{"id":532,"date":"2020-04-13T18:54:47","date_gmt":"2020-04-13T22:54:47","guid":{"rendered":"https:\/\/www.ce.jhu.edu\/lori\/?page_id=532"},"modified":"2023-03-13T17:44:56","modified_gmt":"2023-03-13T21:44:56","slug":"stochastic-homogenization","status":"publish","type":"page","link":"https:\/\/www.ce.jhu.edu\/lori\/stochastic-homogenization\/","title":{"rendered":"Stochastic Homogenization"},"content":{"rendered":"\n

Traditional homogenization considers the effective mechanical properties of a representative volume element (RVE), which assumes that the relevant material behavior is averaged over the macro-scale. However, many failure mechanisms are associated with more localized behavior, requiring that homogenization be performed at an intermediate meso-scale, which means that the material properties vary randomly. For 20 years, the Graham-Brady research group has been applying stochastic homogenization as a way to understand stress localizations and failure.<\/em><\/p>\n\n\n\n\n\n\n\n\n\n\n
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A random field-based method to estimate convergence of apparent properties in computational homogenization<\/h5>\n

This work addresses the challenge of efficiently determining the representative volume size in computational homogenization<\/a> by exploiting the requirement that mechanical response quantities are statistically homogeneous and ergodic random fields. The proposed computational homogenization approach focuses on empirically determining the autocorrelation functions<\/a> of output response quantities through post processing finite element analyses. Once the autocorrelation functions are known, only trivial computations are needed to determine the variance of apparent properties as a function of domain<\/a> size without any additional finite element computation<\/a> by utilizing a simple formula relating the autocorrelation function to the variance of spatially averaged quantities. This approach improves upon the current established approach by circumventing the need to analyze numerous successively larger domains in order to determine convergence of apparent properties, which can be computationally prohibitive. Furthermore, similar previous analytical expressions for the variance of apparent properties are asymptotic<\/a> with respect to domain size while the expression in the proposed approach is exact. After presenting the formulation, the method is first demonstrated for a one-dimensional bar model where analytical expressions can be derived. Then the approach is demonstrated on two numerical examples: (1) a plane strain, linear elastic domain<\/a> with a lognormal<\/a> random field describing its compliance, and (2) a stochastic polycrystalline<\/a> microstructure<\/a> of a Nickel super alloy<\/a> having a crystal plasticity<\/a> model for its constitutive law.<\/p>\n<\/td>\n

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Teferra, K. & Graham-Brady, L. (2018). \u201cA random field-based method to estimate convergence of apparent properties in computational homogenization,\u201d Computer Methods in Applied Mechanics and Engineering<\/em>, 330: 253-270.<\/p>\n<\/td>\n<\/tr>\n
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A Multi-scale Spectral Stochastic Method for Homogenization of Multi-Phase Periodic Composites with Random Material Properties<\/h5>\n

In this work a spectral stochastic computational scheme is proposed that links the global properties of multi\u2010phase periodic composites to the geometry and random material properties of their microstructural components. To propagate the uncertainties associated with the material properties to the microstructural response the scheme benefits from a combination of homogenization theory built into a finite element framework and the spectral representation of uncertainty based on Hermite Chaos where a probabilistic characterization of the solutions to a set of local problems defined on the period cell is first sought. A full stochastic description of the global (effective) properties is then obtained by averaging the solutions to the forgoing set of local problems over the unit cell. A representative subset of results is compared with the results obtained using Monte Carlo simulation to demonstrate the accuracy of the proposed procedure.<\/p>\n

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Tootkaboni, M. & Graham-Brady, L. (2010). \u201cA Multi-scale Spectral Stochastic Method for Homogenization of Multi-Phase Periodic Composites with Random Material Properties,\u201d International Journal of Numerical Methods in Engineering<\/em>, 83(1): 59-90.<\/p>\n<\/td>\n<\/tr>\n

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Simulation of local material properties based on moving-window GMC<\/h5>\n

When analyzing the behavior of composite materials under various loading conditions, the assumption is generally made that the behavior due to randomness in the material can be represented by a homogenized, or effective, set of material properties. This assumption may be valid when considering displacement, average strain, or even average stress of structures much larger than the inclusion size. The approach is less valid, however, when considering either behavior of structures of size at the scale of the inclusions or local stress of structures in general. In this paper, Monte Carlo simulation is used to assess the effects of microstructural randomness on the local stress response of composite materials. In order to achieve these stochastic simulations, the mean, variance and spectral density functions describing the randomly varying elastic properties are required as input. These are obtained here by using a technique known as moving-window generalized method of cells (moving-window GMC). This method characterizes a digitized composite material microstructure by developing fields of local effective material properties. Once these fields are generated, it is straightforward to obtain estimates of the associated probabilistic parameters required for simulation. Based on the simulated property fields, a series of local stress fields, associated with the random material sample under uniaxial tension, is calculated using finite element analysis. An estimation of the variability in the local stress response for the given random composite is obtained from consideration of these simulations.<\/p>\n<\/td>\n

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Graham, L.L. & Baxter, S.C. (2001). \u201cSimulation of local material properties based on moving-window GMC,\u201d Probabilistic Engineering Mechanics<\/em>, 16(4): 295-306.<\/p>\n<\/td>\n<\/tr>\n

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Other relevant publications<\/h5>\n

Yaghoobi, M., Stopka, K.S., McDowell, D.L., Graham-Brady, L., Teferra, K. (2023). \u201cEffect of sample size on the maximum value distribution of fatigue driving forces in metals and alloys,\u201d submitted.<\/p>\n

Teferra, K. & Graham-Brady L. (2019). \u201cThe maximum value distribution of response quantities in computational homogenization,\u201d Journal of Engineering Mechanics<\/em>, 145(5): 06019002.<\/p>\n

Acton, K. & Graham-Brady, L. (2010). \u201cElasto-plastic mesoscale homogenization of composite materials,\u201d Journal of Engineering Mechanics<\/em>, ASCE, 136(5): 613-624.<\/p>\n

Acton, K. & Graham-Brady, L. (2009). \u201cMesoscale modeling of plasticity in composites,\u201d Computer Methods in Applied Mechanics & Engineering<\/em>, 198(9-12): 920-932.<\/p>\n

Tregger, N., Corr, D., Graham-Brady, L. & Shah, S. (2007). \u201cModeling mesoscale uncertainty for concrete in tension,\u201d Computers & Concrete<\/em>, 4: 347-362.<\/p>\n

Tregger, N., Corr, D., Graham-Brady, L. & Shah, S. (2006). \u201cModeling the effect of mesoscale randomness on concrete fracture,\u201d Probabilistic Engineering Mechanics<\/em>, 21(3): 217-225.<\/p>\n

Xu, X.F. & Graham-Brady, L. (2006). \u201cComputational stochastic homogenization of random media elliptic problems using the Fourier Galerkin method,\u201d Finite Elements in Analysis and Design<\/em>, 42(7): 613-622.<\/p>\n

Xu, X.F. & Graham-Brady, L.L. (2005). \u201cA stochastic computational method for evaluation of global and local behavior of random media,\u201d Computer Methods in Applied Mechanics & Engineering<\/em>, 194(42-44): 4362-4385.<\/p>\n

Baxter, S.C., Graham-Brady, L.L. & Xu, X.F. (2005). \u201cModeling the effects of material nonlinearity using moving window micromechanics,\u201d International Journal for Nonlinear Mechanics, <\/em>40: 351-359.<\/p>\n

Graham, L.L., Gurley, K. & Masters, F. (2003). \u201cNon-Gaussian simulation of local material properties based on a moving-window technique,\u201d Probabilistic Engineering Mechanics<\/em>, 18(3): 223-234.<\/p>\n

Graham-Brady, L.L., Siragy, E.F. & Baxter, S.C. (2003). \u201cAnalysis of heterogeneous composite materials using moving-window techniques,\u201d Journal of Engineering Mechanics<\/em>, ASCE, 129(9): 1054-1064.<\/p>\n

Corr, D.J. & Graham, L.L. (2002). \u201cMechanical analysis of concrete with the moving-window generalized method of cells,\u201d ACI Materials Journal<\/em>, 100(2): 156-164.<\/p>\n

Baxter, S.C., Hossain, M.I. & Graham, L.L. (2001). \u201cMicromechanics based random material property fields for particulate reinforced composites,\u201d International Journal of Solids & Structures<\/em>, 30: 9209-9220.<\/p>\n

Baxter, S.C. & Graham, L.L. (2000). \u201cCharacterization of random composite materials using the moving-window generalized method of cells technique,\u201d ASCE Journal of Engineering Mechanics<\/em>, 126(4): 389-397.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"

Traditional homogenization considers the effective mechanical properties of a representative volume element (RVE), which assumes that the relevant material behavior is averaged over the macro-scale. However, many failure mechanisms are associated with more localized behavior, requiring that homogenization be performed at an intermediate meso-scale, which means that the material properties…<\/p>\n

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