Stress-strain relations orientation

However, as mentioned previously, orthotropic laminae are often constructed in such a manner that the principal material coordinates do not coincide with the natural coordinates of the body. This statement is not to be interpreted as meaning that the material itself is no longer orthotropic instead, we are just looking at an orthotropic material in an unnatural manner, i.e., in a coordinate system that is oriented at some angle to the principal material coordinate system. Then, the basic question is given the stress-strain relations In the principal material coordinates, what are the stress-strain relations in x-y coordinates ... 

The stress-strain relations in arbitrary in-plane coordinates, namely Equation (4.5), are useful in the definition of the laminate stiffnesses because of the arbitrary orientation of the constituent laminae. Both Equations (4.4) and (4.5) can be thought of as stress-strain relations for the k layer of a multilayered laminate. Thus, Equation (4.5) can be written as... 

This phenomenon gives us the very important information when we consider the orientation and extensibility of cross-linked molecules under extension. Generally, for the modeling or calculation for the stress-strain relation of cross-hnked molecules, the perfect regular network is adopted a... 

Moreover, we must pay attention to the points that in the cross-linked rubber, the cross-link stops the sliding of molecules and has its own excluded volume. Generally, in the calculation of the stress-strain relation, the four-chain model is used for the cross-link junction and recently the eight-chain model is considered to be more realistic and available. Thus, it is quite reasonable to consider that the bulky excluded volume that a cross-link junction possesses must be a real obstacle for the orientation of molecules, just like the case observed in Figure 18.16B. 

Laminated composites will usually combine laminae with fibres at different orientations. To predict the laminate properties, the stress-strain relations are required for loading a lamina at an angle 0 to the fibre direction and for loading both in-plane and in bending. Composite mechanics for laminated composites is well developed and... 

Since the laminate comprises a number of laminae oriented in different directions with respect to each other, having the same stress-strain relations, the stress-strain equation of the /cth layer of the laminate is as given by Hull [4] as ... 

At finite deformations, equation 59 can be shown to be incorrect because it is not objective i.e., it predicts results which erroneously depend on the orientation of the sample with respect to laboratory coordinates. This error can be eliminated by replacing j/j in equation 59 by the components of a corotational rate-of-strain tensor or the components of one of several possible codeformational rate-of-strain tensors either of these replacements ensures that the unwanted dependence of cy on the instantaneous orientation of a fluid particle in space is removed. If the stress-strain relations are linear within the changing coordinate frame, equation 59 is modified only be replacing y,-j with a different strain rate tensor whose definition is complicated and beyond the scope of this discussion. The corresponding corotational model is that of Goddard and Miller and the codeformational models correspond to those of Lodge or Oldroyd, Walters, and Fredrickson. ... 

It is of special interest to note that this stress-strain relation can also be derived from a consideration of the potential energy of the statistical links in a unidirectional force field. James and Guth [3] and Flory [1] have used the fact that an amount of work of / (1 — cos i) must be performed to rotate a link from a position of 0 = 0 to 0 = i if the orientational energy —/ cos 0 is distributed according to Boltzmann statistics then the mean extension of a link in the direction of force is... 

We will use readily available plastic films to demonstrate stress-strain behavior. Students should be able to relate the physical behavior of thin films to the concepts of orientation and crystallinity. They should be able to explain terms such as cold drawing, yielding, and machine and transverse directions. 

Many years ago De Vries discovered two important relations, when he was studying the orientation in drawing viscose-rayon yams one for the relationship between draw ratio and modulus, the other for the stress-strain correlation in drawn yams. 

In spite of our reluctance to quote numerical values at this point, the effective modulus obtained from the initial portion of the tensile curve ranges from 1 to 5 X 10 dyn cm". Many individual PE crystals have moduli from 3 X 10 to 10 dyn cm" and fracture at forces of about 0.2 dyn. Orientation effects are expected to be present and are presently being investigated. There is no comparable experimental data with which we can directly relate these values. However, moduli are in the range found for bulk specimens but are considerably less than the value of 70 X 10 dyn cm reported by Perkins et al. (8) for ultra-drawn HDPE fibers. The x-ray measurements of the lattice moduli by Ito (9) using an x-ray technique for oriented sheet samples is perhaps the most relevant comparison. He found values of 240 X 10 dyn cm" in a direction parallel to the fiber axis and a value of 4 X 10 dyn cm for the perpendicular direction which would be the closest comparison with our orientation. We are not yet certain whether the initial portion of the stress-strain curve shows nonlinear viscoelastic effects such as found by Chen et al. (4) for springy polypropylene (PP) fibers. 

Testing has shown that transverse fibre debonding for a number of FRP composites is strain related and independent of the type, proportion and orientation of the glass reinforcement. Thus with a knowledge of the material modulus, the stress to initiate transverse fibre debonding may be calculated for any particular material. Available test data indicate that the strain amplitude to debond varies approximately linearly from 0.3% strain at 10 cycles to 0.1% strain at 10 cycles. 

Generally, from spectroscopic data such as frequency position, band shape, intensity and dichroism of specific absorption bands conclusions can be derived in terms of the applied mechanical stress and the state of order and orientation of the polymer under investigation. An extremely powerful method for the study of transient phenomena in polyuMr deformation and relaxation is rheo-optics which describes the relation between stress, strain and an optical quantity (for example birefringence, infrared absorption, light scattering. X-ray diffraction) measured simultaneously with stress and strain as a function of time In a given rheo-optical method therefore, a mechanical test is combined with one of these various types of optical measurements. 

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