arXiv subjects

Davide Bigoni

Publications and source records attributed to Davide Bigoni.

48 records · Page 3Linked to original sources

arXiv2016

Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields. Part II: Singularities, annihilation and invisibility

Notch stress intensity factors and stress intensity factors are obtained analytically for isotoxal star-shaped polygonal voids and rigid inclusions (and also for the corresponding limit cases of star-shaped cracks and stiffeners), when loaded through remote inhomogeneous (self-equilibrated, polynomial) antiplane shear stress in an infinite linear elastic matrix. Usually these solutions show stress singularities at the inclusion corners. It is shown that an infinite set of geometries and loading conditions exist for which not only the singularity is absent, but the stress vanishes ('annihilates') at the corners. Thus the material, which even without the inclusion corners would have a finite stress, remains unstressed at these points in spite of the applied remote load. Moreover, similar conditions are determined in which a star-shaped crack or stiffener leaves the ambient stress completely unperturbed, thus reaching a condition of 'quasi-static invisibility'. Stress annihilation and invisibility define optimal loading modes for the overall strength of a composite and are useful for designing ultra-resistant materials.

cond-mat.mtrl-sci↗

arXiv2016

Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields. Part I: Formulation and full-field solution

An infinite class of nonuniform antiplane shear fields is considered for a linear elastic isotropic space and (non-intersecting) isotoxal star-shaped polygonal voids and rigid inclusions perturbing these fields are solved. Through the use of the complex potential technique together with the generalized binomial and the multinomial theorems, full-field closed-form solutions are obtained in the conformal plane. The particular (and important) cases of star-shaped cracks and rigid-line inclusions (stiffeners) are also derived. Except for special cases (addressed in Part II), the obtained solutions show singularities at the inclusion corners and at the crack and stiffener ends, where the stress blows-up to infinity, and is therefore detrimental to strength. It is for this reason that the closed-form determination of the stress field near a sharp inclusion or void is crucial for the design of ultra-resistant composites.

cond-mat.mtrl-sci↗

arXiv2015

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 channelling 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 and (ii.) its failure is shown to be related to the emergence of discontinuity surfaces; (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 channelling, 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.

math-ph↗

arXiv2015

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

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 behavior 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.

cond-mat.soft↗

arXiv2014

Stress concentration near stiff inclusions: validation of rigid inclusion model and boundary layers by means of photoelasticity

Photoelasticity is employed to investigate the stress state near stiff rectangular and rhombohedral inclusions embedded in a 'soft' elastic plate. Results show that the singular stress field predicted by the linear elastic solution for the rigid inclusion model can be generated in reality, with great accuracy, within a material. In particular, experiments: (i.) agree with the fact that the singularity is lower for obtuse than for acute inclusion angles; (ii.) show that the singularity is stronger in Mode II than in Mode I (differently from a notch); (iii.) validate the model of rigid quadrilateral inclusion; (iv.) for thin inclusions, show the presence of boundary layers deeply influencing the stress field, so that the limit case of rigid line inclusion is obtained in strong dependence on the inclusion's shape. The introduced experimental methodology opens the possibility of enhancing the design of thin reinforcements and of analyzing complex situations involving interaction between inclusions and defects.

cond-mat.soft↗

arXiv2014

Eshelby-like forces acting on elastic structures: theoretical and experimental proof

The Eshelbian (or configurational) force is the main concept of a celebrated theoretical framework associated with the motion of dislocations and, more in general, defects in solids. In a similar vein, in an elastic structure where a (smooth and bilateral) constraint can move and release energy, a force driving the configuration is generated, which therefore is called by analogy 'Eshelby-like' or 'configurational'. This force (generated by a specific movable constraint) is derived both via variational calculus and, independently, through an asymptotic approach. Its action on the elastic structure is counterintuitive, but is fully substantiated and experimentally measured on a model structure that we have designed, realized and tested. These findings open a totally new perspective in the mechanics of deformable mechanisms, with possible broad applications, even at the nanoscale.

cond-mat.soft↗

arXiv2013

Anisotropic effective higher-order response of heterogeneous Cauchy elastic materials

The homogenization results obtained by Bacca et al. (Homogenization of heterogeneous Cauchy-elastic materials leads to Mindlin second-gradient elasticity. Part I: Closed form expression for the effective higher-order constitutive tensor. arXiv:1305.2365 Submitted, 2013), to define effective second-gradient elastic materials from heterogeneous Cauchy elastic solids, are extended here to the case of phases having non-isotropic tensors of inertia. It is shown that the nonlocal constitutive tensor for the homogenized material depends on both the inertia properties of the RVE and the difference between the effective and the matrix local elastic tensors. Results show that: (i.) a composite material can be designed to result locally isotropic but nonlocally orthotropic; (ii.) orthotropic nonlocal effects are introduced when a dilute distribution of aligned elliptical holes and, in the limit case, of cracks is homogenized.

math-ph↗

arXiv2010

Yield criteria for quasibrittle and frictional materials

A new yield/damage function is proposed for modelling the inelastic behaviour of a broad class of pressure-sensitive, frictional, ductile and brittle-cohesive materials. The yield function allows the possibility of describing a transition between the shape of a yield surface typical of a class of materials to that typical of another class of materals. This is a fundamental key to model the behaviour of materials which become cohesive during hardening (so that the shape of the yield surface evolves from that typical of a granular material to that typical of a dense material), or which decrease cohesion due to damage accumulation. The proposed yield function is shown to agree with a variety of experimental data relative to soil, concrete, rock, metallic and composite powders, metallic foams, porous metals, and polymers. The yield function represents a single, convex and smooth surface in stress space approaching as limit situations well-known criteria and the extreme limits of convexity in the deviatoric plane. The yield function is therefore a generalization of several criteria, including von Mises, Drucker-Prager, Tresca, modified Tresca, Coulomb-Mohr, modified Cam-clay, and --concerning the deviatoric section-- Rankine and Ottosen. Convexity of the function is proved by developing two general propositions relating convexity of the yield surface to convexity of the corresponding function. These propositions are general and therefore may be employed to generate other convex yield functions.

math-ph↗

arXiv2010

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part I - small strain formulation

Mechanical densification of granular bodies is a process in which a loose material becomes increasingly cohesive as the applied pressure increases. A constitutive description of this process faces the formidable problem that granular and dense materials have completely different mechanical behaviours (nonlinear elastic properties, yield limit, plastic flow and hardening laws), which must both be, in a sense, included in the formulation. A treatment of this problem is provided here, so that a new phenomenological, elastoplastic constitutive model is formulated, calibrated by experimental data, implemented and tested, that is capable of describing the transition between granular and fully dense states of a given material. The formulation involves a novel use of elastoplastic coupling to describe the dependence of cohesion and elastic properties on the plastic strain. The treatment falls within small strain theory, which is thought to be appropriate in several situations; however, a generalization of the model to large strain is provided in Part II of this paper.

math-ph↗

arXiv2010

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II - the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of `elastoplastic coupling', in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state. The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear --or linear-- dependence of the elastic part of the deformation on the forming pressure in the granular --or dense-- state) makes the use of existing large strain formulations awkward, if even possible. Therfore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its `total' and `plastic' work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.

math-ph↗

arXiv2010

A dynamical interpretation of flutter instability in a continuous medium

Flutter instability in an infinite medium is a form of material instability corresponding to the occurrence of complex conjugate squares of the acceleration wave velocities. Although its occurrence is known to be possible in elastoplastic materials with nonassociative flow law and to correspond to some dynamically growing disturbance, its mechanical meaning has to date still eluded a precise interpretation. This is provided here by constructing the infinite-body, time-harmonic Green's function for the loading branch of an elastoplastic material in flutter conditions. Used as a perturbation, it reveals that flutter corresponds to a spatially blowing-up disturbance, exhibiting well-defined directional properties, determined by the wave directions for which the eigenvalues become complex conjugate. Flutter is shown to be connected to the formation of localized deformations, a dynamical phenomenon sharing geometrical similarities with the well-known mechanism of shear banding occurring under quasi-static loading. Flutter may occur much earlier than shear banding in a process of continued plastic deformation.

math-ph↗

arXiv2009

Yield criteria for quasibrittle and frictional materials: a generalization to surfaces with corners

Convexity of a yield function (or phase-transformation function) and its relations to convexity of the corresponding yield surface (or phase-transformation surface) is essential to the invention, definition and comparison with experiments of new yield (or phase-transformation) criteria. This issue was previously addressed only under the hypothesis of smoothness of the surface, but yield surfaces with corners (for instance, the Hill, Tresca or Coulomb-Mohr yield criteria) are known to be of fundamental importance in plasticity theory. The generalization of a proposition relating convexity of the function and the corresponding surface to nonsmooth yield and phase-transformation surfaces is provided in this paper, together with the (necessary to the proof) extension of a theorem on nonsmooth elastic potential functions. While the former of these generalizations is crucial for yield and phase-transformation condition, the latter may find applications for potential energy functions describing phase-transforming materials, or materials with discontinuous locking in tension, or contact of a body with a discrete elastic/frictional support.

math-ph↗