Numerical methods to determine effective elastic properties

Effective elastic properties for heterogeneous materials are widely discussed in the literature. We discuss and review the application range and accuracy of different numerical approaches to determine those effective elastic values from digitized material samples. Recent examples are used for illustration. Whereas the use of most effective-medium theories is restricted to a limited number of geometries the possible application range of numerical methods is much wider. There are two fundamental different numerical methods to extract effective elastic properties from a given microstructure. Both approaches are sensitive to size effects. Static methods can address this problem by applying uniform stress or uniform displacement boundary conditions. For dynamic wave propagation simulations the ratio of the dominant wavelength to the typical inclusion size is the most crucial factor.

[1]  J. Hudson.  Wave speeds and attenuation of elastic waves in material containing cracks , 1981 .

[2]  M. Ostoja-Starzewski.  Scale effects in materials with random distributions of needles and cracks , 1999 .

[3]  J. Hudson.  Overall properties of a cracked solid , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of multiphase materials , 1963 .

[5]  J. Hudson,et al.  Effective elastic properties of heavily faulted structures , 1999 .

[6]  T. Mukerji,et al.  The Rock Physics Handbook: Contents , 2009 .

[7]  B. Gurevich,et al.  Pore scale numerical modeling of elastic wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers , 2006 .

[8]  S. Shapiro,et al.  Finite-difference modeling of wave propagation on microscale: A snapshot of the work in progress , 2007 .

[9]  M. Schoenberg,et al.  Seismic anisotropy of fractured rock , 1995 .

[10]  A. P. Roberts,et al.  Computation of the linear elastic properties of random porous materials with a wide variety of microstructure , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  Edward J. Garboczi,et al.  An algorithm for computing the effective linear elastic properties of heterogeneous materials: Three-dimensional results for composites with equal phase poisson ratios , 1995 .

[12]  Erik H. Saenger,et al.  A numerical study on reflection coefficients of fractured media , 2007 .

[13]  Keith W. Jones,et al.  Synchrotron computed microtomography of porous media: Topology and transports. , 1994, Physical review letters.

[14]  Erik H. Saenger,et al.  Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid , 2004 .

[15]  Mark Kachanov,et al.  Effective elasticity of fractured rocks: A snapshot of the work in progress , 2006 .

[16]  S. Shapiro,et al.  Modeling the propagation of elastic waves using a modified finite-difference grid , 2000 .

[17]  Christoph H. Arns,et al.  Computation of linear elastic properties from microtomographic images: Methodology and agreement between theory and experiment , 2002 .

[18]  S. Shapiro,et al.  Effective elastic properties of randomly fractured soils: 3D numerical experiments , 2004 .

[19]  E. Saenger,et al.  Numerical considerations of fluid effects on wave propagation: Influence of the tortuosity , 2004 .

[20]  Erik H. Saenger,et al.  Effective Elastic Properties of Fractured Rocks: Dynamic vs. Static Considerations , 2006 .

[21]  Christian Huet,et al.  Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies , 1999 .

[22]  S. Shapiro,et al.  Effects of Parallel Crack Distributions on Effective Elastic Properties - a Numerical Study , 2003 .

[23]  Mark Kachanov,et al.  Effective Elastic Properties of Cracked Solids: Critical Review of Some Basic Concepts , 1992 .

[24]  B. Budiansky,et al.  Elastic moduli of a cracked solid , 1976 .

[25]  J. Labuz,et al.  Experiments on effective elastic modulus of two-dimensional solids with cracks and holes , 1996 .

[26]  J. Hudson,et al.  Effective‐medium theories for fluid‐saturated materials with aligned cracks , 2001 .

[27]  The elastic modulus of media containing strongly interacting antiplane cracks , 1995 .

[28]  Erik H. Saenger,et al.  Seismic effects of viscous Biot‐coupling: Finite difference simulations on micro‐scale , 2005 .

[29]  J. Berryman.  Single‐scattering approximations for coefficients in Biot’s equations of poroelasticity , 1992 .

[30]  M. Biot.  Theory of Propagation of Elastic Waves in a Fluid‐Saturated Porous Solid. I. Low‐Frequency Range , 1956 .

[31]  R. Hill.  Elastic properties of reinforced solids: some theoretical principles , 1963 .