On optimal zeroth-order bounds with application to Hashin–Shtrikman bounds and anisotropy parameters

Nadeau, J. C.; Ferrari, Mauro

doi:10.1016/s0020-7683(00)00393-0

Cited by 29 publications

(15 citation statements)

References 22 publications

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“…In Section 2 the problem is specialized to the tetragonal elastic symmetry: the stationary points are evaluated, together with the conditions for the existence of such points, in terms of three material parameters a 2 , b 2 and b 3 responsible of the degree of anisotropy (for other definitions of anisotropy parameters, see Nadeau and Ferrari, 2001). The usual VoigtÕs contracted representation of stress, strain and elasticity tensors is adopted.…”

Section: Introductionmentioning

confidence: 99%

Extrema of Young’s modulus for elastic solids with tetragonal symmetry

Cazzani

Rovati

2005

International Journal of Solids and Structures

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Section: Introductionmentioning

confidence: 99%

Extrema of Young’s modulus for elastic solids with tetragonal symmetry

Cazzani

Rovati

2005

International Journal of Solids and Structures

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“…The eigenvalues of the symmetric 2nd‐order tensor correspond to the inverse values of the aspect ratios of the average inclusion. Without loss of generality, the comparison material can be chosen to be isotropic . For an isotropic comparison medium with stiffness

ℂ

0 = k 1

double-struckP

1 + k 2

double-struckP

2 and isotropic two‐point statistics, i.e.…”

Section: Nonlinear Homogenization Schemementioning

confidence: 99%

“…Without loss of generality, the comparison material can be chosen to be isotropic. [53] For an isotropic comparison medium with stiffness C 0 ¼ k 1 P 1 þ k 2 P 2 and isotropic two-point statistics, i.e. A ¼ I, the tensor P 0 is given by (see, e.g., [35] )…”

Section: Implications Of Statistical Isotropy and No Long Range Ordermentioning

confidence: 99%

Two‐Scale Modeling of Grain Size and Phase Transformation Effects

Böhlke

Neumann

Rieger

2014

steel research int.

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A nonlinear homogenization scheme is applied to consider two classes of micromechanical problems, i.e. to estimate the grain size dispersion influence and to model phase transformation effects in steels. The macroscopic material behavior is described purely based on micro‐mechanical constitutive models. The coarse graining procedure contains the classical bounds, i.e. mixture theories, as special cases and determines the macroscopic stress and the evolution of the microstructural variables. In the context of the grain size dependent flow behavior of polycrystals, a log‐normal distributed grain size is assumed together with a grain size dependent local plastic behavior. The numerical results are well approximated by a simple analytical expression. It is found that grain size dispersion leads to a decrease of the material strength, especially for small mean diameters around one micron. In the context of phase transformation phenomena, the thermo‐mechanically strongly coupled Greenwood‐Johnson effect is considered. Here, temperature driven phase transformation under external applied stress below the yield stress is modeled that leads to plastic strains. The main result of the present work is, that based on the suggested nonlinear homogenization technique of Hashin‐Shtrikman type a numerically effective modeling approach is given, that can be applied at the Gauss point level of structural finite element simulations. In contrast to FE2 schemes, full three‐dimensional microstructures can be modeled effectively with standard computers. The simulation results presented in this work are in good agreement with the experimental data.

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“…Its eigenvectors are equal to the anisotropy directions of morphologic anisotropy, which for microstructures with ellipsoidal symmetry can be orthotropic, transversely isotropic, or isotropic. Without loss of generality, the comparison material can be chosen to be isotropic (see, e.g., [22]), i.e.,…”

Section: Second-order Boundsmentioning

confidence: 99%

Homogenization of the elastic properties of pyrolytic carbon based on an image processing technique

et al. 2012

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In this work, the linear elastic material properties of differently textured variants of pyrolytic carbon are homogenized from the submicro-to the micro-scale. In high resolution transmission electron microscope (HRTEM) lattice fringe images, the microstructure of pyrolytic carbon manifests itself in terms of projections of graphene layers. According to their orientation distribution, different textures of pyrolytic carbon have been classified. Assuming a von Mises-Fisher distribution for the spatial orientation of single graphene layers, the orientation distribution function of the projected layers in the image plane is analytically found to be a modified Struve function. For each pyrolytic carbon texture, Maximum-likelihood estimates for the mean orientation and the concentration parameter of the von Mises-Fisher distribution are obtained numerically. Hereby, Fourier transformation and appropriate filters are used to determine the probabilities for discrete orientations of the graphene layers directly from HRTEM images. First-and second-order bounds of the linear elastic properties of pyrolytic carbon of the different textures are computed. Elastic constants of graphite and pyrolytic graphite have been used for modeling the elastic behavior of the graphene layers within a continuum mechanical setting. Due to the high anisotropy of all analyzed textures of pyrolytic carbon, the differences even between the second-order bounds are quite large.Notation. A direct tensor notation is preferred throughout the text. If tensor components are used, then the Einstein summation convention is applied. Vectors and 2nd-order tensors are denoted by lowercase and uppercase bold letters, e.g., a and A, respectively. A linear mapping of 2nd-order tensors by a 4th-order tensor is written as A = C[B]. The scalar product

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On optimal zeroth-order bounds with application to Hashin–Shtrikman bounds and anisotropy parameters

Cited by 29 publications

References 22 publications

Extrema of Young’s modulus for elastic solids with tetragonal symmetry

Extrema of Young’s modulus for elastic solids with tetragonal symmetry

Two‐Scale Modeling of Grain Size and Phase Transformation Effects

Homogenization of the elastic properties of pyrolytic carbon based on an image processing technique

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