A micromorphic approach to stress gradient elasticity theory with an assessment of the boundary conditions and size effects

Abstract

Basing on thermodynamics principles, a stress gradient elasticity theory is presented whereby the material is modeled as a centro‐symmetric micromorphic anisotropic material. A principle of minimum complementary energy and a stationarity principle of Hellinger–Reissner type are provided, by which the boundary conditions are uniquely determined in a form consistent with continuum mechanics. For 3‐D solids, these conditions include, beside the three ordinary boundary conditions for the assigned boundary data (displacements and/or tractions), six extra boundary conditions by which a microstructure/continuum compatibility condition is enforced, generally through the vanishing of the normal derivative of the stress, ∂nbold-italicσ=0, on the whole boundary surface. A typical boundary‐value problem for isotropic materials with a single length scale parameter under static loads is discussed, showing that it is governed by the same Navier PDEs of classical elasticity, along with a set of Helmholtz PDEs as constitutive equations. Size effects are shown to arise from two distinct sources, that is, the spatial fluctuating of the body force and the double curl of the Hookean stress field. It is proved that a formulation of stress gradient elasticity theory based on the pure notion of nonsimple material (like the ones existing in the literature) leads to anomalies in the ordinary and extra boundary conditions. A comparison of the present theory with that by Eringen is also discussed. Two examples of stress gradient structural models are analytically worked out, namely, a thick‐walled cylinder and a Kirchhoff–Love circular plate. For comparison, the solutions for strain gradient material are also presented. Graphic illustrations show that in both examples stiffening size effects like with the ancillary strain gradient models are found.

A finite element implementation of the stress gradient theory

“…Applications of stress gradient elasticity have been rather limited up to now. They include prediction of size effects in bending with a comparison with strain gradient and micromorphic approaches [28], and particle size effects in composite materials [33]. The latter reference contains extensions of the Eshelby and Hashin-Shtrikman homogenisation approaches to heterogeneous stress gradient media in a simplified case of isotropic elasticity.…”

Morphology-dependent Hashin–Shtrikman bounds on the effective properties of stress-gradient materials

“…Since then, other similar models have been proposed, which mostly differ by the boundary conditions; see e.g. Polizzotto (2018) and references therein. Such models are relevant for nanocomposites exhibiting "negative" size-effect (Tran et al, 2018) and foams (Hütter et al, 2020).…”

Integral and differential nonlocal micromorphic theory

“…For instance, Yoder et al (2018) considered the discrete beam element model and micropolar elasticity to capture the size-effects in thin-walled lattice structures. Based on micromorphic continua, Polizzotto (2018) analyzed different boundary value problems with the help of the presented stress-gradient generalized continuum theory possessing one length scale parameter.…”

“…The reader is referred to Eringen (1999Eringen ( , 2001 for more details on the micropolar and micromorphic continua. The successful applications of MPT and MMT to different fields have been reported by several researchers up to now (de Borst, 1991;Forest and Sievert, 2003;Dillard et al, 2006;Bigoni and Drugan, 2007;Grammenoudis and Tsakmakis, 2009;Zhang et al, 2018;Yoder et al, 2018;Eremeyev, 2018;Shaat, 2018;Polizzotto, 2018). For instance, Yoder et al (2018) considered the discrete beam element model and micropolar elasticity to capture the size-effects in thin-walled lattice structures.…”

“…Based on micromorphic continua, Polizzotto (2018) analyzed different boundary value problems with the help of the presented stress-gradient generalized continuum theory possessing one length scale parameter.…”