Length scale effects in finite strain micromorphic linear isotropic elasticity: finite element analysis of three-dimensional cubical microindentation

Abstract

A three-dimensional finite strain micromorphic materially linear isotropic elastic model is formulated in two ways for finite element implementation: (i) direct finite strain elasticity; and (ii) rate form with semi-implicit time integration. The model is based upon the finite strain isotropic micromorphic elasticity model proposed by Eringen and Suhubi in 1964. For (i), the direct formulation, the constitutive equations are calculated in the reference configuration, and the resulting stresses are mapped to the current configuration. For (ii), the rate formulation, the constitutive equations are integrated in time in the current configuration using the Truesdell objective stress rates. After formulating the coupled weak form, the balance of linear momentum and the balance of first moment of momentum are linearized to construct the consistent tangent for solution by the Newton–Raphson method. Three-dimensional numerical examples are analysed to compare the two formulations (i) and (ii) for standard finite strain isotropic elasticity, and the formulation (i) is used to demonstrate the elastic length scale effects that come through the higher-order couple stress in the micromorphic theory.

Keywords

length scale micromorphic elasticity finite element microindentation

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References

Eringen

A. C.

Suhubi

E. S.

Nonlinear theory of simple micro-elastic solids, 1. Int. J. Engng Sci., 1964, 2(2), 189–203.

Suhubi

Eringen

Nonlinear theory of simple micro-elastic solids, II. Int. J. Engng Sci., 1964, 2(2), 389–404.

Eringen

Microcontinuum field theories, I: Foundations and solids, 1999 (Springer, Berlin).

Mindlin

Micro-structure in linear elasticity. Arch. Ration. Mech. Anal., 1964, 16, 51–78.

Regueiro

Finite strain micromorphic pressure-sensitive plasticity. J. Eng. Mech., 2009, 135, 178–191.

Regueiro

On finite strain micromorphic elastoplasticity. Int. J. Solids Struct., 2010, 47, 786–800.

Regueiro

Yan

Concurrent multiscale computational modeling for dense dry granular materials interfacing deformable solid bodies. Bifurcations, instabilities and degradations in geomaterials, Springer Series in Geomechanics and Geoengineering, 2011, pp. 251–273 (Springer, Berlin).

Chen

Lee

Connecting molecular dynamics to micromorphic theory, I: Instantaneous and averaged mechanical variables. Phys. A Stat. Mech. Applic., 2003, 322(S), 359–376.

Chen

Lee

Connecting molecular dynamics to micromorphic theory, II: balance laws. Phys. A Stat. Mech. Applic., 2003, 322(S), 377–392.

10.

Garikipati

Couple stresses in crystalline solids: Origins from plastic slip gradients, dislocation core distortions, and three-body interatomic potentials. J. Mech. Phys. Solids, 2003, 51(7), 1189–1214.

11.

Zimmerman

Bammann

Gao

Deformation gradients for continuum mechanical analysis of atomistic simulations. Int. J. Solids Struct., 2009, 46(2), 238–253.

12.

Barthelat

C.-M.

Comi

Espinosa

Mechanical properties of nacre constituents and their impact on mechanical performance. J. Mater. Res., 2006, 21(8), 1977–1986.

13.

Sansour

Unified concept of elastic-viscoplastic cosserat and micromorphic continua. J. Physique IV, 1998, 8(8), 341–348.

14.

Lee

J. D.

Chen

Constitutive relations of micromorphic thermoplasticity. Int. J. Engng Sci., 2003, 41(3-5), 387–399.

15.

Forest

Sievert

Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech., 2003, 160(1-2), 71–111.

16.

Forest

Sievert

Nonlinear microstrain theories. Int. J. Solids Struct., 2006, 43(24), 7224–7245.

17.

Neff

Forest

A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. modelling, existence of minimizers, identification of moduli and computational results. J. Elasticity, 2007, 87(2), 239–276.

18.

Vernerey

Liu

Moran

Multi-scale micromorphic theory for hierarchical materials. J. Mech. Phys. Solids, 2007, 55(12), 2603–2651.

19.

Sansour

Skatulla

Zbib

A formulation for the micromorphic continuum at finite inelastic strains. Int. J. Solids Struct., 2010, 47(11-12), 1546–1554.

20.

Isbuga

Regueiro

Three-dimensional finite element analysis of finite strain micromorphic linear isotropic elasticity. Int. J. Engng Sci., 2011. DOI: 10.1016/j.ijengsci.2011.04.006

21.

Holzapfel

G. A.

Nonlinear solid mechanics: A continuum approach for engineering, 2000 (Wiley, New York).

22.

Eringen

Nonlinear theory of continuous media, 1st edition, 1962 (McGraw-Hill, New York).

23.

Hughes

T. J. R.

The finite element method, 1987 (Prentice-Hall, Englewood Cliffs, New Jersey).

24.

Smith

A. C.

Inequalities between the constants of a linear micro-elastic solid. Int. J. Engng Sci., 1968, 6(2), 65–74.

25.

Zhou

Yang

A microcrack damage model for brittle rocks under uniaxial compression. Mech. Res. Commun., 2010, 37(4), 399–405.

26.

Regueiro

Ebrahimi

Implicit dynamic three-dimensional finite element analysis of an inelastic biphasic mixture at finite strain. Part 1: Application to a simple geomaterial. Comput. Methods Applic. Mech. Engng, 2010, 199, 2024–2049.

27.

Cundall

Strack

A discrete numerical model for granular assemblies. Geotechnique, 1979, 29, 47–65.

28.

Bobko

Ulm

F.-J.

The nano-mechanical morphology of shale. Mech. Mater., 2008, 40(4-5), 318–337.

29.

Bobko

C. P.

Gathier

Ortega

J. A.

Ulm

F.-J.

Borges

Abousleiman

Y. N.

The nanogranular origin of friction and cohesion in shale – A strength homogenization approach to interpretation of nanoindentation results. Int. J. Numerical and Analytical Methods in Geomechanics, 2011, 35. DOI: 10.1002/nag.984