Hooke Voigt notation

Since, in our consideration, both the stress and the strain tensors are symmetrical about the diagonal, there are only six independent stress and strain elements, making many of the elastic compliances equal to each other. This makes it possible to simplify the generalized Hooke s law so that it involves at most 36 elastic compliances or constants, but requires the introduction of a shorthand notation both for stress and for strain for a unique representation that is referred to as the Voigt notation that we state as follows, for stresses and strains ... 

On the basis of the eontracted Voigt notation of stresses and strains, the generalized Hooke s law ean be written out in the form of a set of linear relations, which for the strain-stress relationships are... 

In this Eq. (1.48), the Ax and Au in Fig. 1.25 are replaced by dx and du or in terms of their partials as dx and du, thus indicating the instantaneous change of u with respect to x. Recall that ei = en = e = e x, depending on the type of notation. Nomenclatures may vary in accordance with the original research applications. For instance, the Voigt notation is useftil in calculations involving constitutive models, such as the generalized Hooke s Law, as well as for finite element analysis. 

As discussed in the previous section, strains and stresses are closely linked together through material characteristics and deformation mechanisms. In solid mechanic, it is useful to mathematically express this relation through constitutive laws. Also, as aforementioned, numerical modelling now requires three-dimensional constitutive models. One of the simplest expressions of a constitutive law is the Hooke s law (4) in Voigt notation (Mase and Mase, 1999), for isotropic materials. In this case, assuming the small strain theory, only two... 

In a more complete, tensor form, one would write compliance matrix) to generalize Hooke s law (Equation (1.9b)). Indeed, reducing y and Voigt notation in mathematics, one... 

Using Hooke s law in the Voigt matrix notation, we can calculate the stress state in each coordinate system. We initially assume that the material parameters might differ in the different systems. In the un-primed system, we find a a = Cap ep, in the primed system, the stress is a a = Ca p p - This yields... 

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