As proven in 1971, the stability theories associated with different finite strain measures, corresponding to different values of the parameter m in the Doyle-Ericksen finite strain tensor, are equivalent if a proper stress-dependent transformation of the tangential stiffness moduli is applied. For nonlinear materials the choice of m is immaterial. However, it was not shown which m value is appropriate when the constant small-strain shear modulus is used for soft-in-shear structures buckling at small strain. By extension of an energetic argument recently used for sandwich or lattice columns, it is shown that the correct theory for elastomeric bearings is Haringx's (corresponding to m=-2). The analysis is then extended to the general case of homogenized soft-in-shear layered or fiber composites under bi-axial loading. In that case, the critical load for a constant small-strain shear modulus is found to be intermediate between Engesser- (corresponding to m=2) and Haringx-type theories. A formula for m as a function of the ratio of the initial stresses in the orthotropy directions is derived. It is shown that Biot's solutions for buckling of layered continua (corresponding to m=1) must be transformed to the proper m-value, and that, if a microstructure very soft in shear is homogenized, the standard finite element programs used in commercial software (based on the updated Lagrangian formulation corresponding to m=2), must be generalized for arbitrary m.
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