Strain theory definition

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and die theories relating to them, pressing die analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context. 

The purpose of this chapter is to remind the reader of the basis of the theory of elasticity, to outline some of its principal results and to discuss to what extent the classical theory can be applied to polymeric systems. We shall begin by reviewing the definitions of stress and strain and the compliance and stiffness matrices for linear elastic bodies at small strains. We shall then state several important exact solutions of these equations under idealised loading conditions and briefly discuss the changes introduced if realistic loading conditions are considered. We shall then move on to a discussion of viscoelasticity and its application to real materials. 

In order to respect volumetric strain, a non-associated plastic theory is used. As suggested by Cambou et al. (1989), the unit vector n is calculated in such a way that equation 13 is equivalent to the definition of plastic volume change of equation 15. One can combine the equations to obtain the expression of vector n ... 

There are two principal theories, or models, that attempt to describe what happens during brittle fracture, the Griffith fracture theory and the Irwin model. Both assume that fracture takes place through the presence of preexisting cracks or flaws in the polymer and are concerned with what happens near such a crack when a load is applied. Each leads to the definition of a fracture-toughness parameter and the two parameters are closely related to each other. The Griffith theory is concerned with the elastically stored energy near the crack, whereas the Irwin model is concerned with the distribution of stresses near the crack. Both theories apply strictly only for materials that are perfectly elastic for small strains and are therefore said to describe linear fracture mechanics. 

Fig. 7.8 The dependence of (engineering) stress

The theory outlined above is rigorous only for infinitesimal elastic deformation. Creep of polymeric materials is explicitly concerned with time dependence and implicitly with finite strains and therefore nonlinear behaviour. The nature of the non-linear behaviour is complex and varies not only from material to material but also with direction within a given sample of material, i.e. the non-linearity of behaviour is anisotropic. It is found, therefore, that on a particular definition of strain the behaviour of a sample may appear to be linear in one direction and significantly non-linear in another. Such a phenomenon is demonstrated in results presented below. 

The theory of hydrodynamics similarly describes an ideal liquid behavior making use of the viscosity (see Sect 5.6). The viscosity is the property of a fluid (liquid or gas) by which it resists a change in shape. The word viscous derives from the Latin viscum, the term for the birdlime, the sticky substance made from mistletoe and used to catch birds. One calls the viscosity Newtonian, if the stress is directly proportional to the rate of strain and independent of the strain itself. The proportionality constant is the viscosity, q, as indicated in the center of Fig. 4.157. The definitions and units are listed, and a sketch for the viscous shear-effect between a stationary, lower and an upper, mobile plate is also reproduced in the figure. Schematically, the Newtonian viscosity is represented by the dashpot drawn in the upper left comer, to contrast the Hookean elastic spring in the upper right. 

The experimental data in Table 3.1 show that the calculated values of total angle strain are approximately correct only for cyclopropane, cyclobutane, and cyclopentane. Cyclohexane is definitely not the strained compound Baeyer s theory predicts, and the larger ring compounds are also not very strained. Any chemist today can explain the discrepancy between these calculated and experimental values of strain energy cyclohexane is not planar. In either the chair or boat conformations (Figure 3.12), all bond angles can be approximately 109.5°. In the chair conformation of cyclohexane, all bonds are staggered, and there are no apparent van der Waals repulsions in the molecule. ... 

If we assume that the Bragg model represents accurately the true shape and size of the molecule, the following conclusion, which is of importance for the theory of structure and isomerism, may be drawn Only those compounds can exist which obey the laws of space requirement outlined above, that is, which can be represented without strain by the Bragg molecular models. We arrive at the fact that two carbon atoms, united by primary valences, may not be separated by any distance greater than 1.2-1.6 A, a requirement which places definite limitations on the bridge bonds frequently formulated in organic chemistry. 

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