotted for comparison. The insets show the directions of applied stresses and [in case of the curve (a)] the martensite-phase domains. (From reference 5)...

CA 63,17781 (1965) Proplnt failure characteristics were measured in uniaxial and biaxial stress states for poly butadiene acrylic acid and Nitroplastisol proplnts, and failure conditions were examined over a wide range of temps. The observed failure conditions were compared for various failure criteria, and it was found that a... 

Moreover, in developing and testing the theory, biaxial stress-relaxation experiments were carried out. That is, square sheets were stretched in both directions but in unequal amounts. In all cases, the stress in the major stretch direction relaxed at the same relative rate as that in the minor... 

The maximum permitted value for the combined biaxial stress is kST, where S is the specified minimum yield strength per para. PL-3.7.1(a), T is the temperature derating factor per Table PL-3.7.1(g), and k is defined in (b) and (c) below. 

The [10°] off axis tension specimen shown in Fig 3.23 is another simple specimen similar in geometry to that of the [ 45 ]s tensile test. This test uses a unidirectional laminate with fibers oriented at 10° to the loading direction and the biaxial stress state (i.e. longitudinal, transverse and in-plane shear stresses on the 10° plane) occurs when it is subjected to a uniaxial tension. When this specimen fails under tension, the in-plane shear stress, which is almost uniform through the thickness, is near its critical value and gives the shear strength of the unidirectional fiber composites based on a procedure (Chamis and Sinclair, 1977) similar to the [ 45°]s tensile test. 

A simple bubble machine is devised and successfully applied in characterising lightly crosslinked PE resins for foam expansion. The biaxial stress-strain relationship is deduced from the air injection rate and pressure. The effects of strain rate, temperatnre and crosslinker level on the stress-strain behavionr are investigated. Uniaxial extension experiments are also performed and compared with biaxial extension data. 5 refs. 

Sharma (90) has examined the fracture behavior of aluminum-filled elastomers using the biaxial hollow cylinder test mentioned earlier (Figure 26). Biaxial tension and tension-compression tests showed considerable stress-induced anisotropy, and comparison of fracture data with various failure theories showed no generally applicable criterion at the strain rates and stress ratios studied. Sharma and Lim (91) conducted fracture studies of an unfilled binder material for five uniaxial and biaxial stress fields at four values of stress rate. Fracture behavior was characterized by a failure envelope obtained by plotting the octahedral shear stress against octahedral shear strain at fracture. This material exhibited neo-Hookean behavior in uniaxial tension, but it is highly unlikely that such behavior would carry over into filled systems. 

There are currently no ISO standard methods for biaxial extension and such measurements are rarely made in industrial laboratories. However, biaxial stressing is of value in the consideration of the theory of elasticity and is preferred by many for producing data for input to finite element programmes, as well as being involved in certain practical applications of rubber. The British standard for finite element analysis on rubber19 outlines the two approaches, equibiaxial stretching of a flat sheet and inflation of a flat sheet. The principles of these are illustrated in Figure 8.14. 

Analyze lubricated squeezing flow to determine biaxial extensional viscosity (T)R), which is calculated from biaxial stress (cB) and biaxial extensional strain rate (eB). 

For the c-BN formation a stress threshold was observed in the deposited layers. The h-BN intermediate layer shows a preferred orientation, where the c-axis of the h-BN is parallel to the substrate. Both effects are explained by the compressive biaxial stress induced by the ion bombardment. The mechanism for the conversion of h-BN into c-BN is explained by rather high temperatures originated during thermal spikes (direct h-BN —> c-BN transformation). The stress caused by the bombardment with high energetic ions is considered to be a reason for the growth of the c-BN crystals [190, 191]. A stress within the layer of up to 10 GPa has been observed. This biaxial stress causes a hydrostatic pressure up to the values usual in HP-HT synthesis. 

Equation (15.10) shows that thermal shock induces a biaxial stress field, whose maximum value depends on the elastic properties of the material and the imposed temperature differential. 

In this section, an analytical solution to calculate residual stresses in an FGM disk is discussed, based on simple linear elastic plate theories of classical mechanics, and used for the calculation of residual stresses in a plane stress state. An equi-biaxial stress analysis differs from a plane stress state by simply replacing the Young s modulus A by the corresponding biaxial modulus E = E/( 1 - v). In this way, the residual thermal stress can be calculated in the centre of the FGM disk, far enough away from the free edges where a complex stress state is present. 

A variation of the E2 mode with the thickness of the AIN buffer layer in MOCVD 750 nm GaN/sapphire structures monitoring biaxial stress has been observed (TABLE 6) [31]. A correlation of E2 phonon modes with exciton energies in photoluminescence has been given [30,31]. 

It should be pointed out that the material parameter Ge can be determined in principle more precisely by means of equi-biaxial measurements than by uniaxial measurements. This is due to the fact that the first addend of the Ge-term in Eq. (45) increases linearly with X. This behavior results from the high lateral contraction on the equi-biaxial extension X2=X 2). It postulates a close dependency of the equi-biaxial stress on the tube constraint modu-... 

Figure 46a shows that a reasonable adaptation of the stress contribution of the clusters can be obtained, if the distribution function Eq. (55) with mean cluster size =25 and distribution width b= 0.8 is used. Obviously, the form of the distribution function is roughly the same as the one in Fig. 45b. The simulation curve of the first uniaxial stretching cycle now fits much better to the experimental data than in Fig. 45c. Furthermore, a fair simulation is also obtained for the equi-biaxial measurement data, implying that the plausibility criterion , discussed at the end of Sect. 5.2.2, is also fulfilled for the present model of filler reinforced rubbers. Note that for the simulation of the equi-biaxial stress-strain curve, Eq. (47) is used together with Eqs. (45) and (38). The strain amplification factor X(E) is evaluated by referring to Eqs. (53) and (54) with i= 2= and 3=(l+ ) 2-l.

Ager, J. W. and Drory, M. D. (1993), Quantitative measurement of residual biaxial stress by Raman-spectroscopy in diamond grown on a Ti alloy by chemical-vapor-deposition. Phys. Rev. B, 48(4) 2601-2607. 

Kawagoe M, Kitagawa M (1981) Craze initiation in poly(methyl methacrylate) under biaxial stress. J Polym Sci Polym Phys 19(9) 1423—1433... 

In this section the maximum stress concentration developed by an elliptic hole at arbitrary orientation in an infinite elastic sheet subjected to general biaxial stress loading is considered. In Figure 1, consider an elliptic hole of major axis a, minor axis b, and orientation angle fl with respect to an axis of applied uniaxial tension Si. If the sheet is isotropic, elastic, and infinite, then the major principal stress ([Pg.42]

Figure I. An elliptic flaw in an infinite, isotropic, and elastic medium subjected to uniform biaxial stresses St and SThe major axis of the ellipse is a, minor axis b, ellipticity R = b/a, and orientation angle with respect to St is ft.

A more convenient means of representing general biaxial stress states is obtained by introducing a variable defined as ... 

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