In strain localization problem, the condition for strain localization or loss of hyperbolicity of the governing equation is found by solving eigenvalue problem Bn = wn where B (acoustic tensor) = n.D.n where n is eigen vector, D is the fourth order tensor of material tangent moduli and w is the eigenvalue value.
The acoustic tensor is important not only in strain localization problem, but also in the whole elastodynamics. It enables one to find the velocities and directions of plane waves propagating in anisotropic solids.
In addition to the previous answer:
In fact the term 'acoustic' is given as B has a direct relation to the propagation of planer waves. If we calculate the acoustic tensor (B) using the fourth order elastic stiffness tensor (De), B=n.De.n, then the eigenvalues of B divided by the mass density represents the squares of the elastic wave propagation speed in the direction of n. Therefore, zero determinant of acoustic tensor for a given direction represents the stationary wave with vanishing speed which indicate the formation of spatial discontinuity or localized deformation in an elastic medium.
Few useful references in this context:
Jirásek M. Mathematical analysis of strain localization. Rev Eur Génie Civ 2007;11:977–91.
Petkovski M, Crouch R. Experimental detection of discontinuities in pressure sensitive materials under multiaxial compression. Annu. Conf. Soc. Exp. Mech., Massachusetts: 2007.
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