Shear band

A shear band (or, more generally, a 'strain localization') is a narrow zone of intense shearing strain, usually of plastic nature, developing during severe deformation of ductile materials. As an example, a soil (overconsolidated silty-clay) specimen is shown in Fig. 1, after an axialsymmetric compression test. Initially the sample was cylindrical in shape and, since symmetry was tried to be preserved during the test, the cylindrical shape was maintained for a while during the test and the deformation was homogeneous, but at extreme loading two X-shaped shear bands had formed and the subsequent deformation was strongly localized (see also the sketch on the right of Fig. 1).

Although not observable in brittle materials (for instance glass at room temperature), shear bands or, more generally, ‘localized deformations’ usually develop within a broad range of ductile materials (alloys, metals, granular materials, plastics, polymers, and soils) and even in quasi-brittle materials (concrete, ice, rock, and some ceramics). The relevance of the shear banding phenomena is that they precede failure, since extreme deformations occurring within shear bands lead to intense damage and fracture. Therefore, the formation of shear bands is the key to the understanding of failure in ductile materials, a research topic of great importance for the design of new materials and for the exploiting of existing materials in extreme conditions. As a consequence, localization of deformation has been the focus of an intense research activity since the middle of the 20th century.

Shear band formation is an example of a material instability, corresponding to an abrupt loss of homogeneity of deformation occurring in a solid sample subject to a loading path compatible with continued uniform deformation. In this sense, it may be interpreted as a deformation mechanism ‘alternative’ to a trivial one and therefore a bifurcation or loss of uniqueness of a ‘perfect’ equilibrium path. The distinctive character of this bifurcation is that it may occur even in an infinite body (or under the extreme constraint of smooth contact with a rigid constraint).

Let us consider an infinite body made up of a nonlinear material, quasi-statically deformed in a way that stress and strain may remain homogeneous. The incremental response of this nonlinear material is assumed for simplicity linear, so that it can be expressed as a relation between a stress increment ${\displaystyle {\dot {\sigma }}}$ and a strain increment ${\displaystyle {\dot {\varepsilon }}}$, through a fourth-order constitutive tensor ${\displaystyle \mathbb {C} }$ as

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