Abstract: The problems of mesh dependence and mesh refinement areexhibited in shear bands computations of softening materials withconventional finite element method, and recognized with three numericalcases by a finite element software ABAQUS. Numerical analysis shows themesh-dependence problem does not occur for the thick-walled cylinderswithout shear bands. Couple stress theory is introduced to solve thesenumerical problems as a mesoscale non-local theory and the length scaleparameter \ell is investigated \ell in details. 18-DOF triangle couplestress element, which can pass C0 - 1 patch test, is used for the abovethree numerical cases. Numerical results show that the finite elementmethods of mesoscale couple stress theory are suitable for the shear bandscomputations.