June  2006, 1(2): 259-274. doi: 10.3934/nhm.2006.1.259

On the scaling from statistical to representative volume element in thermoelasticity of random materials


Department of Mechanical Engineering, McGill University, Montreal, QC H3A 2K6, Canada


Department of Mechanical and Industrial Engineering, 1206 W. Green Street, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2906, United States

Received  November 2005 Revised  February 2006 Published  March 2006

Under consideration is the finnite-size scaling of effective thermoelastic properties of random microstructures from a Statistical Volume Element (SVE) to a Representative Volume Element (RVE), without invoking any periodic structure assumptions, but only assuming the microstructure's statistics to be spatially homogeneous and ergodic. The SVE is set up on a mesoscale, i.e. any scale finite relative to the microstructural length scale. The Hill condition generalized to thermoelasticity dictates uniform Neumann and Dirichlet boundary conditions, which, with the help of two variational principles, lead to scale dependent hierarchies of mesoscale bounds on effective (RVE level) properties: thermal expansion and stress coefficients, effective stiffness, and specific heats. Due to the presence of a non-quadratic term in the energy formulas, the mesoscale bounds for the thermal expansion are more complicated than those for the stiffness tensor and the heat capacity. To quantitatively assess the scaling trend towards the RVE, the hierarchies are computed for a planar matrix-inclusion composite, with inclusions (of circular disk shape) located at points of a planar, hard-core Poisson point field. Overall, while the RVE is attained exactly on scales infinitely large relative to the microscale, depending on the microstructural parameters, the random fluctuations in the SVE response may become very weak on scales an order of magnitude larger than the microscale, thus already approximating the RVE.
Citation: Xiangdong Du, Martin Ostoja-Starzewski. On the scaling from statistical to representative volume element in thermoelasticity of random materials. Networks and Heterogeneous Media, 2006, 1 (2) : 259-274. doi: 10.3934/nhm.2006.1.259

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311


Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473


Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190


Vsevolod Laptev. Deterministic homogenization for media with barriers. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 29-44. doi: 10.3934/dcdss.2015.8.29


Monica Conti, Lorenzo Liverani, Vittorino Pata. Correction to "Thermoelasticity with antidissipation" (volume 15, number 8, 2022, 2173–2188). Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2429-2431. doi: 10.3934/dcdss.2022125


Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279


Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951


María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks and Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004


Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91


Luis Caffarelli, Antoine Mellet. Random homogenization of fractional obstacle problems. Networks and Heterogeneous Media, 2008, 3 (3) : 523-554. doi: 10.3934/nhm.2008.3.523


Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks and Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1


Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks and Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689


Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure and Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757


Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.


Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329


Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems and Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025


François Murat, Ali Sili. A remark about the periodic homogenization of certain composite fibered media. Networks and Heterogeneous Media, 2020, 15 (1) : 125-142. doi: 10.3934/nhm.2020006


Arvind Kumar Misra, Rajanish Kumar Rai, Yasuhiro Takeuchi. Modeling the control of infectious diseases: Effects of TV and social media advertisements. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1315-1343. doi: 10.3934/mbe.2018061


Gabriel Montes-Rojas, Pedro Elosegui. Network ANOVA random effects models for node attributes. Journal of Dynamics and Games, 2020, 7 (3) : 239-252. doi: 10.3934/jdg.2020017


Micol Amar. A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 537-556. doi: 10.3934/dcds.2000.6.537

2021 Impact Factor: 1.41


  • PDF downloads (69)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]