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One of the major theoretical riddles in shear banding instabilities is theangle that the shear band chooses spontaneously with respect to the principalstress axis. Here we employ our recent atomistic theory to compute analyticallythe angle in terms of the characteristics of the Eshelby inclusion that modelsfaithfully the eigenfunction of the Hessian matrix that goes soft at theplastic instability. We show that loading protocols that do not conserve volumeresult in shear bands at angles different from 45$^o$ to the strain axis; onlywhen the external strains preserve volume like in pure shear, the shear bandsalign precisely at 45$^o$ to the strain axis. We compute an analytic formulafor the angle of the shear band in terms of the characteristics of the loadingprotocol; quantitative agreement with computer simulations is demonstrated.

Atomistic theory of the shear band direction in amorphous solids