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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Maugin, G. A., 1999, The Thermomechanics of Nonlinear Irreversible Behaviors—An Introduction, World Scientific, Singapura.
Ziegler, H., 1963, “Some Extremum Principles in Irreversible Thermodynamics With Applications to Continuum Mechanics,” Progress in Solid Mechanics, Vol. 42, North-Holland, Amsterdam, pp. 91–191.
Muschik,  W., and Fang,  J., 1989, “Statistical Foundations of Non-equilibrium Contact Quantities. Bridging Phenomenological and Statistical Non-equilibrium Thermodynamics,” Acta Phys. Hung., 66, pp. 39–57.
Lemaitre, J., and Chaboche, J.-L., 1990, Mechanics of Solid Materials, Cambridge University Press, Cambridge, UK.
Maugin, G. A., 1992, The Thermomechanics of Plasticity and Fracture, Cambridge University Press, Cambridge, UK.
Mei,  C. C., Auriault,  J.-L., and Ng,  C.-O., 1996, “Some Applications of the Homogenization Theory,” Adv. Appl. Mech., 32, pp. 277–348.
Dvorak,  G. J., 1997, “Thermomechanics of Heterogeneous Media,” J. Therm. Stresses, 20, pp. 799–817.
Drugan,  W. J., and Willis,  J. R., 1996, “A Micromechanics-Based Nonlocal Constitutive Equation and Estimates of Representative Volume Element Size for Elastic Composites,” J. Mech. Phys. Solids, 44, pp. 497–524.
Gusev,  A. A., 1997, “Representative Volume Element Size for Elastic Composites: A Numerical Study,” J. Mech. Phys. Solids, 45, No. 9, pp. 1449–1459.
Jiang,  M., Ostoja-Starzewski,  M., and Jasiuk,  I., 2001, “Scale-Dependent Bounds on Effective Elastoplastic Response of Random Composites,” J. Mech. Phys. Solids, 49, No. 3, pp. 655–673.
Bažant, Z. P., and Planas, J., 1998, Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton, FL.
Chudnovsky, A., 1977, The Principles of Statistical Theory of the Long Time Strength, Novosibirsk (in Russian).
Krajcinovic, D., 1996, Damage Mechanics, North-Holland, Amsterdam.
Krajcinovic, D., 1997, “Essential Structure of the Damage Mechanics Theories,” Theor. Appl. Mech., T. Tatsumi, E. Watanabe, and T. Kambe, eds., Elsevier Science, Amsterdam, pp. 411–426.
Hill,  R., 1963, “Elastic Properties of Reinforced Solids: Some Theoretical Principles,” J. Mech. Phys. Solids, 11, pp. 357–372.
Hazanov,  S., and Huet,  C., 1994, “Order Relationships for Boundary Conditions Effect in Heterogeneous Bodies Smaller Than the Representative Volume,” J. Mech. Phys. Solids, 41, pp. 1995–2011.
Ziegler, H., 1983, An Introduction to Thermomechanics, North-Holland, Amsterdam.
Ziegler, H., and Wehrli, C., 1987, “The Derivation of Constitutive Relations From Free Energy and the Dissipation Functions,” Adv. Appl. Mech., 25 , Academic Press, New York, pp. 183–238.
Germain,  P., Nguyen,  Quoc Son, and Suquet,  P., 1983, “Continuum Thermodynamics,” ASME J. Appl. Mech., 50, pp. 1010–1020.
Beran, M. J., 1974, “Application of Statistical Theories for the Determination of Thermal, Electrical, and Magnetic Properties of Heterogeneous Materials,” Mechanics of Composite Materials, Vol. 2, G. P. Sendeckyj, ed., Academic Press, San Diego, CA, pp. 209–249.
Nemat-Nasser, S., and Hori, M., 1993, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland, Amsterdam.
Jeulin, D., and Ostoja-Starzewski, M., eds., 2001, Mechanics of Random and Multiscale Microstructures (CISM Courses and Lectures), Springer-Verlag, New York, in press.
Stoyan, D., Kendall, W. S., and Mecke, J., 1987, Stochastic Geometry and its Applications, J. Wiley and Sons, New York.
Dempsey,  J. P., 2000, “Research Trends in Ice Mechanics,” Int. J. Solids Struct., 37, No. 1–2.
Rytov, S. M., Kravtsov, Yu. A., and Tatarskii, V. I., 1989, Principles of Statistical Radiophysics, Vol. 3, Springer-Verlag, Berlin.
Huet,  C., 1990, “Application of Variational Concepts to Size Effects in Elastic Heterogeneous Bodies,” J. Mech. Phys. Solids, 38, pp. 813–841.
Sab,  K., 1992, “On the Homogenization and the Simulation of Random Materials,” Eur. J. Mech. A/Solids, 11, pp. 585–607.
Ostoja-Starzewski,  M., 1994, “Micromechanics as a Basis of Continuum Random Fields,” Appl. Mech. Rev., 47, (No. 1, Part 2), pp. S221–S230.
Ostoja-Starzewski,  M., 1998, “Random Field Models of Heterogeneous Materials,” Int. J. Solids Struct., 35, No. 19, pp. 2429–2455.
Ostoja-Starzewski,  M., 1999, “Scale Effects in Materials With Random Distributions of Needles and Cracks,” Mech. Mater., 31, No. 12, pp. 883–893.
Ostoja-Starzewski,  M., 1999, “Microstructural Disorder, Mesoscale Finite Elements, and Macroscopic Response,” Proc. R. Soc. London, Ser. A, A455, pp. 3189–3199.
Hazanov,  S., 1998, “Hill Condition and Overall Properties of Composites,” Arch. Appl. Mech., 66, pp. 385–394.
Hazanov,  S., 1999, “On Apparent Properties of Nonlinear Heterogeneous Bodies Smaller Than the Representative Volume,” Acta Mech., 134, pp. 123–134.
Huet,  C., 1999, “Coupled Size and Boundary-Condition Effects in Viscoelastic Heterogeneous Composite Bodies,” Mech. Mater., 31, No. 12, pp. 787–829.
Hazanov,  S., and Amieur,  M., 1995, “On Overall Properties of Elastic Heterogeneous Bodies Smaller Than the Representative Volume,” Int. J. Eng. Sci., 33, No. 9, pp. 1289–1301.
Ostoja-Starzewski,  M., 1990, “A Generalization of Thermodynamic Orthogonality to Random Media,” J. Appl. Math. Phys., 41, pp. 701–712.
Ziegler, H., 1968, “A Possible Generalization of Onsager’s Theory,” IUTAM Symp. Irreversible Aspects of Continuum Mechanics, H. Parkus and L. I. Sedov, eds., Springer-Verlag, New York, pp. 411–424.
Ashby, M. F., and Jones, D. R. H., 1986, Engineering Materials, Vol. 2, Pergamon Press, Oxford.
Ostoja-Starzewski,  M., Sheng,  P. Y., and Jasiuk,  I., 1997, “Damage Patterns and Constitutive Response of Random Matrix-Inclusion Composites,” Eng. Fract. Mech., 58, Nos. 5 and 6, pp. 581–606.
Alzebdeh,  K., Al-Ostaz,  A., Jasiuk,  I., and Ostoja-Starzewski,  M., 1998, “Fracture of Random Matrix-Inclusion Composites: Scale Effects and Statistics,” Int. J. Solids Struct., 35, No. 19, pp. 2537–2566.
Hermann, H. J., and Roux, S., eds., 1990, Statistical Models for Fracture of Disordered Media, Elsevier, Amsterdam.
Ostoja-Starzewski,  M., Jasiuk,  I., Wang,  W., and Alzebdeh,  K. 1996, “Composites With Functionally Graded Interfaces: Meso-continuum Concept and Effective Properties,” Acta Mater., 44, No. 5, pp. 2057–2066.
Zbib,  H., and Aifantis,  E. C., 1989, “A Gradient Dependent Flow Theory of Plasticity: Application to Metal and Soil Instabilities,” Appl. Mech. Rev., (Mechanics Pan-America 1989, C. R. Steele, A. W. Leissa, and M. R. M. Crespo da Silva, eds.), 42, No. 11, Part 2, pp. S295–S304.
Lacy,  T. E., McDowell,  D. L., and Talreja,  R., 1999, “Gradient Concepts for Evolution of Damage,” Mech. Mater., 31, No. 12, pp. 831–860.
Ostoja-Starzewski,  M., and Trȩbicki,  J., 1999, “On the Growth and Decay of Acceleration Waves in Random Media,” Proc. R. Soc. London, Ser. A, A455, pp. 2577–2614.


Grahic Jump Location
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).
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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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