Uniaxiality condition

The largest possible value of under uniaxial conditions is 150 ksi (1035 MPa). Thus, the outer layer can be stressed about an additional 149.4 ksi (1030.7 MPa). The corresponding change in the force resultant is obtained from Equation (4.135) as... 

The design of thickwalled components for pulsating pressure is based on the stress calculation with analytical or numerical methods and the determination of the maximum equivalent stress in relation to the admissible stresses at uniaxial conditions. The latter have to be extracted from Woehler-tests with specimen. If the stresses yield too large at load conditions including a safety factor, the design must be optimized by avoiding the major stress concentrations at bore intersections. The avoidance of T-intersections is reducing stresses by factor 2-3 (Fig. Id). 

Strictly speaking, there are no static viscoelastic properties as viscoelastic properties are always time-dependent. However, creep and stress relaxation experiments can be considered quasi-static experiments from which the creep compliance and the modulus can be obtained (4). Such tests are commonly applied in uniaxial conditions for simphcity. The usual time range of quasi-static transient measurements is limited to times not less than 10 s. The reasons for this is that in actual experiments it takes a short period of time to apply the force or the deformation to the sample, and a transitory dynamic response overlaps the idealized creep or relaxation experiment. There is no limitation on the maximum time, but usually it is restricted to a maximum of 10" s. In fact, this range of times is complementary, in the corresponding frequency scale, to that of dynamic experiments. Accordingly, to compare these two complementary techniques, procedures of interconversion of data (time frequency or its inverse) are needed. Some of these procedures are discussed in Chapters 6 and 9. 

Given that a shear band has formed in isotropic material under uniaxial conditions, a simple analysis is available to predict the angle at which it occurs with respect to the... 

Most of this information is already contained by the uniaxiality condition. The only additional effects of the nonpolar condition are that Kj and Kgg =... 

Now we are ready to turn to the main topic of this chapter, which is to address how the various parameters are to be determined from experimental results. To do so, it is convenient to give a uniaxial simplification of the 3D constitutive framework under the assumption of incompressibility, since the basic experiments needed for characterization of the model are performed under uniaxial conditions. The yield stress during a uniaxial test under constant strain rate can be given by... 

In this appendix the 3D constitutive framework is simplified to a simple scalar equation to describe uniaxial conditions. 

A sliding plate rheometer (simple shear) can be used to study the response of polymeric Hquids to extension-like deformations involving larger strains and strain rates than can be employed in most uniaxial extensional measurements (56,200—204). The technique requires knowledge of both shear stress and the first normal stress difference, N- (7), but has considerable potential for characteri2ing extensional behavior under conditions closely related to those in industrial processes. 

Prompt instrumentation is usually intended to measure quantities while uniaxial strain conditions still prevail, i.e., before the arrival of any lateral edge effects. The quantities of interest are nearly always the shock velocity or stress wave velocity, the material (particle) velocity behind the shock or throughout the wave, and the pressure behind the shock or throughout the wave. Knowledge of any two of these quantities allows one to calculate the pressure-volume-energy path followed by the specimen material during the experimental event, i.e., it provides basic information about the material s equation of state (EOS). Time-resolved temperature measurements can further define the equation-of-state characteristics. 

Samples are most frequently shock deformed under laboratory conditions utilizing either explosive or gun-launched flyer (driver) plates. Given sufficient lateral extent and assembly thickness, a sample may be shocked in a onedimensional strain manner such that the sample experiences concurrently uniaxial-strain loading and unloading. Based on the reproducibility of projectile launch velocity and impact planarity, convenience of use, and ability to perform controlled oblique impact (such as for pressure-shear studies) guns have become the method of choice for many material equation-of-state and shock-recovery studies [21], [22]. 

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

The contribution to the stress from electromechanical coupling is readily estimated from the constitutive relation [Eq. (4.2)]. Under conditions of uniaxial strain and field, and for an open circuit, we find that the elastic stiffness is increased by the multiplying factor (1 -i- K ) where the square of the electromechanical coupling factor for uniaxial strain, is a measure of the stiffening effect of the electric field. Values of for various materials are for x-cut quartz, 0.0008, for z-cut lithium niobate, 0.055 for y-cut lithium niobate, 0.074 for barium titanate ceramic, 0.5 and for PZT-5H ceramic, 0.75. These examples show that electromechanical coupling effects can be expected to vary from barely detectable to quite substantial. 

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. 

The reduction in the tensile load capacity of the drill pipe is 311,400 -260,500 = 50,900 lb. That is about 17% of the tensile drill pipe resistance calculated at the minimum yield strength in uniaxial state of stress. For practical purposes, depending upon drilling conditions, a reasonable value of safety factor should be applied. 

Interpretation of data obtained under the conditions of uniaxial extension of filled polymers presents a severe methodical problem. Calculation of viscosity of viscoelastic media during extension in general is related to certain problems caused by the necessity to separate the total deformation into elastic and plastic components [1]. The difficulties increase upon a transition to filled polymers which have a yield stress. The problem on the role and value of a yield stress, measured at uniaxial extension, was discussed above. Here we briefly regard the data concerning longitudinal viscosity. 

Even plastics with fairly linear stress-strain curves to failure, for example short-fiber reinforced TSs (RPs), usually display moduli of rupture values that are higher than the tensile strength obtained in uniaxial tests wood behaves much the same. Qualitatively, this can be explained from statistically considering flaws and fractures and the fracture energy available in flexural samples under a constant rate of deflection as compared to tensile samples under the same load conditions. These differences become less as the... 

Fairly recently, another method for obtaining polymer materials with uniaxial orientation has been developed. It is the directed polymerization i.e. the synthesis of polymers under conditions at which the material attains instanteneously the oriented structure. The formation of crystals from the macromolecules in an extended conformation occurs in those polymerizing systems simultaneously with polymerization22. ... 

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... 

It was therefore with some confidence in the infra-red and Raman spectroscopic methods that a much more complex investigation was carried out on the molecular orientation in one-way drawn PET films which show uniplanar axial orientation 5). In such films the condition of fibre symmetry is removed in two ways (1). There is no longer uniaxial symmetry of the distribution of chain axes. 

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. 

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

The simplest case for modeling particle dissolution is to assume that the particles are monodisperse. Under these conditions, only one initial radius is required in the derivation of the model. Further simplification is possible if the assumption is made that mass transport from a sphere can be approximated by a flat surface or a slab, as was the case for the derivation for the Hixson-Crowell cube root law [70], Using the Nernstian expression for uniaxial flux from a slab (ignoring radial geometry or mass balance), one can derive the expression... 

R > is the mean square end-to-end distance of the polymer chain. Consider a sample stretched by a factor Ay in the Y direction and Az in the Z direction. A general deformation can be easily treated, but for simplicity only uniaxial stretching or isotropic swelling will be examined. Note that under conditions of uniaxial deformation, the volume of the sample changes very little. This change may be ignored, and it becomes convenient to set Az = A and Ax — Ay. ... 

The networks studied were prepared from reactions carried out at different initial dilutions. Aliquots of reaction mixtures were transferred to moulds, which were maintained at the reaction temperature under anhydrous conditions, and were allowed to proceed to complete reaction(32). Sol fractions were removed and shear moduli were determined in the dry and equilibrium-swollen states at known temperatures using uniaxial compression or a torsion pendulum at 1Hz. The procedures used have been described in detail elsewhere(26,32). The shear moduli(G) obtained were interpreted according to Gaussian theory(33 34 35) to give values of Mc, the effective molar mass between junction points, consistent with the affine behaviour expected at the small strains used(34,35). 

Although nearly all creep and stress-relaxation tests are made in uniaxial tension, it is possible to make biaxial tests in which two stresses are applied at 90° to one another, as discussed in Section VI. In a uniaxial test there is a contraction in the transverse direction, but in a biaxial test the transverse contraction is reduced or even prevented. As a result, biaxial creep is less than uniaxial creep--in cquihiaxial loading it is roughly hall as much for equivalent loading conditions. In the linear region the biaxial strain 2 in each direction is (255.256)... 

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