van der Erik Giessen

Discrete dislocation plasticity: a simple planar model

A method for solving small-strain plasticity problems with plastic flow represented by the collective motion of a large number of discrete dislocations is presented. The dislocations are modelled as line defects in a linear elastic medium. At each instant, superposition is used to represent the solution in terms of the infinite-medium solution for the discrete dislocations and a complementary solution that enforces the boundary conditions on the finite body. The complementary solution is nonsingular and is obtained from a finite-element solution of a linear elastic boundary value problem. The lattice resistance to dislocation motion, dislocation nucleation and annihilation are incorporated into the formulation through a set of constitutive rules. Obstacles leading to possible dislocation pile-ups are also accounted for. The deformation history is calculated in a linear incremental manner. Plane-strain boundary value problems are solved for a solid having edge dislocations on parallel slip planes. Monophase and composite materials subject to simple shear parallel to the slip plane are analysed. Typically, a peak in the shear stress versus shear strain curve is found, after which the stress falls to a plateau at which the material deforms steadily. The plateau is associated with the localization of dislocation activity on more or less isolated systems. The results for composite materials are compared with solutions for a phenomenological continuum slip characterization of plastic flow.

Alternative Explanation of Stiffening in Cross-Linked Semiflexible Networks

Strain stiffening of filamentous protein networks is explored by means of a finite strain analysis of a two-dimensional network model of cross-linked semiflexible filaments. The results show that stiffening is caused by nonaffine network rearrangements that govern a transition from a bending-dominated response at small strains to a stretching-dominated response at large strains. Filament undulations, which are key in the existing explanation of stiffening, merely postpone the transition.

Comparison of discrete dislocation and continuum plasticity predictions for a composite material

A two-dimensional model composite with elastic reinforcements in a crystalline matrix subject to macroscopic shear is analyzed using both discrete dislocation plasticity and conventional continuum slip crystal plasticity. In the discrete dislocation formulation, the dislocations are modeled as line defects in a linear elastic medium. At each stage of loading, superposition is used to represent the solution in terms of the infinite medium solution for the discrete dislocations and a complimentary solution that enforces the boundary conditions, which is non-singular and obtained from a linear elastic, finite element solution. The lattice resistance to dislocation motion, dislocation nucleation, and dislocation annihilation are incorporated into the formulation through a set of constitutive rules. Obstacles leading to possible dislocation pile-ups are also accounted for. Results are presented for materials with a single slip system. A reinforcement size effect is exhibited by the discrete dislocation-based analysis whereas the continuum slip results are size independent. The discrete dislocation results have higher average reinforcement stress levels than do the corresponding continuum slip calculations. Averaging of stress fields over windows of increasing size is used to gain insight into the transition from discrete dislocation-controlled to continuum behavior.

A comparison of nonlocal continuum and discrete dislocation plasticity predictions

Abstract Discrete dislocation simulations of two boundary value problems are used as numerical experiments to explore the extent to which the nonlocal crystal plasticity theory of Gurtin (J. Mech. Phys. Solids 50 (2002) 5) can reproduce their predictions. In one problem simple shear of a constrained strip is analyzed, while the other problem concerns a two-dimensional model composite with elastic reinforcements in a crystalline matrix subject to macroscopic shear. In the constrained layer problem, boundary layers develop that give rise to size effects. In the composite problem, the discrete dislocation solutions exhibit composite hardening that depends on the reinforcement morphology, a size dependence of the overall stress–strain response for some morphologies, and a strong Bauschinger effect on unloading. In neither problem are the qualitative features of the discrete dislocation results represented by conventional continuum crystal plasticity. The nonlocal plasticity calculations here reproduce the behavior seen in the discrete dislocation simulations in remarkable detail.

Boundary layers in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity

Simple shear of a constrained strip is analyzed using discrete dislocation plasticity and strain gradient crystal plasticity theory. Both single slip and symmetric double slip are considered. The loading is such that for a local continuum description of plastic flow the deformation state is one of homogeneous shear. In the discrete dislocation formulation the dislocations are all of edge character and are modeled as line singularities in an elastic material. Dislocation nucleation, the lattice resistance to dislocation motion and dislocation annihilation are incorporated into the formulation through a set of constitutive rules. A complementary solution that enforces the boundary conditions is obtained via the finite element method. The discrete dislocation solutions give rise to boundary layers in the deformation field and in the dislocation distributions. The back-extrapolated flow strength for symmetric double slip increases with decreasing strip thickness, so that a size effect is observed. The strain gradient plasticity theory used here is also found to predict a boundary layer and a size effect. Nonlocal material parameters can be chosen to fit some, but not all, of the features of the discrete dislocation results. Additional physical insight into the slip distribution across the strip is provided by simple models for an array of mode II cracks.

Aspects of boundary-value problem solutions with three-dimensional dislocation dynamics

A three-dimensional discrete dislocation dynamics plasticity model is presented. The approach allows realistic boundary conditions on the specimen, as both stress and displacement fields of the dislocations are incorporated in the formulation. Emphasis is placed on various technical details in the formulation as well as on the implementation. The current implementation includes features necessary to model conservative motion of dislocations in presence of surfaces. These include details of the discretization of the evolving dislocation structure, the handling of junction formation and destruction, cross-slip and boundary conditions. Special attention is given to the treatment of dislocations that partly glide out of the material, including the treatment of image forces via the finite-element method.

A discrete dislocation analysis of bending

Bending of a strip in plane strain is analyzed using discrete dislocation plasticity where the dislocations are modeled as line defects in a linear elastic medium. At each stage of loading, superposition is used to represent the solution in terms of the infinite medium solution for the discretedislocationsandacomplementarysolutionthatenforcestheboundaryconditions,which isnon-singularandobtainedfromalinearelastic,finiteelementsolution.Thelatticeresistanceto dislocationmotion, dislocationnucleationanddislocationannihilation areincorporatedintothe formulation through a set of constitutive rules. Solutions for cases with multiple slip systems and with a single slip system are presented. The bending moment versus rotation relation and the evolution of the dislocation structure are outcomes of the boundary value problem solution. The eAects of slip geometry, obstacles to dislocation motion and specimen size on the moment versus rotation response are considered. Also, the evolution of the dislocation structure is studied with emphasis on the role of geometrically necessary dislocations. The dislocation structure that develops leads to well-defined slip bands, with the slip band spacing scaling with the specimen height. # 1999 Published by Elsevier Science Ltd. All rights reserved.

A discrete dislocation analysis of mode I crack growth

Small scale yielding around a plane strain mode I crack is analyzed using discrete dislocation dynamics. The dislocations are all of edge character, and are modeled as line singularities in an elastic material. At each stage of loading, superposition is used to represent the solution in terms of solutions for edge dislocations in a half-space and a complementary solution that enforces the boundary conditions. The latter is non-singular and obtained from a finite element solution. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation are incorporated into the formulation through a set of constitutive rules. A relation between the opening traction and the displacement jump across a cohesive surface ahead of the initial crack tip is also specified, so that crack growth emerges naturally from the boundary value problem solution. Material parameters representative of aluminum are employed. For a low density of dislocation sources, crack growth takes place in a brittle manner; for a low density of obstacles, the crack blunts continuously and does not grow. In the intermediate regime, the average near-tip stress fields are in qualitative accord with those predicted by classical continuum crystal plasticity, but with the local stress concentrations from discrete dislocations leading to opening stresses of the magnitude of the cohesive strength. The crack growth history is strongly affected by the dislocation activity in the vicinity of the growing crack tip.

On the linear elastic properties of regular and random open-cell foam models

Foams can be created from coagulation of gas bubbles in liquid. After removal of cell faces, an open-cell foam remains consisting of a strut framework. In the past, mechanical properties were estimated by a small unit cell consisting of only a few struts. However, the random geometry of the foam can be of importance for the linear elastic properties. Here, large foam unit cells are created using Voronoi techniques. A smooth transition from regular to random geometries is made, showing the strong sensitivity of the mechanical properties from the geometry of the microstructure. Uniaxial global loads are transmitted through chains of highly loaded struts. The deformation of the struts in the foam is a mixture of bending and normal deformation, the ratio of which shown here to be dependent on the magnitude of the density.

Discrete dislocation modeling of fatigue crack propagation

Analyses of the growth of a plane strain crack subject to remote mode I cyclic loading under small-scale yielding are carried out using discrete dislocation dynamics. Cracks along a metal–rigid substrate interface and in a single crystal are studied. The formulation is the same as that used to analyze crack growth under monotonic loading conditions, differing only in the remote stress intensity factor being a cyclic function of time. Plastic deformation is modeled through the motion of edge dislocations in an elastic solid with the lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation being incorporated through a set of constitutive rules. An irreversible relation is specified between the opening traction and the displacement jump across a cohesive surface ahead of the initial crack tip in order to simulate cyclic loading in an oxidizing environment. The cyclic crack growth rate log(da/dN) versus applied stress intensity factor range log(KI) curve that emerges naturally from the solution of the boundary value problem shows distinct threshold and Paris law regimes. Paris law exponents in the range 4 to 8 are obtained for the parameters employed here. Furthermore, rather uniformly spaced slip bands corresponding to surface striations develop in the wakes of the propagating cracks.  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.