Microstructural eects on shear instability and shear band spacing Li Chen, R.C. Batra * Department of Engineering Science and Mechanics, M/C 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Abstract A constitutive relation that accounts for the thermally activated dislocation motion and microstructure interaction is used to study the stability of a homogeneous solution of equations governing the simple shearing deformations of a thermoviscoplastic body. An instability criterion and an upper bound for the growth rate of the in®nitesimal defor- mations superimposed on the homogeneous solution are derived. By adopting Wright and Ockendon's postulate, i.e., the wavelength of the dominant instability mode with the maximum growth rate determines the minimum spacing between shear bands, the shear band spacing is computed. The eect of the initial dislocation density, the nominal strain-rate, and parameters describing the initial thermal activation and the initial microstructure interaction on the shear band spacing are delineated. Ó 2000 Elsevier Science Ltd. All rights reserved. 1. Introduction Adiabatic shear bands are narrow regions of intense plastic deformation that usually form during high strain-rate deformation of several metals and some polymers. Their formation sig- nals a transition from a generally homogeneous deformation to a nonhomogeneous one involving high strain gradients in a narrow region. These shear bands precede shear fractures. Thus there is signi®cant interest in studying their initiation, propagation, width and spacing between adjacent bands. The theoretical/analytical analyses of the problem can broadly be classi®ed into two cate- gories: (i) linear stability analysis, and (ii) numer- ical solution of the coupled set of nonlinear equations. The linear stability analyses (e.g., see [1±7]) are aimed at delineating when a shear band initiates, and the spacing between them. The complete solution of the coupled nonlinear set of equations provides detailed information about the history of various deformation ®elds. Most investigations have employed phenome- nological constitutive relations. It was pointed out in [8±10] that these are valid only within the range of data used to calibrate them. These models do not account for the radically dierent behavior of face-centered-cubic (FCC) and body-centered- cubic (BCC) metals and the grain size. They [8±10] have proposed a constitutive relation that ac- counts for microstructural changes occurring in the body while it is being deformed. A dislocation mechanics based constitutive relation involving only one variable, namely, the total dislocation density has been proposed in [11±13]. This accounts for the eects of dislocation±dislocation www.elsevier.com/locate/tafmec Theoretical and Applied Fracture Mechanics 34 (2000) 155±166 * Corresponding author. Tel.: +1-540-231-6051; fax: +1-540- 231-4574. E-mail address: rbatra@vt.edu (R.C. Batra). 0167-8442/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 0 ) 0 0 0 3 3 - 1