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

A displacement-based formulation for interaction problems between cracks and dislocation dipoles in couple-stress elasticity

Interaction problems of a finite-length crack with plane and antiplane dislocation dipoles in the context of couple-stress elasticity are presented in this study. The analysis is based on the distributed dislocation technique where infinitesimal dislocation dipoles are used as strain nuclei. The stress fields of these area defects are provided for the first time in the framework of couple-stress elasticity theory. In addition, a new rotational defect is introduced to satisfy the boundary conditions of the opening mode problem. This formulation leads to displacement-based hyper-singular integral equations that govern the crack problems, which are solved numerically. It is further shown that this method has several advantages over the slope formulation. Based on the obtained results, it is deduced that in all cases the cracked body behaves in a more rigid way when couple-stresses are considered. The effect of couple-stresses is highlighted in a small zone ahead of the crack-tip and around the dislocation dipole, where the stress level is significantly higher than the classical elasticity prediction. Further, the dependence of the energy release rate and the configurational force exerted on the defect on the characteristic material length and the distance between the defect and the crack-tip is discussed. In the plane problems, couple-stress theory predicts either strengthening or weakening effects while in the antiplane mode a strengthening effect is predicted

On Concentrated Surface Loads and Green's functions in the Toupin-Mindlin theory of Strain-Gradient Elasticity

The two-dimensional Green's functions are derived for the half-plane in the context of the complete Toupin-Mindlin theory of isotropic strain-gradient elasticity. Two types of Green's functions exist for a concentrated force and a concentrated force dipole acting upon the surface of a traction-free half-plane. Our purpose is to examine the possible deviations from the predictions of classical theory of elasticity as well as from the simplified strain-gradient theory, which is frequently utilized in the last decade for the solution of boundary value 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 boundary value problems are attacked with the aid of the Fourier transform and exact full-field solutions are provided. Our results indicate that in all cases the displacement field is bounded and continuous at the point of application of the concentrated loads. The new solutions show therefore a more natural material response. For the concentrated force problem, both displacements and strains are found to be bounded, whereas the strain-gradients exhibit a logarithmic singularity. Thus, in marked contrast with the classical elasticity solution, a finite strain energy is contained within any finite portion of the body. On the other hand, in the case of the concentrated dipole force, the strains are logarithmically singular and the strain gradients exhibit a Cauchy type singularity. The nature of the boundary conditions in strain-gradient elasticity is highlighted through the solution of the pertinent boundary value problems. Finally, based on our analytical solution, the role of edge forces in strain-gradient elasticity is elucidated employing simple equilibrium considerations

Couple-stress effects for the problem of a crack under concentrated shear loading

In this paper, we deal with the plane-strain problem of a semi-infinite crack under concentrated loading in an elastic body exhibiting couple-stress effects. The faces of the crack are subjected to a concentrated shear loading at a distance from the crack tip. This type of loading is chosen since, in principle, shear effects are more pronounced in couple-stress elasticity. The problem involves two characteristic lengths, i.e. the microstructural length and the distance between the point of application of the concentrated shear forces and the crack-tip. The presence of this second characteristic length introduces certain difficulties in the mathematical analysis of the problem: a non-standard Wiener-Hopf equation arises, one that contains a forcing term with unbounded behavior at infinity in the transformed plane. Nevertheless, an analytic function method is employed which circumvents the aforementioned difficulty. For comparison purposes, the case of a semi-infinite crack subjected to a distributed shear load is also treated in the present study. Numerical results for the dependence of the stress intensity factor and the energy release rate upon the ratio of the characteristic lengths are presented

Interaction problems between cracks and crystal defects in constrained Cosserat elasticity

In this work interaction problems between a finite-length crack with plane and antiplane crystal defects in the context of couple-stress elasticity are presented. Two alternative yet equivalent approaches for the formulation of crack problems are discussed based on the distributed dislocation technique. To this aim, the stress fields of climb and screw dislocation dipoles are derived within couple-stress theory and new ‘constrained’ rotational defects are introduced to satisfy the boundary conditions of the opening mode problem. Eventually, all interaction problems are described by single or systems of singular integral equations that are solved numerically using appropriate collocation techniques. The obtained results aim to highlight the deviation from classical elasticity solutions and underline the differences in interactions of cracks with single dislocations and dislocation dipoles. In general, it is concluded that the cracked body behaves in a more rigid way when couple-stresses are considered. Also, the stress level is significantly higher than the classical elasticity prediction. Moreover, the configurational forces acting on the defects are evaluated and their dependence on the characteristic material length of couple-stress theory and the distance between the defect and the crack-tip is discussed. This investigation reveals either a strengthening or a weakening effect in the opening mode problem while in the antiplane mode a strengthening effect is always obtained

Interaction of cracks with dislocations in couple-stress elasticity

In the present work we study the interaction of a finite-length crack with a climb dislocation within the framework of the generalized continuum theory of couple-stress elasticity. Our approach is based on the distributed dislocation technique. 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. These distributions produce both standard stresses and couple stresses in the body. The final results are obtained by numerically solving a system of coupled singular integral equations with both Cauchy and logarithmic kernels. The results for the near-tip fields differ in several respects from the predictions of the classical fracture mechanics. In particular, the present results indicate 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, within a small zone adjacent to the tip, than the one predicted by classical elasticity while the crack-face displacements and rotations are significantly smaller that the respective ones in classical elasticity

On the super-Rayleigh/subseismic elastodynamic indentation problem

The elastodynamic super-Rayleigh/subseismic indentation paradox is examined in this paper. Both the Craggs/Roberts steady-state problem and the Robinson/Thompson transient problem are reconsidered. Certain features of these solutions are discussed from a new point of view, by considering asymptotics at the end of the contact region, the influence of contact inequalities, energetics of the process and existence/uniqueness.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42681/1/10659_2004_Article_BF00044967.pd

Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity

The present study aims at determining the elastic stress and displacement fields around the tips of a finite-length crack in a microstructured solid under remotely applied plane-strain loading (mode I and II cases). The material microstructure is modeled through the Toupin-Mindlin generalized continuum theory of dipolar gradient elasticity. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain tensor (as in classical elasticity) and the gradient of the strain tensor (additional term). A simple but yet rigorous version of the theory is employed here by considering an isotropic linear expression of the elastic strain-energy density that involves only three material constants (the two Lamé constants and the so-called gradient coefficient). First, a near-tip asymptotic solution is obtained by the Knein-Williams technique. Then, we attack the complete boundary value problem in an effort to obtain a full-field solution. Hypersingular integral equations with a cubic singularity are formulated with the aid of the Fourier transform. These equations are solved by analytical considerations on Hadamard finite-part integrals and a numerical treatment. The results show significant departure from the predictions of standard fracture mechanics. In view of these results, it seems that the classical theory of elasticity is inadequate to analyze crack problems in microstructured materials. Indeed, the present results indicate that the stress distribution ahead of the crack tip exhibits a local maximum that is bounded

Distributed dislocation approach for cracks in couple-stress elasticity: shear modes

The distributed dislocation technique proved to be in the past an effective approach in studying crack problems within classical elasticity. The present work aims at extending this technique in studying crack problems within couple-stress elasticity, i.e. within a theory accounting for effects of microstructure. As a first step, the technique is introduced to study finite-length cracks under remotely applied shear loadings (mode II and mode III cases). The mode II and mode III cracks are modeled by a continuous distribution of glide and screw dislocations, respectively, that create both standard stresses and couple stresses in the body. In particular, it is shown that the mode II case is governed by a singular integral equation with a more complicated kernel than that in classical elasticity. The numerical solution of this equation shows that a cracked material governed by couple-stress elasticity behaves in a more rigid way (having increased stiffness) as compared to a material governed by classical elasticity. Also, the stress level at the crack-tip region is appreciably higher than the one predicted by classical elasticity. Finally, in the mode III case the corresponding governing integral equation is hypersingular with a cubic singularity. A new mechanical quadrature is introduced here for the numerical solution of this equation. The results in the mode III case for the crack-face displacement and the near-tip stress show significant departure from the predictions of classical fracture mechanics