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Research Papers

Discrete Averaging Relations for Micro to Macro Transition

[+] Author and Article Information
Chenchen Liu

Department of Mechanical Engineering and
Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104-6315
e-mail: chenchl@seas.upenn.edu

Celia Reina

Department of Mechanical Engineering and
Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104-6315
e-mail: creina@seas.upenn.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 18, 2016; final manuscript received April 29, 2016; published online May 26, 2016. Assoc. Editor: Harold S. Park.

J. Appl. Mech 83(8), 081006 (May 26, 2016) (11 pages) Paper No: JAM-16-1145; doi: 10.1115/1.4033552 History: Received March 18, 2016; Revised April 29, 2016

The well-known Hill's averaging theorems for stresses and strains as well as the so-called Hill–Mandel condition are essential ingredients for the coupling and the consistency between the micro- and macroscales in multiscale finite-element procedures (FE2). We show in this paper that these averaging relations hold exactly under standard finite-element (FE) discretizations, even if the stress field is discontinuous across elements and the standard proofs based on the divergence theorem are no longer suitable. The discrete averaging results are derived for the three classical types of boundary conditions (BC) (affine displacement, periodic, and uniform traction BC) using the properties of the shape functions and the weak form of the microscopic equilibrium equations without further kinematic constraints. The analytical proofs are further verified numerically through a simple FE simulation of an irregular representative volume element (RVE) undergoing large deformations. Furthermore, the proofs are extended to include the effects of body forces and inertia, and the results are consistent with those in the smooth continuum setting. This work provides a solid foundation to apply Hill's averaging relations in multiscale FE methods without introducing an additional error in the scale transition due to the discretization.

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Figures

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Fig. 1

Schematic representation of a FE discretization, where the nodes are colored according to their location, in the interior or at the boundary of the domain

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Fig. 3

Deformed meshes for simple extension mode under affine displacement (a) and periodic BC (b)

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Fig. 4

Deformed meshes for simple shear mode under affine displacement (a) and periodic BC (b)

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Fig. 5

Deformed meshes for simple extension mode (a) and simple shear mode (b) under uniform traction BC

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Fig. 2

Geometry of RVE model (a) and undeformed mesh distribution (b)

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