Microstructural Randomness Versus Representative Volume Element in Thermomechanics

[+] Author and Article Information
M. Ostoja-Starzewski

Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montréal, Québec H3A 2K6, Canada e-mail: martin.ostoja@mcgill.ca

J. Appl. Mech 69(1), 25-35 (Jun 12, 2001) (11 pages) doi:10.1115/1.1410366 History: Received August 31, 2000; Revised June 12, 2001
Copyright © 2002 by ASME
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Passage from a discrete system of tungsten-carbide (black) and cobalt (white) (a) to an intermediate continuum level (b) involving a mesoscale finite element, that serves as input into the macroscale model accounting for spatial nonuniformity. Figures (a) and (b) are generated by a Boolean model of Poisson polygons and a diffusion random function, respectively (45).
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Sampling of paper properties via a gray-scale plot of (a) elastic modulus E lbf/in; (b) breaking strength σmax in lbf/in; (c) strain to failure εmax in percentage and (d) tensile energy absorption TEA lbf/in. All data are for a 25×8 array of 1″ ×1″ specimens tested in the x-(machine) direction. The ranges and assignments of values are shown in the respective insets.
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Thermodynamic orthogonality in (a) the spaces of velocities ǡδ and ensemble average forces 〈Ȳδ〉 on mesoscale δ, with ΔȲδ showing scatter in Ȳδ; (b) the spaces of velocities ǡ≡ǡ and ensemble average forces Y≡Ȳ on macroscale, where the scatter in ǡδ and Ȳδ is absent; (c) ensemble-average velocities 〈ǡδ〉 and forces Ȳδ on mesoscale, with Δǡδ showing scatter in ǡδ. In all the cases, dissipation functions Φ and respective duals Φ*, on mesoscale (parametrized by δ) or macroscale (parametrized by ∞) are shown.
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Antiplane responses of a matrix-inclusion composite, with 35 percent volume fraction of inclusions, at decreasing contrasts: (a) C(i)/C(m)=1, (b) C(i)/C(m)=0.2, (c) C(i)/C(m)=0.05, (d) C(i)/C(m)=0.02. For (bd), the first figure shows response under displacement b.c.’s ε10, while the second one shows response under traction b.c.’s σ10=σ̄1 computed from the first problem.
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Constitutive behavior of a material with elasticity coupled with damage, where ε/εR plays the role of a controllable, time-like parameter of the stochastic process. (a) Stress-strain response of a single specimen Bδ(ω) from Bδ, having a zigzag realization, (b) deterioration of stiffness, (c) evolution of the damage variable D. Curves shown in (ac) indicate the scatter in stress, stiffness and damage at finite scale δ. Assuming spatial ergodicity, this scatter would vanish in the limit (δ≡L/d→∞), whereby unique response curves of continuum damage mechanics would be recovered.




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