Uniaxial tension or compression

as in all following sub-sections, except for equi-biaxial deformation, the principal direction of stretch is the 2-direction, and constancy of volume given by eq. (6.27) applies. 

The dependence of the main-chain stretch on external extension ratios is given by 

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Quotient of the length (/) of a sample under uniaxial tension or compression and its original length (/o)... 

Generally, when testing materials with a nonlinear stress-strain behavior, the tests should be conducted under uniform stress fields, such that the associated damage evolution is also uniform over the gauge section where the material s response is measured. Because the stress field varies with distance from the neutral axis in bending tests, uniaxial tension or compression tests are preferred when characterizing the strength and failure behavior of fiber-reinforced composites. 

Perhaps the simplest deformation that can be applied to a sample is uniaxial tension or compression, shown in Figure 2-la. This is the type of deformation mentioned in the Introduction. However, our previous concept of force is... 

Linear elasticity is the most basic of all material models. Only two material parameters need to be experimentally determined the Young s modulus and the Poisson s ratio. The Young s modulus can be directly obtained from uniaxial tension or compression experiments, and typical values for a few select fluoropolymers at room temperature are presented in Table 11.2. 

The Poisson s ratio can be determined by measuring the transverse strain during uniaxial tension or compression experiments. Due to the small magnitude of the transverse strain, it is difficult to accurately determine the Poisson s ratio. Instead, it is often sufficient to assume a value for the Poisson s ratio of about 0.4. Unless the fluoropolymer component is highly confined, the Poisson s ratio has only very weak influence on the predicted material response. 

It can be seen that uniaxial tension or compression lies on the two axes. Inside the box (outer boundaries) is the elastic range of the material. Yielding is predicted for stress combinations bv the outer line. 

An improvement over the Whitney—Nuismer approach was that by Garbo and Ogonowski [21]. They solved the case of a fastener hole using complex elasticity and recovered the case of an open hole as a special case. They still use a characteristic distance as is done in the Whitney—Nuismer approach, but their method is applied to every single ply. In addition, it allows for any type of combined in-plane loading instead of uniaxial tension or compression. 

Consider a single crystal being subjected to uniaxial tension or compression, as shown in Fig. 6.20. Clearly, the ease with which plastic deformation is activated will depend not only on the ease of dislocation glide for a particular slip system but also the shear stress acting on each system. This is similar to the problem discussed in Section 2.10 (Eq. (2.44)) though one should note the plane normal, the stress direction and the slip direction are not necessarily coplanar, (< +A)5 90°. In other words, slip may not occur in the direction of the maximum shear stress. The resolved shear stress acting on the slip plane in the slip direction is... 

It is useful to note that eqs. (4.19) and (4.20) for uniaxial tension (or compression), in addition to being applicable to uniaxial strain deformation, as stated above, are also applicable to the case of dilatation responding to negative pressure where the basic symmetry of the deformation is maintained. In that case, however, ffii is replaced with tensile stress (negative pressure), n is replaced with e, the dilatation, and Eq, Young s modulus, is replaced with the bulk modulus K(). Moreover, a must be replaced by fi, which represents the reciprocal of the critical athermal cavitation dilatation. 

The relation between simple shear and uniaxial tension or compression... 

Therefore, for small or no rotation, the Cauchy strain can also be used to measure a large strain such as in uniaxial tension or compression testing of specimens where no rotation is involved. In other words, the Cauchy strain is not limited to small stretch it is limited to small rotation. 

It was observed empirically by Hooke that, for many materials under low strain, stress is proportional to strain. Young s modulus may then be defined as the ratio of stress to strain for a material under uniaxial tension or compression, but it should be noted that not all materials (and this includes polymers) obey Hooke s law rigorously. This is particularly so at high values of strain but this section only considers the linear portion of the stress-strain curve. Clearly, reality is more complicated than described previously because the application of stress in one direction on a body results in a strain, not only in that direction, but in the two orthogonal directions also. Thus, a sample subjected to uniaxial tension increases in length, but it also becomes narrower and thinner. This quickly leads the student into tensors and is beyond the scope of this chapter. The subject is discussed elsewhere [21-23]. There are four elastic constants usually used to describe a macroscopically isotropic material. These are Young s modulus, E, shear modulus, G, bulk modulus, K, and Poisson s ratio, v. They are defined in Figure 9.2 and they are related by Equations 9.1-9.3. 

Application of the law of mixtures to fibre-reinforced composites as shown in Section 2.5 is possible after several simplifications and assumptions have been made, which transform the real behaviour of a highly heterogeneous material into that of an elastic and homogeneous one. Such an approach gives only approximate information which may, however, be useful as a general indication on the stress and strain values and on their reciprocal relations. For the simplest case of uniaxial tension or compression these assumptions are ... 

For one-dimensional deformation, the body undergoes uniaxial tension or compression. For a three-dimensional deformation, the body undergoes the shear force. That is, for a three-dimensional deformation, we have shear stress (labeled CTs) and shear strain (labeled y). The shear modulus G and the shear compliance J are defined and related by... 

Of the many theories developed to predict elastic failure, the three most commonly used are the maximum principal stress theory, the maximum shear stress theory, and the distortion energy theory. The maximum (principal) stress theory considers failure to occur when any one of the three principal stresses has reached a stress equal to the elastic limit as determined from a uniaxial tension or compression test. The maximum shear stress theory (also called the Tresca criterion) considers failure to occur when the maximum shear stress equals the shear stress at the elastic limit as determined from a pure shear test. The maximum shear stress is defined as one-half the algebraic difference between the largest and smallest of the three principal stresses. The distortion energy theory (also called the maximum strain energy theory, the octahedral shear theory, and the von Mises criterion) considers failure to have occurred when the distortion energy accumulated in the part under stress reaches the elastic limit as determined by the distortion energy in a uniaxial tension or compression test. 

The form of the stress-strain behaviour depends upon the loading geometry employed. The present analysis will be restricted to simple uniaxial tension or compression when specimens are deformed to an extension ratio A in the direction of the applied stress. It is possible to replace Ai by A but Equation (5.144) requires that A2A3 = 1/Ai and so A2 = A3 = 1/A. Equation (5.142) can therefore be written as... 

In order to simplify the variational formulation of the previous section, two restrictions are introduced here. (1) Elastomeric networks of common interest are generally isotropic in the rest state, and the analysis is restricted to such systems. (2) The analysis is restricted to those deformation fields that are isotropic in a plane which, by convention, is taken to be the xy plane. Thus, cc = a,X = X,F = F. The z direction constitutes t ie th rd princYpal axis and F and X are not restricted. Any combination of equal biaxial tension or compression in the xy plane with uniaxial tension or compression in the z direction is permitted. Unfortunately, pure shear and torsion are excluded from the restricted analysis. 

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