A micromechanics - based nonlocal constitutive equation for composites containing non-spherical voids and inclusions

A generalization of the Hashin-Shtrikman variational formulation is employed to derive a micromechanics-based nonlocal constitutive equation relating the ensemble averages of stress and strain for random linear elastic composites. The analysis builds on that of Drugan and Willis (1996, J Mech Phys Solids 44, 497-524) and Drugan (2000, J Mech Phys Solids 48, 1359-1387), who derived completely explicit results for isotropic matrices containing random distributions of isotropic, non-overlapping identical spheres. Here, it is shown how to generalise this nonlocal constitutive equation to the case of non-spherical inclusions and how to separate the effects of inclusion shape and spatial distribution. The new constitutive model is then used to improve previous results for higher volume fractions of inclusions

A micromechanics-based nonlocal constitutive equation for transversely-isotropic random composites

An explicit, micromechanics-based nonlocal constitutive equation for random linear elastic composite materials exhibiting transversely-isotropic macroscopic behaviour is derived by employing a generalization of the Hashin-Shtrikman variational formulation. A random distribution of isotropic, non-overlapping identical inclusions with fixed spheroidal shape is considered, wherein the inclusions have random spatial location but aligned orientation in an isotropic matrix. The analysis builds on that of Drugan and Willis (1996, J Mech Phys Solids 44, 497-524), here extended to the case of transversely-isotropic composites. New explicit results will be shown, and the effects of inclusion alignment will be highlighted by comparing these results with those previously obtained by Monetto and Drugan (2004, J Mech Phys Solids 52, 359-393) for spheroidal inclusions having random orientation as well as location

On micromechanics-based nonlocal modeling of elastic matrices containing non-spherical heterogeneities

A micromechanics-based nonlocal constitutive equation relating the ensemble averages of stress and strain for a matrix containing a random distribution of non-spherical voids, cracks or inclusions but having macroscopically isotropic behavior is derived. The model of impenetrable particles considered consists of identical particles with fixed spheroidal shape and random orientation. It is shown how the effects of inclusion shape and spatial distribution can be separated. Terms related to inclusion shape reduce to certain intergrals which can be evaluated analytically only in special cases. Terms describing effects of spatial distribution can be obtained explicitly for different statistical models, within the framework of up through two-point statistics. As verification of the formulation, completely explicit expressions are derived for the limiting case of spherical inclusions and for a standard statistical model on the basis of results found in the literature. The new constitutive equation can be used to produce quantitative estimates of the minimum size of a material volume element over which standard local constitutive equations provide a sensible description of the macroscopic constitutive response of the material

Micromechanics-based nonlocal modeling of elastic matrices containing aligned spheroidal inclusions

A micromechanics-based nonlocal constitutive equation relating the ensemble averages of stress and strain for a matrix containing a random distribution of non-spherical hard inclusions having macroscopic transversely-isotropic behavior is derived. The model of impenetrable particles considered consists of identical spheroids having aligned orientations. The analysis builds on and generalises previous papers where elastic composites containing spherical and randomly oriented spheroid-shaped inclusions were analysed. In particular, it is shown how the task of the statistical description of the microstructure can be make reasonable by performing a simple scale transformation. The new constitutive equation is then used to explore nonlocal effects of shape and spatial distribution of inclusions on the anisotropic response of the composite

Nacre: an orthotropic and bimodular elastic material

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Crack tip fields at a ductile single crystal-rigid material interface

Small-scale yielding around a stationary crack along a ductile single crystal–rigid material interface is analyzed. Plane strain conditions are assumed to prevail and geometry changes are neglected. The analyses are carried out using both continuum slip and discrete dislocation plasticity theory for model fcc and bcc crystal geometries having either two or three slip systems. Numerical and analytical asymptotic solutions are presented for continuum slip plasticity theory. Solutions exhibiting both slip bands and kink bands are obtained. The addition of a third slip system to ductile single crystals having two slip systems is found to have a significant effect on the interface crack-tip fields. The results illustrate the role that each of the formulations considered can play in elucidating crack tip fields in single crystals.