From the uniqueness and stability criteria considered in the preceding chapter, local conditions may be derived, which are treated herein. The importance of local conditions lies in the connection to material instabilities, namely, to instabilities which can develop from a point in a continuum and therefore result independent of the boundary conditions. For instance, we will see that loss of ellipticity corresponds to shear band formation. The following five local criteria will be analysed in this chapter:

• Positive definiteness of the constitutive operator (PD)

• Non-singularity of the constitutive operator (NS)

• Strong ellipticity (SE)

• Ellipticity (E)

• Flutter (F)

To begin providing an example of the preceding criteria in the simple case of the infinitesimal theory of isotropic elasticity, we recall the condition of positive definiteness (PD) [Eqs. (2.192)], which is the positiveness of the eigenvalues 2μ and λ + 2μ/3 of the elastic fourth-order tensor. Non-singularity (NS) corresponds to the condition of non-vanishing of these eigenvalues, and in a similar vein, strong ellipticity (SE) corresponds to the positiveness of the eigenvalues μ and λ + 2μ of the acoustic tensor (2.176), whereas ellipticity (E) corresponds to the non-vanishing of the same eigenvalues.