Representation of Hashin–Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials

Fernández, Mauricio; Böhlke, Thomas

doi:10.1007/s10659-018-9679-0

Cited by 20 publications

(36 citation statements)

References 54 publications

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“…Since the elastic modulus bounds of a textured polycrystalline copper (orthotropic texture symmetry) with equiaxed grains are available in Refs. [73,74], it makes possible a quantitative comparison between this work and Refs. [73,74].…”

Section: Comparison Of Quasi-static Velocity From the Soa Model With mentioning

confidence: 76%

“…For statistically isotropic polycrystals, different velocity bounds, including Voigt-Reuss and Hashin-Shtrikman bounds [67], and a rather accurate prediction named self-consistent approach [61] have been developed to predict the effective elastic constants of grains ensemble. For textured polycrystals, Voigt-Reuss bounds [70,71] Hashin-Shtrikman bounds [72][73][74] and self-consistent approach [73,75] have also been reported, from which one can calculate the phase velocity of an incident elastic wave. Although the self-consistent approach in Refs.…”

Section: Quasi-static Phase Velocitymentioning

confidence: 99%

“…Although the self-consistent approach in Refs. [73,75] is still limited to high-symmetry crystallites and certain texture symmetry, a recent study [74] developed Hashin-Shtrikman bounds for polycrystals with arbitrary crystallographic texture and triclinic crystallites. The quasi-static velocities also can be obtained in this work for aggregates of triclinic grains with arbitrary texture symmetry and it counts grain shape impact as well.…”

Section: Quasi-static Phase Velocitymentioning

confidence: 99%

“…The Hashin-Shtrikman bounds for anisotropic aggregates of cubic grains with arbitrary texture symmetry have been derived in Ref. [73] using variational principals while a recent study [74] explicitly represented the Hashin-Shtrikman bounds by second and fourth-order ODCs for polycrystals with arbitrary texture symmetry and crystal symmetry. Since the elastic modulus bounds of a textured polycrystalline copper (orthotropic texture symmetry) with equiaxed grains are available in Refs.…”

Section: Comparison Of Quasi-static Velocity From the Soa Model With mentioning

confidence: 99%

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Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

Guo

2020

Acoustics

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This study extends the second-order attenuation (SOA) model for elastic waves in texture-free inhomogeneous cubic polycrystalline materials with equiaxed grains to textured polycrystals with ellipsoidal grains of arbitrary crystal symmetry. In term of this work, one can predict both the scattering-induced attenuation and phase velocity from Rayleigh region (wavelength >> scatter size) to geometric region (wavelength << scatter size) for an arbitrary incident wave mode (quasi-longitudinal, quasi-transverse fast or quasi-transverse slow mode) in a textured polycrystal and examine the impact of crystallographic texture on attenuation and phase velocity dispersion in the whole frequency range. The predicted attenuation results of this work also agree well with the literature on a textured stainless steel polycrystal. Furthermore, an analytical expression for quasi-static phase velocity at an arbitrary wave propagation direction in a textured polycrystal is derived from the SOA model, which can provide an alternative homogenization method for textured polycrystals based on scattering theory. Computational results using triclinic titanium polycrystals with Gaussian orientation distribution function (ODF) are also presented to demonstrate the texture effect on attenuation and phase velocity behaviors and evaluate the applicability and limitation of an existing analytical model based on the Born approximation for textured polycrystals. Finally, quasi-static phase velocities predicted by this work for a textured polycrystalline copper with generalized spherical harmonics form ODF are compared to available velocity bounds in the literature including Hashin–Shtrikman bounds, and a reasonable agreement is found between this work and the literature.

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Section: Comparison Of Quasi-static Velocity From the Soa Model With mentioning

confidence: 76%

Section: Quasi-static Phase Velocitymentioning

confidence: 99%

Section: Quasi-static Phase Velocitymentioning

confidence: 99%

Section: Comparison Of Quasi-static Velocity From the Soa Model With mentioning

confidence: 99%

See 2 more Smart Citations

Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

Guo

2020

Acoustics

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“…Tensors with a prime are harmonic (completely symmetric and traceless [17]), thus e.g.

V_{i j k l}^{'} = V_{j i k l}^{'} = V_{j i l k}^{'} = V_{k j i l}^{'} = \dots, V_{i i k l}^{'} = 0 .

The number of independent components in a harmonic tensor is

dim false(V_{α_{i}}^{'} false) = 2 α_{i} + 1 .

Cubic base tensors of unequal rank are mutually orthogonal in the following sense [18]

\int_{SO(3)} \frac{{double-struckF}_{α_{i}}^{'} (boldQ)}{2 α_{i} + 1} \otimes \frac{{double-struckF}_{β_{i}}^{'} (boldQ)}{2 β_{i} + 1} d Q = O, \forall α_{i} \neq β_{i} .

Note that the

{α_{i}}

start with index 4, which is due to the fact that a cubic reference tensor of rank 2 is zero. Also cubic texture coefficients with uneven tensor rank vanish up to rank eight [13].…”

Section: Preliminariesmentioning

confidence: 99%

A micro‐mechanically motivated phenomenological yield function for cubic crystal aggregates

Dyck

Böhlke

2020

Z Angew Math Mech

Self Cite

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A micro-mechanically motivated phenomenological yield function, for polycrystalline cubic metals is presented. In the suggested yield function microstructure is taken into account by the crystallographic orientation distribution function in terms of tensorial Fourier coefficients. The yield function is presented in a polynomial form in powers of the stress state. Known group-theoretic results are used to identify isotropic and anisotropic parts in the yield function, whereby anisotropic parts are characterized by tensorial Fourier coefficients. The form of the presented yield function is inspired by the classic, phenomenological von Mises -Hill yield function first published in 1913. For a specific choice of material parameters, both functions coincide, thus a micro-mechanically motivated generalization of the von Mises -Hill yield function is presented. For the given yield function, two dimensional experimental results are sufficient, to identify a three dimensional anisotropic yield behavior. The work concludes with a treatment of the isotropic special case, i.e. a tension-compression split in yield behavior as well as parameter ranges for convexity and shapes of the yield surface. K E Y W O R D Sanisotropic yield function, crystallographic texture, orientation distribution function, plastic anisotropy, tensorial texture coefficients This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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A dual‐scale homogenization method to study the properties of chopped carbon fiber reinforced epoxy resin matrix composites

Zheng

Qiu

et al. 2022

Polymer Composites

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In this work, a dual‐scale homogenization (DSH) procedures including the representative volume element (RVE) and Unit cell (UC) with Halpin–Tsai models in the UC scale and space direction‐sensitive Hotelling expansion (SDSHE) models in RVE scale separately. These two methods are introduced and implemented to predict the response of effective elastic properties in chopped fiber reinforced epoxy matrix structures. A post‐processing scheme of subdomain in UC scale based on Gauss theorem is proposed in SDSHE method, which is used to extract the average strain and stress in UC three‐dimensional structure. In addition, the X‐ray computed tomography was used to determine the fourth order tensor orientation angle and the probability distribution information of chopped fiber length in testing samples. Homogenization properties in RVE scale, and inclusion volume fraction (VF), thickness of interface and orientation tensors are designed in UC scale and compared. The excellent correlations with experiments prove the effectiveness of the proposed models. Furthermore, the algorithm proposed in this paper realizes the homogenization of the two scales in the process of numerical analysis, and it will effectively take into account many problems that cannot be achieved by other algorithms, such as the path strain of the model, the thickness of the interface, and so on.

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Representation of Hashin–Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials

Cited by 20 publications

References 54 publications

Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

A micro‐mechanically motivated phenomenological yield function for cubic crystal aggregates

A dual‐scale homogenization method to study the properties of chopped carbon fiber reinforced epoxy resin matrix composites

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