Kinematics and constitutive relations in the stress-gradient theory: interpretation by homogenization

INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES

卷:193

Article

Huetter, Geralf¹  Sab, Karam²  Forest, Samuel³ 

[1] TU Bergakad Freiberg, Inst Mech & Fluid Dynam, Lampadiusstr 4, D-09596 Freiberg, Germany

[2] Univ Paris Est, Lab Navier, ENPC, IFSTTAR,CNRS UMR 8205, 6 & 8 Ave Blaise Pascal, F-77455 Marne La Vallee 2, France

[3] PSL Univ, MINES ParisTech, Ctr Mat, CNRS UMR 7633, BP 87, F-91003 Evry, France

关键词: Stress-gradient theory;    Generalized continua;    Homogenization;    Negative size effect;   

DOI  :  10.1016/j.ijsolstr.2020.02.014

来源: Elsevier

【 摘 要 】

The stress-gradient theory has a third order tensor as kinematic degree of freedom, which is work-conjugate to the stress gradient. This tensor was called micro-displacements just for dimensional reasons. Consequently, this theory requires a constitutive relation between stress gradient and micro-displacements, in addition to the conventional stress-strain relation. The formulation of such a constitutive relation and identification of the parameters therein is difficult without an interpretation of the micro-displacement tensor. The present contribution presents an homogenization concept from a Cauchy continuum at the micro-scale towards a stress-gradient continuum at the macro-scale. Conventional static boundary conditions at the volume element are interpreted as a Taylor series whose next term involves the stress gradient. A generalized Hill-Mandel lemma shows that the micro-displacements can be identified with the deviatoric part of the first moment of the microscopic strain field. Kinematic and periodic boundary conditions are provided as alternative to the static ones. The homogenization approach is used to compute the stress-gradient properties of an elastic porous material. The predicted negative size effect under uni-axial loading is compared with respective experimental results for foams and direct numerical simulations from literature. (C) 2020 Elsevier Ltd. All rights reserved.

【 授权许可】

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