Panos A. Gourgiotis

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

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.

Stress channelling in extreme couple-stress materials Part I: Strong ellipticity, wave propagation, ellipticity, and discontinuity relations

Materials with extreme mechanical anisotropy are designed to work near a material instability threshold where they display stress channeling and strain localization, effects that can be exploited in several technologies. Extreme couple stress solids are introduced and for the first time systematically analyzed in terms of several material instability criteria: positive-definiteness of the strain energy (implying uniqueness of the mixed b.v.p.), strong ellipticity (implying uniqueness of the b.v.p. with prescribed kinematics on the whole boundary), plane wave propagation, ellipticity, and the emergence of discontinuity surfaces. Several new and unexpected features are highlighted: (i) Ellipticity is mainly dictated by the ‘Cosserat part’ of the elasticity; (ii) its failure is shown to be related to the emergence of discontinuity surfaces; and (iii) ellipticity and wave propagation are not interdependent conditions (so that it is possible for waves not to propagate when the material is still in the elliptic range and, in very special cases, for waves to propagate when ellipticity does not hold). The proof that loss of ellipticity induces stress channeling, folding and faulting of an elastic Cosserat continuum (and the related derivation of the infinite-body Green’s function under antiplane strain conditions) is deferred to Part II of this study.

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 L 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, that is, the microstructural length ℓ and the distance L 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 behaviour at infinity in the transformed plane. Nevertheless, an analytic function method is employed that 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.

Folding and faulting of an elastic continuum

Folding is a process in which bending is localized at sharp edges separated by almost undeformed elements. This process is rarely encountered in Nature, although some exceptions can be found in unusual layered rock formations (called ‘chevrons’) and seashell patterns (for instance Lopha cristagalli). In mechanics, the bending of a three-dimensional elastic solid is common (for example, in bulk wave propagation), but folding is usually not achieved. In this article, the route leading to folding is shown for an elastic solid obeying the couple-stress theory with an extreme anisotropy. This result is obtained with a perturbation technique, which involves the derivation of new two-dimensional Greens functions for applied concentrated force and moment. While the former perturbation reveals folding, the latter shows that a material in an extreme anisotropic state is also prone to a faulting instability, in which a displacement step of finite size emerges. Another failure mechanism, namely the formation of dilation/compaction bands, is also highlighted. Finally, a geophysical application to the mechanics of chevron formation shows how the proposed approach may explain the formation of natural structures.

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

The antiplane strain Greens 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 Greens functions. These Greens 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.

Two-dimensional indentation of microstructured solids characterized by couple-stress elasticity

In this study, we derive general solutions for two-dimensional plane strain contact problems within the framework of the generalized continuum theory of couple-stress elasticity. This theory introduces characteristic material lengths and is able to capture the associated scale effects that emerge from the material microstructure which are often observed in indentation tests used for the material characterization. The contact problems are formulated in terms of singular integral equations using a Green’s function approach. The pertinent Green’s function obtained through the use of integral transforms corresponds to the solution of the two-dimensional Flamant–Boussinesq half-plane problem in couple-stress elasticity. The results show a strong dependence on the microstructural characteristics of the material when this becomes comparable to the characteristic dimension of the problem, which in the case of an indentation test is the contact length/area.

The Cerruti problem in dipolar gradient elasticity

The classical three-dimensional Cerruti problem of an isotropic half-space subjected to a concentrated tangential load on its surface is revisited here in the context of dipolar gradient elasticity. This generalized continuum theory encompasses the analytical possibility of size effects, which are absent in the classical theory, and has proven to be very successful in modelling materials with complex microstructure. The dipolar gradient elasticity theory assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients. In this way, this theory can be viewed as a first-step extension of classical elasticity. The solution method is based on integral transforms and is exact. Of special importance is the behaviour of the new solution near to the point of application of the load where pathological singularities exist in the classical solution (based on the standard theory). The present results show departure from the ones predicted by the classical elasticity theory. Indeed, continuous and bounded displacements are found at the point of application of the load. Such a behaviour of the displacement field is, of course, more natural than the singular behaviour present in the classical solution.

The dynamics of folding instability in a constrained Cosserat medium

Different from Cauchy elastic materials, generalized continua, and in particular constrained Cosserat materials, can be designed to possess extreme (near a failure of ellipticity) orthotropy properties and in this way to model folding in a three-dimensional solid. Following this approach, folding, which is a narrow zone of highly localized bending, spontaneously emerges as a deformation pattern occurring in a strongly anisotropic solid. How this peculiar pattern interacts with wave propagation in the time-harmonic domain is revealed through the derivation of an antiplane, infinite-body Green’s function, which opens the way to integral techniques for anisotropic constrained Cosserat continua. Viewed as a perturbing agent, the Green’s function shows that folding, emerging near a steadily pulsating source in the limit of failure of ellipticity, is transformed into a disturbance with wavefronts parallel to the folding itself. The results of the presented study introduce the possibility of exploiting constrained Cosserat solids for propagating waves in materials displaying origami patterns of deformation. This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’

The Boussinesq problem in dipolar gradient elasticity

The three-dimensional axisymmetric Boussinesq problem of an isotropic half-space subjected to a concentrated normal quasi-static load is studied within the framework of dipolar gradient elasticity involving linear constitutive relations and small strains. Our main concern is to determine possible deviations from the predictions of classical linear elastostatics when a more refined theory is employed to attack the problem. Of special importance is the behavior of the new solution near to the point of application of the load where pathological singularities exist in the classical solution. The use of the theory of gradient elasticity is intended here to model the response of materials with microstructure in a manner that the classical theory cannot afford. A linear version of this theory (as regards both kinematics and constitutive response) results by considering a linear isotropic expression for the strain-energy density that depends on strain gradient terms, in addition to the standard strain terms appearing in classical elasticity and by considering small strains. Through this formulation, a microstructural material constant is introduced, in addition to the standard Lamé constants. The solution method is based on integral transforms and is exact. The present results show significant departure from the predictions of classical elasticity. Indeed, continuous and bounded displacements are predicted at the points of application of the concentrated load. Such a behavior of the displacement field is, of course, more natural than the singular behavior exhibited in the classical solution.

An Approach Based on Integral Equations for Crack Problems in Standard Couple-Stress Elasticity

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 standard couple-stress elasticity (or Cosserat elasticity with constrained rotations), 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 wedge disclinations that 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 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.