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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August 2016
Research-Article
Discrete Averaging Relations for Micro to Macro Transition
Chenchen Liu,
Chenchen Liu
Department of Mechanical Engineering and
Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104-6315
e-mail: chenchl@seas.upenn.edu
Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104-6315
e-mail: chenchl@seas.upenn.edu
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Celia Reina
Celia Reina
Department of Mechanical Engineering and
Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104-6315
e-mail: creina@seas.upenn.edu
Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104-6315
e-mail: creina@seas.upenn.edu
Search for other works by this author on:
Chenchen Liu
Department of Mechanical Engineering and
Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104-6315
e-mail: chenchl@seas.upenn.edu
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
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. Aug 2016, 83(8): 081006 (11 pages)
Published Online: May 26, 2016
Article history
Received:
March 18, 2016
Revised:
April 29, 2016
Citation
Liu, C., and Reina, C. (May 26, 2016). "Discrete Averaging Relations for Micro to Macro Transition." ASME. J. Appl. Mech. August 2016; 83(8): 081006. https://doi.org/10.1115/1.4033552
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