Research Page: Shear Bands

Research Page

Dynamic Shear Band Propagation in Thermoviscoplastic Solids

Solids deforming at high rates often develop narrow layers of intense shearing called shear bands. In some applications such layers are beneficial; for instance, in ultra-high speed metal machining shear bands lead to lower cutting forces and improved surfaces. In other applications shear bands are a nuisance; for instance, in terminal ballistics shear bands lower the forces required for armor penetration. In all cases it is important to understand how shear banding takes place, and what factors promote it.

We are interested in the mathematical characterization of the two-dimensional structure of dynamically propagating shear bands in thermoviscoplastic solids.

The Nature of Shear Bands

The figure shows a shear band in 7039 aluminum alloy. The micrograph was obtained by Leech in a test with an impact velocity of 311 m/s (Metal. Trans. 16 A, 1985), and serves to illustrate the great thinness of shear bands. Typical shear band widths range from 10 to 100 µm. Shear bands are also characterized by high local values of shear strain, of up to 100; ultra-high local shear strain rates, often in excess of 10⁶ per second; local temperature rises of several hundred degrees; and high propagation speeds, sometimes in excess of 1000 m/s.

The realistic modeling of shear bands requires consideration of large plastic deformations, rate sensitivity, hardening, heat convection and conduction, thermal softening and inertia. Fully nonlinear multidimensional solutions to problems of this nature are rare. However, the thinness of shear bands allows for the introduction of a number of approximations which facilitate the analytical characterization of the flow. The systematic use of these approximations results in a much simplified set of boundary layer equations; as expected, the analysis reveals that these equations are valid when R>>1, where R is a generalized Reynolds number. The problem can be further simplified by introducing a space-like and a time-like similarity variable. This reduces from 3 to 2 the number of independent variables in two-dimensional transient problems. The resulting simplified set of field equations is in some cases amenable to semi analytical treatment.

Transient Boundary Layer Solutions

We characterize unstable material behavior by recourse to an analysis of steady state solutions of the boundary layer equations. This development sets the basis for interpreting our transient boundary layer solutions inasmuch as we identify the shear band with those portions of the domain wherein the material, initially stable, has accumulated sufficient deformation to become unstable. Our theory predicts shear bands which propagate at very high, constant speeds, as required by the experimental evidence. The figure shows a series of snapshots of a shear band propagating from a notch in a copper plate impacted by a flat projectile moving at 544 m/s (R=10). The contour lines correspond to constant values of the accumulated plastic work; the shear band proper is the region inclosed by the contour w=5 (i.e., the material is unstable for values of w larger that the critical accumulated plastic work, which in this case equals 5). The tip of the shear band propagates at a constant velocity of 8.8 times the impact velocity. As expected, the shear band is very thin.

A note on the figure: the values of length, time and accumulated plastic work are in units of the characteristic values 1.1 mm, 2 µs and 53 MPa, respectively. We have stretched he vertical axis by a factor of 2 to facilitate visualization. Plots of the attendant fields of temperature, stress and strain rate can also be obtained as part of the solution.

Shear band tip velocities

Shear Band Tip Velocities

The figure shows the predicted dependence of the shear band tip velocity (normalized by the velocity of the impactor) on the Reynolds number, for three different values of the critical accumulated plastic work. The shear band tip velocity increases sharply with R at relatively low values of R, in agreement with the experimental observations of Zhou et al. (J. Mech. Phys. Solids 44, 1996); at high impact velocities (high R), the shear band velocity saturates and approaches a constant value.

Discussion

Our theory underscores the essential differences between propagating shear bands and cracks. The motion of a crack tip involves processes of separation which lie squarely outside the purview of local constitutive theories. As a consequence, the description of crack tip motion requires mechanical postulates which are independent of the constitutive equations. In addition, crack tips carry along with them autonomous singular fields. By contrast, shear banding, as understood here, is strictly a consequence of the constitution of the material. Because dynamic shear bands tend to be very elongated, their leading front may be regarded as a propagating `tip'. The entity thus defined is clearly identifiable by, for instance, optical methods (Zhou et al., 1996, op. cit.), and thus amenable to experimental observation. However, our similarity solutions, as well as the finite element simulations of Needleman (J. Appl. Mech. 56, 1989), show a certain degree of broadening in the developing shear band. In addition, the shear band tip does not carry singular autonomous fields, but is merely a salient feature of an otherwise continuous field. These observations have led some authors to categorize dynamic shear band growth as diffusive and to discard the notion of a shear band tip altogether. However, provided that the physical nature of shear band tips is clearly understood, they can play a useful role, for instance, as a basis for comparisons between theory and experiment.

References

Gioia, G., and Ortiz, M.
The Two-Dimensional Structure of Dynamic Boundary Layers and Shear Bands in Thermoviscoplastic Solids
J. Mech. Phys. Solids 44, 251-292, 1996.
[PDF 1.7 Mb]