Layer-induced elastic anisotropy - part 1: forward relations between constituent parameters and compound medium parameters

Helbig, Klaus

doi:10.1590/s0102-261x1998000200001

Cited by 12 publications

(4 citation statements)

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“…The results of this section are similar to the results of Schoenberg and Muir (1989), Helbig and Schoenberg (1987, Appendix), Helbig (1998), Carcione et al (2012) and Kumar (2013), except that the tensorial form of equation ( 4) requires factors of 2 and √ 2 in several entries of M , B , J and K . This notation allows for a convenient study of rotations, which arise in the study of elasticity tensors expressed in coordinate systems of arbitrary orientations.…”

Section: Equivalent-medium Elasticity Parameterssupporting

confidence: 83%

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On Backus Average for Generally Anisotropic Layers

Bos

Dalton

Slawiński

et al. 2016

J Elast

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In this paper, following the Backus (1962) approach, we examine expressions for elasticity parameters of a homogeneous generally anisotropic medium that is long-wave-equivalent to a stack of thin generally anisotropic layers. These expressions reduce to the results of Backus (1962) for the case of isotropic and transversely isotropic layers.In the over half-a-century since the publications of Backus (1962) there have been numerous publications applying and extending that formulation. However, neither George Backus nor the authors of the present paper are aware of further examinations of the mathematical underpinnings of the original formulation; hence this paper.We prove that-within the long-wave approximation-if the thin layers obey stability conditions then so does the equivalent medium. We examine-within the Backus-average context-the approximation of the average of a product as the product of averages, which underlies the averaging process.In the presented examination we use the expression of Hooke's law as a tensor equation; in other words, we use Kelvin's-as opposed to Voigt's-notation. In general, the tensorial notation allows us to conveniently examine effects due to rotations of coordinate systems.

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Section: Equivalent-medium Elasticity Parameterssupporting

confidence: 83%

“…The subject of Backus average was examined by several researchers, among them, Helbig and Schoenberg (1987), Schoenberg and Muir (1989), Berryman (1997), Helbig (1998Helbig ( , 2000, Carcione et al (2012), Kumar (2013), Brisco (2014), and Danek and Slawinski (2016). However, further venues of investigation remain open.…”

Section: Further Workmentioning

confidence: 99%

On Backus Average for Generally Anisotropic Layers

Bos

Dalton

Slawiński

et al. 2016

J Elast

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“…However, this generalization has been questioned by Hudson and Crampin (1991), who argue that the theory cannot be applied to oblique sets of cracks (even for weak anisotropy), because the structure is no longer 1D, and Backus' assumptions are invalidated (see response in Schoenberg and Muir, 1991). More recently, Helbig (1999) solved the problem of Schoenberg and Muir (1989) without recourse to group theory, but he did not discuss its validity.…”

Section: Introductionmentioning

confidence: 99%

Numerical test of the Schoenberg-Muir theory

et al. 2012

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The Schoenberg-Muir theory states that an equivalent, homogeneous and anisotropic medium can be constructed from a layered medium composed of several thin layers, each anisotropic, under the assumption of stationarity. To test the theory we considered single transversely isotropic layers with different orientations of the symmetry axis and performed numerical simulations of wave propagation with a full-wave solver. The equivalent media have orthorhombic and monoclinic symmetries, respectively. The theory performed very well from the kinematical and dynamical points of view, even for strong anisotropy and layers described by media whose symmetry axes have different orientations.

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“…Part 1 of this paper (Helbig, 1998; henceforth Part 1) describes a matrix algorithm that allows to determine the "compound stiffnesses" of the replacement medium that is, in the long-wavelength approximation, equivalent to a periodic sequence of thin layers. The matrix algorithm works for layers of arbitrary anisotropy, and should have applications in the determination of the properties of industrial laminates like plywood.…”

Section: Introductionmentioning

confidence: 99%

Layer-induced elastic anisotropy Part 2: inversion of compound parameters to constituent parameters

Helbig¹

2000

Rev. Bras. Geof.

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was shown that a medium consisting of a periodic sequence of layers is, in the long-wavelength approximation, equivalent to a homogeneous compound medium with elastic parameters that are generalized averages of the constituents' stiffnesses. Though the matrix-algorithm described in Part 1 works with anisotropic constituents, the most interesting application is to layer sequences with isotropic constituents, i.e., to transversely isotropic (TI) compound media. Part 2 discusses the possibility to obtain information about the (thin-layer) constituents from the properties of the compound medium. Though every periodic sequence of isotropic layers results in a TI medium, the reverse is not true: there are TI media that cannot be "modeled" by a periodic sequence of isotropic layers. Those that can be modeled can be inverted to layer sequences that result in precisely the observed anisotropy. This inversion is not unique, but it constrains the possibilities. The critical tool to determine the possibility of modeling a TI medium is the concept of stability. Unstable compound media -that release energy on being deformed -would not exist. However, for inversion we must insist that not only the compound medium, but also the potential constituents are stable. In preparing a catalog that covers all possible media, instability is the boundary beyond which the calculation becomes meaningless. Inversion means to determine possible causes of the observed anisotropy, ideally the elastic parameters of the constituents and their contribution to the compound medium. This is possible, though under several restrictions: Not all TI media are long-wave equivalent to a periodically layered sequence of isotropic layers. Those that are can be "modeled" by a variety of layer sequences. Every TI medium that can be modeled at all can be modeled by as few as three layers, but the set of all models is a threeparametric manifold. If a TI medium can be modeled by two constituents only, this can be done only in one way, unless the constituents have the same ratio of S-to P-velocity. In that case, the set of possible models forms a one-parametric manifold.

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Layer-induced elastic anisotropy - part 1: forward relations between constituent parameters and compound medium parameters

Cited by 12 publications

References 0 publications

On Backus Average for Generally Anisotropic Layers

On Backus Average for Generally Anisotropic Layers

Numerical test of the Schoenberg-Muir theory

Layer-induced elastic anisotropy Part 2: inversion of compound parameters to constituent parameters

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