Shear bifuraction modes in materials with cubic symmetry

As an approach to the mechanical conditions of the bifurcation of the singl e crystals of metals, the present article considers a three dimensional, in crementally linear, time independant incompressible solid with cubic symmet ry. The possibility of bifurcation into shear bands is studied by the means of continuum mechanics. It involves: i) the rate constitutive law of the m aterial, which depends only on two instantaneous shear moduli mu and mu* ii ) the equilibrium equations For a material with an initial loading sigma. I n this simplified case, all the calculation are analytical, and the results can be easily presented as a function of the ratio mu/mu* Or a related qua ntity f. The bifurcation condition is a single determinant which involves the deviat oric part s of the tenser sigma, moduli mu and mu*, and the vector nu norma l to the shear banding plane. Thus, along a given stress path beginning at s = 0 and extending homothetically s = lambda s degrees there is a whole ra nge of nu determining values lambda(nu) corresponding to the bifurcation in a particular plane; one of them is minimum and gives the limit of bifurcat ion for the considered path. This limit takes the form of an hypersurface i n the five dimensional space of the deviatoric parts of the stress tensors; within it, bifurcation is impossible; outside it can happen on a variety o f shear planes. From the hypotheses of the problem the surface admits 0 as a centre of symmetry. The study of particular cases allows to check results already known in the case of isotropy (mu = mu*): in the case of uniaxial stress along an axis o f symmetry, bifurcation occurs for all values of f if \sigma\ greater than or equal to 2 mu, and even earlier if mu/mu* greater than or equal to 0.5; for shear between two axes of symmetry, the bifurcation condition is simply \tau\ greater than or equal to inf(mu*, 2 mu). The examination of the vari ous cases involving only two non zero components of the tensor sigma shows that shear components are more detrimental from the point of view of shear banding than tensions and compressions. Maxima and minima of resistance to bifurcation have been calculated for var ious values of f; it is found that if the solids can be submitted to any ty pe of loading, the isotropic material is the most secure of all.