J.L. Bassani

Lattice incompatibility and a gradient theory of crystal plasticity

Abstract In the finite-deformation, continuum theory of crystal plasticity, the lattice is assumed to distort only elastically, while generally the elastic deformation itself is not compatible with a single-valued displacement field. Lattice incompatibility is shown to be characterized by a certain skew-symmetry property of the gradient of the elastic deformation field, and this measure can play a natural role in a nonlocal, gradient-type theory of crystal plasticity. A simple constitutive proposal is discussed where incompatibility only enters the instantaneous hardening relations, and thus the incremental moduli, which preserves the classical structure of the incremental boundary value problem.

Latent hardening in single crystals II. Analytical characterization and predictions

Constitutive equations are developed that characterize the multiple-slip behaviour of crystalline materials at low temperature. A matrix of instantaneous hardening moduli that relate the rate of hardening on each slip system to all slip-rates is proposed based upon well-known observations and the latent hardening experiments reported in Part I. In general, these moduli depend on the history of slips. Simulations of various behaviours are presented for FCC single crystals (copper) that are in good agreement with observations. These include, for example, stress-strain curves in a uniaxial loading test, hardening rate variations with respect to initial orientation, latent hardening, tensile overshoot and secondary slips. Numerical calculations are facilitated using an extremum principle and a modified quadratic programming algorithm.

Incompatibility and a simple gradient theory of plasticity

In the continuum theory, at finite strains the crystal lattice is assumed to distort only elastically during plastic flow, while generally the elastic distortion itself is not compatible with a single-valued displacement field. Lattice incompatibility is characterized by a certain skew-symmetry property of the gradient of the elastic deformation field, and this measure can play a natural role in nonlocal theories of plasticity. A simple constitutive proposal is discussed where incompatibility only enters the instantaneous hardening relations. As a result, the incremental boundary value problem for rate-independent and rate-dependent behaviors has a classical structure and rather straightforward modifications of standard finite element programs can be utilized. Two examples are presented in this paper: one for size-scale effects in the torsion of thin wires in the setting of an isotropic J2 flow theory and the other for hardening of microstructures containing small particles embedded in a single crystal matrix.

Plastic Flow of Crystals

Publisher Summary This chapter provides an overview of plastic flow of crystals. Plastic flow under multiple slip, particularly in metallic crystals, has been widely studied by materials scientists and applied mechanicians since the early 1900s. Finite strain crystal plasticity is a rigorous nonlinear continuum theory. With few exceptions, subtleties of this theory such as multislip hardening and non-Schmid effects have not been tested sufficiently in critical experiments. In particular, through the interplay between small secondary slips and hardening, there is now compelling evidence that important effects largely have been ignored. Furthermore, because intermetallic compounds are currently of growing interest in high-temperature applications, non-Schmid effects, which are also important in BCC metals and alloys, should also be considered more seriously. In this chapter, recent studies of multiple-slip interactions and hardening are brought together within a time-independent theory, and their influence on strain localization is explored. It discusses in detail about yield behavior including non-Schmid effects. Flow behavior including non-Schmid effects is elaborated. Concepts of hardening behavior and strain localization are also explained in the chapter.

Latent Hardening in Single Crystals I. Theory and Experiments

Accurate measurements of the initial yield stress on previously latent slip systems as well as a reinterpretation of widely reported experimental observations have led to a new description of single crystal hardening within the framework of the incremental (flow) theory of plasticity. Slip interactions and the history of slips are essential in explaining well-known physical phenomena such as stage II deformation and latent hardening. Guidelines for deriving the set of instantaneous hardening moduli are given in terms of inequality restrictions. Although time-independent behaviour is assumed throughout the present study, these restrictions are expected to apply as well to time-dependent creep behaviour at low to intermediate temperatures. In Part II, a complete constitutive theory is developed with analytical forms given for the instantaneous hardening moduli.

Non-schmid yield behavior in single crystals

Abstract A parently , certain single crystals do not obey Schmids law for slip on individual systems. For example, many intermetallic compounds such as Ni 3Al with the LI 2 structure display yield behaviors that are called “anomalous” in the sense that the critical resolved shear stress on the primary slip system at yield is a function of the orientation of the loading axis and the sense of load. To accommodate such non-Schmid behaviors, a generalization of Schmids law is proposed where stress components other than the Schmid stress also enter the criterion. The general effects of “cross slip” in FCC single crystals as well as the detailed shield behavior of Ni 3Al are well described. Finally, we demonstrate that non-Schmid effects can make it more difficult to activate many slip systems simultaneously.

Cracks on bimaterial and bicrystal interfaces

Abstract The two-dimensional problem of an interface crack between two anisotropic elastic solids is considered. A necessary and sufficient condition for no oscillations in the singular crack-tip fields is derived, and then bicrystals associated with tilt boundaries that satisfy this condition are identified. For a given tilt axis and misorientation the same singularity arises for both geometrically symmetric and asymmetric bicrystals.

Non-associated plastic flow in single crystals

Abstract P lastic flow by shears on well-defined crystallographic planes is associated, most often, with the Schmid yield criterion which is expressed in terms of the resolved shear stress on those planes in the direction of the shears. In this case, for time-independent (low-temperature) flow, the yield function for each slip system is also the potential for the shear and. therefore, flow is said to be associated with that function (i.e. normality). Qin and Bassani (1992. J. Mech. Phys. Solids40, 813) propose a yield criterion to accommodate various non-Schmid behaviors. In this paper it is shown that this criterion leads to a nonassociated flow rule (i.e. non-normality). Time-independent constitutive relations are derived for single crystals undergoing non-associative plastic flow. The tension-compression asymmetry predicted for initial yield persists in strain hardening as well. Strain localization in the form of shear bands is investigated for both a single-slip and a symmetric double slip model. It is shown that non-Schmid effects increase the tendency for localization.

Finite crack on bimaterial and bicrystal interfaces

Abstract A Griffith crack lying along the interface between anisotropic elastic solids is analysed. Explicit solutions for the displacement and stress fields are obtained when the crack-tip fields are non-oscillatory ; i.e. the condition W = 0 is satisfied as derived by Qu and Bassani (J. Mech. Phys. Solids37, 417–433, 1989). In this case, the stress intensity factors are real and the three fracture modes are separable at the crack tip. For interfaces formed by misorienting identical anisotropic solids through a relative tilt, this condition is satisfied if the in-plane and anti-plane deformations decouple in the interface-crack coordinates. Complete crack-tip stress fields are calculated for FCC bicrystals satisfying the decoupling condition. Based on Schmids law, the small-scale-yielding plastic zones surrounding the crack tip are also estimated.

Complex macroscopic plastic flow arising from non-planar dislocation core structures

For a broad range of crystalline materials complex dislocation core structures have a significant effect on macroscopic plastic flow, causing unexpected deformation modes that are strongly influenced by other components of stress in addition to the glide stress on a given slip system and on the sign of stress. In this paper we use atomistic simulations of a screw dislocation in bcc molybdenum to determine the dependence on orientation of the maximum resolved shear stress (in the direction of the Burgers vector) required to move the dislocation. A yield criterion that enters a continuum theory of bcc crystal plasticity and includes non-glide components of stress is developed along the lines of a general framework that was proposed several years ago. The predicted results are in excellent agreement with the atomistic simulations.