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Conference Papers Year : 2018

On stress-gradient materials: formulation and homogenization

Sébastien Brisard
Karam Sab


The stress-gradient model was introduced by Forest and Sab (Mech. Res. Comm. 40, pp. 16–25, 2012) as an alternative to the more popular strain-gradient models. Its formulation was later justified mathematically by Sab, Legoll and Forest (J. Elasticity 123(2), pp. 179–201, 2016). More recently, it has been extended by Forest and Sab to finite deformations (doi:10.1177/1081286517720844). It can therefore be considered as a mature material model that can now be applied to e.g. homogenization issues. The present talk will be divided in two parts. In the first part, I will briefly recall the formulation of the stress-gradient model. While mathematically sound, this model raises some questions pertaining to the physical interpretation of the quantities that are energy-conjugate to the stress and its gradient. I will discuss these questions. I will also introduce a simplified isotropic, linearly elastic material model with only one internal material length. In the second part, I will address homogenization of heterogeneous, stress-gradient materials. More specifically, I will discuss the stress-gradient → classical (Cauchy) homogenization process, which requires the material internal length to be commensurable with the size of the heterogeneities. I will first lay out the general framework (including an extended Hill–Mandel lemma). I will then briefly show the solution to Eshelby's spherical inhomogeneity problem, that is then used to derive Mori–Tanaka estimates of the effective properties stress-gradient composites with spherical inclusions. As expected, size-effects are observed in this application. However, where strain-gradient materials exhibit stiffening size-effects, stress-gradient materials exhibit softening size-effects. This result is general; it shows that the stress-gradient and strain-gradient models are not in duality. Rather, they should be considered as complementary. The results presented in this talk are drawn from a paper submitted to the International Journal of Solids and Structures (Tran, Brisard, Guilleminot and Sab, Mori–Tanaka estimates of the effective properties of stress-gradient composites).
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hal-01758985 , version 1 (05-04-2018)



  • HAL Id : hal-01758985 , version 1


Sébastien Brisard, Vinh Phuc Tran, Karam Sab, Johann Guilleminot. On stress-gradient materials: formulation and homogenization. Encounter of the third kind on generalized continua and microstructures, Apr 2018, Arpino, Italy. ⟨hal-01758985⟩
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