Stress true, defined

The experimental mechanical techniques most commonly used for network characterization are uniaxial extension and compression,and also biaxial strain.A sketch of a rubber sample under extension is shown in Figure 10(a). The nominal stress a is defined as the ratio of the force/to the cross-sectional area Aq of the undeformed specimen, and the strain e as the ratio of the length change AL to the original length Lq. These definitions are given in equations (67) and (68). The deformation is also often expressed in terms of the extension ratio X defined in equation (69). The cross-sectional area of the specimen varies with deformation. A true stress, defined as the ratio of the force to the real deformed area, is also frequently used. 

True stress, o, is defined as o = F/A where A is the actual area of cross section of the member corresponding to the load F. 

One of the simplest criteria specific to the internal port cracking failure mode is based on the uniaxial strain capability in simple tension. Since the material properties are known to be strain rate- and temperature-dependent, tests are conducted under various conditions, and a failure strain boundary is generated. Strain at rupture is plotted against a variable such as reduced time, and any strain requirement which falls outside of the boundary will lead to rupture, and any condition inside will be considered safe. Ad hoc criteria have been proposed, such as that of Landel (55) in which the failure strain eL is defined as the ratio of the maximum true stress to the initial modulus, where the true stress is defined as the product of the extension ratio and the engineering stress —i.e., breaks down at low strain rates and higher temperatures. Milloway and Wiegand (68) suggested that motor strain should be less than half of the uniaxial tensile strain at failure at 0.74 min.-1. This criterion was based on 41 small motor tests. 

A number of materials, however, show stress-strain curves of the shape sketched in Fig. 24.10. After the normal convex first part, the stress-strain curve shows an inversion point, after which the stress increases rapidly with strain. This phenomenon is sometimes called "strain hardening". In this case, a straight line through the origin can intersect the stress-strain curve at two points A and B. This means that only the intersection points A and B are possible conditions. The intermediate intersection point C is unstable. So in this case two parts of the specimen, e.g. a fibre, with different draw ratios and, hence, different cross sections can coexist. If the fibre is stretched, part of the material with a cross section of point A is converted into material with a cross section of point B. In contrast, in Chap. 13 the Considere plot is defined as the tangent line on the true stress-strain curve. 

Yield and necking phenomena can be envisioned usefully with the Considere construction shown in Fig. 11-21. Here the initial conditions are initial gauge length and cross-sectional area /, and /(, respectively and the conditions at any instant in the tensile deformation arc length / and cross-sectional area A, when the force applied is F. The true stress, a, defined as the force divided by the corresponding... 

The relationship between the measured relative retardation (R) and the stress-induced birefringence (An) is given by R = fAn (2), in which t is the sample thickness. The stress optical coefficient C is defined by An = Ct (2), in which t is the true stress (t = f/A f is the force in newtons per square millimeter, and A is the cross-sectional area of the network sample). This coefficient is thus simply the slope of the line in a plot of An versus t. Finally, the optical configuration parameter An is defined by... 

Finally, Figure 14.5d shows a schematic of simple shear deformation. The specimen is clamped between steel blocks. The blocks must move parallel to each other in order to get a shear strain that is uniform along the waisted region. The shear stress is calculated as t = F/A, where F is the force applied to the plane of area A. In this test it is not necessary to distinguish between nominal and true stress because the shear strain does not affect A. The shear strain is defined as y = Ax/y, where Ax is the displacement of planes separated by a distance y. Ax being measured in the direction of the force applied, which is perpendicular to y. As in the case of the compression plane strain test, there is no change in the dimension of the sample along the z axis. 

From a plot of true stress-true strain behavior on logarithmic coordinates K and n can be found where K is determined by extrapolating the curve to unit strain value while the /(is defined by the slope of the plastic region. 

This is the true stress in the network and it is therefore denoted by (Jtrue-Since it is often not easy to measure the cross-sectional area of the deformed network, an engineering stress is also defined. In the engineering stress the original cross-sectional area LyoL o is used instead of the... 

Engineering stress and strain are easy to calculate and are used widely in engineering practice. However, engineering stress-strain curves generally depend on the shape of the specimen. A more accurate measure of intrinsic material performance is plots of true stress vs. true strain. True stress Ot is defined as the ratio of the measured force (F) to the instantaneous cross-sectional area (A) at a given elongation, that is,... 

As the force is the product of the cross-sectional area A and the true stress [Pg.236]

As already mentioned in the previous section, the extension that takes place before a polymer yields can be significant, which implies that there is a change in area perpendicular to the stress. It is therefore necessary in defining the yield stress to take account of the differenee between the true stress and the nominal stress (see section 6.3.2). For polymers that behave in the manner illustrated by curve (c) in fig. 6.2 the yield stress can then be defined as the true stress at the maximum observed load. For polymers that behave in the manner illustrated by curve (d) in fig. 6.2, such that there is merely a change in slope rather than a drop in load, the yield point is usually defined as the point of intersection of the tangents to the initial and final parts of the curve. 

Using the original cross-sectional area is not realistic when considering the stress in the plastic deformation region, it being better defined as the force divided by the current cross-sectional area of the specimen. This is known as the true stress. 

Using the defining expressions of the true stresses acting on the cube faces,... 

Equation (7) indicates that the cross-sectional area decreases exponentially with true strain. Now, we are able to define true stress as... 

In practice, the significance of this decision is that we must be careful to distinguish between true stresses a, which relate to the stress tensor, and the convenience of using nominal stresses /, which are defined as the force per unit area of unstrained cross-section. This distinction will be exemplified further when the theory is developed for simple tension (Section 2.4.3 below). 

Equation (11.3) defines a geometric condition for the true stress-strain curve, corresponding to the simple construction due to Considere shown in Figure 11.3. The ultimate stress is obtained when the tangent to the true stress-strain curve da/(U is given by the line from the point A = 0 on the extension axis. The angle a in Figure 11.3 is defined by... 

Yield stress may be regarded most simply as the minimum stress at which permanent strain is produced when the stress is subsequently removed. Although this deformation is satisfactory for metals, where there is a clear distinction between elastic recoverable definition and plastic irrecoverable deformation, in polymers the distinction is not so straightforward. In many cases, such as the tensile tests discussed above, yield coincides with the observation of a maximum load in the load-elongation curve. The yield stress then can be defined as the true stress at the maximum observed load (Figure 11.8(a)). Because this stress is achieved at a comparatively low elongation of the sample, it is often adequate to use the engineering definition of the yield stress as the maximum observed load divided by the initial cross-sectional area. 

During plastic deformation, the specimen s cross-sectional area changes significantly. Therefore, the nominal stress differs from the true stress that is defined as the quotient between the external load F Al) and the current... 

Stress a. Most commonly defined as engineering stress , the ratio of the applied load P to the original cross-sectional area A , i.e., a = PIA. The true stress Of or instantaneous stress is sometimes used and is defifted as the applied load P per instantaneous cross-sectional area A, i.e., Ot = PIA. 

Stress is defined as the force on a material divided by the cross sectional area over which it initially acts (engineering stress). When stress is calculated on the actual cross section at the time of the observed failure instead of the original cross sectional area it is called true stress. The engineering stress is reported and used practically all the time. 

Figure 2.54. Illustration of a true stress vs. strain curve and comparison of stress-strain curves for various materials. UTS = ultimate strength, and YS = yield strength. The tensile strength is the point of rupture, and the offset strain is typically 0.2% - used to determine the yield strength for metals without a well-defined yield point.Reproduced with permission from Cardarelli, F. Materials Handbook, 2nd ed.. Springer New York, 2008. Copyright 2008 Springer Science Business Media.

In what follows, nominal stress is defined as the load on the sample, registered in tension, divided by the initial cross-sectional area of the sample. True stress is the load divided by the actual cross-section of the sample as it narrows under extension. When a sample is subjected to a tensile force or load at a constant rate of strain, the stress measured as a function of strain can show certain unusual features (Figure 9.1). 

Define true stress and true strain and write an appropriate equation for each. 

In an electroelastic boundary-value problem (BVP), the stress state at a point of die dielectric depends on the mechanical and the electrostatic variables. Similarly to die elastic problem, we can think of a trae stress o,y which is in equilibrium with the external (volume and surfece) mechanical forces that are appUed to the elastomer (Dorfinann and Ogden 2005 McMeeking and Landis 2005 Suo et al. 2008). This quantity is often referred to as the total (true) stress to which a total nominal stress can be associated employing again Eq. 1. The adjective total refers to the properties for which the two stresses eontain both mechanical and electrical information (often in a coupled way) at each point of the deformed elastomer, defined via constitutive equations. 

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