On the Lack of Symmetry in Materials

In this paper we examine the loss of symmetry in mechanics of mate rial To this end we rst review the so called Bromwich bounds of eigenvalues in linear algebra For illustration we revisit the hierarchy of di erent failure diagnos tics when the material properties loose symmetry Subsequently we examine the lack of symmetry in the stress and strain measures which appears in nite deforma tion analysis and in micropolar continua For de niteness we evaluate maximum and minimum values of the non symmetric second order tensors which no longer coincide with the principal eigenvalues and we discuss the special format of the trace invariants For geometric visualization we generalize Mohr s circle and the underlying transformation relations which account for the loss of symmetry in two dimensions To conclude we consider two examples of shear failure which serve as model problems to address the intricate di erences of symmetric and non symmetric stress and deformation measures