35 research outputs found

    Stress channelling in extreme couple-stress materials Part II: Localized folding vs faulting of a continuum in single and cross geometries

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    The antiplane strain Green's functions for an applied concentrated force and moment are obtained for Cosserat elastic solids with extreme anisotropy, which can be tailored to bring the material in a state close to an instability threshold such as failure of ellipticity. It is shown that the wave propagation condition (and not ellipticity) governs the behaviour of the antiplane strain Green's functions. These Green's functions are used as perturbing agents to demonstrate in an extreme material the emergence of localized (single and cross) stress channelling and the emergence of antiplane localized folding (or creasing, or weak elastostatic shock) and faulting (or elastostatic shock) of a Cosserat continuum, phenomena which remain excluded for a Cauchy elastic material. During folding some components of the displacement gradient suffer a finite jump, whereas during faulting the displacement itself displays a finite discontinuity

    Macroscopic stress and strain in a doubly periodic array of dislocation dipoles

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    It is known that in two-dimensional periodic arrays of dislocations the summation of the periodic image fields is conditionally convergent. This is due to the long-range character of the elastic fields of dislocations. As a result, the stress field obtained for a doubly periodic array of dislocation dipoles may contain a spurious constant stress that depends on the adopted summation scheme. In the present work, we provide, based on micromechanical considerations, a simple physical explanation of the origin of the conditional convergence of lattice sums of image interactions. In this context, the spurious stresses are found in a closed form for an arbitrary elastic anisotropy, and this is achieved without using the stress field of an individual dislocation. An alternative procedure is also developed where the macroscopic spurious stresses are determined using the solution of the Eshelby's inclusion problem

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

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    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

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

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    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

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

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    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

    Finite element simulation of a gradient elastic half-space subjected to thermal shock on the boundary

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    The influence of the microstructure on the macroscopical behavior of complex materials is disclosed under thermal shock conditions. The thermal shock response of an elastic half-space subjected to convective heat transfer at its free surface from a fluid undergoing a sudden change of its temperature is investigated within the context of the generalized continuum theory of gradient thermoelasticity. This theory is employed to model effectively the material microstructure. This is a demanding initial boundary value problem which is solved numerically using a higher-order finite element procedure. Simulations have been performed for different values of the microstructural parameters showing that within the gradient material the thermoelastic pulses are found to be dispersive and smoother than those within a classical elastic solid, for which the solution is retrieved as a special case. Energy type stability estimates for the weak solution have been obtained for both the fully and weakly coupled thermoelastic systems. The convergence characteristics of the proposed finite element schemes have been verified by several numerical experiments. In addition to the direct applicative significance of the obtained results, our solution serves as a useful benchmark for modeling more complicated problems within the framework of gradient thermoelasticity

    Analysis of the tilted flat punch in couple-stress elasticity

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    In the present paper we explore the response of a half-plane indented by a tilted flat punch with sharp corners in the context of couple-stress elasticity theory. Contact conditions arise in a number of modern engineering applications ranging from structural and geotechnical engineering to micro and nanotechnology. As the contact scales reduce progressively the effects of the microstructure upon the macroscopic material response cannot be ignored. The generalized continuum theory of couple-stress elasticity introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying material microstructure. The problem under investigation is interesting for three reasons: Firstly, the indentor's geometry is simple so that benchmark results may be extracted. Secondly, important deterioration of the macroscopic results may emerge in the case that a tilting moment is applied on the indentor inadvertently or in the case that the flat punch itself is not self-aligning so that asymmetrical contact pressure distributions arise on the contact faces. Thirdly, the voluntary application of a tilting moment on the flat punch during an experiment gives rise to potential capabilities of the flat punch for the determination of the material microstructural characteristic lengths. The solution methodology is based on singular integral equations which have resulted from a treatment of the mixed boundary value problem via integral transforms and generalized functions. The results show significant departure from the predictions of classical elasticity revealing that valuable information may be deducted from the indentation of a tilted punch of a microstructured solid

    Interaction of cracks with dislocations in couple-stress elasticity

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    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

    Analysis of the tilted flat punch in couple-stress elasticity

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    This paper was accepted for publication in the journal International Journal of Solids and Structures and the definitive published version is available at https://doi.org/10.1016/j.ijsolstr.2016.01.017.In the present paper we explore the response of a half-plane indented by a tilted flat punch with sharp corners in the context of couple-stress elasticity theory. Contact conditions arise in a number of modern engineering applications ranging from structural and geotechnical engineering to micro and nanotechnology. As the contact scales reduce progressively the effects of the microstructure upon the macroscopic material response cannot be ignored. The generalized continuum theory of couple-stress elasticity introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying material microstructure. The problem under investigation is interesting for three reasons: Firstly, the indentor's geometry is simple so that benchmark results may be extracted. Secondly, important deterioration of the macroscopic results may emerge in the case that a tilting moment is applied on the indentor inadvertently or in the case that the flat punch itself is not self-aligning so that asymmetrical contact pressure distributions arise on the contact faces. Thirdly, the voluntary application of a tilting moment on the flat punch during an experiment gives rise to potential capabilities of the flat punch for the determination of the material microstructural characteristic lengths. The solution methodology is based on singular integral equations which have resulted from a treatment of the mixed boundary value problem via integral transforms and generalized functions. The results show significant departure from the predictions of classical elasticity revealing that valuable information may be deducted from the indentation of a tilted punch of a microstructured solid

    Steady-state propagation of a mode II crack in couple stress elasticity

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    The present work deals with the problem of a semi-infinite crack steadily propagating in an elastic body subject to plane-strain shear loading. It is assumed that the mechanical response of the body is governed by the theory of couple-stress elasticity including also micro-rotational inertial effects. This theory introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying microstructure and has proved to be very effective for modeling complex microstructured materials. It is assumed that the crack propagates at a constant sub-Rayleigh speed. An exact full field solution is then obtained based on integral transforms and the Wiener–Hopf technique. Numerical results are presented illustrating the dependence of the stress intensity factor and the energy release rate upon the propagation velocity and the characteristic material lengths in couple-stress elasticity. The present analysis confirms and extends previous results within the context of couple-stress elasticity concerning stationary cracks by including inertial and micro-inertial effects
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