A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations

We propose a deformation theory of strain gradient crystal plasticity that accounts for the density of geometrically necessary dislocations by including, as an independent kinematic variable, Nye's dislocation density tensor [1953. Acta Metallurgica 1, 153-162]. This is accomplished in the same fashion as proposed by Gurtin and co-workers (see, for instance, Gurtin and Needleman [2005. J. Mech. Phys. Solids 53, 1-31]) in the context of a flow theory of crystal plasticity, by introducing the so-called defect energy. Moreover, in order to better describe the strengthening accompanied by diminishing size, we propose that the classical part of the plastic potential may be dependent on both the plastic slip vector and its gradient; for single crystals, this also makes it easier to deal with the “higher-order” boundary conditions. We develop both the kinematic formulation and its static dual and apply the theory to the simple shear of a constrained strip (example already exploited in Shu et al. [2001. J. Mech. Phys. Solids 49, 1361-1395], Bittencourt et al. [2003. J. Mech. Phys. Solids 51, 281-310], Niordson and Hutchinson [2003. Euro J. Mech. Phys. Solids 22, 771-778], Evers et al. [2004. J. Mech. Phys. Solids 52, 2379-2401], and Anand et al. [2005. J. Mech. Phys. Solids 53, 1789-1826]) to investigate what sort of behaviour the new model predicts. The availability of the total potential energy functional and its static dual allows us to easily solve this simple boundary value problem by resorting to the Ritz method.     

A finite-deformation,gradient theory of single-crystal plasticity with free energy dependent on densities of geometrically necessary dislocations

This paper develops a finite-deformation, gradient theory of single crystal plasticity. The theory is based on a system of microscopic force balances, one balance for each slip system, derived from the principle of virtual power, and a mechanical version of the second law that includes, via the microscopic forces, work performed during plastic flow. When combined with thermodynamically consistent constitutive relations the microscopic force balances become flow rules for the individual slip systems. Because these flow rules are in the form of partial differential equations requiring boundary conditions, they are nonlocal. The chief new ingredient in the theory is a free energy dependent on (geometrically necessary) edge and screw dislocation-densities as introduced in Gurtin [Gurtin, 2006. The Burgers vector and the flow of screw and edge dislocations in finite-deformation plasticity. Journal of Mechanics and Physics of Solids 54, 1882].     

A finite-deformation,gradient theory of single-crystal plasticity with free energy dependent on the accumulation of geometrically necessary dislocations

This paper develops a gradient theory of single-crystal plasticity based on a system of microscopic force balances, one balance for each slip system, derived from the principle of virtual power, and a mechanical version of the second law that includes, via the microscopic forces, work performed during plastic flow. When combined with thermodynamically consistent constitutive relations the microscopic force balances become nonlocal flow rules for the individual slip systems in the form of partial differential equations requiring boundary conditions. Central ingredients in the theory are geometrically necessary edge and screw dislocations together with a free energy that accounts for work hardening through a dependence on the accumulation of geometrically necessary dislocations.     

Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector

This paper discusses boundary conditions appropriate to a theory of single-crystal plasticity (Gurtin, J. Mech. Phys. Solids 50 (2002) 5) that includes an accounting for the Burgers vector through energetic and dissipative dependences on the tensor G=curlH^p, with H^p the plastic part in the additive decomposition of the displacement gradient into elastic and plastic parts. This theory results in a flow rule in the form of N coupled second-order partial differential equations for the slip-rates , and, consequently, requires higher-order boundary conditions. Motivated by the virtual-power principle in which the external power contains a boundary-integral linear in the slip-rates, hard-slip conditions in which

(A)

on a subsurface S_hard of the boundary

for all slip systems α are proposed. In this paper we develop a theory that is consistent with that of (Gurtin, 2002), but that leads to an external power containing a boundary-integral linear in the tensor , a result that motivates replacing (A) with the microhard condition

(B)

on the subsurface S_hard.

We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across S_hard.What is most important, we establish uniqueness for the underlying initial/boundary-value problem associated with (B); since the conditions (A) are generally stronger than the conditions (B), this result indicates lack of existence for problems based on (A). For that reason, the hard-slip conditions (A) would seem inappropriate as boundary conditions.Finally, we discuss conditions at a grain boundary based on the flow of the Burgers vector at and across the boundary surface.     

Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations

In this study we develop a gradient theory of small-deformation single-crystal plasticity that accounts for geometrically necessary dislocations (GNDs). The resulting framework is used to discuss grain boundaries. The grains are allowed to slip along the interface, but growth phenomenona and phase transitions are neglected. The bulk theory is based on the introduction of a microforce balance for each slip system and includes a defect energy depending on a suitable measure of GNDs. The microforce balances are shown to be equivalent to nonlocal yield conditions for the individual slip systems, yield conditions that feature backstresses resulting from energy stored in dislocations. When applied to a grain boundary the theory leads to concomitant yield conditions: relative slip of the grains is activated when the shear stress reaches a suitable threshold; plastic slip in bulk at the grain boundary is activated only when the local density of GNDs reaches an assigned threshold. Consequently, in the initial stages of plastic deformation the grain boundary acts as a barrier to plastic slip, while in later stages the interface acts as a source or sink for dislocations. We obtain an exact solution for a simple problem in plane strain involving a semi-infinite compressed specimen that abuts a rigid material. We view this problem as an approximation to a situation involving a grain boundary between a grain with slip systems aligned for easy flow and a grain whose slip system alignment severely inhibits flow. The solution exhibits large slip gradients within a thin layer at the grain boundary.     

Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation

A strain gradient dependent crystal plasticity approach is used to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. Material points are considered as aggregates of grains, subdivided into several fictitious grain fractions: a single crystal volume element stands for the grain interior whereas grain boundaries are represented by bi-crystal volume elements, each having the crystallographic lattice orientations of its adjacent crystals. A relaxed Taylor-like interaction law is used for the transition from the local to the global scale. It is relaxed with respect to the bi-crystals, providing compatibility and stress equilibrium at their internal interface. During loading, the bi-crystal boundaries deform dissimilar to the associated grain interior. Arising from this heterogeneity, a geometrically necessary dislocation (GND) density can be computed, which is required to restore compatibility of the crystallographic lattice. This effect provides a physically based method to account for the additional hardening as introduced by the GNDs, the magnitude of which is related to the grain size. Hence, a scale-dependent response is obtained, for which the numerical simulations predict a mechanical behaviour corresponding to the Hall-Petch effect. Compared to a full-scale finite element model reported in the literature, the present polycrystalline crystal plasticity model is of equal quality yet much more efficient from a computational point of view for simulating uniaxial tension experiments with various grain sizes.     

X-ray microdiffraction and strain gradient crystal plasticity studies of geometrically necessary dislocations near a Ni bicrystal grain boundary

We compare experimental measurements of inhomogeneous plastic deformation in a Ni bicrystal with crystal plasticity simulations. Polychromatic X-ray microdiffraction, orientation imaging microscopy and scanning electron microscopy, were used to characterize the geometrically necessary dislocation distribution of the bicrystal after uniaxial tensile deformation. Changes in the local crystallographic orientations within the sample reflect its plastic response during the tensile test. Elastic strain in both grains increases near the grain boundary. Finite element simulations were used to understand the influence of initial grain orientation and structural inhomogeneities on the geometrically necessary dislocations arrangement and distribution and to understand the underlying materials physics.     

Dislocation pile-ups in bicrystals within continuum dislocation theory

Within continuum dislocation theory the plastic deformation of bicrystals under a mixed deformation of plane constrained uniaxial extension and shear is investigated with regard to the nucleation of dislocations and the dislocation pile-up near the phase boundaries of a model bicrystal with one active slip system within each single crystal. For plane uniaxial extension, we present a closed-form analytical solution for the evolution of the plastic distortion and of the dislocation network in the case of symmetric slip planes (i.e. for twins), which exhibits an energetic as well as a dissipative threshold for the dislocation nucleation. The general solution for non-symmetric slip systems is obtained numerically. For a combined deformation of extension and shear, we analyze the possibility of linearly superposing results obtained for both loading cases independently. All solutions presented in this paper also display the Bauschinger effect of translational work hardening and a size effect typical to problems of crystal plasticity.     

A continuum model for initiation and evolution of deformation twinning

Within continuum dislocation theory the plastic deformation of a single crystal with one active slip system under plane-strain constrained shear is investigated. By introducing a twinning shear into the energy of the crystal, we show that in a certain range of straining the formation of deformation twins becomes energetically preferable. An energetic threshold for the onset of twinning is determined. A rough analysis qualitatively describes not only the evolving volume fractions of twins but also their number during straining. Finally, we analyze the evolution of deformation twins and of the dislocation network at non-zero dissipation. We present the corresponding stress-strain hysteresis, the evolution of the plastic distortion, the twin volume fractions and the dislocation densities.     

A gradient theory of small-deformation, single-crystal plasticity that accounts for GND-induced interactions between slip systems

This paper develops a gradient theory of single-crystal plasticity based on a system of microscopic force balances, one balance for each slip system, derived from the principle of virtual power, and a mechanical version of the second law that includes, via the microscopic forces, work performed during plastic flow. When combined with thermodynamically consistent constitutive relations the microscopic force balances become nonlocal flow rules for the individual slip systems in the form of partial differential equations requiring boundary conditions. Central ingredients in the theory are densities of (geometrically necessary) edge and screw dislocations, densities that describe the accumulation of dislocations, and densities that characterize forest hardening. The form of the forest densities is based on an explicit kinematical expression for the normal Burgers vector on a slip plane.     

Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations

The higher-order stress work-conjugate to slip gradient in single crystals at small strains is derived based on the self-energy of geometrically necessary dislocations (GNDs). It is shown that this higher-order stress changes stepwise as a function of in-plane slip gradient and therefore significantly influences the onset of initial yielding in polycrystals. The higher-order stress based on the self-energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5-32] and applied to single-slip-oriented 2D and 3D model crystal grains of size D. It is thus found that the self-energy of GNDs gives a D^-1-dependent term for the averaged resolved shear stress in such a model grain under yielding. Using published experimental data for several polycrystalline metals, it is demonstrated that the D^-1-dependent term successfully explains the grain size dependence of initial yield stress and the dislocation cell size dependence of flow stress in the submicron to several-micron range of grain and cell sizes.     

A Finite Element approach with patch projection for strain gradient plasticity formulations

Several strain gradient plasticity formulations have been suggested in the literature to account for inherent size effects on length scales of microns and submicrons. The necessity of strain gradient related terms render the simulation with strain gradient plasticity formulation computationally very expensive because quadratic shape functions or mixed approaches in displacements and strains are usually applied. Approaches using linear shape functions have also been suggested which are, however, limited to regular meshes with equidistanced Finite Element nodes. As a result the majority of the simulations in the literature deal with plane problems at small strains. For the solution of general three dimensional problems at large strains an approach has to be found which has to be computationally affordable and robust.     

Plane constrained uniaxial extension of a single crystal strip

Within continuum dislocation theory the plane constrained uniaxial extension of a single crystal strip deforming in single or double slip is analyzed. For the single and symmetric double slip, the closed-form analytical solutions are found which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. Numerical solutions for the non-symmetric double slip are obtained by finite element procedures.     

The Burgers vector and the flow of screw and edge dislocations in finite-deformation single-crystal plasticity

We consider finite plasticity based on the decomposition F=F^eF^p of the deformation gradient F into elastic and plastic distortions F^e and F^p. Within this framework the macroscopic Burgers vector may be characterized by the tensor field . We derive a natural convected rate for G associated with evolution of F^p and as our main result show that, for a single-crystal,

•

temporal changes in G—as characterized by its convected time derivative—may be decomposed into temporal changes in distributions of screw and edge dislocations on the individual slip systems.

We discuss defect energies dependent on the densities of these distributions and show that corresponding thermodynamic forces are macroscopic counterparts of classical Peach-Koehler forces.     

Analysis of deformation localization based on proposed theory of ultrasonic wave velocity propagating in plastically deformed solids

The localization of plastic deformation is discussed as “stationary discontinuity” characterized by a vanishing velocity of an acceleration wave derived using the author’s proposed theory of ultrasonic wave velocities propagating in plastically deformed solids. To formulate the proposed theory, the elasto-plastic coupling effect was introduced to consider the elastic stiffness degradation due to the plastic deformation. The driving force of the deformation localization is caused by the yield vertex effect, which introduces a pronounced softening of the shear modulus, and geometrical softening due to double slip caused by lattice rotations. In the present paper, it is examined theoretically and experimentally that the diagonal terms of the introduced elasto-plastic coupling tensor represent a slight hardening followed by a pronounced softening of the elastic modulus induced by the point defect development caused by cross slides among dislocations at multiple slip stages similar to the yield vertex effects. The off-diagonal terms represent geometrical softening induced by lattice rotations such as texture evolution. Then, based on the coincidence of the onset strains between localization and acceleration waves of vanishing velocity, the diagrams of diffuse necking, localized necking and forming limit are analyzed by applying the proposed acoustic tensor, which is based on the generalized Christoffel tensor derived by the author, and solving cut off conditions of the quasi-longitudinal wave to determine the onset strains of deformation localization and localization modes. As a result, diagrams of diffuse necking, localized necking and forming limit were obtained. Moreover, the localization modes were determined and distinguished as the SH-mode, SV-mode, tearing mode and splitting mode.     

Effect of defect energy on strain-gradient predictions of confined single-crystal plasticity

Gurtin recently proposed a strain-gradient theory for crystal plasticity in which the gradient effect originates from a defect energy that characterizes energy storage due to the presence of a net Burgers vector. Here we consider a number of different possibilities for this energy: specifically, working within a simple two-dimensional framework, we compare predictions of the theory with results of discrete-dislocation simulations of stress relaxation in thin films. Our objective is to investigate which specific defect energies are capable of capturing the size-dependent response of such systems for different crystal orientations.     

A three-dimensional crystal plasticity model for duplex Ti–6Al–4V

A rate dependent crystal plasticity model for the α/β Ti–Al alloy Ti–6Al–4V with duplex microstructure is developed and presented herein. Duplex Ti–6Al–4V is a dual-phase alloy consisting of an hcp structured matrix primary α-phase and secondary lamellar α + β domains that are composed of alternating layers of secondary α laths and bcc structured residual β laths. The model accounts for distinct three-dimensional slip geometry for each phase, anisotropic and length scale dependent slip system strengths, the non-planar dislocation core structure of prismatic screw dislocations in the primary α-phase, and crystallographic texture. The model is implemented in the general purpose finite element code (ABAQUS, 2005. Ver 6.5, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI) via a UMAT subroutine.     

A finite deformation theory of higher-order gradient crystal plasticity

For higher-order gradient crystal plasticity, a finite deformation formulation is presented. The theory does not deviate much from the conventional crystal plasticity theory. Only a back stress effect and additional differential equations for evolution of the geometrically necessary dislocation (GND) densities supplement the conventional theory within a non-work-conjugate framework in which there is no need to introduce higher-order microscopic stresses that would be work-conjugate to slip rate gradients. We discuss its connection to a work-conjugate type of finite deformation gradient crystal plasticity that is based on an assumption of the existence of higher-order stresses. Furthermore, a boundary-value problem for simple shear of a constrained thin strip is studied numerically, and some characteristic features of finite deformation are demonstrated through a comparison to a solution for the small deformation theory. As in a previous formulation for small deformation, the present formulation applies to the context of multiple and three-dimensional slip deformations.