Exact determination of the volume of an inclusion in a body having constant shear modulus

Thaler, Andrew E.; Milton, Graeme W.

doi:10.1088/0266-5611/30/12/125008

Cited by 4 publications

(6 citation statements)

References 42 publications

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“…This section generalizes the ideas developed in [49], where it was shown how Hill's exact relation in the theory of composites, could be used to derive exact identities satisfied by the "Dirichlet-to-Neumann map" of a body Ω containing two elastically isotropic materials with the same shear modulus: in particular, these identities allow one to exactly deduce the volume fractions occupied by the phases from boundary measurements. The key idea was to apply (non-local) boundary conditions on the boundary tractions and displacements on the boundary ∂Ω of Ω in such a way that they mimic the body placed in an appropriate infinite medium with appropriate sources outside.…”

Section: Links Between Green's Functions Of Different Physical Problemsmentioning

confidence: 79%

“…Depending on the nature of the exact relations, boundary field equalities may involve the volume fraction of one phase in a body containing two phases. and thus allow the volume fraction to be exactly determined (e.g., see [49]).…”

Section: Links Between Green's Functions Of Different Physical Problemsmentioning

confidence: 99%

“…These constraints on the flux n · J(x) for the prescribed u(x) are an example of a boundary field equality. Another example of a boundary field equality is that given in [49] for an elastic body containing two isotropic elastic phases having the same shear modulus, where the boundary field equality involves the volume fraction (and thus may be used in an inverse way to determine this volume fraction).…”

Section: Introductionmentioning

confidence: 99%

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Exact relations for Green’s functions in linear PDE and boundary field equalities: a generalization of conservation laws

Milton

Onofrei

2019

Res Math Sci

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Contents 1 Introduction 2 A brief review of exact relations in composites 3 Functional framework 4 The central theorem 5 Exact identities satisfied by the Green's function 6 Links between Green's functions of different physical problems 7 Exact identities satisfied by the DtN map and Boundary Field Equalities: A generalization of conservation laws 8 A alternative view of some Boundary Field Equalities 9 Conjectured extension of exact relations to some non-linear minimization problems 10 Conclusions

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Section: Links Between Green's Functions Of Different Physical Problemsmentioning

confidence: 79%

Section: Links Between Green's Functions Of Different Physical Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exact relations for Green’s functions in linear PDE and boundary field equalities: a generalization of conservation laws

Milton

Onofrei

2019

Res Math Sci

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“…Shear modulus (G) is a physical quantity that describes a material's resistance to shear deformation. It measures the material's tendency to deform under shear stress [15] . The shear modulus of Al2O3 differs significantly from that of the steel matrix, potentially leading to different shear strains between inclusions and the matrix under stress.…”

Section: Crystallographic Parametersmentioning

confidence: 99%

Impact of la on steel inclusions: a first-principles investigation into elastic properties and anisotropy

Song,

Li,

Qiao

et al. 2024

J. Phys.: Conf. Ser.

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This study employed first-principles calculations based on density functional theory to explore the La-induced modifications in the elastic properties of inclusions at the microscopic level. Initially, through formation energy calculations, stable La2O3, La2O2S, and LaAlO3 inclusions were identified after the treatment with the rare earth element La. Subsequently, the elastic moduli and anisotropy of both the steel matrix and the inclusions were computed. The results revealed a substantial difference in the average elastic moduli between the common inclusions MnS, Al2O3 and the steel matrix, indicating their non-ductile nature. Conversely, La-modified inclusions, such as La2O2S and LaAlO3, exhibited closer average elastic moduli to the steel matrix with reduced anisotropy, mitigating stress concentration. This elucidates one of the reasons behind the enhanced fatigue performance of La-doped steel.

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“…Regularized inversion schemes and stable reconstruction algorithms to recover µ and its moments from data on the effective complex permittivity were developed in Cherkaev (2001Cherkaev ( , 2004; Cherkaev and Ou (2008); Bonifasi-Lista and Cherkaev ( 2009 Other approaches to the volume fraction bounds include Engström (2005); Milton (2012); Thaler and Milton (2014) based on estimates for higher order moments and on variational bounds, as well as direct inversion of known formulas or mixing rules Bergman and Stroud (1992); Levy and Cherkaev (2013) for effective properties of composites with specific structure, however, an advantage of the methods discussed here, is their applicability without a priori assumption about the microgeometry.…”

Section: Spectral Measure Computations For Uniaxial Polycrystalline M...mentioning

confidence: 99%

Stieltjes Functions and Spectral Analysis in the Physics of Sea Ice

Golden

Murphy²,

Cherkaev³

2022

Preprint

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Abstract. Polar sea ice is a critical component of Earth’s climate system. As a material it is a multiscale composite with temperature dependent millimeter-scale brine microstructure, and centimeter-scale polycrystalline microstructure which is largely determined by how the ice was formed. The surface layer of the polar oceans can be viewed as a granular composite of ice floes in a sea water host, with floe sizes ranging from centimeters to tens of kilometers. A principal challenge in modeling sea ice and its role in climate is how to use information on smaller scale structure to find the effective or homogenized properties on larger scales relevant to process studies and coarse-grained climate models. That is, how do you predict macroscopic behavior from microscopic laws, like in statistical mechanics and solid state physics? Also of great interest in climate science is the inverse problem of recovering parameters controlling small scale processes from large scale observations. Motivated by sea ice remote sensing, the analytic continuation method for obtaining rigorous bounds on the homogenized coefficients of two phase composites was applied to the complex permittivity of sea ice, which is a Stieltjes function of the ratio of the permittivities of ice and brine. Integral representations for the effective parameters distill the complexities of the composite microgeometry into the spectral properties of a self-adjoint operator like the Hamiltonian in quantum physics. These techniques have been extended to polycrystalline materials, advection diffusion processes, and ocean waves in the sea ice cover. Here we discuss this powerful approach in homogenization, highlighting the spectral representations and resolvent structure of the fields that are shared by the two component theory and its extensions. Spectral analysis of sea ice structures leads to a random matrix theory picture of percolation processes in composites, establishing parallels to Anderson localization and semiconductor physics, which then provides new insights into the physics of sea ice.

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Exact determination of the volume of an inclusion in a body having constant shear modulus

Cited by 4 publications

References 42 publications

Exact relations for Green’s functions in linear PDE and boundary field equalities: a generalization of conservation laws

Exact relations for Green’s functions in linear PDE and boundary field equalities: a generalization of conservation laws

Impact of la on steel inclusions: a first-principles investigation into elastic properties and anisotropy

Stieltjes Functions and Spectral Analysis in the Physics of Sea Ice

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