Discrete micromechanics of elastoplastic crystals

The theoretical prediction of elastoplastic behaviour of single crystals is a basic problem which is central to the prediction of the overall behaviour of the crystal aggregate. It is generally well known that quasi‐static and isothermal plastic deformation in single crystals arises almost solely from slip on specific crystallographic planes, and that this process occurs when the resolved shear stress on a critical slip system reaches a certain maximum value. What is not obvious is how one can identify the specific slip systems activated by a given load increment, since the process usually involves selection from a pool of linearly dependent slip systems. In this paper we use small‐deformation multisurface plasticity theory to phrase properly the problem of crystal slips at infinitesimal increments. We then describe an ‘ultimate’ algorithm for systematically identifying the active slip systems at finite increments. We arrive at the following major conclusions when the ultimate algorithm is applied to f.c.c. crystals: For perfectly plastic crystals the combination of active slip systems may or may not be unique; however, the imposition of the discrete Kuhn‐Tucker conditions is sufficient to determine the (unique) final crystal stresses. For Taylor hardening crystals in which active and latent slip systems harden by the same amount, the discrete Kuhn‐Tucker conditions are also sufficient to make the mathematical problem of crystal stress integration well posed, i.e. the final stresses can be determined uniquely albeit the combination of active slip systems may not be unique. To illustrate the latter point, an accurate return‐mapping algorithm for perfectly plastic and Taylor hardening crystals is described and tested against the ultimate algorithm to demoustrate numerically that it is possible to generate different combinations of active slip systems and yet produce identical final stresses.