Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics

We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold—where the body is stress free—is a Weitzenb?ck manifold, that is, a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan’s moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields, assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance.     

Dislocations and disclinations: continuously distributed defects in elasto-plastic crystalline materials

The paper deals with elasto-plastic models for crystalline materials with defects, dislocations coupled with disclinations. The behaviour of the material is described with respect to an anholonomic configuration, endowed with a non-Riemannian geometric structure. The geometry of the material structure is generated by the plastic distortion, which is an incompatible second-order tensor, and by the so-called plastic connection which is metric compatible, with respect to the metric tensor associated with the plastic distortion. The free energy function is dependent on the second-order elastic deformation and on the state of defects. The tensorial measure of the defects is considered to be the Cartan torsion of the plastic connection and the disclination tensor. When we restrict to small elastic and plastic distortions, the measures of the incompatibility as well as the dislocation densities reduced to the classical ones in the linear elasticity. The constitutive equations for macroforces and the evolution equations for the plastic distortion and disclination tensor are provided to be compatible with the free energy imbalance principle.     

Recasting theory of elasticity with micro-finite elements

In the classical theory of elasticity,a body is initially modeled as a homogeneous and dense assemblage of constituent "material particles".The kernel concept of elastic deformation is the displacement of the particle that associates the current configuration with the reference one.In this paper,we exploit an alternative constituent "micro-finite element",and use the stretch of the element as the essential quality to recast the theory of elasticity.It should be realized that such a treatment means that the elastic body can be modeled as a finite covering of elements and consequently characterized by a manifold.The recasting of the elasticity theory becomes more feasible for dealing with defects and topological evolution.     

A three dimensional field formulation,and isogeometric solutions to point and line defects using Toupin's theory of gradient elasticity at finite strains

We present a field formulation for defects that draws from the classical representation of the cores as force dipoles. We write these dipoles as singular distributions. Exploiting the key insight that the variational setting is the only appropriate one for the theory of distributions, we arrive at universally applicable weak forms for defects in nonlinear elasticity. Remarkably, the standard, Galerkin finite element method yields numerical solutions for the elastic fields of defects that, when parameterized suitably, match very well with classical, linearized elasticity solutions. The true potential of our approach, however, lies in its easy extension to generate solutions to elastic fields of defects in the regime of nonlinear elasticity, and even more notably for Toupin's theory of gradient elasticity at finite strains (Toupin Arch. Ration. Mech. Anal., 11 (1962) 385). In computing these solutions we adopt recent numerical work on an isogeometric analytic framework that enabled the first three-dimensional solutions to general boundary value problems of Toupin's theory (Rudraraju et al. Comput. Methods Appl. Mech. Eng., 278 (2014) 705). We first present exhaustive solutions to point defects, edge and screw dislocations, and a study on the energetics of interacting dislocations. Then, to demonstrate the generality and potential of our treatment, we apply it to other complex dislocation configurations, including loops and low-angle grain boundaries.     

A crystal plasticity model that incorporates stresses and strains due to slip gradients

This work is concerned with incorporating the kinematic and stress effects of excess dislocations in a constitutive model for the elastoplastic behavior of crystalline materials. The foundation of the model is a three term multiplicative decomposition of the deformation gradient in which the two classical terms of plastic and elastic deformation are included along with an additional term for long range strain due to the collective effects of excess dislocations. The long range strain is obtained from an assumed density of Volterra edge dislocations and is directly related to gradients in slip. A new material parameter emerges which is the size the region about a continuum point that contributes to long range strains.Using Hookean elasticity, the stress at a point is linearly related to the sum of the elastic plus the long range strain fields. However, the driving force for slip is postulated to be due only to the elastic stress so that the long range stress is a back stress in the constitutive relationship for plastic deformation. A consistent balance of the total deformation rate with the three proposed mechanisms of deformation leads to a set of differential equations that can be solved for the elastic stress, rotation and pressure which then implicitly defines the material state and equilibrium stress. Results from the simulation of a tapered tensile specimen demonstrate that the constitutive model exhibits isotropic and kinematic type hardening effects as well as changes in the pattern of plastic deformation and necking when compared to a material without slip gradient effects.     

Defects in flexoelectric solids

A solid is said to be flexoelectric when it polarizes in proportion to strain gradients. Since strain gradients are large near defects, we expect the flexoelectric effect to be prominent there and decay away at distances much larger than a flexoelectric length scale. Here, we quantify this expectation by computing displacement, stress and polarization fields near defects in flexoelectric solids. For point defects we recover some well known results from strain gradient elasticity and non-local piezoelectric theories, but with different length scales in the final expressions. For edge dislocations we show that the electric potential is a maximum in the vicinity of the dislocation core. We also estimate the polarized line charge density of an edge dislocation in an isotropic flexoelectric solid which is in agreement with some measurements in ice. We perform an asymptotic analysis of the crack tip fields in flexoelectric solids and show that our results share some features from solutions in strain gradient elasticity and piezoelectricity. We also compute the energy release rate for cracks using simple crack face boundary conditions and use them in classical criteria for crack growth to make predictions. Our analysis can serve as a starting point for more sophisticated analytic and computational treatments of defects in flexoelectric solids which are gaining increasing prominence in the field of nanoscience and nanotechnology.     

Discrete Crystal Elasticity and Discrete Dislocations in Crystals

This article is concerned with the development of a discrete theory of crystal elasticity and dislocations in crystals. The theory is founded upon suitable adaptations to crystal lattices of elements of algebraic topology and differential calculus such as chain complexes and homology groups, differential forms and operators, and a theory of integration of forms. In particular, we define the lattice complex of a number of commonly encountered lattices, including body-centered cubic and face-centered cubic lattices. We show that material frame indifference naturally leads to discrete notions of stress and strain in lattices. Lattice defects such as dislocations are introduced by means of locally lattice-invariant (but globally incompatible) eigendeformations. The geometrical framework affords discrete analogs of fundamental objects and relations of the theory of linear elastic dislocations, such as the dislocation density tensor, the equation of conservation of Burgers vector, Kröner's relation and Mura's formula for the stored energy. We additionally supply conditions for the existence of equilibrium displacement fields; we show that linear elasticity is recovered as the Γ-limit of harmonic lattice statics as the lattice parameter becomes vanishingly small; we compute the Γ-limit of dilute dislocation distributions of dislocations; and we show that the theory of continuously distributed linear elastic dislocations is recovered as the Γ-limit of the stored energy as the lattice parameter and Burgers vectors become vanishingly small.     

Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of the equations of motion, viewing motion as a curve in an infinite-dimensional Banach manifold of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies, where the residual stresses result from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two-dimensional curved annulus embedded in a sphere.     

变形体非协调理论的理性理论

本文介绍Noll和Wang等人用理性力学观点建立的变形体非协调理论(即缺陷的连续统理论)。这理论表明,本构方程完全确定物质流形的几何结构。因而,几何结构是理论的自然结果,而不是作为理论出发点的最初假设。      

Qualitative Properties of Local Random Invariant Manifolds for SPDEs with Quadratic Nonlinearity

The qualitative properties of local random invariant manifolds for stochastic partial differential equations with quadratic nonlinearities and multiplicative noise is studied by a cut off technique. By a detail estimates on the Perron fixed point equation describing the local random invariant manifold, the structure near a bifurcation is given.     

Elastic precursor decay and radiation from nonuniformly moving dislocations

Precursor decay in plate impact experiments on single crystals is re-examined from the viewpoint of the elastodynamics of moving dislocations. Superposition of solutions for many dislocations set in motion by an incident plane wave is used to relate the decay of the wave amplitude at the front of the plane wave to the density and velocity of dislocations at the wavefront. The resulting precursor decay relation is the same as the one derived from an elastic/visco-plastic model of the material, except for a small correction due to differences between the effects of forward and backward propagating dislocations. Motivated by this added support for the validity of the precursor decay equation, the values used for the quantities in this equation are re-examined. Recent experimental results and the elastodynamics analysis are interpreted as indicating that the commonly-used values of dislocation velocity are probably satisfactory, but that the values used for dislocation density are several orders of magnitude too small near the lapped surfaces of the crystal. These large dislocation densities are identified as the probable dominant cause of the lower-than-predicted precursor amplitudes that are recorded in experiments. More accurate experimental data and inclusion of the non-linear elasticity effects are essential in determining whether or not the observed precursor decay in the bulk of the specimen can be explained by the motion of dislocations present initially. Calculations of energy radiated from screw and edge dislocations that start from rest and move thereafter at constant velocity confirm that dislocation drag forces due to continuum elasticity effects are small for dislocation velocities less than, say, 80% of the elastic shear wave speed. At supersonic speeds the continuum drag effects become so large that sustained supersonic motion of dislocations appears unlikely.     

Singularity-free dislocation dynamics with strain gradient elasticity

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.     

Structure of kinematic and force fields in the Riemannian continuum model

In this paper, we consider a non-Euclidean continuum model for which the structure of defects in the material is characterized by an internal metric and scalar curvature. It is shown that the irrotational displacement field for points of this medium is composed of elastic displacements (in the absence of defects) and the field which characterizes the deviation of the internal geometry of the model from Euclidean geometry. The corresponding components of the internal stresses are the sum of elastic stresses and the self-equilibrated stresses determined by the scalar curvature. The exact solution for the vortex field of dislocations is constructed, and conditions of the existence of a nonzero stress field parametrized by a scalar curvature in the absence of external forces are formulated.     

Sh surface Waves in a Homogeneous Gradient-Elastic Half-Space with Surface Energy

The existence of SH surface waves in a half-space homogeneous material (i.e. anti-plane shear wave motions which decay exponentially with the distance from the free surface) is shown to be possible within the framework of the generalized linear continuum theory of gradient elasticity with surface energy. As is well-known such waves cannot be predicted by the classical theory of linear elasticity for a homogeneous half-space, although there is experimental evidence supporting their existence. Indeed, this is a drawback of the classical theory which is only circumvented by modelling the half-space as a layered structure (Love waves) or as having non-homogeneous material properties. On the contrary, the present study reveals that SH surface waves may exist in a homogeneous half-space if the problem is analyzed by a continuum theory with appropriate microstructure. This theory, which was recently introduced by Vardoulakis and co-workers, assumes a strain-energy density expression containing, besides the classical terms, volume strain-gradient and surface-energy gradient terms. This revised version was published online in August 2006 with corrections to the Cover Date.     

Geometry of constraint manifolds and wave propagation in internally constrained elastic bodies

The implications of the non-Euclidean structure of constraint manifolds on differentiation of the stress in internally constrained elastic bodies are examined, and the equations governing propagation of acceleration waves in such bodies are deduced differentiating the reactive stress consistently with the assumption that it does no work in any admissible motion. This yields a treatment of the subject in which the presence of internal constraints imposes restrictions on the set of possible amplitudes of waves but the condition for local existence of waves, that amplitudes must satisfy, is of the same type as that for bodies free from internal constraints, in the sense that it depends on the properties of the response map of the material and is independent of reactive stress.     

Problems of the Flamant–Boussinesq and Kelvin Type in Dipolar Gradient Elasticity

This work studies the response of bodies governed by dipolar gradient elasticity to concentrated loads. Two-dimensional configurations in the form of either a half-space (Flamant–Boussinesq type problem) or a full-space (Kelvin type problem) are treated and the concentrated loads are taken as line forces. Our main concern is to determine possible deviations from the predictions of plane-strain/plane-stress classical linear elastostatics when a more refined theory is employed to attack the problems. Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities and discontinuities exist in the classical solutions. The use of the theory of gradient elasticity is intended here to model material microstructure and incorporate size effects into stress analysis in a manner that the classical theory cannot afford. A simple but yet rigorous version of the generalized elasticity theories of Toupin (Arch. Ration. Mech. Anal. 11:385–414, 1962) and Mindlin (Arch. Ration. Mech. Anal. 16:51–78, 1964) is employed that involves an isotropic linear response and only one material constant (the so-called gradient coefficient) additional to the standard Lamé constants (Georgiadis et al., J. Elast. 74:17–45, 2004). This theory, which can be viewed as a first-step extension of the classical elasticity theory, assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients. The solution method is based on integral transforms and is exact. The present results show departure from the ones of the classical elasticity solutions (Flamant–Boussinesq and Kelvin plane-strain solutions). Indeed, continuous and bounded displacements are predicted at the points of application of the loads. Such a behavior of the displacement fields is, of course, more natural than the singular behavior present in the classical solutions.      

An approach based on distributed dislocations and disclinations for crack problems in couple-stress elasticity

The technique of distributed dislocations proved to be in the past an effective approach in studying crack problems within classical elasticity. The present work is intended to extend this technique in studying crack problems within couple-stress elasticity, i.e. within a theory accounting for effects of microstructure. This extension is not an obvious one since rotations and couple-stresses are involved in the theory employed to analyze the crack problems. Here, the technique is introduced to study the case of a mode I crack. Due to the nature of the boundary conditions that arise in couple-stress elasticity, the crack is modeled by a continuous distribution of climb dislocations and constrained wedge disclinations (the concept of ‘constrained wedge disclination’ is first introduced in the present work). These distributions create both standard stresses and couple stresses in the body. In particular, it is shown that the mode-I case is governed by a system of coupled singular integral equations with both Cauchy-type and logarithmic kernels. The numerical solution of this system shows that a cracked solid governed by couple-stress elasticity behaves in a more rigid way (having increased stiffness) as compared to a solid governed by classical elasticity. Also, the stress level at the crack-tip region is appreciably higher than the one predicted by classical elasticity.