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Abstract

In this paper, volume-average relations related to the multilevel modelling process in continuum mechanics are analysed and the concept of average consistency is investigated both analytically and numerically. These volume averages are used in the computational homogenization technique, where a transition of the mechanical properties from the local, microscopic, to the global, macroscopic, length scale is obtained. The representative volume element (RVE) is used as a reference placement and the solution, in terms of volume-averaged stress, will depend on which boundary conditions are chosen for the RVE. Three types of boundary conditions – periodic, affine and anti-periodic – are analysed with respect to the average consistence for the kinematical and stress relations used in continuum mechanics. The inconsistence is quantified by introducing the inconsistence ratio. It is shown analytically that some average stress relations are fulfilled, assuming the periodic boundary condition and anti-periodic traction vector, whereas the average relations connected to the deformation are in general not average consistent. The inconsistence is investigated in a plane model using the finite element technique. The numerical investigation has shown that the inconsistence ratios related to the deformation are also average consistent in the examples considered.

Keywords

Volume-average relations computational homogenization continuum mechanics periodic boundary conditions representative volume element

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References

Hill

. Elastic properties of reinforced solids; some theoretical principles. J Mech Phys Solids 1963; 11: 357–372.

Hill

. On constitutive macro-variables for heterogeneous solids in finite strain. Proc R Soc Lond A 1972; 326: 131–147.

Mandel

. Plasticité classique et viscoplasticité. Vienna: Springer, 1971 (CISM lecture Notes, Udine, Italy).

Havner

. A discrete model for the prediction of subsequent yield surfaces in polycrystalline plasticity. Int J Solid Struct 1971; 7: 719–730.

Suquet

. Elements of homogenization for inelastic solid mechanics. In: Sanchez-Palencia

Zaoui

(eds) Homogenization techniques for composite mechanics. Berlin, New York: Springer, 1987, 193–278.

Miehe

Schröder

Schotte

. Computational homogenization analysis in finite plasticity formulation of texture development in polycrystalline materials. Comput Meth Appl Mech Eng 1999; 171: 387–418.

van der Sluis

Schreurs

PJG

Brekelmans

WAM

. Overall behaviour of heterogeneous elastoviscoplastic materials: effects of microstructural analysis. Comput Meth Appl Mech Eng 1991; 39: 735–752.

Kristensson

Ahadi

. Numerical study of localization in soil systems. Comp Geotech 2005; 32: 600–612.

Nemat-Nasser

Hori

. Micromechanics: overall properties of heterogeneous materials. , Amsterdam: Elsevier, 1993.

10.

Havner

. Finite plastic deformation of crystalline solids. New York: Cambridge University Press, 1992.

11.

Nemat-Nasser

. Averaging theorems in finite deformation plasticity. Mech Mater1999; 31: 493–523.

12.

Cree

. Changes in the dimensions of elastic solids due to given systems of forces. Cambridge Phil Soc Trans 1892; 15: 313–337.

13.

Gurtin

. The linear theory of elasticity. In: Gurtin

(ed.) Mechanics of solids, Volume II. Berlin, Springer, 1984.

14.

Truesdell

. A first course in rational continuum mechanics. New York: Academic Press Inc., 1991.

15.

Hill

. On macroscopic effects of heterogeneity in elastoplastic media at finite strain. Math Proc Camb Phil Soc 1984; 95: 481–494.

16.

Cook

Malkus

Plesha

. Concepts and applications of finite element method. 3rd ed. New York: John Wiley & Sons, 1989.

17.

Ahadi

Lidström

. On the solution of linear equations under constraints. Lund: Division of Mechanics, Lund University, 2012.

18.

Bowen

Wang

C-C

. Introduction to vectors and tensors, vol. 1. New York and London: Plenum Press, 1976.

19.

Signorini

. Sopra alcune questioni di statica dei sistemi continui. Ann Scuola Norm Pisa 1933; 2: 231–257.

20.

Signorini

. Sulle proprietà di media comuni a tutti I sistemi continui. Rend Accad. Ital 1941; 2: 728–734.

Analysis of volume-average relations in continuum mechanics

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