Abstract
In this chapter our objective will be to review the concepts of stress and strain and to present the equations which relate these quantities, and if necessary other variables, for various types of material behavior. Some general comments will be made on the methods of solution of the equations, but a description of specific solutions useful for fracture prediction will be deferred to in the next chapter. For this reason, the reader with background in the analysis of stress and strain and the equations for elastic and plastic deformation can proceed to Chap. 3. The treatment in this chapter is based on a “homogeneous” and “isotropic” continuum. That is, the properties do not vary with location and are the same in all directions. Although individual grains in a ceramic or metal may not be isotropic, the assumption of isotropic behavior is justified for a randomly oriented polycrystalline aggregate if the dimensions over which stress and strain change appreciably are large compared to those of the individual grains. A similar argument applies to noncrystalline materials if grain size is replaced by the dimension over which significant microstructural changes occur. However, for small regions the assumption of a homogeneous isotropic continuum is less realistic. In subsequent chapters, solutions based on this concept will be applied to predict stresses at the tips of sharp cracks and the growth of small voids. In both cases the dimensions involved may be smaller than the grain size. This approach is justified by expediency, since more realistic calculations are vastly more complicated, but its limitations should not be overlooked.
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- 1.
This ignores the possible presence of distributed moments on the faces of the unit cube. Although a large amount of literature has appeared on this topic, the distributed moment does not appear to be important in practice.
- 2.
The development of yield criteria and the equations of plasticity are treated in many texts. For example, Hill [3] gives a thorough discussion with references to the papers of Tresca and later workers in this field that we will mention.
- 3.
Instability, comparable to necking in the tension test, limits the usefulness of pressurized specimens, and thin-walled tubes loaded in torsion will buckle at a certain shear strain. Fortunately, it is possible to obtain shear stress–shear strain data from solid circular rods in torsion.
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Dharan, C.K.H., Kang, B.S., Finnie, I. (2016). Stress, Strain, and the Basic Equations of Solid Mechanics. In: Finnie's Notes on Fracture Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2477-6_2
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